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Proposal_for_a_tool_to_design_masonry_double_curved_shells

1. Dual grid zeta 0 TJ van Swinderen August 2009 89 Main report Structural Design Lab TU Delft JP Lf fi VILLI LS J Li fy PEL ELS ff VEELS DET LLL ra LIM ELIT TLS LS WELLE j TLS ni eff LY Al i INA i i 7 ERPS PL EL FTP S AID LUL PLE LEP ptt SGI RED Ds Vogt typ bor tf Bn Pritt Pe Pera ee ear Sn EAO W Papeete EFS Er x a T wid PE a eee PPP PEEPLES eel mbna Prenen dennen Dr en an ee nd Teee en qissasi ue ape tt td Sm ee SEE Figure 5 26 The lines on the surface of the shell which are used as the centerlines for the bricks to be placed on Above the 3D view is shown at the left the topview EETL et AH muide REE Tren EEn dt LET et ht ze i T gt T i zE ann RETE See yt BESPREKEN SapT RM SE VEJE p T rn A TF aes Y Lyyprrtto mty cy aan pl men whi dl ieee Lyrae Tr Sch Dr TE PEEL F F al ae ren tM H Tar en rd s ar Am HES P T foie re Tass er 335 A er en as XE kili p ne gn ya 5 F TI ITF we J y fe aE HF ye waited ae SEEE EEFE Tta iF zE Ir ter ge T 1 En Licht land ITE bl drf Shi nend ted SHEER Erde HEHE rd rated FEE Ene Ad EEDE feli EEE oA Te Et d nt Ji 7 5 wak E 7 F Ee 7 EEE mk CE 7 F Ae eert ALA 3 EEn Zanki It E
2. Chilton J 2000 The Engineers Contribution to Contemporary Architecture Heinz Isler Pottman H Asperl A Hofer M amp Kilian A 2007 Architectural Geometry Bentley Reas C amp Fry B 2007 Processing A Programming Handbook for Visual Designers and Artists Kanellos A 2007 Topological Self Organisation Using a particle spring system simulation to generate structural space filling lattices internet Available at http eprints ucl ac uk 4985 Accessed September 2008 Horssen W T van amp Burgh A H P van der 1999 Inleiding Matrixrekening en Lineaire Optimalisering Cremona L 2007 Graphical Statics Graphical Calculus and Reciprocal Figures in Graphical Statics Kahn L Attributed to Louis Kahn 1903 1974 August 2009 Structural Design Lab TU Delft ARTICLES 10 11 12 13 14 15 16 Block P and Ochsendorf J 2007 Thrust Network Analysis A new methodology for three dimensional equilibrium Journal of the International Association for Shell and Spatial Structures no 48 pp 167 173 Maxwell J C 1864 On Reciprocal Figures and Diagrams of Forces Kilian A and Ochsendorf J 2005 Particle Spring Systems for structural form finding Journal of the international association for shell and spatial structures no 48 pp 77 84 Chak D Galbraith M and Kilian A 2002 CatenaryCAD An Architectural Design Tool Final Pr
3. TU Delft local matrix for every node 2 A global matrix is formed consisting of a combination of the local matrices 3 The global matrix is solved as is explained before for the local matrix in 4 5 3 and as is explained in Appendix C 4 5 5 An example Consider the figure shown in Figure 4 17 The structure consists of four normal nodes 1 4 and three foundation nodes 5 7 First we consider the four nodes separately and create the local objective functions Node Nodes around node 2 3 4 and 5 Ni K N1 Z Tom 2 K N1 z t K N1 Zijt K N1 A In which K K are H Ho Hu Hs K _ G _ H Py Ma H INI Ee E E arn En H y Hok er Ms min H P ses Kasa En H P Ha k Kew Ms a H P ele ti Node 2 Nodes around node 2 3 4 and 6 Ny K N2 Zj a l Z T K N2 Z Jt K N2 Z K N2 Z 53 Main report Structural Design Lab TU Delft Generate global matrix to be solved Zi Zj Kii Kaia Ke Kani Ks i 0 z fN 3 Kiia Kona Kian Kaw 0 Kee ns z ND 4 l total Kin Kowa Kaia Kays 0 0 Koig z In3 5 ORT Kani Kaya Kiya 0 0 0 fN4 Ze via N Figure 4 18 Global matrix to solve for r and z values TJ van Swinderen August 2009 54 Main report In which KK and K are Ha Ha Huy Ha K _ C _ H Ho Ha z 2 N2 P E P 2 2 C H Kip THP 2 21 2 k G 2 Hs gt N P HP 2 23 2 K _ Ci a H eN P H P 2 24 2 K En
4. 5 6 2 Relocate points Processing is working in 2D mouse locations while the model is in 3D In other words coordinates can only be expressed in X and Y location of the mouse while the vectors nodes of the 3D model have a X Y and Z coordinate This creates a problem a node can be selected but after relocating the point the new coordinates can not be determined since it is based on the X and Y coordinate of the mouse at the moment of releasing the node after relocating it To make sure this method is not a complex procedure the following is assumed regarding moving in 3D environment Two procedures replace the 3D relocating To be able to use these procedures a node has to be selected by clicking on it in the primal grid The two procedures which only work when a node is selected in the primal grid are Adapt the X and Y coordinate by dragging a node in the primal grid Adapt the Z coordinate using the UP and DOWN keys 73 Main report Structural Design Lab TU Delft Class Node Description A point defined by three coordinates X Y and Z Characteristics X Y and Z coordinates Node number i in primal grid and polygon of dual grid Procedures Contains To check if a node is selected when the mouse is clicked AdaptLoading To adapt the loading when the or key is pressed while a node is selected Display DisplayPG The same as Display but without using the Z coordinate
5. D w m ai Mt IOSDSOTOSIOMMIGSSSGTOSSISSGSISGSGSSGSGGG990005 ml a a e a an a S I I I San a tun st N t st H i mr O N ieeconococonNnoocomoocososcssssososcser M HI a ona a LN 1 i i I I m darn st o i co OW CO st NANN mn Ee leeovoooo e000mMa00000R0R000000090000R000070 00 8 H a ona a ur 1 i I 1 G l a mn DO m lt i D oom on 3 cowmcooonnNocosoncssccsscssscsscasscan on on en O a ona a I i 1 i om 5 ru 00 H wT mw N S omoocoornmsoconNscscccsssccassoccaacon 5 Ol a ona a lt lt l l A 1a ano H oO 1 oO am m I tODSDISAMIGIOMSGISGSGSGIIISGSGSGSG99GG000959060 a Ia da a w i 1 i w o m com N R IOSDSSGTNIGIGOOSGISGSSGGG999G09900990099005 S go Th 7 2 i wdc wo Ss I HANN Co E LOSOTATOSISOMOGSGSOSSGSGGSSG9G99090090900595005 5 af SNS S l l gt un 0 O ot Qe WN st on LOOTNTOOSOSTPTOSCSOSSIISOSGISISGSGSG9999000999000 D m wt l ona 3 B Yp om O I am oH cd om l ag wt EES omNsoooconsocccsssscsssscscossoocoason x 4 mi ana a S HNM i i i 1 2 eo PS DO lend st co D OO Hdi INON 2 rs ae ot IsptOSGOMAGDSSISGSGSSG9G9990090090900599005 v Ear Nong q U 0 Lu Pi i S Ke a C Legen on eo 7 IH co bo E oS lowrooooTscococoecececocoeoecocsooscocsososcsss F m aa e Ii Oo a dida a g S me x LU gug pod l o VO EN Su LE uno a Ti nm Do e D OOMGE aj u C SC END Et H GG MH Ue ry Opp 5 u a ba 9 ok EEO Om
6. Using the geometry of the primal grid and the dual grid together with the nodal loading and the boundary conditions the minimum and maximum height of each point this problem can be solved using a one step linear optimization More about this optimization and finding the solution can be found in 4 5 The analysis is performed using several checks The network lines have to lie within the brick thickness which is represented by the minimum and maximum height of each node The stress in the structure should be lower than the maximum allowed stress The active stress must be determined by dividing the force in a network line with the masonry area related with that line The maximum allowable angle between bricks so that the bounding layer does not exceed the maximum thickness Or in other words to avoid gaps in the masonry pattern 44 4 Nodal loading The weights attributed to the loaded nodes come from distributing the dead load of the 3D area around those nodes In addition to this self weight loads such as asymmetric live loads can be applied An extra point of attention in this research all loads are applied in the same direction as is the case for gravitational loading It is recommended to expand this option by making it possible to add loading in any direction The loading should be divided into a horizontal and vertical part The vertical 45 Main report Structural Design Lab TU Delft Figure 4
7. Choose the rectangular or spherical model 2 Select the wished dimensions 3 Approximate the imported file with the parameter model Adapt the force network until a satisfying approximation has been obtained Tab 2 Design and analysis Adapt force network Change scalefactor 2 Move nodes 3 Change start variables YES mii i i ee Ce emi i i Se Tab 3 Masonry pattern l Set brickstone dimensions Brick length width and height Spacing of brick 2 Select to display or hide the pattern Adapt a variable Brick dimensions 2 Force network model Tab 4 Export end Display and select the layer to export The force network 2 The surface either a curves or a polygon model 3 The masonry pattern 2 Select file format to export as DXF to open in AutoCAD MEL to open in Maya RVB to open in Rhinoceros RB to open in SketchUP Figure 5 17 Workflow case 2 Shape is known check if masonry is possible T J van Swinderen August 2009 Structural Design Lab TU Delft POSITION IN THE DESIGN PROCESS The application is used during the conceptual design stage of the design process The whole process consists of several stages and the period of time it takes can be from a year up to several years Figure 5 14 The conceptual phase of the design is located in the beginning of t
8. DisplaySelected The same as Display but a different color is used to highlight the node Figure 5 5 Characteristics of Node class Description Connection between two nodes Characteristics Coordinates of the two nodes XI YI ZI X2 Y2 Z2 Two numbers i and j which are the node ID numbers Procedures getLength3D To obtain the length of the line to create the brick pattern getLengthPG To obtain the length of the line to form the polygon in the dual grid getAngle3DX To obtain the angle of the line with the X axis to rotate the bricks with getAngle3DY To obtain the angle of the line with the Y axis to rotate the bricks with getAnglePG To obtain angle in the primal grid to create the dual grid obtain To check if a line is clicked when the mousebutton is clicked update To update all coordinates after moving a node display displayPG Figure 5 6 Characteristics of Line class TJ van Swinderen August 2009 74 Main report 5 6 3 Change scale factor This is a factor that influences the outcome of the Thrust Network Analysis TNA and therefore is relevant for the application The relation between the primal and dual grid is for one determined by the scale factor If the primal grid consists of polygons formed by more than three sides the dual grid has not got an unique solution In this case the scale factor determines the scale of the dual grid and as a res
9. Figure 4 19 A Catmull Rom spline Figure 4 20 Catmull Rom spline derivation Figure 4 21 The effect of tension variable T TJ van Swinderen August 2009 56 Main report The steps to solve this matrix with an interation are Take a random internal node and calculate r by choosing a random value for the z values 2 Use the values for r and z in the calculation of the next random node 3 If the value for on the of the Z values is out of boundary adapt r until it is OK and restart the iteration Continue this process of iteration until all values for r are equal and the Z values are within their boundaries Global constraints The constraints belonging to the objective function are E LZ SZ 2 7 27 Solution The global matrix that has to be solved is shown in Figure 4 18 After taking all steps of the Simplex method Appendix C the final result is obtained This linear optimization formula is not integrated in the current version of the script The matrix is formed and the boundary conditions are calculated Both can be exported and seen in a text file 5 11 T J van Swinderen August 2009 Structural Design Lab TU Delft 4 6 Theory of Catmull Rom splines This theory is applied to determine the height and angles of each individual brick in the brick pattern THEORY The theory uses the coordinates and tangent of several points to determine the curve between these points This curve
10. His buildings were mostly roofed with thin shell vaults constructed of brick and ceramic tiles These forms were cheaper than reinforced concrete and did not require ribs and beams Several essays which embrace the technical and philosophical aspects of his work were written by Torrecillas 1997 In these essays explanation of deflection and stresses in double curved vaults are found 4 Quote by Eladio Dieste in Arquitectura y Construction Architec ture and Construction Torrecillas 1997 5 Quote by Pedreschi in The Engineers Contribution to Contempo rary Architecture 2 6 Pandeo de Laminas de Doble Curvatura Deflection in double curvature vaults English version Eladio Dieste 1943 1996 Calculation Methods T J van Swinderen August 2009 Structural Design Lab TU Delft ADVANTAGES OF REINFORCED BRICK STRUCTURES Brick is lighter in weight than concrete reducing the cost of the supporting structure or foundation Shorter construction period in comparison with concrete because there is no need for hardening of the brick and only a short time of hardening of the bonding layer Brick has good environmental properties its hygroscopic nature helps to control humidity Brickwork is easier to shape into double curvature forms since the material does not have to be deformed However form work is needed just as it is needed for in situ concrete Steel grids are also not complex to shape but
11. fire resistance and insulation 6 2 Last aspect for discussion is the construction and production of the structure There are two options in situ or prefab but which is best to use for these brick 3D models 6 3 Several technological developments discovered during research in the brick industry are shortly discussed 6 4 The last paragraph shows an interesting alternative that was encountered during the course of the research 6 5 T J van Swinderen August 2009 PRACTICAL ASPECTS OF BRICK S Structural Design Lab TU Delft 6 1 Brick fabrication and production 6 1 Non standard brick forms In a brick factory everything happens automatic and on an assembly line All machines for pouring cutting and transport are restricting dramatic changes in size and shape of the brick To keep the costs for the material low it is therefore needed to stay within the limits given by the machines in the factories The ranges of the dimensions which are used in the application are Width or length 160 280 mm Height 75 120 mm Depth 50 90mm The costs for non standard bricks are up to five times higher than for standard brick shapes and dimensions This is caused by the need for manual labor to produce non standard bricks Moreover this process requires more time 6 1 2 Texture As mentioned in 3 4 there are three possible ways of fabricating brick regarding the texture The choice of
12. in which the theory of Catmull Rom splines is used 4 6 and 4 7 Finally in the last paragraph 4 8 a list of limitations is presented in addition to the ones presented earlier 3 7 An important remark regarding the Thrust Network Analysis The majority of the material presented in this chapter is based on the article by Philippe Block and John Ochsendorf who are the authors and creators of this theory 10 T J van Swinderen August 2009 THEORIES AND METHODS Structural Design Lab TU Delft 4 1 Theory of force polygons Two descriptions of a force polygon are A closed polygon whose sides are vectors representing the forces acting on a body in equilibrium 2 The graphical representation of the internal and external forces of a structure POLYGON The first term to discuss is polygon This word derives from the Greek moAvc many and ywvia gonia meaning knee or angle Accordingly a polygon is many angles Today a polygon is more usually understood in terms of many sides An appropriate description of the term polygon is A plane figure bounded by three or more straight line segments Figure 4 1 These line segments are from here further called sides and the points where two lines meet are the polygon s vertices If the object is not closed it is referred to as polyline one line consisting of several line elements connected to each other When the obj
13. C Z H eN P HoP 2 26 2 Node 3 Nodes around node 3 2 4 and 7 Iy Ln ZJF Kia Z T K N3 2 K N3 Za K N3 In which Kik and K are 32 H Ha Ha Hg H K oF _ H H H 37 3 N3 p Z P 3 3 C H Kip THP 3 31 3 K C Z H gt N P H P 3 32 3 K C En H eN P H P 3 34 3 K C Ea oN P HP 3 37 3 Node 4 Nodes around node 4 2 and 3 Na ya Da ae Kona ei On ZK w Z T j van Swinderen August 2009 7 Structural Design Lab TU Delft In which K K and K are H HHA C H Hy H Kaas a 4 4 GB kin E El ig E G B o H HE o GQ H DoS E H oe All local objective functions are known Next step is to combine them and create the global objective functions and the belonging constraints Global objective function The global objective function is the combination of all separate objective functions of all nodes where the individual r for each node has to be the same r oa Ini Ero Es Ta And as a result Foral Im K 1 NI Z Ky vj Z K 3 NI Zi HK yi 24 Ks vi Z5 And Foa Ina Kiwa Ky yo 22 K3 2 25 HK 2 Zi Ko 2 Zo And Poa Pys Kin Zi Ko 3 Zo tK m3 Za HK 3 24 Ky 3 27 And r total f wa K 1 N4 Z K w4 2 Z HK 74 Zi Ka va Za This is shown in matrix form in Figure 4 18 55 Main report Structural Design Lab TU Delft
14. Due to the bending forces the element is also loaded by a shear force Only vertical support forces are active Figure 3 9 An example of bending based elements are straight beams The loading on a beam is distributed to the support points by bending and shear forces in the element This bending moment is a combination of tension and compression in the cross section of the element Force system based on axial forces These structures can either be based on compressive forces or tensile forces In general the amount of material needed to be able to withstand the axial forces is less than for bending forces Besides that an important difference with the bending system is the additional horizontal support forces Figure 3 10 A disadvantage of normal force structures is their shape Especially when the function of the structure is a building In these type of structures the volume and useful floor area are most important Though the normal force structures are often curved and especially near the supports the floor area can hardly be used Another disadvantage is the buckling behaviour If the compressive force reaches a certain level the element might fail due to buckling T J van Swinderen August 2009 Structural Design Lab TU Delft 3 3 2 Theory of form active structures Form active structures are structures in which the loading is taken by the form or the shape of the structure In general they are non rigid flexib
15. Heyman J 1977 Equilibrium of Shell Structures Oxford Clarendon Press 15 Billington D P 1982 Thin Shell Concrete Structures New York McGraw Hill Book Co 3 TU Delft Structural Design Lab Main report Figure 3 21 Possibilities with Processing and Java 23 i E is Figure 3 22 Results of application of algorithms Image courtesy of A Killian 32 August 2009 T J van Swinderen Main report 3 6 Computational design Processing JAVA and algorithms 3 6 1 The software Processing Processing is an open source project initiated by Casey Reas and Benjamin Fry both formerly of the Aesthetics and Computation Group at the MIT Media Lab It is a programming language and integrated development environment IDE built for the electronic arts and visual design communities which aims to teach the basics of computer programming in a visual context and to serve as the foundation for electronic sketchbooks One of the stated aims of Processing is to act as a tool to get non programmers started with programming through the instant gratification of visual feedback Figure 3 21 The language builds on the graphical capabilities of the Java programming language simplifying features and creating new ones Reasons to use Processing Besides the fact of designing with bricks interactivity is the aspect which has to distinguish this application from other available programs To assure the
16. Next step is to transform these lines into polygons These polygons have to be closed to assure equilibrium lf the primal grid consists of polygons existing of more than three lines the scale factor C is variable It will be set to 0 and can be changed manually afterwards The dual grid is shown in the right lower corner of the user interface This aspect is working in the current version of the prototype when a polygon is made of 4 lines When it has only 3 lines the script to form polygons and the dual grid is not working Extra features When a node is selected in the primal grid the corresponding polygon in the dual grid is highlighted in orange This enhances the ability to see what the effect is of any changes made to a node 5 7 6 Brick An assumption is that only standard brick shapes and sizes 77 TU Delft Main report Structural Design Lab PATTERN Primal grid Dua grid zeta 1 0 THE HETWORK MODEL PATTERN Primal grid Suallertd Duar grid THE NETWORK MODEL PATTERN HIDE Li E SHOW LAY ECT FILETYPE T AD COHF Primal grid Dua grid 2 col THE NETWORK MODEL PATTERN Model not exported yet Primal grid Figure 5 11 The four layers created by the application From top to bottom the force network the lines surface the polygon surface and the brick pattern T J van Swinderen August 2009 78 Main report are used which can be
17. advantages and disadvantages of it 3 1 1 Types and dimensions of brick 3 1 2 Structural information such as strength and bonding properties 3 1 3 Brick in free form designs 3 1 4 3 1 1 Introduction to brick In Figure 3 1 some pictures of brick and brick structures are shown Brick performs best when it is subjected to compression force because the tensile strength of brick is very low Vertical straight elements such as walls for houses warehouses factories and garden barriers are mainly subjected to vertical loading assuming the horizontal loading such as wind is transferred using the floor system or an other structure Therefore brick is in general only used in these type of elements An exception are arches although this type of structure is regarded as two dimensional as well height and length and can therefore be regarded as a flat plane as well When bricks are combined into a structural element such as a wall a bonding layer is used between them The combination of bricks and bonding is referred to as masonry 18 ADVANTAGES The use of materials such as brick and stone increases the thermal mass of a building giving increased comfort in the heat of summer and the cold of winter In general brick will not require painting and so can provide a structure with reduced life cycle costs Nevertheless an appropriate sealing of brick will reduce potential burst and failure due
18. application is interactive there are two requirements The ability to adapt the shape in a simple and fast way 2 The new shape has to be analysed and results must be given fast so that the concepts can be altered real time during a meeting Processing offers the possibilities for both requirements and for that reason it has been chosen to use during this research and to design the new application with 3 6 2 Programming language Java This is one of the three modes available in Processing and is the most flexible one allowing complete Java programs to be written from inside the Processing Environment as long as they re still subclasses of PApplet This mode is for advanced users only and is not really recommended Using this mode means that any additional tabs will no longer be inner classes meaning that you ll have to do extra work to make them communicate properly with the host PApplet It is not necessary to use this mode just to get features of the Java language T J van Swinderen August 2009 Structural Design Lab TU Delft public class MyDemo extends PApplet void setup size 200 200 nostroke fill 0 102 153 204 void draw background 255 rect width mouseX height mouseY 50 50 rect mouseX mouseY 50 50 A good example is the digital hanging chain modelling tool created by Axel Kilian CADenary This modelling program has been made in Processing The program
19. as free form and create renderings and digital models The traditional straight forward designs such as factory halls and houses are functional and useful for application in residential and industrial areas But due to their everyday appearance they tend to be less interesting for more expressive functions and fields of architecture Moreover buildings become higher highrise spans bigger column free space and fagades more complex challenging appearance all to satisfy the wishes of the project initiator and to comply with the rules and need for sustainability However designing free form structures creates a challenge for both the architect and the structural engineer The engineer designs structures using basic structural mechanics and sometimes with the help of physical models Due to the increasing complexity the engineer needs to use finite element methods and algorithms to calculate the complex and non standard elements and structures such as double curved surfaces T J van Swinderen August 2009 Structural Design Lab TU Delft As a result the communication with the architect tends to be more intense and proves to be more difficult and time consuming the methods and algorithms need to be designed tested and finally applied The resulting shape and structure might not satisfy the architect As a result the design and model have to be changed and the structure has to be adapted to this new design This wh
20. concrete However the shell designs in this research are bounded to compression only and therefore the freedom and range of possible shapes and designs is reduced significantly One of the earliest goals for this research was free form designing However during the research process this principle had to be moderated and eventually dropped Two early made choices are the cause The choice to use the Thrust Network Analysis which is based on compression only structures and 2 The assumption of no reinforcement which again implies compression only structures Due to this the end product is not actually producing free form designs The possible angles and curvatures are restricted and so the range of shapes is not as big as hoped for Instead the designs are double curved shapes FUNCTION OF THE SHELL DESIGN An assumption was to take pavilions into account only One of the reasons for this was the ability to neglect building physics in the design process 103 Main report Structural Design Lab TU Delft T J van Swinderen August 2009 104 Main report THE TOOL One of the aims of the research was to create a stand alone application with the following abilities e To import and start with an earlier designed shape in 3D software such as Rhinoceros e To export and use the result in the further design process The tool performs a force flow analysis of a shell shape The stability such as buckling beh
21. continue with step 3a until the whole line is filled with bricks Step 4 The result is a masonry pattern of rings of brick Figure 4 30 right One disadvantage is gaps between the rings when the curvature of the surface is too high Figure 4 31 Another disadvantage are the top rings Their radius is too small to be able to fit bricks in it and still make sure the angle between the bricks is not too big Figure 4 32 Therefore either a closing element is needed or the top of the structure is left open The last option is not recommended because it has a bad influence on the force flow and moreover it is often not desired when the shell covers a space When a closing element is used this element does have influence on the force flow since it has adds a loading to the top ring nodes This loading has to be added in the analysis ie N CORRECT Figure 4 32 Problem of the top rings with small radius The solution is a closing element that replaces these rings T J van Swinderen August 2009 Structural Design Lab TU Delft 4 8 Limitations When nodes are located to close in the primal grid the result of the dual grid is unrealistic An exact value for this limitation has not been obtained Therefore it is recommended to use the program and change the primal grid with a common sense When the dual grid is not realistic the results of the analysis should not be used The grid is li
22. easily made by brick producers Moreover only one brick shape is used to generate the masonry pattern Flowchart Figure 5 10 Extra features The brick pattern can be displayed or hidden in the 3D model by toggling a button in Tab 3 Brick pattern In this research only one color is used for the brick A good addition would be the possibility to use more colors so that patterns can be created using certain color schemes This has therefore been added as a recommendation for further research to extend the ability to adjust the appearance of the brick facade Chapter 8 T J van Swinderen August 2009 Structural Design Lab TU Delft 5 8 Export options 5 8 Stresses and forces in structure There is an option is to generate a text file in which the information of all lines and nodes is given This information consists of Node number Node coordinates Nodal loading Line lengths Line stress The loadings in the lines connected to foundation nodes are separated from the other lines When designing the structure the foundation is an important aspect especially for shell structures To design the foundation the relevant forces must be known The force has a value and a direction The force should be divided into its horizontal and vertical component 5 8 2 Export final model Several layers are created when the application is used so that the results are available for all actors that are acti
23. exported as several file formats There are four layers to export The force network Surface created with lines Surface created with polygons he The masonry pattern And there are four file formats to export the data to be used in these software programs AutoCAD 2 Maya 3 Rhinoceros 4 SketchUP 5 3 Display the 3D model The 3D model consists of several layers which can all be selected and displayed These layers are The force network white lines The surface displayed with curves green blue lines The surface displayed as polygons black planes white lines The masonry pattern red cubes How to display them switch between them export them and adapt them is explained in the user manual Appendix F 69 Main report Structural Design Lab TU Delft TJ van Swinderen August 2009 70 Main report 5 4 Variables to adapt The application offers several possibilities to control the model and its outcome Nodal loading for each individual node in the network 5 4 1 2 Load case 5 4 2 3 Relocate nodes in the 3D model and primal grid 5 4 3 4 Adapt the relation between the primal and dual grid scale factor 5 4 4 5 4 1 Nodal loading This value should be determined by calculating the surface area transferred to the node This is a rough approximation since it is a complex matter to predict what the area for each node will be I
24. is available online 25 3 6 3 Algorithms An algorithm is a sequence of finite instructions steps often used for calculation and data processing It is a method in which a list of well defined steps for completing a task is run through The transition from one step to the next is not necessarily deterministic some algorithms incorporate randomness These are known as probabilistic algorithms Nowadays algorithms are most of the times connected to the use of the computer because of the speed they offer That way the result from an algorithm is obtained fast When looking at algorithms from this way they are essential to the way computers process information The analysis and study of algorithms is a discipline of computer science and is often practiced abstractly without the use of a specific programming language or implementation In this sense algorithms analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm and not on the specifics of any particular implementation Three examples of the results images and analysis that are obtained when algorithms are used are shown in figure 3 22 16 CADenary form finding project developed in Processing by Axel Kilian using a Particle Spring library developed by Simon Greenwold 12 33 Main report Structural Design Lab TU Delft TJ van Swinderen August 2009 34 Main report 3 7 Overview of choices and ass
25. is described by a formula consisting of the coordinates of the four points and a factor tau The curve always passed through these points and the curve is adapted according to the position of each of the four points The last remark is the reason to use Catmull Rom splines when a position of a point is changed the curve is changed as well and still passes all points FORMULAE To calculate the angle for each individual brick in the pattern the theory of Catmull Rom splines is used 15 Catmull Rom splines are a family of cubic interpolating splines formulated in such way that the tangent at each point p called a is calculated using the tangent of the previous and next point on the spline p a and p _ a_ Figure 4 19 Consider a single Catmull Rom segment p s Suppose it is defined by four control points P p P and p Figure 4 20 The tangents for the non border points p_ and p are Pat Dap pt PP Note that the tangent at the border points p and p _ is not clearly defined For those points one of the two angles is not known The tangent is set to Po t p p P Pend Pan 7 Dna omer 7 Pat The formula for the curve segment between p and pis pls c Herude ww te u 57 Main report Structural Design Lab TU Delft Surface section in Y TOPVIEW direction Zx Surface section in X direction Figure 4 22 Point P which is the centerpoint
26. is frozen for instance by using a mirror or taking a photo and the resulting model is rotated 180 degrees In the new model only compression forces are present in the structure instead of tensile forces Figure 3 12 To make the shape even more realistic weights can be added to the chains which represent nodal forces in the compression structure By playing with the length and connections of the chains he could make an architectural design This method of designing is the exact opposite of the traditional way of structural design Normally the form of the building is given and the structure is determined according to that while in this case the structure is determined and the form follows the shape of this structure Relevance of Gaudi s work for this research Gaudi designed by adapting the shape and size of the structure instead of the dimensions of the elements He rather changes the form according to his wishes and the possibilities of the structure form finding instead of calculating dimensioning and adapting structural elements of one fixed model This aspect is one of the objectives of this research Pier Luigi Nervi This Italian engineer 1891 1979 is renowned for his reinforced concrete structures The hangars and halls he designed in the end of the 1930 s such as the Palazzetto dello sport Figure 3 13 are good examples of his common used ribbed structures Relevance of Nervi s work for this res
27. lt M2 SS er Uw o OAN on st NO In 0 DO A NN MN sf IO In DO DO AN MM st WO gt EE 2220r UZHNMNONO ODHAHHHAHAHHAHAHHAHNNNNNNNNNNAMAMMAMMA 92 30 Part of the text file of the matrix that needs to be solved August 2009 T J van Swinderen Figure 5 Main report Structural Design Lab TU Delft Primal grid P Heights bd Notepad P Objective functions R t lt t Notepad File Edit Format View Help File Edit Format View Help NODE HEIGHTS AND BOUNDARIES NODE OBJECTIVE FUCTIONS R Node number Lower boundary m Height z m Upper boundary m Node number Lower boundary Final R Upper boundary a 1 608 1 617 1 625 1 0 728 0 752 0 777 2 2 626 2 634 2 643 2 0 729 0 751 0 774 3 3 145 3 154 3 162 3 1 102 1 133 1 163 4 3 145 3 154 3 162 4 0 961 0 988 1 015 5 2 626 2 634 2 643 5 1 003 1 033 1 063 6 1 608 1 617 1 625 6 0 707 0 731 0 755 7 2 647 2 655 2 664 7 0 898 0 925 0 953 8 4 511 4 519 4 527 8 0 760 0 781 0 802 9 5 548 5 556 5 565 9 1 043 1 069 1 096 10 5 548 5 556 5 565 10 0 840 0 861 0 882 11 4 511 4 519 4 527 11 0 755 0 775 0 796 12 2 647 2 655 2 664 12 1 090 1 123 1 157 13 3 182 3 190 3 198 13 0 784 0 806 0 829 14 5 586 5 995 5 603 14 0 657 0 674 0 690 15 7 002 7 010 7 019 15 0 771 0 788
28. model b Import a model and approximate it with a parameter model Main report Structural Design Lab Start with a node with no neighbouring foundation nodes In this case 2 5 8 or 10 gt node 10 a A gt B a1 61 TU Delft a2 177 a3 299 a Start with a line keep angle the same b Connect next clockwise to end of it c Connect next clockwise to end of 2nd d Find intersection between 1st and last line and calculate scaling factor of the lines to form the polygon to use in the other steps of the process b A B u RESULT Force polygon d A Angles are the same and have to stay the same Length proportion is known BUT can still change This will be seen when regarding the other nodes NODE 10 Length 2 Length5 14 0657 14 1419 1 00 gt 1 005 Length 8 14 9754 1 065 Figure 0 1 Example of a reciprocal relationship Left is the network shown primal grid en at the right is shown how the reciprocal figure of node 10 is determined T J van Swinderen August 2009 2 Main report The interface is designed to be user friendly Figure 0 2 top picture At the top of the interface the four tabs are located that represent the four steps Several layers are created when the application is used so that the results are available for all actors that are active in the continuation of the design process For the structural e
29. only be adapted for the complete shell It is advised to make this possible only for selected areas of the shell as well More regarding this aspect is found in the report of P Block 16 7 ANGLE CHECK The angle between the bricks must be implemented as a check whether the masonry pattern is realistic 4 7 This check consists of two parts Check of the angle of the selected brick and the neighbouring bricks within the centerline 2 Check of the angle of the selected brick and the neighbouring brick in the neighbouring line of bricks These checks are to be implemented in future research 8 GAPS INTHE MASONRY PATTERN The gaps between the lines of brick in the linear pattern must be repositioned so that the gaps in perpendicular direction are closed Figure 5 33 This extra step is to be implemented in future research TJ van Swinderen August 2009 Structural Design Lab TU Delft 95 Main report Structural Design Lab TU Delft Figure 6 1 Color variety in brick T J van Swinderen August 2009 96 Main report CHAPTER Introduction In this chapter the aspects concerning brick structures in the building practice are discussed During and after the design process an important aspect is the production and fabrication of brick Which shapes are possible which colors which textures etc 6 1 Another aspects are the characteristics of brick concerning building physics like water
30. order analysis is not regarded The Gaussian curvature should be positive in the whole structure This limits the range of possible heights for every node since it is dependent on the height of surrounding nodes This check has not been implemented in the prototype of the application The tensile forces in the hoop direction of the surface are not analysed and checked 35 Main report Structural Design Lab TU Delft Figure 4 I Examples of polygons Figure 4 2 Polygon classified by number of sides DOO RAZI equilateral square regular regular regular regular triangle pentagon hexagon heptagon octagon Figure 4 3 Polygon classified by convexity convex polygon concave polygon Figure 4 4 Polygon classified by symmetry T j van Swinderen August 2009 36 Main report CHAPTER Introduction In this chapter the theories and methods that are used in the research are presented One of the basic concepts is the force polygon 4 1 Once this concept is clear the theory of the Thrust Network Analysis is discussed After an introduction 4 2 an explanation is given why and how this theory is useful for this research 4 3 The next step is to show and explain the steps taken in this theory 4 4 One of the steps is to solve a linear optimization problem for which the Simplex method is used 4 5 The next two paragraphs focus on the surface and brick pattern generation
31. pull wires together and clamp crown pre stress ends of wires anchored NEEN push wires apart and insert block to hold pre stress Remo Pedreschi valley pre stress c The reinforcement is tensioned by pulling the wires towards each other in the middle of the span and anchoring them Image courtesy of R Pedreschi 2 T J van Swinderen August 2009 Structural Design Lab TU Delft Figure 3 7 Reinforcement in the designs of Eladio Dieste CORTE 14 Nec ldming Aem Tanev 2 sar zac Een 406 en NEEN Ver lamina Es aren a teo enia junta ce le losa A AD jonta nO met mn ae eed rien HI TAFU Sep final 15 20 30 40 Salombres oS Sap SO 2 12 L 170 D g 142 L170 ag t2 Lodi Sg 42 L 17 9 42 1 170 a Crown reinforcement scheme seen from the top Image courtesy of R Pedreschi 2 Dieste Archive b Crown reinforcement installed after the bricks have been placed but before they are tensioned in the middle After the tensioning a covering layer is poured over the reinforcement too protect it Image courtesy of R Pedreschi 2 20 Main report The technique enables the vault to be reinforced to resist these secondary stresses The form of the vault and its ability to cantilever long distances as used in Massaro make it suitable for large canopy type structures providing shelter rather than enclosure REIN
32. research the objective function for the linear optimization problem is r the inverse of the scale factor C This variable r either has to be minimized or maximized to obtain the envelope in which the model has to be placed The result of this objective function is respectively the shallowest and deepest solution still within the limits for a chosen combination of primal and dual grid 4 5 3 Solution for one point Determination of the general problem for one node The objective function is to maximize or minimize the value of r where r is described as l a EO FO Pr C ZZ ZZ Ea an i P j P k P l Next step is to introduce a new variable in this case K to simplify the process to find the solution All values in T J van Swinderen August 2009 Structural Design Lab TU Delft the description of K are given on forehand and are constant during the linear optimization process They can be altered after the optimization has been performed r K 2 K z K 2 K z 8 In which the constants K are described as il Hy Ha _ Hi C K P P C H K PHP K Cc oe Hi Do m K Q 5 H P H P Determination of the boundary conditions constraints Formula 7 is divided into two constraints a the lt constraints and b the 2 constraint These constraints are applied to the considered node land to the nodes connected to the force network lines corresponding
33. that they can be used in the continuation of the script Description of the problem At some positions of the surface the pattern script is failing Figure 5 27 This is caused by the shape of the primal grid This shape is a result of the decision to equalize the length of the lines in the 3D force network As a result the related primal grid has angled lines The pattern script fails where the related nodes for the Catmull Rom theory are obtained Figure 5 28 In the yellow blocks the starting point of the failure is shown The remaining of the bricks of that masonry line is a continuation of this failure at the beginning Moreover the aspect of assuming a certain value for a point at the border of the surface is questionable The best situation would be when these border points are related to the angle of the surrounding surface around these points This has not been scripted and so is not part of the current version of the program Solution to this problem Review the script by an expert in scripting and add the missing aspects as mentioned above T J van Swinderen August 2009 Structural Design Lab TU Delft 4 PERFORMING THE ANALYSIS A matrix is generated using the theory explained in Appendix C First the matrix values are determined Figure 5 30 Another matrix is created with the boundaries for the height of all nodes Figure 5 31 After the analysis has been completed the values for the inv
34. the solving of two matrix problems 4 2 3 of 16 The script of this solution method is an aspect that needs to be researched further 87 Main report Structural Design Lab TU Delft Scalefactor L1 L2 Figure 5 21 Step 3 of the loop to form the dual grid the scalefactor of the related polygons is determined Figure 5 20 Step 2 of the loop to form the dual grid the related polygons of the selected polygon are searched for Scale in all directions Scalefactor 1 3547 Figure 5 24a Scaling of a polygon in direction Figure 5 24b Scaling of a polygon in all directions Primal grid Figure 5 25 Screenshot From left to right 3D primal grid and dual grid In the last figure is displayed that there are different scale factors TJ van Swinderen August 2009 88 Main report Structural Design Lab TU Delft Scalefactor L1 L2 11 733 14 9754 0 7835 New length 0 7835 14 9754 11 733 gt Figure 5 22 Step 4 of the loop to form the dual grid the scalefactor x2 x3 newx3 is applied to the whole polygon in this case for polygon 0 new x2 newx1 Figure 5 23 Step 5 of the loop to form the dual grid the polygon is translated to the right position in this case polygon 10 is translated Comparison of different results of scaling Scalefactor 1 3547 Figure 5 24c Comparison of scaling in and all directions EE en
35. though the lines connected to a node in the dual grid form a polygon of forces in the primal grid This two way relation is known as a reciprocal relationship The mechanical property of reciprocal diagrams is expressed in the following theorem by professor Maxwell If forces represented in magnitude by the lines of a figure be made to act between the extremities of the corresponding lines of the reciprocal figure then the points of the reciprocal figure will all be in equilibrium under the action of these forces The steps to visualize this reciprocal relationship are For every node in the primal grid excluding the foundation nodes obtain the related lines 2 Make a polygon of these lines by adding one to the end of the other Start with any line and add the next as it 4 Maxwell Professor Clerk 1864 Philosophical Magazine p 258 11 T J van Swinderen August 2009 Structural Design Lab TU Delft is the next in clockwise order around the node in the primal grid 3 If any of the lines of the regarded node is not connected to a foundation node Close the force polygon by finding the intersection point of the two lines that are not connected yet 4 Place and scale all the polygons Determine the range of possible angles for the foundation lines and set it at the midpoint of this range A more detailed explanation including pictures and drawings is found in Appendix A 4 4 3 Solve the problem
36. vectors and it is complex to generate the correct line elements Moreover the surfaces imported as OBJ files are polygon surfaces and have no relation with a structural system force distribution or network Therefore it is advised to use the imported model only to approximate it with a parameter model and perform the analysis on this parameter approximation model As a result additional steps have to be taken after importing before an analysis can be done and a pattern generated T J van Swinderen August 2009 Structural Design Lab TU Delft 5 6 Procedures Several procedures are used in the application An analysis method the Simplex algorithm 5 6 1 Method to relocate points of the 3D model 5 6 2 Possibility to adapt the scale factor 5 6 3 Create a force network to begin with 5 6 4 One more procedure is the generation of the masonry pattern Due to its significant contribution and relevance for the research it is discussed in another paragraph 5 9 5 6 1 Implementation of the Simplex method With the Simplex method the linear optimization problem is solved The results are used to calculate the forces in the line elements of the force network model The constants of the optimization problem are a combination of characteristics of the network the lengths of the line elements in the primal and dual grids and the loading of each node 4 4 and 4 5 Flowchart Figure 5 4
37. with node i As a result the amount of constraints is dependent on the number of lines connected to the node In formula noc 2 1 nol In this formula noc is the Number of Constraints and nol is the Number of Lines e g a node with three lines has 8 constraints and a node with four lines has 10 constraints More information about how to make these formulas suitable for solving is found in Appendix C Generate the matrix Objective function P KZ PK Z KZ KZ Constraints Glz z bizz 5 Main report Structural Design Lab TU Delft Figure 4 16 Final matrices when regarding only one node Minimize r Maximize ro I E E E E I I I K Z FK z FRZ K zi K Zi KZ Ky Ze Ki Zi a Minimizing r b Maximizing r Figure 4 17 3D model used in the example to show the Simplex method for more nodes On the right the primal grid topview is shown T J van Swinderen August 2009 52 Main report The result written in matrix notation are found in Figure 4 15 Solving the matrix The important steps to solve this matrix are Write all constraints in lt form This is achieved by introducing an extra variable y 2 Write all constraints in form This is achieved by introducing an extra variable x 3 Minimize all y variables until they are all 0 After this they can be eliminated from the matrix 4 Solve the remaining matrix until r has either been maximized or minimized Exact details
38. 0 805 16 7 002 7 010 7 019 16 1 002 1 023 1 045 17 5 586 5 595 5 603 17 0 878 0 900 0 922 18 3 182 3 190 3 198 18 0 806 0 830 0 853 19 3 182 3 190 3 198 19 0 719 0 739 0 760 20 5 586 5 595 5 603 20 1 045 1 071 1 097 21 7 002 7 010 7 019 21 i 171 1 196 1 222 22 7 002 7 010 7 019 22 1 107 1 131 1 155 23 5 586 5 595 5 603 23 0 837 0 858 0 879 24 3 182 3 190 3 198 24 1 136 1 170 1 203 25 2 647 2 655 2 664 25 0 693 0 714 0 735 26 4 511 4 519 4 527 26 0 904 0 929 0 954 27 5 548 5 556 5 565 27 0 772 0 791 0 811 28 5 548 5 556 5 565 28 0 806 0 826 0 847 29 4 511 4 519 4 527 29 0 943 0 969 0 994 30 2 647 2 655 2 664 30 0 855 0 881 0 908 31 1 608 1 617 1 625 31 0 715 0 739 0 763 32 2 626 2 634 2 643 32 0 676 0 697 0 718 33 3 145 3 154 3 162 33 1 268 1 303 1 339 34 3 145 3 154 3 162 34 1 032 1 061 1 090 35 2 626 2 634 2 643 35 0 922 0 949 0 977 E 2 608 eee oe 36 0 634 0 656 0 677 38 0 Q 0 The lowest minimum r is 0 6344559 39 Q Q 0 The highest minimum r is 1 2679008 40 Q 0 Q The lowest maximum r is 0 6769748 41 0 Q Q The highest maximum r is 1 3390043 42 0 0 0 43 Q 0 0 The minimum value of the top boundary R values 0 6769748 is LOWER than the maximum
39. 12 Valency and the influence of the scalefactor Image courtesy of Philippe Block Hy Pr 12 A pe scale Z unique reciprocal multiple reciprocal solution solutions a Valency is 3 and the dual grid is set b Valency is 4 and the dual grid has several solutions The scalefactor determines the final result E Mn a e E n a a a an nn En iy a eg ne c Decreasing the scale factor C of the dual grid means overall lower horizontal forces in the system and hence a deeper solution for the same set of applied loads TJ van Swinderen August 2009 46 Main report part is still used to solve the problem though the horizontal part has to be added in the force polygon If and how this influences the reciprocal relationship and the final result is a topic for further research 44 5 Scalefactor C and its influence on the solution The user can manually change the force distribution by changing the scale between the primal and dual grid This scale factor is not always of importance This depends on the valency of the grid A grid where the nodes have three lines connected to them has valency 3 The same way a grid where the nodes have four lines connected to them has valency 4 When a grid has valency 4 or higher it is regarded as an indeterminate grid In this case the scalefactor has influence on the result of the analysis Figure 4 12 Decreasing the scale factor of th
40. 3D force network model after selecting it Adapt the nodal loading 5 7 2 Lines Characteristics A line is given a red color when it is overloaded or is in tension When one of the two nodes it is connected to is selected it is highlighted in orange Flowchart Figure 5 6 This aspect is not working in the current version of the prototype because the analysis is not performed completely yet due to the problems with scaling the dual grid correct Constructor 75 Main report Class Polygon Description Several nodes connected by lines to form a closed polygon Characteristics Coordinates of a certain amount of nodes XI YI Xn 1 Yn l Xn Yn Identification number i which is the number of the node belonging to the polygon ID s of the lines forming the polygon IDI IDn 1 IDn Procedures update To update the coordinates and position of the polygon display Figure 5 7 Characteristics of Polygon class Step 2 Compose Dual Grid The arraylists Node and Line elements forming the Primal grid are needed The scalefactor is set to 1 0 at start unless the primal grid only has one solution Create polygon for all nodes except foundation nodes Find all connected lines of every node Connect these lines in a clockwise order putting the next line to the end of the last line Close the polygon by finding the intersection of the two lines that are not connected y
41. 44 0 0 0 NO solution is possible 45 0 Q 0 46 0 0 0 47 0 0 0 48 0 0 0 49 0 0 0 50 0 0 0 51 0 0 0 52 0 0 a 53 0 Q Q 54 0 Q Q 55 Q Q 0 56 0 Q 0 57 0 0 0 58 0 0 0 59 0 0 i 60 0 Q e Figure 5 32 Text file of the final results for R the inverse of the Figure 5 31 Text file of the height limits for all nodes scalefactor C TJ van Swinderen August 2009 93 Main report Structural Design Lab TU Delft 1 SETUP FORCE NETWORK MODEL 2 DESIGH AND ANALYSIS 3 MASONRY PATTERN 4 EXPORT TEXT AND MODELS LENGTH OF BRICK SET TO 210 MM 100 0 mm DEPTH OF BRICK SET TO 100 MM HEIGHT OF BRICK SET TO 50 MM THICKHESS OF THE BONDING LAYER SET TO 5 MM DO HOT SHOW BRICKS B ERICK ON HETWORKLINES BRICK SHELL Figure 5 33 Gaps in perpendicular direction of the brick lines TJ van Swinderen August 2009 Main report 5 EXPORT LAYERS OF THE MODEL It is possible to export several layers in the current version of the prototype but in some occasions the model is not positioned in the origin of the axis and it is rotated This has to be reviewed in future research 6 NODAL LOADING In the current version of the prototype the initial load on every node is set to a certain value In a realistic situation this value should be linked to the surface and thickness of the area surrounding the node The thickness of the shell surface can
42. Analysis TNA 10 In this research an interactive design tool has been developed with which masonry shells can be rapidly designed and analysed The new tool uses the TNA theory which is based on three dimensional equilibrium of force networks After designing and analysing the conceptual shape of the shell it has to be materialised Following examples by Eladio Dieste whose designs are the inspiration to use brick as building material the tool is able to generate masonry patterns and takes into account manufacturability constraints for this material THEORIES Thrust Network Analysis The most relevant theory is the Thrust Network Analysis which makes use of force polygons and the reciprocal relationship This analysis performs a check of the force flow Stability checks such as buckling behaviour and displacements are not regarded Force polygons When the sides of the polygon are representations of forces in a network the resulting polygon is regarded as a force polygon The lengths of the sides represent the magnitude of the corresponding force If the polygon is closed the body on which these forces act is in equilibrium T J van Swinderen August 2009 Structural Design Lab TU Delft Reciprocal relationship The relation between the primal and dual grid can be described as lines connected to a node in the dual grid form a polygon of forces in the primal grid and vice versa This two way relation is kno
43. BSTRACT l INTRODUCTION 2 RESEARCH QUESTION AND TOPICS 2 l 2 2 Research question Research topics 3 INFORMATION FROM THE LITERATURE STUDY 3 1 3 2 3 3 3 4 3 5 3 6 3 7 The building material brick 3 1 1 Introduction to brick 3 1 2 Types and dimensions of brick 3 1 3 Structural information 3 1 4 Brick in double curved designs The masonry structures of Eladio Dieste 3 2 1 Free standing barrel vault J22 Gaussian vaults Form active structures 3 3 1 Theory of force distribution systems 332 Theory of form active structures 3 3 3 Engineers designing form active structures Shell structures 3 4 Thin shell structures 3 4 2 Shell geometry 3 4 3 Structural mechanics of shells Structural analysing methods Computational design Processing JAVA and algorithms 3 6 The software Processing 3 6 2 Programming language Java 3 6 3 Algorithms Overview of choices and assumptions and limitations 4 THEORIES AND METHODS 4 4 2 4 3 Theory of force polygons Introduction of the Thrust Network Analysis The benefit for this research T j van Swinderen August 2009 Structural Design Lab TU Delft 3 3 3 I5 I7 I7 19 19 21 23 23 23 23 27 27 29 29 3 33 33 33 33 35 37 37 4 4 Main report 4 4 4 5 4 6 4 7 4 8 4 3 1 Architecture 4 3 2 Mechanics Explanation of steps 4 4 Construct the primal grid 4 4 2 Generate the dual grid 4 4 3 Sol
44. E eS f2S s 22 EEE en en en en EE Figure 5 27 Failure of the masonry pattern script at several locations in the pattern men gE Zi T Ese H 1 en eee bal peer fb ib a E eee an I TE A i KE Ee Es ks LET H gee OAU Reap Arb lt b aF it Ps os wudaass PETA EE f ret er yi eet PERIN TE few Figure 5 28 In yellow the beginning of the failure of the masonry pattern is shown TJ van Swinderen August 2009 Main report 3 GENERATION OF THE MASONRY PATTERN The masonry pattern procedure is partly working in the current version of the prototype Procedure to create the masonry pattern Draw lines every brickwidth Figure 5 26 2 Approximate the length of these lines using the coordinates of the force network and assuming straight lines between these known points 3 Determine the amount of bricks per line by dividing the length with the brick length Procedure to find coordinates and angles of one brick this procedure is run for every line of bricks Determine the coordinates of the center point of the brick using the coordinates of the brick before it 2 Determine the angles of the brick using the theory of Catmull Rom splines by using the coordinates in the derivative of the Catmull Rom formula 3 Store the values so
45. FORCEMENT AND PRESTRESSING Crown top of the vault reinforcement Consists of looped prestressing wires Figure 3 7 They are placed on the top of the vault once the bricks have been installed and the reinforcement between the bricks grouted in place Each end of the loop is embedded in reinforced anchorages tied with steel rods to the vaults The central part of the loop remains free and rests on top of the vault The distance between the two sides of the loop is critical Once the anchorages have sufficient strength the loops are pinched together at the middle point causing them to stretch and generate inward reactions at the anchor points pre compressing the vault The top of the vault is then covered with a light layer of concrete to cover and protect the cables Valley reinforcement Consists of two overlapping loops The ends of each loop are anchored into concrete and tied in the vault A specially developed jack is placed between the overlapping cables that pushes the ends of the loop apart stretching the cables Once the required extension has been reached a steel block is placed between the two loops to maintain the separation of the cables When the jack is relaxed the cables tighten onto the steel block locking in the prestress force 3 2 2 Gaussian vaults The Gaussian vault Figure 3 6 has evolved out of the barrel vault extending the use of the catenary to shallower and longer spanning vaults The shape is also u
46. Further research at the Massachusetts Institute of Technology M I T into the theory of force polygons 3D line networks and the Thrust Line Analysis has resulted in an enhanced theory the Thrust Network Analysis In short this is the 3D I Block Philippe 2005 Available at http web mit edu masonry interactive T hrust T J van Swinderen August 2009 Structural Design Lab TU Delft application of the Thrust Line Analysis instead of regarding only two neighbouring nodes and lines the network is extended to three dimensions This is discussed in the next paragraphs 39 Main report Structural Design Lab TU Delft Figure 4 9 Theory of the Thrust Line Analysis method Image courtesy of Philippe Block left Two possible compression only equilibrium shapes for a random set of loads and right an interactive thrust line application the user can adapt the geometry by dragging control points and the structural feedback in the form of a thrust line is updated in real time The magnitudes of the forces in the system are visualized in the accompanying funicular polygon right TJ van Swinderen August 2009 40 Main report 4 2 Introduction of the Thrust Network Analysis The Thrust Network Analysis TNA is a new methodology for three dimensional equilibrium calculations The theory presents a methodology for generating compression only vaulted surfaces and networks Two important aspects are The
47. Proposal for a tool to design masonry double curved shells Analysis of conceptual models and generation of a masonry pattern T J van Swinderen eee eee see eee cee eee eee eee eee ee eee eee eee eee eee eee eee eee ee eee ee eee eee eee eee ee ee eee er eee reese eeeeeeeeee ere eee eee eeeeeeeee eee eee eee eeeee eee eee e ee eeeeeeeee eee eee eee eee eee eeee eee eeeee TUDelft Delft University of Technology eee se ee ee es eee se ee eee eee ee ee ee ee es ee es eee ee ee es ee ee ee es ee e ee ee ee eo ee ee ee ee ee ee ee se ee ee ee ee es eee se ee ee es ee ee ee ee ee ee ee es ee eee eee ee ee ee ee ee ee se ee se ee ee ee eee see es eee es ee es eee es eee eee eee sees T J van Swinderen eee eee ese es eeseeeseseeseseeeeseseseeseseeeeseeseseseeseseeseeseeseeseseeeeseseseeseseeeeeseesseeseeeeseeseesesseeeeseeseee Delft August 2009 Delft University of Technology U ft Faculty Civil Engineering and Geosciences Masters Degree Building Engineering Delft University of Technology Specialization Structural Design Main report E E The interest for structural design and computational design started during the course CT5251 Special Structures and the design project AR0651 XXL Design Engineering The combination of special shapes structures and the digital world forms the basis for the decision to choose the field of computational design for the graduation project The final results of the Master s the
48. S TJ van Swinderen August 2009 12 Main report CHAPTER Introduction In order to have an adequate level of information and knowledge regarding the topics involved with this research a literature study has been performed during the first phase of the research Several topics have been researched Brick structures Shell structures Free form design Typology of structures Form active structures Computational design Geometry descriptions Analysis methods The building practice The information of these topics that was considered useful in the continuation of the research is presented in this chapter Following the reasoning as presented in the Chapter 2 2 2 the main topics are The material brick 3 1 Eladio Dieste and his masonry structures 3 2 Form active structures 3 3 Shell structures 3 4 Methods to analyse shell structures 3 5 Computational design tools 3 6 In the end a list of the decisions assumptions and limitations that have effect on the outcome of the research is given 3 7 T J van Swinderen August 2009 Structural Design Lab TU Delft INFORMATION FROM THE LITERA 3 1 The building material brick In this paragraph information regarding the building material brick and the masonry structures made with it is presented It is divided into subparagraphs General introduction of brick including
49. TNA makes use of four main elements in the process of analysis 3 Force networks representing possible forces in equilibrium in the structure 2 Reciprocal diagrams visualizing the proportional relationship of the horizontal forces in the network and providing a high level of control for the user to manipulate the force distributions in the system 3 The use of envelopes boundaries defining the solution space 4 Linear optimization resulting in fast computation of results The main requirements for the new application are 2 2 Interactivity Fast results Compression only structures amp he Double curvature models Each of these requirements is shortly discussed to see if the main elements of TNA as mentioned above can be used to fulfill these requirements INTERACTIVITY This is taken care of by point 2 the reciprocal relationship of the diagrams In the new application it is possible To adapt the position of nodes in the 3D model To change the loading Jo add or remove lines So in three ways interactivity is assured The ability to adapt the force distribution by adapting the scale factor 4 4 5 and directly related the reciprocal figure 2 The ability to change the force network the 3D model or the primal grid 3 The ability to change the loading for each node FAST RESULTS The results are available fast due to the use of a fast linear optimization met
50. TWORK Moreover the application requires the ability to create a suitable force network for the imported shape s BRICK PATTERN The script to generate the brick pattern needs more research In the current situation each brick is treated as an individual element while it is intended that all of them form one unity The method to calculate the angles of each brick individually has to be looked at further to improve the resulting pattern The pattern has to be optimised to assure there are no gaps IMPROVE THE FREEDOM OF DESIGN AND FORM FINDING To extend the freedom of designing and form finding several options should be added and integrated e The option to use more than one shape of brick in a design and to use non standard shaped bricks e The option to use more than one color of brick in a design so that color patterns can be applied IMPROVE THE FUNCTIONALITY OF THE APPLICATION The script code of the application must be looked at by a professional tool developer so that the performance of the application is improved T J van Swinderen August 2009 Structural Design Lab TU Delft Moreover the performance of the application is improved when this option is implemented e The ability to move around the model in the X Y plane TECHNICAL ASPECTS TENSILE FORCES Compression only structures restrict the freedom in form finding especially in free form design To make it possible to design shapes in which tensi
51. a mould could be produced 6 3 2 On site with formwork Every brick has to be positioned in the designed pattern on site To be able to do this wooden formwork is needed since the structure will not be able to resist any loading even not its own weight until it is completed The buildings of Eladio Dieste were build using this technique Figure 6 2 This gives the impression as if the structure has to be built twice the designed brick structure and the form work With new techniques of laying bricks though like the robot technique in Switzerland 6 3 3 it might not be needed to use form work for the whole structure Another option is to create a surface with the application on which the brick pattern in shown When this pattern and surface is approximated with a plastic mould the bricks can be placed on their correct positions This surface has to be hold into position very precisely though and the technique to create 3D plastic moulds is still very expensive 6 3 3 Laying patterns using a robot system Gramazio amp Kohler 19 together with the Architektur und Digitale Fabrikation at the E T H Zurich have done and are still in the process of developing a robot system to lay bricks in a designed pattern There is a working model and several test projects are produces with it Figure 6 3 More information is found in the next paragraph I http www crhclayproducts com 99 Main report Struc
52. aY 6 STEPS OF THE DESIGN AND DEVELOPMENT PROCESS Figure 5 15 The yellow highlighted area is the position of the application in the design process TJ van Swinderen August 2009 Structural Design Lab TU Delft Tab Input start Generate a parameter model Choose the rectangular or spherical model 2 Select the wished dimensions 3 Adapt the dimensions until a satisfying shell is displayed Tab 2 Design and analysis Adapt force network Change scalefactor G 2 Move nodes 3 Change start variables Tab 3 Masonry pattern l Set brickstone dimensions Brick length width and height Spacing of brick 2 Select to display or hide the pattern Adapt a variable Brick dimensions 2 Force network model Tab 4 Export end Display and select the layer to export The force network 2 The surface either a curves or a polygon model 3 The masonry pattern 2 Select file format to export as DXF to open in AutoCAD MEL to open in Maya mii i nn en M RVB to open in Rhinoceros RB to open in SketchUP eN OE NN J Figure 5 16 Workflow case the wish is a masonry shell 82 Main report 5 10 Workflow of the application Tab Input start 0 Create a shape in 3d software and export it as obj file Import the shape as obj file 2 Generate a parameter model
53. age silos for grain and other bulk materials The rise of a silo is one half of the span 21 Main report a L ze e en 7 hell fh te if Ws te i orks i a Indoor Tennis Center Heimberg Switzerland 1978 A shi structure designed by Heinz Isler ns Figure 3 8 Examples of form active structures Structural Design Lab TU Delft a l b Roof of the Olympic Stadium Munchen Germany 972 A membrane steel structure designed by Frei Otto Figure 3 9 Example of a force distribution based on bending Image courtesy of C Hartsuijker Toegepaste Mechanica Deel page 395 T J van Swinderen August 2009 Figure 3 10 Example of a force distribution based on axial forces Image courtesy of C Hartsuijker Toegepaste Mechanica Deel page 646 22 Main report 3 3 Form active structures The principle of Form active structures is used all around the world The designs are often characterized by curvature and have an uncommon special and spatial appearance Figure 3 8 First a short introduction is presented regarding force distribution systems 3 3 1 Theory of force distribution systems Two force distribution systems are considered System mainly based on bending forces System mainly based on axial forces Force system based on bending force These structures are characterized by a combination of tension and compression in the cross section of an element
54. and the grid is translated to the right top corner of the User Interface aS 2 Adapt color when a node or line is selected If a node is selected highlight this node in red highlight the connected Lines in orange If a line is selected highlight it in yellow Figure 5 8 Flowchart of Primal grid Class Brick Description 3D cube Characteristics Width W Height H Depth D Color colorBrick Coordinates of base X Y and Z Procedures update display Figure 5 10 Characteristics of the Brick class 76 Main report Create a line between two Nodes Add the numbers of these nodes as identification of the line during analysis Methods Select a line in the 3D or primal grid and show the corresponding information 5 7 3 Polygons Characteristics A polygon is highlighted in orange if the corresponding node is selected in the primal grid Flowchart Figure 5 7 Constructor Create a polygon in the dual grid using the X and Y coordinates of all nodes belonging to the lines included in the polygon Add a number to it to correspond to the node number in the primal grid Methods Move let the polygon be moved to the right position in the dual grid using the procedure to create the dual grid Scale the polygon so that the dual grid is correct 5 7 4 The Primal grid T As mentioned before the primal grid is the planar projection of the f
55. and the maximum allowed angle between two bricks to determine the coordinates for the next brick and continue with step 3a until the whole line is filled with bricks Step 4 The result is a masonry pattern of long lines of brick Figure 4 30 left One disadvantage is big openings when the curvature of the surface is too high Figure 4 31 4 7 2 Spherical pattern This pattern consists of elliptical rings which form the centre line for the brick STEPS Step Create rings above each other with a spacing of one brick height These form the centre lines of the bricks Step 2 Determine the length of the rings and divide it by the brick length The amount of bricks is now known Step 3 Loop through all rings Step 3a 63 Main report Structural Design Lab TU Delft a Rectangular masonry pattern b Spherical masonry pattern Figure 4 30 The two types of brick patterns STEP 1 STEP 2 STEP 3 Place brick lines next to each other Rotate bricks Adjust position of bricks or adjust curvature WJ SOD Brick depth Brick depth Figure 4 31 Proposed solution to the problem of gaps in the pattern TJ van Swinderen August 2009 64 Main report Determine the X and Y coordinate of the first bricks and calculate the corresponding height and angles using the theory of 4 6 Step 3b Use the current X and Y coordinate to determine the coordinates for the next brick and
56. arch It should be investigated before using the output of the new application in practice since it is an important aspect to secure the stability of a shell 10 Coenders J L 2007 Dictaat CT525 1 Structural Design Spe cial structures 2 ed 1 page 131 140 29 Main report Structural Design Lab TU Delft Dome thickness Internal forces Membrane plane Axis of revolution Meridians Continuous inclined reactions Figure 3 20 Shell like behaviour meridional and hoop forces TJ van Swinderen August 2009 30 Main report 3 5 Structural analysing methods Several theories and methods to analyse shell structures exist One is discussed in this paragraph The membrane theory This theory is applied in two types of methods a Analytical methods b Graphical methods THE MEMBRANE THEORY The membrane theory is the basis of the current methodology of dome structural analysis and provides a reasonable approximation for thin shelled domes The predominantly load case is most often its own weight The membrane theory assumes the following Applied loads are resisted by internal forces within the surface which has no stiffness against bending therefore internal forces are either pure tension or pure compression 2 On a symmetrically and uniformly loaded dome internal forces act perpendicular to each other in the meridional and latitudinal hoop directions 3 Internal forces ar
57. aviour and the displacements have not been considered FEASIBILITY The final goal of every design is to build and use it Therefore it is important to consider the aspect of feasibility From the point of view of masonry and shells the following aspects have to be looked at e Constructing the double curved shell e Technical developments and alternatives Constructing the shell To be able to construct a shell formwork is required Laying bricks is labor intensive especially if the pattern is not common Therefore specialised bricklayers are needed to create the masonry shell Developments and alternatives Recent development a robot laying bricks An alternative is to combine materials concrete as structural network and brick as cover in and or outside the concrete structure The construction process of masonry shells is more labour intensive expensive and time consuming in comparison with traditional brickwork as used in vertical wall type structures However the result has a unique character it stands out from its surroundings and it is better looking The latter is a subjective matter and is therefore not a valid argument in the decision whether to use this construction and designing method T J van Swinderen August 2009 Structural Design Lab TU Delft REFLECTION The prototype is a good start and has the potential to be developed in what it was meant to be Research has been done in whic
58. brick increases structural requirements especially in areas regularly subjected to earthquakes 3 1 2 Types and dimensions of brick TYPES The three main types of brick that are used in the brick industry are 16 Figure 3 2 Box shaped in Dutch vormbak A sanded rather smooth stone with a regular surface of which one of the flat sides has been flattened by striking Hand shaped in Dutch handvorm Similar to the box shaped stone but the form is less straight and the surface is more irregular Cord press in Dutch strengpers A smooth sometimes perforated stone with cut flat sides and with surfaces that are varying between flat till sanded and smooth till very rough T J van Swinderen August 2009 Structural Design Lab TU Delft DIMENSIONS The dimensions of bricks are standardized 18 An internationally acknowledged modular size has been set Table 3 1 and Figure 3 3 Due to technological developments and the constantly increasing demands and requests from costumers and project initiators the demand for uncommon and differentiating structures has increased These uncommon forms are constructed with non standard dimensions and shapes for the brick Table 3 1 Standard dimensions brickstones Generally the solution to create an uncommon structure with brick is found in the appearance of the stone the texture and by the use of non standard shapes In combination with different colored stone
59. created by the application From top to bottom the force network the lines surface the polygon surface and the brick pattern Figure I I CADenary form finding project developed in Processing by Axel Kilian 18 using a Particle Spring library developed by Simon Greenwold Image courtesy of Axel Killian Figure 1 2 The theories and results of research into masonry by Philippe Block and John Ochsendorf 18 Image courtesy of Philippe Block Main report INTRODUCTION The building design process is a long and complex process and many parties are active in it One of the first phases is the conceptual design stage Two of all participating actors during this stage are the structural engineer and the architect Recently the use of the computer in the design process increased significantly both in the field of the architectural design as in the field of engineering In some situations these processes are combined for instance when the design has a complex shape or when the geometry of the surface is complicated In these cases the process of form finding is based on architectural design and structural analysis The increase in use of the computer during the design process is observed in the work field of both the structural engineer and the architect The engineer performs finite element calculations with the computer while the architect uses the computer to design complex shapes often resulting in designs referred to
60. d 14 FULL LINES Lines which will keep the same angle in the dual reciprocal grid G DOTTED LINES Lines connected with a foundation node Their angles can change in the dual grid depending on equilibrium requirements im 16 Figure 5 18 Imaginary primal grid used to support the explanation of several aspects of the script TJ van Swinderen August 2009 84 Main report 5 11 Current status of the script Important topics for further research and development Generation of the polygons Generation of the dual grid Generation of the masonry pattern Performing the analysis Exporting the model Nodal loading Angle check of the brick stones et Oo se al Gaps in the masonry pattern These topics are explained in further detail The explanation consists of a short description of the procedure as it has been designed and is intended to work followed by the problems with this script and finally the proposed solution is given In most situations the explanation is supported by an example These examples are all based on an imaginary example primal grid Figure 5 18 T J van Swinderen August 2009 Structural Design Lab TU Delft GENERATION OF THE POLYGONS Procedure to generate the polygons All the nodes are regarded using a loop At this point of the script all nodes are still independent of each other and so for each node the same procedure is performed After a n
61. dual grid are visible In other situations for instance when files are exported these diagrams can be turned off This assures a good and orderly UI The latest version of the user interface Figure 5 2 is divided into four tabs displayed in the top of the Ul These tabs correspond with the four groups as mentioned in 5 1 Setup force network Analysis and design the conceptual shape Brick pattern aN Export The four tabs are briefly discussed A more detailed description is found in the user manual Appendix F TAB SETUP FORCE NETWORK MODEL Two options are available to setup the network model Create a parameter model A obj file can be imported and than manually approximated with a parameter model TAB 2 CONTROL The main functions are to control the appearance of the 3D model and adapt the result dependant variables At the left side information about how to pan rotate and zoom the 3D model is displayed Several buttons to show the top side or front view of the 3D model The option to display or hide the loading and adapt the applied loadcase At the right side of the UI the primal grid top and in the dual grid bottom are displayed TAB 3 MASONRY PATTERN The dimension of the brick is chosen Adapting one of the T J van Swinderen August 2009 Structural Design Lab TU Delft values results in a new pattern of the brick TAB 4 EXPORT A layer can be
62. e coplanar that is the membrane has zero thickness 4 The membrane plane is located along the centre line of the actual dome thickness thus the lines of thrust must also follow the centre line The last assumption which constrains the line of thrust to a two dimensional plane along the centre line of the dome merits discussion In reality many lines of thrust may lie within the finite thickness of the dome all viable solutions for a stable structure Furthermore the lines of thrusts in the meridional and hoop directions may not coincide The thickness of the structure only defines a permissible region within which a membrane solution must be found Therefore the membrane theory operates on the lower bound principle if one solution is found by assuming the line of thrust at the centre line that achieves stability and equilibrium then the structure will also find its own solution II Heyman J 1996 Arches Vaults and Buttress Hampshire Great Britain Variorum 2 Heyman J 1995 The Stone Skeleton Cambridge Cambridge University Press 13 Heyman J 1996 Arches Vaults and Buttress Hampshire Great Britain Variorum T J van Swinderen August 2009 Structural Design Lab TU Delft The membrane theory remains the basis of most modern engineering methods that model the behavior of domes ANALYTICAL METHODS Analytical methods utilize geometry and calculus to calculate the change of internal forces
63. e dual grid means overall lower horizontal forces in the system and hence a deeper higher solution for the same set of applied loads Following the same reasoning a higher scale factor causes a shallower shell and thus higher horizontal forces T J van Swinderen August 2009 Structural Design Lab TU Delft 4 5 Linear optimization theory the Simplex method Most of the information mentioned in this paragraph has been found in a book in which the theory of matrices and linear optimisation are explained 7 Also two websites have been used one dealing with linear optimization problems 21 and the other with Simplex method 22 In the last paragraph an example is given of these steps applied to an example network 4 5 5 4 5 1 General information To make an analysis according to the theory of the TNA a one step linear optimization is used First of all the theory of linear optimization will be shortly discussed Linear programming and solving There are several techniques and theories dealing with linear optimization problems A first distinction is the Graphical and Analytical method The first one gives clear results but performs best with just one or two variables in the objective function When this function becomes more extensive and when the problem increases in number of steps needed to solve it the Analytical methods are better In this research the Simplex method is used Simplex Method In this me
64. earch The ribs can be seen as force networks which is one of the aspects of interest for the new application T J van Swinderen August 2009 Structural Design Lab TU Delft Felix Candela Candela 1910 1997 originates from Spain though the majority of his designs are found in Mexico Candela s major contribution to structural engineering was the development of thin shells made out of reinforced concrete Reinforced concrete is extremely efficient in a dome or shell like shape He tried to solve problems by the simplest means possible In regard to shell design he tended to rely on the geometric properties of the shell for analysis instead of complex mathematical means Figure 3 14 Relevance for this research One of the aims and assumptions of this research are thin shell structures and if possible with no tensile forces since reinforcement is not desirable Moreover one of the objectives of this research is to investigate the use of geometry to create the brick pattern and force network Candela s designs can be a good reference in using geometry to design structures even though his designs are not actually free form Figure 3 14 Los Manantiales Restaurant in Xochimilco Mexico City A thin concrete hyper shell designed by Felix Candela Heinz Isler This engineer and architect from Switzerland 1926 present is famous for his designs of concrete shells Similar as Gaudi observation of t
65. ect consists of less than three sides it can not form a closed figure and is therefore not a polygon Polygons can be classified in different means By number of sides Figure 4 2 2 By convexity Figure 4 3 3 By symmetry Figure 4 4 For this research the relevant type is the simple polygon For this type the sides of the polygon do not cross themselves The number of nodes depends on the number of lines connected to a node in the network The symmetry and convexity of the polygon is determined by the composition of the lines network 4 4 37 Main report Structural Design Lab TU Delft Figure 4 5 The length of the line represent the magnitude of the force Figure 4 6 Constructing of one force polygon when one value is in it known A 0 vertex B Common point of two rays Common point of two line segments ag Fig S7 7NC Pac AN Fae STNG 5 Figure 4 7 2D example a node in a network with three lines connected Figure 4 8 Closed force polygon the related node is in equilibrium to it T j van Swinderen August 2009 38 Main report FORCE POLYGON If the sides of the polygon are representations of forces in a network the resulting closed polygon is regarded as a force polygon The lengths of the sides represent the magnitude of the corresponding force Figure 4 5 Reading back the second description of a force polygon as given above a body in equilibrium is represented by a clos
66. ed polygon When regarding a node as a body a force polygon can be constructed for each node This polygon consists of the loading and the forces in the line elements connected to the node Since equilibrium is required the polygon has to be closed Since the magnitude of one of the lines is known in this case the force the polygon can be constructed Figure 4 6 2D EXAMPLE The theory is now explained using an example in 2D space Figure 4 7 An arch consists of several stones Each stone is represented by a node located in the centre of the stone The material in between two nodes is shown as a line To each node a force is applied which in reality is for instance the own weight of the stone increased with a certain live load The force polygon can now be created using three lines the direction of the loading and the direction of the two lines connected to the node Figure 4 8 When the polygon is closed the node is in equilibrium Though the arch consists of more than one node and a line is connected to two nodes Therefore the polygons of all nodes have to be combined into one diagram When the polygons of all nodes are in equilibrium the whole arch is in equilibrium The resulting diagram gives a clear insight in the size of the forces in all lines Moreover the support forces are shown when regarding the polygons of the border nodes This method has been documented in the Thrust Line Analysis theory 3D APPLICATION
67. ed with the Gaussian curvature the product of two main surface curves through a point on the surface Figure 3 19 The formula for the Gaussian curvature K is K kK K With x principal curvature K principal curvature 2 d i And K su ds R With 09 Difference in angle Os Difference in distance on the curve of main curvature R Radius of the curve distance to centerpoint Three cases are to be considered K gt 0 gt Synclastic surface the two curvatures K and K are in the same direction 2 K lt 0O gt Anticlastic surface the two curvatures K and K are in different direction Zeroclastic one of the two curvatures is zero T J van Swinderen August 2009 Structural Design Lab TU Delft The interesting aspect the three cases of Gaussian curvature are when the curvature is synclastic positive the surface is shell like and is therefore desirable in this research When the curvature is anticlastic negative there are tensile forces in the meridional direction Finally when the curvature is zeroclastic the structure is not double curved and considered a ruled surface 3 4 3 Structural mechanics of shells FORCE FLOW IN SHELLS Shell like behaviour is characterized by mainly axial compressive forces and little bending moments Two important forces are Figure 3 20 Meridional forces Hoop forces BUCKLING OF SHELLS This aspect has not been regarded in the rese
68. elft reinforcement can be introduced to increase the tensile strength For this research it is assumed brick can not take any tensile forces and that no reinforcement is applied Therefore the tensile strength has been neglected and is set to 0 N mm 3 1 4 Brick in double curved designs An example of the possibilities of masonry in double curved structures are the designs of Eladio Dieste 2 His methods to analyse and design masonry structures are remarkable and a good source of information for this research 3 2 Another engineer well known for his brick and stone structures is Louis Kahn Figure 3 4 top picture 9 However his work is not as slender and curved as the work of Eladio Dieste Therefore his work is of less interest for this research and will not be mentioned further on Main report Structural Design Lab TU Delft Figure 3 5 Brick structures designed and engineered by Eladio Dieste Say mdk a Church of Christ the Worker Atlantida Uruguay 960 BEE E b Casa Dieste Montevideo Uruguay 1961 ns mn c Monumento homenaje en rotonda Salto Uruguay 1976 ede at ineen f Roof of Julio Herrera amp Obes Warehouses Montevideo Uruguay d Chacineria Fenix Salto Uruguay 978 Completed in 1979 YN mets TJ van Swinderen August 2009 18 Main report 3 2 The masonry structures of Eladio Dieste T
69. ells To clarify and describe the research completely the title is extended with To analyse concepts and generate a brick pattern The research question accompanying the title is formulated as Is it possible to improve the speed and quality of the conceptual phase of the design process with an interactive tool offering the ability to design and analyse masonry double curved shells To guarantee a good final result the research question is divided into several topics These topics are presented in the next paragraph 2 2 T J van Swinderen August 2009 Structural Design Lab TU Delft RESEARCH QUESTION ANOT 2 2 Research topics An essential term in the research question is masonry shells The tool is created to design these type of structures This is only possible when information is available regarding masonry and brick and regarding shells Next step is to integrate this information in a new application For this purpose information is required regarding the design of a software application Moreover certain theories and algorithms are needed to implement the information about brick and shells and to make a well performing and interactive application The last step is to make sure the results of the application can be applied in practice Therefore aspects such as building a brick structure and new technologies in the brick industry are looked at The results together with the general informa
70. enhances the chances of a rapid process of finding and determining the conceptual shape Parametric design of the pattern The brick shapes are placed according to several desired characteristics for the structure which are symbolized by a number of variables and parameters Shape and size of the brick Dimensions of the structure Final result The solution is an application consisting of three parts First of all it rapidly analyses the model The input is either a model based on user specified parameters or an architectural shape which is approximated with a parameter model Secondly the application is interactive the shape is manually adaptable and the application instantly performs the analysis and displays the new result And finally the application will generate a masonry pattern The application and the result of the analysis are presented in user friendly interface Main report CHAPTER Introduction In this chapter the research question 2 1 and related research topics 2 2 are presented The research question is the basis of the research This question is subdivided into several topics that are each used to find the answer of an aspect related to the research question When the answer of all topics are combined the main question should be answered to a satisfactory level 2 1 Research question The main title of the thesis is Proposal for a tool to design masonry double curved sh
71. erse of the scalefactor are put in a matrix as well Figure 5 32 Description of the problem The problem should be solved using a matrix optimization 16 The consensus was that this could be achieved by an iteration as well but it was proven this was not a right assumption In combination with the problems to obtain the lengths of the branches in the dual grid this results in an incomplete analysis Solution to this problem The solution is to use the method as described by P Block 16 The script of this solution method is an aspect that needs to be researched further 9 I a 5 I a MN M YJ l wO MN Mm mn PAPSSSSSSSSSISIMSSGSISGSTMHGGGIMGGGGSGGGG05G HI a Ha a 5 a l l l En m N N wmm i ite nw ite p POSS SSSSSISIONSGIGOMOSIIIONGIISGSGSGSGG9GGG00 3 4i a ad a a 5 aa i u com et 5 O E S co AN Is mn pel eens i nm PPSSSSSSSIONGIIIMMMBSSGONGSSGGGGGG99000 ao EN nn Se N a 3 dig 3 Cc eaten are inte a En 9 20 eee oe a J39 co ne Seek st ro eS a o PPSSSSSSIIMISOANMNGBIIMIGGGIGGGGGGGGGG QO 7 dl a ono a 5 I 1 l 1 5 i a nor et p 2 I SODSDSSOSSOTISOSOONGNSGSGGOMSISGGSGSS9990009956 U mn u Ca C 5 bemi HI a ona a z gt i i i i a S F ae A A ISBDSDSSGONGIGOMANGSGONGSSGSGSGSSGSG900095956 st EN an n A a aaa a i i 1 i N wo co a Mm N Mm 20 IOSDSSSSTFOSISOSOHMGSIOMSGSGGSGGGG9G900990005 en Ps a m o H I a Ha a Q i i I Ls ol
72. es lecture m 6 matrices pivot html Accessed October 2008 TJ van Swinderen August 2009 Structural Design Lab TU Delft 23 Dr Hossein Arsham 1994 The Classical Simplex Method Online Updated 2009 Available at http home ubalt edu ntsbarsh Business stat opre partlV htm Accessed October 2008 COMPUTATIONAL DESIGN 24 Fry B and Reas C 2001 Processing Online Updated 14 May 2009 Available at http processing org Accessed September 2008 May 2009 25 Kilian A 1998 Designexplorer Online Updated 2006 Available at http designexplorer net Accessed May 2008 26 Masonry Research Group M I T 2006 Masonry at MIT Online Available at http web mit edu masonry Accessed July September 2008 13
73. et 2 Reposition the polygons Find the lines that are common in two polygons Place these polygons next to eachother so that the lines match up 3 Rescale the polygons Scale the polygons so that the common lines have the same length 4 Determine foundation nodes and lines Determine the range of possible positions for the angle of the lines connected to a foundation node Use the midpoint of this range as angle The new 5 Display the polygons scalefactor G after YES clea T T nnn Scalefactor adapted to adapt the y model Scale the final figure NO with this NO YES ode selected in primal grid set color to white Set color to orange for polygon Display the polygons using the display procedure of this class l l l l the corresponding l l l l l menen a a a a a a a nennen ennn nennen 7 Figure 5 9 Flowchart of Dual grid T J van Swinderen August 2009 Structural Design Lab TU Delft Step Compose Primal Grid All elements of the arraylists Node and Line the 3D force network model are needea Loop through all elements in arraylists Node and Line Use the display primal grid procedure The only difference with the normal display procedure is the fact that the Z values are not used
74. ext brick is determined according to the coordinates of the first brick Correct pattern NO Adapt force network model Back to tab or 2 to adapt force network y Create network Create brick pattern for networklines Fill the network lines with brick stones with the dimensions as inputted Create the used brick and one layer of the number of rows thickness as input by the user 2 Find length of every line 3 Determine amount of bricks and place them vertically 4 Rotate the line of bricks according to the angle of the line in the 3D model YES Go to Tab 4 Export August 2009 Awe mn nnn nn nnn nn nnn nn nn nn nn ann nn a nn nn a a an nn an a a eee TU Delft 80 Main report All file formats make use of points coordinates lines or a combination of them This explains why in all programs except AutoCAD the script editor of the program is needed to open the exported file It is possible to export in the current version of the prototype but the model is not positioned in the origin of the axis and it is rotated Figure 5 13 Failing of the masonry pattern script procedure T J van Swinderen August 2009 Structural Design Lab TU Delft 5 9 Masonry pattern To visualize the masonry shell a pattern is created Flow chart Figure 5 12 The technique to generate the patterns is explai
75. from one side of the element to another side of an infinitesimally small element of the dome The sum of these forces must establish equilibrium in directions tangential to the meridians normal to the dome surface and parallel to the latitudes for the entire dome structure Though formulae for membrane analysis of domes was introduced as early as 1858 it was not until 1926 when a mathematical theory describing the behavior of dome shells of revolution became simplified enough for practical use GRAPHICAL METHODS Graphical analysis provides a visual method of solving for structural equilibrium through the knowledge of building geometry and forces Information obtained from a graphical analysis include meridional and hoop forces along the arc of a dome horizontal thrusts and the deviation of the thrust line from the line of the assumed membrane in cases where no tensile capabilities of the dome structure are assumed Possible for shells made by surfaces of revolution subjected to its own weight More insight in flow of forces Easy way to construct a polygon of forces This polygon then represents the corrected line of thrust whereby the hoop forces correct the line of thrust of the load to coincide with the system line of the shell The graphical method gives a very good result compared with a FEM calculation Also it gives a good understanding of the mechanical behavior besides only the numerical result 14
76. global matrix The exact steps are found in the flowchart of the Simplex method and in Appendix C The result consists of the possible range of height for all nodes and the range for the scalefactor C 4 Calculate forces Using the height for the nodes and the scalefactor the lenths of the lines in the dual and primal grid are known These lengths are an equivalent of the horizontal force Using the following formula gives the forces Fh Ldual 3 Solve global matrix Figure 5 4 Flowchart of the Simplex method procedure T J van Swinderen August 2009 72 Main report 5 5 Input options There are two options Create a parameter model 5 5 1 2 Import a model 5 5 2 Flowchart Figure 5 3 5 5 1 Create a parameter model Five parameters are used all manually adaptable to generate a Shell shape The model can either be based on A rectangular grid Aspherical grid The common parameters are Shell height Base width Base depth For the rectangular grid the two other parameters are Number of rows in X and Y direction to determine the density of the grid and the amount of compression lines And for the spherical grid Number of bays and rings density of the grid and the amount of compression lines 5 5 2 Import a model The only possible file format to import is OB This format is not the optimal file format since it only consists of
77. h elements and options are required for the application and an overview of this has been presented It has been attempted to give the basic information regarding brick and analysis theories TNA Simplex method Catmull Rom splines and to integrate these in a new application The user interface is developed and is considered to be easy in use and clear in displaying its content However there is one important remark concerning the script it needs more development to make the tool complete before it can be used in practice At the start of the research the level of scripting knowledge was low and as a consequence a big part of the research time was invested in obtaining this knowledge Nevertheless in the end the knowledge is still not at that level with which an application can be developed in a smooth and fast process Unfortunately as a result certain aspects were not scripted and integrated in this prototype 105 Main report Structural Design Lab TU Delft T J van Swinderen August 2009 106 Main report 7 2 Conclusions The conclusions have been split into two categories Programming aspects Technical aspects PROGRAMMING ASPECTS DOUBLE CURVED SHELLS With the prototype double curved masonry shells are analysed and a brick pattern is generated The range of possible shell shapes is not as big as was aimed for at the start of the research The analysis part of the program is partly working in the current ve
78. he architect and design stage After the initiation of the design Engage a list of demands and wishes is made Research The architect makes some sketches according to these wishes Architect and he is now interested in an engineering design Design At this moment it is time to use the application Figure 5 15 yellow section Together with the architect a meeting is attended during which the conceptual design is completed WORKFLOW OF THE APPLICATION The application is used in two situations The architect knows he is going to use a masonry shell and knows the rough dimensions 2 The architect wants to create a shell structure and wonders if it is possible in masonry The difference between these situations is mainly the starting model The other steps of the work flows are the same for both cases Case Figure 5 16 The workflow starts in Tab with a parameter model The starting model is a parameter model of which the parameters are adjustable according to the wishes of the architect Case 2 Figure 5 17 The workflow starts in Tab with an import model obj format This model is manually approximated with a parameter model 83 Main report Structural Design Lab TU Delft PRIMAL GRID 3 RED NUMBERS nodes Nodes which will form polygons in the dual reciprocal grid 12 E i GREEN LETTERS polygons D Polygons open areas in the network which will form nodes in the dual reciprocal gri
79. he natural world where most structures have organic shapes with double curvature is very important to Isler He tries to use mathematical formula as few as possible and therefore he approaches the challenges of each new structure by using physical modelling 25 Main report Structural Design Lab TU Delft Figure 3 15 Physical modelling by Heinz Isler Figure 3 17 Examples of physical modelling with for instance panties and soap bubbles during assignment of the course CT525 Special Structures TJ van Swinderen August 2009 26 Main report to determine the form and to investigate the stability of the structure After experimenting with pneumatic forms to create shell shapes he discovered the in his opinion best method to create these shell forms the reversed hanging membrane model The method of inversed hanging membrane models Isler This theory has a lot of similarities with the hanging chain model by Gaudi with one difference instead of creating a grid a full surface a shell is produced Figure 3 15 The most commonly constructed type of Isler s shells is the bubble shell Figure 3 16 With these shells he moved away from the traditional geometric described shells so that equations could be derived to calculate the forces and stresses within them and developed a method of form finding based on physical models Two projects realized with it are the Gr tzingen Open Ai
80. he resistant virtues of the structure that we make depend on their form it is through their form that they are stable and not because of an awkward accumulation of materials There is nothing more noble and elegant from an intellectual viewpoint than this resistance through form The work of Eladio Dieste is the inspiration for this research Double curved structures are commonly constructed with steel timber or concrete However Dieste used the material brick reinforced with steel and the result is remarkable Figure 3 5 Eladio Dieste 1917 2000 both architect and engineer was born in Spain However most of his life he has lived and worked in Uruguay He is famous for his reinforced brick construction techniques His work is based on the catenary form of the arch He relies on the great strength of brick in compression and modifies the geometry to improve stability against buckling This results in structures that behave as traditional vaults but have a lightness that defies tradition and positions them firmly in the 20 century In buildings which are enclosed Dieste often refuses to use the walls to support the roof He did not use models for his designs unlike for instance Frei Otto A particular innovation was his Gaussian vault 3 2 2 a thin shell structure for roofs in single thickness brick that derives its stiffness and strength from a double curvature catenary arch form that resists buckling failure
81. hod the Simplex method Performing the 3 Block Philippe 2007 Journal of the international association for shell and spatial structures J IASS nr 47 p 169 10 4l Main report Structural Design Lab TU Delft Figure 4 10 Relationship between force network primal grid and dual grid Image courtesy of Philippe Block Relationship between compression shell G its planar projection primal grid T and the reciprocal diagram dual grid I to determine equilibrium TJ van Swinderen August 2009 42 Main report analysis and finding the solution in this research is achieved by a one step linear optimization This optimization is performed rapidly and so the results are available rapid COMPRESSION ONLY STRUCTURES The theory of TNA is based on compression only structures If tensile forces are present in the network it is not possible to obtain the primal and dual grid One remark all loads are applied in the same direction in this case vertically as is the case for gravitational loading Due to this wind loads are not taking into account for now A result of the assumption of compression only structures is that the surface of the shape can not curl back onto itself The 3D force networks represent load paths throughout a structure T his observation is important since this means that in theory only these network paths have to be completely of brick and all planes between and inside the network lines can be ope
82. ific brick shape In the case of a double curved surface this is even more complex and can even be considered impossible Therefore it is important that several sizes of bricks can be used and if needed some unique shaped closing bricks This is possible due to the cutting brick machines which are becoming faster and more accurate 6 4 3 Robot laying brick stones in a pattern In Switzerland at the University of Zurich E T H the department of Architecture together with Gramazio amp Kohler 20 is currently researching the use of robots to lay patterns of bricks Figure 6 4 A disadvantage of the system in its current state is the requirement of having to lay the bricks in the right order to assure the robot lays the right brick at the right position Considering the current capacities of the system it is not a possibility to use for construction T J van Swinderen August 2009 Structural Design Lab TU Delft 6 5 Interesting alternative STRUCTURE OF CONCRETE WITH A COVER OF BRICK Instead of a complete brick structure concrete is used for the load bearing network structure with an external layer of brick in a certain pattern A German company GTecz has developed a new type of fluid Ultra High Performance Concrete 21 They developed a technique called Membrane Concrete Grid Shell MBGT with which networks shells of liquid concrete are created by using membrane in double layers Figure 6 5 This alternati
83. ighttop 59 Main report Structural Design Lab TU Delft C1 C1 X1 X coord imo te X Coorong ce AC Figure 4 25 Determine x values for the third and second point Y coordinate C1 C2 X1 A COOLE De Figure 4 26 Determine height of point P using the four heights z0 z3 TJ van Swinderen August 2009 60 Main report Structural Design Lab TU Delft 2 Check if there are enough neighbouring nodes values to calculate height and angles of the Point If not set a value for the missing points Figure 4 23 3 Determine the four z values of the column lines around the Cell in which the Point is located Figure 4 24 4 Determine the x values of the second and third point Figure 4 25 These values are used to determine the tx value which is needed to calculate the height for point P 5 With the four z values step3 and the two x values step4 the z value of the Point is calculated Figure 4 26 In addition the angle of the surface in both directions must be calculated and checked These angles are the angles of the brick in that Point Figure 4 27 This theory is implemented to create the lines and surface of the shape The step to calculate the angles of each individual brick with this theory has not been implemented yet In the current versi
84. ilding is designed by the architect the appearance of the structure such as the facade is part of the design as well Therefore the possibility to design and adapt the pattern of brick stones is a relevant research aspect to implement in the application It offers the ability to control the appearance of the structure An example is the work of the Austrian designer Erwin Hauer He created patterns and concrete elements by varying the size position rotation and repetition of one element Figure 1 4 T J van Swinderen August 2009 Structural Design Lab TU Delft GOAL The aim is to design a tool that offers the possibility to enhance the design process during the conceptual design phase of a masonry double curved shape The chosen shape is the shell Figure 1 5 which has a curved shape and is used as for instance covering structure of a space The tool assures a better communication between architect and engineer so that the design of the model is finalized timely With the application the user is able to design concepts for masonry double curved shells The tool offers the ability to interactively adapt and finalize this model according to the wishes of the user The tool generates masonry patterns according to certain specified variables and parameters Interactivity of the application is the key aspect to include The architect and engineer have to be able to work together and adapt the shape and get results fast This
85. ilibrium of the structure is analysed Aspects such as buckling behaviour and displacements are not taken into account 43 Main report Structural Design Lab Figure 4 1 I The two way relation between primal and dual grid Image courtesy of Philippe Block TJ van Swinderen August 2009 TU Delft 44 Main report 4 4 Explanation of steps The important steps of the TNA for this research are Construct the primal grid 4 4 1 2 Construct the dual grid 4 4 2 3 Solve the problem 4 4 3 Two other aspects concerning the TNA are the application of loading 4 4 4 and the function of the scalefactor 4 4 5 4 4 1 Construct the primal grid The primal grid is the planar vertical projection of a three dimensional grid or force network of a compression shell If to compare it with an event in nature imagine the sun exactly above the center of the grid the shadow of the three dimensional network on the ground is the primal grid 4 4 2 Generate the dual grid The lines connected to a node in the primal grid form a polygon of forces in the dual grid Figure 4 11 A definition of polygon of forces is The sides of a force polygon represent in magnitude and direction a system of forces in equilibrium 8 The direction angle of each line remains the same the length representing the force and position change To relation between the dual grid and primal grid is exactly the same
86. in the primal grid is H and in the dual grid H The relation between Hy and the corresponding horizontal force Fe is scale factor In formula this is shown as BY Hj B H 4 B H By using these relations in 3 and afterwards rearranging it so that it becomes a function of the branch lengths in both grids the new formula is Hi H Hil i g H H H dle ij i i ij ae a 5 In 5 the new variable r is the inverse of the unknown scale of the dual grid By introducing constant C 5 can be rewritten as GZO ZAO ZEG zekr 16 49 Main report Structural Design Lab TU Delft Figure 4 14 Boundary conditions of a node and its neighbouring nodes Image courtesy of Philippe Block Matrix 1 Maximize Minimize Constraints Figure 4 15 Start matrix when regarding only one node TJ van Swinderen August 2009 50 Main report The constants C are Hi Ha H H H H ij i i H C H H C Z ik H H C Z il H 2 The lower and upper boundaries of the height for each node To actually approximate the model the height of every node needs to have some ability to be shifted up or down This way the force path actual model and forces in it can be controlled Therefore a lower and upper boundary is needed for each node Figure 4 14 The solution has to lie within these boundaries In formula EAR A 7 The objective function In this
87. k model Change scalefactor 2 Move nodes 3 Change start variables Tab 3 Masonry pattern Step Set brickstone dimensions Step 2 Create a shell pattern Depending on whether the force network model is linear or spherical Linear pattern or 2 Spherical pattern For Step 3 Adapt pattern Adapt the force network model or 2 Adapt the dimensions of the brick stone For 2 Tab 4 Export end Select one of four layers Forces textfile Possible file formats File which contains the forces for DXF to open in AutoCAD each line and the foundation forces MEL to open in Maya for each foundation node RVB to open in Rhinoceros TXT CRB to open in SketchUP J to open in Notepad or Word J Figure 5 1 Main flow chart of the new application eN J T j van Swinderen August 2009 66 Main report CHAPTER Introduction In this chapter the prototype of the new application is presented The theories of Chapter 4 are the basis of the application First the main flowchart of the application is explained 5 1 To operate and use the application and display the results a user Interface Ul has been created 5 2 The main content of the application is the 3D model There are several options to display it 5 3 To make the application interactive it is relevant the user can affect the re
88. ked in the same way If the structure fails in any load case the shape has to be changed and again checked for all load cases 5 4 3 Relocate nodes To adapt the shape of the 3D model it is possible to change the position of network nodes Simply click and drag a node in the 3D model and see the shape being changed The new shape the primal grid and dual grid are instantly updated The analysis and brick pattern will not be updated though until the node has been released again 5 6 2 5 4 4 Adapt scale factor The function and importance of the scale factor is discussed earlier Chapter 4 However because this variable is manually adaptable it is once again mentioned here The scale factor determines the relation and value of the forces in the network beams When the value becomes smaller the steepness of the line element increases which results in a higher shell structure and lower horizontal and higher vertical forces Figure 4 12 c Following the same reasoning a higher scale value results in a lower structure higher horizontal forces and lower vertical forces 7 Main report Structural Design Lab TU Delft Tab Input Start Import eee a Step 0 Location of file Save the OBJ file in the Sketch folder Step Show import UI group Enter filename in textfield without Manual obj 2 Push button Import Linear Spherical Linear force
89. le forces are active more research needs to be done in improving the application and theories BUCKLING OF SHELL STRUCTURES This aspect is not looked at in the research It should be researched before actually using an output of the application since it is an important factor to check before constructing and using a shell NODAL LOADING The nodal loading is determined manually To improve the results and make them more realistic the nodal loading needs to be determined by the loading of the surface area related to the specific node SHELL THICKNESS When a shape is not buildable with one layer of bricks a situation may exist where it can be built but with more layers in certain areas The implementation of this option will be an improvement for the application BUILDING PHYSICS To be able to use the models proposed by the application in other functions besides pavilions such as roofing of internal climate spaces one needs the building physics during the design process For this the building physics of brick structures and the related codes and demands needs to be integrated in the application 109 Main report CHAPTER REFERENCES BOOKS I 2 3 4 5 6 7 8 9 T J van Swinderen Coenders J L 2007 Dictaat CT525 1 Structural Design Special structures 2 ed Pedreschi R 2000 The Engineers Contribution to Contemporary Architecture Eladio Dieste
90. le shaped in a certain way and secured at the ends A form active structure can support itself and is usually used to span and cover a space They are governed by axial forces either tensile or compressive stresses There are four types cable structures tent structures pneumatic structures and arch structures The first three are mainly tensile form active structures and therefore of less interest for this research The arch structures are interesting because they are compressive form active structures Figure 3 11 They behave best when their shape is as a mirrored cable under load To become more familiar with these type of structures and how the theory has been applied in the building history some engineers and architects and the method they use are given in the next paragraph 3 3 3 Engineers designing form active structures Several engineers and architects and their work have been looked at to obtain more insight of form active structures such as common shapes and the historical developments throughout time Information is given about Antoni Gaudi hanging chain models method Pier Luigi Nervi ribbed shell structures Felix Candela thin concrete shell structures Heinz Isler inversed hanging cloth models Frei Otto tensile structures MODELLING COMPRESSION BASED STRUCTURES Antoni Gaudi Antoni Gaudi 1891 1979 is a Spanish engineer and architect His work is inspired a lot by nature and wa
91. les are based on vertex and line models TECHNICAL ASPECTS THRUST NETWORK ANALYSIS The Thrust Network Analysis is a good theory to make a force flow analysis of compression only network structures Therefore it is useful for the application It offers a fast method to make an accurate analysis of the force flow for the conceptual design stage A disadvantage of the theory is the compression only boundary To extend the possibilities of the application the theory should be researched into the option of adding tension CATMULL ROM SPLINES This theory is used to create the surface and pattern of the shell It uses the location of four neighbouring nodes to determine the location and angle of a specific point It is a fast method and in combination with the factor t the curvature is limited Therefore this theory is perfect to use in this research 107 Main report Structural Design Lab TU Delft T J van Swinderen August 2009 108 Main report 7 3 Recommendations The recommendations have been split into two categories Programming aspects Technical aspects PROGRAMMING ASPECTS FILE FORMATS Import To increase the functionality of the application it should become possible to import more file formats for instance NURBS models from Rhinoceros or Maya Export An aspect that is not working in all cases is to export the model without rotation and placed in the centre of the coordinate system FORCE NE
92. lication Current status of the script 6 PRACTICAL ASPECTS OF BRICK STRUCTURES 6 6 2 6 3 6 4 6 5 Brick fabrication and production 6 1 1 Non standard brick forms 6 1 2 Texture 6 1 3 Color 6 1 4 Bonding material Building physics characteristics Building the structures 6 3 Prefab elements in fabric 6 3 2 On site with formwork 6 3 3 Laying patterns using a robot system Developments in the brick industry 6 4 Glue as bonding material 6 4 2 Cutting brick stones with a computer 6 4 3 Robot laying brick stones in a pattern Interesting alternative 7 CONCLUSION AND RECOMMENDATIONS Lal 7 2 7 3 Discussion Conclusions Recommendations 8 REFERENCES T j van Swinderen August 2009 Structural Design Lab TU Delft 79 79 79 8l 8l 83 85 97 97 97 97 97 99 99 99 99 99 99 IOI Ka Ka 101 Ka 103 103 107 109 Main report INTRODUCTION The design of a double curved shell is time consuming because the shapes are often complex the architect is very specific in the desired shape and the engineer has to perform an elaborate analysis To decrease the time and increase the quality of the process of making a conceptual design the communication process between architect and engineer has to be improved Philippe Block and John Ochsendorf active at the Masonry Research department of the M I T 25 have proposed a new theory to analyse shells with the Thrust Network
93. mited to consist of maximum 225 nodes 15x15 grid to assure a smooth performance and to limit the size of the matrix to calculate stresses In theory the matrix to calculate the stresses and forces can be of unlimited size However the analysis takes longer with increasing amount of nodes The dimensions of the grid are limited to 15 0x15 0x4 0m This assures that the limited grid size is still applicable and realistic The angle between the bricks must be implemented as a check whether the masonry pattern is realistic This is not implemented yet in the the current prototype The gap between the lines of brick in the linear pattern must be repositioned so that gaps in the perpendicular direction are closed This is not implemented yet in the the current prototype 65 Main report Structural Design Lab TU Delft Tab Input start Options to generate networkmodel Create a parameter network model with five parameters or 2 Import a OBJ file and approximate it with a parameter network model Tab 2 Analysis Step Compose Primal Grid Step 2 Compose Dual Grid Step 3 Analyse network For Compose local objective functions 2 Compose global problem 3 Solve global matrix 4 Calculate forces NO Step 4 Design conceptual networ
94. n These open planes can then be used to create brick patterns to comply with the architectural wishes DOUBLE CURVATURE MODELS The Thrust Network Analysis is developed especially for three dimensional models which often have double curved surfaces In the next two paragraphs the benefits from architectural point of view 4 3 1 and from engineering point of view 4 3 2 are explained 4 3 1 Architecture As long as the compression only force network is in equilibrium the resulting model form is realistic and applicable Architectural freedom is created due to the ability to change the shape of the shell and with it the brick pattern The ability to adapt the brick pattern is a recommendation for further development of the application Two additions that would significantly increase the possibilities of the application are The implementation of being able to use more than one brick shape and of irregular non rectangular shapes The possibility to use colored bricks and create color patterns T J van Swinderen August 2009 Structural Design Lab TU Delft These additions are not implemented during this research and are therefore recommendations for further research Chapter 7 4 3 2 Mechanics Due to the use of force diagrams the primal and dual grid the Thrust Network Analysis gives a clear graphical representation of forces in the system Figure 4 10 and 4 1 Only the force flow and equ
95. n is determined by the curvature of the surface the direction in which the curvature is smallest is the main direction The curvature is determined by the height width ratio The wider the base plan the shallower the surface which means a smaller curvature In other words the main direction is determined by the longer side of the base plan This direction is important because it determines in which direction the grid is generated CHECK OF PATTERN MAXIMUM ANGLE BETWEEN BRICKS The brick pattern has to be realistic Therefore the bricks can not cross each other On the other hand the gap between two bricks is also limited to 16 mm because otherwise the bonding layer becomes too thick The maximum angle depends on the height of the brick Figure 4 29 This angle must be determined and a warning has to be given when the angle is too big This is not yet incorporated in the prototype T J van Swinderen August 2009 Structural Design Lab TU Delft STEPS Step Create lines next to each other with a spacing of one brick width These form the centre lines of the bricks Step 2 Determine the length of the lines and divide it by the brick length The amount of bricks is now known Step 3 Loop through all main direction lines Step 3a Determine the X and Y coordinate of the first brick and calculate the corresponding height and angles using the theory of 4 6 Step 3b Use the current X and Y coordinate
96. n the current version of the program the value is set to a certain value without using the area The loading of each individual node can be altered according to the wishes of the user Another relevant aspect regarding nodal loading is the thickness of the shell When a shape is not buildable with a shell thickness of one layer of bricks there might bea situation in which it can be build if certain areas have a thickness of two or more layers Therefore two aspects the nodal loading and the element thickness have to be manually adaptable In the current version of the prototype the initial load on every node is set to a certain value In a realistic situation this value should be linked to the surface and thickness of the area surrounding the node The thickness of the shell surface can only be adapted for the complete shell It is advised to make this possible only for selected areas of the shell as well 54 2 Load case A certain load case such as snow load can be applied on the shell surface In the application this aspect is obtained by adding a load to all nodes at once instead of adapting the load of one individual node 5 4 1 The shape remains the same while the loading is changed The analysis is performed with the new loading If the network is T J van Swinderen August 2009 Structural Design Lab TU Delft still in equilibrium the shape is able to resist the load case All load cases can be chec
97. ned earlier 4 7 To generate a perfect pattern closing the exact boundaries of the base plane non standard brick shapes are needed However an assumption is to only use one size of a standard brick and so this is not being researched further Non standard forms will bean improvement forthe application and has therefore been added to the recommendations Chapter 8 5 9 1 Range of dimensions for brick The range of allowed dimensions of the standard brick is determined by the factory and the machinery where the bricks are produced After contact with brick producers 17 the following ranges have been set Length 160 280 mm Depth 75 120 mm Height 50 90 mm The exact dimensions of the brick are determined with the sliders in tab 3 Masonry pattern The masonry pattern procedure is partly working in the current version of the prototype At some positions of the surface the pattern script is failing Figure 5 13 This is caused by the decision to equalize the length of the lines in the 3d force network As a result the related primal grid has inclined lines resulting in curved lines from left to right and top to bottom The script where the related nodes for the Catmull Rom theory are obtained fails in certain positions due to this curvature 8l Main report BUILD HOYVsASsY 6 STEPS OF THE DESIGN AND DEVELOPMENT PROCESS Figure 5 14 Design process build up in 6 stages BUILD HOYvsass
98. network Spherical force network Three sliders Three sliders W Width base X axes W Width base X axes D Depth base Y axes D Depth base Y axes H Height structure in Z direction H Height structure in Z direction Two knobs Two knobs x Number of networklines in X Ribs Number of ribs slices vertical Y Number of networklines in Y Rings Number of rings horizontal Create elements Node and Line When changing any aspect of the network Calculate all coordinates according to the value of the parameters and create Nodes 2 Create a rectangular grid of Lines L Show model Using the display procedures of the classes Node and Line we e u M M M M a a i i Ce Figure 5 3 Flowchart of tab l Setup networkmodel Step 3 Analyse network Compose local objective functions Using a loop compose the local matrix for each node except the foundation nodes Use the nodal loading and lengths of the related lines in both the primal and dual grid to obtain the K values 2 Compose global problem Collect all K values for each node and add them Use these K values in the objective function which consists of the K values of Jall nodes Now collect all boundary conditions for the height of every node and use them as constraints The global matrix is finished Use the Simplex method to solve the
99. ngineer the force network model layer is of importance for the architect the surface models and the contractor might be interested in the masonry pattern layer Figure 0 2 Companies use different software programs Rhinoceros Maya 3DMax AutoCAD and SketchUP are most commonly used To assure widespread use the tool supports a wide range of export formats such as dxf for AutoCAD and 3DMax rvb for Rhinoceros mel for Maya rb for SketchUP DISCUSSION One of the main goals was to create an interactive tool with which the architect and engineer can rapidly generate a conceptual shape for a masonry shell Using the theory of TNA in combination with Catmull Rom splines has provided the right conditions to design a first prototype for this tool The tool performs a force flow analysis of a shell shape The stability such as buckling behaviour and the displacements are not considered To improve the results and increase the range of possible shapes to analyse research should be done into expanding the TNA to make it suitable for tensile forces and into designing the tool to let it form find the force network and masonry pattern automatically instead of manually by the user Finally the script code must be looked at by a professional tool developer so that the performance of the application is improved T J van Swinderen August 2009 Structural Design Lab TU Delft Figure 0 2 The four layers
100. ode has been selected in the loop the following steps are taken Find the lines related to the node by checking whether the node number is one of the two node numbers of a line This is done for all lines by looping through the array of all lines 2 List the related lines according to their angle 3 Start with the first line of the list as obtained in step 2 and connect the second line of the list to the end of the first line Figure 5 19 step a b 4 Continue as step 3 until all lines have been connected Figure 5 19 step c It is expected that the endpoint of the last line and the starting point of the first line do not share the same coordinates The result is either an open polygon or one where lines intersect This is not acceptable for application in the TNA Therefore the point of intersection of the first and last line has to be determined This is done as last step of the procedure to generate the polygon 5 Find the coordinates of the point of intersection of the first and last line to ensure a closed polygon Figure 5 19 step d Description of the problem The script is not performing well when a combination of polygons of 3 and 4 sides is used In the current version of the program it is advised to only use grids consisting of polygons of 4 sides The script to form the polygons performs well and therefore the reason why the script does not work for polygons for a grid of a combination of polygon
101. of a brick It is characterisied by a Cell and 3 coordinates x y and z TOPVIEW Unknown nodes Figure 4 23 Unknown value for specific nodes at the border of the grid TJ van Swinderen August 2009 58 Main report In which Co Pin Ni nt a FOER c 2T p 7T 3 p 3 27 p oe 6 F Pig rr rep a Pea The first derivative of this function determines the angle of the curve in that point PARAMTER TAU t The parameter t is known as tension and it affects how sharply the curve bends at the interpolated control points Figure 4 21 In general it is set to 0 5 Though when using a brick pattern the curvature of the Figure 4 24 Determine z0 zl z2 and z3 T J van Swinderen August 2009 Structural Design Lab TU Delft surface is limited by the angle between the brick The lower the curvature the higher the chance the shell is buildable Therefore it is better to lower the value of t The value has been set to 0 25 and can be altered only in the script code APPLICATION OF CATMULL ROM SPLINES IN 3D GRID To explain the theory some variables are introduced Figure 4 22 A brick is regarded as Point which is the center point of the brick A polygon in the grid is regarded as Cell The location of a Cell is a combination of a row and column Steps Find the Cell in which the Point is located Figure 4 22 r
102. of four main groups which individually consist of their own elements and procedures The flow chart for each individual group is given in the corresponding paragraph Start input 5 5 2 Analysis create the primal and dual grid and analyse the model 5 6 and 5 7 Generate the masonry pattern 5 7 4 Export specific layers 5 8 The important steps in the script are Analysis step 2 of the four steps 2 Surface generation and masonry pattern step 3 of the four steps In the current version of the prototype the script of both of these steps contains errors These errors and bugs are mentioned at each step that is explained in the previous and this chapter 67 Main report Structural Design Lab TU Delft 2 CONTROL OVER THE NETWORK MODEL 3 BRICK PATTERN 4 EXPORT THE 30 MODEL IMPORT AN OBJ FILE CREATE A PARAMETER MODEL Primal grid Dual grid zeta 1 0 Figure 5 2 Current user interface TJ van Swinderen August 2009 68 Main report 5 2 User interface A relevant aspect to assure a clear and easy use of the application is the user interface Ul The appearance has to be clear Using it has to be easy and in a logical sense Finally the results have to be presented in such a way that they are well comprehensible The application uses several schemes and diagrams that should all be well visible when the force network is determined and analysed the primal and
103. of this procedure can be found in Appendix C The results for one single node are not special in the sense that heights of the nodes either go to maximum or minimum allowed height Figure 4 16 The explanation for this is the fact that only one node is regarded As a result the height values can obtain any value and so also the minimum and maximum boundary values The node is not dependent on other nodes In other words when maximizing r the deepest solution is obtained which is reached when node i has maximum height and all other nodes the minimum values Following this reasoning the result when minimizing r which results in the shallowest result is minimum height for node i and maximum height for the other nodes This will change when the whole network is regarded 4 5 4 4 5 4 Solution for all points Difference with solving for one point In this case all nodes are considered instead of only one Consequence is that certain nodes become dependent on others A global matrix is created which consists of all local matrices per single node The steps needed for this are shown below Another aspect that has to be considered is the fact that all foundation points or in practical point of view the points that are connected to the ground have only one possible height Additional steps A loop is needed to run through all nodes and create a T J van Swinderen August 2009 Structural Design Lab
104. oject Report internet Available at http designexplorer net newscreens cadenarytool final_paper pdf Accessed July 2008 Kilian A 2004 Linking Hanging Chain Models to Fabrication For ACADIA conference Catmull E and Rom R 1974 A class of local interpolating splines In Computer Aided Geometric Design R E Barnhill and R F Reisenfeld Eds Academic Press New York pp 31 7 326 Block P 2009 Thrust Network Analysis Exploring Three dimensional Equilibrium Main report Structural Design Lab TU Delft T J van Swinderen August 2009 112 Main report WORLD WIDE WEB BRICK AND MASONRY 17 Wienerberger Online Updated 2009 http www wienerberger nl Accessed June 2008 18 The International Union of Bricklayers and Allied Craftworkers Online Updated 2009 Available at http www bacweb org Accessed June 2008 19 Koninklijk Verbond van Nederlands Baksteenfabrikanten Online Updated 2009 Available at http www knb baksteen nl Accessed June 2008 20 Gramazio amp Kohler Online Updated 2009 Available at http www gramaziokohler com Accessed July 2008 21 GTecz Online Updated 2009 Available at http www gtecz com Accessed May 2009 THEORIES AND METHODS 22 Richland Community College 2003 Gauss Jordan Elimination Through Pivoting Online Updated 7 January 2009 Available at http people richland edu jam
105. ole process of optimising may take a long time Computational design The computer technology and capacity increased significantly last several decades Therefore the computer is also used more often as a tool in specific phases of the design process which is referred to as computational design Technological development and research into the use and abilities of materials create a wider range of possible shapes designs and structures and make it possible to optimise designs even more One of the results is the tendency of structures becoming more slender and at the same time more complex To decrease the time of the design process the communication between the architect and the structural engineer has to be faster smoother and better One of the possible solutions is an application which performs the analysis faster and makes it possible to adapt and optimise the model This way the process of designing the conceptual model is reduced to one meeting between the architect and engineer Several of these digital applications for free form design are already developed These applications are mainly based on physical models or material characteristics Examples are CADenary the hanging model program of Axel Killian 18 Figure I 1 and the research into masonry structures and applications of John Ochsendorf and Philippe Block 18 Figure 1 2 Brick An important design aspect is the building material used for the structure Concrete s
106. on of the prototype the angles are calculated using the known values of the angles between the nodes and interpolating between them As a result the brick pattern is not following the exact surface everywhere i Figure 4 27 Angles of brick T j van Swinderen August 2009 6l Main report Structural Design Lab TU Delft a Rectangular grid b Spherical grid Figure 4 28 Appearance of the two pattern techniques A Yyoiey APM Figure 4 29 Angle between two bricks TJ van Swinderen August 2009 62 Main report 4 7 Masonry pattern generation Two types of patterns can be generated depending on the force network that has been chosen A linear pattern 4 7 1 Aspherical pattern 4 7 2 Which pattern is chosen by the application depends mainly on the shape of the base plan when the plan is rectangular it is convenient to use the linear force network and the linear pattern However when the plan is elliptical the spherical force network and spherical pattern are more convenient The appearance of the two options is quite different even when the common dimension parameters the base plan width base plan depth and height of the shell are the same Figure 4 28 4 7 Linear rectangular pattern This pattern consists a grid of lines in X and Y direction There are two directions The main direction The secondary direction MAIN DIRECTION The main directio
107. orce network of the 3D model it consists of the horizontal reflections of the force network line elements on the horizontal ground plan X Y plane It can be compared with the shadows of a 3D frame on the ground when it is being lighted from above by a diffuse light source Flowchart Figure 5 8 Steps The direction and length of the line elements of the force network are used to determine the horizontal length of them in the primal grid The primal grid is shown in the right top corner of the user interface T J van Swinderen August 2009 Structural Design Lab TU Delft Extra features It is possible to select a node in the primal grid if it for instance needs to be repositioned The selected node is highlighted including the lines connected to it This is also applied to the 3D model When a beam is overloaded the corresponding line gets a red color 5 7 5 The Dual grid I Starting point is the primal grid The dual grid is the reciprocal figure of the primal grid A reciprocal figure is defined as Two plane figures are reciprocal when they consist of an equal number of lines so that corresponding lines in the two figures are parallel and corresponding lines which converge to a point in one figure form a closed polygon in the other 10 More information is presented earlier in the report 4 4 Flowchart Figure 5 9 Steps All lines related to a node in the primal grid are collected
108. ow it is finished and being used ET A In combination with the glass facade it creates a nice composition b Sydney Opera House Australia 1973 Concrete frame amp precast concrete ribbed roof Engineered by Arup SSS Een re ee at LIS ar A LDS ZITA Ay Figure 3 19 Shell geometry Gaussian curvature c Roof of the Central Library during construction Troms Norway From top to bottom anti synclastic surface negative curvature 1970 Designed by Gunnar Bogeberg Haugen synclastic surface positive curvature and flat surface zero curvature TJ van Swinderen August 2009 28 Main report ideal thin shell must be capable of developing both tension and compression to be able to deal with deformations and point loadings 3 4 2 Shell geometry The surface of a shell is described using geometry This reduces the complexity of the process of generating the brick pattern To be able to describe the geometry of a surface it is relevant to investigate the options to do this Below is shown how a surface can be classified by its curvature the method of Gaussian curvature THE DEFINITION OF GAUSSIAN CURVATURE Every point of a surface has two principal curvatures The curvatures are found using the formula below with the use of the radius of the curve With the curvature is measured how the surface bends by different amounts in different directions at a certain point The result is often present
109. primal grid which is the planar projection of a three dimensional grid of a compression shell 2 The dual grid which is the reciprocal figure of the primal grid When this relation is used in a linear optimization method in this case the Simplex method it provides a graphical and intuitive method adopting the same advantages of techniques such as graphic statics but offering a viable extension to fully three dimensional problems THRUST LINE ANALYSIS This new theory is based on the Thrust Line Analysis which is a powerful graphical method for calculating the range of lower bound equilibrium solutions of compression only systems Figure 4 9 and 4 1 A disadvantage of this analysis method is the limitation of application for 2D cases only such as arches while a shell structure has a three dimensional force distribution FIELDS OF APPLICATION The TNA theory is applied in several situations For the analysis of vaulted historical structures specifically in unreinforced masonry To design new vaulted structures The last category is interesting for this final thesis research More information regarding the benefit of the TNA for this research is found in 4 3 2 Block Philippe 2007 Journal of the international association for shell and spatial structures J IASS nr 47 p 167 10 T J van Swinderen August 2009 Structural Design Lab TU Delft 4 3 The benefit for this research The
110. r Theater in Baden Wurttemberg Germany and the Heimberg Tennis Center in Berne Switzerland Relevance for this research His projects are good examples of thin shell structures and his inversed hanging membrane method is useful in the research since it deals with form finding of full shell structures which is one of the objectives of this research Figure 3 16 A service station in Deitingen S d Switzerland A bubble shell designed by Heinz Isler T J van Swinderen August 2009 Structural Design Lab TU Delft 3 4 Shell structures An essential difference between a shell structure and a plate structure is that in the undeformed state the shell structure has curvature while plate structures are flat Thin shells are focused on axial forces and little bending which is caused by the curvature of the surface The supporting conditions of plates and beams are mainly determined by vertical forces while for shells also horizontal forces are active Membrane action in a shell is primarily caused by in plane forces but there may be secondary forces resulting from flexural deformations Where a flat plate acts similar to a beam with bending and shear stresses shells are analogous to a cable which resists loads through tensile stresses or an arch which resists loading through compressive stresses The shell shape intents to eliminate tensile forces in the structure DIFFICULTIES The structural behaviour of irregula
111. r curved surfaces which have shell like behaviour is difficult to predict Figure 3 19 Especially the buckling behaviour and 2 order deformations are complex to determine More information about the difficulties in both designing and constructing a shell is found in literature Shells are very efficient in carrying load However a big disadvantage of shells is their brittle behaviour A shell gives less warning signals when it is close to failing compared to other structures For instance steel grids will first deform and a concrete structure will crack and the reinforcement will deform 3 4 1 Thin shell structures Thin shell structures are light weight constructions and commonly based on the form active structure theory If it consists of prefabricated element they are typically curved and assembled into large structures on site Typical applications are fuselages of air planes boat hulls and roof structures in building A thin shell is defined as a shell with a thickness which is relatively small compared to its other dimensions and in which deformations are not large compared to thickness The 9 Holgate A The art of structural design 1986 about the Syd ney Opera House 27 Main report Structural Design Lab TU Delft Figure 3 18 Shell structures a Korkeasaari Island Lookout Tower Helsinki Finland Timber grid shell Designed by Avanto Architects d Same roof as in picture a though n
112. ree connected network lines is considered When using the program it is also possible a node is connected with more than three lines As mentioned in the last paragraph the next step is to determine The objective function 2 The constraints To be able to explain the content in a clear manor first the constraints will be shown There are two constraining aspects l static equilibrium in every node and 2 a lower and upper boundary for the height of every node Static equilibrium in every node for the applied loading A description of static equilibrium is The sum of all forces acting on the object in static equilibrium must add to zero In formula form YF 0 1 The force situation of a node in the 3D diagram G consists of the applied loading P and the forces in the network lines connected to the node Figure 4 13 When considering vertical equilibrium formula 1 transforms in Fi E F P 0 gt FO E F P 2 However in this research the horizontal forces are most relevant because the relation between the primal and dual grid is based on these forces Therefore the vertical forces of 2 are now expressed in the horizontal components T J van Swinderen August 2009 Structural Design Lab TU Delft Next step is to use the lengths of the branches in the primal and dual grid and the relation between the dual grid lengths and the horizontal force First of all the length of the branch
113. rmats The transformation of the applied theories Chapter 4 into scripts to be able to use in the application Topic 3 Design of the user interface The outcome and results of the application have to be understandable and shown in a clear way This assures a proper handling of the application during a meeting between the engineer and architect As a result the process of form finding and obtaining a conceptual design should be better and faster The results are shown in Chapter 5 Topic 3 Analysis methods and algorithms This topic is investigated during the research and design stage Relevant theories and methods are T J van Swinderen August 2009 Structural Design Lab TU Delft A theory to calculate and analyse the models and designs A method to generate brick patterns The results are shown in Chapters 4 and 5 Topic 4 Brick structures in building practice This topic is looked upon during every stage of the final thesis because it is important to make sure the final design is realistic and buildable Related aspects are Brick fabrication and production Building physics Constructing the structure e Prefab elements connected on the building site e Allin situ The results are shown in Chapter 6 together with several other aspects that were encountered during the research process Main report Structural Design Lab TU Delft Figure 3 1 Examples of brick structures ST SED s OASI
114. rsion of the tool due to problems with the generation of the dual grid and with performing the analysis To make the analysis step fully functional the theory regarding the use of several matrices 16 has to be implemented The script to generate the masonry pattern is partly working but fails at certain places of the pattern This is solved by reviewing the script by an professional INTERACTIVE The application offers a good interactivity between the users and the application and the results A shell based on five parameters can be created The user has the ability to change the position of nodes in the model and adapt analysis variables such as the loading Several layers can be exported such as the masonry pattern for the producer the surface model for architectural renderings and the force network for further analysis by the engineer CONNECTION WITH OTHER SOFTWARE The ability to import files is reached to a certain extent where the ability to export is incorporated nearly completely Import In the current version of the application it is possible to import an obj file When it is imported it can be used as shape to approximate with a parametric model Export Each layer created and used in the application is exportable To assure the layers are usable in other programs such as T J van Swinderen August 2009 Structural Design Lab TU Delft AutoCAD and Maya several file formats are available The export fi
115. s regarded as uncommon for the period of time he lived in Some of his projects are the Sagrada Familia and the Colonia Guell both located in Barcelona Spain He is famous in the engineering world due to the method he used frequently in his work the hanging chain model 23 Main report Structural Design Lab TU Delft Figure 3 11 An example of a compressive form active structure The Salginatobel bridge Switzerland Robert Maillart 1929 1930 oe dela meike eeu of he wappert a asair Sug hrae ah dotlrn ee a F Tit Figure 3 12 Hanging chain models created by Antoni Gaudi to analyse and design structures e g the Sagrada Familia Barcelona Spain deal rn eed at bee sap ee antal recitals Alaran b a mdr ENE k at uA ek eee t ga sei he ene zel de Predict rm e gi r E E i ta E en gt ip en Pe LE ke i Figure 3 13 Palazetto dello Sport a ribbed concrete shell designed by Pier Luige Nervi Rome Italy 1957 TJ van Swinderen August 2009 24 Main report The method of the hanging chain model Gaudi First chains are hung on a wooden frame The chains can also be connected to each other to create a grid This model will now find equilibrium under the loading of its own weight When the shape is not moving anymore it has found its equilibrium and only tensile forces are acting in the chains To obtain the compression only model the model
116. s and within each of them a procedure is followed see below Procedure within a loop applied to all polygons A polygon is selected example polygon 8 belonging to node 8 Figure 5 18 2 The polygons connected to the selected polygon are searched for example polygons 7 9 and 10 Figure 5 20 3 The cmmon line of each related polygon is scaled to the length of the corresponding line of the selected polygon The length of the lines is calculated according to the Euclidean length which is based on the theorem of Pythagoras The scalefactor is the product of these two lengths example polygons 8 and 10 are regarded The scalefactor is Ll L2 Figure 5 21 4 The polygon related to the common scaled line is scaled by the corresponding scalefactor Figure 5 22 5 The related polygon is moved to the right position of the selected polygon Figure 5 23 Description of the problem The final dual grid consists of polygons with several scalefactors This is caused by the fact that a polygon of four and more sides is scaled either in one Figure 5 24a or in all directions Figure 5 24b The resulting new shape of the polygon is different Figure 5 24c The way of scaling in combination with the choice to loop through the polygons using their D number causes the dual grid to have different scalefactors Figure 5 25 Solution to this problem The solution is to use the method as described by P Block 16 using
117. s this gives a lot of possibilities regarding the appearance of a building or structure Besides using brick in designs for industrial and commercial buildings architects also want to give private housing a more unique and varying appearance by using shape variations of the brick Some information regarding non standard brick shapes is found in Chapter 6 TEXTURE The texture of bricks is mainly determined by the method of production Alternatively the steps during the process of giving the final shape to the stone and the post processing gives the brick a special texture Main report Brick buildings in Dhaka Bangladesh Design by Louis Kahn Crawford Municipal Art Gallery Cork Ireland 2000 Design by Erick van Egeraat Architects Smk we Aaiya ger Kann af s aie er EErEE NE t en Figure 3 4 Examples of free form designs in combination with masonry TJ van Swinderen August 2009 Structural Design Lab TU Delft Main report 3 1 3 Structural information LIMIT STATE ANALYSIS OF MASONRY The limit state analysis of masonry assumes the following Masonry particularly the mortar joints in masonry has no tensile strength 2 Compressive stress levels from loads applied to the structure are low relatively to the maximum allowed compressive crushing stress of masonry Therefore material strength properties are not likely to be determining in failure anal
118. s with less or more than 4 sides 85 Main report Structural Design Lab TU Delft Choose any other node related to an earlier inspected node as long as it i Is NO foundation node In this case node 8 a 10 ai 8 a Start with a line corresponding to one of a2 129 the already inspected lines in earlier steps a3 227 Do not change the angle In this case line 8 10 b Connect next clockwise to end of it In this case line 8 7 c Connect next clockwise to end of 2nd In this case line 8 9 d Find intersection between 1st and last line and calculate scaling factor of the lines to form the polygon to use in the other steps of the process RESULT Force polygon Angles are the same and have to stay the same Length proportion is known BUT can still change This will be seen when regarding the other nodes Length9 Length 10 17 2057 11 733 1 466 1 00 Figure 5 19 Procedure to construct a polygon in this case for node 8 of figure 5 18 TJ van Swinderen August 2009 Main report has not been found It is thought that it should be found at the point in the script where the dual grid is formed and the related polygons are scaled see 2 Generation of the dual grid Solution to this problem Review the script by an expert in scripting TJ van Swinderen August 2009 Structural Design Lab TU Delft 2 GENERATION OF THE DUAL GRID The dual grid is formed using several loop
119. sed by Torroja and Candela It is mainly used for large single story sheds used as warehouses gymnasia and workshops There are two generic 8 Dieste E Cascaras autoportantes de directriz catenaria sin timpanos Free standing vaults of catenary directrix without tym panum T J van Swinderen August 2009 Structural Design Lab TU Delft variations of the Gaussian vault thought they share similar geometric and structural roots the long span shallow vaulted roofs and the tall curved shells Long span shallow vaulted roofs Normally supported on a concrete frame or load bearing walls The main structural problem is not the axial stresses themselves but the thrust induced in such a slender structure leading to a tendency to failure by buckling Solutions to prevent buckling due to own weight and too low span thickness ratio Reduce the compressive stresses by increasing the rise of the vault Increase the cross section to make the vault stiffer Best is to add arched ribs Dieste used both methods but in a unexpected way he used hollow clay blocks to reduce weight to approximately two thirds of an equivalent solid concrete vault and the shape of the structure is manipulated to provide increased resistance to buckling without increasing the thickness This shape can be described using a family of catenary curves of varying rises Tall curved shells These structures are used for large horizontal stor
120. sis is presented in two parts The main report explaining all theories and information regarding the research 2 The appendices including the user manual This main report consists of The introduction to the problem and a presentation of the motivation for this research Chapter l The formulation of the research question and the topics that have been researched Chapter 2 The results of and information gathered from the literature study Chapter 3 The theories and methods used in the research the Thrust Network Analysis and Catmull Rom splines Chapter 4 Information about the new application such as the user interface procedures and options Chapter 5 Aspects concerning brick in building practice Chapter 6 Discussion conclusions and recommendations Chapter 7 Chapter 8 contains the references The user manual of the application more detailed explanation of relevant theories and other informative documents are listed in the second report Appendices T J van Swinderen August 2009 Structural Design Lab TU Delft Before presenting the results would like to thank the members of my graduation committee for their time assistance and for the knowledge they shared with me During the research process and the consultations they were of great help to me in advising guiding and commenting Tom van Swinderen August 2009 Main report TABLE OF CONI PREFACE A
121. sult and outcome of the application Therefore the user has to be able to change and adapt certain variables such as the nodal loading the load case and the position of the nodes 5 4 In the following paragraphs the main flow chart is explained more in detail The input options 5 5 The procedures within the application 5 6 The main elements of the model and used as input for the procedures 5 7 The output options 5 8 The next contribution to this chapter is description regarding the relation between the masonry pattern and force network 5 9 More information regarding the application is found in the user manual Appendix F Finally the last paragraph of this chapter deals with the place of the application within the workflow and within the design process to explain how the application is used in practice 5 10 In the most recent version of the report the current status of the application and its script is included at the end of the chapter 5 11 In the other paragraphs the text concerning the status of the discussed aspect is highlighted in red T J van Swinderen August 2009 THE NEW APPLICATION Structural Design Lab TU Delft 5 1 Main flowchart for the new application To explain the steps of the application in a clear way flowcharts are used First of all an overview of the steps of the application is presented in the main flowchart Figure 5 1 It consists
122. teel and timber are commonly used Main report Figure 1 3 Several brick structures by Eladio Dieste a Roof of Julio Herrera amp Obes Warehouses Montevideo Uruguay Completed in 1979 x ees Bis A a Light diffusing wall design 3 in church in Liesing Vienna Austria 1952 AAR FFF aI Fe AAA ATA volet II IIs volet IIS ERRANS BF III II II IID 53 3 RPRRRRRR b Example used in the Showroom of Knoll Internacional de Mexico Mexico City 1950 Figure 4 Concrete pattern designs by Erwin Hauer T J van Swinderen August 2009 Structural Design Lab TU Delft rs pe ne es n 24 KE Pant Ene af Arts b Church of Christ the Worker Atlantida close to Montevideo Uruguay This is a photo during construction It was completed in 1960 a A thin concrete shell Service station in Deitingen S d Switzerland designed by Heinz Isler b The Eden Project Cornwall UK Steel grid shells by Nicholas Grimshaw Figure 1 5 Example of a shell shape structure Main report as material for the structural elements for most structures and buildings In several parts of the world among others in the Netherlands another material is also used and applied regularly brick It is used in dwellings and as facade material although mainly in vertical elements such as walls A lot of knowledge has been gained by the building industr
123. texture in this research is of small importance Other aspects like the color and shape of the bricks and actual design of the structure are of bigger influence on the actual appearance and so the attention will be focused on these aspects 6 1 3 Color The color of a structure and the patterns which can be made with it are of big influence on the actual appearance Figure 6 1 Brick can be delivered in several colors e g white yellow red and brown The choice of color for instance a light or darker one affects the appearance of a building but even more when several colors are applied in one structure in other words applying color patterns As long as the 97 Main report Structural Design Lab TU Delft FN RASH Be a Figure 6 2 Wooden formwork to construct the structures of Eladio Figure 6 3 Brick patterns created by a robot laying the stones 1 9 Dieste TJ van Swinderen August 2009 98 Main report chosen colors of the brick are commonly used it should not be a problem to include several colors in the design of one structure 6 1 4 Bonding material As mentioned in 3 4 there is a new bonding material making its entrance in the building practice glue Bonding layers are a lot thinner this way and the appearance of the building changes One of the functions of bonding material though is to apply curvature in brick structures If this layer is made thinner the angle of curva
124. the grid has a bigger mesh than that of a masonry structure since the masonry has a grid of bricks while the steel structure has a grid of big planes between the steel grid Two generic forms of vaulted reinforced brick structures that Dieste is well known for are The free standing barrel vault 3 2 1 2 The Gaussian vaults 3 2 2 3 2 1 Free standing barrel vault Dieste describes this vault as free standing catenary shells without tympanums Figure 3 5 c and d In most of the buildings the structure underneath is minimal in order to leave the roof floating hovering above the floor below The barrel vault in one form can be seen as an extension of the arch as a series of connected arches running along a line of supporting walls These walls provide both horizontal and vertical reaction When the longitudinal span exceeds three times the transverse span the dominant structural action is bending But Dieste s vaults have a much higher longitudinal to transverse span ratio they are designed to allow both arch action and bending action to develop Dieste has developed the mathematical theory to calculate the additional stresses in the transverse section of the roof 7 Quote by Dieste in The Engineer s Contribution to Contemporary Architecture 2 page 28 Main report Figure 3 6 Example of a Gaussian vault by Eladio Dieste Be a looped pre stressing steel ia en
125. thod the theory of Gauss Jordan GJ is used It deals with solving LP problems with pivoting Pivoting uses row operations known as Gauss Jordan row operations sep to change one matrix entry the pivot to 1 and then to change all other entries in the pivots column into zero s More detailed information regarding the GJ theory is added in the appendices Appendix B However the Simplex method has some disadvantages For example it requires that all variables be non negative gt 0 also all other constraints must be in lt form with non negative right hand side RHS values More information about the steps and actual application of the Simplex method is found in the appendices Appendix C Two important variables within the Simplex method are the objective function and the constraints of the problem 47 Main report Figure 4 13 Equilibrium of one node in the force network Image courtesy of Philippe Block TJ van Swinderen August 2009 Structural Design Lab TU Delft 48 Main report The objective function contains the variable which needs to be optimized This optimization can be either to minimize or to maximize a variable The objective function consists of several variables and certain constraints may be assigned to these variables 4 5 2 Implementation in this research To explain the implementation in a good way an assumption has been made This is the fact that a node with th
126. tion about brick are used as checks and limits for the analysis and result of the application In the end the following five topics have been assigned as topics for research a The material brick b Shell structures 2 Design of a new software application Theories and algorithms 4 Brick structures in building practice Main report Structural Design Lab TU Delft TJ van Swinderen August 2009 10 Main report Topic Brick and shell structures a The material brick This topic is researched during the literature study Included aspects are Characteristics of brick such as strength and dimensions A comparison and advantages of brick over other materials Possibilities in regard to non standard brick shapes b Shell structures This topic is researched during the literature study Included aspects are Force distribution systems with the emphasis on form active structures Structural mechanics of shells such as geometry and curvature A Spanish architect and engineer Eladio Dieste designed structures in which topics a and b are combined Therefore his work is researched as well The useful information regarding this topic is presented in Chapter 3 Topic 2 Design of a new software application This topic is investigated during the research and design stage Relevant aspects are Software related aspects such as the ability to import and export models in certain file fo
127. to frost damage Main report Structural Design Lab TU Delft Figure 3 2 Brick types Box shaped Vormbak Handshaped Handvorm Cord press Strengpers 1 Length 2 Width 3 Height 4 Bed 5 Face 6 Header Figure 3 3 Dimensions and variables of brick T J van Swinderen August 2009 14 Main report The appearance can when well designed create an impression of stability and durability Brick is very heat resistant and thus will provide good fire protection DISADVANTAGES The costs for maintenance of masonry are higher than for steel timber and concrete structures To replace one brick is almost impossible and therefore it is difficult to repair damage to the bonding or to any brick Moreover masonry is more fragile for damages than other materials since masonry structures consist of many elements and as a result many connections that can be damaged while for instance steel grids are made out of several big elements and a concrete shell can even be considered as one big element Extreme weather may cause degradation of the surface due to frost damage If clay based brick is used care should be taken to select bricks suitable for the climate in question Masonry must be built upon a firm foundation usually reinforced concrete to avoid potential settling and cracking If expansive soils such as clay are present this foundation may need to be quite elaborate The high self weight of
128. tural Design Lab TU Delft Figure 6 4 A brick pattern upper picture layed by the robot bottom picture created by the E T H and Gramazio amp Kohler 20 Image Figure 6 5 Concrete structures A new technique created by Glecz courtesy of Gramazio amp Kohler MBGT Membrane Concrete Grid Shell Image courtesy of GTecz T J van Swinderen August 2009 100 Main report 6 4 Developments in the brick industry One of the reasons for the increased possibilities of masonry and structures made of brick are technological developments and research Three of these developments are discussed Glue as bonding material 6 4 1 2 Cutting brick stones in special shapes 6 4 2 3 A robot laying bricks in a computerized pattern 6 4 3 6 4 1 Glue as bonding material Glue offers more strength as bonding material and for that reason could be of interest for this research It needs a thinner bonding layer about 2 5 mm instead of 10 15mm which is normal for mortar bonding On the other hand the bonding layer is needed to create the curvature in the structure Moreover the strength of the masonry is determined by the weakest element of the combination of brick and bonding The bonding layer has a comparable compressive strength as brick Due to these reasons glue is not considered as bonding material 6 4 2 Cutting brick stones with a computer Curved surfaces are difficult to make with only one spec
129. ture per brick also decreases This is exactly what is not wanted for this research As a consequence this development will not be taken into consideration in this research 6 2 Building physics characteristics The main function for the models designed with the application are for now pavilions and other non internal climate buildings Though when a model is intended for an internal climate space the aspect of building physics is very important Aspects such as water penetration fire resistance and insulation have to be checked and included in the design Therefore to make the application better and more useful it is recommended to research how the building physics can be integrated in the application T J van Swinderen August 2009 Structural Design Lab TU Delft 6 3 Building the structures 6 3 Prefab elements in fabric In the factory prefab elements of brick are produced in the same way as is done for concrete prefab elements One of the options is to create a mould made out of for instance plastic or concrete in which the bricks are being placed This way also even reinforcement could be added which would increase the possibilities of shapes and models An example from building practice is the AKA Blade System used by CRH and other producers This system does not apply brick as a load bearing element though It is only used as facade and appearance element Therefore it is only used as an example of how
130. ult also determines the outcome of the force distribution Therefore the scale factor has to be adjustable The selected factor is directly linked to the scale of the dual grid shown in the user interface 5 6 4 Starting force network Block experienced the problem of finding a suitable force network for a random shape and therefore this was one of the recommendations for further research of the Thrust Network Analysis article 10 Since in this research this aspect has not been solved the need for an algorithm to find a starting 3D force network for an imported model remains a recommendation for further research T J van Swinderen August 2009 Structural Design Lab TU Delft 5 7 Main elements The input elements for the procedures 5 6 are specific elements These elements are the stones with which the application and the corresponding models and diagrams are build and analyzed The basic elements are Nodes 5 7 1 Lines 5 7 2 Polygons 5 7 3 Primal grid T 5 7 4 Dual grid I 5 7 5 Bricks 5 7 6 5 7 1 Nodes Characteristics When a node is selected it is highlighted in red including the loading Flowchart Figure 5 5 Constructor Create a 3D point by determining the X Y and Z coordinate Add a number to it for identifaction in the primal and dual grids Methods Select a point in the 3D force network model or primal grid Move a point in the
131. umptions and limitations DECISIONS A method inspired on graphical analysis is used to analyse the model the Thrust Network Analysis 10 Only shell type structures are regarded The Gaussian curvature has to be positive The program Processing is used to create a stand alone application which makes use of the JAVA scripting language ASSUMPTIONS The masonry can not be loaded by any tensile forces because the tensile strength of the masonry is neglected Reinforcement is not used and applied Compression only structures are designed The maximum compressive strength of the masonry cross section is 30 N mm When the stress in the network is higher a warning must be given in the application OBJECTIVES T J van Swinderen The new application has to be interactive after any change in the design the analysis and pattern have to adapt instantly The new application has to be able to perform a partial structural analysis of a double curved shell However only the force flow is checked The new application has to be able to create a brick pattern The new application has to be a stand alone application with the ability to export the result so that it can be used in the continuation of the design process August 2009 Structural Design Lab TU Delft LIMITATIONS The buckling behaviour of the shell shape is not regarded during the analysis The displacements and accompanying second
132. ve combines the new possibilities of concrete with the classical appearance of masonry and therefore is an interesting option 10I Main report CHAPTER Introduction In this chapter the conclusions and recommendations are presented The current results of the research are satisfying considering the aims and goals Chapter and 2 However several aspects concerning the application are up for discussion 7 1 After this discussion the conclusions concerning the achievements are given 7 2 Before the application can be applied in the design process more research is required Therefore several recommendations regarding aspects for further research are advised 7 3 T J van Swinderen August 2009 CONCLUSION AND RECOMMEND Structural Design Lab TU Delft 7 1 Discussion One of the main goals was to create an interactive tool with which the architect and engineer can rapidly generate a conceptual shape for a masonry shell Using the theory of Thrust Network Analysis 10 in combination with Catmull Rom splines has provided the right conditions to design a first prototype for this tool SHELL SHAPE Shell structures have a long history in technical development materials and designs Numerous shells have been designed and the material used to build them ranges from concrete and steel to timber The range of buildable shapes is big especially when several materials are combined such as reinforced
133. ve in the continuation of the design process For the structural engineer the force network model layer is of importance for the architect the surface models and the contractor might be interested in the masonry pattern layer Figure 5 11 The force network mode for the structural engineer The surface model consisting of curves for the architect The surface model consisting of polygons for the architect The brick pattern for the brick producer and contractor The 3D model can be exported as four file formats AutoCAD DXF file Autodesk Maya MEL file Rhinoceros RVB file SketchUP RB file 79 Main report Figure 5 12 Flowchart of the Masonry pattern procedure Tab or 2 Finalized conceptual force network model Structural Design Lab Wrong pattern ee ee T J van Swinderen Tab 3 Masonry pattern Set the dimensions of the brick stone Three sliders W Width base X axes D Depth base Y axes H Height structure in Z direction Create shell Create a shell of bricks Create surface model according to the force network model The theory of Catmull Rom is applied 2 Approximate the length of the curves of the surface Determine the number of bricks needed for each curve 3 Determine position and angles of the first brick using Catmull Rom formula The position of the n
134. ve the problem 4 4 4 Nodal loading 4 4 5 Scalefactor C and its influence on the solution Linear optimization theory the Simplex method 4 5 1 General information 4 5 2 Implementation in this research 4 5 3 Solution for one point 4 5 4 Solution for all points 4 5 5 An example Theory of Catmull Rom splines Masonry pattern generation 4 7 Linear rectangular pattern 4 7 2 Spherical pattern Limitations 5 THE NEW APPLICATION 5 5 2 Bas 5 4 5 5 5 6 5 7 Main flowchart for the new application User interface Display the 3D model Variables to adapt 5 4 Nodal loading 542 Load case 5 4 3 Relocate nodes 5 4 4 Adapt scale factor Input options 5 5 Create a parameter model 5972 Import a model Procedures 5 6 Implementation of the Simplex method 5 6 2 Relocate points 5 6 3 Change scale factor 5 6 4 Starting force network Main elements 5 7 1 Nodes 5 7 2 Lines 5 7 3 Polygons 5 7 4 The Primal grid T 5 7 5 The Dual grid I 5 7 6 Brick TJ van Swinderen August 2009 Structural Design Lab TU Delft 43 43 45 45 45 45 45 47 47 47 49 5 53 53 57 63 63 63 65 67 67 69 69 7 7 7 7 71 73 73 73 73 73 73 75 75 75 75 75 77 I7 77 77 Main report 5 8 5 9 5 10 5 11 Export options 5 8 Stresses and forces in structure 5 8 2 Export final model Masonry pattern 5 9 Range of dimensions for brick Workflow of the app
135. wn as a reciprocal relationship The mechanical property of reciprocal diagrams is expressed in the following theorem by Maxwell 8 If forces represented in magnitude by the lines of a figure be made to act between the extremities of the corresponding lines of the reciprocal figure then the points of the reciprocal figure will all be in equilibrium under the action of these forces Figure 0 1 Catmull Rom splines To generate the shell surface create the masonry pattern and make the application interactive the theory of Catmull Rom splines has been used The theory creates curves using four points and the angle of the curve in these points The curve passes through all points The factor t determines the curvature of the line The curvature of the masonry pattern is limited by the allowed angles between the bricks To make sure the range of possible shapes is as big as possible the factor t has been set lower than the commonly used 0 5 In the current prototype it is 0 2 THE TOOL It has been created in Processing 24 an open source programming language and environment The new application consists of four main steps 1 Setup of the initial force network model 2 Analysis and design of the network model 3 Generate the masonry pattern which can either be a linear pattern or a spherical pattern 4 Export the final model The initial force network model setup in step has two options a Create a parameter
136. y about brick and how to apply it in practice 17 In general bricks are made out of clay It is a natural material and therefore considered to be a sustainable material After demolition it can be re used as filling in for instance concrete This is a positive characteristic of brick taking into account all recent climate problems Brick performs well in building physics the thermal insulation is good Another advantage of brick is the freedom it gives for a structure to be built out of small elements Theoretically all these elements can have a different size colour and surface However up till now brick is mainly being used in straight vertical walls The question raises why brick is almost never considered as a material for double curved structures such as shells In other parts of the world for instance in Uruguay the Spanish architect and engineer Eladio Dieste created double curved structures with a very slender appearance Figure 3 Besides their beautiful appearance another remarkable aspect is the time in which they were built between the 1950 s and 1980 s long before the era of the introduction of the computer in the design process The techniques he used are therefore interesting for further research on how to use them in the present time 2 Architectural wishes As mentioned before the communication with the architect is a requirement to assure a smooth and optimised design process Not only the shape of the bu
137. ysis However the active stresses are one of the checks for the analysis as performed in this research 4 4 3 Sliding failure does not occur BONDING There are rules of bonding which have some exceptions These specify the overlap between courses that is visible outside the wall and also the overlap which must be made within the wall for walls which are more than half a brick thick The maximum width of the bonding layer in this research has been set to 16 mm COMPRESSIVE STRENGTH Masonry has a high compressive strength but is much lower in tensile strength twisting or stretching unless it is reinforced Brickwork arches can span great distances and carry considerable loads The compressive strength of brick ranges from 7 140 N mm depending on the type of brick and function it is used for For load bearing walls in residence buildings the strength is set to 25 30 N mm These values are used in this research to check the active stresses in the structure TENSILE STRENGTH Brickwork like unreinforced concrete has little tensile strength and therefore performs best when the whole structure is in compression Where required steel Heyman J 1995 The Stone Skeleton Cambridge Cambridge University Press 2 http www knb baksteen nl publicaties publicatie 62 htm 19 3 http www knb baksteen nl publicaties publicatie_61 htm 19 T J van Swinderen August 2009 Structural Design Lab TU D

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