GBTUL 1.0β
Contents
1. 60 65 70 75 80 85 90 95 100 110 120 130 140 1 220 230 240 250 300 350 400 450 500 950 1000 50 0 6 N 55 22 Figure 3 5 GBTUL 1 02 Overview of Screen 3 Member graphic representation window see 29 View Plan see 30 Log uniform see 31 and Fig 3 6 b Lengths see 32 Button NEXT see 33 Screen 3 Member Analysis dialogue boxes Numerical solution Number of Finite Elements 10 Eigensystem solver Cholesky Stodola Modal boundary conditions C F c V Use Numerical Solution Plot Member Number of GBT finite elements Eigensystem solver see 35 Modal boundary conditions see 36 Loading see 37 40 and Lengths Log uniform Number of lengths 100 From L 1 00 ToL f 100 00 ox e b see 31 Figure 3 6 GBTUL 1 0 Screen 3 a Numerical Solution tab and b Log Uniform dialogue box Screen 3 Member Analysis dialogue boxes Numerical solution N x Numerical solution My x Qy c Loading minor axis bending moment see 39 d Loading bimoment see 40 Figure 3 7 GBTUL 1 02 Screen 3 Dialogue boxes concerning the loading for the Numerical Solution 4 Output Data Screens 4 and Text Files The results of the analyses performed are represented graphically in Scre2. Conditions Loads Solver Halfwaves Analytical Uniform Cholesky Fact User provided Uniform or Cholesky Fact or Automatically Non Uniform Stodola Method computed Numerical Notes The same S S for all deformation modes To be provided independently for i major and ii minor axis bending iii torsional and distortional modes and iv local plate modes 3 Resulting from i unequal end moments and ii distributed or iii point loads A set of numbers from 1 to the maximum value specified by the user The user is able to provide an arbitrarily long list of member lengths L values so that the code produces a curve describing the evolution of buckling load parameter or natural angular frequency with L as well as the corresponding P vs L modal participation diagrams The buckling or vibration modes are represented by means of either i 3D deformed configurations of the entire member including interactive visualisation tools or ii 2D deformed configurations of any given cross section Furthermore the code output data is also saved in formatted text files which makes it very easy to process them by means of other spreadsheet applications e g Microsoft Excel The GBTUL 1 02 graphic interface involves the sequence of four screens shown in Figure 2 2 while the first three deal with data input the fourth one provides the result output This sequence is closely related to th
3. the member end support conditions the code covers i pinned simply supported S S ii cantilever C F iii fixed C C and iv fixed pinned C S members moreover it is possible to specify different support conditions for the various deformation modes e g for bending and torsion As for the pre buckling loadings they may stem from combinations of i end moments or axial forces ii distributed and iii concentrated forces the transverse loads acting upon the shear center However no effect of applied loading i e reduction of stiffness frequency due to geometrically non linear work of the acting stresses is accounted for in vibration analyses Moreover two types of member analysis are available i the Analytical Solution only for simply supported S S members under uniform loads and ii the Numerical Solution always applicable and involving a discretisation of the member length into GBT Based finite elements see Table 2 1 Whenever possible the Analytical Solution should be used because of the lower computational time required and its simpler inputs with relation to an equivalent Numerical Solution gt a b Figure 2 1 Examples of member cross sections a open handled by GBTUL 1 0 and b closed not handled by GBTUL 1 0 Table 2 1 Differences between Analytical and Numerical Solutions Type of Support Pre Buckling Eigensystem Number of Solution
4. 606809060 Figure 3 1 Two examples of natural nodes and walls numbering natural nodes 7 8 9 10 11 walls Rack sections template Define the geometry of the Rack sections by entering the web bw flanges bf inner lips b11 and outer lips 012 widths and numbers of intermediate nodes INodes the inner and outer lips angles 901 and 02 and the thickness t For a Return lips section enter 02 180 Z sections template Define the geometry of the Z section by entering the web bw flanges bf and lips 61 widths and numbers of intermediate nodes INodes the lip angle 8 and the thickness t For plain Z sections just enter b1 0 I T sections template Define the geometry of the I section by entering the web bw and flanges of1 top and bf2 bottom widths numbers of intermediate nodes INodes and thicknesses tw tf1 and t 2 For T sections just enter bf 2 0 Angle sections template Define the geometry of the angle section by entering the web bw and flange of widths and numbers of intermediate nodes INodes and the thickness t Plate section template Define the geometry of the plate section by entering its height h the thickness t as well as the number of intermediate nodes INodes Consider at least 2 intermediate nodes in the plate Besides note that a minimum slenderness of h t 5 is expected at least for an accurate analysis of loca
5. GBTUL 1 07 Buckling and Vibration Analysis of Thin Walled Members USER MANUAL Rui Bebiano Nuno Silvestre Dinar Camotim Department of Civil Engineering and Architecture DECivil IST Technical University of Lisbon Portugal 2010 1 Introduction The code GBTUL 1 02 which implements recently developed formulations of the Generalized Beam Theory GBT performs linear buckling bifurcation and vibration analyses of elastic thin walled members The objective of the Program Tutorial is to provide concise descriptions of all the commands data entries and results outputs available at the GBTUL graphical user interface GBTUL 1 0 acronym for GBT at the Technical University of Lisbon is a freeware code meant to provide the users with a graphical and easy to use structural analysis tool Being based on GBT it allows the users to benefit from the technique s unique modal features For more information including access to the electronic forms of the manuals referenced above visit the program website http www civil ist utl pt gbt 2 Program Scope and Structure The code GBTUL more specifically its 1 08 version performs buckling bifurcation and vibration analyses of elastic thin walled members with arbitrary open cross sections i e cross sections containing closed cells cannot be handled see Figure 2 1 Moreover each wall can be made of one or more isotropic or specially orthotropic materials Concerning
6. analysis The tabs are intended for the user to review or change input data provided before and not to proceed to the next screen rather this should be done by clicking button NEXT As for the fifth and last screen it displays some information about the authors of the program 2 Material model Allows the user to specify the orthotropic material properties i longitudinal Exx and transversal Ess elastic modulli ii Poisson s ratios Uxs and Usx iii distortion modulus Gxs and iv volumetric mass density ro Different materials can be specified in further lines each one should be given a different reference number Mat For isotropic members e g steel beams it is easier to use the Isotropic template see 3 For buckling analyses a unit value may be assigned to ro 3 4 5 6 Isotropic material template Prompts the user for the mechanical properties defining the isotropic material i elastic modulus E ii Poisson s ratio u and iii volumetric mass density ro For buckling analyses a unit value may be assigned to ro Natural nodes In this table the user introduces the X horizontal and Y vertical coordinates of the natural nodes i e the points defining the ends and intersections of the plates that compose the member cross section For an N plated cross section the number of natural nodes is always equal to N 1 and each one should be defined by its own ref
7. cification of distinct support conditions to four types of GBT modes mode 2 major axis bending mode 3 minor axis bending modes 4 and the distortional ones 4 D and the local plate modes LP The four types of support conditions available are simply supported S S clamped free C F cantilever clamped clamped C C and clamped supported C S For more information on this modal support conditions see GBT Theoretical Reference Part 5 Loading axial force Numerical Solution Enter the parameters defining the applied constant axial force N and distributed longitudinal load nx both positive for compression Loading major axis bending moment Numerical Solution Enter the parameters defining the applied left My1 and right My2 end moments the transversal uniformly distributed load py and one transversal point load Qy acting at the longitudinal coordinate x aL where parameter a is also to be provided a e g for a mid span point load enter 0 5 The transversal loads act over the shear centre axis 39 Loading minor axis bending moment Numerical Solution Enter the parameters defining the applied left Mz1 and right Mz2 end moments the transversal uniformly distributed load pz and one transversal point load Qz acting at the longitudinal coordinate x aL where parameter a is also to be provided a e g for a mid span point load enter 0 5 The transversal loads act ove
8. e performance of a GBT analysis see GBT Theoretical Reference 1 while Screens 1 2 and 3 concern the inputs associated with the cross section analysis deformation mode selection and member analysis ii Screen 4 displays the sought buckling or vibration solution Outputs oN e Type of analysis e Deformation mode e Lengths e A L L P L curves e Member material display and e Applied Loads gt e Section geometry selection e Support Conditions e 2D 3D configurations Figure 2 2 GBTUL structure of the graphic interface 3 Input Data Screens 1 3 3 1 Screen 1 Cross section Analysis The first screen of GBTUL 1 0 prompts the user for the data concerning i the material ii the cross section geometry and nodal discretisation and iii the type of analysis pretended i e buckling or vibration The interface includes templates corresponding to several usual cross section geometries e g C Z I sections which the user should use whenever possible since the Natural Nodes and Section Walls tables are intended for general type cross sections Figures 3 2 3 3 present respectively a general overview of the Screen 1 and the dialogue boxes related in both cases all the interface objects are identified and the corresponding usage is explained next Notes concerning the features of Screen 1 1 Screen tabs The first four tabs correspond to each of the four screens involved in the procedure of a GBTUL
9. elements for Numerical Solution defined at Screen 3 43 GBT modes Use this tool to select a sub set of deformation modes from within those selected at Screen 2 to be included on the 2D or 3D graphical representation The numbers of the modes to be considered can be entered directly on the dialogue box see Fig 4 2 a This enables for instance to isolate the contribution of a given single mode to the member deformed configuration 44 Scale Enter a scale factor for the 2D or 3D representation 45 46 47 48 49 2D Plot Pressing this button yields the 2D configuration see Fig 4 2 b related with the member see 41 and cross section see 42 selected Moreover the two radio buttons below allow the choice between the in plane in plane or longitudinal warping cross section displacement fields The undeformed cross section is represented in yellow while the deformed in red 3D Plot Pressing the button creates on a separate window an interactive interface see Fig 4 2 c where the whole member is displayed and which contains several viewing tools The deformed configuration can also be plotted either with opaque surfaces Surface or with a network of lines Net Plot options Several options concerning the visualization of the buckling load or frequency vs length plot see 48 e Scale Allows the choice between 3 possible scales for Logarithmic Log Bi logarithmic Log log o
10. en 4 as i plots of buckling or vibration curves providing the variation of the buckling load parameter or natural frequency with the member length L ii modal participation diagrams and iii 2D or 3D representations of the member buckling or vibration modes In addition all the data are also recorded in formatted text files making it possible any further processing 4 1 Screen 4 Results Screen 4 the last of the GBTUL 1 02 analysis procedure presents graphical outputs of the analyses consisting of 1 buckling load or vibration frequency vs length curves and ii 2D or 3D configurations and iii modal participation diagrams of the buckling or vibration modes Figures 4 1 4 2 a and 4 2 b c depict respectively a general overview of Screen 4 the Mode Selection dialogue box and the 2D and 3D representation windows Notes concerning the features of Screen 4 41 Case Selection Use the lt and gt buttons to select the member length top buttons and buckling or vibration mode bottom buttons to which refer i the summary of results and ii the 2D or 3D representations 42 Cross Section Use the lt and gt buttons to select the member cross section to which corresponds the 2D graphical representation The cross section is identified by its x L coordinate and the number of cross sections available is equal to 1 plus the number of intervals for Analytical Solution or the number of finite
11. erence number Nodes The nodes should be numbered successively and according to their order of placement in the cross section see Figure 3 1 for branched sections see GBT Theoretical Reference Part 3 Section walls In this table the user enters the data defining each of the cross section walls namely e Wall The wall reference number e Node I The reference number of the initial node of the wall e Node J The reference number of the end node of the wall e Order The order of the wall for branched sections see GBT Theoretical Reference Part 3 For unbranched sections insert 0 for all the walls e Mat The reference number of the wall material e Inodes The number of intermediate nodes of the discretization e Tick The wall uniform thickness The walls as well as the natural nodes should be defined in a consistent and manner i e numbered according to their placement in the cross section geometry as Figure 3 1 shows C U sections template Define the geometry of the lipped channel by entering the web bw flanges bf and lips 61 widths and numbers of intermediate nodes INodes the lip angle 0 and the thickness t For plain channels U sections just enter b1 0 For a Hat section enter 0 90 A cross section is said to be unbranched if no more than 2 walls share any of its nodes e g C Z U sections and branched otherwise e g I T sections
12. ird screen of GBTUL the user i chooses the type of solution analytical or numerical see the GBT Theoretical Reference Part 5 and ii specifies the member lengths loading and support conditions The specific details associated with the two types of solution are introduced on the corresponding tabs Analytical Solution and Numerical Solution on the left side of the screen Figures 3 5 and 3 6 a b show respectively a general overview of the Screen 3 the Numerical Solution tab and the Log Uniform dialogue box in all cases the interface objects are identified and explained in the following notes Notes concerning the features of Screen 3 22 Analytical Numerical tabs These tabs prompt the user for the data required to perform the Member Analysis i e the resolution of the member equilibrium equations by the analytical or the numerical procedure respectively The analytical solution is applicable only for simply supported members subjected in the case of a buckling analysis to uniform loadings The numerical solution which involves the longitudinal discretization of the member into GBT based beam finite elements see 34 is always applicable See Table 2 1 for a comparison between the features of these two procedures Only the tab associated to the procedure to be used need to be filled by the user it is important to check the checkbox at the bottom left corner of the tab to confirm the ch
13. l deformation 12 13 14 15 Section graphic representation window Plots the cross section as well as some related items see 13 for the user to visualize confirm the data provided Visualization tools Check the checkboxes for the following items to be displayed Natural Nodes i e the Nodes numbers Intermediate nodes Coordinate System x y the one associated with the natural nodes Wall Numbers i e the Walls numbers Wall Orders and Materials i e the Mat numbers Analysis Type Select the type of analysis desired Stability i e linear buckling or Vibration i e free vibration NEXT Move on to the Screen 2 to proceed to the GBT Mode Selection inputs screen Screen 1 Cross Section Analysis at GBTUL 1 0b Cross Section Analysis GBT Mode Selection Member Analysis Results About Material Model input see 2 Screen tabs see 1 Material Model 2 Isotropic y Ess Uxs Usx Gxs ro 21000 21000 0 3 03 8076 92 7 8E 08 Isotropic material Clear Table template see 3 and Fig 3 3 a Section graphic representation Natural Nodes Clears all the Clear Table table cells x window see 12 ow wo oo Natural Nodes Section Walls Clear Table Wall Node Node J Order Tick a _Clear Table b h 1 0 0 1 2 0 0 1 input see 4 2 0 1 z Section Walls a4 Visualisation tool
14. lection The second screen of GBTUL 1 08 shows the results of the cross section analysis i e 1 the most relevant geometrical properties e g area inertia moments etc ii the stiffness matrices linear and geometrical and iii the deformed configurations of the GBT deformation modes Moreover this screen allows the user to select the set of GBT deformation modes to be included in the analysis Figure 3 4 shows a general overview of Screen 2 where the objects are identified and explained in the following notes Notes concerning the features of Screen 2 16 Cross section geometrical properties A list of eleven cross section geometrical properties only available for isotropic members Area A major and minor inertia moments 11 12 warping G and torsional J constants center of mass x cg y cg and shear center x sc y sc coordinates and asymmetry factors b1 b2 17 GBT stiffness matrices By pressing the buttons the GBT stiffness matrices are displayed in separate dialogue boxes see the GBT Theoretical Reference Part 2 for physical meaning of the matrices 18 Mode graphic representation window Plots the in plane or out of plane displacement fields associated with the selected GBT deformation mode see 19 The undeformed cross section is represented in yellow while the modal configuration does in red 19 Mode visualization tools The mode representation window see 18 displays the configuratio
15. list and ii the lowest and i112 the highest length values Lengths On this text box the user writes the list of lengths of the members to be analyzed at least 2 values should be given The 33 34 35 36 37 38 values should be written in ascending order By default a list of 63 lengths is provided For a list of lengths equally spaced on a logarithmic scale use the Log Uni form tool see 31 and Fig 3 6 b NEXT Move on to perform the main analyses and then Screen 4 with the output of the results Number of GBT finite elements Numerical Solution Enter the number of GBT based beam finite elements of the uniform longitudinal discretization While the default number 10 has proved enough for accurate single to three halfwave mode estimates a higher number should be provided if modes exhibiting higher number of halfwaves are expected For more information on this finite element see GBT Theoretical Reference Part 5 Eigensystem Solver Numerical Solution The two mathematical procedures to solve the GBT finite element eigensystem are i the complete solution of the eigensystem by using the Cholesky factorization or ii the simplified solution by the Stodola method faster but less accurate The user is recommended to use the first option except for larger problems Modal Boundary Conditions Numerical Solution By checking the corresponding radio buttons this tool allows the spe
16. n of the mode which number appears on the choice box by using the lt and gt buttons the user may choose the mode to be shown Furthermore the buttons in plane displacements and warping displacements allows the choice between in plane and out of plane displacements 20 Mode selection tools By default all the available deformation modes see the GBT Theoretical Reference Part 4 for comments on the number of GBT modes obtained are to be considered In order to choose a subset of these modes the user may write their number directly on the text box or use the button Pick mode button to pick the mode currently displayed at the graphic window 21 NEXT Move on to the Screen 3 to proceed to the Member Analysis inputs screen Cross section geometrical pro perties see 16 GBT stiffness Matrices see 17 Mode visualization tools see 19 Screen 2 GBT Mode Selection a GBTUL 1 0b Cross Section Analysis GBT Mode Selection Member Analysis Results About Cross section Geometrical Properties KA bl GBT Matrices 2 GBT Mode Mode Selection JF 3 G in plane displacements Pick Mode Pick All Warping displacements Figure 3 4 GBTUL 1 02 Overview of Screen 2 Mode graphic representation window see 18 Mode selection tools see 20 Button NEXT see 21 3 3 Screen 3 Member Analysis In the th
17. ned and used as input to most data processing applications Those files created in the folder GBTUL calc Output_Files are the following i Matrices txt contains i the displacement values u v and w at each cross section node for each deformation mode and 12 the components of the GBT matrices stiffness and mass ii Results txt includes iij a list of the eigenvalues buckling load parameters or natural frequencies associated with every member length and eigenvector buckling or vibration mode as well as ii2 the corresponding modal participation factors and iiz the number of half waves they exhibit only for the analytical solution iii Mafunes txt contains the longitudinal amplitude functions Ax and their derivatives 4 x associated with every deformation mode included in the analysis for all member lengths and buckling or vibration modes determined These functions are defined by their values at a finite set of cross sections along the member length See GBT Theoretical Reference Case Selection see 41 Cross Section see 42 GBT modes see 43 and Fig 4 2 a Scale see 44 2D Plot see 45 and Fig 4 2 b 3D Plot see 46 and Fig 4 2 c Plot options see 47 Screen 4 Results a GBTUL 1 0b Cross Section Analysis GBT Mode Selection Member Analysis Results About Case Selection Copy Length Defo
18. oice 23 Loading Analytical Solution Enter the load parameters defining the pre buckling consisting of a combination of uniform axial force N positive for compression major My positive for compression on the upper part of section and minor Mz positive for compressions on the left part of section axis bending moment and bimoment B note that the bending moments act about the principal axes of the cross section and not necessarily about x and y axes 24 Number of halfwaves Analytical Solution Enter the maximum number of longitudinal halfwaves to be exhibited by the sinusoidal buckling or vibration mode e g if this number is 3 the resulting 25 26 27 28 29 30 31 32 modes will exhibit between 1 and 3 halfwaves For single halfwave buckling or fundamental vibration modes enter 1 Number of intervals Analytical Solution This entry concerns only the 2D and 3D graphical representations to be shown in Screen 4 it defines the number of longitudinal intervals that define i the number of cross sections available for 2D representations and ii the quality of the 3D representation The default value 10 is fairly enough for the 3D representation of single or two halfwave modes but might be too low for modes exhibiting higher number of halfwaves Plot Member Plots a representation of the member i e its supporting conditions and loading for the condi
19. r rectangular Rectangular When changing the scale the Change limits checkbox must be unchecked e Change Limits Allows the user to change the top and bottom limits of the plot The checkbox Change limits must be on After writing the limits values the user must press button Update to update the plot e Multiple plots When this checkbox is checked several curves corresponding to all the buckling or vibration modes calculated are plotted simultaneously Otherwise only the one related to the first mode obtained is shown e Show Markers If this is checked the points i e solutions obtained used to trace the curve are represented by markers on the plot Buckling or frequency curves The curves are traced by straight line segments linking the points corresponding to the solutions obtained there will be as many points as the number of lengths provided Several plot options are available see 47 Summary of results Presents the main results concerning the member selected see 41 namely i the buckling load or frequency and ii the modal participation percentages of the 3 most important GBT modes 50 Modal participation diagram The GBT modal participation diagram P L associated to the resulting buckling or vibration modes For more information on modal participation diagrams see GBT Theoretical Reference Part 5 4 2 Text Files All the results are saved into three formatted text files which can be ope
20. r the shear centre axis 40 Loading bimoment Numerical Solution Enter the value of the applied bimoment Only uniform bimoments are dealt with by GBTUL 1 0 3 4 DOS like interface window After pressing button NEXT on Screen 3 the program starts to perform the analyses for all the members lengths During this process a DOS like interface appears providing the user with some information about the analyses being performed in the case of the Numerical Solution always much more time consuming that the Analytical Solution this information includes i the estimated time that the process will take and ii some results already available namely the first buckling loads or vibration frequencies of the members already analyzed Analytical Numeri cal tabs see 22 Loading see 23 Number of halfwaves see 24 Number of intervals see 25 Plot member see 26 Solution checkbox see 27 Number of eigen modes see 28 s SBF 1 0b Screen 3 Member Analysis Cross SectioM analysis GBT Mode Selection Member Analysis Results About 2 Analytical solution Loading Number of halfwaves Number of intervals V Use Analytical Solution Numerical solution Mz B foo oo From 1 to 10 Plot Member Number of Eigemodes ES Fay Lengths 2 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 22 35 40 45 5 170
21. rmed Configuration Cross Section GBT Modes 2D Plan 3D Plan in plane Surface Results of buckling analysis of member with L 1 Mode 1 Buckling local Ib 608 68083 Modal participations 7 of the most important GBT modes Warping C Net 2 oes P7 65 0737 P9 24 905 P13 4 2173 Oth 5 804 Copy Plot options Scale Log Log log C Rectangular 2 Change Limits Lmin i Se nax 100 Pb min V Multiple plots Updat Show markers _ Update Figure 4 1 GBTUL 1 02 Overview of Screen 4 Buckling load or frequency vs length curves see 48 Summary of results see 49 Modal participation diagram see 50 Screen 4 Results Dialogue Box and Graphic Windows aJ 2D Deformed Configuration a GBT 3D Ouput 7 File Change View Mode Selection Pick All 1234567891011 121314 15161718 CORTONA Ce Or 0 0 O O a see 43 b see 45 c see 46 Figure 4 2 GBTUL 1 02 Screen 4 a Modal selection dialogue box and b 2D and c 3D graphic representation window
22. s see 13 0 1 input see 5 7 PON y Natural Nodes Wall Numbers l Intermediate Nodes Wall Orders Templates 2 F Coordinate System xy I Material Analysis Type see 14 Cross Section T Analysis Mode Templates see Stability Analysis cu uT 6 11 and C Vibration Analysis Button NEXT Rack Z Angle Rectangular Figs 3 3 b f see 15 Figure 3 2 GBTUL 1 02 Overview of Screen 1 Screen 1 Cross Section Analysis dialogue boxes Material Model Isotropic Young s Modulus E Poisson s Ratio u Mass density ro Cancel a Isotropic template see 3 Template Z Seccion al Template C U Seccions Dimensions INodes m m Cancel b C U sections template see 6 Template I T Seccions ps Dimensions INodes C M n E m Dimensions INodes 7 1 i 1 4 Ae 90 Cancel A d Z section template see 8 e I T section template see 9 Template Rack seccion Dimensions INodes c Rack section template see 7 Template Angle Seccions Dimensions INodes i L M z Cancel f Angle section template see 10 Figure 3 3 GBTUL 1 08 Dialogue boxes of Screen 1 a Isotropic material template and b f cross section geometry templates 3 2 Screen 2 GBT Mode Se
23. tions specified Solution checkbox Confirms the user s choice between Analytical or Numerical solution it is necessary to check the due checkbox in addition to fill the tab and plot the member Number of Eigenmodes Defines the number of buckling or vibration modes to be determined for each length For the first critical buckling or fundamental vibration mode enter 1 Member Graphic Representation Window Represents graphically the members to be analyzed including the supporting conditions and pre buckling loading For the Numerical Solution the points corresponding to end nodes of the finite elements are also presented The representation can refer to either x y or x z plans see 30 View plan Allows the choice of the view plan to be represented on the graphical window above see 29 i x y corresponds to longitudinal and major axis and represents the support conditions and loading pertaining to major axis bending while ii x z is the same for minor axis bending After changing the view plan one must press the Plot Member button for the view to be plotted Log uniform This tool easily generates a list of lengths that are equally spaced on a logarithmic scale the one usually used in load or frequency versus length curves In the corresponding dialogue box see Fig 3 6 b the entries ii Number of lengths iii From L and iii To L prompt respectively for i2 the number of lengths in the
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