Lecture 03a Assignment 1
Contents
1. then use Accessories Communications Remote desktop connection Alogin box will appear Type kontor uio no and press Connect You will then be faced with a normal UiO login box Log in to UiO with your UiO user and password Click on the Linux penguin icon to open a Linux login box log in on this with your UIO user and password then you will get a Linux window Proceed as described above e You can also purchase a student MATLAB CD and licence at the bookshop MEK4540 201 1 3a 8 U10 Department of Mathematics University of Oslo MATLAB some basic principles Use the help function to find out how each type of command works MATLAB also contains some tutorials for beginners MATLAB is specially designed for matrix operations It uses scalars vectors and matrices Matrix operations are simple to write e g Invert matrix A to give matrix B B inv A Transpose matrix A to give matrix B B transpose A or B A Pre multiply vector V by matrix A to give vector W W A V Basic input file is filename m This can call up subroutine files but this facility is not needed for this assignment Comments ignored by program are preceded by sign can be separate line or end of line The result of each line will be written to the command window unless it ends with sign Values of varables are retained unless set to zero by clear command MEK4540 201 1 3a 92. whether you are using the tensor shear stram or the engineering shear strain NB This part should be solved first without using MATLAB but the results can be checked with MATLAB 1n connection with Part d MEK4540 2011 3a 2 U10 Department of Mathematics University of Oslo Assignment no 1 contd d Study the MATLAB script provided and modify it to calculate the A B and D matrices for laminates consisting of the above plies im the three lay ups Q 0 45 45 0 un 0 45 45 90 and Qu By ctr 43 0 Comment on the differences identifying any special properties displayed by the lay ups Note The MATLAB script may also be used to check the individual ply stiffness matrices Q calculated in part c e Calculate the total thickness of each of the lamunates 1 and 11 Modify the same MATLAB script to calculate the strams for each of these laminates when they are exposed to a tensile stress 1n the 0 direction of 1 MPa which must first be converted to a force per unit length Hence find the effective E modulus Ex and Poisson 5 ratio Vy for each of these two laminates where the x axis lies parallel to the 0 mees By applying a tensile stress 1n the 90 direction find the effective E and v and verify that v vy By Ey 5amilarly by applying a shear stress find Gy e5 between the results for the two laminates Comment on the differer MEK4540 201 1 3a 3 U10 Department of Mathema
3. K4540 201 1 3a 6 U10 Department of Mathematics University of Oslo The MATLAB script provided 2 A 1 3 1 3 20 Set matrices to zero B 1 3 1 3 0 D 1 3 1 3 20 for i 1 length R Calculate T and Q for each ply k Assemble A B and D Transformation matrix T matrix T 1 1 2cos R i 2 T 2 2 cos R 1 2 T 1 2 2s5in R 1 2 T 2 1 2sin R 1 2 T 3 1 2 sin R i cos R 1 T 3 2 2sin R 1i O cos R 1 T 1 3 22 sin R i cos R i T 2 3 2 2 sin R 1i cos R i T 3 3 2cos R 1 2 sin R 1 2 Qk i1 inv T Q T Stiffness matrix for tensor strains following rotation XI 3 1 0kC 3 1 2 Convert back to engineering shear strains A ATQk iJ ChCi 1 hC1 Calculate A matrix B B 5 Qk i9 h ic1 2 hCi 2 Calculate B matrix 3 D D41 3 0k oL EC CIX1 3 h i 3 Calculate D matrix ern Print ABD matrices A B D 36 Assemble and print total stiffness matrix for the laminate This gives the relation between stresses and strains with engineering shear strains atot A B B DII Places contributions A B D with B D in rows below A B MEK4540 201 1 3a 7 U10 Department of Mathematics University of Oslo How to run MATLAB at UiO e f you are logged on to a Linux machine in the Science Faculty at UIO just write the command matlab or matlab amp e f you are using a Windows PC somewhere else e g your own laptop Make sure you have an internet connection
4. U10 Department of Mathematics University of Oslo MATLAB some basic principles Open MAILAB You will see an empty command window Use File Open to open the m file e You can edit this file as you wish e Save it then press the green arrow button to run it e Results will appear in the command window e You can copy any part of the command window and paste into a text file Further notes MATLAB script files are simply text files They can be edited either within MATLAB as above or by use of a text editor such as Notepad or Wordpad in Windows e Much information is available from the web page of MAT1110 including a user manual in Norwegian but also some links to internet websites MEK4540 201 1 3a 10
5. UiO Department of Mathematics University of Oslo MEK4540 Komposittmaterialer og konstruksjoner Composite materials and structures Introduction to Assignment no 1 MEK4540 2011 3a 1 U10 Department of Mathematics University of Oslo MEK 4540 Autumn 2011 Compulsory Assignment no 1 To be submitted by 11 15 on 4 October 2011 as described below A glass fibre laminate consi ts of the following materials Glass fibres E modulus E 72 4 GPa Possson 5 ratio v 027 Density p 2 54 gem Vinylester E modulus E 3 4 GFa Possson 5 ratio vy 0 35 Density p 11g en The fibre fabric in each unidirectional ply weighs 150 zm After a burn off test on 7 5 g of laminate 5 6 g of glass is left a Calculate the density of the composite the m volume fraction and the thickness of a single ply contamimng 150 g m of reinforcement b Estimate Er Er Grr and vrr Use the simple combination laws and where appropriate also the Halpin Isai formulae Explain the differences c With the elastic properties derived in b above taking the Halpi Grr set up the stiffness matrix Q 1n the L T axes Use the transformation matrix T as defined in the lectures or the matrices T1 and T as defined in the text book to calculate the stiffness matrices O in directions 0 90 45 and 45 Explain briefly how these matrices express the relationship between stress and strain for a single ply and indicate clearly
6. s sent by email must be m either MS or preferably PDF format The MATLAB script that is developed to solve parts d e and f is to be handed in together with the solution Even if the solution is handed in on paper the script must be sent in digital format MATLAB input file by email to brianha at math ui1o no Brian Hayman 30 08 2011 MEK4540 2011 3a 5 U10 Department of Mathematics University of Oslo The MATLAB script provided 1 Script for Assignment 1 MEK4540 Autumn 2011 Brian Hayman U10 2011 76 The material properties and laminate lay ups must be replaced with the correct data 75 clear all Sets all variables to zero Material data to be replaced by correct values EL 40000 ET 00 vLT O 25 GLT 2800 t 0 5 Thickness of one UD ply Ply orientations to be replaced by correct lay ups which have more plies than this R 45 0 45 3 14159 180 ply angles in radians alternative is R2 45 0 45 rad h 1 5 t 0 5 rt O 5 t 1 5 t z coordinates for top and bottom surfaces of plies as defined in AB amp C Fig 6 5 Compliance 5 matrix 5 1 1 1 EL S5 2 2 1 ET 5 1 2 vLT EL 5 3 3 1 GLT Stiffness Q matrix Q inv s The procedure shown here uses tensor shear strains You may keep to engineering shear strains if you prefer but then you need the T1 and T2 matrices as defined in AB amp Q 3 3 2 0 3 3 convert from engineering to tensor shear strains ME
7. tics University of Oslo Assignment no 1 contd f Laminate 1 15 subjected simultaneously to stresses cy 300 MPa c 140 MPa and ty 65 MPa By extending the MATLAB script calculate the direct and shear stress components parallel and perpendicular to the fibre direction 1n each ply Hence calculate also the Tsai Wu factor Fry and the inverse of the Tsai Wu strength ratio index 1 R in each ply and deduce whether the Tsa1 Wu failure criterion is violated given the following ply strength properties Tensile strength 1n fibre direction Cyr 490 MPa Compressive strength in fibre direction c 463 MPa Tensile strength 1n transverse direction c5 1 70MPa Compressive strength in transverse direction Cy 175 MPa In plane shear senpia fy 189 MPa 05 Note Since there are no bendinz mo necessary to consider i ply in udi of consider only the first t two plies deformati i 1n part 1 1t 1s only i ections 0 and 45 Thus it is suffi MEK4540 2011 3a 4 U1O Department of Mathematics University of Oslo Assignment no 1 contd Important Submission of solutions Solutions are to be submitted by 11 15 on 04 10 2011 by one of the following means directly by email to brianha at math wic no placed in Brian 5 mail pigeon hole on the 7th Floor e delivered by hand at lecture on 4 October Solutions may be either handwritten or typed using suitable word text processing software Solution
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