FEA CODE WITH MATAB Finite Element Analysis of an Arch ME 5657 FINITE EEMENT METHOD Submitted by: APAY BURAK DEMIRYUREK
This report summarizes the finite element analysis of an arch-beam with using matlab. The geometry and material properties of arch-beam given is shown below Following assumptions are made in FEA of the beam - Global coordinate system X-Y is located at left hand side support. So that center of arch is located at R cos(90-φ /). - Height of the arch beam changes linearly along its arch length. An average beam height is taken into consideration for each element. Such that, h avg = h i + h j h i = height at i th Node h j = height at j th Node - External load P applied on element by using its horizontal distance from the origin (X=0, Y=0), Vertical distance variations from the original point application to elements are ignored. - Elements and nodes are numbered starting from the left hand side support. Such that, first element is numbered as 1 and node 1 at i th end at j th end. - Counterclockwise Moments, and X-Y directions shown above are considered as + ve directions.
Structure Stiffness Matrix in ocal Coordinates where ; X = EA k local = X 0 0 X 0 0 0 Y Z 0 Y Z 0 Z T 0 Z W X 0 0 X 0 0 0 Y Z 0 Y Z [ 0 Z W 0 Z T ] Y = 1EI z (1 + φ y ) 3 Z = 6EI z (1 + φ y ) T = (4 + φ y)ei z (1 + φ y ) W = ( φ y)ei z (1 + φ y ) Since, the height of the arch beam changes linearly along its arch length in order to use above stiffness matrix, an average beam height is taken into consideration. Such that, Consistent Nodal Force Vector h i = height at i th Node h avg = h i + h j h j = height at j th Node To determine consistent nodal forces due to self-weight, following integral statement is used. {R b } = F b A [N] T dx 0 To obtain 6 by 1 local consistent nodal force vector following shape functions are used to evaluate axial force, and Vertical force End moment couples. [N bar ] = [1 x [N beam ] = [1 3 ( x ) + ( x )3 x(1 ( x ) + (x ) 3 ( x ) ( x )3 x ( ( x ) + (x ) ) ] Based on the assumption of constant area along the element, and given constant material unit weight above integrals are computed symbolically and following load vectors are obtained. x ] {r bar } = F bx A [ ] {r beam } = F by A [ 1 1]
Combining these two consistent nodal force vectors. Following consistent nodal force vector due to body force is obtained {r b } = F bx A F by A F by A 1 F bx A F by A [ F bya 1 ] U 1 V 1 Θ 1 U V Θ Consistent nodal forces due to applied point load, Following local consistent nodal load vector gives the forces at nodes {r e } = b P x P y b ( + a) [ 3 ab P y a P x P y a ( + b) 3 P y a b ] U 1 V 1 Θ 1 U V Θ
DOF-solution and convergence By using matlab script provided in Appendix, for 4,8,16,3 and 64 elements convergence of DOFs evaluated. Following graphs for different mesh sizes are obtained.
For given criterion, %1 difference in deflections from previous case, mid-point deflections of different mesh size analyses are compared. Given criterion is satisfied when # Elements = 64 Stresses for converged case (N=64) For calculation of stresses along the beam, Eqn s..9-3 to.9-6 from Cook. Et al. are used. To compute normal stresses along the element following equation is implemented into matlab code To compute bending stresses at top of the section # Elements u (m) v (m) Δu (%) Δv (%) 4 5.68E-05-1.9E-05 - - 8 6.14E-05 -.6E-05 0.08089 0.367057 16 6.33E-05 -.8E-05 0.09941 0.097515 3 6.38E-05 -.9E-05 0.00811 0.07599 64 6.39E-05 -.9E-05 0.00054 0.007143 σ N = E u u 1 σ b = E h avg [( 6 + 1x 3 ) v 1 + ( 4 + 6x ) θ 1 + ( 6 1x 3 ) v + ( + 6x ) θ ] Bending stresses are computed at ith node of each element, where x=0. To compute shear stresses following expression is implemented. τ = c V A = cei A [(1 3 ) v 1 + ( 6 ) θ 1 + ( 1 3 ) v + ( 6 ) θ ]
# Elements u (m) v (m) Δu (%) Δv (%) 4 5.68E-05-1.9E-05 - - 8 6.14E-05 -.6E-05 0.08089 0.367057 16 6.33E-05 -.8E-05 0.09941 0.097515 3 6.38E-05 -.9E-05 0.00811 0.07599 64 6.39E-05 -.9E-05 0.00054 0.007143 18 6.4E-05 -.9E-05 0.000516 0.001797 56 6.4E-05 -.9E-05 0.00019 0.000451 As number of elements used in modeling of the beam increased, deflections at nodes converges to exact solution. One can obtain exact solution by using infinitely many elements to model the structure, however it is not efficient for computation. Thus optimum number of elements should be determined. Optimum number of elements can be determined for an acceptable accuracy, in our case %1 difference in displacements from previous case.
APPENDIX Material Properties Modulus of Elasticity (E) (kn/m ) 00000000 Poisson's Ratio(ν) 0 Shear Modulus(G) 100000000 Thermal Expansion C. (α) 0 Unit Weight(γ kn/m 3 ) 68.67 Section Properties Tapered or Not (1 or 0) 1 Section width(m) 0.1 0.1 Section height(m) 0.03 0.01 Cross Sectional Area(m ) 0.003 0.001 Moment of Inertia(z-z) (m 4 ).5E-07 8.33E-09 Moment of Inertia(y-y) (m 4 ) 0.000005 8.33E-07 Shear Area(y-y) 0.005 0.000833 Shear Area(z-z) 1.875E-07 6.94E-09 External oad Central Magnitude Angle (from eft End of Structure) X -10 0 0.1444 4-Elements Structure Geometry Number of Elements 64 Number of Nodes 65 inear or Curved(0 / 1) 1 Curved Beam R (m) 0.5 Central Angle ( Degrees) 90 First Element Starts at 0 0 ast Element Ends at 0.7071068 0 ast Element Not Curved 0 0 Boundary Conditions Node U V Theta 1 0 0 0 5 0 0 0
8-Elements Structure Mesh Element No X1 Y1 X Y A I i-thnode j-th Node 1 0 0 0.161 0.108386 0.19509 0.0075 1.73307E-07 1 0.16117 0.108386 0.353553 0.146447 0.19509 0.005 9.4919E-08 3 3 0.3535534 0.146447 0.544895 0.108386 0.19509 0.00175 4.46615E-08 3 4 4 0.5448951 0.108386 0.707107 5.55E-17 0.19509 0.0015 1.676E-08 4 5 Structure Geometry Number of Elements 8 Number of Nodes 9 inear or Curved(0 / 1) 1 Curved Beam R (m) 0.5 Central Angle ( Degrees) 90 First Element Starts at 0 0 ast Element Ends at 0.7071068 0 ast Element Not Curved 0 0 Boundary Conditions Node U V Theta 1 0 0 0 9 0 0 0 Structure Mesh Element No X1 Y1 X Y A I i-thnode j-th Node 1 0 0 0.075768 0.06181 0.098017 0.00875 1.98031E-07 1 0.0757683 0.06181 0.161 0.108386 0.098017 0.0065 1.5073E-07 3 3 0.16117 0.108386 0.56008 0.136839 0.098017 0.00375 1.11637E-07 3 4 4 0.56008 0.136839 0.353553 0.146447 0.098017 0.0015 7.9964E-08 4 5 5 0.3535534 0.146447 0.451099 0.136839 0.098017 0.001875 5.49316E-08 5 6 6 0.4510986 0.136839 0.544895 0.108386 0.098017 0.00165 3.57585E-08 6 7 7 0.5448951 0.108386 0.631339 0.06181 0.098017 0.001375.16634E-08 7 8 8 0.6313385 0.06181 0.707107 5.55E-17 0.098017 0.00115 1.1865E-08 8 9 16 Elements Structure Geometry Number of Elements 16 Number of Nodes 17 inear or Curved(0 / 1) 1 Curved Beam R (m) 0.5 Central Angle ( Degrees) 90 First Element Starts at 0 0 ast Element Ends at 0.7071068 0 ast Element Not Curved 0 0
Boundary Conditions Node U V Theta 1 0 0 0 17 0 0 0 Structure Mesh Element No X1 Y1 X Y A I i-thnode j-th Node 1 0 0 0.036357 0.0395 0.049068 0.00938.118E-07 1 0.0363567 0.0395 0.075768 0.06181 0.049068 0.00813 1.85394E-07 3 3 0.0757683 0.06181 0.117855 0.087407 0.049068 0.00688 1.61757E-07 3 4 4 0.117855 0.087407 0.161 0.108386 0.049068 0.00563 1.40E-07 4 5 5 0.16117 0.108386 0.08411 0.14917 0.049068 0.00438 1.0685E-07 5 6 6 0.084111 0.14917 0.56008 0.136839 0.049068 0.00313 1.03054E-07 6 7 7 0.56008 0.136839 0.304545 0.144039 0.049068 0.00188 8.794E-08 7 8 8 0.3045448 0.144039 0.353553 0.146447 0.049068 0.00063 7.3114E-08 8 9 9 0.3535534 0.146447 0.4056 0.144039 0.049068 0.001938 6.06099E-08 9 10 10 0.4056 0.144039 0.451099 0.136839 0.049068 0.001813 4.96195E-08 10 11 11 0.4510986 0.136839 0.498696 0.14917 0.049068 0.001688 4.0045E-08 11 1 1 0.4986957 0.14917 0.544895 0.108386 0.049068 0.001563 3.17891E-08 1 13 13 0.5448951 0.108386 0.5895 0.087407 0.049068 0.001438.47538E-08 13 14 14 0.589518 0.087407 0.631339 0.06181 0.049068 0.001313 1.88416E-08 14 15 15 0.6313385 0.06181 0.67075 0.0395 0.049068 0.001188 1.39547E-08 15 16 16 0.67075 0.0395 0.707107 5.55E-17 0.049068 0.001063 9.9955E-09 16 17 3 Elements Structure Geometry Number of Elements 3 Number of Nodes 33 inear or Curved(0 / 1) 1 Curved Beam R (m) 0.5 Central Angle ( Degrees) 90 First Element Starts at 0 0 ast Element Ends at 0.7071068 0 ast Element Not Curved 0 0 Boundary Conditions Node U V Theta 1 0 0 0 33 0 0 0
64 Elements Structure Mesh Element No X1 Y1 X Y A I i-thnode j-th Node 1 0 0 0.017774 0.0169 0.04541 0.00969.1804E-07 1 0.0177739 0.0169 0.036357 0.0395 0.04541 0.00906.04559E-07 3 3 0.0363567 0.0395 0.055704 0.04805 0.04541 0.00844 1.91643E-07 3 4 4 0.0557037 0.04805 0.075768 0.06181 0.04541 0.00781 1.7983E-07 4 5 5 0.0757683 0.06181 0.09650 0.075311 0.04541 0.00719 1.67466E-07 5 6 6 0.09650 0.075311 0.117855 0.087407 0.04541 0.00656 1.5618E-07 6 7 7 0.117855 0.087407 0.139776 0.098441 0.04541 0.00594 1.45413E-07 7 8 8 0.1397758 0.098441 0.161 0.108386 0.04541 0.00531 1.3515E-07 8 9 9 0.16117 0.108386 0.185108 0.11719 0.04541 0.00469 1.5386E-07 9 10 10 0.1851085 0.11719 0.08411 0.14917 0.04541 0.00406 1.1610E-07 10 11 11 0.084111 0.14917 0.3063 0.13146 0.04541 0.00344 1.0788E-07 11 1 1 0.30633 0.13146 0.56008 0.136839 0.04541 0.0081 9.8931E-08 1 13 13 0.56008 0.136839 0.80188 0.141035 0.04541 0.0019 9.1015E-08 13 14 14 0.80188 0.141035 0.304545 0.144039 0.04541 0.00156 8.3544E-08 14 15 15 0.3045448 0.144039 0.390 0.145844 0.04541 0.00094 7.6488E-08 15 16 16 0.390196 0.145844 0.353553 0.146447 0.04541 0.00031 6.98407E-08 16 17 17 0.3535534 0.146447 0.378087 0.145844 0.04541 0.001969 6.3590E-08 17 18 18 0.378087 0.145844 0.4056 0.144039 0.04541 0.001906 5.7743E-08 18 19 19 0.4056 0.144039 0.46919 0.141035 0.04541 0.001844 5.306E-08 19 0 0 0.469186 0.141035 0.451099 0.136839 0.04541 0.001781 4.7097E-08 0 1 1 0.4510986 0.136839 0.475043 0.13146 0.04541 0.001719 4.3114E-08 1 0.4750435 0.13146 0.498696 0.14917 0.04541 0.001656 3.78614E-08 3 3 0.4986957 0.14917 0.51998 0.11719 0.04541 0.001594 3.37349E-08 3 4 4 0.519983 0.11719 0.544895 0.108386 0.04541 0.001531.99197E-08 4 5 5 0.5448951 0.108386 0.567331 0.098441 0.04541 0.001469.64036E-08 5 6 6 0.5673309 0.098441 0.5895 0.087407 0.04541 0.001406.31743E-08 6 7 7 0.589518 0.087407 0.610605 0.075311 0.04541 0.001344.0197E-08 7 8 8 0.6106048 0.075311 0.631339 0.06181 0.04541 0.00181 1.7575E-08 8 9 9 0.6313385 0.06181 0.651403 0.04805 0.04541 0.00119 1.50856E-08 9 30 30 0.651403 0.04805 0.67075 0.0395 0.04541 0.001156 1.8817E-08 30 31 31 0.67075 0.0395 0.689333 0.0169 0.04541 0.001094 1.09037E-08 31 3 3 0.689339 0.0169 0.707107 5.55E-17 0.04541 0.001031 9.1395E-09 3 33 Structure Geometry Number of Elements 64 Number of Nodes 65 inear or Curved(0 / 1) 1 Curved Beam R (m) 0.5 Central Angle ( Degrees) 90 First Element Starts at 0 0 ast Element Ends at 0.7071068 0 ast Element Not Curved 0 0 Boundary Conditions Node U V Theta 1 0 0 0 65 0 0 0
Structure Mesh Element No X1 Y1 X Y A I i-thnode j-th Node 1 0 0 0.008783 0.00857 0.017 0.00984.1503E-07 1 0.0087831 0.00857 0.017774 0.0169 0.017 0.00953.14617E-07 3 3 0.0177739 0.0169 0.06967 0.05051 0.017 0.009.07876E-07 3 4 4 0.06967 0.05051 0.036357 0.0395 0.017 0.00891.0177E-07 4 5 5 0.0363567 0.0395 0.045938 0.0406 0.017 0.00859 1.94819E-07 5 6 6 0.0459376 0.0406 0.055704 0.04805 0.017 0.0088 1.88501E-07 6 7 7 0.0557037 0.04805 0.065649 0.05539 0.017 0.00797 1.83E-07 7 8 8 0.0656493 0.05539 0.075768 0.06181 0.017 0.00766 1.7678E-07 8 9 9 0.0757683 0.06181 0.086055 0.068873 0.017 0.00734 1.7037E-07 9 10 10 0.0860546 0.068873 0.09650 0.075311 0.017 0.00703 1.64595E-07 10 11 11 0.09650 0.075311 0.107104 0.08149 0.017 0.0067 1.5895E-07 11 1 1 0.1071043 0.08149 0.117855 0.087407 0.017 0.00641 1.5344E-07 1 13 13 0.117855 0.087407 0.18748 0.093059 0.017 0.00609 1.48057E-07 13 14 14 0.187477 0.093059 0.139776 0.098441 0.017 0.00578 1.4801E-07 14 15 15 0.1397758 0.098441 0.150933 0.103551 0.017 0.00547 1.37671E-07 15 16 16 0.150937 0.103551 0.161 0.108386 0.017 0.00516 1.3665E-07 16 17 17 0.16117 0.108386 0.173606 0.11943 0.017 0.00484 1.778E-07 17 18 18 0.1736059 0.11943 0.185108 0.11719 0.017 0.00453 1.301E-07 18 19 19 0.1851085 0.11719 0.196713 0.1111 0.017 0.004 1.18379E-07 19 0 0 0.196715 0.1111 0.08411 0.14917 0.017 0.00391 1.13855E-07 0 1 1 0.084111 0.14917 0.0197 0.18335 0.017 0.00359 1.09448E-07 1 0.0197 0.18335 0.3063 0.13146 0.017 0.0038 1.05157E-07 3 3 0.30633 0.13146 0.44003 0.13498 0.017 0.0097 1.00979E-07 3 4 4 0.44008 0.13498 0.56008 0.136839 0.017 0.0066 9.6913E-08 4 5 5 0.56008 0.136839 0.6807 0.139085 0.017 0.0034 9.958E-08 5 6 6 0.68074 0.139085 0.80188 0.141035 0.017 0.0003 8.911E-08 6 7 7 0.80188 0.141035 0.9348 0.14686 0.017 0.0017 8.53735E-08 7 8 8 0.93481 0.14686 0.304545 0.144039 0.017 0.00141 8.17411E-08 8 9 9 0.3045448 0.144039 0.316771 0.14509 0.017 0.00109 7.813E-08 9 30 30 0.3167711 0.14509 0.390 0.145844 0.017 0.00078 7.47883E-08 30 31 31 0.390196 0.145844 0.34183 0.14696 0.017 0.00047 7.14649E-08 31 3 3 0.34188 0.14696 0.353553 0.146447 0.017 0.00016 6.8414E-08 3 33 33 0.3535534 0.146447 0.36584 0.14696 0.017 0.001984 6.51163E-08 33 34 34 0.36584 0.14696 0.378087 0.145844 0.017 0.001953 6.088E-08 34 35 35 0.378087 0.145844 0.390336 0.14509 0.017 0.0019 5.91554E-08 35 36 36 0.3903357 0.14509 0.4056 0.144039 0.017 0.001891 5.63164E-08 36 37 37 0.4056 0.144039 0.414759 0.14686 0.017 0.001859 5.35698E-08 37 38 38 0.4147587 0.14686 0.46919 0.141035 0.017 0.00188 5.09139E-08 38 39 39 0.469186 0.141035 0.439034 0.139085 0.017 0.001797 4.83473E-08 39 40 40 0.4390343 0.139085 0.451099 0.136839 0.017 0.001766 4.58685E-08 40 41 41 0.4510986 0.136839 0.463104 0.13498 0.017 0.001734 4.34758E-08 41 4 4 0.463104 0.13498 0.475043 0.13146 0.017 0.001703 4.11679E-08 4 43 43 0.4750435 0.13146 0.48691 0.18335 0.017 0.00167 3.89431E-08 43 44 44 0.4869098 0.18335 0.498696 0.14917 0.017 0.001641 3.67999E-08 44 45 45 0.4986957 0.14917 0.510394 0.1111 0.017 0.001609 3.47369E-08 45 46 46 0.5103943 0.1111 0.51998 0.11719 0.017 0.001578 3.754E-08 46 47 47 0.519983 0.11719 0.533501 0.11943 0.017 0.001547 3.0845E-08 47 48 48 0.5335009 0.11943 0.544895 0.108386 0.017 0.001516.90131E-08 48 49 49 0.5448951 0.108386 0.556174 0.103551 0.017 0.001484.755E-08 49 50 50 0.556174 0.103551 0.567331 0.098441 0.017 0.001453.55698E-08 50 51 51 0.5673309 0.098441 0.578359 0.093059 0.017 0.0014.39554E-08 51 5 5 0.5783591 0.093059 0.5895 0.087407 0.017 0.001391.4104E-08 5 53 53 0.589518 0.087407 0.60000 0.08149 0.017 0.001359.0933E-08 53 54 54 0.600005 0.08149 0.610605 0.075311 0.017 0.00138 1.955E-08 54 55 55 0.6106048 0.075311 0.6105 0.068873 0.017 0.00197 1.81766E-08 55 56 56 0.6105 0.068873 0.631339 0.06181 0.017 0.00166 1.68941E-08 56 57 57 0.6313385 0.06181 0.641457 0.05539 0.017 0.00134 1.56733E-08 57 58 58 0.6414575 0.05539 0.651403 0.04805 0.017 0.00103 1.4518E-08 58 59 59 0.651403 0.04805 0.661169 0.0406 0.017 0.00117 1.3411E-08 59 60 60 0.661169 0.0406 0.67075 0.0395 0.017 0.001141 1.3665E-08 60 61 61 0.67075 0.0395 0.68014 0.05051 0.017 0.001109 1.13777E-08 61 6 6 0.6801398 0.05051 0.689333 0.0169 0.017 0.001078 1.0443E-08 6 63 63 0.689339 0.0169 0.69834 0.00857 0.017 0.001047 9.561E-09 63 64 64 0.698337 0.00857 0.707107 5.55E-17 0.017 0.001016 8.73009E-09 64 65