finite element analysis of shear wall and pipe interestion problem

CE 529a

FINITE ELEMENT ANALYSIS COURSE PROJECT

AYUSHI SRIVASTAVA

9664835820

Contents

1. Theory

2. Shear wall – node and element label plots

2.1 Shear wall – node and element labels

For Shear loading

2.2 shear wall – node and element labels

For thermal loading

3. Shell model – node and element

Label plots

4. Shear wall – deformation plot under

Shear loading

5. Shear wall – deformation plot under thermal

Loading

6. Shell model – deformation plot

Under pressure loading

7. Discussion of results

7.1 Shear wall

7.1.1 Static Shear loading case

7.1.2 Static thermal loading case

7.2 Shell model

8. Shear wall – stress plots

8.1 Stress plots for Shear force case

8.2 Stress plots for thermal load case

9. Shell model – stress plots

9.1 Stress plots for bottom surface

9.2 Stress plots for top surface

10. List of codes

10.1 Shear wall

10.1.1 Shear wall – concentrated force case

10.1.2 Shear wall – thermal load case

1. THEORY

This project is basically a computer implementation of finite element method using ABAQUS

and MATLAB. In this project, a particular class of finite element models were considered, the

3D plate/ shell elements. These elements are a special type of shell element created from plate

elements that are essentially flat except for a small amount of warpage. Two elements are used

for the assembly of this element – a membrane component to model the in-plane performance of

the structure and a bending component to model the out-of-plane bending and torsion. Each

component has its own stiffness matrix, and these components are combined or assembled to

obtain the stiffness matrix for the entire plate/ shell element.

For analysis purposes, we use Gauss quadrature on a 4 noded Lagrange Isoparametric

element. For the shell problem, we use Flat element approximation. A shell element can be

subdivided into three parts, viz., membrane element, plate element and a drilling element. These

three parts each have a stiffness matrix which has to be calculated and assembled together.For

the membrane part, firstly, we define the material properties, for eg, E matrix, etc and A matrix,

shape function and its derivatives. After this, we create Jacobian matrix and its inverse. Then,

we create the B matrix and C matrix. We use 2x2 Gauss points to generate the stiffness matrix

for the membrane part.

Similarly for the plate part, we calculate each of the above for bending and transverse

shear. For bending we are using 2X2 Gauss points, whereas for the transverse shear we use 1X1

Gauss point to avoid shear lock.

Gauss quadrature

These points are chosen so as to integrate cetain degree polynomials exactly.

Shear Force Calculation

The Shear Force on the wall can be calculated by the formula

F=P/L

Where, P= The force which is applied on the wall

L=The length of the side of the wall to which force is applied.

Pressure force calculation

The pressure force used in the bending part is shown below

Thermal force calculation

Thermal loading is applied to the shear wall for analysis. Thermal loading is applied to

the shear wall for analysis.

{GFther}8x1 =

=

Where; t: Thickness

[D]8x3=[A][B][C]

Stress calculation

ϭ=εE where ϭ – stress

E – Young’s Modulus

For thermal stress calculation,

ϭthermal = E(ε-εthermal)

For Von-mises stress calculation,

3. Shear wall – Node and element label plots

3.1 Shear wall – Node and element labels for concentrated force

Node Labels – Shear Load(Top Edge)

Element Labels – Shear Load

3.2 Shear wall – Node and element labels for thermal load case

Node Labels – Thermal load case

Element Labels – Thermal load case

4. Shell intersection model – Node and element label plots

Node Labels

Element Labels

5. Shear wall – Deformation plot under concentrated force loading

Shear wall deformation plot (concentrated load) from Abaqus

Shear wall deformation plot (concentrated load) from Matlab

6. Shear wall – Deformation plot under thermal loading

Shear wall deformation plot (Thermal load) from Abaqus

Shear wall deformation plot (Thermal load) from Matlab

7. Shell intersection model – Deformation plot under pressure loading

Shell Intersection model deformation plot from Abaqus

8. Discussion of results

8.1 Shear wall

8.1.1 Static pressure loading case

PROGRAM Dx Dy

MATLAB 15.5324 -5.54696

ABAQUS 15.76 -5.58588

PERCENTAGE ERROR 1.44 0.69

8.1.2 Static thermal loading case

PROGRAM Dx Dy

MATLAB 0.29413 0.84120

ABAQUS 0.302049 0.838056

PERCENTAGE ERROR 2.62 0.375

9. Shear wall – Stress Plots

9.1 Stress plots for concentrated force case

Von Mises from Abaqus

Von Mises from Matlab

Principal Stress 1 from Abaqus

Principal Stress 1 from Matlab

9.2 Stress plots for thermal load case

Von Mises from Abaqus

Von Mises from Matlab

10. Shell Intersection Model – Stress Plots

10.1 Stress plots for bottom surface

Von Mises @ bottom from Abaqus

Principal Stress 1 @ bottom from Abaqus

Principal Stress 2 @ bottom from Abaqus

10.2 Stress plots for top surface

Von Mises @ top from Abaqus

Principal Stress 1 @ top from Abaqus

Principal Stress 2 @ top from Abaqus

12. List of Codes

12.1 Shear Wall

12.1.1 Shear Wall – Concentrated Force Case

CreateBCmatrix_membrane

%This code creates (4X4)B and (4X8)C Matrices

function [B,C] =

CreateBCmatrix_membrane(dhdr,dhdn,ajacinv,x_loc,y_loc,dkdr,dkdn)

B =

[ajacinv(1,1),ajacinv(1,2),0,0;ajacinv(2,1),ajacinv(2,2),0,0;0,0,ajacinv(1,1),a

jacinv(1,2);0,0,ajacinv(2,1),ajacinv(2,2)];

for i = 1:4,

if i < 4

j=i+1;

xlocal(i,1) = x_loc(j,1)-x_loc(i,1);

ylocal(i,1) = y_loc(j,1)-y_loc(i,1);

else if i==4

j=1;

xlocal(i,1) = x_loc(j,1)- x_loc(i,1);

ylocal(i,1) = y_loc(j,1)- y_loc(i,1);

end

end

end;

for i=1:4,

if i<4

j=i+1;

l(i,1) = sqrt(xlocal(i,1)^2+ylocal(i,1)^2);

else if i==4

j=1;

l(i,1) = sqrt(xlocal(i,1)^2+ylocal(i,1)^2);

end

end

end

for i=1:4,

if i<4

j=i+1;

c(i,1) = ylocal(i,1)/l(i,1);

s(i,1) = -xlocal(i,1)/l(i,1);

else if i==4

j=1;

c(i,1)=ylocal(i,1)/l(i,1);

s(i,1)=-xlocal(i,1)/l(i,1);

end

end

end;

C = zeros(4,12);

C=[dhdr(1,1),0,1/8*(-

dkdr(1,1)*l(1,1)*c(1,1)+dkdr(4,1)*l(4,1)*c(4,1)),dhdr(2,1),0,1/8*(-



dkdr(4,1)*l(4,1)*c(4,1)+dkdr(3,1)*l(3,1)*c(3,1));dhdn(1,1),0,1/8*(-

dkdn(1,1)*l(1,1)*c(1,1)+dkdn(4,1)*l(4,1)*c(4,1)), dhdn(2,1),0,1/8*(-

dkdn(2,1)*l(2,1)*c(2,1)+dkdn(1,1)*l(1,1)*c(1,1)),dhdn(3,1),0,1/8*(-

dkdn(3,1)*l(3,1)*c(3,1)+dkdn(2,1)*l(2,1)*c(2,1)),dhdn(4,1),0,1/8*(-

dkdn(4,1)*l(4,1)*c(4,1)+dkdn(3,1)*l(3,1)*c(3,1));0,dhdr(1,1),1/8*(-

dkdr(1,1)*l(1,1)*s(1,1)+dkdr(4,1)*l(4,1)*s(4,1)),0,dhdr(2,1),1/8*(-



dkdr(4,1)*l(4,1)*s(4,1)+dkdr(3,1)*l(3,1)*s(3,1));0,dhdn(1,1),1/8*(-

dkdn(1,1)*l(1,1)*s(1,1)+dkdn(4,1)*l(4,1)*s(4,1)),0,dhdn(2,1),1/8*(-



dkdn(4,1)*l(4,1)*s(4,1)+dkdn(3,1)*l(3,1)*s(3,1))];

end

CreateEA_membrane

%This code creates E (Youngs Modulus) and A Matrices for the membrane element

function [E,A]=CreateEA_membrane(young,poisson)

E=young/(1-poisson^2)*[1,poisson,0;poisson,1,0;0,0,(1-poisson)/2];

A = [1 0 0 0; 0 0 0 1; 0 1 1 0];

end

CreateJacobian

%Generates the Jacobian Matrix

function [ajac]=CreateJacobian(dhdr,dhdn,x_loc,y_loc)

ajac=zeros(2,2);

ajac=[dhdr'*x_loc,dhdr'*y_loc;dhdn'*x_loc,dhdn'*y_loc];

end

Jacinv

%Returns the inverse of the Jacobian Matrix

function [ajacinv,det,c]=jacinv(ajac)

c(1,1) = ajac(2,2);

c(1,2) = -ajac(2,1);

c(2,1) = -ajac(1,2);

c(2,2of th = ajac(1,1);

det = 0.0;

for i = 1:2,

det = det + ajac(1,i)*c(1,i);

end;

for i = 1:2,

for j = 1:2,

ajacinv(i,j) = c(j,i)/det;

end;

end;

end

CreateShapeFunc

function [shapeF,dhdr,dhdn] = CreateShapeFunc(r,n)

% CREATE THE SHAPE FUNCTION EVALUATED AT (r,n) AND THEIR DERIVATIVES

% r: Ksi n: Eta

shapeF=zeros(4,1);

dhdr=zeros(4,1);

dhdn=zeros(4,1);

% INCLUDE THE 4 SHAPE FUNCTIONS AND DERIVATIVES

% CALCULATE THE SHAPE FUNCTIONS

shapeF(1,1)=((1-r)*(1-n))/4;

shapeF(2,1)=((1+r)*(1-n))/4;

shapeF(3,1)=((1+r)*(1+n))/4;

shapeF(4,1)=((1-r)*(1+n))/4;

% CALCULATE THE DERIVATIVES OF THE SHAPE FUNCTION W.R.T r

dhdr(1,1)=-0.25*(1-n);

dhdr(2,1)=0.25*(1-n);

dhdr(3,1)=0.25*(1+n);

dhdr(4,1)=-0.25*(1+n);

% CALCULATE THE DERIVATIVES OF THE SHAPE FUNCTION W.R.T n

dhdn(1,1)=-0.25*(1-r);

dhdn(2,1)=-0.25*(1+r);

dhdn(3,1)=0.25*(1+r);

dhdn(4,1)=0.25*(1-r);

end

Stiff_membrane

function [akloc_m,fel_m,amloc_m] =

stiff_membrane(young,poisson,density,x_loc,y_loc,z_loc,dTemp,coefExp,Thickness)

msize = 12;

fel_m = zeros(msize,1);

akloc_m = zeros(msize,msize);

amloc_m = zeros(msize,msize);

% ZERO ELEMENT THERMAL FORCE VECTOR.

felTher = zeros(msize,1);

% CREATE THE [E] Elasticity and [A] MATRIX FOR THE ELEMENT.

[E,A]=CreateEA_membrane(young,poisson);

% NUMBER OF GAUSS POINT TO USE FOR THE NUMERICAL INTEGRATION.

nGP=3;

gamma=1e5;

% THERMAL STRAIN.

ThermalStrain=zeros(3,1);

ThermalStrain(1:2,1)=coefExp*dTemp;

% CALCULATE ELEMENT STIFFNESS MATRIX, MASS MATRIX AND FORCE VECTOR.

for i=1:nGP,

[r,wr]=GaussPoint(nGP,i);

for j=1:nGP,

[n,wn]=GaussPoint(nGP,j);

% Create the Shape functions and their derivatives

[shapeF,dhdr,dhdn]=CreateShapeFunc(r,n);

[dkdr,dkdn]=CreateMidSideShapeFunc(r,n);

% Create the jacobian and its inverse

[ajac]=CreateJacobian(dhdr,dhdn,x_loc,y_loc);

[ajacinv,det,c]=jacinv(ajac);

% Create B and C of membrane

[B,C]=CreateBCmatrix_membrane(dhdr,dhdn,ajacinv,x_loc,y_loc,dkdr,dkdn);

% Create extra stiffness matrix for constraint

G=RotDispCoupling(dhdr,dhdn,dkdr,dkdn,B,C,ajacinv,x_loc,y_loc,shapeF);

% Local Stiffness Matrix

Kms=Thickness*wr*wn*C'*B'*A'*E*A*B*C*det;

Kmc=gamma*Thickness*wr*wn*G*det;

akloc_m=akloc_m+Kms+Kmc;

end

end

Loc2GlobTrans

function [akglob,felglob] = Loc2GlobTrans(akloc,felloc,lambda,amloc)

% [L] 24x24 TRANSFORMATION MATRIX

L = eye(24,24);

L = blkdiag(lambda,lambda,lambda,lambda,lambda,lambda,lambda,lambda);

% CALCULATE ELEMENT STIFFNESS MATRIX IN GLOBAL COORDINATE SYSTEM

akglob = L'*akloc*L;

% CALCULATE ELEMENT MASS MATRIX IN GLOBAL COORDINATE SYSTEM

%amglob = L'*amloc*L;

% CALCULATE ELEMENT FORCE VECTOR IN GLOBAL COORDINATE SYSTEM

felglob = L'*felloc;

end

RotDispCoupling

function [ G ]

=RotDispCoupling(dhdr,dhdn,dkdr,dkdn,B,C,ajacinv,x_loc,y_loc,shapeF)

G1 =[0,0,shapeF(1),0,0,shapeF(2),0,0,shapeF(3),0,0,shapeF(4)];

A1= [0,-1/2,1/2,0];

G = ((A1*B*C)-G1)'*((A1*B*C)-G1);

FEA3DDKQ_create_input

function FEA3DDKQ_create_input

node=load('FEA3DDKQ_node.txt');

element=load('FEA3DDKQ_element.txt');

num_ele = size(element,1);

coord = node(:,2:4);

lotogo = zeros(num_ele,4);

% Load Local Node Numbering Connectivity Data

% from Abaqus to FEA

lotogo(:,1) = element(:,2);




% Adding the Boundary Restraint Conditions

% ----------------------------------------

% jj = 1 means that DOF is free to move

% jj = 0 means the DOF is fixed or restrained

%

jj = ones(size(coord,1),6);

jj(:,3:5)=0;

jj(23,1:6)=0;

jj(164,1:6)=0;

jj(165,1:6)=0;

jj(166,1:6)=0;

jj(167,1:6)=0;

jj(168,1:6)=0;

jj(169,1:6)=0;

jj(16,1:6)=0;

jj(93,1:6)=0;

jj(94,1:6)=0;

jj(95,1:6)=0;

jj(96,1:6)=0;

jj(97,1:6)=0;

jj(7,1:6)=0;

jj(44,1:6)=0;

jj(45,1:6)=0;

jj(1,1:6)=0;

jj(24,1:6)=0;

jj(25,1:6)=0;

jj(2,1:6)=0;

%

% Set Restraints to Zero for Nodes at the "offset"

% offset = -25;

% ind = (abs(coord(:,1)-offset)<1e-6);

% %

% jj(ind,:) = 0;

% Material Properties

%young = 2.9E+7;

young =4.2E+6;

poisson = 0.3;

density = .0003;

coefExp = 7.6E-6;

% Dynamic Analysis Parameters

ntimeStep = 200;

dt = .005;

AlfaDamp = 0.08;

BetaDamp = 0.002;

% Temperature Variation

dTemp = 60;

% Thickness of the Shell

Thickness = 6;

Pressure = 4500;

ifornod = [20 150 149 148 147 146 145 144 143 142 141 21]; ifordir = [1 1 1 1 1 1

1 1 1 1 1 1]; forval = [727272.727 1454545.45 1454545.45 1454545.45 1454545.45

1454545.45 1454545.45 1454545.45 1454545.45 1454545.45 1454545.45 727272.727];

save FEA_input_data.mat coord lotogo jj young poisson density ifornod ifordir

forval ntimeStep dt AlfaDamp BetaDamp dTemp coefExp Thickness Pressure;

clear

12.1.2 Shear Wall – Thermal Load Case

Stiff_membrane

function [akloc_m,fel_m,amloc_m] =

stiff_membrane(young,poisson,density,x_loc,y_loc,z_loc,dTemp,coefExp,Thickness)

msize = 12;

fel_m = zeros(msize,1);

akloc_m = zeros(msize,msize);

amloc_m = zeros(msize,msize);

% ZERO ELEMENT THERMAL FORCE VECTOR.

felTher = zeros(msize,1);

% CREATE THE [E] Elasticity and [A] MATRIX FOR THE ELEMENT.


% NUMBER OF GAUSS POINT TO USE FOR THE NUMERICAL INTEGRATION.

nGP=3;

gamma=1e5;

% THERMAL STRAIN.

ThermalStrain=zeros(3,1);

ThermalStrain=[(coefExp*dTemp);(coefExp*dTemp);0];

% CALCULATE ELEMENT STIFFNESS MATRIX, MASS MATRIX AND FORCE VECTOR.

for i=1:nGP,

[r,wr]=GaussPoint(nGP,i);

for j=1:nGP,

[n,wn]=GaussPoint(nGP,j);




% Create the jacobian and its inverse



% Create B and C of membrane

[B,C] =

CreateBCmatrix_membrane(dhdr,dhdn,ajacinv,x_loc,y_loc,dkdr,dkdn);

G=RotDispCoupling(dhdr,dhdn,dkdr,dkdn,B,C,ajacinv,x_loc,y_loc,shapeF);

% Local Stiffness Matrix

Kms=Thickness*wr*wn*C'*B'*A'*E*A*B*C*det;

Kmc=gamma*Thickness*wr*wn*G*det;

akloc_m=akloc_m+Kms+Kmc;

% Thermal Force Vector

D = [A*B*C];

fel_m=fel_m+Thickness*[D'*E*ThermalStrain*wr*wn*det];

end

end

Stress

function

[sig]=stress(young,poisson,x_loc,y_loc,z_loc,disp,r,n,dTemp,coefExp,Thickness)

sig = zeros(3,3); % 1st column @ bottom surface (z=-h/2),

% 2nd column @ midsurface (z=0), and

% 3rd column @ top surface (z=h/2)

sig_m = zeros(3,1);

sig_b = zeros(3,1);

% ZERO STRESSES - SIGMA_Von_Mises, SIGMA_1, SIGMA_2.

sig_out = zeros(3,3); % 1st column @ bottom surface (z=-h/2),

% 2nd column @ midsurface (z=0), and

% 3rd column @ top surface (z=h/2)

sigVon = zeros(1,3); % Von-Mises Stress

sig_P = zeros(2,3); % Principal Stresses, 1st row is SIGMA_1, 2nd row is

SIGMA_2

% ZERO DISPLACEMENT VECTORS

disp_m = zeros(12,1);

disp_p = zeros(12,1);

%

%Thermal Strain

ThermalStrain = zeros(3,1);

%COMPUTE STRESSES OF MEMBRANE

%CREATE THE [E] Elasticity and [A] MATRIX FOR THE ELEMENT





% CREATE JACOBIAN AND ITS INVERSE



% CREATE B AND C MATRICES

[B,C]=CreateBCmatrix_membrane(dhdr,dhdn,ajacinv,x_loc,y_loc,dkdr,dkdn);

%CREATE ELEMENT DISPLACEMENT VECTOR OF MEMBRANE

disp_m=[disp(1,1);disp(2,1);disp(6,1);disp(7,1);disp(8,1);disp(12,1);disp(13,1);d

isp(14,1);disp(18,1);disp(19,1);disp(20,1);disp(24,1)];

%CALCULATE THE 3x1 STRESS VECTOR

sig_m=E*(A*B*C*disp_m-ThermalStrain);

%COMPUTE STRESSES OF BENDING

%CREATE THE [E] Elasticity and [A] MATRIX FOR THE ELEMENT

%CALL THE FUNCTION THAT CREATE THE DERIVATIVES OF SHAPE FUNCTIONS

%CREATE THE INVERSE OF JACOBIAN TO FORMULATE B_b

%CREATE THE B AND C MATRICES

%CREATE ELEMENT DISPLACEMENT VECTOR OF BENDING

%Calculate the 3*1 stress vector @ z = h/2 (h: shell thickness)

% use E not E_b (E_b is for stiff calc.)

% CALCULATE THE STRESS VECTOR AT BOTTOM, MID, AND TOP SURFACE OF THE SHELL

ELEMENT THICKNESS.

sig_out(:,1) = sig_m - sig_b; % Stress @ z = -h/2

sig_out(:,2) = sig_m; % Stress @ z = 0

sig_out(:,3) = sig_m + sig_b; % Stress @ z = +h/2

%

%

% % CALCULATE THE VON-MISES STRESS COMPONENT (IF REQUIRED).

for i=1:3

sig(1,:)=sqrt(0.5*((sig_out(1,i)-

sig_out(2,i))^2+(sig_out(1,i))^2+(sig_out(2,i))^2)+3*(sig_out(3,i))^2);

end

% % CALCULATE THE PRINCIPAL STRESS COMPONENTS (IF REQUIRED).

for i =1:3

sig(2,:)=0.5*(sig_out(1,i)+sig_out(2,i))+sqrt(((sig_out(1,i)-

sig_out(2,i))/2)^2+sig_out(3,i)^2);

end

for i=1:3

sig(3,:)=0.5*(sig_out(1,i)+sig_out(2,i))-sqrt(((sig_out(1,i)-

sig_out(2,i))/2)^2+sig_out(3,i)^2);

end

% % RETURN THE VON-MISES STRESS AND PRINCIPAL STRESSES (IF REQUIRED).

%sig= [sigVon;sig_P];

end

FEA3DDKQ_create_input

function FEA3DDKQ_create_input

node = load('FEA3DDKQ_node.txt');

element = load('FEA3DDKQ_element.txt');

num_ele = size(element,1);

coord = node(:,2:4);

lotogo = zeros(num_ele,4);

% Load Local Node Numbering Connectivity Data

% from Abaqus to FEA





% Adding the Boundary Restraint Conditions

% ----------------------------------------

% jj = 1 means that DOF is free to move

% jj = 0 means the DOF is fixed or restrained

%

jj = ones(size(coord,1),6);

jj(:,3:5)=0;

jj(23,1:6)=0;

jj(164,1:6)=0;

jj(165,1:6)=0;

jj(166,1:6)=0;

jj(167,1:6)=0;

jj(168,1:6)=0;

jj(169,1:6)=0;

jj(16,1:6)=0;

jj(93,1:6)=0;

jj(94,1:6)=0;

jj(95,1:6)=0;

jj(96,1:6)=0;

jj(97,1:6)=0;

jj(7,1:6)=0;

jj(44,1:6)=0;

jj(45,1:6)=0;

jj(1,1:6)=0;

jj(24,1:6)=0;

jj(25,1:6)=0;

jj(2,1:6)=0;

%

% Set Restraints to Zero for Nodes at the "offset"

% offset = -25;

% ind = (abs(coord(:,1)-offset)<1e-6);

% %

% jj(ind,:) = 0;

% Material Properties

%young = 2.9E+7;

young =4.2E+6;

poisson = 0.3;

density = .0003;

coefExp = 7.6E-6;

% Dynamic Analysis Parameters

ntimeStep = 200;

dt = .005;

AlfaDamp = 0.08;

BetaDamp = 0.002;

% Temperature Variation

dTemp = 60;

% Thickness of the Shell

Thickness = 6;

Pressure = 4500;

% ----------------------------------------

% Concentrated Forces

% ----------------------------------------

% force directions (1 - x, 2 - y, 3 - z)

% ifornod - Global nodes at which forces are applied

% ifordir - Direction of forces

% forval - Force values

% end

ifornod = [20]; ifordir = [1]; forval = [0];

% ----------------------------------------

save FEA_input_data.mat coord lotogo jj young poisson density ifornod ifordir

forval ntimeStep dt AlfaDamp BetaDamp dTemp coefExp Thickness Pressure;

clear

finite element analysis of shear wall and pipe interestion problem

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