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Jen Ming Victor Shih 994751648

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Advanced Aerospace Structures Project 2

Results

ii) Physical Coordinates of all the nodes (16 elements arranges along the length of beam)

xg = x y

0 0

0.2500 0.0156

0.5000 0.0313

0.7500 0.0469

1.0000 0.0625

1.2500 0.0781

1.5000 0.09381.7500 0.1094

2.0000 0.1250

2.2500 0.1406

2.5000 0.1563

2.7500 0.1719

3.0000 0.1875

3.2500 0.2031

3.5000 0.2188

3.7500 0.2344

4.0000 0.2500

4.0000 0.5000

3.7500 0.5000

3.5000 0.5000

3.2500 0.5000

3.0000 0.5000

2.7500 0.5000

2.5000 0.5000

2.2500 0.5000

2.0000 0.5000

1.7500 0.5000

1.5000 0.5000

1.2500 0.5000

1.0000 0.50000.7500 0.5000

0.5000 0.5000

0.2500 0.5000

0 0.5000

xg is the physical global coordinates of all the nodes in the

beam, the nodal are numbered starting at the bottom left of

the beam and are subsequently numbered along the border

of the beam in a counter-clockwise direction

the code uses a tailored fetch algorithm to generate element

nodal coordinates from xg

figure1 shows how the beam is broken down

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iii) Displacement of the nodes along the right edge of the beam:

U = x y

-0.0000 -0.0023 (bottom)

0.0002 -0.0023 (top)

iv) Contour Plot of the strains in the beam:

Figure 1: x direction strain is shown to vary across the height of the beam

Figure 2: y direction strain is shown to vary along the length of the beam

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Figure 3: y-direction strain mesh plot to emphasize area of greatest change

Figure 4: xy-direction strain has an oscillatory effect along the length of the beam

v) Contour Plot of von Mises equivalent stress:

Figure 5: von-Mises equivalent stress varies across the length of the beam

Figure 6: mesh plot of the von-Mises equivalent stress showing the variation along the rest of the beam

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MATLAB Code

%% Advanced Aerospace Structures% Assignment 4: Finite Element Solution to Tapered Cantilever Beam% Isoparametric Element (15 QuadralateralElemets)% Victor Shih

clc;clearall;closeall;

% assume plane stresst = 0.1;E = 75e9;nu = 0.3;D = (E/(1-nu^2))*[1 nu 0;...nu 1 0;...

0 0 (1-nu)/2];

%% FORMING NODES xgelems = 16;nodes = 2*(elems+1);cx = 0:4/elems:4;cyt = zeros(1,elems+1) + 0.5;

cyb = zeros(1,elems+1);fori = 1:elems+1cyb(i) = 0.25/4*cx(i);end

xg = horzcat([cx;cyb],[cx(elems+1:-1:1);cyt])'figure(1)

plot(xg(:,1),xg(:,2), '-ob')holdonaxis([-0.05 4.05 -0.05 0.55])gridon

% view(2)

%% FORM GLOBAL STIFFNESS MATRIX and GLOBAL FORCE VECTOR

Kg = zeros(2*nodes);fg = zeros(2*nodes,1);%element loopfori = 1:elems

x = form_x(i,elems,nodes,xg);K = form_K(D,x,t);L = form_fetch(i,nodes);

Kg = Kg + L'*K*L;

ifi> 12load = [0, -1e3];

N = [1 0 0 0 0 0 1 0;...0 1 0 0 0 0 0 1];

f = (load*N)';

fg = fg + L'*f;endend

%% CALCULATE DISPLACEMTN dg%remove boundary nodesK = Kg(3:2*nodes-2,3:2*nodes-2);f = fg(3:2*nodes-2);%calculate displacementd = K\f;

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%put back boundarydg = zeros(2*nodes,1);dg(3:2*nodes-2) = d;

U = zeros(nodes,2);fori = 1:nodesU(i,1) = dg(i*2-1);

U(i,2) = dg(i*2);endU(nodes/2:nodes/2+1,:)newxg = xg + U;plot(newxg(:,1),newxg(:,2), '-or')xlabel('x');ylabel('y');title('16 ISOPARAMETRIC ELEMENT FEM DEFORMATION')holdoff

%% CALCULATE STRAIN AND STRESS%element loope_x = zeros(2,2*elems);e_y = zeros(2,2*elems);e_xy = zeros(2,2*elems);sv = zeros(2,2*elems);x_gauss = zeros(2,2*elems);

y_gauss = zeros(2,2*elems);fori = 1:elems

x = form_x(i,elems,nodes,xg);

%map physical coordinate of qausspts[x_sub, y_sub] = form_x_gauss(x);

x_gauss(:,i*2-1:i*2) = x_sub;y_gauss(:,i*2-1:i*2) = y_sub;

%1st Point (point a')xi1 = -1/sqrt(3);xi2 = -1/sqrt(3);[H1,detJ1] = func_hesssion(xi1, xi2, x);

%3rd Point (point b')xi1 = 1/sqrt(3);xi2 = -1/sqrt(3);

[H2,detJ2] = func_hesssion(xi1, xi2, x);%4th Point (point c')

xi1 = 1/sqrt(3);xi2 = 1/sqrt(3);[H3,detJ3] = func_hesssion(xi1, xi2, x);

%2nd Point (point d')xi1 = -1/sqrt(3);xi2 = 1/sqrt(3);[H4,detJ4] = func_hesssion(xi1, xi2, x);

L = form_fetch(i,nodes);

d = L*dg;

%% STRAIN at gauss pointe_a = H1*d;e_b = H2*d;

e_c = H3*d;e_d = H4*d;

e_x(:,i*2-1:i*2) = [e_a(1) e_d(1); e_b(1) e_c(1)];e_y(:,i*2-1:i*2) = [e_a(2) e_d(2); e_b(2) e_c(2)];e_xy(:,i*2-1:i*2) = [e_a(3) e_d(3); e_b(3) e_c(3)];

%% STRESS at gauss points_a = D*H1*d;s_b = D*H2*d;s_c = D*H3*d;s_d = D*H4*d;

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s_av = sqrt( s_a(1)^2 - s_a(1)*s_a(2) + s_a(2)^2 + 3*s_a(3)^2 );s_bv = sqrt( s_b(1)^2 - s_b(1)*s_b(2) + s_b(2)^2 + 3*s_b(3)^2 );s_cv = sqrt( s_c(1)^2 - s_c(1)*s_c(2) + s_c(2)^2 + 3*s_c(3)^2 );s_dv = sqrt( s_d(1)^2 - s_d(1)*s_d(2) + s_d(2)^2 + 3*s_d(3)^2 );

sv(:,i*2-1:i*2) = [s_avs_dv; s_bvs_cv];

end

%ex-gauss pt plotfigure(2)plot(xg(:,1),xg(:,2), '-ob')axis([-0.05 4.05 -0.05 0.55])xlabel('x');ylabel('y');title('\epsilon_x')holdongridon[C,h] = contour(x_gauss,y_gauss,e_x);clabel(C,h);

holdoff

%ey-gauss pt plotfigure(3)plot(xg(:,1),xg(:,2), '-ob')axis([-0.05 4.05 -0.05 0.55])xlabel('x');ylabel('y');title('\epsilon_y')holdongridon[C,h] = contour(x_gauss,y_gauss,e_y);clabel(C,h);holdoff

% % %exy-gauss pt plot% figure(4)% plot(xg(:,1),xg(:,2), '-ob')% hold on

% grid on% [C,h] = contour(x_gauss,y_gauss,e_xy);% clabel(C,h);% axis([0 4 0 0.5])% hold off

%Von-Mise Equivalent stress - gauss pt plotfigure(5)plot(xg(:,1),xg(:,2), '-ob')xlabel('x');ylabel('y');title('\sigma_v')axis([-0.05 4.05 -0.05 0.55])holdongridon[C,h] = contour(x_gauss,y_gauss,sv);clabel(C,h);

holdoff

function x = form_x(i,elems,nodes,xg)

L = zeros(4, 2*(elems+1));L(1,nodes-i+1) = 1; %fetch node aL(2,i) = 1; %fetch node bL(3,i+1) = 1; %fetch node cL(4,nodes-i) = 1; %fetch node d

x = L*xg;

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end

function [H,detJ] = func_hesssion(xi1, xi2, x)

J = 0.25*[ xi2-1, 1-xi2, 1+xi2, -xi2-1;...

xi1-1, -xi1-1, 1+xi1, 1-xi1]*x;

G = 0.25*[ xi2-1, 1-xi2, 1+xi2, -xi2-1;...xi1-1, -xi1-1, 1+xi1, 1-xi1];

detJ = J(1,1)*J(2,2)-J(1,2)*J(2,1);

Hv = J\G;

H = [Hv(1,1) 0 Hv(1,2) 0 Hv(1,3) 0 Hv(1,4) 0;...0 Hv(2,1) 0 Hv(2,2) 0 Hv(2,3) 0 Hv(2,4);...

Hv(2,1) Hv(1,1) Hv(2,2) Hv(1,2) Hv(2,3) Hv(1,3) Hv(2,4) Hv(1,4)];

function [x_gauss, y_gauss] = form_x_gauss(x)

%point axi1 = -1/sqrt(3);xi2 = -1/sqrt(3);N = form_shape(xi1,xi2);

xa = x(:,1)'*N;ya = x(:,2)'*N;

%point bxi1 = 1/sqrt(3);xi2 = -1/sqrt(3);N = form_shape(xi1,xi2);

xb = x(:,1)'*N;yb = x(:,2)'*N;

%point cxi1 = 1/sqrt(3);xi2 = 1/sqrt(3);N = form_shape(xi1,xi2);

xc = x(:,1)'*N;yc = x(:,2)'*N;

%point dxi1 = -1/sqrt(3);xi2 = 1/sqrt(3);N = form_shape(xi1,xi2);

xd = x(:,1)'*N;yd = x(:,2)'*N;

x_gauss = [xaxd; xb xc];

y_gauss = [yayd; ybyc];

function K = form_K(D,x,t)

%Gauss Quadrature%1st Point (point a')xi1 = -1/sqrt(3);xi2 = -1/sqrt(3);[H1,detJ1] = func_hesssion(xi1, xi2, x);%3rd Point (point b')xi1 = 1/sqrt(3);

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xi2 = -1/sqrt(3);[H2,detJ2] = func_hesssion(xi1, xi2, x);%4th Point (point c')xi1 = 1/sqrt(3);xi2 = 1/sqrt(3);[H3,detJ3] = func_hesssion(xi1, xi2, x);%2nd Point (point d')

xi1 = -1/sqrt(3);xi2 = 1/sqrt(3);[H4,detJ4] = func_hesssion(xi1, xi2, x);

K1 = detJ1*H1'*D*H1;K2 = detJ2*H2'*D*H2;K3 = detJ3*H3'*D*H3;K4 = detJ4*H4'*D*H4;

K = t*(K1+K2+K3+K4);

function N = form_shape(xi1,xi2)

Na = 1/4*(1-xi1)*(1-xi2);Nb = 1/4*(1+xi1)*(1-xi2);

Nc = 1/4*(1+xi1)*(1+xi2);Nd = 1/4*(1-xi1)*(1+xi2);

N = [Na;Nb;Nc;Nd];

function L = form_fetch(i,nodes)

L = zeros(8, nodes*2);%fetch node aL(1,2*nodes-2*i+1) = 1;L(2,2*nodes-2*i+2) = 1;

%fetch node bL(3,2*i-1) = 1;L(4,2*i) = 1;

%fetch node cL(5,2*i+1) = 1;L(6,2*i+2) = 1;

%fetch node dL(7,2*nodes-2*i-1) = 1;L(8,2*nodes-2*i) = 1;

end