Solutions to MATLAB Guide to Finite Elements Reduced Version

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Solutions Manual for "MATLAB Guide to Finite Elements - Second Edition,"
(Reduced Version)
Book · January 2007
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Solutions Manual 
 
(Reduced Version) 
 
To Accompany the Book 
 
MATLAB Guide to Finite Elements 
An Interactive Approach 
Second Edition 
 
By 
 
Peter I. Kattan 
 
 The solutions to all the problems are given below with each command ending 
with a semi-colon to suppress the output except at key locations where the result is 
given like the global stiffness matrix, the displacements, and the reactions. 
 
Problem 2.1: 
 
» k1=SpringElementStiffness(200); 
» k2=SpringElementStiffness(250); 
» K=zeros(3,3); 
» K=SpringAssemble(K,k1,1,2); 
» K=SpringAssemble(K,k2,2,3) 
 
K = 
 
 200 -200 0 
 -200 450 -250 
 0 -250 250 
 
» k=K(2,2); 
» f=[10]; 
» u=k\f 
 
u = 
 
 0.0222 
 
» U=[0 ; u ; 0]; 
» F=K*U 
 
F = 
 
 -4.4444 
 10.0000 
 -5.5556 
 
» u1=[0;u]; 
» f1=SpringElementForces(k1,u1); 
» u2=[u ; 0]; 
» f2=SpringElementForces(k2,u2); 
 
Problem 2.2: 
 
» k1=SpringElementStiffness(170); 
» k2=SpringElementStiffness(170); 
» k3=SpringElementStiffness(170); 
» k4=SpringElementStiffness(170); 
» K=zeros(4,4); 
» K=SpringAssemble(K,k1,1,2); 
» K=SpringAssemble(K,k2,2,3); 
» K=SpringAssemble(K,k3,2,3); 
» K=SpringAssemble(K,k4,3,4) 
 
K = 
 
 170 -170 0 0 
 -170 510 -340 0 
 0 -340 510 -170 
 0 0 -170 170 
 
» k=K(2:4,2:4); 
» f=[0 ; 0 ; 25]; 
» u=k\f 
 
u = 
 
 0.1471 
 0.2206 
 0.3676 
 
» U=[0;u]; 
» F=K*U 
 
F = 
 
 -25.0000 
 0.0000 
 0.0000 
 25.0000 
 
» u1=[0;U(2)]; 
» f2=SpringElementForces(k1,u1); 
» u2=[U(2);U(3)]; 
» f2=SpringElementForces(k2,u2); 
» u3=[U(2);U(3)]; 
» f3=SpringElementForces(k3,u3); 
» u4=[U(3);U(4)]; 
» f4=SpringElementForces(k4,u4); 
 
Problem 3.1: 
 
» E=70e6; 
» A=0.005; 
» L1=1; 
» L2=2; 
» L3=1; 
» k1=LinearBarElementStiffness(E,A,L1); 
» k2=LinearBarElementStiffness(E,A,L2); 
» k3=LinearBarElementStiffness(E,A,L3); 
» K=zeros(4,4); 
» K=LinearBarAssemble(K,k1,1,2); 
» K=LinearBarAssemble(K,k2,2,3); 
» K=LinearBarAssemble(K,k3,3,4) 
 
K = 
 
 350000 -350000 0 0 
 -350000 525000 -175000 0 
 0 -175000 525000 -350000 
 0 0 -350000 350000 
 
» k=K(2:4,2:4); 
» f=[-10 ; 0 ; 15]; 
» u=k\f 
 
u = 
 
 1.0e-003 * 
 
 0.0143 
 0.1000 
 0.1429 
 
» U=[0;u]; 
» F=K*U 
 
F = 
 
 -5.0000 
 -10.0000 
 -0.0000 
 15.0000 
 
» u1=[0;U(2)]; 
» sigma1=LinearBarElementStresses(k1,u1,A); 
» u2=[U(2);U(3)]; 
» sigma2=LinearBarElementStresses(k2,u2,A); 
» u3=[U(3);U(4)]; 
» sigma3=LinearBarElementStresses(k3,u3,A); 
 
Problem 3.2: 
 
» E=210e6; 
» L=3/10; 
» A1=0.002+(0.01*0.15/3); 
» A2=0.002+(0.01*0.45/3); 
» A3=0.002+(0.01*0.75/3); 
» A4=0.002+(0.01*1.05/3); 
» A5=0.002+(0.01*1.35/3); 
» A6=0.002+(0.01*1.65/3); 
» A7=0.002+(0.01*1.95/3); 
» A8=0.002+(0.01*2.25/3); 
» A9=0.002+(0.01*2.55/3); 
» A10=0.002+(0.01*2.85/3); 
» k1=LinearBarElementStiffness(E,A1,L); 
» k2=LinearBarElementStiffness(E,A2,L); 
» k3=LinearBarElementStiffness(E,A3,L); 
» k4=LinearBarElementStiffness(E,A4,L); 
» k5=LinearBarElementStiffness(E,A5,L); 
» k6=LinearBarElementStiffness(E,A6,L); 
» k7=LinearBarElementStiffness(E,A7,L); 
» k8=LinearBarElementStiffness(E,A8,L); 
» k9=LinearBarElementStiffness(E,A9,L); 
» k10=LinearBarElementStiffness(E,A10,L); 
» K=zeros(11,11); 
» K=LinearBarAssemble(K,k1,1,2); 
» K=LinearBarAssemble(K,k2,2,3); 
» K=LinearBarAssemble(K,k3,3,4); 
» K=LinearBarAssemble(K,k4,4,5); 
» K=LinearBarAssemble(K,k5,5,6); 
» K=LinearBarAssemble(K,k6,6,7); 
» K=LinearBarAssemble(K,k7,7,8); 
» K=LinearBarAssemble(K,k8,8,9); 
» K=LinearBarAssemble(K,k9,9,10); 
» K=LinearBarAssemble(K,k10,10,11) 
 
K = 
 
 1.0e+007 * 
 
 Columns 1 through 7 
 
 0.1750 -0.1750 0 0 0 0 0 
 -0.1750 0.4200 -0.2450 0 0 0 0 
 0 -0.2450 0.5600 -0.3150 0 0 0 
 0 0 -0.3150 0.7000 -0.3850 0 0 
 0 0 0 -0.3850 0.8400 -0.4550 0 
 0 0 0 0 -0.4550 0.9800 -0.5250 
 0 0 0 0 0 -0.5250 1.1200 
 0 0 0 0 0 0 -0.5950 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 
 Columns 8 through 11 
 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 -0.5950 0 0 0 
 1.2600 -0.6650 0 0 
 -0.6650 1.4000 -0.7350 0 
 0 -0.7350 1.5400 -0.8050 
 0 0 -0.8050 0.8050 
 
» k=K(1:10,1:10); 
» f=[-18;0;0;0;0;0;0;0;0;0]; 
» u=k\f 
 
u = 
 
 1.0e-004 * 
 
 -0.4582 
 -0.3554 
 -0.2819 
 -0.2248 
 -0.1780 
 -0.1385 
 -0.1042 
 -0.0739 
 -0.0469 
 -0.0224 
 
It is clear from the results above that the displacement at the free end is 
-0.04582 mm which is very close to that obtained in Example 3.2. However,the result 
obtained in this problem is more accurate because we have used more elements in the 
discretization. 
 
 
Problem 3.3: 
 
» E=200e6; 
» A=0.01; 
» L=2; 
» k1=LinearBarElementStiffness(E,A,L); 
» k2=SpringElementStiffness(1000); 
» K=zeros(3,3); 
» K=LinearBarAssemble(K,k1,1,2); 
» K=SpringAssemble(K,k2,2,3) 
 
K = 
 
 1000000 -1000000 0 
 -1000000 1001000 -1000 
 0 -1000 1000 
 
» k=K(2,2); 
» f=[25]; 
» u=k\f 
 
u = 
 
 2.4975e-005 
 
» U=[0;u;0]; 
» F=K*U 
 
F = 
 
 -24.9750 
 25.0000 
 -0.0250 
 
» u1=[0;u]; 
» sigma1=LinearBarElementStresses(k1,u1,A); 
» u2=[u;0]; 
» f2=SpringElementForces(k2,u2); 
 
Problem 4.1: 
 
» E=210e6; 
» L=3/2; 
» A1=0.002+(0.01*0.75/3); 
» A2=0.002+(0.01*2.25/3); 
» k1=QuadraticBarElementStiffness(E,A1,L); 
» k2=QuadraticBarElementStiffness(E,A2,L); 
» K=zeros(5,5); 
» K=QuadraticBarAssemble(K,k1,1,3,2); 
» K=QuadraticBarAssemble(K,k2,3,5,4) 
 
K = 
 
 1.0e+006 * 
 
 1.4700 -1.6800 0.2100 0 0 
 -1.6800 3.3600 -1.6800 0 0 
 0.2100 -1.6800 4.5733 -3.5467 0.4433 
 0 0 -3.5467 7.0933 -3.5467 
 0 0 0.4433 -3.5467 3.1033 
 
» k=K(1:4,1:4); 
» f=[-18 ; 0 ; 0 ; 0]; 
» u=k\f 
 
u = 
 
 1.0e-004 * 
 
 -0.4211 
 -0.2782 
 -0.1353 
 -0.0677 
 
Thus it is clear that the displacement at the free end is -0.4211 x 10-4 m or 
-0.04211 mm which is very close to that obtained in Example 3.2 which was -0.04517 
mm and that obtained in the solution of Problem 3.2 which was -0.04582 mm. 
 
Problem 4.2: 
 
» E=70e6; 
» A=0.001; 
» L=4; 
» k1=SpringElementStiffness(2000); 
» k2=QuadraticBarElementStiffness(E,A,L); 
» K=zeros(4,4); 
» K=SpringAssemble(K,k1,1,2); 
» K=QuadraticBarAssemble(K,k2,2,4,3) 
 
K = 
 
 1.0e+004 * 
 
 0.2000 -0.2000 0 0 
 -0.2000 4.2833 -4.6667 0.5833 
 0 -4.6667 9.3333 -4.6667 
 0 0.5833 -4.6667 4.0833 
 
» k=K(2:4,2:4); 
» f=[0 ; 10 ; 5]; 
» u=k\f 
 
u = 
 
 0.0075 
 0.0079 
 0.0081 
 
» U=[0 ; u]; 
» F=K*U 
 
F = 
 
 -15.0000 
 0.0000 
 10.0000 
 5.0000 
 
» u1=[0 ; U(2)]; 
» f1=SpringElementForces(k1,u1); 
» u2=[U(2) ; U(4) ; U(3)]; 
» sigma2=QuadraticBarElementStresses(k2,u2,A); 
 
Problem 5.1: 
 
» E=210e6; 
» A=0.005; 
» L1=PlaneTrussElementLength(0,0,5,7); 
» L5=PlaneTrussElementLength(0,0,5,-7); 
» L9=PlaneTrussElementLength(0,0,5,-7); 
» theta1=atan(7/5)*180/pi; 
» theta2=0; 
» theta3=270; 
» theta4=0; 
» theta5=360-theta1; 
» theta6=0; 
» theta7=270; 
» theta8=0; 
» theta9=theta5; 
» k1=PlaneTrussElementStiffness(E,A,L1,theta1); 
» k2=PlaneTrussElementStiffness(E,A,5,theta2); 
» k3=PlaneTrussElementStiffness(E,A,7,theta3); 
» k4=PlaneTrussElementStiffness(E,A,5,theta4); 
» k5=PlaneTrussElementStiffness(E,A,L5,theta5); 
» k6=PlaneTrussElementStiffness(E,A,5,theta6); 
» k7=PlaneTrussElementStiffness(E,A,7,theta7); 
» k8=PlaneTrussElementStiffness(E,A,5,theta8); 
» k9=PlaneTrussElementStiffness(E,A,L9,theta9); 
» K=zeros(12,12); 
» K=PlaneTrussAssemble(K,k1,1,2); 
» K=PlaneTrussAssemble(K,k2,1,3); 
» K=PlaneTrussAssemble(K,k3,2,3); 
» K=PlaneTrussAssemble(K,k4,3,5); 
» K=PlaneTrussAssemble(K,k5,2,5); 
» K=PlaneTrussAssemble(K,k6,2,4); 
» K=PlaneTrussAssemble(K,k7,4,5); 
» K=PlaneTrussAssemble(K,k8,5,6); 
» K=PlaneTrussAssemble(K,k9,4,6) 
 
K = 
 
 1.0e+005 * 
 
 Columns 1 through 7 
 
 2.5124 0.5773 -0.4124 -0.5773 -2.1000 0 0 
 0.5773 0.8082 -0.5773 -0.8082 0 0 0 
 -0.4124 -0.5773 2.9247 0.0000 -0.0000 -0.0000 -2.1000 
 -0.5773 -0.8082 0.0000 3.1165 -0.0000 -1.5000 0 
 -2.1000 0 -0.0000 -0.0000 4.2000 0.0000 0 
 0 0 -0.0000 -1.5000 0.0000 1.5000 0 
 0 0 -2.1000 0 0 0 2.5124 
 0 0 0 0 0 0 -0.5773 
 0 0 -0.4124 0.5773 -2.1000 0 -0.0000 
 0 0 0.5773 -0.8082 0 0 -0.0000 
 0 0 0 0 0 0 -0.4124 
 0 0 0 0 0 0 0.5773 
 
 Columns 8 through 12 
 
 0 0 0 0 0 
 0 0 0 0 0 
 0 -0.4124 0.5773 0 0 
 0 0.5773 -0.8082 0 0 
 0 -2.1000 0 0 0 
 0 0 0 0 0 
 -0.5773 -0.0000 -0.0000 -0.4124 0.5773 
 2.3082 -0.0000 -1.5000 0.5773 -0.8082 
 -0.0000 4.6124 -0.5773 -2.1000 0 
 -1.5000 -0.5773 2.3082 0 0 
 0.5773 -2.1000 0 2.5124 -0.5773 
 -0.8082 0 0 -0.5773 0.8082 
 
» k=K(3:10,3:10); 
» f=[20 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0]; 
» u=k\f 
 
u = 
 
 1.0e-003 * 
 
 0.2083 
 -0.0333 
 0.0106 
 -0.0333 
 0.1766 
 0.0107 
 0.0212 
 -0.0516 
 
» U=[0 ; 0 ; u ; 0 ; 0]; 
» F=K*U 
 
F = 
 
 -8.8889 
 -9.3333 
 20.0000 
 0.0000 
 0.0000 
 0 
 0.0000 
 0 
 0 
 -0.0000 
 -11.1111 
 9.3333 
 
» u1=[U(1) ; U(2) ; U(3) ; U(4)]; 
» u2=[U(1) ; U(2) ; U(5) ; U(6)]; 
» u3=[U(3) ; U(4) ; U(5) ; U(6)]; 
» u4=[U(5) ; U(6) ; U(9) ; U(10)]; 
» u5=[U(3) ; U(4) ; U(9) ; U(10)]; 
» u6=[U(3) ; U(4) ; U(7) ; U(8)]; 
» u7=[U(7) ; U(8) ; U(9) ; U(10)]; 
» u8=[U(9) ; U(10) ; U(11) ; U(12)]; 
» u9=[U(7) ; U(8) ; U(11) ; U(12)]; 
» sigma1=PlaneTrussElementStress(E,L1,theta1,u1); 
» sigma2=PlaneTrussElementStress(E,5,theta2,u2); 
» sigma3=PlaneTrussElementStress(E,7,theta3,u3); 
» sigma4=PlaneTrussElementStress(E,5,theta4,u4); 
» sigma5=PlaneTrussElementStress(E,L5,theta5,u5); 
» sigma6=PlaneTrussElementStress(E,5,theta6,u6); 
» sigma7=PlaneTrussElementStress(E,7,theta7,u7); 
» sigma8=PlaneTrussElementStress(E,5,theta8,u8); 
» sigma9=PlaneTrussElementStress(E,L9,theta9,u9); 
 
Problem 5.2: 
 
» E=70e6; 
» A=0.01; 
» L1=PlaneTrussElementLength(0,0,4,3); 
» L2=PlaneTrussElementLength(0,0,4,0); 
» L3=PlaneTrussElementLength(0,0,4,-4); 
» theta1=atan(3/4)*180/pi; 
» theta2=0; 
» theta3=360-atan(4/4)*180/pi; 
» k1=PlaneTrussElementStiffness(E,A,L1,theta1); 
» k2=PlaneTrussElementStiffness(E,A,L2,theta2); 
» k3=PlaneTrussElementStiffness(E,A,L3,theta3); 
» k4=SpringElementStiffness(3000); 
» K=zeros(9,9); 
» K=PlaneTrussAssemble(K,k1,1,4); 
» K=PlaneTrussAssemble(K,k2,2,4); 
» K=PlaneTrussAssemble(K,k3,3,4); 
» K=SpringAssemble(K,k4,7,9) 
 
K = 
 
 1.0e+005 * 
 
 Columns 1 through 7 
 
 0.8960 0.6720 0 0 0 0 -0.8960 
 0.6720 0.5040 0 0 0 0 -0.6720 
 0 0 1.7500 0 0 0 -1.7500 
 0 0 0 0 0 0 0 
 0 0 0 0 0.6187 -0.6187 -0.6187 
 0 0 0 0 -0.6187 0.6187 0.6187 
 -0.8960 -0.6720 -1.7500 0 -0.6187 0.6187 3.2947 
 -0.6720 -0.5040 0 0 0.6187 -0.6187 0.0533 
 0 0 0 0 0 0 -0.0300 
 
 Columns 8 through 9 
 
 -0.6720 0 
 -0.5040 0 
 0 0 
 0 0 
 0.6187 0 
 -0.6187 0 
 0.0533 -0.0300 
 1.1227 0 
 0 0.0300 
 
» k=K(7:9,7:9); 
» f=[0 ; 0 ; 10]; 
» u=k\f 
 
u = 
 
 0.0000 
 -0.0000 
 0.0034 
 
» U=[0 ; 0 ; 0 ; 0 ; 0 ; 0 ; u]; 
» F=K*U 
 
F = 
 
 -2.6489 
 -1.9866 
 -5.3645 
 0 
 -1.9866 
 1.9866 
 -0.0000 
 -0.0000 
 10.0000 
 
» u1=[U(1) ; U(2) ; U(7) ; U(8)]; 
» u2=[U(3) ; U(4) ; U(7) ; U(8)]; 
» u3=[U(5) ; U(6) ; U(7) ; U(8)]; 
» u4=[U(7); U(9)]; 
» sigma1=PlaneTrussElementStress(E,L1,theta1,u1); 
» sigma2=PlaneTrussElementStress(E,L2,theta2,u2);» sigma3=PlaneTrussElementStress(E,L3,theta3,u3); 
» f4=SpringElementForces(k4,u4); 
 
Problem 6.1: 
 
» E=200e6; 
» A=0.003; 
» L1=SpaceTrussElementLength(0,0,-3,0,5,0); 
» L2=SpaceTrussElementLength(-3,0,0,0,5,0); 
» L3=SpaceTrussElementLength(0,0,3,0,5,0); 
» L4=SpaceTrussElementLength(4,0,0,0,5,0); 
» theta1x=acos(0/L1)*180/pi; 
» theta1y=acos(5/L1)*180/pi; 
» theta1z=acos(3/L1)*180/pi; 
» theta2x=acos(3/L2)*180/pi; 
» theta2y=acos(5/L2)*180/pi; 
» theta2z=acos(0/L2)*180/pi; 
» theta3x=acos(0/L3)*180/pi; 
» theta3y=acos(5/L3)*180/pi; 
» theta3z=acos(-3/L3)*180/pi; 
» theta4x=acos(-4/L4)*180/pi; 
» theta4y=acos(5/L4)*180/pi; 
» theta4z=acos(0/L4)*180/pi; 
» k1=SpaceTrussElementStiffness(E,A,L1,theta1x,theta1y,theta1z); 
» k2=SpaceTrussElementStiffness(E,A,L2,theta2x,theta2y,theta2z); 
» k3=SpaceTrussElementStiffness(E,A,L3,theta3x,theta3y,theta3z); 
» k4=SpaceTrussElementStiffness(E,A,L4,theta4x,theta4y,theta4z); 
» K=zeros(15,15); 
» K=SpaceTrussAssemble(K,k1,1,5); 
» K=SpaceTrussAssemble(K,k2,2,5); 
» K=SpaceTrussAssemble(K,k3,3,5); 
» K=SpaceTrussAssemble(K,k4,4,5) 
 
K = 
 
 1.0e+005 * 
 
 Columns 1 through 7 
 
 0.0000 0.0000 0.0000 0 0 0 0 
 0.0000 0.7566 0.4540 0 0 0 0 
 0.0000 0.4540 0.2724 0 0 0 0 
 0 0 0 0.2724 0.4540 0.0000 0 
 0 0 0 0.4540 0.7566 0.0000 0 
 0 0 0 0.0000 0.0000 0.0000 0 
 0 0 0 0 0 0 0.0000 
 0 0 0 0 0 0 0.0000 
 0 0 0 0 0 0 -0.0000 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 -0.0000 -0.0000 -0.0000 -0.2724 -0.4540 -0.0000 -0.0000 
 -0.0000 -0.7566 -0.4540 -0.4540 -0.7566 -0.0000 -0.0000 
 -0.0000 -0.4540 -0.2724 -0.0000 -0.0000 -0.0000 0.0000 
 
 Columns 8 through 14 
 
 0 0 0 0 0 -0.0000 -0.0000 
 0 0 0 0 0 -0.0000 -0.7566 
 0 0 0 0 0 -0.0000 -0.4540 
 0 0 0 0 0 -0.2724 -0.4540 
 0 0 0 0 0 -0.4540 -0.7566 
 0 0 0 0 0 -0.0000 -0.0000 
 0.0000 -0.0000 0 0 0 -0.0000 -0.0000 
 0.7566 -0.4540 0 0 0 -0.0000 -0.7566 
 -0.4540 0.2724 0 0 0 0.0000 0.4540 
 0 0 0.3657 -0.4571 -0.0000 -0.3657 0.4571 
 0 0 -0.4571 0.5714 0.0000 0.4571 -0.5714 
 0 0 -0.0000 0.0000 0.0000 0.0000 -0.0000 
 -0.0000 0.0000 -0.3657 0.4571 0.0000 0.6381 -0.0031 
 -0.7566 0.4540 0.4571 -0.5714 -0.0000 -0.0031 2.8412 
 0.4540 -0.2724 0.0000 -0.0000 -0.0000 -0.0000 0.0000 
 
 Column 15 
 
 -0.0000 
 -0.4540 
 -0.2724 
 -0.0000 
 -0.0000 
 -0.0000 
 0.0000 
 0.4540 
 -0.2724 
 0.0000 
 -0.0000 
 -0.0000 
 -0.0000 
 0.0000 
 0.5448 
 
» k=K(13:15,13:15); 
» f=[15 ; 0 ; -20]; 
» u=k\f 
 
u = 
 
 1.0e-003 * 
 
 0.2351 
 0.0003 
 -0.3671 
 
» U=[0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; u]; 
» F=K*U 
 
F = 
 
 0.0000 
 16.6471 
 9.9883 
 -6.4151 
 -10.6919 
 -0.0000 
 -0.0000 
 -16.6862 
 10.0117 
 -8.5849 
 10.7311 
 0.0000 
 15.0000 
 -0.0000 
 -20.0000 
 
» u1=[U(1) ; U(2) ; U(3) ; U(13) ; U(14) ; U(15)]; 
» u2=[U(4) ; U(5) ; U(6) ; U(13) ; U(14) ; U(15)]; 
» u3=[U(7) ; U(8) ; U(9) ; U(13) ; U(14) ; U(15)]; 
» u4=[U(10) ; U(11) ; U(12) ; U(13) ; U(14) ; U(15)]; 
» sigma1=SpaceTrussElementStress(E,L1,theta1x,theta1y,theta1z,u1); 
» sigma2=SpaceTrussElementStress(E,L2,theta2x,theta2y,theta2z,u2); 
» sigma3=SpaceTrussElementStress(E,L3,theta3x,theta3y,theta3z,u3); 
» sigma4=SpaceTrussElementStress(E,L4,theta4x,theta4y,theta4z,u4); 
 
Problem 7.1: 
 
» E=200e6; 
» I=70e-5; 
» L1=3.5; 
» L2=2; 
» k1=BeamElementStiffness(E,I,L1); 
» k2=BeamElementStiffness(E,I,L2); 
» K=zeros(6,6); 
» K=BeamAssemble(K,k1,1,2); 
» K=BeamAssemble(K,k2,2,3) 
 
K = 
 
 1.0e+005 * 
 
 0.3918 0.6857 -0.3918 0.6857 0 0 
 0.6857 1.6000 -0.6857 0.8000 0 0 
 -0.3918 -0.6857 2.4918 1.4143 -2.1000 2.1000 
 0.6857 0.8000 1.4143 4.4000 -2.1000 1.4000 
 0 0 -2.1000 -2.1000 2.1000 -2.1000 
 0 0 2.1000 1.4000 -2.1000 2.8000 
 
» k=[K(2,2) K(2,4) K(2,6) ; K(4,2) K(4,4) K(4,6) ; K(6,2) K(6,4) 
K(6,6)]; 
» f=[0 ; -15 ; 0]; 
» u=k\f 
 
u = 
 
 1.0e-004 * 
 
 0.2273 
 -0.4545 
 0.2273 
 
» U=[0 ; u(1) ; 0 ; u(2) ; 0 ; u(3)]; 
» F=K*U 
 
F = 
 
 -1.5584 
 0 
 -3.2143 
 -15.0000 
 4.7727 
 0 
 
» u1=[U(1) ; U(2) ; U(3) ; U(4)]; 
» u2=[U(3) ; U(4) ; U(5) ; U(6)]; 
» f1=BeamElementForces(k1,u1); 
» f2=BeamElementForces(k2,u2); 
» BeamElementShearDiagram(f1,L1); 
» BeamElementShearDiagram(f2,L2); 
» BeamElementMomentDiagram(f1,L1); 
» BeamElementMomentDiagram(f2,L2); 
 
Problem 7.2: 
 
» E=210e6; 
» I=50e-6; 
» L1=3; 
» L2=3; 
» L3=4; 
» k1=BeamElementStiffness(E,I,L1); 
» k2=BeamElementStiffness(E,I,L2); 
» k3=BeamElementStiffness(E,I,L3); 
» K=zeros(8,8); 
» K=BeamAssemble(K,k1,1,2); 
» K=BeamAssemble(K,k2,2,3); 
» K=BeamAssemble(K,k3,3,4) 
 
K = 
 
 1.0e+004 * 
 
 Columns 1 through 7 
 
 0.4667 0.7000 -0.4667 0.7000 0 0 0 
 0.7000 1.4000 -0.7000 0.7000 0 0 0 
 -0.4667 -0.7000 0.9333 0 -0.4667 0.7000 0 
 0.7000 0.7000 0 2.8000 -0.7000 0.7000 0 
 0 0 -0.4667 -0.7000 0.6635 -0.3063 -0.1969 
 0 0 0.7000 0.7000 -0.3063 2.4500 -0.3937 
 0 0 0 0 -0.1969 -0.3937 0.1969 
 0 0 0 0 0.3937 0.5250 -0.3937 
 
 Column 8 
 
 0 
 0 
 0 
 0 
 0.3937 
 0.5250 
 -0.3937 
 1.0500 
 
» k=[K(4,4) K(4,6) K(4,8) ; K(6,4) K(6,6) K(6,8) ; K(8,4) K(8,6) 
K(8,8)]; 
» f=[7.5 ; -15 ; 15]; 
» u=k\f 
 
u = 
 
 0.0006 
 -0.0012 
 0.0020 
 
» U=[0 ; 0 ; 0 ; u(1) ; 0 ; u(2) ; 0 ; u(3)]; 
» F=K*U 
 
F = 
 
 3.9946 
 3.9946 
 -8.4783 
 7.5000 
 7.7242 
 -15.0000 
 -3.2405 
 15.0000 
 
» u1=[U(1) ; U(2) ; U(3) ; U(4)]; 
» u2=[U(3) ; U(4) ; U(5) ; U(6)]; 
» u3=[U(5) ; U(6) ; U(7) ; U(8)]; 
» f1=BeamElementForces(k1,u1); 
» f2=BeamElementForces(k2,u2); 
» f3=BeamElementForces(k3,u3); 
» f1=f1-[-15 ; -7.5 ; -15 ; 7.5]; 
» f3=f3-[-15 ; -15 ; -15 ; 15]; 
» BeamElementShearDiagram(f1,L1); 
» BeamElementShearDiagram(f2,L2); 
» BeamElementShearDiagram(f3,L3); 
» BeamElementMomentDiagram(f1,L1); 
» BeamElementMomentDiagram(f2,L2); 
» BeamElementMomentDiagram(f3,L3); 
 
Problem 7.3: 
 
» E=70e6; 
» I=40e-6; 
» k1=BeamElementStiffness(E,I,3); 
» k2=BeamElementStiffness(E,I,3); 
» k3=SpringElementStiffness(5000); 
» K=zeros(7,7); 
» K=BeamAssemble(K,k1,1,2); 
» K=BeamAssemble(K,k2,2,3); 
» K=SpringAssemble(K,k3,3,7) 
 
K = 
 
 1.0e+003 * 
 
 1.2444 1.8667 -1.2444 1.8667 0 0 0 
 1.8667 3.7333 -1.8667 1.8667 0 0 0 
 -1.2444 -1.8667 7.4889 0 -1.2444 1.8667 -5.0000 
 1.8667 1.8667 0 7.4667 -1.8667 1.8667 0 
 0 0 -1.2444 -1.8667 1.2444 -1.8667 0 
 0 0 1.8667 1.8667 -1.8667 3.7333 0 
 0 0-5.0000 0 0 0 5.0000 
 
» k=[K(3:4,3:4) K(3:4,6) ; K(6,3:4) K(6,6)]; 
» f=[-10 ; 0 ; 0]; 
» u=k\f 
 
u = 
 
 -0.0016 
 -0.0002 
 0.0009 
 
» U=[0 ; 0 ; u(1) ; u(2) ; 0 ; u(3) ; 0]; 
» F=K*U 
 
F = 
 
 1.5225 
 2.4913 
 -10.0000 
 0.0000 
 0.6920 
 0 
 7.7855 
 
» u1=[U(1) ; U(2) ; U(3) ; U(4)]; 
» u2=[U(3) ; U(4) ; U(5) ; U(6)]; 
» u3=[U(3) ; U(7)]; 
» f1=BeamElementForces(k1,u1); 
» f2=BeamElementForces(k2,u2); 
» f3=SpringElementForces(k3,u3); 
» BeamElementShearDiagram(f1,3); 
» BeamElementShearDiagram(f2,3); 
» BeamElementMomentDiagram(f1,3); 
» BeamElementMomentDiagram(f2,3); 
 
Problem 8.1: 
 
» E=210e6; 
» A=4e-2; 
» I=4e-6; 
» L=4; 
» k1=PlaneFrameElementStiffness(E,A,I,L,90); 
» k2=PlaneFrameElementStiffness(E,A,I,L,0); 
» K=zeros(9,9); 
» K=PlaneFrameAssemble(K,k1,1,2); 
» K=PlaneFrameAssemble(K,k2,2,3) 
 
K = 
 
 1.0e+006 * 
 
 Columns 1 through 7 
 
 0.0002 0.0000 -0.0003 -0.0002 -0.0000 -0.0003 0 
 0.0000 2.1000 0.0000 -0.0000 -2.1000 0.0000 0 
 -0.0003 0.0000 0.0008 0.0003 -0.0000 0.0004 0 
 -0.0002 -0.0000 0.0003 2.1002 0.0000 0.0003 -2.1000 
 -0.0000 -2.1000 -0.0000 0.0000 2.1002 0.0003 0 
 -0.0003 0.0000 0.0004 0.0003 0.0003 0.0017 0 
 0 0 0 -2.1000 0 0 2.1000 
 0 0 0 0 -0.0002 -0.0003 0 
 0 0 0 0 0.0003 0.0004 0 
 
 Columns 8 through 9 
 
 0 0 
 0 0 
 0 0 
 0 0 
 -0.0002 0.0003 
 -0.0003 0.0004 
 0 0 
 0.0002 -0.0003 
 -0.0003 0.0008 
 
» k=[K(4:7,4:7) K(4:7,9) ; K(9,4:7) K(9,9)]; 
» f=[0 ; 0 ; 15 ; 20 ; 0 ]; 
» u=k\f 
 
u = 
 
 0.1865 
 0.0000 
 -0.0298 
 0.1865 
 0.0149 
 
» U=[0 ; 0 ; 0 ; u(1:4) ; 0 ; u(5)]; 
» F=K*U 
 
F = 
 
 -20.0000 
 -4.6875 
 46.2501 
 0.0000 
 0 
 15.0000 
 20.0000 
 4.6875 
 -0.0000 
 
» u1=[U(1) ; U(2) ; U(3) ; U(4) ; U(5) ; U(6)]; 
» u2=[U(4) ; U(5) ; U(6) ; U(7) ; U(8) ; U(9)]; 
» f1=PlaneFrameElementForces(E,A,I,L,90,u1); 
» f2=PlaneFrameElementForces(E,A,I,L,0,u2); 
» PlaneFrameElementAxialDiagram(f1,L); 
» PlaneFrameElementAxialDiagram(f2,L); 
» PlaneFrameElementShearDiagram(f1,L); 
» PlaneFrameElementShearDiagram(f2,L); 
» PlaneFrameElementMomentDiagram(f1,L); 
» PlaneFrameElementMomentDiagram(f2,L); 
 
Problem 8.2: 
 
» E=210e6; 
» A=1e-2; 
» I=9e-5; 
» L1=PlaneFrameElementLength(0,0,2,3); 
» L2=5; 
» L3=L1; 
» theta1=atan(3/2)*180/pi; 
» theta2=0; 
» theta3=360-theta1; 
» k1=PlaneFrameElementStiffness(E,A,I,L1,theta1); 
» k2=PlaneFrameElementStiffness(E,A,I,L2,theta2); 
» k3=PlaneFrameElementStiffness(E,A,I,L3,theta3); 
» K=zeros(12,12); 
» K=PlaneFrameAssemble(K,k1,1,2); 
» K=PlaneFrameAssemble(K,k2,2,3); 
» K=PlaneFrameAssemble(K,k3,3,4) 
 
K = 
 
 1.0e+005 * 
 
 Columns 1 through 7 
 
 1.8256 2.6658 -0.0726 -1.8256 -2.6658 -0.0726 0 
 2.6658 4.0471 0.0484 -2.6658 -4.0471 0.0484 0 
 -0.0726 0.0484 0.2097 0.0726 -0.0484 0.1048 0 
 -1.8256 -2.6658 0.0726 6.0256 2.6658 0.0726 -4.2000 
 -2.6658 -4.0471 -0.0484 2.6658 4.0653 -0.0030 0 
 -0.0726 0.0484 0.1048 0.0726 -0.0030 0.3609 0 
 0 0 0 -4.2000 0 0 6.0256 
 0 0 0 0 -0.0181 -0.0454 -2.6658 
 0 0 0 0 0.0454 0.0756 0.0726 
 0 0 0 0 0 0 -1.8256 
 0 0 0 0 0 0 2.6658 
 0 0 0 0 0 0 0.0726 
 
 Columns 8 through 12 
 
 0 0 0 0 0 
 0 0 0 0 0 
 0 0 0 0 0 
 0 0 0 0 0 
 -0.0181 0.0454 0 0 0 
 -0.0454 0.0756 0 0 0 
 -2.6658 0.0726 -1.8256 2.6658 0.0726 
 4.0653 0.0030 2.6658 -4.0471 0.0484 
 0.0030 0.3609 -0.0726 -0.0484 0.1048 
 2.6658 -0.0726 1.8256 -2.6658 -0.0726 
 -4.0471 -0.0484 -2.6658 4.0471 -0.0484 
 0.0484 0.1048 -0.0726 -0.0484 0.2097 
 
» k=K(4:9,4:9); 
» f=[20 ; -12.5 ; -10.417 ; 0 ; -12.5 ; 10.417]; 
» u=k\f 
 
u = 
 
 0.0013 
 -0.0009 
 -0.0005 
 0.0012 
 0.0008 
 0.0003 
 
» U=[0 ; 0 ; 0 ; u ; 0 ; 0 ; 0]; 
» F=K*U 
 
F = 
 
 2.0283 
 8.4058 
 8.0296 
 20.0000 
 -12.5000 
 -10.4170 
 -0.0000 
 -12.5000 
 10.4170 
 -22.0283 
 16.5942 
 15.1226 
 
» u1=[U(1) ; U(2) ; U(3) ; U(4) ; U(5) ; U(6)]; 
» u2=[U(4) ; U(5) ; U(6) ; U(7) ; U(8) ; U(9)]; 
» u3=[U(7) ; U(8) ; U(9) ; U(10) ; U(11) ; U(12)]; 
» f1=PlaneFrameElementForces(E,A,I,L1,theta1,u1); 
» f2=PlaneFrameElementForces(E,A,I,L2,theta2,u2); 
» f3=PlaneFrameElementForces(E,A,I,L3,theta3,u3); 
» f2=f2-[0 ; -12.5 ; -10.417 ; 0 ; -12.5 ; 10.417]; 
» PlaneFrameElementAxialDiagram(f1,L1); 
» PlaneFrameElementAxialDiagram(f2,L2); 
» PlaneFrameElementAxialDiagram(f3,L3); 
» PlaneFrameElementShearDiagram(f1,L1); 
» PlaneFrameElementShearDiagram(f2,L2); 
» PlaneFrameElementShearDiagram(f3,L3); 
» PlaneFrameElementMomentDiagram(f1,L1); 
» PlaneFrameElementMomentDiagram(f2,L2); 
» PlaneFrameElementMomentDiagram(f3,L3); 
 
Problem 8.3: 
 
» E1=70e6; 
» A1=1e-2; 
» I=1e-5; 
» E2=2500; 
» A2=10; 
» L2=5; 
» L1=4; 
» theta1=0; 
» theta2=atan(3/4)*180/pi; 
» k1=PlaneFrameElementStiffness(E1,A1,I,L1,theta1); 
» k2=PlaneTrussElementStiffness(E2,A2,L2,theta2); 
» K=zeros(8,8); 
» K=PlaneFrameAssemble(K,k1,1,2); 
» K=PlaneTrussAssemble(K,k2,1,4) 
 
K = 
 
 1.0e+005 * 
 
 Columns 1 through 7 
 
 1.7820 0.0240 0 -1.7500 0 0 -0.0320 
 0.0240 0.0193 0.0026 0 -0.0013 0.0026 -0.0240 
 0 0.0026 0.0070 0 -0.0026 0.0035 0 
 -1.7500 0 0 1.7500 0 0 0 
 0 -0.0013 -0.0026 0 0.0013 -0.0026 0 
 0 0.0026 0.0035 0 -0.0026 0.0070 0 
 -0.0320 -0.0240 0 0 0 0 0.0320 
 -0.0240 -0.0180 0 0 0 0 0.0240 
 
 Column 8 
 
 -0.0240 
 -0.0180 
 0 
 0 
 0 
 0 
 0.0240 
 0.0180 
 
» k=K(1:3,1:3); 
» f=[0 ; -10 ; 0]; 
» u=k\f 
 
u = 
 
 0.0001 
 -0.0056 
 0.0021 
 
» U=[u ; 0 ; 0 ; 0 ; 0 ; 0]; 
» F=K*U 
 
F = 
 
 -0.0000 
 -10.0000 
 -0.0000 
 -13.0903 
 0.1822 
 -0.7290 
 13.0903 
 9.8178 
 
» u1=[U(1) ; U(2) ; U(3) ; U(4) ; U(5) ; U(6)]; 
» u2=[U(1) ; U(2) ; U(7) ; U(8)]; 
» f1=PlaneFrameElementForces(E1,A1,I,L1,theta1,u1); 
» f2=PlaneTrussElementForce(E2,A2,L2,theta2,u2); 
» PlaneFrameElementAxialDiagram(f1,L1); 
» PlaneFrameElementShearDiagram(f1,L1); 
» PlaneFrameElementMomentDiagram(f1,L1); 
 
Problem 9.1: 
 
» E=210e6; 
» G=84e6; 
» I=20e-5; 
» J=5e-5; 
» L1=GridElementLength(4,0,0,3); 
» L2=GridElementLength(4,0,0,-3); 
» theta1=180+atan(3/4)*180/pi; 
» theta2=180-atan(3/4)*180/pi; 
» k1=GridElementStiffness(E,G,I,J,L1,theta1); 
» k2=GridElementStiffness(E,G,I,J,L2,theta2); 
» K=zeros(9,9); 
» K=GridAssemble(K,k1,1,2); 
» K=GridAssemble(K,k2,1,3) 
 
K = 
 
 1.0e+004 * 
 
 Columns 1 through 7 
 
 0.8064 0.0000 -1.6128 -0.4032 0.6048 -0.8064 -0.4032 
 0.0000 2.5267 0.0000 -0.6048 0.5510 -0.8467 0.6048 
 -1.6128 0.0000 4.3613 0.8064 -0.8467 1.0450 0.8064 
 -0.4032 -0.6048 0.8064 0.4032 -0.6048 0.8064 0 
 0.6048 0.5510 -0.8467 -0.6048 1.2634 -1.5725 0 
 -0.8064 -0.8467 1.0450 0.8064-1.5725 2.1806 0 
 -0.4032 0.6048 0.8064 0 0 0 0.4032 
 -0.6048 0.5510 0.8467 0 0 0 0.6048 
 -0.8064 0.8467 1.0450 0 0 0 0.8064 
 
 Columns 8 through 9 
 
 -0.6048 -0.8064 
 0.5510 0.8467 
 0.8467 1.0450 
 0 0 
 0 0 
 0 0 
 0.6048 0.8064 
 1.2634 1.5725 
 1.5725 2.1806 
 
» k=K(1:3,1:3); 
» f=[-10 ; 0 ; 0]; 
» u=k\f 
 
u = 
 
 -0.0048 
 0.0000 
 -0.0018 
 
» U=[u ; 0 ; 0 ; 0 ; 0 ; 0 ; 0]; 
» F=K*U 
 
F = 
 
 -10.0000 
 0.0000 
 0 
 5.0000 
 -13.8905 
 20.0000 
 5.0000 
 13.8905 
 20.0000 
 
» u1=[U(1) ; U(2) ; U(3) ; U(4) ; U(5) ; U(6)]; 
» u2=[U(1) ; U(2) ; U(3) ; U(7) ; U(8) ; U(9)]; 
» f1=GridElementForces(E,G,I,J,L1,theta1,u1); 
» f2=GridElementForces(E,G,I,J,L2,theta2,u2); 
 
Problem 10.1: 
 
» E=210e6; 
» G=84e6; 
» A=2e-2; 
» Iy=10e-5; 
» Iz=20e-5; 
» J=5e-5; 
» k1=SpaceFrameElementStiffness(E,G,A,Iy,Iz,J,0,0,0,0,5,0); 
» k2=SpaceFrameElementStiffness(E,G,A,Iy,Iz,J,0,0,4,0,5,4); 
» k3=SpaceFrameElementStiffness(E,G,A,Iy,Iz,J,4,0,4,4,5,4); 
» k4=SpaceFrameElementStiffness(E,G,A,Iy,Iz,J,4,0,0,4,5,0); 
» k5=SpaceFrameElementStiffness(E,G,A,Iy,Iz,J,0,5,0,0,5,4); 
» k6=SpaceFrameElementStiffness(E,G,A,Iy,Iz,J,0,5,4,4,5,4); 
» k7=SpaceFrameElementStiffness(E,G,A,Iy,Iz,J,4,5,4,4,5,0); 
» k8=SpaceFrameElementStiffness(E,G,A,Iy,Iz,J,0,5,0,4,5,0); 
» K=zeros(48,48); 
» K=SpaceFrameAssemble(K,k1,1,5); 
» K=SpaceFrameAssemble(K,k2,2,6); 
» K=SpaceFrameAssemble(K,k3,3,7); 
» K=SpaceFrameAssemble(K,k4,4,8); 
» K=SpaceFrameAssemble(K,k5,5,6); 
» K=SpaceFrameAssemble(K,k6,6,7); 
» K=SpaceFrameAssemble(K,k7,7,8); 
» K=SpaceFrameAssemble(K,k8,5,8); 
» k=K(25:48,25:48); 
» f=[0;0;0;0;0;0;0;0;0;0;0;0;-15;0;0;0;0;0;0;0;0;0;0;0]; 
» u=k\f 
 
u = 
 
 -0.0004 
 0.0000 
 -0.0006 
 0.0000 
 -0.0004 
 0.0000 
 -0.0021 
 0.0000 
 -0.0006 
 0.0000 
 -0.0004 
 0.0002 
 -0.0021 
 0.0000 
 0.0006 
 0.0000 
 -0.0004 
 0.0002 
 -0.0004 
 0.0000 
 0.0006 
 0.0000 
 -0.0004 
 0.0000 
 
» U=[0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;u]; 
» F=K*U 
 
F = 
 
 1.1599 
 2.5054 
 1.0091 
 2.6719 
 0.3008 
 -3.2737 
 6.3324 
 5.7484 
 1.0091 
 2.6719 
 0.3008 
 -17.6937 
 6.3481 
 -5.7484 
 -1.0091 
 -2.6719 
 0.3019 
 -17.7439 
 1.1596 
 -2.5054 
 -1.0091 
 -2.6719 
 0.3019 
 -3.2733 
 0 
 0.0000 
 0 
 0.0000 
 0.0000 
 0.0000 
 0 
 0.0000 
 0 
 0.0000 
 0.0000 
 0.0000 
 -15.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0 
 0 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 
» 
u1=[U(1);U(2);U(3);U(4);U(5);U(6);U(25);U(26);U(27);U(28);U(29);U(30)
]; 
» 
u2=[U(7);U(8);U(9);U(10);U(11);U(12);U(31);U(32);U(33);U(34);U(35);U(
36)]; 
» 
u3=[U(13);U(14);U(15);U(16);U(17);U(18);U(37);U(38);U(39);U(40);U(41)
;U(42)]; 
» 
u4=[U(19);U(20);U(21);U(22);U(23);U(24);U(43);U(44);U(45);U(46);U(47)
;U(48)]; 
» 
u5=[U(25);U(26);U(27);U(28);U(29);U(30);U(31);U(32);U(33);U(34);U(35)
;U(36)]; 
» 
u6=[U(31);U(32);U(33);U(34);U(35);U(36);U(37);U(38);U(39);U(40);U(41)
;U(42)]; 
» 
u7=[U(37);U(38);U(39);U(40);U(41);U(42);U(43);U(44);U(45);U(46);U(47)
;U(48)]; 
» 
u8=[U(25);U(26);U(27);U(28);U(29);U(30);U(43);U(44);U(45);U(46);U(47)
;U(48)]; 
» f1=SpaceFrameElementForces(E,G,A,Iy,Iz,J,0,0,0,0,5,0,u1); 
» f2=SpaceFrameElementForces(E,G,A,Iy,Iz,J,0,0,4,0,5,4,u2); 
» f3=SpaceFrameElementForces(E,G,A,Iy,Iz,J,4,0,4,4,5,4,u3); 
» f4=SpaceFrameElementForces(E,G,A,Iy,Iz,J,4,0,0,4,5,0,u4); 
» f5=SpaceFrameElementForces(E,G,A,Iy,Iz,J,0,5,0,0,5,4,u5); 
» f6=SpaceFrameElementForces(E,G,A,Iy,Iz,J,0,5,4,4,5,4,u6); 
» f7=SpaceFrameElementForces(E,G,A,Iy,Iz,J,4,5,4,4,5,0,u7); 
» f8=SpaceFrameElementForces(E,G,A,Iy,Iz,J,0,5,0,4,5,0,u8); 
 
Problem 11.1: 
 
» E=210e6; 
» NU=0.3; 
» t=0.025; 
» k1=LinearTriangleElementStiffness(E,NU,t,0,0,0.25,0.125,0,0.25,1); 
» k2=LinearTriangleElementStiffness(E,NU,t,0,0,0.5,0,0.25,0.125,1); 
» 
k3=LinearTriangleElementStiffness(E,NU,t,0.5,0.25,0,0.25,0.25,0.125,1
); 
» 
k4=LinearTriangleElementStiffness(E,NU,t,0.5,0,0.5,0.25,0.25,0.125,1)
; 
» K=zeros(10,10); 
» K=LinearTriangleAssemble(K,k1,1,5,4); 
» K=LinearTriangleAssemble(K,k2,1,2,5); 
» K=LinearTriangleAssemble(K,k3,3,4,5); 
» K=LinearTriangleAssemble(K,k4,2,3,5) 
 
K = 
 
 1.0e+007 * 
 
 Columns 1 through 7 
 
 0.3462 0.1875 0.0288 -0.0072 0 0 -0.0288 
 0.1875 0.6274 0.0072 0.2632 0 0 -0.0072 
 0.0288 0.0072 0.3462 -0.1875 -0.0288 -0.0072 0 
 -0.0072 0.2632 -0.1875 0.6274 0.0072 -0.2632 0 
 0 0 -0.0288 0.0072 0.3462 0.1875 0.0288 
 0 0 -0.0072 -0.2632 0.1875 0.6274 0.0072 
 -0.0288 -0.0072 0 0 0.0288 0.0072 0.3462 
 0.0072 -0.2632 0 0 -0.0072 0.2632 -0.1875 
 -0.3462 -0.1875 -0.3462 0.1875 -0.3462 -0.1875 -0.3462 
 -0.1875 -0.6274 0.1875 -0.6274 -0.1875 -0.6274 0.1875 
 
 Columns 8 through 10 
 
 0.0072 -0.3462 -0.1875 
 -0.2632 -0.1875 -0.6274 
 0 -0.3462 0.1875 
 0 0.1875 -0.6274 
 -0.0072 -0.3462 -0.1875 
 0.2632 -0.1875 -0.6274 
 -0.1875 -0.3462 0.1875 
 0.6274 0.1875 -0.6274 
 0.1875 1.3846 0 
 -0.6274 0 2.5096 
 
» k=[K(3:6,3:6) K(3:6,9:10) ; K(9:10,3:6) K(9:10,9:10)]; 
» f=[9.375 ; 0 ; 9.375 ; 0 ; 0 ; 0]; 
» u=k\f 
 
u = 
 
 1.0e-005 * 
 
 0.6928 
 0.0714 
 0.6928 
 -0.0714 
 0.3271 
 0.0000 
 
» U=[0;0;u(1:4);0;0;u(5:6)]; 
» F=K*U 
 
F = 
 
 -9.3750 
 -3.7540 
 9.3750 
 0.0000 
 9.3750 
 0.0000 
 -9.3750 
 3.7540 
 0 
 0.0000 
 
» u1=[U(1) ; U(2) ; U(9) ; U(10) ; U(7) ; U(8)]; 
» u2=[U(1) ; U(2) ; U(3) ; U(4) ; U(9) ; U(10)]; 
» u3=[U(5) ; U(6) ; U(7) ; U(8) ; U(9) ; U(10)]; 
» u4=[U(3) ; U(4) ; U(5) ; U(6) ; U(9) ; U(10)]; 
sig1=LinearTriangleElementStresses(E,NU,t,0,0,0.25,0.125,0,0.25,1,u1)
; 
» 
sig2=LinearTriangleElementStresses(E,NU,t,0,0,0.5,0,0.25,0.125,1,u2); 
» 
sig3=LinearTriangleElementStresses(E,NU,t,0.5,0.25,0,0.25,0.25,0.125,
1,u3); 
» 
sig4=LinearTriangleElementStresses(E,NU,t,0.5,0,0.5,0.25,0.25,0.125,1
,u4); 
» s1=LinearTriangleElementPStresses(sig1); 
» s2=LinearTriangleElementPStresses(sig2); 
» s3=LinearTriangleElementPStresses(sig3); 
» s4=LinearTriangleElementPStresses(sig4); 
 
Problem 11.2: 
 
» E=70e6; 
» NU=0.25; 
» t=0.02; 
» k1=LinearTriangleElementStiffness(E,NU,t,0,0,0.3,0.3,0,0.3,1); 
» k2=LinearTriangleElementStiffness(E,NU,t,0,0,0.3,0,0.3,0.3,1); 
» k3=LinearTriangleElementStiffness(E,NU,t,0.3,0,0.6,0.3,0.3,0.3,1); 
» k4=LinearTriangleElementStiffness(E,NU,t,0.3,0,0.6,0,0.6,0.3,1); 
» k5=LinearTriangleElementStiffness(E,NU,t,0.6,0,0.9,0.3,0.6,0.3,1); 
» k6=LinearTriangleElementStiffness(E,NU,t,0.6,0,0.9,0,0.9,0.3,1); 
» k7=LinearTriangleElementStiffness(E,NU,t,0,0.3,0.3,0.6,0,0.6,1); 
» k8=LinearTriangleElementStiffness(E,NU,t,0,0.3,0.3,0.3,0.3,0.6,1); 
» 
k9=LinearTriangleElementStiffness(E,NU,t,0.6,0.3,0.9,0.6,0.6,0.6,1); 
» 
k10=LinearTriangleElementStiffness(E,NU,t,0.6,0.3,0.9,0.3,0.9,0.6,1); 
» k11=LinearTriangleElementStiffness(E,NU,t,0,0.6,0.3,0.9,0,0.9,1); 
» k12=LinearTriangleElementStiffness(E,NU,t,0,0.6,0.3,0.6,0.3,0.9,1); 
» 
k13=LinearTriangleElementStiffness(E,NU,t,0.3,0.6,0.6,0.9,0.3,0.9,1); 
» 
k14=LinearTriangleElementStiffness(E,NU,t,0.3,0.6,0.6,0.6,0.6,0.9,1); 
» 
k15=LinearTriangleElementStiffness(E,NU,t,0.6,0.6,0.9,0.9,0.6,0.9,1); 
» 
k16=LinearTriangleElementStiffness(E,NU,t,0.6,0.6,0.9,0.6,0.9,0.9,1); 
K=zeros(32,32); 
K=LinearTriangleAssemble(K,k1,1,6,5); 
K=LinearTriangleAssemble(K,k2,1,2,6); 
K=LinearTriangleAssemble(K,k3,2,7,6); 
K=LinearTriangleAssemble(K,k4,2,3,7); 
K=LinearTriangleAssemble(K,k5,3,8,7);K=LinearTriangleAssemble(K,k6,3,4,8); 
K=LinearTriangleAssemble(K,k7,5,10,9); 
K=LinearTriangleAssemble(K,k8,5,6,10); 
K=LinearTriangleAssemble(K,k9,7,12,11); 
K=LinearTriangleAssemble(K,k10,7,8,12); 
K=LinearTriangleAssemble(K,k11,9,14,13); 
K=LinearTriangleAssemble(K,k12,9,10,14); 
K=LinearTriangleAssemble(K,k13,10,15,14); 
K=LinearTriangleAssemble(K,k14,10,11,15); 
K=LinearTriangleAssemble(K,k15,11,16,15); 
» K=LinearTriangleAssemble(K,k16,11,12,16) 
 
K = 
 
 1.0e+006 * 
 
 Columns 1 through 7 
 
 1.0267 0 -0.7467 0.1867 0 0 0 
 0 1.0267 0.2800 -0.2800 0 0 0 
 -0.7467 0.2800 2.0533 -0.4667 -0.7467 0.1867 0 
 0.1867 -0.2800 -0.4667 2.0533 0.2800 -0.2800 0 
 0 0 -0.7467 0.2800 2.0533 -0.4667 -0.7467 
 0 0 0.1867 -0.2800 -0.4667 2.0533 0.2800 
 0 0 0 0 -0.7467 0.2800 1.0267 
 0 0 0 0 0.1867 -0.2800 -0.4667 
 -0.2800 0.1867 0 0 0 0 0 
 0.2800 -0.7467 0 0 0 0 0 
 0 -0.4667 -0.5600 0.4667 0 0 0 
 -0.4667 0 0.4667 -1.4933 0 0 0 
 0 0 0 -0.4667 -0.5600 0.4667 0 
 0 0 -0.4667 0 0.4667 -1.4933 0 
 0 0 0 0 0 -0.4667 -0.2800 
 0 0 0 0 -0.4667 0 0.1867 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 
 Columns 8 through 14 
 
 0 -0.2800 0.2800 0 -0.4667 0 0 
 0 0.1867 -0.7467 -0.4667 0 0 0 
 0 0 0 -0.5600 0.4667 0 -0.4667 
 0 0 0 0.4667 -1.4933 -0.4667 0 
 0.1867 0 0 0 0 -0.5600 0.4667 
 -0.2800 0 0 0 0 0.4667 -1.4933 
 -0.4667 0 0 0 0 0 0 
 1.0267 0 0 0 0 0 0 
 0 2.0533 -0.4667 -1.4933 0.4667 0 0 
 0 -0.4667 2.0533 0.4667 -0.5600 0 0 
 0 -1.4933 0.4667 3.0800 -0.9333 -0.7467 0.2800 
 0 0.4667 -0.5600 -0.9333 3.0800 0.1867 -0.2800 
 0 0 0 -0.7467 0.1867 3.0800 -0.4667 
 0 0 0 0.2800 -0.2800 -0.4667 3.0800 
 0.2800 0 0 0 0 -1.4933 0.4667 
 -0.7467 0 0 0 0 0.4667 -0.5600 
 0 -0.2800 0.1867 0 0 0 0 
 0 0.2800 -0.7467 0 0 0 0 
 0 0 -0.4667 -0.2800 0.2800 0 0 
 0 -0.4667 0 0.1867 -0.7467 0 0 
 0 0 0 0 0 -0.2800 0.1867 
 0 0 0 0 0 0.2800 -0.7467 
 0 0 0 0 0 0 -0.4667 
 0 0 0 0 0 -0.4667 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 
 Columns 15 through 21 
 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 -0.4667 0 0 0 0 0 
 -0.4667 0 0 0 0 0 0 
 -0.2800 0.1867 0 0 0 0 0 
 0.2800 -0.7467 0 0 0 0 0 
 0 0 -0.2800 0.2800 0 -4667 0 
 0 0 0.1867 -0.7467 -0.4667 0 0 
 0 0 0 0 -0.2800 0.1867 0 
 0 0 0 0 0.2800 -0.7467 0 
 -1.4933 0.4667 0 0 0 0 -0.2800 
 0.4667 -0.5600 0 0 0 0 0.1867 
 2.0533 -0.4667 0 0 0 0 0 
 -0.4667 2.0533 0 0 0 0 0 
 0 0 2.0533 -0.4667 -1.4933 0.4667 0 
 0 0 -0.4667 2.0533 0.4667 -0.5600 0 
 0 0 -1.4933 0.4667 3.0800 -0.4667 -0.7467 
 0 0 0.4667 -0.5600 -0.4667 3.0800 0.2800 
 0 0 0 0 -0.7467 0.2800 3.0800 
 0 0 0 0 0.1867 -0.2800 -0.9333 
 -0.2800 0.2800 0 0 0 0 -1.4933 
 0.1867 -0.7467 0 0 0 0 0.4667 
 0 0 -0.2800 0.1867 0 0 0 
 0 0 0.2800 -0.7467 0 0 0 
 0 0 0 -0.4667 -0.5600 0.4667 0 
 0 0 -0.4667 0 0.4667 -1.4933 0 
 0 0 0 0 0 -0.4667 -0.5600 
 0 0 0 0 -0.4667 0 0.4667 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 -0.4667 
 
 Columns 22 through 28 
 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 00 0 0 0 0 
 0 0 0 0 0 0 0 
 0.2800 0 -0.4667 0 0 0 0 
 -0.7467 -0.4667 0 0 0 0 0 
 0 -0.2800 0.1867 0 0 0 0 
 0 0.2800 -0.7467 0 0 0 0 
 0 0 0 -0.2800 0.2800 0 -0.4667 
 0 0 0 0.1867 -0.7467 -0.4667 0 
 0.1867 0 0 0 0 -0.5600 0.4667 
 -0.2800 0 0 0 0 0.4667 -1.4933 
 -0.9333 -1.4933 0.4667 0 0 0 0 
 3.0800 0.4667 -0.5600 0 0 0 0 
 0.4667 2.0533 -0.4667 0 0 0 0 
 -0.5600 -0.4667 2.0533 0 0 0 0 
 0 0 0 1.0267 -0.4667 -0.7467 0.2800 
 0 0 0 -0.4667 1.0267 0.1867 -0.2800 
 0 0 0 -0.7467 0.1867 2.0533 -0.4667 
 0 0 0 0.2800 -0.2800 -0.4667 2.0533 
 0.4667 0 0 0 0 -0.7467 0.1867 
 -1.4933 0 0 0 0 0.2800 -0.2800 
 -0.4667 -0.2800 0.2800 0 0 0 0 
 0 0.1867 -0.7467 0 0 0 0 
 
 Columns 29 through 32 
 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 -0.4667 0 0 
 -0.4667 0 0 0 
 -0.5600 0.4667 0 -0.4667 
 0.4667 -1.4933 -0.4667 0 
 0 0 -0.2800 0.1867 
 0 0 0.2800 -0.7467 
 0 0 0 0 
 0 0 0 0 
 -0.7467 0.2800 0 0 
 0.1867 -0.2800 0 0 
 2.0533 -0.4667 -0.7467 0.2800 
 -0.4667 2.0533 0.1867 -0.2800 
 -0.7467 0.1867 1.0267 0 
 0.2800 -0.2800 0 1.0267 
 
» k=[K(3:8,3:8) K(3:8,11:16) K(3:8,19:24) K(3:8,27:32) ; K(11:16,3:8) 
K(11:16,11:16) K(11:16,19:24) K(11:16,27:32) ; K(19:24,3:8) 
K(19:24,11:16) K(19:24,19:24) K(19:24,27:32) ; K(27:32,3:8) 
K(27:32,11:16) K(27:32,19:24) K(27:32,27:32)]; 
» f=[0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;-20]; 
» u=k\f 
 
u = 
 
 1.0e-003 * 
 
 -0.0200 
 -0.0225 
 -0.0291 
 -0.0581 
 -0.0305 
 -0.0854 
 -0.0008 
 -0.0173 
 -0.0072 
 -0.0585 
 -0.0077 
 -0.0867 
 0.0001 
 -0.0176 
 0.0010 
 -0.0639 
 0.0064 
 -0.0960 
 0.0207 
 -0.0199 
 0.0346 
 -0.0635 
 0.0356 
 -0.1167 
 
» U=[0;0;u(1:6);0;0;u(7:12);0;0;u(13:18);0;0;u(19:24)]; 
» F=K*U 
 
F = 
 
 18.8054 
 1.0788 
 0 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 1.3366 
 9.2538 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.9105 
 0.2247 
 0 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 -21.0525 
 9.4427 
 0.0000 
 0 
 0 
 0.0000 
 0.0000 
 -20.0000 
 
Problem 11.3: 
 
» E=200e6; 
» NU=0.3; 
» t=0.01; 
» k1=LinearTriangleElementStiffness(E,NU,t,0,0.4,0,0,0.7,0.4,1); 
» k2=LinearTriangleElementStiffness(E,NU,t,0.7,0.4,0,0,0.7,0,1); 
» k3=SpringElementStiffness(4000); 
» k4=SpringElementStiffness(4000); 
» K=zeros(10,10); 
» K=LinearTriangleAssemble(K,k1,1,3,2); 
» K=LinearTriangleAssemble(K,k2,2,3,4); 
» K=SpringAssemble(K,k3,6,9); 
» K=SpringAssemble(K,k4,8,10) 
 
K = 
 
 1.0e+006 * 
 
 Columns 1 through 7 
 
 1.3010 -0.7143 -0.6279 0.3846 -0.6731 0.3297 0 
 -0.7143 2.1429 0.3297 -0.2198 0.3846 -1.9231 0 
 -0.6279 0.3297 1.3010 0 0 -0.7143 -0.6731 
 0.3846 -0.2198 0 2.1429 -0.7143 0 0.3297 
 -0.6731 0.3846 0 -0.7143 1.3010 0 -0.6279 
 0.3297 -1.9231 -0.7143 0 0 2.1469 0.3846 
 0 0 -0.6731 0.3297 -0.6279 0.3846 1.3010 
 0 0 0.3846 -1.9231 0.3297 -0.2198 -0.7143 
 0 0 0 0 0 -0.0040 0 
 0 0 0 0 0 0 0 
 
 Columns 8 through 10 
 
 0 0 0 
 0 0 0 
 0.3846 0 0 
 -1.9231 0 0 
 0.3297 0 0 
 -0.2198 -0.0040 0 
 -0.7143 0 0 
 2.1469 0 -0.0040 
 0 0.0040 0 
 -0.0040 0 0.0040 
 
» k=K(1:8,1:8); 
» f=[0 ; 17.5 ; 0 ; 17.5 ; 0 ; 0 ; 0 ; 0]; 
» u=k\f 
 
Warning: Matrix is close to singular or badly scaled. 
 Results may be inaccurate. RCOND = 2.544804e-017. 
 
u = 
 
 0.0002 
 0.0044 
 0.0002 
 0.0044 
 0.0002 
 0.0044 
 0.0002 
 0.0044 
 
» U=[u(1:8) ; 0 ; 0]; 
» F=K*U 
 
F = 
 
 0 
 17.5000 
 0 
 17.5000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 -17.5000 
 -17.5000 
 
» u1=[U(1) ; U(2) ; U(5) ; U(6) ; U(3) ; U(4)]; 
» u2=[U(3) ; U(4) ; U(5) ; U(6) ; U(7) ; U(8)]; 
» u3=[U(6) ; U(9)]; 
» u4=[U(8) ; U(10)]; 
» 
sigma1=LinearTriangleElementStresses(E,NU,t,0,0.4,0,0,0.7,0.4,1,u1); 
» 
sigma2=LinearTriangleElementStresses(E,NU,t,0.7,0.4,0,0,0.7,0,1,u2); 
» s1=LinearTriangleElementPStresses(sigma1); 
» s2=LinearTriangleElementPStresses(sigma2); 
» f3=SpringElementForces(k3,u3); 
» f4=SpringElementForces(k4,u4); 
 
Problem 12.1: 
 
» E=210e6; 
» NU=0.3; 
» t=0.025; 
» k1=QuadTriangleElementStiffness(E,NU,t,0,0,0.25,0.125,0,0.25,1); 
» k2=QuadTriangleElementStiffness(E,NU,t,0,0,0.5,0,0.25,0.125,1); 
» 
k3=QuadTriangleElementStiffness(E,NU,t,0.25,0.125,0.5,0.25,0,0.25,1); 
» 
k4=QuadTriangleElementStiffness(E,NU,t,0.25,0.125,0.5,0,0.5,0.25,1); 
» K=zeros(26,26); 
» K=QuadTriangleAssemble(K,k1,1,7,11,4,9,6); 
» K=QuadTriangleAssemble(K,k2,1,3,7,2,5,4); 
» K=QuadTriangleAssemble(K,k3,7,13,11,10,12,9); 
» K=QuadTriangleAssemble(K,k4,7,3,13,5,8,10) 
 
K = 
 
 1.0e+007 * 
 
 Columns 1 through 7 
 
 0.3462 0.1875 0.0385 -0.0096 -0.0096 0.0024 -0.4615 
 0.1875 0.6274 0.0096 0.3510 -0.0024 -0.0877 -0.2500 
 0.0385 0.0096 1.0000 0 0.0385 -0.0096 -0.5385 
 -0.0096 0.3510 0 2.3750 0.0096 0.3510 0.2500 
 -0.0096 -0.0024 0.0385 0.0096 0.3462 -0.1875 0 
 0.0024 -0.0877 -0.0096 0.3510 -0.1875 0.6274 0 
 -0.4615 -0.2500 -0.5385 0.2500 0 0 1.8462 
 -0.2500 -0.8365 0.2500 -1.5385 0 0 0 
 0 0 -0.5385 -0.2500 -0.4615 0.2500 0.0769 
 0 0 -0.2500 -1.5385 0.2500 -0.8365 0 
 -0.0385 -0.0096 0 0 0 0 -0.3846 
 0.0096 -0.3510 0 0 0 0 0.2500 
 0.1154 0.0625 0 0 0.1154 -0.0625 -0.4615 
 0.0625 0.2091 0 0 -0.0625 0.2091 -0.2500 
 0 0 0 0 -0.0385 0.00960 
 0 0 0 0 -0.0096 -0.3510 0 
 0 0 0 0 0 0 -0.0769 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0.0096 0.0024 0 0 0 0 0 
 -0.0024 0.0877 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0.0096 -0.0024 0 
 0 0 0 0 0.0024 0.0877 0 
 
 Columns 8 through 14 
 
 -0.2500 0 0 -0.0385 0.0096 0.1154 0.0625 
 -0.8365 0 0 -0.0096 -0.3510 0.0625 0.2091 
 0.2500 -0.5385 -0.2500 0 0 0 0 
 -1.5385 -0.2500 -1.5385 0 0 0 0 
 0 -0.4615 0.2500 0 0 0.1154 -0.0625 
 0 0.2500 -0.8365 0 0 -0.0625 0.2091 
 0 0.0769 0 -0.3846 0.2500 -0.4615 -0.2500 
 3.3462 0 0.7019 0.2500 -0.1346 -0.2500 -0.8365 
 0 1.8462 0 0 0 -0.4615 0.2500 
 0.7019 0 3.3462 0 0 0.2500 -0.8365 
 0.2500 0 0 0.8462 0 0 0 
 -0.1346 0 0 0 0.9712 0 0 
 -0.2500 -0.4615 0.2500 0 0 1.3846 0 
 -0.8365 0.2500 -0.8365 0 0 0 2.5096 
 0 -0.3846 -0.2500 0 0 0 0 
 0 -0.2500 -0.1346 0 0 0 0 
 0 0 0 -0.3846 -0.2500 -0.4615 0.2500 
 -0.7019 0 0 -0.2500 -0.1346 0.2500 -0.8365 
 0 -0.0769 0 0 0 -0.4615 -0.2500 
 0 0 -0.7019 0 0 -0.2500 -0.8365 
 0 0 0 -0.0385 -0.0096 0.1154 -0.0625 
 0 0 0 0.0096 -0.3510 -0.0625 0.2091 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0.1154 0.0625 
 0 0 0 0 0 0.0625 0.2091 
 
 Columns 15 through 21 
 
 0 0 0 0 0 0 0.0096 
 0 0 0 0 0 0 0.0024 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 -0.0385 -0.0096 0 0 0 0 0 
 0.0096 -0.3510 0 0 0 0 0 
 0 0 -0.0769 0 0 0 0 
 0 0 0 -0.7019 0 0 0 
 -0.3846 -0.2500 0 0 -0.0769 0 0 
 -0.2500 -0.1346 0 0 0 -0.7019 0 
 0 0 -0.3846 -0.2500 0 0 -0.0385 
 0 0 -0.2500 -0.1346 0 0 -0.0096 
 0 0 -0.4615 0.2500 -0.4615 -0.2500 0.1154 
 0 0 0.2500 -0.8365 -0.2500 -0.8365 -0.0625 
 0.8462 0 0 0 -0.3846 0.2500 0 
 0 0.9712 0 0 0.2500 -0.1346 0 
 0 0 1.8462 0 0.0769 0 -0.4615 
 0 0 0 3.3462 0 0.7019 0.2500 
 -0.3846 0.2500 0.0769 0 1.8462 0 0 
 0.2500 -0.1346 0 0.7019 0 3.3462 0 
 0 0 -0.4615 0.2500 0 0 0.3462 
 0 0 0.2500 -0.8365 0 0 -0.1875 
 0 0 -0.5385 -0.2500 -0.5385 0.2500 0.0385 
 0 0 -0.2500 -1.5385 0.2500 -1.5385 0.0096 
 -0.0385 0.0096 0 0 -0.4615 -0.2500 -0.0096 
 -0.0096 -0.3510 0 0 -0.2500 -0.8365 -0.0024 
 
 Columns 22 through 26 
 
 -0.0024 0 0 0 0 
 0.0877 0 0 0 0 
 0 0 0 0 0 
 0 0 0 0 0 
 0 0 0 0.0096 0.0024 
 0 0 0 -0.0024 0.0877 
 0 0 0 0 0 
 0 0 0 0 0 
 0 0 0 0 0 
 0 0 0 0 0 
 0.0096 0 0 0 0 
 -0.3510 0 0 0 0 
 -0.0625 0 0 0.1154 0.0625 
 0.2091 0 0 0.0625 0.2091 
 0 0 0 -0.0385 -0.0096 
 0 0 0 0.0096 -0.3510 
 0.2500 -0.5385 -0.2500 0 0 
 -0.8365 -0.2500 -1.5385 0 0 
 0 -0.5385 0.2500 -0.4615 -0.2500 
 0 0.2500 -1.5385 -0.2500 -0.8365 
 -0.1875 0.0385 0.0096 -0.0096 -0.0024 
 0.6274 -0.0096 0.3510 0.0024 -0.0877 
 -0.0096 1.0000 0 0.0385 0.0096 
 0.3510 0 2.3750 -0.0096 0.3510 
 0.0024 0.0385 -0.0096 0.3462 0.1875 
 -0.0877 0.0096 0.3510 0.1875 0.6274 
 
» k=[K(3:10,3:10) K(3:10,13:20) K(3:10,23:26) ; K(13:20,3:10) 
K(13:20,13:20) K(13:20,23:26) ; K(23:26,3:10) K(23:26,13:20) 
K(23:26,23:26)]; 
» f=[0 ; 0 ; 3.125 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 12.5 ; 0 ; 0 ; 0 ; 0 
; 0 ; 0 ; 0 ; 3.125 ; 0]; 
» u=k\f 
 
u = 
 
 1.0e-005 * 
 
 0.3500 
 0.0590 
 0.7006 
 0.0415 
 0.1653 
 0.0172 
 0.5286 
 0.0288 
 0.3454 
 0.0000 
 0.7080 
 0.0000 
 0.1653 
 -0.0172 
 0.5286 
 -0.0288 
 0.3500 
 -0.0590 
 0.7006 
 -0.0415 
 
» U=[0;0;u(1:8);0;0;u(9:16);0;0;u(17:20)]; 
» F=K*U 
 
F = 
 
 -3.4469 
 -1.5335 
 0 
 0.0000 
 3.1250 
 0.0000 
 0 
 0.0000 
 0.0000 
 0.0000 
 -11.8562 
 0.0000 
 0.0000 
 0.0000 
 12.5000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0 
 -3.4469 
 1.5335 
 0.0000 
 0.0000 
 3.1250 
 0 
 
» u1=[U(1) ; U(2) ; U(13) ; U(14) ; U(21) ; U(22) ; U(7) ; U(8) ; 
U(17) ; U(18) ; U(11) ; U(12)]; 
» u2=[U(1) ; U(2) ; U(5) ; U(6) ; U(13) ; U(14) ; U(3) ; U(4) ; U(9) 
; U(10) ; U(7) ; U(8)]; 
» u3=[U(13) ; U(14) ; U(25) ; U(26) ; U(21) ; U(22) ; U(19) ; U(20) ; 
U(23) ; U(24) ; U(17) ; U(18)]; 
» u4=[U(13) ; U(14) ; U(5) ; U(6) ; U(25) ; U(26) ; U(9) ; U(10) ; 
U(15) ; U(16) ; U(19) ; U(20)]; 
» 
sigma1=QuadTriangleElementStresses(E,NU,t,0,0,0.25,0.125,0,0.25,1,u1)
; 
» 
sigma2=QuadTriangleElementStresses(E,NU,t,0,0,0.5,0,0.25,0.125,1,u2); 
» 
sigma3=QuadTriangleElementStresses(E,NU,t,0.25,0.125,0.5,0.25,0,0.25,
1,u3); 
» 
sigma4=QuadTriangleElementStresses(E,NU,t,0.25,0.125,0.5,0,0.5,0.25,1
,u4); 
» s1=QuadTriangleElementPStresses(sigma1); 
» s2=QuadTriangleElementPStresses(sigma2); 
» s3=QuadTriangleElementPStresses(sigma3); 
» s4=QuadTriangleElementPStresses(sigma4); 
 
Problem 13.1: 
 
» E=210e6; 
» NU=0.3; 
» h=0.025; 
» 
k1=BilinearQuadElementStiffness(E,NU,h,0,0,0.125,0,0.125,0.125,0,0.12
5,1); 
» 
k2=BilinearQuadElementStiffness(E,NU,h,0.125,0,0.25,0,0.25,0.125,0.12
5,0.125,1); 
» 
k3=BilinearQuadElementStiffness(E,NU,h,0.25,0,0.375,0,0.375,0.125,0.2
5,0.125,1); 
» 
k4=BilinearQuadElementStiffness(E,NU,h,0.375,0,0.5,0,0.5,0.125,0.375,
0.125,1); 
» 
k5=BilinearQuadElementStiffness(E,NU,h,0,0.125,0.125,0.125,0.125,0.25,0,0.25,1); 
» 
k6=BilinearQuadElementStiffness(E,NU,h,0.125,0.125,0.25,0.125,0.25,0.
25,0.125,0.25,1); 
» 
k7=BilinearQuadElementStiffness(E,NU,h,0.25,0.125,0.375,0.125,0.375,0
.25,0.25,0.25,1); 
» 
k8=BilinearQuadElementStiffness(E,NU,h,0.375,0.125,0.5,0.125,0.5,0.25
,0.375,0.25,1); 
» K=zeros(30,30); 
» K=BilinearQuadAssemble(K,k1,1,2,7,6); 
» K=BilinearQuadAssemble(K,k2,2,3,8,7); 
» K=BilinearQuadAssemble(K,k3,3,4,9,8); 
» K=BilinearQuadAssemble(K,k4,4,5,10,9); 
» K=BilinearQuadAssemble(K,k5,6,7,12,11); 
» K=BilinearQuadAssemble(K,k6,7,8,13,12); 
» K=BilinearQuadAssemble(K,k7,8,9,14,13); 
» K=BilinearQuadAssemble(K,k8,9,10,15,14); 
 
 
K = 
 
 1.0e+007 * 
 
 Columns 1 through 7 
 
 0.2596 0.0937 -0.1587 -0.0072 0 0 0 
 0.0937 0.2596 0.0072 0.0288 0 0 0 
 -0.1587 0.0072 0.5192 0 -0.1587 -0.0072 0 
 -0.0072 0.0288 0 0.5192 0.0072 0.0288 0 
 0 0 -0.1587 0.0072 0.5192 0 -0.1587 
 0 0 -0.0072 0.0288 0 0.5192 0.0072 
 0 0 0 0 -0.1587 0.0072 0.5192 
 0 0 0 0 -0.0072 0.0288 0 
 0 0 0 0 0 0 -0.1587 
 0 0 0 0 0 0 -0.0072 
 0.0288 -0.0072 -0.1298 0.0937 0 0 0 
 0.0072 -0.1587 0.0937 -0.1298 0 0 0 
 -0.1298 -0.0937 0.0577 0 -0.1298 0.0937 0 
 -0.0937 -0.1298 0 -0.3173 0.0937 -0.1298 0 
 0 0 -0.1298 -0.0937 0.0577 0 -0.1298 
 0 0 -0.0937 -0.1298 0 -0.3173 0.0937 
 0 0 0 0 -0.1298 -0.0937 0.0577 
 0 0 0 0 -0.0937 -0.1298 0 
 0 0 0 0 0 0 -0.1298 
 0 0 0 0 0 0 -0.0937 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 
 Columns 8 through 14 
 
 0 0 0 0.0288 0.0072 -0.1298 -0.0937 
 0 0 0 -0.0072 -0.1587 -0.0937 -0.1298 
 0 0 0 -0.1298 0.0937 0.0577 0 
 0 0 0 0.0937 -0.1298 0 -0.3173 
 -0.0072 0 0 0 0 -0.1298 0.0937 
 0.0288 0 0 0 0 0.0937 -0.1298 
 0 -0.1587 -0.0072 0 0 0 0 
 0.5192 0.0072 0.0288 0 0 0 0 
 0.0072 0.2596 -0.0937 0 0 0 0 
 0.0288 -0.0937 0.2596 0 0 0 0 
 0 0 0 0.5192 0 -0.3173 0 
 0 0 0 0 0.5192 0 0.0577 
 0 0 0 -0.3173 0 1.0385 0 
 0 0 0 0 0.0577 0 1.0385 
 0.0937 0 0 0 0 -0.3173 0 
 -0.1298 0 0 0 0 0 0.0577 
 0 -0.1298 0.0937 0 0 0 0 
 -0.3173 0.0937 -0.1298 0 0 0 0 
 -0.0937 0.0288 0.0072 0 0 0 0 
 -0.1298 -0.0072 -0.1587 0 0 0 0 
 0 0 0 0.0288 -0.0072 -0.1298 0.0937 
 0 0 0 0.0072 -0.1587 0.0937 -0.1298 
 0 0 0 -0.1298 -0.0937 0.0577 0 
 0 0 0 -0.0937 -0.1298 0 -0.3173 
 0 0 0 0 0 -0.1298 -0.0937 
 0 0 0 0 0 -0.0937 -0.1298 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 
 Columns 15 through 21 
 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 -0.1298 -0.0937 0 0 0 0 0 
 -0.0937 -0.1298 0 0 0 0 0 
 0.0577 0 -0.1298 -0.0937 0 0 0 
 0 -0.3173 -0.0937 -0.1298 0 0 0 
 -0.1298 0.0937 0.0577 0 -0.1298 -0.0937 0 
 0.0937 -0.1298 0 -0.3173 -0.0937 -0.1298 0 
 0 0 -0.1298 0.0937 0.0288 -0.0072 0 
 0 0 0.0937 -0.1298 0.0072 -0.1587 0 
 0 0 0 0 0 0 0.0288 
 0 0 0 0 0 0 -0.0072 
 -0.3173 0 0 0 0 0 -0.1298 
 0 0.0577 0 0 0 0 0.0937 
 1.0385 0 -0.3173 0 0 0 0 
 0 1.0385 0 0.0577 0 0 0 
 -0.3173 0 1.0385 0 -0.3173 0 0 
 0 0.0577 0 1.0385 0 0.0577 0 
 0 0 -0.3173 0 0.5192 0 0 
 0 0 0 0.0577 0 0.5192 0 
 0 0 0 0 0 0 0.2596 
 0 0 0 0 0 0 -0.0937 
 -0.1298 0.0937 0 0 0 0 -0.1587 
 0.0937 -0.1298 0 0 0 0 0.0072 
 0.0577 0 -0.1298 0.0937 0 0 0 
 0 -0.3173 0.0937 -0.1298 0 0 0 
 -0.1298 -0.0937 0.0577 0 -0.1298 0.0937 0 
 -0.0937 -0.1298 0 -0.3173 0.0937 -0.1298 0 
 0 0 -0.1298 -0.0937 0.0288 0.0072 0 
 0 0 -0.0937 -0.1298 -0.0072 -0.1587 0 
 
 Columns 22 through 28 
 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0.0072 -0.1298 -0.0937 0 0 0 0 
 -0.1587 -0.0937 -0.1298 0 0 0 0 
 0.0937 0.0577 0 -0.1298 -0.0937 0 0 
 -0.1298 0 -0.3173 -0.0937 -0.12980 0 
 0 -0.1298 0.0937 0.0577 0 -0.1298 -0.0937 
 0 0.0937 -0.1298 0 -0.3173 -0.0937 -0.1298 
 0 0 0 -0.1298 0.0937 0.0577 0 
 0 0 0 0.0937 -0.1298 0 -0.3173 
 0 0 0 0 0 -0.1298 0.0937 
 0 0 0 0 0 0.0937 -0.1298 
 -0.0937 -0.1587 0.0072 0 0 0 0 
 0.2596 -0.0072 0.0288 0 0 0 0 
 -0.0072 0.5192 0 -0.1587 0.0072 0 0 
 0.0288 0 0.5192 -0.0072 0.0288 0 0 
 0 -0.1587 -0.0072 0.5192 0 -0.1587 0.0072 
 0 0.0072 0.0288 0 0.5192 -0.0072 0.0288 
 0 0 0 -0.1587 -0.0072 0.5192 0 
 0 0 0 0.0072 0.0288 0 0.5192 
 0 0 0 0 0 -0.1587 -0.0072 
 0 0 0 0 0 0.0072 0.0288 
 
 Columns 29 through 30 
 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 -0.1298 -0.0937 
 -0.0937 -0.1298 
 0.0288 -0.0072 
 0.0072 -0.1587 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 0 0 
 -0.1587 0.0072 
 -0.0072 0.0288 
 0.2596 0.0937 
 0.0937 0.2596 
 
» k=[K(3:10,3:10) K(3:10,13:20) K(3:10,23:30) ; K(13:20,3:10) 
K(13:20,13:20) K(13:20,23:30) ; K(23:30,3:10) K(23:30,13:20) 
K(23:30,23:30)]; 
» f=[0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 4.6875 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 
9.375 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 4.6875 ; 0]; 
» u=k\f 
 
u = 
 
 1.0e-005 * 
 
 0.1768 
 0.0552 
 0.3500 
 0.0548 
 0.5284 
 0.0536 
 0.7071 
 0.0535 
 0.1648 
 0.0000 
 0.3496 
 0.0000 
 0.5287 
 0.0000 
 0.7071 
 0.0000 
 0.1768 
 -0.0552 
 0.3500 
 -0.0548 
 0.5284 
 -0.0536 
 0.7071 
 -0.0535 
 
» U=[0;0;u(1:8);0;0;u(9:16);0;0;u(17:24)]; 
» F=K*U 
 
F = 
 
 -4.9836 
 -1.2580 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 4.6875 
 0.0000 
 -8.7829 
 0.0000 
 0.0000 
 0.0000 
 0 
 0.0000 
 0.0000 
 0.0000 
 9.3750 
 0.0000 
 -4.9836 
 1.2580 
 0.0000 
 0 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 4.6875 
 0.0000 
 
» u1=[U(1) ; U(2) ; U(3) ; U(4) ; U(13) ; U(14) ; U(11) ; U(12)]; 
» u2=[U(3) ; U(4) ; U(5) ; U(6) ; U(15) ; U(16) ; U(13) ; U(14)]; 
» u3=[U(5) ; U(6) ; U(7) ; U(8) ; U(17) ; U(18) ; U(15) ; U(16)]; 
» u4=[U(7) ; U(8) ; U(9) ; U(10) ; U(19) ; U(20) ; U(17) ; U(18)]; 
» u5=[U(11) ; U(12) ; U(13) ; U(14) ; U(23) ; U(24) ; U(21) ; U(22)]; 
» u6=[U(13) ; U(14) ; U(15) ; U(16) ; U(25) ; U(26) ; U(23) ; U(24)]; 
» u7=[U(15) ; U(16) ; U(17) ; U(18) ; U(27) ; U(28) ; U(25) ; U(26)]; 
» u8=[U(17) ; U(18) ; U(19) ; U(20) ; U(29) ; U(30) ; U(27) ; U(28)]; 
» 
sigma1=BilinearQuadElementStresses(E,NU,0,0,0.125,0,0.125,0.125,0,0.1
25,1,u1); 
» 
sigma2=BilinearQuadElementStresses(E,NU,0.125,0,0.25,0,0.25,0.125,0.1
25,0.125,1,u2); 
» 
sigma3=BilinearQuadElementStresses(E,NU,0.25,0,0.375,0,0.375,0.125,0.
25,0.125,1,u3); 
» 
sigma4=BilinearQuadElementStresses(E,NU,0.375,0,0.5,0,0.5,0.125,0.375
,0.125,1,u4); 
» 
sigma5=BilinearQuadElementStresses(E,NU,0,0.125,0.125,0.125,0.125,0.2
5,0,0.25,1,u5); 
» 
sigma6=BilinearQuadElementStresses(E,NU,0.125,0.125,0.25,0.125,0.25,0
.25,0.125,0.25,1,u6); 
» 
sigma7=BilinearQuadElementStresses(E,NU,0.25,0.125,0.375,0.125,0.375,
0.25,0.25,0.25,1,u7); 
» 
sigma8=BilinearQuadElementStresses(E,NU,0.375,0.125,0.5,0.125,0.5,0.2
5,0.375,0.25,1,u8); 
» s1=BilinearQuadElementPStresses(sigma1); 
» s2=BilinearQuadElementPStresses(sigma2); 
» s3=BilinearQuadElementPStresses(sigma3); 
» s4=BilinearQuadElementPStresses(sigma4); 
» s5=BilinearQuadElementPStresses(sigma5); 
» s6=BilinearQuadElementPStresses(sigma6); 
» s7=BilinearQuadElementPStresses(sigma7); 
» s8=BilinearQuadElementPStresses(sigma8); 
 
Problem 13.2: 
 
» E=70e6; 
» NU=0.25; 
» h=0.02; 
» k1=BilinearQuadElementStiffness(E,NU,h,0,0,0.3,0,0.3,0.3,0,0.3,1); 
» 
k2=BilinearQuadElementStiffness(E,NU,h,0.3,0,0.6,0,0.6,0.3,0.3,0.3,1)
; 
» 
k3=BilinearQuadElementStiffness(E,NU,h,0.6,0,0.9,0,0.9,0.3,0.6,0.3,1)
; 
» 
k4=BilinearQuadElementStiffness(E,NU,h,0,0.3,0.3,0.3,0.3,0.6,0,0.6,1)
; 
» 
k5=BilinearQuadElementStiffness(E,NU,h,0.6,0.3,0.9,0.3,0.9,0.6,0.6,0.
6,1); 
» 
k6=BilinearQuadElementStiffness(E,NU,h,0,0.6,0.3,0.6,0.3,0.9,0,0.9,1)
; 
» 
k7=BilinearQuadElementStiffness(E,NU,h,0.3,0.6,0.6,0.6,0.6,0.9,0.3,0.
9,1); 
» 
k8=BilinearQuadElementStiffness(E,NU,h,0.6,0.6,0.9,0.6,0.9,0.9,0.6,0.
9,1); 
» K=zeros(32,32); 
» K=BilinearQuadAssemble(K,k1,1,2,6,5); 
» K=BilinearQuadAssemble(K,k2,2,3,7,6); 
» K=BilinearQuadAssemble(K,k3,3,4,8,7); 
» K=BilinearQuadAssemble(K,k4,5,6,10,9); 
» K=BilinearQuadAssemble(K,k5,7,8,12,11); 
» K=BilinearQuadAssemble(K,k6,9,10,14,13); 
» K=BilinearQuadAssemble(K,k7,10,11,15,14); 
» K=BilinearQuadAssemble(K,k8,11,12,16,15) 
 
K = 
 
 1.0e+006 * 
 
 Columns 1 through 7 
 
 0.6844 0.2333 -0.4044 -0.0467 0 0 0 
 0.2333 0.6844 0.0467 0.0622 0 0 0 
 -0.4044 0.0467 1.3689 0 -0.4044 -0.0467 0 
 -0.0467 0.0622 0 1.3689 0.0467 0.0622 0 
 0 0 -0.4044 0.0467 1.3689 0 -0.4044 
 0 0 -0.0467 0.0622 0 1.3689 0.0467 
 0 0 0 0 -0.4044 0.0467 0.6844 
 0 0 0 0 -0.0467 0.0622 -0.2333 
 0.0622 -0.0467 -0.3422 0.2333 0 0 0 
 0.0467 -0.4044 0.2333 -0.3422 0 0 0 
 -0.3422 -0.2333 0.1244 0 -0.3422 0.2333 0 
 -0.2333 -0.3422 0 -0.8089 0.2333 -0.3422 0 
 0 0 -0.3422 -0.2333 0.1244 0 -0.3422 
 0 0 -0.2333 -0.3422 0 -0.8089 0.2333 
 0 0 0 0 -0.3422 -0.2333 0.0622 
 0 0 0 0 -0.2333 -0.3422 -0.0467 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 
 Columns 8 through 14 
 
 0 0.0622 0.0467 -0.3422 -0.2333 0 0 
 0 -0.0467 -0.4044 -0.2333 -0.3422 0 0 
 0 -0.3422 0.2333 0.1244 0 -0.3422 -0.2333 
 00.2333 -0.3422 0 -0.8089 -0.2333 -0.3422 
 -0.0467 0 0 -0.3422 0.2333 0.1244 0 
 0.0622 0 0 0.2333 -0.3422 0 -0.8089 
 -0.2333 0 0 0 0 -0.3422 0.2333 
 0.6844 0 0 0 0 0.2333 -0.3422 
 0 1.3689 0 -0.8089 0 0 0 
 0 0 1.3689 0 0.1244 0 0 
 0 -0.8089 0 2.0533 -0.2333 -0.4044 0.0467 
 0 0 0.1244 -0.2333 2.0533 -0.0467 0.0622 
 0.2333 0 0 -0.4044 -0.0467 2.0533 0.2333 
 -0.3422 0 0 0.0467 0.0622 0.2333 2.0533 
 0.0467 0 0 0 0 -0.8089 0 
 -0.4044 0 0 0 0 0 0.1244 
 0 0.0622 -0.0467 -0.3422 0.2333 0 0 
 0 0.0467 -0.4044 0.2333 -0.3422 0 0 
 0 -0.3422 -0.2333 0.0622 0.0467 0 0 
 0 -0.2333 -0.3422 -0.0467 -0.4044 0 0 
 0 0 0 0 0 0.0622 -0.0467 
 0 0 0 0 0 0.0467 -0.4044 
 0 0 0 0 0 -0.3422 -0.2333 
 0 0 0 0 0 -0.2333 -0.3422 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 
 Columns 15 through 21 
 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 -0.3422 -0.2333 0 0 0 0 0 
 -0.2333 -0.3422 0 0 0 0 0 
 0.0622 -0.0467 0 0 0 0 0 
 0.0467 -0.4044 0 0 0 0 0 
 0 0 0.0622 0.0467 -0.3422 -0.2333 0 
 0 0 -0.0467 -0.4044 -0.2333 -0.3422 0 
 0 0 -0.3422 0.2333 0.0622 -0.0467 0 
 0 0 0.2333 -0.3422 0.0467 -0.4044 0 
 -0.8089 0 0 0 0 0 0.0622 
 0 0.1244 0 0 0 0 -0.0467 
 1.3689 0 0 0 0 0 -0.3422 
 0 1.3689 0 0 0 0 0.2333 
 0 0 1.3689 0 -0.8089 0 0 
 0 0 0 1.3689 0 0.1244 0 
 0 0 -0.8089 0 2.0533 0.2333 -0.4044 
 0 0 0 0.1244 0.2333 2.0533 0.0467 
 -0.3422 0.2333 0 0 -0.4044 0.0467 2.0533 
 0.2333 -0.3422 0 0 -0.0467 0.0622 -0.2333 
 0.0622 0.0467 0 0 0 0 -0.8089 
 -0.0467 -0.4044 0 0 0 0 0 
 0 0 0.0622 -0.0467 -0.3422 0.2333 0 
 0 0 0.0467 -0.4044 0.2333 -0.3422 0 
 0 0 -0.3422 -0.2333 0.1244 0 -0.3422 
 0 0 -0.2333 -0.3422 0 -0.8089 0.2333 
 0 0 0 0 -0.3422 -0.2333 0.1244 
 0 0 0 0 -0.2333 -0.3422 0 
 0 0 0 0 0 0 -0.3422 
 0 0 0 0 0 0 -0.2333 
 
 Columns 22 through 28 
 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0.0467 -0.3422 -0.2333 0 0 0 0 
 -0.4044 -0.2333 -0.3422 0 0 0 0 
 0.2333 0.0622 -0.0467 0 0 0 0 
 -0.3422 0.0467 -0.4044 0 0 0 0 
 0 0 0 0.0622 0.0467 -0.3422 -0.2333 
 0 0 0 -0.0467 -0.4044 -0.2333 -0.3422 
 -0.0467 0 0 -0.3422 0.2333 0.1244 0 
 0.0622 0 0 0.2333 -0.3422 0 -0.8089 
 -0.2333 -0.8089 0 0 0 -0.3422 0.2333 
 2.0533 0 0.1244 0 0 0.2333 -0.3422 
 0 1.3689 0 0 0 0 0 
 0.1244 0 1.3689 0 0 0 0 
 0 0 0 0.6844 -0.2333 -0.4044 0.0467 
 0 0 0 -0.2333 0.6844 -0.0467 0.0622 
 0.2333 0 0 -0.4044 -0.0467 1.3689 0 
 -0.3422 0 0 0.0467 0.0622 0 1.3689 
 0 -0.3422 0.2333 0 0 -0.4044 -0.0467 
 -0.8089 0.2333 -0.3422 0 0 0.0467 0.0622 
 -0.2333 0.0622 0.0467 0 0 0 0 
 -0.3422 -0.0467 -0.4044 0 0 0 0 
 
 Columns 29 through 32 
 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 0 0 0 0 
 -0.3422 -0.2333 0 0 
 -0.2333 -0.3422 0 0 
 0.1244 0 -0.3422 -0.2333 
 0 -0.8089 -0.2333 -0.3422 
 -0.3422 0.2333 0.0622 -0.0467 
 0.2333 -0.3422 0.0467 -0.4044 
 0 0 0 0 
 0 0 0 0 
 -0.4044 0.0467 0 0 
 -0.0467 0.0622 0 0 
 1.3689 0 -0.4044 0.0467 
 0 1.3689 -0.0467 0.0622 
 -0.4044 -0.0467 0.6844 0.2333 
 0.0467 0.0622 0.2333 0.6844 
 
» k=[K(3:8,3:8) K(3:8,11:16) K(3:8,19:24) K(3:8,27:32) ; K(11:16,3:8) 
K(11:16,11:16) K(11:16,19:24) K(11:16,27:32); K(19:24,3:8) 
K(19:24,11:16) K(19:24,19:24) K(19:24,27:32) ; K(27:32,3:8) 
K(27:32,11:16) K(27:32,19:24) K(27:32,27:32)]; 
» f=[zeros(22,1) ; 0 ; -20]; 
» u=k\f 
 
u = 
 
 1.0e-003 * 
 
 -0.0299 
 -0.0284 
 -0.0402 
 -0.0753 
 -0.0386 
 -0.1102 
 0.0015 
 -0.0203 
 -0.0068 
 -0.0800 
 -0.0123 
 -0.1088 
 -0.0021 
 -0.0185 
 -0.0023 
 -0.0824 
 0.0047 
 -0.1224 
 0.0307 
 -0.0260 
 0.0489 
 -0.0758 
 0.0565 
 -0.1589 
 
» U=[0;0;u(1:6);0;0;u(7:12);0;0;u(13:18);0;0;u(19:24)]; 
» F=K*U 
 
F = 
 
 17.6570 
 3.4450 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 7.4806 
 7.0314 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 -7.9321 
 6.7416 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 -17.2054 
 2.7819 
 0 
 0 
 0.0000 
 0.0000 
 0.0000 
 -20.0000 
 
» u1=[U(1) ; U(2) ; U(3) ; U(4) ; U(11) ; U(12) ; U(9) ; U(10)]; 
» u2=[U(3) ; U(4) ; U(5) ; U(6) ; U(13) ; U(14) ; U(11) ; U(12)]; 
» u3=[U(5) ; U(6) ; U(7) ; U(8) ; U(15) ; U(16) ; U(13) ; U(14)]; 
» u4=[U(9) ; U(10) ; U(11) ; U(12) ; U(19) ; U(20) ; U(17) ; U(18)]; 
» u5=[U(13) ; U(14) ; U(15) ; U(16) ; U(23) ; U(24) ; U(21) ; U(22)]; 
» u6=[U(17) ; U(18) ; U(19) ; U(20) ; U(27) ; U(28) ; U(25) ; U(26)]; 
» u7=[U(19) ; U(20) ; U(21) ; U(22) ; U(29) ; U(30) ; U(27) ; U(28)]; 
» u8=[U(21) ; U(22) ; U(23) ; U(24) ; U(31) ; U(32) ; U(29) ; U(30)]; 
» 
sigma1=BilinearQuadElementStresses(E,NU,0,0,0.3,0,0.3,0.3,0,0.3,1,u1)
; 
» 
sigma2=BilinearQuadElementStresses(E,NU,0.3,0,0.6,0,0.6,0.3,0.3,0.3,1
,u2); 
» 
sigma3=BilinearQuadElementStresses(E,NU,0.6,0,0.9,0,0.9,0.3,0.6,0.3,1
,u3); 
» 
sigma4=BilinearQuadElementStresses(E,NU,0,0.3,0.3,0.3,0.3,0.6,0,0.6,1
,u4); 
» 
sigma5=BilinearQuadElementStresses(E,NU,0.6,0.3,0.9,0.3,0.9,0.6,0.6,0
.6,1,u5); 
» 
sigma6=BilinearQuadElementStresses(E,NU,0,0.6,0.3,0.6,0.3,0.9,0,0.9,1
,u6); 
» 
sigma7=BilinearQuadElementStresses(E,NU,0.3,0.6,0.6,0.6,0.6,0.9,0.3,0
.9,1,u7); 
» 
sigma8=BilinearQuadElementStresses(E,NU,0.6,0.6,0.9,0.6,0.9,0.9,0.6,0
.9,1,u8); 
» s1=BilinearQuadElementPStresses(sigma1); 
» s2=BilinearQuadElementPStresses(sigma2); 
» s3=BilinearQuadElementPStresses(sigma3); 
» s4=BilinearQuadElementPStresses(sigma4); 
» s5=BilinearQuadElementPStresses(sigma5); 
» s6=BilinearQuadElementPStresses(sigma6); 
» s7=BilinearQuadElementPStresses(sigma7); 
» s8=BilinearQuadElementPStresses(sigma8); 
 
Problem 13.3: 
 
» E=200e6; 
» NU=0.3; 
» h=0.01; 
» 
k1=BilinearQuadElementStiffness(E,NU,h,0,0,0.35,0,0.35,0.4,0,0.4,1); 
» 
k2=BilinearQuadElementStiffness(E,NU,h,0.35,0,0.7,0,0.7,0.4,0.35,0.4,
1); 
» k3=SpringElementStiffness(4000); 
» k4=SpringElementStiffness(4000); 
» k5=SpringElementStiffness(4000); 
» K=zeros(15,15); 
» K=BilinearQuadAssemble(K,k1,4,5,2,1); 
» K=BilinearQuadAssemble(K,k2,5,6,3,2); 
» K=SpringAssemble(K,k3,8,13); 
» K=SpringAssemble(K,k4,10,14); 
» K=SpringAssemble(K,k5,12,15) 
 
K = 
 
 1.0e+006 * 
 
 Columns 1 through 7 
 
 1.0616 -0.3571 -0.7251 0.0275 0 0 0.1943 
 -0.3571 0.9341 -0.0275 0.0275 0 0 0.0275 
 -0.7251 -0.0275 2.1232 0 -0.7251 0.0275 -0.5308 
 0.0275 0.0275 0 1.8681 -0.0275 0.0275 -0.3571 
 0 0 -0.7251 -0.0275 1.0616 0.3571 0 
 0 0 0.0275 0.0275 0.3571 0.9341 0 
 0.1943 0.0275 -0.5308 -0.3571 0 0 1.0616 
 -0.0275 -0.4945 -0.3571 -0.4670 0 0 0.3571 
 -0.5308 0.3571 0.3885 0 -0.5308 -0.3571 -0.7251 
 0.3571 -0.4670 0 -0.9890 -0.3571 -0.4670 -0.0275 
 0 0 -0.5308 0.3571 0.1943 -0.0275 0 
 0 0 0.3571 -0.4670 0.0275 -0.4945 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 
 Columns 8 through 14 
 
 -0.0275 -0.5308 0.3571 0 0 0 0 
 -0.4945 0.3571 -0.4670 0 0 0 0 
 -0.3571 0.3885 0 -0.5308 0.3571 0 0 
 -0.4670 0 -0.9890 0.3571 -0.4670 0 0 
 0 -0.5308 -0.3571 0.1943 0.0275 0 0 
 0 -0.3571 -0.4670 -0.0275 -0.4945 0 0 
 0.3571 -0.7251 -0.0275 0 0 0 0 
 0.9381 0.0275 0.0275 0 0 -0.0040 0 
 0.0275 2.1232 0 -0.7251 -0.0275 0 0 
 0.0275 0 1.8721 0.0275 0.0275 0 -0.0040 
 0 -0.7251 0.0275 1.0616 -0.3571 0 0 
 0 -0.0275 0.0275 -0.3571 0.9381 0 0 
 -0.0040 0 0 0 0 0.0040 0 
 0 0 -0.0040 0 0 0 0.0040 
 0 0 0 0 -0.0040 0 0 
 
 Column 15 
 
 0 
 0 
 0 
 0 
 0 
 0 
 0 
 0 
 0 
 0 
 0 
 -0.0040 
 0 
 0 
 0.0040 
 
» k=K(1:12,1:12); 
» f=[0 ; 8.75 ; 0 ; 17.5 ; 0 ; 8.75 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0]; 
» u=k\f 
 
Warning: Matrix is close to singular or badly scaled. 
 Results may be inaccurate. RCOND = 4.354270e-017. 
 
u = 
 
 -0.0003 
 0.0029 
 -0.0003 
 0.0029 
 -0.0003 
 0.0029 
 -0.0003 
 0.0029 
 -0.0003 
 0.0029 
 -0.0003 
 0.0029 
 
» U=[u(1:12);0;0;0]; 
» F=K*U 
 
F = 
 
 0.0000 
 8.7500 
 0.0000 
 17.5000 
 0.0000 
 8.7500 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0 
 0.0000 
 -11.6553 
 -11.6894 
 -11.6553 
 
» u1=[U(7) ; U(8) ; U(9) ; U(10) ; U(3) ; U(4) ; U(1) ; U(2)]; 
» u2=[U(9) ; U(10) ; U(11) ; U(12) ; U(5) ; U(6) ; U(3) ; U(4)]; 
» u3=[U(8) ; U(13)]; 
» u4=[U(10) ; U(14)]; 
» u5=[U(12) ; U(15)]; 
» 
sigma1=BilinearQuadElementStresses(E,NU,0,0,0.35,0,0.35,0.4,0,0.4,1,u
1); 
» 
sigma2=BilinearQuadElementStresses(E,NU,0.35,0,0.7,0,0.7,0.4,0.35,0.4
,1,u2); 
» s1=BilinearQuadElementPStresses(sigma1); 
» s2=BilinearQuadElementPStresses(sigma2); 
» f3=SpringElementForces(k3,u3); 
» f4=SpringElementForces(k4,u4); 
» f5=SpringElementForces(k5,u5); 
 
Problem 14.1: 
 
» E=200e6; 
» NU=0.3; 
» h=0.01; 
» k1=QuadraticQuadElementStiffness(E,NU,h,0,0,0.7,0,0.7,0.4,0,0.4,1); 
» k2=SpringElementStiffness(4000); 
» k3=SpringElementStiffness(4000); 
» k4=SpringElementStiffness(4000); 
» K=zeros(19,19); 
» K=QuadraticQuadAssemble(K,k1,6,8,3,1,7,5,2,4); 
» K=SpringAssemble(K,k2,12,17); 
» K=SpringAssemble(K,k3,14,18); 
» K=SpringAssemble(K,k4,16,19) 
 
K = 
 
 1.0e+006 * 
 
 Columns 1 through 7 
 
 1.5034 -0.6746 -1.0266 0.3541 0.6450 -0.0092 -1.1129 
 -0.6746 2.4762 0.2808 -0.1343 0.0092 0.8632 0.3541 
 -1.0266 0.2808 2.9506 0 -1.0266 -0.2808 0 
 0.3541 -0.1343 0 2.8327 -0.3541 -0.1343 -0.6349 
 0.6450 0.0092 -1.0266 -0.3541 1.5034 0.6746 -0.6820 
 -0.0092 0.8632 -0.2808 -0.1343 0.6746 2.4762 -0.1587 
 -1.1129 0.3541 0 -0.6349 -0.6820 -0.1587 3.0630 
 0.2808 -3.3895 -0.6349 0 -0.1587 -1.7387 0 
 -0.6820 0.1587 0 0.6349 -1.1129 -0.3541 0.5268 
 0.1587 -1.7387 0.6349 0 -0.2808 -3.3895 0 
 0.6560 -0.0092 -0.6479 -0.1587 0.6650 0.2778 -1.1129 
 0.0092 1.2796 -0.1587 -0.4518 0.2778 1.0952 -0.3541 
 -0.6479 0.1587 0.3984 0 -0.6479 -0.1587 0 
 0.1587 -0.4518 0 -1.6606 -0.1587 -0.4518 0.6349 
 0.6650 -0.2778 -0.6479 0.1587 0.6560 0.0092 -0.6820 
 -0.2778 1.09520.1587 -0.4518 -0.0092 1.2796 0.1587 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
 
 Columns 8 through 14 
 
 0.2808 -0.6820 0.1587 0.6560 0.0092 -0.6479 0.1587 
 -3.3895 0.1587 -1.7387 -0.0092 1.2796 0.1587 -0.4518 
 -0.6349 0 0.6349 -0.6479 -0.1587 0.3984 0 
 0 0.6349 0 -0.1587 -0.4518 0 -1.6606 
 -0.1587 -1.1129 -0.2808 0.6650 0.2778 -0.6479 -0.1587 
 -1.7387 -0.3541 -3.3895 0.2778 1.0952 -0.1587 -0.4518 
 0 0.5268 0 -1.1129 -0.3541 0 0.6349 
 7.0720 0 3.1844 -0.2808 -3.3895 0.6349 0 
 0 3.0630 0 -0.6820 -0.1587 0 -0.6349 
 3.1844 0 7.0720 -0.1587 -1.7387 -0.6349 0 
 -0.2808 -0.6820 -0.1587 1.5034 0.6746 -1.0266 -0.3541 
 -3.3895 -0.1587 -1.7387 0.6746 2.4802 -0.2808 -0.1343 
 0.6349 0 -0.6349 -1.0266 -0.2808 2.9506 0 
 0 -0.6349 0 -0.3541 -0.1343 0 2.8367 
 0.1587 -1.1129 0.2808 0.6450 -0.0092 -1.0266 0.3541 
 -1.7387 0.3541 -3.3895 0.0092 0.8632 0.2808 -0.1343 
 0 0 0 0 -0.0040 0 0 
 0 0 0 0 0 0 -0.0040 
 0 0 0 0 0 0 0 
 
 Columns 15 through 19 
 
 0.6650 -0.2778 0 0 0 
 -0.2778 1.0952 0 0 0 
 -0.6479 0.1587 0 0 0 
 0.1587 -0.4518 0 0 0 
 0.6560 -0.0092 0 0 0 
 0.0092 1.2796 0 0 0 
 -0.6820 0.1587 0 0 0 
 0.1587 -1.7387 0 0 0 
 -1.1129 0.3541 0 0 0 
 0.2808 -3.3895 0 0 0 
 0.6450 0.0092 0 0 0 
 -0.0092 0.8632 -0.0040 0 0 
 -1.0266 0.2808 0 0 0 
 0.3541 -0.1343 0 -0.0040 0 
 1.5034 -0.6746 0 0 0 
 -0.6746 2.4802 0 0 -0.0040 
 0 0 0.0040 0 0 
 0 0 0 0.0040 0 
 0 -0.0040 0 0 0.0040 
 
» k=K(1:16,1:16); 
» f=[0 ; 5.8333 ; 0 ; 23.3333 ; 0 ; 5.8333 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 
0 ; 0 ; 0 ; 0]; 
» u=k\f 
 
Warning: Matrix is close to singular or badly scaled. 
 Results may be inaccurate. RCOND = 1.316376e-017. 
 
u = 
 
 -0.0002 
 0.0029 
 -0.0002 
 0.0029 
 -0.0002 
 0.0029 
 -0.0002 
 0.0029 
 -0.0002 
 0.0029 
 -0.0002 
 0.0029 
 -0.0002 
 0.0029 
 -0.0002 
 0.0029 
 
» U=[u;0;0;0]; 
» F=K*U; 
 
 
F = 
 
 0 
 5.8333 
 0.0000 
 23.3333 
 0.0000 
 5.8333 
 0.0000 
 0.0000 
 0 
 0 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 -11.6419 
 -11.7162 
 -11.6419 
 
» u1=[U(11) ; U(12) ; U(15) ; U(16) ; U(5) ; U(6) ; U(1) ; U(2) ; 
U(13) ; U(14) ; U(9) ; U(10) ; U(3) ; U(4) ; U(7) ; U(8)]; 
» u2=[U(12) ; U(17)]; 
» u3=[U(14) ; U(18)]; 
» u4=[U(16) ; U(19)]; 
» 
sigma1=QuadraticQuadElementStresses(E,NU,0,0,0.7,0,0.7,0.4,0,0.4,1,u1
); 
» s1=QuadraticQuadElementPStresses(sigma1); 
» f2=SpringElementForces(k2,u2); 
» f3=SpringElementForces(k3,u3); 
» f4=SpringElementForces(k4,u4); 
 
Problem 15.1: 
 
» E=210e6; 
» NU=0.3; 
» 
k1=TetrahedronElementStiffness(E,NU,0,0,0,0.025,0,0,0.025,0.5,0,0.025
,0.5,0.25); 
» 
k2=TetrahedronElementStiffness(E,NU,0,0,0,0.025,0,0,0.025,0.5,0.25,0,
0,0.25); 
» 
k3=TetrahedronElementStiffness(E,NU,0.025,0,0,0.025,0.5,0.25,0,0,0.25
,0.025,0,0.25); 
» 
k4=TetrahedronElementStiffness(E,NU,0,0,0,0,0.5,0,0,0.5,0.25,0.025,0.
5,0); 
» 
k5=TetrahedronElementStiffness(E,NU,0,0,0,0,0.5,0.25,0,0,0.25,0.025,0
.5,0.25); 
» 
k6=TetrahedronElementStiffness(E,NU,0,0,0,0.025,0.5,0.25,0.025,0.5,0,
0,0.5,0.25); 
» K=zeros(24,24); 
» K=TetrahedronAssemble(K,k1,1,2,4,8); 
» K=TetrahedronAssemble(K,k2,1,2,8,5); 
» K=TetrahedronAssemble(K,k3,2,8,5,6); 
» K=TetrahedronAssemble(K,k4,1,3,7,4); 
» K=TetrahedronAssemble(K,k5,1,7,5,8); 
» K=TetrahedronAssemble(K,k6,1,8,4,7) 
 
K = 
 
 1.0e+008 * 
 
 Columns 1 through 7 
 
 4.7300 -0.0841 0.1683 -4.7132 0.1346 -0.0673 -0.0017 
 -0.0841 1.3773 -0.0084 0.1178 -1.3520 0.0034 0.0505 
 0.1683 -0.0084 1.3983 -0.1010 0.0050 -1.3478 0 
 -4.7132 0.1178 -0.1010 4.7216 -0.1683 0 0 
 0.1346 -1.3520 0.0050 -0.1683 1.3647 0 0 
 -0.0673 0.0034 -1.3478 0 0 1.3731 0 
 -0.0017 0.0505 0 0 0 0 2.3642 
 0.0337 -0.0059 0.0034 0 0 0 -0.0841 
 0 0.0050 -0.0017 0 0 0 0.1683 
 -0.0017 -0.0841 0.0673 -0.0017 0.0337 -0.0673 -2.3558 
 -0.0841 -0.0059 0.0034 0.0505 -0.0059 0.0034 0.0337 
 0.1010 0.0050 -0.0017 -0.1010 0.0050 -0.0017 -0.0673 
 -0.0151 0.0337 -0.0673 0.0017 -0.0337 0.1683 0 
 0.0505 -0.0194 0.0135 -0.0505 0.0059 -0.0034 0 
 -0.1010 0.0118 -0.0488 0.1683 -0.0050 0.0017 0 
 0 0 0 -0.0067 0 -0.1010 0 
 0 0 0 0 -0.0067 0.0050 0 
 0 0 0 -0.0673 0.0034 -0.0236 0 
 -0.0017 0.0505 0.1010 0 0 0 -0.0067 
 0.0337 -0.0059 -0.0084 0 0 0 0 
 0.0673 -0.0084 -0.0017 0 0 0 -0.1010 
 0.0034 -0.0841 -0.1683 -0.0017 0.0337 0.0673 0 
 -0.0841 0.0118 -0.0084 0.0505 -0.0059 -0.0084 0 
 -0.1683 -0.0084 0.0034 0.1010 -0.0084 -0.0017 0 
 
 Columns 8 through 14 
 
 0.0337 0 -0.0017 -0.0841 0.1010 -0.0151 0.0505 
 -0.0059 0.0050 -0.0841 -0.0059 0.0050 0.0337 -0.0194 
 0.0034 -0.0017 0.0673 0.0034 -0.0017 -0.0673 0.0135 
 0 0 -0.0017 0.0505 -0.1010 0.0017 -0.0505 
 0 0 0.0337 -0.0059 0.0050 -0.0337 0.0059 
 0 0 -0.0673 0.0034 -0.0017 0.1683 -0.0034 
 -0.0841 0.1683 -2.3558 0.0337 -0.0673 0 0 
 0.6857 -0.0084 0.0505 -0.6731 0 0 0 
 -0.0084 0.6983 -0.1010 0 -0.6731 0 0 
 0.0505 -0.1010 2.3726 0 0 0 0 
 -0.6731 0 0 0.6983 -0.0168 0 0 
 0 -0.6731 0 -0.0168 0.7236 0 0 
 0 0 0 0 0 2.3726 0 
 0 0 0 0 0 0 0.6983 
 0 0 0 0 0 0 -0.0168 
 0 0 0 0 0 -2.3558 0.0337 
 0 0 0 0 0 0.0505 -0.6731 
 0 0 0 0 0 -0.1010 0 
 0 -0.0673 0.0017 -0.0505 0.1683 -0.0017 0.0505 
 -0.0067 0.0034 -0.0337 0.0059 -0.0050 0.0337 -0.0059 
 0.0050 -0.0236 0.1683 -0.0034 0.0017 -0.0673 0.0034 
 0 0 -0.0151 0.0505 -0.1010 -0.0017 -0.0841 
 0 0 0.0337 -0.0194 0.0118 -0.0841 -0.0059 
 0 0 -0.0673 0.0135 -0.0488 0.0673 0.0034 
 
 Columns 15 through 21-0.1010 0 0 0 -0.0017 0.0337 0.0673 
 0.0118 0 0 0 0.0505 -0.0059 -0.0084 
 -0.0488 0 0 0 0.1010 -0.0084 -0.0017 
 0.1683 -0.0067 0 -0.0673 0 0 0 
 -0.0050 0 -0.0067 0.0034 0 0 0 
 0.0017 -0.1010 0.0050 -0.0236 0 0 0 
 0 0 0 0 -0.0067 0 -0.1010 
 0 0 0 0 0 -0.0067 0.0050 
 0 0 0 0 -0.0673 0.0034 -0.0236 
 0 0 0 0 0.0017 -0.0337 0.1683 
 0 0 0 0 -0.0505 0.0059 -0.0034 
 0 0 0 0 0.1683 -0.0050 0.0017 
 0 -2.3558 0.0505 -0.1010 -0.0017 0.0337 -0.0673 
 -0.0168 0.0337 -0.6731 0 0.0505 -0.0059 0.0034 
 0.7236 -0.0673 0 -0.6731 -0.1010 0.0050 -0.0017 
 -0.0673 2.3642 -0.0841 0.1683 0 0 0 
 0 -0.0841 0.6857 -0.0084 0 0 0 
 -0.6731 0.1683 -0.0084 0.6983 0 0 0 
 -0.1010 0 0 0 4.7216 -0.1683 0 
 0.0050 0 0 0 -0.1683 1.3647 0 
 -0.0017 0 0 0 0 0 1.3731 
 0.1010 -0.0017 0.0337 0 -4.7132 0.1346 -0.0673 
 0.0050 0.0505 -0.0059 0.0050 0.1178 -1.3520 0.0034 
 -0.0017 0 0.0034 -0.0017 -0.1010 0.0050 -1.3478 
 
 Columns 22 through 24 
 
 0.0034 -0.0841 -0.1683 
 -0.0841 0.0118 -0.0084 
 -0.1683 -0.0084 0.0034 
 -0.0017 0.0505 0.1010 
 0.0337 -0.0059 -0.0084 
 0.0673 -0.0084 -0.0017 
 0 0 0 
 0 0 0 
 0 0 0 
 -0.0151 0.0337 -0.0673 
 0.0505 -0.0194 0.0135 
 -0.1010 0.0118 -0.0488 
 -0.0017 -0.0841 0.0673 
 -0.0841 -0.0059 0.0034 
 0.1010 0.0050 -0.0017 
 -0.0017 0.0505 0 
 0.0337 -0.0059 0.0034 
 0 0.0050 -0.0017 
 -4.7132 0.1178 -0.1010 
 0.1346 -1.3520 0.0050 
 -0.0673 0.0034 -1.3478 
 4.7300 -0.0841 0.1683 
 -0.0841 1.3773 -0.0084 
 0.1683 -0.0084 1.3983 
 
» k=[K(7:12,7:12) K(7:12,19:24) ; K(19:24,7:12) K(19:24,19:24)]; 
» f=[0 ; 3.125 ; 0 ; 0 ; 6.25 ; 0 ; 0 ; 6.25 ; 0 ; 0 ; 3.125 ; 0]; 
» u=k\f 
 
u = 
 
 1.0e-005 * 
 
 0.0185 
 0.6710 
 0.1485 
 0.0091 
 0.6699 
 0.1489 
 0.0183 
 0.5809 
 0.0319 
 0.0074 
 0.5795 
 0.0317 
 
» U=[0;0;0;0;0;0;u(1:6);0;0;0;0;0;0;u(7:12)]; 
» F=K*U 
 
F = 
 
 -51.1925 
 -3.0565 
 -4.4842 
 51.2090 
 -6.3185 
 -3.0368 
 0.0000 
 3.1250 
 0.0000 
 0.0000 
 6.2500 
 0.0000 
 -29.2553 
 -6.3185 
 4.6493 
 29.2388 
 -3.0565 
 2.8717 
 0.0000 
 6.2500 
 0 
 0.0000 
 3.1250 
 0 
 
» u1=[U(1) ; U(2) ; U(3) ; U(4) ; U(5) ; U(6) ; U(10) ; U(11) ; U(12) 
; U(22) ; U(23) ; U(24)]; 
» u2=[U(1) ; U(2) ; U(3) ; U(4) ; U(5) ; U(6) ; U(22) ; U(23) ; U(24) 
; U(13) ; U(14) ; U(15)]; 
» u3=[U(4) ; U(5) ; U(6) ; U(22) ; U(23) ; U(24) ; U(13) ; U(14) ; 
U(15) ; U(16) ; U(17) ; U(18)]; 
» u4=[U(1) ; U(2) ; U(3) ; U(7) ; U(8) ; U(9) ; U(19) ; U(20) ; U(21) 
; U(10) ; U(11) ; U(12)]; 
» u5=[U(1) ; U(2) ; U(3) ; U(19) ; U(20) ; U(21) ; U(13) ; U(14) ; 
U(15) ; U(22) ; U(23) ; U(24)]; 
» u6=[U(1) ; U(2) ; U(3) ; U(22) ; U(23) ; U(24) ; U(10) ; U(11) ; 
U(12) ; U(19) ; U(20) ; U(21)]; 
» 
sigma1=TetrahedronElementStresses(E,NU,0,0,0,0.025,0,0,0.025,0.5,0,0.
025,0.5,0.25,u1); 
» 
sigma2=TetrahedronElementStresses(E,NU,0,0,0,0.025,0,0,0.025,0.5,0.25
,0,0,0.25,u2); 
» 
sigma3=TetrahedronElementStresses(E,NU,0.025,0,0,0.025,0.5,0.25,0,0,0
.25,0.025,0,0.25,u3); 
» 
sigma4=TetrahedronElementStresses(E,NU,0,0,0,0,0.5,0,0,0.5,0.25,0.025
,0.5,0,u4); 
» 
sigma5=TetrahedronElementStresses(E,NU,0,0,0,0,0.5,0.25,0,0,0.25,0.02
5,0.5,0.25,u5); 
» 
sigma6=TetrahedronElementStresses(E,NU,0,0,0,0.025,0.5,0.25,0.025,0.5
,0,0,0.5,0.25,u6); 
» s1=TetrahedronElementPStresses(sigma1); 
» s2=TetrahedronElementPStresses(sigma2); 
» s3=TetrahedronElementPStresses(sigma3); 
» s4=TetrahedronElementPStresses(sigma4); 
» s5=TetrahedronElementPStresses(sigma5); 
» s6=TetrahedronElementPStresses(sigma6); 
 
 
Problem 16.1 
 
 
>> k1 = 
LinearBrickElementStiffness(210e6,0.3,0,0,0.025,0,0,0,0,0.25,0,0,0.25,0.025,
0.25,0,0.025,0.25,0,0,0.25,0.25,0,0.25,0.25,0.025); 
 
>> k2 = 
LinearBrickElementStiffness(210e6,0.3,0.25,0,0.025,0.25,0,0,0.25,0.25,0,0.25
,0.25,0.025,0.5,0,0.025,0.5,0,0,0.5,0.25,0,0.5,0.25,0.025); 
 
>> K = zeros(36,36); 
 
>> K = LinearBrickAssemble(K,k1,1,2,3,4,5,6,7,8) 
 
K = 
 
 1.0e+007 * 
 
 Columns 1 through 14 
 
 2.3446 -2.1931 -1.1134 1.1386 1.0545 -1.1554 -0.5861 
0.5104 0.0421 0.0210 -0.0042 -0.0084 0.0084 0.0042 
 -2.1931 2.3446 1.1386 -1.1134 -1.1554 1.0545 0.5104 -
0.5861 0.0210 0.0421 -0.0084 -0.0042 0.0042 0.0084 
 -1.1134 1.1386 2.3446 -2.1931 -0.5861 0.5104 1.0545 -
1.1554 0.0042 0.0084 -0.0421 -0.0210 0.0210 0.0421 
 1.1386 -1.1134 -2.1931 2.3446 0.5104 -0.5861 -1.1554 
1.0545 0.0084 0.0042 -0.0210 -0.0421 0.0421 0.0210 
 1.0545 -1.1554 -0.5861 0.5104 2.3446 -2.1931 -1.1134 
1.1386 -0.0084 -0.0042 0.0210 0.0421 -0.0421 -0.0210 
 -1.1554 1.0545 0.5104 -0.5861 -2.1931 2.3446 1.1386 -
1.1134 -0.0042 -0.0084 0.0421 0.0210 -0.0210 -0.0421 
 -0.5861 0.5104 1.0545 -1.1554 -1.1134 1.1386 2.3446 -
2.1931 -0.0210 -0.0421 0.0084 0.0042 -0.0042 -0.0084 
 0.5104 -0.5861 -1.1554 1.0545 1.1386 -1.1134 -2.1931 
2.3446 -0.0421 -0.0210 0.0042 0.0084 -0.0084 -0.0042 
 0.0421 0.0210 0.0042 0.0084 -0.0084 -0.0042 -0.0210 -
0.0421 2.3446 -2.1931 -1.1554 1.0545 1.1386 -1.1134 
 0.0210 0.0421 0.0084 0.0042 -0.0042 -0.0084 -0.0421 -
0.0210 -2.1931 2.3446 1.0545 -1.1554 -1.1134 1.1386 
 -0.0042 -0.0084 -0.0421 -0.0210 0.0210 0.0421 0.0084 
0.0042 -1.1554 1.0545 2.3446 -2.1931 -0.5861 0.5104 
 -0.0084 -0.0042 -0.0210 -0.0421 0.0421 0.0210 0.0042 
0.0084 1.0545 -1.1554 -2.1931 2.3446 0.5104 -0.5861 
 0.0084 0.0042 0.0210 0.0421 -0.0421 -0.0210 -0.0042 -
0.0084 1.1386 -1.1134 -0.5861 0.5104 2.3446 -2.1931 
 0.0042 0.0084 0.0421 0.0210 -0.0210 -0.0421 -0.0084 -
0.0042 -1.1134 1.1386 0.5104 -0.5861 -2.1931 2.3446 
 -0.0210 -0.0421 -0.0084 -0.0042 0.0042 0.0084 0.0421 
0.0210 -0.5861 0.5104 1.1386 -1.1134 -1.1554 1.0545 
 -0.0421 -0.0210 -0.0042 -0.0084 0.0084 0.0042 0.0210 
0.0421 0.5104 -0.5861 -1.1134 1.1386 1.0545 -1.1554 
 -0.4207 -0.0841 -0.0421 -0.2103 0.0841 0.4207 0.2103 
0.0421 -0.4207 -0.0841 0.4207 0.0841 -0.2103 -0.0421 
 0.0841 0.4207 0.2103 0.0421 -0.4207 -0.0841 -0.0421 -
0.2103 0.0841 0.4207 -0.0841 -0.4207 0.0421 0.2103 
 0.0421 0.2103 0.4207 0.0841 -0.2103 -0.0421 -0.0841 -
0.4207 0.4207 0.0841 -0.4207 -0.0841 0.2103 0.0421 
 -0.2103 -0.0421 -0.0841 -0.4207 0.0421 0.2103 0.4207 
0.0841 -0.0841 -0.4207 0.0841 0.4207 -0.0421 -0.2103 
 -0.0841 -0.4207 -0.2103 -0.0421 0.4207 0.0841 0.0421 
0.2103 -0.2103 -0.0421 0.2103 0.0421 -0.4207 -0.0841 
 0.4207 0.0841 0.0421 0.2103 -0.0841-0.4207 -0.2103 -
0.0421 0.0421 0.2103 -0.0421 -0.2103 0.0841 0.4207 
 0.2103 0.0421 0.0841 0.4207 -0.0421 -0.2103 -0.4207 -
0.0841 0.2103 0.0421 -0.2103 -0.0421 0.4207 0.0841 
 -0.0421 -0.2103 -0.4207 -0.0841 0.2103 0.0421 0.0841 
0.4207 -0.0421 -0.2103 0.0421 0.2103 -0.0841 -0.4207 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 
 Columns 15 through 28 
 
 -0.0210 -0.0421 -0.4207 0.0841 0.0421 -0.2103 -0.0841 
0.4207 0.2103 -0.0421 0 0 0 0 
 -0.0421 -0.0210 -0.0841 0.4207 0.2103 -0.0421 -0.4207 
0.0841 0.0421 -0.2103 0 0 0 0 
 -0.0084 -0.0042 -0.0421 0.2103 0.4207 -0.0841 -0.2103 
0.0421 0.0841 -0.4207 0 0 0 0 
 -0.0042 -0.0084 -0.2103 0.0421 0.0841 -0.4207 -0.0421 
0.2103 0.4207 -0.0841 0 0 0 0 
 0.0042 0.0084 0.0841 -0.4207 -0.2103 0.0421 0.4207 -
0.0841 -0.0421 0.2103 0 0 0 0 
 0.0084 0.0042 0.4207 -0.0841 -0.0421 0.2103 0.0841 -
0.4207 -0.2103 0.0421 0 0 0 0 
 0.0421 0.0210 0.2103 -0.0421 -0.0841 0.4207 0.0421 -
0.2103 -0.4207 0.0841 0 0 0 0 
 0.0210 0.0421 0.0421 -0.2103 -0.4207 0.0841 0.2103 -
0.0421 -0.0841 0.4207 0 0 0 0 
 -0.5861 0.5104 -0.4207 0.0841 0.4207 -0.0841 -0.2103 
0.0421 0.2103 -0.0421 0 0 0 0 
 0.5104 -0.5861 -0.0841 0.4207 0.0841 -0.4207 -0.0421 
0.2103 0.0421 -0.2103 0 0 0 0 
 1.1386 -1.1134 0.4207 -0.0841 -0.4207 0.0841 0.2103 -
0.0421 -0.2103 0.0421 0 0 0 0 
 -1.1134 1.1386 0.0841 -0.4207 -0.0841 0.4207 0.0421 -
0.2103 -0.0421 0.2103 0 0 0 0 
 -1.1554 1.0545 -0.2103 0.0421 0.2103 -0.0421 -0.4207 
0.0841 0.4207 -0.0841 0 0 0 0 
 1.0545 -1.1554 -0.0421 0.2103 0.0421 -0.2103 -0.0841 
0.4207 0.0841 -0.4207 0 0 0 0 
 2.3446 -2.1931 0.2103 -0.0421 -0.2103 0.0421 0.4207 -
0.0841 -0.4207 0.0841 0 0 0 0 
 -2.1931 2.3446 0.0421 -0.2103 -0.0421 0.2103 0.0841 -
0.4207 -0.0841 0.4207 0 0 0 0 
 0.2103 0.0421 7.8974 -7.8301 -3.9319 3.9151 3.9151 -
3.9319 -1.9744 1.9407 0 0 0 0 
 -0.0421 -0.2103 -7.8301 7.8974 3.9151 -3.9319 -3.9319 
3.9151 1.9407 -1.9744 0 0 0 0 
 -0.2103 -0.0421 -3.9319 3.9151 7.8974 -7.8301 -1.9744 
1.9407 3.9151 -3.9319 0 0 0 0 
 0.0421 0.2103 3.9151 -3.9319 -7.8301 7.8974 1.9407 -
1.9744 -3.9319 3.9151 0 0 0 0 
 0.4207 0.0841 3.9151 -3.9319 -1.9744 1.9407 7.8974 -
7.8301 -3.9319 3.9151 0 0 0 0 
 -0.0841 -0.4207 -3.9319 3.9151 1.9407 -1.9744 -7.8301 
7.8974 3.9151 -3.9319 0 0 0 0 
 -0.4207 -0.0841 -1.9744 1.9407 3.9151 -3.9319 -3.9319 
3.9151 7.8974 -7.8301 0 0 0 0 
 0.0841 0.4207 1.9407 -1.9744 -3.9319 3.9151 3.9151 -
3.9319 -7.8301 7.8974 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
 
 Columns 29 through 36 
 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 00 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 
 
>> K = LinearBrickAssemble(K,k2,5,6,7,8,9,10,11,12) 
 
K = 
 
 1.0e+008 * 
 
 Columns 1 through 14 
 
 0.2345 -0.2193 -0.1113 0.1139 0.1054 -0.1155 -0.0586 
0.0510 0.0042 0.0021 -0.0004 -0.0008 0.0008 0.0004 
 -0.2193 0.2345 0.1139 -0.1113 -0.1155 0.1054 0.0510 -
0.0586 0.0021 0.0042 -0.0008 -0.0004 0.0004 0.0008 
 -0.1113 0.1139 0.2345 -0.2193 -0.0586 0.0510 0.1054 -
0.1155 0.0004 0.0008 -0.0042 -0.0021 0.0021 0.0042 
 0.1139 -0.1113 -0.2193 0.2345 0.0510 -0.0586 -0.1155 
0.1054 0.0008 0.0004 -0.0021 -0.0042 0.0042 0.0021 
 0.1054 -0.1155 -0.0586 0.0510 0.2345 -0.2193 -0.1113 
0.1139 -0.0008 -0.0004 0.0021 0.0042 -0.0042 -0.0021 
 -0.1155 0.1054 0.0510 -0.0586 -0.2193 0.2345 0.1139 -
0.1113 -0.0004 -0.0008 0.0042 0.0021 -0.0021 -0.0042 
 -0.0586 0.0510 0.1054 -0.1155 -0.1113 0.1139 0.2345 -
0.2193 -0.0021 -0.0042 0.0008 0.0004 -0.0004 -0.0008 
 0.0510 -0.0586 -0.1155 0.1054 0.1139 -0.1113 -0.2193 
0.2345 -0.0042 -0.0021 0.0004 0.0008 -0.0008 -0.0004 
 0.0042 0.0021 0.0004 0.0008 -0.0008 -0.0004 -0.0021 -
0.0042 0.2345 -0.2193 -0.1155 0.1054 0.1139 -0.1113 
 0.0021 0.0042 0.0008 0.0004 -0.0004 -0.0008 -0.0042 -
0.0021 -0.2193 0.2345 0.1054 -0.1155 -0.1113 0.1139 
 -0.0004 -0.0008 -0.0042 -0.0021 0.0021 0.0042 0.0008 
0.0004 -0.1155 0.1054 0.2345 -0.2193 -0.0586 0.0510 
 -0.0008 -0.0004 -0.0021 -0.0042 0.0042 0.0021 0.0004 
0.0008 0.1054 -0.1155 -0.2193 0.2345 0.0510 -0.0586 
 0.0008 0.0004 0.0021 0.0042 -0.0042 -0.0021 -0.0004 -
0.0008 0.1139 -0.1113 -0.0586 0.0510 0.4689 -0.4386 
 0.0004 0.0008 0.0042 0.0021 -0.0021 -0.0042 -0.0008 -
0.0004 -0.1113 0.1139 0.0510 -0.0586 -0.4386 0.4689 
 -0.0021 -0.0042 -0.0008 -0.0004 0.0004 0.0008 0.0042 
0.0021 -0.0586 0.0510 0.1139 -0.1113 -0.2269 0.2193 
 -0.0042 -0.0021 -0.0004 -0.0008 0.0008 0.0004 0.0021 
0.0042 0.0510 -0.0586 -0.1113 0.1139 0.2193 -0.2269 
 -0.0421 -0.0084 -0.0042 -0.0210 0.0084 0.0421 0.0210 
0.0042 -0.0421 -0.0084 0.0421 0.0084 0.0844 -0.1198 
 0.0084 0.0421 0.0210 0.0042 -0.0421 -0.0084 -0.0042 -
0.0210 0.0084 0.0421 -0.0084 -0.0421 -0.1113 0.1265 
 0.0042 0.0210 0.0421 0.0084 -0.0210 -0.0042 -0.0084 -
0.0421 0.0421 0.0084 -0.0421 -0.0084 -0.0376 0.0552 
 -0.0210 -0.0042 -0.0084 -0.0421 0.0042 0.0210 0.0421 
0.0084 -0.0084 -0.0421 0.0084 0.0421 0.0468 -0.0796 
 -0.0084 -0.0421 -0.0210 -0.0042 0.0421 0.0084 0.0042 
0.0210 -0.0210 -0.0042 0.0210 0.0042 -0.0379 -0.0063 
 0.0421 0.0084 0.0042 0.0210 -0.0084 -0.0421 -0.0210 -
0.0042 0.0042 0.0210 -0.0042 -0.0210 0.0105 0.0463 
 0.0210 0.0042 0.0084 0.0421 -0.0042 -0.0210 -0.0421 -
0.0084 0.0210 0.0042 -0.0210 -0.0042 0.0416 0.0076 
 -0.0042 -0.0210 -0.0421 -0.0084 0.0210 0.0042 0.0084 
0.0421 -0.0042 -0.0210 0.0042 0.0210 -0.0093 -0.0425 
 0 0 0 0 0 0 0 
0 0 0 0 0 0.0008 0.0004 
 0 0 0 0 0 0 0 
0 0 0 0 0 0.0004 0.0008 
 0 0 0 0 0 0 0 
0 0 0 0 0 -0.0021 -0.0042 
 0 0 0 0 0 0 0 
0 0 0 0 0 -0.0042 -0.0021 
 0 0 0 0 0 0 0 
0 0 0 0 0 -0.0421 -0.0084 
 0 0 0 0 0 0 0 
0 0 0 0 0 0.0084 0.0421 
 0 0 0 0 0 0 0 
0 0 0 0 0 0.0042 0.0210 
 0 0 0 0 0 0 0 
0 0 0 0 0 -0.0210 -0.0042 
 0 0 0 0 0 0 0 
0 0 0 0 0 -0.0084 -0.0421 
 0 0 0 0 0 0 0 
0 0 0 0 0 0.0421 0.0084 
 0 0 0 0 0 0 0 
0 0 0 0 0 0.0210 0.0042 
 0 0 0 0 0 0 0 
0 0 0 0 0 -0.0042 -0.0210 
 
 Columns 15 through 28 
 
 -0.0021 -0.0042 -0.0421 0.0084 0.0042 -0.0210 -0.0084 
0.0421 0.0210 -0.0042 0 0 0 0 
 -0.0042 -0.0021 -0.0084 0.0421 0.0210 -0.0042 -0.0421 
0.0084 0.0042 -0.0210 0 0 0 0 
 -0.0008 -0.0004 -0.0042 0.0210 0.0421 -0.0084 -0.0210 
0.0042 0.0084 -0.0421 0 0 0 0 
 -0.0004 -0.0008 -0.0210 0.0042 0.0084 -0.0421 -0.0042 
0.02100.0421 -0.0084 0 0 0 0 
 0.0004 0.0008 0.0084 -0.0421 -0.0210 0.0042 0.0421 -
0.0084 -0.0042 0.0210 0 0 0 0 
 0.0008 0.0004 0.0421 -0.0084 -0.0042 0.0210 0.0084 -
0.0421 -0.0210 0.0042 0 0 0 0 
 0.0042 0.0021 0.0210 -0.0042 -0.0084 0.0421 0.0042 -
0.0210 -0.0421 0.0084 0 0 0 0 
 0.0021 0.0042 0.0042 -0.0210 -0.0421 0.0084 0.0210 -
0.0042 -0.0084 0.0421 0 0 0 0 
 -0.0586 0.0510 -0.0421 0.0084 0.0421 -0.0084 -0.0210 
0.0042 0.0210 -0.0042 0 0 0 0 
 0.0510 -0.0586 -0.0084 0.0421 0.0084 -0.0421 -0.0042 
0.0210 0.0042 -0.0210 0 0 0 0 
 0.1139 -0.1113 0.0421 -0.0084 -0.0421 0.0084 0.0210 -
0.0042 -0.0210 0.0042 0 0 0 0 
 -0.1113 0.1139 0.0084 -0.0421 -0.0084 0.0421 0.0042 -
0.0210 -0.0042 0.0210 0 0 0 0 
 -0.2269 0.2193 0.0844 -0.1113 -0.0376 0.0468 -0.0379 
0.0105 0.0416 -0.0093 0.0008 0.0004 -0.0021 -0.0042 
 0.2193 -0.2269 -0.1198 0.1265 0.0552 -0.0796 -0.0063 
0.0463 0.0076 -0.0425 0.0004 0.0008 -0.0042 -0.0021 
 0.4689 -0.4386 -0.0376 0.0468 0.0844 -0.1113 0.0425 -
0.0076 -0.0463 0.0063 0.0021 0.0042 -0.0008 -0.0004 
 -0.4386 0.4689 0.0552 -0.0796 -0.1198 0.1265 0.0093 -
0.0416 -0.0105 0.0379 0.0042 0.0021 -0.0004 -0.0008 
 -0.0376 0.0552 1.0242 -1.0023 -0.5045 0.5054 0.3907 -
0.3936 -0.1953 0.1983 -0.0042 -0.0021 0.0004 0.0008 
 0.0468 -0.0796 -1.0023 1.0242 0.5054 -0.5045 -0.3936 
0.3907 0.1983 -0.1953 -0.0021 -0.0042 0.0008 0.0004 
 0.0844 -0.1198 -0.5045 0.5054 1.0242 -1.0023 -0.1995 
0.1899 0.3923 -0.3928 -0.0004 -0.0008 0.0042 0.0021 
 -0.1113 0.1265 0.5054 -0.5045 -1.0023 1.0242 0.1899 -
0.1995 -0.3928 0.3923 -0.0008 -0.0004 0.0021 0.0042 
 0.0425 0.0093 0.3907 -0.3936 -0.1995 0.1899 1.0242 -
1.0023 -0.5087 0.4970 0.1139 -0.1113 -0.0586 0.0510 
 -0.0076 -0.0416 -0.3936 0.3907 0.1899 -0.1995 -1.0023 
1.0242 0.4970 -0.5087 -0.1113 0.1139 0.0510 -0.0586 
 -0.0463 -0.0105 -0.1953 0.1983 0.3923 -0.3928 -0.5087 
0.4970 1.0242 -1.0023 -0.0586 0.0510 0.1139 -0.1113 
 0.0063 0.0379 0.1983 -0.1953 -0.3928 0.3923 0.4970 -
0.5087 -1.0023 1.0242 0.0510 -0.0586 -0.1113 0.1139 
 0.0021 0.0042 -0.0042 -0.0021 -0.0004 -0.0008 0.1139 -
0.1113 -0.0586 0.0510 0.2345 -0.2193 -0.1155 0.1054 
 0.0042 0.0021 -0.0021 -0.0042 -0.0008 -0.0004 -0.1113 
0.1139 0.0510 -0.0586 -0.2193 0.2345 0.1054 -0.1155 
 -0.0008 -0.0004 0.0004 0.0008 0.0042 0.0021 -0.0586 
0.0510 0.1139 -0.1113 -0.1155 0.1054 0.2345 -0.2193 
 -0.0004 -0.0008 0.0008 0.0004 0.0021 0.0042 0.0510 -
0.0586 -0.1113 0.1139 0.1054 -0.1155 -0.2193 0.2345 
 -0.0042 -0.0210 0.0084 0.0421 0.0210 0.0042 -0.0421 -
0.0084 0.0421 0.0084 -0.0210 -0.0042 0.0210 0.0042 
 0.0210 0.0042 -0.0421 -0.0084 -0.0042 -0.0210 0.0084 
0.0421 -0.0084 -0.0421 0.0042 0.0210 -0.0042 -0.0210 
 0.0421 0.0084 -0.0210 -0.0042 -0.0084 -0.0421 0.0421 
0.0084 -0.0421 -0.0084 0.0210 0.0042 -0.0210 -0.0042 
 -0.0084 -0.0421 0.0042 0.0210 0.0421 0.0084 -0.0084 -
0.0421 0.0084 0.0421 -0.0042 -0.0210 0.0042 0.0210 
 -0.0210 -0.0042 0.0421 0.0084 0.0042 0.0210 -0.0210 -
0.0042 0.0210 0.0042 -0.0421 -0.0084 0.0421 0.0084 
 0.0042 0.0210 -0.0084 -0.0421 -0.0210 -0.0042 0.0042 
0.0210 -0.0042 -0.0210 0.0084 0.0421 -0.0084 -0.0421 
 0.0084 0.0421 -0.0042 -0.0210 -0.0421 -0.0084 0.0210 
0.0042 -0.0210 -0.0042 0.0421 0.0084 -0.0421 -0.0084 
 -0.0421 -0.0084 0.0210 0.0042 0.0084 0.0421 -0.0042 -
0.0210 0.0042 0.0210 -0.0084 -0.0421 0.0084 0.0421 
 
 Columns 29 through 36 
 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 0 0 0 0 0 0 0 
0 
 -0.0421 0.0084 0.0042 -0.0210 -0.0084 0.0421 0.0210 -
0.0042 
 -0.0084 0.0421 0.0210 -0.0042 -0.0421 0.0084 0.0042 -
0.0210 
 -0.0042 0.0210 0.0421 -0.0084 -0.0210 0.0042 0.0084 -
0.0421 
 -0.0210 0.0042 0.0084 -0.0421 -0.0042 0.0210 0.0421 -
0.0084 
 0.0084 -0.0421 -0.0210 0.0042 0.0421 -0.0084 -0.0042 
0.0210 
 0.0421 -0.0084 -0.0042 0.0210 0.0084 -0.0421 -0.0210 
0.0042 
 0.0210 -0.0042 -0.0084 0.0421 0.0042 -0.0210 -0.0421 
0.0084 
 0.0042 -0.0210 -0.0421 0.0084 0.0210 -0.0042 -0.0084 
0.0421 
 -0.0421 0.0084 0.0421 -0.0084 -0.0210 0.0042 0.0210 -
0.0042 
 -0.0084 0.0421 0.0084 -0.0421 -0.0042 0.0210 0.0042 -
0.0210 
 0.0421 -0.0084 -0.0421 0.0084 0.0210 -0.0042 -0.0210 
0.0042 
 0.0084 -0.0421 -0.0084 0.0421 0.0042 -0.0210 -0.0042 
0.0210 
 -0.0210 0.0042 0.0210 -0.0042 -0.0421 0.0084 0.0421 -
0.0084 
 -0.0042 0.0210 0.0042 -0.0210 -0.0084 0.0421 0.0084 -
0.0421 
 0.0210 -0.0042 -0.0210 0.0042 0.0421 -0.0084 -0.0421 
0.0084 
 0.0042 -0.0210 -0.0042 0.0210 0.0084 -0.0421 -0.0084 
0.0421 
 0.7897 -0.7830 -0.3932 0.3915 0.3915 -0.3932 -0.1974 
0.1941 
 -0.7830 0.7897 0.3915 -0.3932 -0.3932 0.3915 0.1941 -
0.1974 
 -0.3932 0.3915 0.7897 -0.7830 -0.1974 0.1941 0.3915 -
0.3932 
 0.3915 -0.3932 -0.7830 0.7897 0.1941 -0.1974 -0.3932 
0.3915 
 0.3915 -0.3932 -0.1974 0.1941 0.7897 -0.7830 -0.3932 
0.3915 
 -0.3932 0.3915 0.1941 -0.1974 -0.7830 0.7897 0.3915 -
0.3932 
 -0.1974 0.1941 0.3915 -0.3932 -0.3932 0.3915 0.7897 -
0.7830 
 0.1941 -0.1974 -0.3932 0.3915 0.3915 -0.3932 -0.7830 
0.7897 
 
 
 
>> k = K(13:36,13:36) 
 
k = 
 
 1.0e+008 * 
 
 Columns 1 through 14 
 
 0.4689 -0.4386 -0.2269 0.2193 0.0844 -0.1113 -0.0376 
0.0468 -0.0379 0.0105 0.0416 -0.0093 0.0008 0.0004 
 -0.4386 0.4689 0.2193 -0.2269 -0.1198 0.1265 0.0552 -
0.0796 -0.0063 0.0463 0.0076 -0.0425 0.0004 0.0008 
 -0.2269 0.21930.4689 -0.4386 -0.0376 0.0468 0.0844 -
0.1113 0.0425 -0.0076 -0.0463 0.0063 0.0021 0.0042 
 0.2193 -0.2269 -0.4386 0.4689 0.0552 -0.0796 -0.1198 
0.1265 0.0093 -0.0416 -0.0105 0.0379 0.0042 0.0021 
 0.0844 -0.1198 -0.0376 0.0552 1.0242 -1.0023 -0.5045 
0.5054 0.3907 -0.3936 -0.1953 0.1983 -0.0042 -0.0021 
 -0.1113 0.1265 0.0468 -0.0796 -1.0023 1.0242 0.5054 -
0.5045 -0.3936 0.3907 0.1983 -0.1953 -0.0021 -0.0042 
 -0.0376 0.0552 0.0844 -0.1198 -0.5045 0.5054 1.0242 -
1.0023 -0.1995 0.1899 0.3923 -0.3928 -0.0004 -0.0008 
 0.0468 -0.0796 -0.1113 0.1265 0.5054 -0.5045 -1.0023 
1.0242 0.1899 -0.1995 -0.3928 0.3923 -0.0008 -0.0004 
 -0.0379 -0.0063 0.0425 0.0093 0.3907 -0.3936 -0.1995 
0.1899 1.0242 -1.0023 -0.5087 0.4970 0.1139 -0.1113 
 0.0105 0.0463 -0.0076 -0.0416 -0.3936 0.3907 0.1899 -
0.1995 -1.0023 1.0242 0.4970 -0.5087 -0.1113 0.1139 
 0.0416 0.0076 -0.0463 -0.0105 -0.1953 0.1983 0.3923 -
0.3928 -0.5087 0.4970 1.0242 -1.0023 -0.0586 0.0510 
 -0.0093 -0.0425 0.0063 0.0379 0.1983 -0.1953 -0.3928 
0.3923 0.4970 -0.5087 -1.0023 1.0242 0.0510 -0.0586 
 0.0008 0.0004 0.0021 0.0042 -0.0042 -0.0021 -0.0004 -
0.0008 0.1139 -0.1113 -0.0586 0.0510 0.2345 -0.2193 
 0.0004 0.0008 0.0042 0.0021 -0.0021 -0.0042 -0.0008 -
0.0004 -0.1113 0.1139 0.0510 -0.0586 -0.2193 0.2345 
 -0.0021 -0.0042 -0.0008 -0.0004 0.0004 0.0008 0.0042 
0.0021 -0.0586 0.0510 0.1139 -0.1113 -0.1155 0.1054 
 -0.0042 -0.0021 -0.0004 -0.0008 0.0008 0.0004 0.0021 
0.0042 0.0510 -0.0586 -0.1113 0.1139 0.1054 -0.1155 
 -0.0421 -0.0084 -0.0042 -0.0210 0.0084 0.0421 0.0210 
0.0042 -0.0421 -0.0084 0.0421 0.0084 -0.0210 -0.0042 
 0.0084 0.0421 0.0210 0.0042 -0.0421 -0.0084 -0.0042 -
0.0210 0.0084 0.0421 -0.0084 -0.0421 0.0042 0.0210 
 0.0042 0.0210 0.0421 0.0084 -0.0210 -0.0042 -0.0084 -
0.0421 0.0421 0.0084 -0.0421 -0.0084 0.0210 0.0042 
 -0.0210 -0.0042 -0.0084 -0.0421 0.0042 0.0210 0.0421 
0.0084 -0.0084 -0.0421 0.0084 0.0421 -0.0042 -0.0210 
 -0.0084 -0.0421 -0.0210 -0.0042 0.0421 0.0084 0.0042 
0.0210 -0.0210 -0.0042 0.0210 0.0042 -0.0421 -0.0084 
 0.0421 0.0084 0.0042 0.0210 -0.0084 -0.0421 -0.0210 -
0.0042 0.0042 0.0210 -0.0042 -0.0210 0.0084 0.0421 
 0.0210 0.0042 0.0084 0.0421 -0.0042 -0.0210 -0.0421 -
0.0084 0.0210 0.0042 -0.0210 -0.0042 0.0421 0.0084 
 -0.0042 -0.0210 -0.0421 -0.0084 0.0210 0.0042 0.0084 
0.0421 -0.0042 -0.0210 0.0042 0.0210 -0.0084 -0.0421 
 
 Columns 15 through 24 
 
 -0.0021 -0.0042 -0.0421 0.0084 0.0042 -0.0210 -0.0084 
0.0421 0.0210 -0.0042 
 -0.0042 -0.0021 -0.0084 0.0421 0.0210 -0.0042 -0.0421 
0.0084 0.0042 -0.0210 
 -0.0008 -0.0004 -0.0042 0.0210 0.0421 -0.0084 -0.0210 
0.0042 0.0084 -0.0421 
 -0.0004 -0.0008 -0.0210 0.0042 0.0084 -0.0421 -0.0042 
0.0210 0.0421 -0.0084 
 0.0004 0.0008 0.0084 -0.0421 -0.0210 0.0042 0.0421 -
0.0084 -0.0042 0.0210 
 0.0008 0.0004 0.0421 -0.0084 -0.0042 0.0210 0.0084 -
0.0421 -0.0210 0.0042 
 0.0042 0.0021 0.0210 -0.0042 -0.0084 0.0421 0.0042 -
0.0210 -0.0421 0.0084 
 0.0021 0.0042 0.0042 -0.0210 -0.0421 0.0084 0.0210 -
0.0042 -0.0084 0.0421 
 -0.0586 0.0510 -0.0421 0.0084 0.0421 -0.0084 -0.0210 
0.0042 0.0210 -0.0042 
 0.0510 -0.0586 -0.0084 0.0421 0.0084 -0.0421 -0.0042 
0.0210 0.0042 -0.0210 
 0.1139 -0.1113 0.0421 -0.0084 -0.0421 0.0084 0.0210 -
0.0042 -0.0210 0.0042 
 -0.1113 0.1139 0.0084 -0.0421 -0.0084 0.0421 0.0042 -
0.0210 -0.0042 0.0210 
 -0.1155 0.1054 -0.0210 0.0042 0.0210 -0.0042 -0.0421 
0.0084 0.0421 -0.0084 
 0.1054 -0.1155 -0.0042 0.0210 0.0042 -0.0210 -0.0084 
0.0421 0.0084 -0.0421 
 0.2345 -0.2193 0.0210 -0.0042 -0.0210 0.0042 0.0421 -
0.0084 -0.0421 0.0084 
 -0.2193 0.2345 0.0042 -0.0210 -0.0042 0.0210 0.0084 -
0.0421 -0.0084 0.0421 
 0.0210 0.0042 0.7897 -0.7830 -0.3932 0.3915 0.3915 -
0.3932 -0.1974 0.1941 
 -0.0042 -0.0210 -0.7830 0.7897 0.3915 -0.3932 -0.3932 
0.3915 0.1941 -0.1974 
 -0.0210 -0.0042 -0.3932 0.3915 0.7897 -0.7830 -0.1974 
0.1941 0.3915 -0.3932 
 0.0042 0.0210 0.3915 -0.3932 -0.7830 0.7897 0.1941 -
0.1974 -0.3932 0.3915 
 0.0421 0.0084 0.3915 -0.3932 -0.1974 0.1941 0.7897 -
0.7830 -0.3932 0.3915 
 -0.0084 -0.0421 -0.3932 0.3915 0.1941 -0.1974 -0.7830 
0.7897 0.3915 -0.3932 
 -0.0421 -0.0084 -0.1974 0.1941 0.3915 -0.3932 -0.3932 
0.3915 0.7897 -0.7830 
 0.0084 0.0421 0.1941 -0.1974 -0.3932 0.3915 0.3915 -
0.3932 -0.7830 0.7897 
 
 
 
>> f = [0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 4.6875 ; 0 ; 0 ; 
4.6875 ; 0 ; 0 ; 4.6875 ; 0 ; 0 ; 4.6875 ; 0 ; 0] 
 
f = 
 
 0 
 0 
 0 
 0 
 0 
 0 
 0 
 0 
 0 
 0 
 0 
 0 
 4.6875 
 0 
 0 
 4.6875 
 0 
 0 
 4.6875 
 0 
 0 
 4.6875 
 0 
 0 
 
>> u = k\f 
Warning: Matrix is close to singular or badly scaled. 
 Results may be inaccurate. RCOND = 1.551156e-017. 
 
u = 
 
 1.0e+008 * 
 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 0.0000 
 1.5729 
 1.5729 
 1.5729 
 1.5729 
 1.5729 
 1.5729 
 1.5729 
 1.5729 
 
 
 
 
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