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1. The Figure 1 shows the axial force F, in each of the 21 member pin-connected

ID: 2074788 • Letter: 1

Question

1. The Figure 1 shows the axial force F, in each of the 21 member pin-connected truss. The system of equations can be obtained by using the mechanical equilibrium principle of the system with 21 equations as follows. Solve the system of equation to find the unknown axial forces. Use (i) Jacobi method. (ii) Gauss-Seidel method, (iii) SOR with o (iv) SOR with > 1, and (v) MATLAB . Comment on the suitability of methods used in (i) to (iv) for solving this problem. 13 bF-bF0.7433F0 -F -0.7433 F-5000 1216 F21-cF. = 10000 sin 600 Consider. a=0.7071, b=0.9806. c =0.1961, d=0 669 5000 N 60° 10000 N 8000 N Figure 1

Explanation / Answer

In Jacobi method we first find the variable in terms of other variables, like if we have three simulatneous equations in x,y and z.

a1x+b1y+c1z=d1…………………(1)

a2x+b2y+c2z=d2…………………(2)

a3x+b3y+c3z=d3…………………(3)

we write x, y and z as,

x = (1/a1)*((d1) – (b1)y – (c1)z)…………....(4)

y = (1/b2)*((d2) – (a2)x – (c2)z)…………….(5)

z = (1/c3)*((d3) – (a3)x – (b3)y)…………….(6)

and then we iterate with intial guesses as x0=0,y0=0 and z0=0. We substitute these values in eq.(4),(5) and (6) to get new set of values of x,y and z. The more the number of iterations the greater is the accuracy of result.

Following is the MATLAB code.

-----------------------------------------------------------------------------------------------------------------------------------------------------------

clc;clear all;
% intial value of given constants
a=0.7071;b=0.9806;c=0.1961;d=0.669;
% Finding the each force term ( in terms of other forces)
f1=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
-(a*F3);
f2=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
-(a*F3)+F6;
f3=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
-(1/a)*F5;
f5=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
-(a*F7);
f4=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
(a*F7)+F8;
f6=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
-(a*F7)+F10;
f7=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
-(1/a)*F9;
f10=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
-(a*F11);
f11=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
-(1/a)*F12;
f12=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
F16;
f13=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
0;
f9=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
(b*F14+a*F15-a*F11-F9);
f8=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
(c*F14+a*F11+a*F15+F13);

f14=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
((1/c)*(F17+d*F19+c*F18));
f15=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
(-(1/a)*F17);
f16=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
(F20-a*F15);
f17=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
(-(c*F18-c*F14+d*F19));
f18=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
(1000*cosd(60));
f19=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
(-(1/d)*F21);
f20=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
(-(-5000+0.7433*F19));
f21=@(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)...
(1000*sind(60)+c*F18);

% Initial guesses for all forces
F1=0;F2=0;F3=0;F4=0;F5=0;F6=0;F7=0;F8=0;F9=0;F10=0;F11=0;F12=0;F13=0;F14=0;
F15=0;F16=0;F17=0;F18=0;F19=0;F20=0;F21=0;

% Iterating the solution for 5 iterations only
for i=1:5
f1(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f2(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f3(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f4(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f5(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f6(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f7(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f8(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f9(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f10(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f11(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f12(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f13(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f14(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f15(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f16(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f17(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f18(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f19(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f20(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
f21(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21)
  
  
a1= f1(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a2=f2(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a3= f3(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a4=f4(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a5=f5(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a6=f6(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a7=f7(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a8=f8(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a9=f9(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a10=f10(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a11=f11(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a12=f12(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a13=f13(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a14=f14(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a15=f15(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a16=f16(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a17=f17(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a18=f18(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a19=f19(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a20=f20(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);
a21=f21(F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,F11,F12,F13,F14,F15,F16,F17,F18,F19,F20,F21);

F1=a1;F2=a2;F3=a3;F4=a4;F5=a5;F6=a6;F7=a7;F8=a8;F9=a9;F10=a10;F11=a11;F12=a12;F13=a13;F14=a14;
F15=a15;F16=a16;F17=a17;F18=a18;F19=a19;F20=a20;F21=a21;
end
% display results
F1
F2
F3
F4
F5
F6
F7
F8
F9
F10
F11
F12
F13
F14
F15
F16
F17
F18
F19
F20
F21

-----------------------------------------------------------------------------------------------------------------------------------------------------------

The result I am getting for 5 iterations (although you are encouraged to go for more number of iterations for more accurate result) is as follow ;

F1 =
0
F2 =
0
F3 =
0
F4 =
-1.2583e+03
F5 =
490.3000
F6 =
490.3000
F7 =
6.6791e+03
F8 =
-5.8660e+03
F9 =
8.8568e+03
F10 =
5000
F11 =
-7.0711e+03
F12 =
5.8642e+03
F13 =
0
F14 =
-4.4162e+03
F15 =
-1.6078e-13
F16 =
6.9372e+03
F17 =
866.0254
F18 =
500.0000
F19 =
-1.4411e+03
F20 =
6.0711e+03
F21 =
964.0754

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