Truss Example

1
Truss Example
Solving Linear Systems with Matlab
2
Find the axial forces in each of the members of
this truss
Problem
Ax
Ay1
Ay2
100
3
the ASCII files
Ex1StructPoints.dat
Ex1StructConnect.dat
joint x-coord y-coord 1 0 0 2
0.5 0 3 1 0 4 1.5
0 5 2 0 6 0.5 0.5 7
1.5 0.5 8 1 1
member joint1 joint2 1 1 6 2
1 2 3 2 6 4 6 8
5 3 6 6 2 3 7 3
8 8 7 8 9 3 7 10
3 4 11 4 7 12 5
7 13 4 5
4
Skills needed to solve this problem
5
Understanding the linear system
Joint 8
å F F4 F7 F8 0
å Fx F4,x F7,x F8,x 0
å Fy F4,y F7,y F8,y 0
F4
F8
F7
6
Understanding the linear system
The Unit Vector
F
e
F
Fx
ex
F
Fy
ey
F
7
Understanding the linear system
Joint 8
å F F4 F7 F8 0
å Fx e4,x F4 e7,x F7 e8,x F8 0
å Fy e4,y F4 e7,y F7 e8,y F8 0
F4
F8
F7
8
Understanding the linear system
-Ax 0 0 0 0 0 0 0 -Ay1 0 100 0 -Ay2 0 0 0
0 0 0 ex,4-5 ex,5-4 0 0 0 0 0 0 ey,4-5 ey,5-4 0 0
0
ex,1-2 ex,2-1 0 0 0 0 0 0 ey,1-2 ey,2-1 0 0 0 0 0
0
ex,1-6 0 0 0 0 ex,6-1 0 0 ey,1-6 0 0 0 0 ey,6-1 0
0
0 ex,2-6 0 0 0 ex,6-2 0 0 0 ey,2-6 0 0 0 ey,6-2 0
0
0 0 0 0 0 ex,6-8 0 ex,8-6 0 0 0 0 0 ey,6-8 0 ey,8-
6
0 0 ex,3-6 0 0 ex,6-3 0 0 0 0 ey,3-6 0 0 ey,6-3 0
0
0 ex,2-3 ex,3-2 0 0 0 0 0 0 ey,2-3 ey,3-2 0 0 0 0
0
0 0 ex,3-8 0 0 0 0 ex,8-3 0 0 ey,3-8 0 0 0 0 ey,8-
3
0 0 0 0 0 0 ex,7-8 ex,8-7 0 0 0 0 0 0 ey,7-8 ey,8-
7
0 0 ex,3-7 0 0 0 ex,7-3 0 0 0 ey,3-7 0 0 0 ey,7-3
0
0 0 ex,3-4 ex,4-3 0 0 0 0 0 0 ey,3-4 ey,4-3 0 0 0
0
0 0 0 ex,4-7 0 0 ex,7-4 0 0 0 0 ey,4-7 0 0 ey,7-4
0
0 0 0 0 ex,5-7 0 ex,7-5 0 0 0 0 0 ey,5-7 0 ey,7-5
0
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13
Joints 1-8 x-components

Joints 1-8 y-components
9
Understanding the linear system
-Ax 0 0 0 0 0 0 0 -Ay1 0 100 0 -Ay2 0 0 0
ex,1-2 ex,2-1 0 0 0 0 0 0 ey,1-2 ey,2-1 0 0 0 0 0
0
ex,1-6 0 0 0 0 ex,6-1 0 0 ey,1-6 0 0 0 0 ey,6-1 0
0
0 ex,2-6 0 0 0 ex,6-2 0 0 0 ey,2-6 0 0 0 ey,6-2 0
0
0 0 0 0 0 ex,6-8 0 ex,8-6 0 0 0 0 0 ey,6-8 0 ey,8-
6
0 0 ex,3-6 0 0 ex,6-3 0 0 0 0 ey,3-6 0 0 ey,6-3 0
0
0 ex,2-3 ex,3-2 0 0 0 0 0 0 ey,2-3 ey,3-2 0 0 0 0
0
0 0 ex,3-8 0 0 0 0 ex,8-3 0 0 ey,3-8 0 0 0 0 ey,8-
3
0 0 0 0 0 0 ex,7-8 ex,8-7 0 0 0 0 0 0 ey,7-8 ey,8-
7
0 0 ex,3-7 0 0 0 ex,7-3 0 0 0 ey,3-7 0 0 0 ey,7-3
0
0 0 ex,3-4 ex,4-3 0 0 0 0 0 0 ey,3-4 ey,4-3 0 0 0
0
0 0 0 ex,4-7 0 0 ex,7-4 0 0 0 0 ey,4-7 0 0 ey,7-4
0
0 0 0 0 ex,5-7 0 ex,7-5 0 0 0 0 0 ey,5-7 0 ey,7-5
0
0 0 0 ex,4-5 ex,5-4 0 0 0 0 0 0 ey,4-5 ey,5-4 0 0
0
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13

10
Understanding the linear system
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13
0 0 0 0 0 0 0 0 100 0 0 0 0
0 0 ex,4-5 ex,5-4 0 0 0 0 0 ey,4-5 0 0 0
ex,2-1 0 0 0 0 0 0 ey,2-1 0 0 0 0 0
0 0 0 0 ex,6-1 0 0 0 0 0 ey,6-1 0 0
ex,2-6 0 0 0 ex,6-2 0 0 ey,2-6 0 0 ey,6-2 0 0
0 0 0 0 ex,6-8 0 ex,8-6 0 0 0 ey,6-8 0 ey,8-6
0 ex,3-6 0 0 ex,6-3 0 0 0 ey,3-6 0 ey,6-3 0 0
ex,2-3 ex,3-2 0 0 0 0 0 ey,2-3 ey,3-2 0 0 0 0
0 ex,3-8 0 0 0 0 ex,8-3 0 ey,3-8 0 0 0 ey,8-3
0 0 0 0 0 ex,7-8 ex,8-7 0 0 0 0 ey,7-8 ey,8-7
0 ex,3-7 0 0 0 ex,7-3 0 0 ey,3-7 0 0 ey,7-3 0
0 ex,3-4 ex,4-3 0 0 0 0 0 ey,3-4 ey,4-3 0 0 0
0 0 ex,4-7 0 0 ex,7-4 0 0 0 ey,4-7 0 ey,7-4 0
0 0 0 ex,5-7 0 ex,7-5 0 0 0 0 0 ey,7-5 0

11
Programming the solution
Problem Solving Outline
Extract data from ASCII files
Find unit vector components
Compile all equilibrium equations from all joints
Eliminate unnecessary equations
Solve equations by Gaussian elimination
12
Programming the solution
Extract data from ASCII files
Ex1StructPoints.dat
Ex1StructConnect.dat
joint x-coord y-coord 1 0 0 2
0.5 0 3 1 0 4 1.5
0 5 2 0 6 0.5 0.5 7
1.5 0.5 8 1 1
member joint1 joint2 1 1 6 2
1 2 3 2 6 4 6 8
5 3 6 6 2 3 7 3
8 8 7 8 9 3 7 10
3 4 11 4 7 12 5
7 13 4 5
13
Programming the solution
Extract data from ASCII files
Ex1StructPoints.dat
Ex1StructPointsMod.dat
joint x-coord y-coord 1 0 0 2
0.5 0 3 1 0 4 1.5
0 5 2 0 6 0.5 0.5 7
1.5 0.5 8 1 1
1 0 0 2 0.5 0 3 1
0 4 1.5 0 5 2 0 6
0.5 0.5 7 1.5 0.5 8 1
1
14
Programming the solution
Extract data from ASCII files
Ex1StructConnect.dat
Ex1StructConnectMod.dat
member joint1 joint2 1 1 6 2
1 2 3 2 6 4 6 8
5 3 6 6 2 3 7 3
8 8 7 8 9 3 7 10
3 4 11 4 7 12 5
7 13 4 5
1 1 6 2 1 2 3 2
6 4 6 8 5 3 6 6
2 3 7 3 8 8 7
8 9 3 7 10 3 4 11
4 7 12 5 7 13 4 5
15
Programming the solution
Extract data from ASCII files
load Ex1StructJointsMod.dat load
Ex1StructConnectMod.dat Matrix with
coordinates of joints coord Ex1StructJointsMod(
,23) Connectivity matrix C
Ex1StructConnectMod(,23)
16
Programming the solution
Extract data from ASCII files
load Ex1StructJointsMod.dat load
Ex1StructConnectMod.dat Matrix with
coordinates of joints coord Ex1StructJointsMod(
,23) Connectivity matrix C
Ex1StructConnectMod(,23)
17
Programming the solution
Find unit vector components
num_joints size(Ex2StructJointsMod,1)
Distance matrix i X j for i
1num_joints for j 1num_joints
dist(i,j) ((coord(i,1)-coord(j,1))2
(coord(i,2)-coord(j,2))2)(1/2)
end end
18
Programming the solution
Find unit vector components
Unit vector matrices (x-component and
y-component) iXj for i 1num_joints
for j 1num_joints if (dist(i,j) 0)
x_comp(i,j) 0 y_comp(i,j) 0
else x_comp(i,j)
(coord(j,1)-coord(i,1))/dist(i,j)
y_comp(i,j) (coord(j,2)-coord(i,2))/dist(i,j)
end end end
19
Programming the solution
Compile all equilibrium equations from all joints
num_members size(Ex1StructConnectMod,1)
Matrix construction for linear system for m
1num_members Mat( C(m,1) , m) x_comp(
C(m,1) , C(m,2) ) Mat( C(m,2) , m)
x_comp( C(m,2) , C(m,1) ) Mat(
C(m,1)num_joints, m) y_comp( C(m,1) , C(m,2)
) Mat( C(m,2)num_joints, m) y_comp( C(m,2)
, C(m,1) ) end Equilibrium constant for net
force at each joint eq_const zeros(2num_joints,
1) eq_const(num_joints2)100
20
Programming the solution
Compile all equilibrium equations from all joints
num_members size(Ex1StructConnectMod,1)
Matrix construction for linear system for m
1num_members Mat( C(m,1) , m) x_comp(
C(m,1) , C(m,2) ) Mat( C(m,2) , m)
x_comp( C(m,2) , C(m,1) ) Mat(
C(m,1)num_joints, m) y_comp( C(m,1) , C(m,2)
) Mat( C(m,2)num_joints, m) y_comp( C(m,2)
, C(m,1) ) end Equilibrium constant for net
force at each joint eq_const zeros(2num_joints,
1) eq_const(num_joints2)100
21
Programming the solution
Compile all equilibrium equations from all joints
num_members size(Ex1StructConnectMod,1)
Matrix construction for linear system for m
1num_members Mat( C(m,1) , m) x_comp(
C(m,1) , C(m,2) ) Mat( C(m,2) , m)
x_comp( C(m,2) , C(m,1) ) Mat(
C(m,1)num_joints, m) y_comp( C(m,1) , C(m,2)
) Mat( C(m,2)num_joints, m) y_comp( C(m,2)
, C(m,1) ) end Equilibrium constant for net
force at each joint eq_const zeros(2num_joints,
1) eq_const(num_joints3)100
22
Programming the solution
Eliminate unnecessary equations
Elimination of unnecessary equations
rows 210, 12, 1420 AA Mat( rows
, ) rhs eq_const( rows , )
23
Programming the solution
Eliminate unnecessary equations
Elimination of unnecessary equations
rows 210, 12, 1420 AA Mat( rows
, ) rhs eq_const( rows , )
24
Programming the solution
Solve equations by Gaussian elimination
Solve for forces AA\rhs
25
Programming the solution
The Complete Code
clear load Ex1StructJointsMod.dat load
Ex1StructConnectMod.dat num_joints
size(Ex1StructJointsMod,1) num_members
size(Ex1StructConnectMod,1) Matrix with
coordinates of joints coord Ex1StructJointsMod(
,23) Connectivity matrix C
Ex1StructConnectMod(,23) Distance
matrix i X j for i 1num_joints for
j 1num_joints dist(i,j)
((coord(i,1)-coord(j,1))2
(coord(i,2)-coord(j,2))2)(1/2) end end
Unit vector matrices (x-component and
y-component) iXj for i 1num_joints
for j 1num_joints if (dist(i,j) 0)
x_comp(i,j) 0 y_comp(i,j) 0
else
x_comp(i,j) (coord(j,1)-coord(i,1))/dis
t(i,j) y_comp(i,j) (coord(j,2)-coord(i
,2))/dist(i,j) end
end end Matrix construction for linear
system for m 1num_members Mat( C(m,1)
, m) x_comp( C(m,1) , C(m,2) ) Mat( C(m,2)
, m) x_comp( C(m,2) , C(m,1) ) Mat(
C(m,1) num_joints , m) y_comp( C(m,1) ,
C(m,2) ) Mat( C(m,2) num_joints , m)
y_comp( C(m,2) , C(m,1) ) end Equilibrium
constant for net force at each joint eq_const
zeros(2num_joints,1) eq_const(num_joints3)100
Elimination of unnecessary equations
rows 28, 1012, 1416 AA Mat(
rows , ) rhs eq_const( rows , )
Solve for forces AA\rhs
26
Programming the solution
The Matlab Output
ans
-70.7107 50.0000 0 -70.7107
0 50.0000 100.0000 -70.7107 0
50.0000 0 -70.7107 50.0000