/2
L
L
Rdx = 0 =
/2
L
L
{AE [C1 + 2C2
x] – 18000 + 300x}dx = 0 (8)
/2
22
12
0
2 18000 300
22
L
xx
AE C x C x
= 0
2
2
12
/2
2
L
L
x
AE
2
12
22
LL
CC
– 18000
2
L
+ 150
2
2
L
= 0
2
2
12
22
LL
2
2
2
L
Simplifying (10)
AE
2
12
22
LL
CC
– 9000 L + 150
2
2
L
= 0 (11.1)
AE
2
12
3
24
LL
CC
– 9000 L + 150
2
3
4
L
= 0 (11.2)
Subtracting (11.1) from (11.2)
2
L
2
L
L
6.66
3.642 10–4
13.32
8.099 10–4
19.98
1.384 10–3
u(0) = 0 satisfied
3 Ci’s. So need error to go to zero at 3 points.
3
L
2
3
L
A(x) E
du
dx
– P = 0
21
AA
3.61
Exact solution
u =
1
AE
0
x
P(x) dx
1
AE
0
u =
1
AE
3
5
45 3
x
x
+ C
u(0) = 0 C = 0
3
15
x
AE
Collocation method:
Substituting for R using Equation (3) into (4)
AE
2
1 2 3
23
33
LL
C C C
– 45 + 5
2
3
L
= 0
22
2 2 2
L L L
1
1
3
3
27
81
1
3
AE C1 L +
5
9
AE C2 L2 +
19
27
AE C3 L3 – 15 L +
95
81
L3 = 0
45
AE
128
2 3 4 2 4
1 2 3
2 3 4 2 4
L L L L L
AE
34
5 3 5
1 2 3
3 45
3 2 5 3
LL
C C C L L L
= 0
45
AE


0
Need 3 equations
Let W1 = x, W2 = x2, W3 = x3
0
L
{AE [C1 + 2C2x + 3C3x2] – 45 + 5x2} x dx = 0
0
L
{AE [C1 + 2C2x + 3C3x2] – 45 + 5x2} x2 dx = 0
0
4
3
2
3
L
Cx
Cx
x x x
L
1
0
2 3 4 2 4
AE
5
4
3 3 5
3
2
1
0
3
25
45
3 4 5 4 5
L
Cx
Cx
x x x
C
= 0
6
5
4 4 6
3
2
1
0
3
25
4 5 6 4 6
L
Cx
Cx
x x x
Simplifying
AE
4
3
22
3
2
1
3
25
45
2 3 4 2 4
CL
CL
LL
C
L4 = 0
5
4
3
3
2
1
3
3 2 5
CL
CL
L
AE
6
5
46
4
3
2
1
245 5
4 5 2 4 6
CL
CL
LL
CL
= 0
Solve for C1 – C3 using Mathcad
MAXIMUM VALUES
Figure 4 Free and deformed states
Try W 21 22 A36 steel based on critical buckling member 7. Assumed Le = 2.1 L
(conservative).
3.65
Figure P3–65 Truss bridge (FS = 3.0)
or 4.5 4.5
5
16
 square tube
3.66
132
3.67
Try one of these cross sections – 1) a square solid bar, 3.25 in. 3.25 in., 2) a 6 6
1
2
in. structural square tube, 3) a W 8 35 wide flange section. Any of these made of A
36 steel. The critical force of – 120,000 lb is in element EG. So Johnson buckling
formula dictates the section selected.
3.68
Figure P3–68 Howe scissors roof truss (FS = 2.0)
3
3.69
3.73
The recommended sqaure box section is 2 in. by 2 in. by ¼ in. thick.
3.74
135
3.75
Nodal Displacements
136
The three largest displacements are shown on one side of the truss with the mirrored side of the
truss having the same displacements
Principal Stress Plot