the problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) /...
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![Page 1: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/1.jpg)
The Problem
![Page 2: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/2.jpg)
Elements and Nodes
144 in 3
4
5
3
45
192 in 108 in
Plane Truss
2
1
3
1
2
10 kips
100 kips
(AE/L) = (AE/L) = AE/L1 2
Global System Degrees of Freedom
4P , X 4
3P , X 3 5P , X 5
6P , X 6
2P , X 2
1P ,
1 u1
Local Element Degrees of Freedom
X 1
f , u 4 u4f , u
3 u3f , u2 u2f , u
![Page 3: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/3.jpg)
sin1 = (-12 - 0) / (20) = -0.6cos1 = (16 - 0) / (20) = 0.8sin2 = (12 - 0) / (15) = 0.8cos2 = (9 - 0) / (15) = 0.6
Sines and Cosines
![Page 4: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/4.jpg)
3 4 1 2
S1 AEL
1
0.64 0.48 0.64 0.48
0.48 0.36 0.48 0.36
0.64 0.48 0.64 0.48
0.48 0.36 0. 48 0.36
3
4
1
2
S2 AEL
2
0.36 0.48 0.36 0.48
0. 48 0.64 0.48 0.64
0.36 0.48 0.36 0.48
0.48 0.64 0.48 0.64
1
2
5
6
1 2 5 6
Element Matrices [S]
![Page 5: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/5.jpg)
K Sii1
NM
System Stiffness Matrix
![Page 6: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/6.jpg)
3 4 1 2
S1 AEL
1
0.64 0.48 0.64 0.48
0.48 0.36 0.48 0.36
0.64 0.48 0.64 0.48
0.48 0.36 0. 48 0.36
3
4
1
2
S2 AEL
2
0.36 0.48 0.36 0.48
0. 48 0.64 0.48 0.64
0.36 0.48 0.36 0.48
0.48 0.64 0.48 0.64
1
2
5
6
1 2 5 6
Element Matrices [S]
![Page 7: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/7.jpg)
Summing Element Stiffnesses
.64 -.48 -.64 .48
-.48 .36 .48 -.36
-.64 .48 .64 -.48
.48 -.36 -.48 .36
![Page 8: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/8.jpg)
Summing Element Stiffnesses
.64+.36 -.48+.48 -.64 .48 -.36 -.48
-.48+.48 .36+.64 .48 -.36 -.48 -.64
-.64 .48 .64 -.48
.48 -.36 -.48 .36
-.36 -.48 .36 .48
-.48 -.64 .48 .64
![Page 9: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/9.jpg)
KAEL
0.64 0.36 0.48 0.48 0.64 0.48 0.36 0.48
0.48 0.48 0.36 0.48 0.48 0.36 0.48 0.64
0.64 0.48 0.64 0.48 0 0
0.48 0.36 0.48 0.36 0 0
0.36 0.48 0 0 0.36 0.48
0.48 0.64 0 0 0.48 0.64
1
2
3
4
5
6
1 2 3 4 5 6
Two Matrix Contributions
![Page 10: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/10.jpg)
1 2 3 4 5 6
K AEL
1.00 0.00 0.64 0.48 0.36 0.48
0.00 1.00 0.48 0.36 0.48 0.64
0.64 0.48 0.64 0.48 0 0
0.48 0.36 0.48 0.36 0 0
0.36 0. 48 0 0 0.36 0.48
0.48 0.64 0 0 0.48 0.64
1
2
3
4
5
6
Final [K]
![Page 11: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/11.jpg)
P1
P2
P3
P4
P5
P6
AE
L
1.00 0.00 0.64 0.48 0.36 0.48
0.00 1.00 0.48 0.36 0.48 0.64
0.64 0.48 0.64 0.48 0 0
0.48 0.36 0.48 0.36 0 0
0.36 0.48 0 0 0.36 0.48
0.48 0.64 0 0 0.48 0.64
X1
X2
X3
X4
X5
X6
Final Equation P=KX
![Page 12: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/12.jpg)
System Stiffness Matrices
1
2 4
53
6
7 1
7 6
32
5
4
Node Numbering Scheme 1 Node Numbering Scheme 2
KScheme 1
X X X X X X 0 0 0 0 0 0 0 0
X X X X X X 0 0 0 0 0 0 0 0
X X X X X X X X 0 0 0 0 0 0
X X X X X X X X 0 0 0 0 0 0
X X X X X X X X X X 0 0 0 0
X X X X X X X X X X 0 0 0 0
0 0 X X X X X X X X X X 0 0
0 0 X X X X X X X X X X 0 0
0 0 0 0 X X X X X X X X X X
0 0 0 0 X X X X X X X X X X
0 0 0 0 0 0 X X X X X X X X
0 0 0 0 0 0 X X X X X X X X
0 0 0 0 0 0 0 0 X X X X X X
0 0 0 0 0 0 0 0 X X X X X X
KScheme 2
X X X X 0 0 0 0 0 0 0 0 X X
X X X X 0 0 0 0 0 0 0 0 X X
X X X X X X 0 0 0 0 X X X X
X X X X X X 0 0 0 0 X X X X
0 0 X X X X X X X X 0 0 0 0
0 0 X X X X X X X X 0 0 0 0
0 0 0 0 X X X X X X X X 0 0
0 0 0 0 X X X X X X X X 0 0
0 0 0 0 X X X X X X X X X X
0 0 0 0 X X X X X X X X X X
0 0 X X X X 0 0 X X X X X X
0 0 X X X X 0 0 X X X X X X
X X X X 0 0 0 0 0 0 X X X X
X X X X 0 0 0 0 0 0 X X X X
![Page 13: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/13.jpg)
Solving the System of Equations
![Page 14: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/14.jpg)
Modify for Known Loads
Elements and Nodes
144 in 3
4
5
3
45
192 in 108 in
Plane Truss
2
1
3
1
2
10 kips
100 kips
(AE/L) = (AE/L) = AE/L1 2
Global System Degrees of Freedom
4P , X
4
3P , X3 5P , X5
6P , X
6
2P , X2
1P ,
1 u1
Local Element Degrees of Freedom
X 1
f , u 4 u4f , u
3 u3f , u2 u2f , u
10
100
P3
P4
P5
P6
AE
L
1.00 0.00 0.64 0.48 0.36 0.48
0.00 1.00 0.48 0.36 0.48 0.64
0.64 0.48 0.64 0.48 0 0
0.48 0.36 0.48 0.36 0 0
0.36 0.48 0 0 0.36 0.48
0. 48 0.64 0 0 0.48 0.64
X1
X2
X3
X4
X5
X6
![Page 15: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/15.jpg)
Modify for Boundary Conditions
Elements and Nodes
144 in 3
4
5
3
45
192 in 108 in
Plane Truss
2
1
3
1
2
10 kips
100 kips
(AE/L) = (AE/L) = AE/L1 2
Global System Degrees of Freedom
4P , X4
3P , X3 5P , X5
6P , X 6
2P , X2
1P ,
1 u1
Local Element Degrees of Freedom
X 1
f , u 4 u4f , u
3 u3f , u2 u2f , u
10
100
P3
P 4
P5
P6
AE
L
1.00 0.00 0.64 0.48 0.36 0.48
0.00 1.00 0.48 0.36 0.48 0.64
0.64 0.48 0.64 0.48 0 0
0.48 0.36 0.48 0.36 0 0
0.36 0.48 0 0 0.36 0.48
0. 48 0.64 0 0 0.48 0.64
X1
X 2
0
0
0
0
![Page 16: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/16.jpg)
Modify to Ease Solution
10
100
0
0
0
0
AE
L
1.00 0.00 0.64 0.48 0.36 0.48
0.00 1.00 0.48 0.36 0.48 0.64
0 0 L / AE 0 0 0
0 0 0 L / AE 0 0
0 0 0 0 L / AE 0
0 0 0 0 0 L / AE
X1
X2
X3
X4
X5
X6
![Page 17: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/17.jpg)
Return Symmetry
10 ( 0.64AE / L)(X3) (0.48AE / L)(X4) ( 0.36AE / L)(X5 ) ( 0.48AE / L)(X6 )
100 (0.48AE / L)(X3) ( 0.36AE / L)(X4 ) ( 0.48AE / L)(X5 ) ( 0.64AE / L)(X6 )
0
0
0
0
AE
L
1.00 0.00 0 0 0 0
0.00 1.00 0 0 0 0
0 0 L / AE 0 0 0
0 0 0 L / AE 0 0
0 0 0 0 L / AE 0
0 0 0 0 0 L / AE
X1
X2
X3
X4
X5
X6
![Page 18: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/18.jpg)
Modified Equations
10
100
0
0
0
0
AE
L
1.00 0.00 0 0 0 0
0.00 1.00 0 0 0 0
0 0 L / AE 0 0 0
0 0 0 L / AE 0 0
0 0 0 0 L / AE 0
0 0 0 0 0 L / AE
X1
X2
X3
X4
X5
X6
![Page 19: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/19.jpg)
Recap
![Page 20: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/20.jpg)
Initial Matrix
0
0
0
0
0
0
AE
L
1.00 0.00 0.64 0.48 0.36 0.48
0.00 1.00 0.48 0.36 0.48 0.64
0.64 0.48 0.64 0.48 0 0
0.48 0.36 0.48 0.36 0 0
0.36 0.48 0 0 0.36 0.48
0. 48 0.64 0 0 0.48 0.64
X1
X2
X3
X4
X5
X6
![Page 21: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/21.jpg)
Loads
10
100
0
0
0
0
AE
L
1.00 0.00 0.64 0.48 0.36 0.48
0.00 1.00 0.48 0.36 0.48 0.64
0.64 0.48 0.64 0.48 0 0
0.48 0.36 0.48 0.36 0 0
0.36 0.48 0 0 0.36 0.48
0.48 0.64 0 0 0. 48 0.64
X1
X2
X3
X4
X5
X6
![Page 22: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/22.jpg)
Boundary Conditions
10
100
0
0
0
0
AE
L
1.00 0.00 0.64 0. 48 0.36 0.48
0.00 1.00 0.48 0.36 0.48 0.64
0 0 L / AE 0 0 0
0 0 0 L / AE 0 0
0 0 0 0 L /AE 0
0 0 0 0 0 L / AE
X1
X2
X3
X4
X5
X6
![Page 23: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/23.jpg)
Symmetry
10
100
0
0
0
0
AE
L
1.00 0.00 0 0 0 0
0.00 1.00 0 0 0 0
0 0 L / AE 0 0 0
0 0 0 L / AE 0 0
0 0 0 0 L / AE 0
0 0 0 0 0 L / AE
X1
X2
X3
X4
X5
X6
![Page 24: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/24.jpg)
Solution
10 = AE/L X1100 = AE/L X2 0 = AE/L X3 0 = AE/L X4 0 = AE/L X5 0 = AE/L X6
X
10LAE
100LAE
0
0
0
0
![Page 25: The Problem. sin 1 = (-12 - 0) / (20) = -0.6 cos 1 = (16 - 0) / (20) = 0.8 sin 2 = (12 - 0) / (15) = 0.8 cos 2 = (9 - 0) / (15) = 0.6 Sines](https://reader035.vdocument.in/reader035/viewer/2022062221/56649d695503460f94a47789/html5/thumbnails/25.jpg)
Force Calculation (f=sbX)
f i (EAT) i
1
1
AE
L i
1 1
1 1
cosi sin i 0 0
0 0 cosi sini
X1i
X2i
X3i
X4i
t i AEL
iX3i X1i cosi X4i X2i sin i (EA)iT
{(X3i-X1i) cosi + (X4v-X2v) sini} is simply the change in length
t1 = AE/L {(10L/AE - 0)(0.8) + (100L/AE - 0)(-0.6)} + (0)t2 = AE/L {(0 - 10L/AE - 0)(0.6) + (0 - 100L/AE - 0)(0.8)} + (0)
t1 = f2 = -f1 = -52 kipst2 = f4 = -f3 = -86 kips