Question

Consider the frame in Fig. 1, the node and element numbers as well as the material and geometrical characteristics of the bea

60 kN 4 m 15 kN/m E 210 GPa I 104 m A 102 m2 5 kN 6 m 8 m Fig.1 e*s

Consider the frame in Fig. 1, the node and element numbers as well as the material and geometrical characteristics of the beam elements are also displayed on the same figure. The frame is subjected to two concentrated loads at nodes 2 and 3 and a uniform distributed load over beam 3. The frame is fixed at nodes 1 and 5. A global coordinate system is established with origin at node 1 and x-y axes positively directed to the right and up respectively. 1. Calculate the stiftness matrix for each element in the global coordinate system. Express you results in terms of the global displacement vector. (Show step by step calculations) the global displacement vector. (Show step by step calculations) using the Gauss-Elimination method (show steps of the calculation process). (Show step by step cal- 2. Calculate the work done by the concentrated and distributed loads. Express you results in terms of 3. Assemble the stiffness matrices then solve for the nodal displacement (u, vi8) forl siS number of nodes culations) 4. Calculate the reactions at the supports. (Show step by step calculations)
60 kN 4 m 15 kN/m E 210 GPa I 104 m A 102 m2 5 kN 6 m 8 m Fig.1 e*s
0 0
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Answer #1

Nite 計10T) 6C1 0 CS Scanned with CamScanner

E=210*10^6;
I=1*10^-4;
A=1*10^-2;

k 1=

10^5 *

[3.5000 0 0 -3.5000 0 0
0 0.0117 0.0350 0 -0.0117 0.0350
0 0.0350 0.1400 0 -0.0350 0.0700
-3.5000 0 0 3.5000 0 0
0 -0.0117 -0.0350 0 0.0117 -0.0350
0 0.0350 0.0700 0 -0.0350 0.1400]


T 1=

0 1 0 0 0 0
-1 0 0 0 0 0
0 0 1 0 0 0
0 0 0 0 1 0
0 0 0 -1 0 0
0 0 0 0 0 1


Kele1 =

10^5 *

0.0117 0 -0.0350 -0.0117 0 -0.0350
0 3.5000 0 0 -3.5000 0
-0.0350 0 0.1400 0.0350 0 0.0700
-0.0117 0 0.0350 0.0117 0 0.0350
0 -3.5000 0 0 3.5000 0
-0.0350 0 0.0700 0.0350 0 0.1400


k2(local) =

10^5 *

3.7123 0 0 -3.7123 0 0
0 0.0139 0.0394 0 -0.0139 0.0394
0 0.0394 0.1485 0 -0.0394 0.0742
-3.7123 0 0 3.7123 0 0
0 -0.0139 -0.0394 0 0.0139 -0.0394
0 0.0394 0.0742 0 -0.0394 0.1485


T 2=

0.7071 0.7071 0 0 0 0
-0.7071 0.7071 0 0 0 0
0 0 1.0000 0 0 0
0 0 0 0.7071 0.7071 0
0 0 0 -0.7071 0.7071 0
0 0 0 0 0 1.0000



Kele(2) =

10^5 *

1.8631 1.8492 -0.0278 -1.8631 -1.8492 -0.0278
1.8492 1.8631 0.0278 -1.8492 -1.8631 0.0278
-0.0278 0.0278 0.1485 0.0278 -0.0278 0.0742
-1.8631 -1.8492 0.0278 1.8631 1.8492 0.0278
-1.8492 -1.8631 -0.0278 1.8492 1.8631 -0.0278
-0.0278 0.0278 0.0742 0.0278 -0.0278 0.1485


k 3=

10^5 *

3.7123 0 0 -3.7123 0 0
0 0.0139 0.0394 0 -0.0139 0.0394
0 0.0394 0.1485 0 -0.0394 0.0742
-3.7123 0 0 3.7123 0 0
0 -0.0139 -0.0394 0 0.0139 -0.0394
0 0.0394 0.0742 0 -0.0394 0.1485


T3 =

0.7071 -0.7071 0 0 0 0
0.7071 0.7071 0 0 0 0
0 0 1.0000 0 0 0
0 0 0 0.7071 -0.7071 0
0 0 0 0.7071 0.7071 0
0 0 0 0 0 1.0000



Kele(3) =

10^5 *

1.8631 -1.8492 0.0278 -1.8631 1.8492 0.0278
-1.8492 1.8631 0.0278 1.8492 -1.8631 0.0278
0.0278 0.0278 0.1485 -0.0278 -0.0278 0.0742
-1.8631 1.8492 -0.0278 1.8631 -1.8492 -0.0278
1.8492 -1.8631 -0.0278 -1.8492 1.8631 -0.0278
0.0278 0.0278 0.0742 -0.0278 -0.0278 0.1485


k 4=

10^5 *

3.5000 0 0 -3.5000 0 0
0 0.0117 0.0350 0 -0.0117 0.0350
0 0.0350 0.1400 0 -0.0350 0.0700
-3.5000 0 0 3.5000 0 0
0 -0.0117 -0.0350 0 0.0117 -0.0350
0 0.0350 0.0700 0 -0.0350 0.1400


T 4=

0 1 0 0 0 0
-1 0 0 0 0 0
0 0 1 0 0 0
0 0 0 0 1 0
0 0 0 -1 0 0
0 0 0 0 0 1

Kele(4) =

10^5*

0.0117 0 -0.0350 -0.0117 0 -0.0350
0 3.5000 0 0 -3.5000 0
-0.0350 0 0.1400 0.0350 0 0.0700
-0.0117 0 0.0350 0.0117 0 0.0350
0 -3.5000 0 0 3.5000 0
-0.0350 0 0.0700 0.0350 0 0.1400

Kglobal=10^5 *

0.0117 0 -0.0350 -0.0117 0 -0.0350 0 0 0 0 0 0 0 0 0

0 3.5000 0 0 -3.5000 0 0 0 0 0 0 0 0 0 0

-0.0350 0 0.1400 0.0350 0 0.0700 0 0 0 0 0 0 0 0 0

-0.0117 0 0.0350 1.8748 1.8492 0.0072 -1.8631 -1.8492 -0.0278 0 0 0 0 0 0

0 -3.5000 0 1.8492 5.3631 0.0278 -1.8492 -1.8631 0.0278 0 0 0 0 0 0

-0.0350 0 0.0700 0.0072 0.0278 0.2885 0.0278 -0.0278 0.0742 0 0 0 0 0 0

0 0 0 -1.8631 -1.8492 0.0278 3.7262 0 0.0557 -1.8631 1.8492 0.0278 0 0 0

0 0 0 -1.8492 -1.8631 -0.0278 0 3.7262 0 1.8492 -1.8631 0.0278 0 0 0

0 0 0 -0.0278 0.0278 0.0742 0.0557 0 0.2970 -0.0278 -0.0278 0.0742 0 0 0

0 0 0 0 0 0 -1.8631 1.8492 -0.0278 1.8748 -1.8492 0.0072 -0.0117 0 0.0350

0 0 0 0 0 0 1.8492 -1.8631 -0.0278 -1.8492 5.3631 -0.0278 0 -3.5000 0

0 0 0 0 0 0 0.0278 0.0278 0.0742 0.0072 -0.0278 0.2885 -0.0350 0 0.0700

0 0 0 0 0 0 0 0 0 -0.0117 0 -0.0350 0.0117 0 -0.0350

0 0 0 0 0 0 0 0 0 0 -3.5000 0 0 3.5000 0

0 0 0 0 0 0 0 0 0 0.0350 0 0.0700 -0.0350 0 0.1400召 3 0 $0乙 36 3 Scanned ith Cann Scanne 24

F+Ff=Kglobal*d

F=Nodal forces

Ff=lump vector

d=nodal displacement

d=

0
0
0
-0.0608
-0.0002
0.0078
-0.0530
-0.0083
-0.0075
-0.0449
-0.0001
0.0117
0
0
0


F =

43.6514
72.0911
-158.1780
5.0000
0.0000
0.0000
0.0000
-60.0000
0.0000
-0.0000
0.0000
0.0000
11.3486
47.9089
-75.0930



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