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Find nodal displacement and element stress for bar as shown in Fig. Take E = 200GPa

Subject: Finite Element Analysis

Topic: One Dimensional Problems

Difficulty: Medium


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1 Answer
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a) Number of elements and nodes,

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b) Element matrix Equation is given by,

[K]e=AEhe[1111]

Forelement1,A=250mm2,he=300mm
AEhe=2502105300=167.67103
[K]1=103[167.67167.67167.67167.67]

Forelement2,A=400mm2,he=200mm
AEhe=4002105200=400103
[K]2=103[400400400400]

Forelement3,A=400mm2,he=200mm
AEhe=4002105200=400103
[K]3=103[400400400400]

c) Elemental matrix
[K]=103[167.67167.6700167.67167.67+40040000400400+40040000400400]

d) Global Matrix Equation,
[K]{u}={P}
103[167.67167.6700167.67567.674000040080040000400400]{u1u2u3u4}={P1P2P3P4}

e) Imposing boundary condition,
u1=0,u4=3.5mm,P3=75103N
P2=0 -for balancing,

Frame the equation,
103(167.67u2)P1 -(1)
103(567.67u2400u3)=0 -(2)
103(400u2+800u34003.5)=75103 -(3)
103(400u3+4003.5)=P4 -(4)

Solving equation (2) and (3) we get,
u2=0.10389mmu3=0.1474mm
Substituting in equation (1) and (4), we get,
P1=17.41kNP4=57.56
Px=P1+P2+P3+P+4=17.41+0+7557.56=0 -verified

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