Answer to Question #255147 in Mechanical Engineering for Gani

Question #255147

In the spring system as shown in the figure below, each spring constant is ki=k2=1000 N/mm, k3=500 N/mm, k4=200 N/mm, and the applied load is F2=500 N, F300 N. (1) Find the stiffness equation of the spring system (KU=F). (2) Find the displacement of node 2 and node 3


1
Expert's answer
2021-10-24T01:17:53-0400

1)

Stiffness matrix for element 1 same as element 2

[k]1or2=1000100010001000N/mm[k]^{1or2}=\begin{vmatrix} 1000 & -1000 \\ -1000 & 1000 \end{vmatrix}N/mm

Stiffness matrix for element 3

[k]3=500500500500N/mm[k]^3=\begin{vmatrix} 500 & -500 \\ -500 & 500 \end{vmatrix}N/mm

Stiffness matrix for element 4

[k]1=200200200200N/mm[k]^1=\begin{vmatrix} 200 & -200 \\ -200 & 200 \end{vmatrix}N/mm

Global stiffness matrix

[k]=1000100000010002000100000010001500500000500700200000200200N/mm[k]=\begin{vmatrix} 1000 & -1000 & 0 & 0 & 0\\ -1000 & 2000 & -1000 & 0 & 0\\ 0 & -1000 & 1500 & 500 & 0\\ 0 & 0 & -500 & 700 & -200\\ 0 & 0 & 0 & -200 & 200 \end{vmatrix}N/mm


Let u1,u2,u3,u4,u5u_1,u_2,u_3,u_4,u_5 be the displacement at the nodes, then the stiffness equation becomes;

1000100000010002000100000010001500500000500700200000200200u1u2u3u4u5\begin{vmatrix} 1000 & -1000 & 0 & 0 & 0\\ -1000 & 2000 & -1000 & 0 & 0\\ 0 & -1000 & 1500 & 500 & 0\\ 0 & 0 & -500 & 700 & -200\\ 0 & 0 & 0 & -200 & 200 \end{vmatrix}\begin{vmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \\ u_5 \\ \end{vmatrix}

=R150003000=\begin{vmatrix} R_1 \\ 500 \\ 0 \\ 300 \\ 0 \end{vmatrix}


2)

10002000100001000150000500u2u3u4=5000300\begin{vmatrix} -1000 & 2000 & -1000 \\ 0 & -1000 & 1500 \\ 0 & 0 & -500 \\ \end{vmatrix}\begin{vmatrix} u_2 \\ u_3 \\ u_4 \\ \end{vmatrix}=\begin{vmatrix} 500 \\ 0 \\ 300 \end{vmatrix}


u2=1.7mmu_2=1.7mm

u3=0.9mmu_3=0.9mm


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