- #1
psuaero
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Homework Statement
Determine the constants [tex]c_{1}[/tex],[tex]c_{2}[/tex],[tex]c_{3}[/tex],[tex]c_{4}[/tex]
of the column displacement equation where k=[tex]\sqrt{\frac{P}{EI}}[/tex]
In lab we subjected 3 specimens to a compressive load in a pinned-pinned and clamped clamped configurations. I have to compare theory results to experimental.
Homework Equations
displacement: [tex]w(x)=c_{1}sin(kx)+c_{2}cos(kx)+c_{3}x+c_{4}[/tex]
The Attempt at a Solution
I know i have the following boundary conditions for pinned pinned:
[tex]w(0)=0 , EI*w''(0)=0 , w(L)=0 , EI*w''(L)=0[/tex]
and for clamped clamped
[tex]w(0)=0 , w'(0)=0 , w(L)=0 , w'(L)=0[/tex]
I found that
[tex]w'(x)=kc_1cos(kx)-kc_2sin(kx)+c_3[/tex]
[tex]w''(x)=-k^2c_1sin(kx)-k^2c_2cos(kx)[/tex]
using boundary conditions for simply supported I arrive at system of equations:
[tex]w(0)\rightarrow c_2+c_4=0[/tex]
[tex]w(L)\rightarrow c_1sin(kL)+c_2cos(kL)+c_3L+c_4=0[/tex]
[tex]w''(0) \rightarrow -k^2c_2=0[/tex]]
[tex]w''(L)\rightarrow -k^2c_1sin(kL)-k^2c_2cos(kL)=0[/tex]
using clamped clamped boundary conditions
[tex]w(0)\rightarrow c_2+c_4=0[/tex]
[tex]w'(0)\rightarrow c_1k+c_3=0[/tex]
[tex]w(L)\rightarrow c_1sin(kL)+c_2cos(kL)+c_3L+c_4=0[/tex]
[tex]w'(L)\rightarrow c_1kcos(kL)-c_2ksin(kL)+c_3=0[/tex]
I tried putting the above in matrix form and solve simultaneously but only achieved the trivial solution. I was thinking of finding the determinate of the matrices and plotting them but not sure if that would provide the correct solution. any suggestions?