# Determining displacement of columns equation coefficients

## Homework Statement

Determine the constants $$c_{1}$$,$$c_{2}$$,$$c_{3}$$,$$c_{4}$$

of the column displacement equation where k=$$\sqrt{\frac{P}{EI}}$$

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: $$w(x)=c_{1}sin(kx)+c_{2}cos(kx)+c_{3}x+c_{4}$$

## The Attempt at a Solution

I know i have the following boundary conditions for pinned pinned:
$$w(0)=0 , EI*w''(0)=0 , w(L)=0 , EI*w''(L)=0$$

and for clamped clamped
$$w(0)=0 , w'(0)=0 , w(L)=0 , w'(L)=0$$

I found that
$$w'(x)=kc_1cos(kx)-kc_2sin(kx)+c_3$$

$$w''(x)=-k^2c_1sin(kx)-k^2c_2cos(kx)$$

using boundary conditions for simply supported I arrive at system of equations:
$$w(0)\rightarrow c_2+c_4=0$$

$$w(L)\rightarrow c_1sin(kL)+c_2cos(kL)+c_3L+c_4=0$$

$$w''(0) \rightarrow -k^2c_2=0$$]

$$w''(L)\rightarrow -k^2c_1sin(kL)-k^2c_2cos(kL)=0$$

using clamped clamped boundary conditions
$$w(0)\rightarrow c_2+c_4=0$$

$$w'(0)\rightarrow c_1k+c_3=0$$

$$w(L)\rightarrow c_1sin(kL)+c_2cos(kL)+c_3L+c_4=0$$

$$w'(L)\rightarrow c_1kcos(kL)-c_2ksin(kL)+c_3=0$$

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?