Determining displacement of columns equation coefficients

[psuaero]

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?

 

[KataKoniK]

Thank you for your post. It seems like you are on the right track with your approach. Here are some suggestions to help you find the constants c_{1},c_{2},c_{3},c_{4}:

1. For the pinned-pinned boundary conditions, you have correctly identified the four boundary conditions. However, when you plug in w''(0), you should get -k^2c_2=0 instead of -k^2c_1=0. This is because the second derivative of w(x) with respect to x is -k^2c_1sin(kx)-k^2c_2cos(kx), and when x=0, only the second term remains.

2. For the clamped-clamped boundary conditions, you have correctly identified the four boundary conditions as well. However, when you plug in w'(0), you should get c_1k+c_3=0 instead of c_1k=0. This is because the first derivative of w(x) with respect to x is c_1cos(kx)-c_2sin(kx)+c_3, and when x=0, only the first term remains.

3. Once you have corrected those mistakes, you can put the equations in matrix form and solve them simultaneously. You should get a non-trivial solution for the constants c_{1},c_{2},c_{3},c_{4}. If you are still getting the trivial solution, make sure you are using the correct equations and boundary conditions.

4. Another approach you can try is to plot the equations and boundary conditions on a graph. This can help you visually identify the values of the constants c_{1},c_{2},c_{3},c_{4} that satisfy all the conditions.

I hope these suggestions help you solve the problem. Good luck with your experiment and comparison of theory and experimental results!