Slope and deflection diagrams using conjugate beam method

[musicmar]

Homework Statement

Construct the slope and deflection diagrams. I've attached the problem with the original diagram (problem #1).

The Attempt at a Solution

Considering the number of diagrams required, I thought it would be best to attach a photo of my work.
I drew the shear and moment diagrams for the original beam, with equations for V and M as follows:
V = {150 (0<x<3), -75x (3<x<5)
M = {150x (0<x<3), -75/2*x² + 375x (3<x<5)

I then drew the conjugate beam, which is essentially a mirror image of the original beam. I then loaded it with M/EI, where M is the moment from the original beam. Now, I know I need to draw the shear and moment diagrams for the conjugate beams, and these will be the slope and deflection diagrams. I'm just not sure how to go about this, considering the complicated loading.
Also, for another problem, I am told to find the max deflection. Is there a way to do this without constructing the shear and moment diagrams(for the conjugate beam)?

Thank you.

 

[afreiden]

Use an "integration" method, rather than a "cutting" method.

1) Get your reactions in you conjugate beam.

2) Integrate to get the slope (makes no difference if you start at the left end or the right end). You should find that you get zero at the left end of your conjugate beam, and something nonzero (equal and opposite to your reaction) at the right end of your conjugate beam. So, in the actual beam, the slope is zero at the fixed end and nonzero at the free end (as expected).

3) Integrate (2) to get the deflection.
You will find that you again get zero at the left end of your conjugate beam, and something nonzero at the right end of your conjugate beam. So, in the actual beam, the deflection is zero at the fixed end and nonzero at the free end (as expected).

Hope that helps,

 

[musicmar]

I got R_C'y=1425/EI and M_C'=4387.5/EI .
I also found the equations for V and M of the conjugate beam. Do I need to consider end conditions to find integration constants? As it stands right now, my shear diagram goes down to -900/EI at x=3, and with the equation I have for 3<x<5, it would go towards the axis. However, the reaction I found at C' would need the diagram to end at -1425/EI at x=5.

Attached is more of my work.
Thank you.

 

[afreiden]

To answer your question regarding "integration constants" -- yes, I would expect to see some constant values in your equations. My advice would be to consult a statics textbook if you aren't familiar with the "integration" method for finding shear and moment diagrams.

Some more advice, just to help you out, because I've been in your position:
Show a lot more work.

The grader can't give you partial credit if they cannot follow your steps. For example, I have no idea how you got your equations because you don't show the steps that you took in order to arrive at them. Where are the integrals? I expected to see an integral sign "∫" somewhere in your work.

 

[DeeNos]

I would like to first commend you on your attempt at solving the problem using the conjugate beam method. It shows that you have a good understanding of the concept and are applying it correctly.

To draw the slope and deflection diagrams using the conjugate beam method, you can follow these steps:

1. Draw the conjugate beam: As you have correctly mentioned, the conjugate beam is a mirror image of the original beam. It is important to note that the conjugate beam should have the same supports and boundary conditions as the original beam.

2. Calculate the bending moment at each point along the conjugate beam: To do this, you can use the equations you have already derived for the bending moment of the original beam. Simply substitute the distance along the conjugate beam for the distance along the original beam in the equations.

3. Draw the shear and moment diagrams for the conjugate beam: Using the calculated bending moment values, you can draw the shear and moment diagrams for the conjugate beam. These diagrams will be the slope and deflection diagrams for the original beam.

4. Calculate the maximum deflection: To find the maximum deflection, you can use the equation Δmax = 5wL^4 / 384EI, where w is the distributed load, L is the length of the beam, E is the modulus of elasticity, and I is the moment of inertia. You can substitute the values from the original beam to calculate the maximum deflection.

I hope this helps you in constructing the slope and deflection diagrams using the conjugate beam method. Keep up the good work!