Solving beam deflection problems using the moment-deflection approach and using the Euler-Bernoulli approach
by Nasser M. Abbasi
November 2009
Links
Introduction
These are problems in beam deflection showing how to use Mathematica to solve them.
Problem 1
This is problem 9-3, page 551, from bok Problem Solvers, strength of materials and mechanics of materials by REA. I show here how to solve this problem using Mathematica.
Start by setting up the moment deflection equation for the Euler beam E I y''[x]=M(x) , this equation is found for both halves of the beam resulting in 2 solutions, 
and 
.
On the right side, boundary condition is that 
and on the left side 
, then we need an additional 2 boundary conditions, for this, use the continuity conditions at L/2 where we set 
and 
and now we have the complete solutions.
For the left half of beam, it is easily found that 
and for the right half 
Find 
Use first boundary condition 
Find 
Use boundary condition 
We see from the above, that we have 2 additional constants of integrations to solve for. c2 and c4. To solve for these, we use the continuity conditions at L/2
All the integration of constants now found, we can now plot moment diagrams, shear diagram, and deflection diagrams
Assign some values for E, I, L and w and do the plots. PLot the deflection
Plot the bending moment
Plot the shear diagram
Solving the same problem directly from the Euler-Bernoulli 
order ODE
The above approach (using the Moment-deflection ODE) is a standard approach to solve deflection beam problems. However, we can also use the 4th order Euler beam equation direclty as follows. One needs to make sure that the load on the RHS of this ODE is the load per unit length only, i.e. w in this problem. Notice the use of the unit step function since w only acts on half the beam.
Assign some values for E, I, L and w and do the plots. PLot the deflection
We see that we obtain the same result as earlier.
Plot the bending moment
Plot the shear diagram
Problem 2
In this problem. I set up a cantilever beam and first solved it using the Java beam applet that can be accessed here
and then I solved this problem using Mathematica below, showing that I get the same answer as shown
The boundary conditions to use are: y[L] == 0, y''[0] == 0, y'''[0] == 0, y'[L] == 0
Out[8]=
Assign same values as shown above for E, I, L and w and do the plots.
In[17]:=
Out[19]=
Plot the bending moment
In[23]:=
Out[23]=
Plot the shear diagram
In[25]:=
Out[25]=
