# Beam deflection solution

• Laurry
In summary, the deflection curve of the beam in picture can be solved by using the approach of integrating four times and solving for the constants from boundary conditions of deflection and moment being zero at the supports. The shear force and bending moment can also be determined using the equations of equilibrium and a free body diagram. The slope and deflection of the beam can be calculated using integrals, with the constants of integration evaluated by applying the appropriate boundary conditions.

#### Laurry

Can the deflection curve of the beam in picture be solved by using the following
approach:

https://dl.dropboxusercontent.com/u/104865119/beam2.PNG [Broken]

1) EI w''''(x) = q

2) Integrate four times, solve the constants from boundary conditions (deflection and moment zero at the supports)

Thanks,

Laurry

Last edited by a moderator:
Laurry said:
Can the deflection curve of the beam in picture be solved by using the following
approach:

https://dl.dropboxusercontent.com/u/104865119/beam2.PNG [Broken]

1) EI w''''(x) = q

2) Integrate four times, solve the constants from boundary conditions (deflection and moment zero at the supports)

Thanks,

Laurry
The shear force and bending moment for this beam don't depend on the elastic properties of the material or the moment of inertia of the cross section.

You can determine the reactions at the supports by using the equations of equilibrium, and then construct a free body diagram of the beam, from which you can then construct the shear force and bending moment diagrams.

Once you have constructed the bending moment diagram, then the slope and deflection of the beam can be calculated by the following integrals:

$$θ(x)=\int_0^x \frac{M(ξ)}{EI}\,dξ$$ and

$$δ(x)=\int_0^x θ(ξ) \,dξ$$ where ξ is a dummy coordinate measured along the length of the beam.

The appropriate constants of integration are added to the results of each integration. These constants can be evaluated by applying the appropriate boundary conditions for the beam. The deflections will be zero at the supports, but since the beam overhangs the supports at each end, the bending moment may not necessarily be zero at each support. The bending moments must be zero at the free ends of the beam, however.