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Strategy to analyze beam deflections (Statically determinant case)

Steps in Strategy:

Step 1

Construct a free body diagram of the beam showing the loads acting

on the beam as well as support reactions at each end of the beam.

Incorrect FBD's will always result in incorrect equations of equilibrium

since you are writing the equations of equilibrium based on your FBD's.

Step 2

Write the equations of equilibrium for the beam.

For systems in the x˗y plane the equations of equilibrium are:

→∑ Fx = 0, ↑∑ Fy = 0, and ccw ∑ M_A = 0 where point A is any

convenient point to sum moments about. ccw refers to counterclockwise

although you could take clockwise as positive

Step 3

Solve for the support reactions.

Step 4

Recall that the differential equation for beam deflections is:

EI d²y/dx² = M(x)

So you will need to determine the bending moment distribution, M(x), along

the axis, x, of the beam.

You have two options to determine the bending moment distribution.

Option 1: Pass sections along the beam where the bending moment changes

and sum moments at the end of each section to determine the bending moment

distribution in each section.

Option 2: Determine the distribution of shearing force by integration of

dV/dx = ˗ w(x). Then integrate dM/dx = V(x) to obtain the moment

distribution, M(x).

Step 5

Double integration of EI d²y/dx² yields the deflection, y(x), of the beam. It also

introduces two constants of integration which must be determined from boundary

conditions at supports and from continuity conditions at different sections

along the beam where the bending moment changes.

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