SM 143 Bending moment and Shear force diagrams

Importance of SFD & BMD

• They provide a visual representation of the change in internal shear and bending moment in the structure/beam.

• Used to design the beam.

• Helps find the maximum shear and moment, so that the structure can resist the applied loading.

Method of sections

Cut frame at point x, which exposes it’s V_x, N_x and M_x internal forces.

Solve with the 3 equilibrium equations in terms of x

V(x) or M(x) should also be a term in the equation representing the internal force at x. M(x) equation should take moment from x.

• From 0 < x < m do normally. From m < x < x_max include all previous forces from 0 to m, then add the forces of m to x_max part in terms of (x - m) normally

flashcard needs revision.

Method of Integration

Start with w(x) of each respective boundary WITHOUT using FBD. Then integrate to find V(x) and M(x). Plug in calculated values to get Cs.

Internal force graph relationships

Integrate to go down list: (going from w(x) ←→ V(x) swaps signs)

-w(x)
V(x)

M(x)

Boundary points / section boundaries of diagrams

Boundary points made at:

• Point loads

• External moments

• Change in distributed load

• Supports

When V/M formulas do NOT include these points (< NOT <), however, DO include ends of object (x = 0 & x = max)

Finding V & M max

Mmax is where V(x) cuts the x-axis. When V = 0, M = Mmax

Vmax is where w(x) cuts the x-axis. When w = 0, V = Vmax

Finding C constant

Insert boundary value as x where y value is known.