Beam Analysis: Reactions, Shear and Moment Diagrams

What does a beam analysis give you?

Sizing a beam starts with three things: how much force goes into the supports, and how shear force and bending moment vary along the beam. This tool works them out for simply supported and cantilever beams carrying several loads and draws the shear force (V) and bending moment (M) diagrams. If you enter the elastic modulus and the second moment of area it also gives the deflection curve.

How are the support reactions found?

Two equilibrium equations are enough: the vertical forces sum to zero and the moments about one support sum to zero. A distributed load is replaced by its resultant. For a uniform load the resultant is W = w × length, acting at the middle. For a triangular load it acts at two thirds of the base, and for a trapezoidal load at x̄ = x₁ + (x₂ − x₁) × (w₁ + 2 × w₂) / (3 × (w₁ + w₂)).

• Simply supported: R_B = (Σ W × x̄ + Σ M) / L, R_A = Σ W − R_B
• Cantilever: R = Σ W, fixed end moment M_A = −(Σ W × x̄ + Σ M)

Reading the diagrams

Shear force is the sum of the forces to the left of the section. It jumps under a point force and slopes under a distributed load. Moment is the area under the shear diagram: dM / dx = V. That is why the moment peaks where the shear force crosses zero. A concentrated moment makes the moment diagram jump. The diagram is drawn on the tension side: a positive moment puts the bottom fibre in tension.

Example

A simply supported beam spanning 6 m carries a 10 kN point load 2 m from the left support and 4 kN/m over the whole span. The resultant of the distributed load is 24 kN at midspan. The right reaction is (10 × 2 + 24 × 3) / 6 = 15.33 kN and the left one is 34 − 15.33 = 18.67 kN. Shear crosses zero at x = 2.17 m, where the maximum moment is 29.39 kN·m.

Deflection

Deflection comes from integrating the moment diagram twice: E × I × y″ = −M. The tool does this in closed form for each load segment, with no numerical approximation. You can take the second moment of area from the section properties tool.

Limits

The tool covers single span beams of constant section. Overhanging, continuous and fixed-fixed beams need other methods. The moment and shear are those of the loads you entered: load factors, load combinations and the resistance check of the section come from the relevant standard.