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Beam calculator: support reactions and shear and moment diagrams

Pick the layout - simply supported beam (pin and roller anywhere, with overhangs) or cantilever with a fixed end - set the length and add loads: point forces, moments, uniformly distributed load on a segment. The calculator finds support reactions with equilibrium equations, draws shear and bending moment diagrams with labelled values and the moment extremum, and writes the solution segment by segment - free, no sign-up.

Loads
The x axis runs along the beam from x = 0 to x = L. Force P and intensity q are positive downward, moment M is positive clockwise; a negative value flips the direction. Q > 0 when the forces to the left of the section sum upward; M > 0 when the bottom fibre is in tension. Units: m, kN, kN·m, kN/m.
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How to draw shear force and bending moment diagrams

Bending of a beam is the most common assignment in strength of materials and applied mechanics. The procedure is the same for every layout.

  1. Support reactions. Remove the supports and replace them with reactions. For a pin and a roller write two moment equations, one about each support; they give R_A and R_B directly, no system needed. Check with the vertical force sum, ΣF_y = 0. For a cantilever take the vertical reaction from ΣF_y = 0 and the fixed-end moment from ΣM = 0.
  2. Segments. Segment boundaries are the points where forces and moments act, the supports, and the start and end of a distributed load.
  3. Method of sections. On each segment cut the beam and look at the left part. Shear Q is the sum of vertical forces to the left of the cut (upward +), bending moment M is the sum of their moments about the cut (tension at the bottom +).
  4. Diagrams. Q is constant on a segment without load and changes linearly with slope −q under uniform load q. M is a straight line without q and a parabola under q, bulging toward the load.
  5. Moment extremum. Where Q passes through zero, M reaches a maximum or minimum, since M′(x) = Q(x). Solve Q(z) = 0 and substitute into M(z).

Checking the diagrams

  • Under a point force the Q diagram jumps by the force; under a point moment the M diagram jumps by the moment.
  • At a free end Q and M are zero unless an external force or moment is applied there.
  • The derivative of M is Q and the derivative of Q is −q, which fixes the direction of the parabola on the M diagram.
  • M_max is usually the input of the next step: choosing a section from σ = M/W.

Sign convention and diagram side

In civil engineering the M diagram is drawn on the tension side, so a positive moment is plotted downward. In mechanical engineering it is drawn on the compression side, upward. The switch above the Calculate button flips the plotting side, the moment values stay the same.

What the calculator does

Simply supported beam: pin and roller anywhere, including overhangs on one or both sides. Cantilever with the fixed end on the left or right. Loads: any number of point forces, moments and uniformly distributed loads on segments. Output: reactions with numbers substituted, Q and M diagrams labelled at characteristic points, M_max and Q_max with their position, solution by segments and a link to your calculation.

FAQ

How do I find the support reactions of a beam?
Write moment equations about each support: ΣM_A = 0 gives R_B, ΣM_B = 0 gives R_A. Check with the vertical force sum ΣF_y = 0. For a cantilever the reaction equals the sum of the loads and the fixed-end moment equals the moment of the loads about the fixed end.
Where is the maximum on the M diagram?
Either at a section under a point force or at a support, or inside a segment with distributed load where the shear force Q is zero. The calculator finds such points and labels the value of M there.
Why does the Q diagram jump while the M diagram only has a kink?
A point force changes Q by a jump equal to the force. M is the integral of Q, so it is continuous and only changes slope where Q jumps. M jumps only under a point moment.
Which side should the M diagram be drawn on?
In civil engineering on the tension side (positive moment downward), in mechanical engineering on the compression side (positive moment upward). The switch only changes the plotting side, the values stay the same.
Is it suitable for a homework assignment?
Yes, for statically determinate beams: simply supported with overhangs and cantilevers. Enter your data and compare reactions, diagrams, M_max and the critical section. The solution by segments is given as formulas with numbers substituted.