Engineering Calc Tools

Mechanics of Materials

Beam Solver (Indeterminate Beams and Deflection)

Reactions, shear, moment, slope and deflection for any beam: continuous, propped, fixed–fixed, hinged or on springs.

Tool no.
ECT-S25
Rev
1.0
Updated
Oct 2026
Method
Direct stiffness method
Examples

Given

0 pin · 6 roller · 0 fixed · 4 hinge (internal hinge) · 3 spring 5000 (stiffness in force/length).

point 3 10 (at x, size) · udl 0 6 4 (from, to, intensity) · lin 0 6 0 9 (from, to, start and end intensity) · moment 2 15 (counterclockwise +; add cw).

Answers

Maximum deflection = —
Runs in your browser. Nothing you enter is uploaded.Stiffness method · Euler–Bernoulli beams
How to use this solver

Enter the length, list the supports by position, and list the loads. Positions are measured from the left end. The solver works for determinate beams and for continuous beams, propped cantilevers and fixed–fixed beams that statics alone can’t solve.

How it works

Differential equationEI v″ = M(x),  dM/dx = V,  dV/dx = −w
Beam element[k] = (EI/h³)[12, 6h, −12, 6h; 6h, 4h², −6h, 2h²; …]
Determinacydegree = reactions − 2 − internal hinges

Worked example

Two equal 5 m spans, 12 kN/m everywhere (the default)

  1. Three supports give one more reaction than statics can find: the beam is indeterminate to degree 1.
  2. Compatibility (zero deflection at the middle support) gives the middle reaction 1.25wL = 75 kN, and 0.375wL = 22.5 kN at each end.
  3. The largest moment is the hogging moment over the middle support, wL²/8 = −37.5 kN·m.

Frequently asked questions

What sign convention is used?

Loads are entered positive downward. Reactions are + upward, couples + counterclockwise, deflection + upward, and bending moment + when it bends the beam concave up (sagging).

Does it include shear deformation or axial loads?

No. It uses Euler–Bernoulli beam theory, which is what statics and mechanics of materials courses use. Horizontal reactions are zero because there are no axial loads.