InsureCalcENGINEERING
STRUCTURAL ENGINEERING · FREE TOOL

Beam analysis,
made visible.

Build a load case. Follow the forces. Understand the deflection.

01 / STRUCTURAL TOOLS

Four support arrangements.
Point, uniform and moment loads.
SI units throughout.

Explore the method
Results ready

InsureCalc · Beam analysis report

LOAD & SUPPORT MODELSIMPLY SUPPORTED · L = 6 m
Applied loads Support reactionsLoads shown schematically
MAX |SHEAR|——
MAX |BENDING MOMENT|——
MAX |DEFLECTION|——

Shear force

V · kN

Bending moment

M · kN·m

Elastic deflection

v · mm

Inspect a section

At a point force or couple, left/right values show the jump. All diagram ordinates are positive upward; positive M is sagging.

Calculation details & equilibrium

Calculations run in your browser. Save and import your own load cases.

THE ENGINEERING BEHIND THE RESULT

Clear assumptions.
Traceable results.

Version 1.0.0 · Constant-EI beam analysis

What this calculator analyses

A straight, prismatic beam under static loads in one bending plane. The model uses linear elastic Euler–Bernoulli theory, constant E and I, ideal supports, and small displacements. It calculates elastic response, rather than member strength or a code-compliance decision. Self-weight is included only when you add it as a distributed load.

How the solution works

The calculator builds the bending-moment function from the applied forces and couples. It integrates M / EI to obtain rotation and deflection, enforcing the displacement, rotation and free-end conditions of the selected support arrangement. Load effects are superimposed. Peaks are evaluated at load boundaries and stationary points, including both sides of internal jumps.

dM / dx = V · dV / dx = −w · EI d²v / dx² = M

Here x increases from support A to end B, downward w is positive, and vertical deflection v is positive upward. A positive applied couple acts clockwise and causes an upward jump in the moment diagram. External support couples are reported counterclockwise positive. Their signs differ from the internal end moments where required by equilibrium.

Units that stay consistent

Enter length in metres, point forces in kN, line loads in kN/m, applied couples in kN·m, E in GPa, and I in millions of mm⁴. For example, I = 80 means 80 × 10⁶ mm⁴ = 0.000080 m⁴. With E = 200 GPa, EI = 16,000 kN·m². Exported deflections are in millimetres and rotations in milliradians.

A worked example: uniform load on a simple beam

Take L = 6 m, w = 10 kN/m over the entire span, E = 200 GPa and I = 80 × 10⁶ mm⁴. The total load is 60 kN. Symmetry gives RA = RB = 30 kN. Maximum sagging moment is wL² / 8 = 45 kN·m. Maximum downward deflection is 5wL⁴ / (384EI) = 10.547 mm at midspan. The reported signed deflection is −10.547 mm.

Choose stiffness for the actual problem

Shear deformation, cracking, creep, yielding, axial-force interaction, support settlement and second-order effects are outside this model. For reinforced concrete, gross-section EI may underestimate deflection after cracking. Use stiffness values justified for your analysis and assess the applicability of the assumptions.

Reference reading