Beam Analysis Calculator with Shear Force and Bending Moment Diagrams
A beam with several point and uniform loads on four support types: reactions, shear force, bending moment and deflection diagrams.
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Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.
How this calculator works
Choose how the beam is supported: simply supported, cantilever, fixed at both ends, or a propped cantilever (fixed at one end and on a roller at the other). Enter the span, the stiffness E and I, and the loads, one per line. A point load is written P, its position from the left end in metres, and its size in kN. A uniform load is written U, its start and end positions, and its size in kN per metre. Downward loads are positive.
The calculator finds the reactions, then the shear force and bending moment at every point along the beam, and the deflection by integrating the curvature twice. Fixed and propped beams are statically indeterminate, so the extra reactions are found from the force method, by making the deflection and slope at the supports match. The diagrams show the largest value and where it occurs. Add the section modulus and yield strength to get the bending stress and the factor of safety.
Formulas used
- \frac{dV}{dx} = -w,\quad \frac{dM}{dx} = V
- EI\,\frac{d^2 y}{dx^2} = M(x)
- \sigma_{max} = M_{max}/Z
Worked example
With the default numbers in the calculator above:
- The loads read: point load 20 kN at 1.5 m; uniform load 10 kN/m from 0 m to 3 m
- Reactions: A 25 kN, B 25 kN
- Largest shear force: 25 kN
- Largest bending moment: 26.25 kN m
- Largest deflection: 5.449 mm
- Span to deflection ratio: 550.5
- Maximum bending stress: 131.3 MPa
- Factor of safety against first yield: 1.905
Change the numbers in the calculator to see the full working, every step, for your own case.
Common questions
What sign convention is used?
Loads act downward when positive. Bending moment is positive when the beam sags (the underside is in tension). Deflection is shown positive downward.
How do I enter a load that is not at one point?
Use a uniform load U with a start and end, and break a varying load into a few uniform steps, or use point loads at the centres of each part.
Is the self-weight included?
No. Add it as a uniform load over the whole span.
How accurate is it?
The values come from numerical integration with 1000 steps and agree with the textbook formulas to better than 0.01 percent.
Related calculators
Further reading: Hibbeler, Mechanics of Materials; Gere and Goodno, Mechanics of Materials; Shigley, Mechanical Engineering Design.
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