Calculator
For any supports, internal hinges and any set of loads — determinate or not. Each diagram is exact along the whole beam, not interpolated between points.
- Supports
- Pin, roller, fixed, hinges
- Loads
- Point, distributed, moments
- Units
- US and SI
Calculation sheet
Beam reactions and diagrams
ttcivilstructural.com/resources/beam
The beam
Loads and supports
Shear is positive when the part of the beam to the left of the cut is pushed up. Moment is positive when the beam sags, with tension at the bottom.
Reactions
| Support | Position (ft) | Force (kip) | Moment (kip·ft) |
|---|---|---|---|
| Pin | 0 | 15.2 upward | — |
| Roller | 20 | 16.8 upward | — |
The reactions add up to the load: 32 = 32 kip.
Largest values
- Shear
- 16.8 kip at x = 20 ft
- Sagging moment
- 96 kip·ft at x = 12 ft
- Hogging moment
- none
- Deflection
- 0.2801 in downward, at x = 10.21 ft
- Bending stress
- 12.96 ksi
- Shear stress
- 2.629 ksi
Shear divided by the shear area entered. Elastic stresses from M ÷ S and the shear rule above — not a check of strength.
Deflection by span
| Span (ft) | Deflection (in) | Length ÷ deflection |
|---|---|---|
| 0 – 20 | 0.2801 downward | L/857 |
Values at a point
- Shear, V
- 3.2 kip
- Bending moment, M
- 92 kip·ft
- Deflection
- -0.28 in
- Slope
- -0.0001192 rad
- Bending stress
- 12.42 ksi
Move over a diagram, or type a position.
The calculation runs in your browser. Nothing you enter is sent to us.
Method
How the beam is solved
The whole beam is solved as one piece, by the method of initial parameters. The unknowns are the slope and the deflection at the left end, the reaction at each support and the rotation at each hinge; the equations are the support conditions, zero moment at each hinge and the equilibrium of the beam.
Every diagram is then a sum of polynomials, exact at every point. There is no mesh and nothing is interpolated, so a load next to a support, or a short and heavy distributed load, is solved as precisely as any other. A statically indeterminate beam is solved the same way as a determinate one.
Largest values are found where they occur — the moment where the shear crosses zero, the deflection where the slope does — not picked from the nearest plotted point.
What it assumes
- A straight beam with one section along its whole length.
- Linear elastic material and small deflections (Euler–Bernoulli bending; shear deformation is not included).
- Supports that do not settle or rotate, other than as their type allows.
- Vertical loads and moments only: no axial force, which is why a pin and a roller give the same answer.
What it does not do
- It does not check strength, lateral-torsional buckling, web crippling or any other limit state of a design code.
- It does not combine or factor loads: enter the loads of the combination you are checking.
- It does not compare deflection with a limit. It gives the length-to-deflection ratio of each span for you to compare.
- It does not solve two supports closer together than a thousandth of the beam’s length, nor a hinge hard against the hinge or the support beside it: it says so and gives no numbers.
- For concrete, the deflection is that of the uncracked section entered: cracking and creep are not modelled.
- The beam’s own weight is not a load unless a steel shape is chosen and its box is ticked. Otherwise add it to the loads.
Signs and units
- Loads are entered as magnitudes with a direction; a downward load is gravity.
- Shear is positive when the part of the beam to the left of the cut is pushed up.
- Moment is positive when the beam sags (tension at the bottom) and is drawn above the axis.
- Deflection is drawn the way the beam moves: down is down.
- In the tables, deflection is positive upward, and slope is in radians, positive counter-clockwise.
- Lengths, forces and moments are in feet and kips or in metres and kilonewtons; the section is in inches or millimetres.
Use it as an engineer would
This calculator is an aid for people qualified to judge its results. It is not an engineering opinion on any structure, and it comes with no warranty. Check the results by independent means: a decision in a design rests on your own check, never on this calculator. Tercero Tablada Civil and Structural Engineering Inc. did not prepare, review or seal what is computed with it, and accepts no responsibility for its results or for how they are used.
Produced with the beam calculator of Tercero Tablada Civil and Structural Engineering Inc. — ttcivilstructural.com/resources/beam