Calculator
One simply supported span, a pin at one end and a roller at the other, carrying point loads and distributed loads. Beside the diagrams: what each load does on its own, and the section a deflection limit and a bending stress ask for.
- Supports
- Pin and roller, at the ends
- Loads
- Point and distributed
- Section needed
- I for L/n, S for a stress
Calculation sheet
Single-span beam, simply supported
ttcivilstructural.com/resources/single-span-beam
The span
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 | 13.6 upward | — |
| Roller | 24 | 11.6 upward | — |
The reactions add up to the load: 25.2 = 25.2 kip.
Largest values
- Shear
- 13.6 kip at x = 0 ft
- Sagging moment
- 84.1 kip·ft at x = 9.5 ft
- Hogging moment
- none
- Deflection
- 0.5674 in downward, at x = 11.67 ft
- Bending stress
- 15.6 ksi
- Shear stress
- 2.787 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 – 24 | 0.5674 downward | L/508 |
Load by load
| Load | Left support (kip) | Right support (kip) | Moment at x = 9.5 ft (kip·ft) | Deflection at x = 11.67 ft (in) |
|---|---|---|---|---|
| Point load 1 | 4 | 2 | 29 | -0.1702 |
| Distributed load 1 | 9.6 | 9.6 | 55.1 | -0.3972 |
| All loads | 13.6 | 11.6 | 84.1 | -0.5674 |
Each load alone on the span; the rows add up to the last one, though each value is rounded on its own and the last figure of a sum may differ. The moment is read at the section where the moment of all the loads is largest, or at midspan when it is zero all along. Positive: an upward reaction, a sagging moment. The deflection is read where that of all the loads is largest, and is positive upward.
Section needed
- Moment of inertia needed
- 367.4 in⁴ the section entered has 518 in⁴
- Section modulus needed
- Enter a bending stress limit to see the section modulus needed.
From the largest deflection and E, and from the largest moment: elastic mechanics only. No design code, load factor or safety factor is applied, and comparing each value with the section is left to you.
Values at a point
- Shear, V
- -2 kip
- Bending moment, M
- 81.6 kip·ft
- Deflection
- -0.5669 in
- Slope
- 0.0002556 rad
- Bending stress
- 15.13 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 span is solved
This page runs the engine of Beam reactions and diagrams on one arrangement of supports: a pin at the left end of the span, a roller at the right, nothing between them. Such a span is statically determinate — equilibrium alone fixes its two reactions — and shear, moment and deflection follow as exact polynomials in x, with each largest value taken where it occurs.
Load by load is superposition, which holds because the span is linear. Each load is placed on the span by itself and solved again; its row gives the two reactions it causes, the moment it causes at the section of the largest total moment, and the deflection it causes at the point of the largest total deflection. So the rows add up to the totals. They are not, in general, each load’s own largest values: those may occur at other sections.
The two values under Section needed are mechanics and nothing more. Deflection is proportional to 1 ÷ EI, so the moment of inertia that holds the largest deflection to L/n comes from one solution and the modulus E, whether or not an I was entered. The section modulus is the largest moment divided by the bending stress you type. Each is printed beside what the section entered has; the comparison is yours. With a steel shape’s own weight included, both are for the weight of the shape chosen, not of a heavier one they may point to.
Three cases to check it against by hand, L being the span.
Uniform load w over the whole span: reactions wL/2; largest moment wL²/8 and largest deflection 5wL⁴/(384EI), both at midspan.
Point load P at a from the left support and b from the right: reactions Pb/L and Pa/L; largest moment Pab/L, under the load.
Load rising from zero at the left support to w at the right: reactions wL/6 and wL/3; largest moment wL²/(9√3), at L/√3 from the left.
What it assumes
- A straight beam of a single span and constant section, simply supported: pinned at one end, on a roller at the other.
- Supports that neither settle nor resist rotation.
- Small deflections of an elastic material (Euler–Bernoulli): no shear deformation, no cracking, no creep.
- Vertical loads in the plane of bending; no axial force.
- Linear behaviour: the effects of the loads add.
What it does not do
- It checks nothing against a design code and gives no verdict: the needed values are mechanics for the limit and the stress you choose.
- One span only, with no overhang, fixed end, hinge or applied moment: the form links to the calculator that solves those, on this same beam.
- It uses the loads as entered, service or factored: a deflection limit and a stress may call for different combinations.
- Own weight counts only for a steel shape with its box ticked, and is that shape’s. For another section, add it as a load.
Signs and units
- Each load is a size and a direction; down is gravity.
- Positive shear pushes the left side of a cut upward. Positive moment sags the span, and plots above the axis.
- In the load-by-load table the values are signed: an upward load subtracts, and every column adds to its total, within the rounding of each value.
- L/n is the deflection allowed, the span over n: n = 360 on 24 ft is 0.8 in.
- Feet and kips or metres and kilonewtons; E and stresses in ksi or MPa; I in in⁴ or 10⁶ mm⁴, S in in³ or 10³ mm³.
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 single-span beam calculator of Tercero Tablada Civil and Structural Engineering Inc. — ttcivilstructural.com/resources/single-span-beam