Calculator
One span or several, up to eight, entered one by one: each with its own stiffness, on supports that may settle. The live load can be taken in every arrangement; each diagram is then the band between its two extremes, and each largest value names the spans loaded to cause it.
- Ends
- Simple support, fixed or free
- Live load
- In every arrangement
- Method
- Direct stiffness
Calculation sheet
Continuous beam over several supports
ttcivilstructural.com/resources/continuous-beam
The continuous beam
Loads and supports · uniform loads: dead + live
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. Each band runs from the lowest value to the highest among the arrangements.
Supports
Beside a value: a span with its live load on, one with it off, in the arrangement that gives the value.
| Support | Position (ft) | Reaction, largest (kip) | Reaction, other extreme (kip) | Moment over it, largest (kip·ft) | Moment over it, other extreme (kip·ft) |
|---|---|---|---|---|---|
| 1 | 0 | 21.17 upward | 5.581 upward | 0 | 0 |
| 2 | 24 | 65.43 upward | 24.82 upward | -166.9 | -55 |
| 3 | 54 | 65.43 upward | 24.82 upward | -166.9 | -55 |
| 4 | 78 | 21.17 upward | 5.581 upward | 0 | 0 |
Span by span
Beside a value: a span with its live load on, one with it off, in the arrangement that gives the value.
| Span (ft) | Largest sagging moment (kip·ft) | Deflection from its supports (in) | Length ÷ deflection |
|---|---|---|---|
| 1 0 – 24 | 106.8 at x = 10.08 ft | 0.3986 downward, at x = 11.08 ft | L/723 |
| 2 24 – 54 | 111 at x = 39 ft | 0.5695 downward, at x = 39 ft | L/632 |
| 3 54 – 78 | 106.8 at x = 67.92 ft | 0.3986 downward, at x = 66.92 ft | L/723 |
Largest values
- Shear
- 33.27 kip at x = 24 ft live load on spans 1 and 2
- Sagging moment
- 111 kip·ft at x = 39 ft live load on span 2
- Hogging moment
- 166.9 kip·ft at x = 54 ft live load on spans 2 and 3
- Deflection
- 0.5695 in downward, at x = 39 ft live load on span 2
- Bending stress
- 24.54 ksi
- Shear stress
- 4.592 ksi
Shear divided by the shear area entered. Elastic stresses from M ÷ S and the shear rule above — not a check of strength.
Values at a point
- Shear, V
- -1.772 to 1.772 kip
- Bending moment, M
- 4.637 to 111 kip·ft
- Deflection
- -0.5695 to 0.09732 in
- Slope
- -0.0003915 to 0.0003915 rad
- Bending stress
- 0.6819 to 16.32 ksi
Move over a diagram, or type a position.
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Method
How the continuous beam is solved
The beam is solved by the direct stiffness method, with one beam element for each span. Every joint, an end of a span, has two displacements: its deflection and its rotation. At a support the deflection is known — zero, or the settlement entered — and at a fixed end so is the rotation. The rotations that remain are the unknowns, with one equation each: the end moments of the spans meeting at that support add up to zero.
The loads on a span enter those equations as its fixed-end moments, in closed form: a point load P, a from one end and b from the other, gives Pab²/L² at the first, and a load varying linearly over any part of the span gives that expression integrated. A settlement enters the same way: a span with one support settling δ more than the other, its ends held against rotation, takes 6kEIδ/L² at each end, k being its stiffness factor. An end span whose end is free is a cantilever from its support; statics gives the moment it brings there, and it adds no unknown.
With its two end moments known, each span is a simply supported beam under its own loads and those two moments, and shear, moment, slope and deflection inside it are closed forms worked out at the point asked for. Nothing is interpolated between joints. A largest value that falls between two loads or supports is located on the polynomial itself, where its derivative changes sign, and worked out there.
For a beam, this is the finite element method, with a span for an element. And one element per span loses nothing. A mesh is approximate when its elements cannot take the true deflected shape. A beam element of constant stiffness can: between two joints an unloaded span bends into a cubic, which is the element’s own shape, and what the loads add inside the span is put back in closed form. Beam reactions and diagrams reaches the same result with no elements at all, as its page says; on a beam that both can describe, the two calculators agree, except where a value is so small that the other page prints it as 0.
Taking the live load in every arrangement — pattern loading — means each span’s live load is on that span or off it, whatever the others carry: 2ⁿ arrangements for n spans with live load, 256 for eight. Dead loads and settlements are in all of them.
The beam being linear, the arrangements are not solved one by one. It is solved once under what is always there and once under each span’s live load alone. At any point the highest value of a response is the first of those plus every span’s effect that raises it there, and the lowest is the first plus every effect that lowers it: the extremes of all the arrangements, exactly.
Each largest value over the arrangements comes with the spans whose live load is on when it occurs. A span whose live load has no effect on a value is not listed with it: no load on another span puts shear or moment in an overhang, for one.
Four results to verify by hand: equal spans L of one stiffness EI, on simple supports.
Two spans under a uniform load w: −wL²/8 over the middle support; reactions 3wL/8, 5wL/4 and 3wL/8; largest sagging moment 9wL²/128, at 3L/8 from each end.
Three spans under w: −wL²/10 over each interior support; reactions 2wL/5, 11wL/10, 11wL/10 and 2wL/5; largest sagging moments 2wL²/25 in the end spans and wL²/40 in the middle one.
Two unloaded spans whose middle support settles by δ: a sagging moment 3EIδ/L² over it; each end reaction rises by 3EIδ/L³ and the middle one falls by 6EIδ/L³.
Two spans with a live load w in every arrangement: largest sagging moment 49wL²/512, at 7L/16 from an end, with that span alone loaded; −wL²/8 over the middle support with both.
What it assumes
- A straight beam with no internal hinge, continuous over every support between two spans. Each span has one flexural stiffness along its length: the section’s EI times that span’s factor.
- Every joint between two spans is a simple support. Each end of the beam is on a simple support, fixed, or free.
- Rigid supports: one moves only by the settlement entered for it, and a simple support does not restrain rotation.
- Small deflections of a linear elastic material in bending (Euler–Bernoulli), with shear deformation left out.
- Vertical loads only, and so no axial force. The effects of loads add, which is what allows the arrangements to be combined.
What it does not do
- It applies no design code and no load factor, and gives no verdict. Dead and live loads are used as entered, and an arrangement is their plain sum.
- A span’s live load is all on or all off: no part of it is taken on its own, and a moving load is not followed along the beam.
- One stiffness for each span, and the factor is yours: a section that changes within a span is not modelled, and no factor is derived for a cracked or a deeper one.
- No internal hinges and no applied moments. Beam reactions and diagrams has both, and the form links to it with this beam when that calculator can hold it.
- No spring supports. A settlement is a length you enter, not one the page works out, and it needs E and I to become forces.
- Stresses are of the section entered: M/S in bending and, for shear, the rule printed with the value — V over a shear area, 1.5V/A for a rectangle. Where any span’s stiffness factor is not 1, that span is another section and no stress is printed at all.
- A span under 1/1000 of the whole length, or a stiffness factor outside 0.001 to 1,000, is refused: the form marks the entry and no result is shown.
- With a steel shape and its own-weight box ticked, that weight is a dead load on every span. For any other section, enter it as dead load.
Signs and units
- Each span’s two uniform loads, dead and live, are positive downward: a minus sign turns one upward. The further loads have a direction of their own, and each is marked dead or live.
- A further load is placed from the left end of its own span, not of the beam. In the results x runs along the whole beam, from its left end.
- A settlement is entered as a length downward, in inches or millimetres; the support’s deflection is then that length, downward.
- A sagging moment is positive, so the hogging moment that gravity loads usually cause over an interior support is negative; an arrangement of the live load, or an overhang, can make the moment there positive. Shear is positive when the forces to the left of a cut add up to an upward one.
- The deflection of a span is measured from the straight line through its two supports, so that a settlement is not counted as bending; an overhang’s is measured from its support. The deflection diagram, the largest deflection and the one at a point are the movement of the beam itself, a settlement included. Where a deflection is printed with a sign, upward is positive; a slope is positive counter-clockwise, in radians.
- With the live load in every arrangement a diagram is a band. Its upper edge is the highest value at each point and its lower edge the lowest; the two edges, and two points of one edge, are in general from different arrangements.
- Spans and positions in feet or metres, loads in kips or kilonewtons; a settlement, like the section, in inches or millimetres. A stiffness factor has no unit.
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 continuous beam calculator of Tercero Tablada Civil and Structural Engineering Inc. — ttcivilstructural.com/resources/continuous-beam