What is the difference between a pin, a roller and a fixed support?

A roller holds vertical movement only, so the beam can rotate and slide along its own axis. A pin holds vertical and horizontal movement but still allows rotation, so it gives no moment reaction. A fixed support holds all three — vertical, horizontal and rotation — which is why it returns a moment as well as a force. The choice changes the answer substantially: on the same 8 m beam under the same 10 kN/m, fixing both ends cuts the peak bending moment from 80 to 53.3 kN.m and moves it from mid-span to the supports.

Where does the maximum bending moment occur on a beam?

Wherever the shear force crosses zero. That is the single most useful rule for reading the two diagrams together: the bending moment is the integral of the shear, so it reaches a local maximum or minimum exactly where the shear changes sign. On a simply supported beam under a uniform load that point is mid-span; on a cantilever the maximum moment is at the fixed support; on a continuous beam there is one such point in each span plus a hogging peak over each interior support.

How do I know if my beam deflection is acceptable?

Compare it against the span/ratio limit your design code sets. Typical values are L/250 for total deflection of a floor beam, L/300 for live load alone, L/360 where brittle finishes are supported, L/180 for a roof with no ceiling, and L/500 for glass or sensitive equipment. Always use the limit your governing code states rather than a general figure. If the beam fails a deflection check the fix is a stiffer section — more second moment of area — not a stronger one, because deflection does not depend on yield strength.

What is a statically indeterminate beam?

A beam with more unknown support reactions than the three equations of plane equilibrium can solve. A single span on a pin and a roller is determinate; add a third support, or fix an end, and the reactions now depend on the relative stiffness of the spans as well as on the geometry. Hand methods need compatibility conditions as well as equilibrium, which is why StructureCalcs uses the direct stiffness method instead — to the person using it, a continuous beam is no harder to enter than a simple one.

Can a beam carry axial force as well as bending?

Yes, and StructureCalcs shows it. If a point load is applied at an angle rather than straight down, its horizontal component travels along the beam as axial force and appears in a fourth diagram alongside shear, moment and deflection. The supports then return a horizontal reaction as well as a vertical one. A member carrying both bending and axial load is a beam-column, and its design check is a combined one rather than bending alone.

How to analyze a beam: supports, loads, and the four diagrams

A visual reference. Every figure on this page is a real model solved by the same engine that runs the Beam Calculator and drawn by the same code that draws the calculator’s canvas — so what you see here is exactly what the tool will give you. For full hand-worked derivations, see the ten beam worked examples.

What a beam analysis has to produce

A beam carries load by bending. Analysis answers four questions about it, and a complete answer is four diagrams plus the support reactions — not a single number.

Figure 1.The reference beam used throughout this page: 8 m, pinned at the left, roller at the right, carrying 10 kN/m. The purple arrows are the solved support reactions; the dashed line is the deflected shape, exaggerated so it is visible.

Reactions — what the supports push back with. Their sum must balance the applied load exactly; that check is the first thing to do on any result.
Shear force — the internal force trying to slide one part of the beam past the next. It governs web thickness and shear connections.
Bending moment — the internal couple curving the beam. It governs the section size, and it is the diagram most design work starts from.
Deflection — how far the beam actually moves. Strength keeps it from breaking; deflection keeps it usable.

For this beam the engine gives a peak moment of 80.0 kN·m at mid-span and a peak deflection of 26.67 mm — the textbook wL²/8 and 5wL⁴/384EI, which is a useful reminder that the classical formulas are special cases of what the solver does in general.

Supports: what each one holds

A support is defined by what it prevents. The same beam under the same load gives three different answers depending on the restraint at its ends — which is why picking the support type correctly matters more than almost any other modeling decision.

Roller — holds vertical movement only. The beam is free to rotate and to slide along the beam axis. Used where thermal expansion has to be allowed, such as one end of a bridge girder on a sliding bearing.
Pin — holds vertical and horizontal movement, but allows rotation. It gives no moment reaction. A single bolt group in a steel beam-to-column connection is usually modeled as a pin.
Fixed — holds all three: vertical, horizontal and rotation. It therefore returns a moment reaction. A fully welded beam-to-column joint, or a beam cast into a concrete wall.

Figure 2.The same span and the same 10 kN/m, with the left end FIXED instead of pinned. The fixed end now returns a moment of 80.0 kN·m, and the reactions are no longer equal — the stiffer end attracts more load.

Figure 3.Fixed at both ends. The end moments rise to 53.3 kN·m each and the mid-span sagging moment drops to a third of the simply supported value — the same load, carried far more efficiently, at the cost of needing joints that can actually deliver the fixity.

Fixing both ends of the reference beam cuts its peak moment from 80.0 to 53.3 kN·m. That is the whole argument for continuity, and also its warning: the analysis only delivers it if the real connection does.

Loads: what each one does to the diagrams

Four load types cover almost everything a beam sees, and each leaves a signature on the diagrams that is worth being able to recognize on sight.

Figure 4.All four on one beam: a 20 kN point load at 2 m, an 8 kN/m uniform load from 4–7 m, a load rising to 12 kN/m from 8–11 m, and a 25 kN·m applied moment at 11.5 m.

Point load — a force at one location: a column landing on a transfer beam, a machine bolted to a floor. It puts a vertical step in the shear diagram and a kink in the moment diagram.
Uniform load (UDL) — constant intensity over a length: slab self-weight, a floor build-up. Shear varies linearly under it; the moment goes parabolic.
Varying load — trapezoidal or triangular: soil or water pressure, wind that grows with height. Shear goes parabolic, the moment cubic.
Applied moment — a pure couple with no net force, from an eccentric bracket or a fixed-end connection carried in. It puts a step in the moment diagram and leaves the shear untouched.

An area load is the same thing expressed as a pressure (kN/m²) times a tributary width. Give the calculator the pressure and the width at each end and it converts to the equivalent line load for you, which saves the arithmetic step where mistakes usually happen.

Reading the shear force diagram

Figure 5.Shear for the reference beam. It starts at the left reaction, falls linearly under the uniform load, and crosses zero at mid-span — which is exactly where the moment peaks.

A point load (or a support reaction) makes the shear jump, by its own magnitude.
A uniform load makes it fall linearly; a varying load, parabolically.
Where nothing is applied, the shear is constant.
Where the shear crosses zero, the bending moment is at a local maximum or minimum.

That last rule is the one worth internalizing: it tells you where to look on the moment diagram without reading it, and it is a fast sanity check that a result is self-consistent. Hover (or tap, on a phone) anywhere on the calculator’s chart to read off the exact value and position.

Reading the bending moment diagram

Figure 6.Moment for the reference beam — the parabola a uniform load always produces, peaking at 80.0 kN·m at mid-span. Sagging is plotted above the axis here; the calculator has a Flip control if your convention is the other way up.

At a pin or roller the moment is zero — those supports cannot resist one.
At a fixed support it is not zero: the fixed-end moment is a reaction.
Under a point load the diagram kinks but stays continuous.
An applied moment makes it step.
Under a uniform load it is parabolic; the peak sits where the shear crosses zero.

Sign convention is the usual source of confusion, and it is a convention, not a fact — different textbooks draw sagging up or down. StructureCalcs plots sagging above the axis by default and lets you flip it, so you can match whatever your course or office uses. The flip carries through to the PDF and the report builder, so the paper never disagrees with the screen.

The axial diagram, and loads that are not vertical

Not every load on a beam points straight down. A hanger rod at an angle, a raking prop, a cable tie-back, wind on a sloping member — all of them push along the beam as well as across it. Give a point load an angle and the calculator resolves it: the vertical part bends the beam as before, and the horizontal part travels along it as axial force.

Figure 7.A 25 kN load at 35° from vertical, at mid-span. The pin now returns a horizontal reaction as well as a vertical one, drawn outboard along the beam’s own axis, and the vertical reactions have dropped by cos 35°.

Figure 8.The axial force diagram for that beam: constant TENSION from the pin to the load, zero beyond it. The load pushes right at mid-span while the pin holds left at the end, so that stretch of beam is being pulled apart; past the load there is nothing to resist, because a roller cannot deliver a horizontal force. Tension is positive here, matching the truss, frame and section tools.

On a beam with no tilted load this diagram is flat zero, so the calculator only shows it when there is something to show. Worth knowing it exists: a beam that is also carrying axial load is a beam-column, and its design check is a combined one, not bending alone.

Deflection, and the limits that govern it

Figure 9.Deflection for the reference beam, peaking at 26.67 mm. Down is negative throughout the site.

Strength stops a beam breaking; deflection stops it being annoying. Excessive movement cracks plaster, jams doors, makes floors feel bouncy, and upsets sensitive equipment — none of which is a collapse, and all of which is a defect. Codes express the limit as a fraction of the span:

L/250 — total deflection of a floor beam (a common AS 4100 / Eurocode figure)
L/300 — live load only
L/360 — floors carrying brittle finishes
L/180 — roof beams with no ceiling below
L/500 — supporting glass or sensitive equipment

Always check the limit that actually applies to your project; those are typical values, not a substitute for the code. For the 8 m reference beam, L/250 is 32 mm against 26.67 mm actual — comfortable. When it is not comfortable the fix is a stiffer section (more I), not a stronger one; deflection does not care about yield strength.

Determinate, indeterminate, and why it stops mattering

A beam is statically determinate when equilibrium alone gives the reactions — a single span on a pin and a roller, or a cantilever. Add a support, or fix an end, and there are more unknowns than equations: the beam is indeterminate, and the answer now depends on stiffness as well as geometry.

Figure 10.Two 6 m spans over three supports, carrying 12 kN/m. The middle support takes far more than its share, and a hogging moment of 54.0 kN·m appears over it — a moment that does not exist if the two spans are built as separate simply supported beams.

Continuity is worth having: the hogging moment over the interior support pulls the mid-span sagging moments down, so the same load needs a smaller section. It is also why continuous beams cannot be solved by equilibrium — and why every serious tool, including this one, uses the direct stiffness method instead. To the person using the calculator the distinction disappears: add spans in the properties table, put supports where they belong, and the solver assembles and inverts the global stiffness matrix without being asked.

Figure 11.A cantilever: fixed at the left, free at the right, 20 kN at the tip. Maximum moment at the support, maximum deflection at the free end, hogging throughout — the mirror image of the simply supported case in every respect.

Sign conventions used here

Loads — negative acts downward. This trips up nearly everyone once; enter gravity as a negative number.
Shear — positive shear acts upward on the left face of a cut.
Moment — positive is sagging (concave up). Flip the plot if your convention differs.
Axial — positive is tension, matching the truss, frame and section tools.
Deflection and settlement — negative is downward.

These match most textbooks. If you are comparing against a specific one, check its convention first — a result that looks wrong by exactly a minus sign almost always is not wrong.

Where to go next

Work the topics on this page through in full:

Simply supported beam with a UDL — where wL²/8 comes from, derived.
Cantilever with a tip load and a UDL — fixed-end behavior and superposition.
Two-span continuous beam — the three-moment equation against the solver.
Beam with support settlement — the case where geometry alone is not enough.

Then take it further:

Beam Calculator — build your own; results update as you type.
Frame Calculator — when the beam and its columns act together, and sway matters.
Section Calculator — draw a cross-section and get the I your beam analysis needs.
Understanding steel sections — choosing from the 2,600-section library.
Report Builder — put the diagrams and tables into a calculation report.

Frequently asked questions

What is the difference between a pin, a roller and a fixed support?: A roller holds vertical movement only, so the beam can rotate and slide along its own axis. A pin holds vertical and horizontal movement but still allows rotation, so it gives no moment reaction. A fixed support holds all three — vertical, horizontal and rotation — which is why it returns a moment as well as a force. The choice changes the answer substantially: on the same 8 m beam under the same 10 kN/m, fixing both ends cuts the peak bending moment from 80 to 53.3 kN.m and moves it from mid-span to the supports.
Where does the maximum bending moment occur on a beam?: Wherever the shear force crosses zero. That is the single most useful rule for reading the two diagrams together: the bending moment is the integral of the shear, so it reaches a local maximum or minimum exactly where the shear changes sign. On a simply supported beam under a uniform load that point is mid-span; on a cantilever the maximum moment is at the fixed support; on a continuous beam there is one such point in each span plus a hogging peak over each interior support.
How do I know if my beam deflection is acceptable?: Compare it against the span/ratio limit your design code sets. Typical values are L/250 for total deflection of a floor beam, L/300 for live load alone, L/360 where brittle finishes are supported, L/180 for a roof with no ceiling, and L/500 for glass or sensitive equipment. Always use the limit your governing code states rather than a general figure. If the beam fails a deflection check the fix is a stiffer section — more second moment of area — not a stronger one, because deflection does not depend on yield strength.
What is a statically indeterminate beam?: A beam with more unknown support reactions than the three equations of plane equilibrium can solve. A single span on a pin and a roller is determinate; add a third support, or fix an end, and the reactions now depend on the relative stiffness of the spans as well as on the geometry. Hand methods need compatibility conditions as well as equilibrium, which is why StructureCalcs uses the direct stiffness method instead — to the person using it, a continuous beam is no harder to enter than a simple one.
Can a beam carry axial force as well as bending?: Yes, and StructureCalcs shows it. If a point load is applied at an angle rather than straight down, its horizontal component travels along the beam as axial force and appears in a fourth diagram alongside shear, moment and deflection. The supports then return a horizontal reaction as well as a vertical one. A member carrying both bending and axial load is a beam-column, and its design check is a combined one rather than bending alone.