The Beam Analysis tool calculates support reactions, shear force diagrams (SFD), and bending moment diagrams (BMD) for simply supported and continuous beams. It handles multiple simultaneous loads of different types and displays results both graphically and numerically.

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For a given beam configuration and loading, the tool computes:

• Support reactions — vertical forces (and moments for fixed supports) at each support
• Shear force diagram — the variation of internal shear force along the span
• Bending moment diagram — the variation of internal bending moment along the span
• Maximum and minimum values — peak shear and moment with their locations
• Deflection — the deflected shape and maximum deflection (requires elastic modulus E and moment of inertia I)

The tool supports the most common boundary conditions in structural engineering:

• Simply supported — pinned support at one end, roller support at the other. This is the most common configuration for bridge girders and floor beams.
• Cantilever — fixed at one end, free at the other. Used for overhanging brackets, retaining wall stems, and similar elements.
• Continuous spans — multiple spans sharing interior supports. The tool solves for reactions and diagrams using the principle of consistent deformations.

Three load types can be applied individually or in any combination:

• Point load — a concentrated force at a specific location along the span, defined by its magnitude and distance from the left support.
• Uniform distributed load (UDL) — a load of constant intensity applied over the full span, defined by its magnitude per unit length (e.g., kip/ft or kN/m).
• Partial distributed load — a distributed load applied over a portion of the span, defined by its intensity and the start and end positions.

Multiple loads of different types can be combined — for example, a beam carrying a UDL from self-weight plus two point loads from framing members above. The tool superposes all loads and returns a single set of results for the combined loading.

The tool uses standard structural engineering sign conventions:

• Downward loads are entered as positive values.
• Positive shear: the left portion of a cut section has an upward net force.
• Positive bending moment: the beam is in a sagging condition (concave up), which places the bottom fiber in tension.

Consider a simply supported beam with a span of L = 20 ft carrying a single point load P = 10 kips at the midspan.

Step 1 — Reactions. By symmetry, each support carries half the total load:

R_A = R_B = P / 2 = 10 / 2 = 5 kips

Step 2 — Shear force diagram. Starting from the left support:

• At x = 0 (just right of A): V = +5 kips (upward reaction from A)
• At x = 10 ft (just left of load): V = +5 kips (no change between supports)
• At x = 10 ft (just right of load): V = +5 − 10 = −5 kips (load drops shear by P)
• At x = 20 ft (just left of B): V = −5 kips
• At x = 20 ft (just right of B): V = −5 + 5 = 0 ✓

Step 3 — Bending moment diagram. For a simply supported beam with a central point load, the moment increases linearly from zero at each support to a maximum at midspan:

M_max = P × L / 4 = 10 × 20 / 4 = 50 kip·ft at x = 10 ft

Step 4 — Maximum deflection. For a simply supported beam under a central point load, the peak deflection at midspan is:

Δ_max = P L³ / (48 E I)

For example, using a W16×40 (I_x = 518 in⁴) and E = 29,000 ksi, with P = 10 kips and L = 20 ft = 240 in:

Δ = (10 × 240³) / (48 × 29,000 × 518) = 138,240,000 / 721,056,000 = 0.192 in

• For beams with multiple loads, enter each load separately. The tool automatically superposes all applied loads.
• To model a simply supported span with a cantilever overhang, use the continuous span configuration and set the overhang span with a free end condition.
• Check that the sum of reactions equals the sum of applied loads as a quick sanity check on your inputs.
• For deflection calculations, consistent units are essential — if load is in kips and span in inches, E must be in ksi and I in in⁴.

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