Complete Guide to Beam Analysis

This beam calculator solves shear force, bending moment, reaction forces, and deflection for any beam configuration. It automatically draws the shear and moment diagrams (SFD and BMD) with full step-by-step working for simply supported beams, cantilever beams, and fixed-fixed beams.

Beam Types Explained

Understanding the three main beam types is the foundation of structural beam analysis and beam design.

1. Simply Supported Beam

A simply supported beam rests on two supports: a pin (resists vertical and horizontal forces) and a roller (resists only vertical force). It cannot resist moments at the supports. Common uses: floor joists, bridges, lintels.

Key formula: For a central point load P, max moment M = PL/4 and max deflection δ = PL³/48EI.

2. Cantilever Beam

A cantilever beam is fixed at one end and free at the other. The fixed end resists vertical force, horizontal force, AND moment. Common uses: balconies, diving boards, aircraft wings, signposts.

Key formula: For a tip point load P, max moment M = PL at the wall, max deflection δ = PL³/3EI at the tip.

3. Fixed-Fixed Beam

A fixed-fixed beam (or built-in beam) is rigidly clamped at both ends. Both ends resist forces and moments. Stiffer than simply supported — same load produces ¼ the deflection. Common uses: built-in slabs, frame structures.

Key formula: For a central point load P, max moment M = PL/8 at the supports, max deflection δ = PL³/192EI at midspan.

How to Calculate Shear Force and Bending Moment

Here is the step-by-step method that every shear force and bending moment calculator uses internally, demonstrated with a worked example.

Beam Deflection Formulas Reference Table

The most commonly used beam deflection and beam equations, organized by support condition and loading.

Beam Type	Max Deflection (δmax)	Max Moment (Mmax)	Location
Simply supported, central point load P	PL³ / 48EI	PL / 4	Midspan
Simply supported, UDL w	5wL⁴ / 384EI	wL² / 8	Midspan
Cantilever, tip point load P	PL³ / 3EI	PL	Tip / Wall
Cantilever, UDL w	wL⁴ / 8EI	wL² / 2	Tip / Wall
Fixed-fixed, central point load P	PL³ / 192EI	PL / 8	Midspan / Supports
Fixed-fixed, UDL w	wL⁴ / 384EI	wL² / 12	Midspan / Supports

Worked Examples — All Beam Calculations

1. How to Calculate Reaction Forces for a Simply Supported Beam

For any simply supported beam, use two equilibrium equations: ΣF_y = 0 (vertical forces balance) and ΣM = 0 about one support (moments balance about that point). Take moments about the support whose reaction you don't yet know — that reaction has zero moment arm and drops out.

Example: Beam L = 8 m, point load 40 kN at x = 3 m from A. ΣM_A = 0: R_B(8) − 40(3) = 0 → R_B = 15 kN. Then R_A = 40 − 15 = 25 kN.

2. How to Draw a Shear Force Diagram Step by Step

Start from the left end at V = 0. Move right along the beam — at every point load or reaction, the shear jumps vertically by that force amount (up for upward forces, down for downward). Between point loads with no UDL, shear is constant (horizontal line). Under a UDL, shear varies linearly with slope = −w.

Example: L = 6 m, R_A = 10 kN, 20 kN load at x = 3 m, R_B = 10 kN. SFD: jumps up to +10 at A, stays +10 until x = 3, jumps down to −10, stays −10 until x = 6, jumps up to 0 at B. V_max = 10 kN.

3. How to Draw a Bending Moment Diagram Step by Step

Moment at any point = area under the shear diagram up to that point. Start at M = 0 at a pin or roller support. For point loads, the BMD is triangular (linear segments). For UDL, the BMD is parabolic (curve). Maximum moment occurs where shear crosses zero.

Example: Same beam as above. At x = 0, M = 0. At x = 3 m, M = +10 × 3 = +30 kN·m (area of rectangle). At x = 6 m, M = 30 + (−10 × 3) = 0. M_max = 30 kN·m at midspan.

4. How to Calculate Maximum Deflection of a Cantilever Beam

For a cantilever with point load P at the tip: δ_max = PL³ / (3EI). For UDL w: δ_max = wL⁴ / (8EI). Make sure units are consistent — convert E to N/m², I to m⁴, and L to m.

Example: Cantilever L = 3 m, P = 10 kN at tip, E = 200 GPa = 200×10⁹ N/m², I = 1000 cm⁴ = 1000×10⁻⁸ m⁴ = 1×10⁻⁵ m⁴. δ = (10000 × 27) / (3 × 200×10⁹ × 1×10⁻⁵) = 270000 / 6×10⁶ = 0.045 m = 45 mm.

5. How to Calculate Bending Stress in a Beam

Bending stress σ = M·c / I, where M is bending moment at the section, c is the distance from the neutral axis to the outermost fiber (half the section depth for symmetric sections), and I is the moment of inertia about the neutral axis. Use consistent SI units (Pa).

Example: M = 30 kN·m = 30,000 N·m, c = 100 mm = 0.1 m, I = 1000 cm⁴ = 1×10⁻⁵ m⁴. σ = 30000 × 0.1 / 1×10⁻⁵ = 300 MPa.

6. How to Find the Moment Diagram for a Cantilever Beam

For a cantilever with tip load P, the moment is maximum at the fixed wall and zero at the free end. M(x) = −P(L − x), measured from the wall. The diagram is a triangle peaking at the wall with magnitude PL.

Example: L = 4 m, P = 5 kN at tip. M_max at wall = −5 × 4 = −20 kN·m (negative because the beam curves downward — hogging).

7. What is the Superposition Method for Beams?

The superposition principle states that for a linear elastic structure, the total response (deflection, moment, shear, stress) under combined loads equals the algebraic sum of responses from each load applied separately. This works because beam equations are linear in load.

Example: A simply supported beam carries a central point load AND a UDL. Calculate δ from each load separately using table formulas, then add: δ_total = PL³/48EI + 5wL⁴/384EI.

8. How to Calculate Beam Size for a Given Load

Find the required moment of inertia I from the allowable deflection limit (e.g., L/360), then choose a section from steel tables that has at least that I. Also check bending stress σ = Mc/I ≤ σ_allow.

Example: Simply supported L = 5 m with UDL 10 kN/m, allow δ = L/360 = 13.9 mm. Rearrange: I_req = 5wL⁴ / (384·E·δ) = 5(10000)(5⁴) / (384 × 200×10⁹ × 0.0139) ≈ 2.93×10⁻⁵ m⁴ = 2930 cm⁴.

9. How to Use the Moment Calculator

The moment calculator computes M = F × d, where F is the force and d is the perpendicular distance (moment arm) from the pivot. To reverse, F = M / d. Make sure F is perpendicular to d — otherwise use F·d·sin(θ).

Example: F = 100 kN, d = 2 m → M = 200 kN·m. Reverse: M = 500 N·m, d = 0.25 m → F = 500/0.25 = 2000 N.

10. How to Find Shear Force at Any Point on a Beam

Cut the beam at the point of interest and sum all vertical forces on one side: V = ΣF_y (left of cut). Upward forces are positive. This gives the shear at that location.

Example: Same beam (R_A = 10 kN, load 20 kN at midspan). At x = 1.5 m (left of load): V = +10 kN. At x = 4.5 m (right of load): V = 10 − 20 = −10 kN.

Beam Load Quick Reference Table

This beam load calculator reference table shows approximate max moment and deflection for simply supported steel beams (E = 200 GPa, I = 1000 cm⁴) under uniform distributed loads.

Span L (m)	UDL w (kN/m)	Mmax = wL²/8 (kN·m)	δmax (mm)
3	5	5.6	2.6
3	10	11.3	5.3
3	20	22.5	10.5
5	5	15.6	20.3
5	10	31.3	40.7
5	20	62.5	81.4
6	10	45.0	84.4
8	10	80.0	266.7
10	5	62.5	325.5
10	10	125.0	651.0

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