Simply Supported Beam Calculator — Point, Moment, Uniform & Varying Loads

Sign convention used

• Coordinate x: measured from the left support toward the right.
• Loads: downward point loads and distributed loads are entered as positive.
• Shear force V(x): positive when it acts upward on the left face of the cut section (equivalently, when it tends to rotate the segment clockwise).
• Bending moment M(x): positive (sagging) when the beam is concave up and the top fibers are in compression. Negative (hogging) moments put the bottom fibers in compression.
• Concentrated moments: a positive M produces sagging, a negative M produces hogging.
• Slope θ(x): positive when the deflected centerline rotates counter-clockwise (upward to the right).
• Deflection y(x): reported with sign from the differential equations of the elastic curve. For typical positive downward loading, sagging deflections at mid-span will appear with a negative sign; the calculator also reports |y|max as an algebraic extremum.

Use the sign of V(x), M(x), θ(x), and y(x) together with the charts to interpret the beam response.

Global units

Input parameters

Point loads (P at a)

P (lbf)	a (from left) (ft)	Action
Unit: lbf	Unit: ft

+ Add load

Concentrated moments (M at a)

M (lbf·in)	a (from left) (ft)	Action
Unit: lbf·in	Unit: ft

+ Add moment

Positive M produces sagging (top fibers in compression). Reactions from a positive M are R₁ = +M/L and R₂ = −M/L.

UDL segments (intensity w from s to e)

w (lbf/ft)	s (start) (ft)	e (end) (ft)	Action
Unit: lbf/ft	Unit: ft	Unit: ft

+ Add UDL

For a uniform load over [s, e], total load W = w·(e−s) acting at x̄ = (s+e)/2.

VDL segments (linearly varying load w₁→w₂ from s to e)

w₁ (at s) (lbf/ft)	w₂ (at e) (lbf/ft)	s (start) (ft)	e (end) (ft)	Action
Unit: lbf/ft	Unit: lbf/ft	Unit: ft	Unit: ft

+ Add VDL

Trapezoidal load: total W = (w₁+w₂)/2 · (e−s). Centroid \(x̄ = s + (e-s)\frac{w_1+2w_2}{3(w_1+w_2)}\).

• Use dot “.” as decimal separator.
• Constraints: 0 ≤ aᵢ ≤ L; 0 ≤ sⱼ < eⱼ ≤ L; 0 ≤ x ≤ L.
• Sign convention: downward loads positive; positive moments cause sagging.

Results

Inputs used in calculation

Charts

FAQ

Worked example (multiple point loads + UDL)

This example shows how to reproduce a full beam analysis using the calculator above: reactions, shear, bending moment, slope, deflection, and bending stress.

Show the worked example

Problem

A timber beam AB of span 3 m, width 200 mm and height 100 mm supports three concentrated loads as shown. The elastic modulus is 8 GPa and the density is 600 kg/m³.

• Goal Find max deflection, max shear, max bending moment, mid-span deflection/slope, and end reactions.
• Sign convention Downward loads positive; sagging moments positive.

Step 1 — Inputs

Parameter	Value	Unit
Timber width, b	100	mm
Timber height, H	200	mm
Span, L	3000	mm
Mid-span position, x	1500	mm
Elastic modulus, E	8	GPa
Beam type	Simply supported with multiple point loads + UDL

Step 2 — Section properties (rectangle)

Compute the rectangular section properties using: Solid Rectangle — Sectional Properties Calculator.

Parameter	Value	Unit
Second moment, Ixx	66,666,668	mm⁴
Section modulus, Sxx	666,666.688	mm³

For bending stress at the extreme fiber, use \(c=H/2=50\;\mathrm{mm}\).

Step 3 — Enter the case into this calculator

Beam self-weight UDL: \(w = (M g)/L = 36 \cdot 9.81 / 3 = 117.7\;\mathrm{N/m}\).

Point loads

#	Location (m)	Magnitude (kN)
1	0.5	10
2	1.5	5
3	2.5	10

Distributed loads

#	Start (m)	w (N/m)	End (m)	w (N/m)
1	0	117.7	3	117.7

Beam & material

Parameter	Symbol	Value	Unit
Beam length	L	3	m
Report position	x	1.5	m
Elastic modulus	E	8	GPa
Extreme fiber distance	c	50	mm
Second moment	I	66,666,668	mm⁴

Tip: set your global units to m, kN, kN·m (or use N/m and convert consistently).

Step 4 — Results (reference)

Parameter	Value	Unit
Reaction Force R₁	12676.5	N
Reaction Force R₂	12676.5	N
Shear @ x	2500.0	N
Max Shear	12676.5	N
Moment @ x	8882.4	N·m
Max Moment	8882.4	N·m
Slope at left, θ₁	-0.988	degree
Slope at right, θ₂	0.988	degree
Deflection @ x	-15.662	mm
Max Deflection	-15.662	mm
Bending Stress @ x	6.7	MPa

Related: Rectangle section properties · Beam calculators overview

Formula library for common load cases

These closed-form equations match the calculator’s solver (Macaulay/singularity functions) and are useful for hand checks.

Macaulay bracket: \(\langle x-a\rangle^n = 0\) for \(x < a\) and \(\langle x-a\rangle^n = (x-a)^n\) for \(x \ge a\).

A) Point load P at position a

$$R_1=\frac{P(L-a)}{L},\qquad R_2=\frac{Pa}{L}$$ $$V(x)=R_1 - P\langle x-a\rangle^{0}$$ $$M(x)=R_1\,x - P\langle x-a\rangle^{1}$$ $$\theta(x)=\theta_1+\frac{R_1 x^{2}}{2EI}-\frac{P}{2EI}\,\langle x-a\rangle^{2}$$ $$y(x)=\theta_1\,x+\frac{R_1 x^{3}}{6EI}-\frac{P}{6EI}\,\langle x-a\rangle^{3}$$ $$\sigma(x)=\dfrac{M(x)\,c}{I}$$

B) Linearly varying (trapezoidal / triangular) load w₁→w₂ on [s,e]

Let \(\ell=e-s\), \(k=\dfrac{w_2-w_1}{\ell}\), \(W=\dfrac{w_1+w_2}{2}\ell\).
$$\bar{x}=s+\ell\,\frac{w_1+2w_2}{3(w_1+w_2)}$$ $$R_2=\frac{W\bar{x}}{L},\qquad R_1=W-R_2$$ $$V(x)=R_1-\Big[w_1(\langle x-s\rangle-\langle x-e\rangle)+\frac{k}{2}\big(\langle x-s\rangle^2-\langle x-e\rangle^2\big)\Big]$$ $$M(x)=R_1x-\Big[\frac{w_1}{2}\big(\langle x-s\rangle^2-\langle x-e\rangle^2\big)+\frac{k}{6}\big(\langle x-s\rangle^3-\langle x-e\rangle^3\big)\Big]$$ $$\theta(x)=\theta_1+\frac{R_1x^2}{2EI}-\frac{1}{EI}\Big[\frac{w_1}{6}\big(\langle x-s\rangle^3-\langle x-e\rangle^3\big)+\frac{k}{12}\big(\langle x-s\rangle^4-\langle x-e\rangle^4\big)\Big]$$ $$y(x)=\theta_1x+\frac{R_1x^3}{6EI}-\frac{1}{EI}\Big[\frac{w_1}{24}\big(\langle x-s\rangle^4-\langle x-e\rangle^4\big)+\frac{k}{60}\big(\langle x-s\rangle^5-\langle x-e\rangle^5\big)\Big]$$ $$\sigma(x)=\dfrac{M(x)c}{I}$$

C) Concentrated moment M₀ at position a

$$R_1=-\frac{M_0}{L},\qquad R_2=\frac{M_0}{L}$$ $$V(x)=R_1$$ $$M(x)=R_1x+M_0\langle x-a\rangle^{0}$$ $$\theta(x)=\theta_1+\frac{R_1x^2}{2EI}+\frac{M_0}{EI}\langle x-a\rangle^{1}$$ $$y(x)=\theta_1x+\frac{R_1x^3}{6EI}+\frac{M_0}{2EI}\langle x-a\rangle^{2}$$ $$\sigma(x)=\dfrac{M(x)c}{I}$$

Note: Use the same sign convention as the calculator (sagging positive; downward loads positive).

Related beam calculators

• Beam Deflection Calculators – overview
• Simply Supported Beam – single point load
• Simply Supported Beam – two point loads
• Simply Supported Beam – uniform distributed load
• Simply Supported Beam – multiple distributed loads
• Simply Supported Beam – applied end/midspan moments
• Cantilever Beam – mixed loads
• Fixed–Fixed Beam – mixed loads
• Column Buckling Calculator

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