Simply supported beam with a central point load

A beam rests on two supports, one at each end, and carries a single force F at mid-span. This is the textbook case of a beam that sags between two supports.

Typical examples:

a plank or board on two trestles with a weight in the middle
a shaft between two bearings with a pulley or gear at mid-span
a crossbar of a test rig loaded at its centre

Calculator

Enter the span and the load, choose a cross-section and a material. Results, verdicts and diagrams update as you type. Lengths are in millimetres; decimal comma or point both work.

Case

Cross-section

Span and load

Span L

Point load F

Solid rectangle dimensions

Width b

Height h

Material

Required safety factor on yield

Results

Maximum deflection	f	1.102	mm
Span-to-deflection ratio	L/f	1814
Maximum bending moment	M_max	500	N·m
Maximum bending stress	σ_max	20.83	MPa
Left support reaction	R_A	500	N
Right support reaction	R_B	500	N
Maximum shear stress	τ_max	0.3125	MPa
Maximum combined stress (von Mises)	σ_eq,max	20.83	MPa

OK — Bending stress vs yield strength: σ_max = 20.83 MPa, fy = 235 MPa, safety factor 11.28 ≥ required 1.5.

OK — Combined stress (von Mises) vs yield strength: σ_eq,max = 20.83 MPa, fy = 235 MPa, safety factor 11.28 ≥ required 1.5.

Deflection is given without a verdict: the admissible deflection depends on what the beam is for. Compare f with your own requirement.

Diagrams

Sign conventions: dM/dx = V; sagging moment positive (cantilevers hog, so M < 0); deflection negative downward. σ_eq is the largest value over the section height at each position.

Section and material

Solid rectangle: I = 720000 mm⁴, c = 30 mm, A = 2400 mm².
Structural steel S235: E = 210 GPa, fy = 235 MPa (guaranteed minimum). Source: fy: EN 10025-2, minimum ReH for nominal thickness ≤ 16 mm. E: EN 1993-1-1, 3.2.6.

Assumptions

Euler–Bernoulli beam: plane sections stay plane, small deflections and rotations.
Linear elastic, homogeneous and isotropic material.
Constant cross-section along the span.
Self-weight neglected.
Shear deformation neglected (valid for slender beams, span much larger than section height).
Not checked: lateral-torsional buckling, local buckling of thin walls, fatigue, stress concentrations.
Shear stress by the Jourawski formula: mean value across the width (approximate for round sections).
Equivalent stress by the von Mises criterion, scanned over the section height at each position.
Local stresses under concentrated loads and at the supports are not included.

References: Classical beam theory (Euler–Bernoulli) — Double integration of the elastic curve equation E·I·y″ = M(x), with the support boundary conditions. Classical strength of materials, Jourawski (Zhuravskii) shear formula — τ(y) = V·Q(y) / (I·b(y)), mean shear stress across the width b at height y. Von Mises yield criterion — σ_eq = √(σ² + 3τ²) for a normal stress σ combined with a shear stress τ. Modules beam.simply_supported.center_load v1, beam.combined_stress v1.

Engineering aid only. Results are indicative and must be checked by the user before any design decision.

Formulas

Deflection and bending moment are both largest at mid-span, under the load.

Maximum deflection, at mid-span: f_{max} = \frac{F L^3}{48 E I}
Maximum bending moment, at mid-span: M_{max} = \frac{F L}{4}
Support reactions: R_A = R_B = \frac{F}{2}
Deflected shape, for 0 ≤ x ≤ L/2 (symmetric): y(x) = -\frac{F x \left(3L^2 - 4x^2\right)}{48 E I}
Maximum bending stress (c: distance from the neutral axis to the extreme fiber): \sigma_{max} = \frac{M_{max}\, c}{I}
Shear stress over the section (Jourawski): \tau(y) = \frac{V\, Q(y)}{I\, b(y)}
Combined stress (von Mises): \sigma_{eq} = \sqrt{\sigma^2 + 3\,\tau^2}

Symbols: L span, F point load, w load per unit length, E Young's modulus, I second moment of area of the section, c distance from the neutral axis to the extreme fiber, V shear force, M bending moment, Q first moment of area, b section width at height y.

Assumptions

Euler–Bernoulli beam: plane sections stay plane, small deflections and rotations.
Linear elastic, homogeneous and isotropic material.
Constant cross-section along the span.
Self-weight neglected.
Shear deformation neglected (valid for slender beams, span much larger than section height).
Not checked: lateral-torsional buckling, local buckling of thin walls, fatigue, stress concentrations.

Other beam cases

Reference: Classical beam theory (Euler–Bernoulli) — Double integration of the elastic curve equation E·I·y″ = M(x), with the support boundary conditions. Module beam.simply_supported.center_load, version 1.