Rectangular Beam Section Formulas

Beam calculations depend on how a member is supported and where its load acts. This reference compares four elementary cases for a prismatic rectangular section: a cantilever or simply supported beam, each with either a concentrated load or a uniform line load. It gives the peak bending moment, maximum elastic deflection, and extreme-fiber bending stress for each case.

Beam formula reference table

MECHANITEC / ENGINEERING REFERENCE

Rectangular beam section formulas

Elastic bending and deflection equations for four common rectangular-beam load cases.

1–4 of 4 matching rows (4 total)

Rectangular beam section formulas — page 1 of 1
Support and loading	Maximum bending moment	Maximum deflection	Maximum bending stress	Example stress (MPa)
Cantilever; point load P at free end	P L	P L^3 / (3 E I)	6 P L / (b h^2)	3.75; b=20 mm, h=40 mm, P=100 N, L=200 mm
Cantilever; uniform line load w over full span	w L^2 / 2	w L^4 / (8 E I)	3 w L^2 / (b h^2)	3.75; b=20 mm, h=40 mm, w=1 N/mm, L=200 mm
Simply supported; center point load P	P L / 4	P L^3 / (48 E I)	3 P L / (2 b h^2)	0.9375; b=20 mm, h=40 mm, P=100 N, L=200 mm
Simply supported; uniform line load w over full span	w L^2 / 8	5 w L^4 / (384 E I)	3 w L^2 / (4 b h^2)	0.9375; b=20 mm, h=40 mm, w=1 N/mm, L=200 mm

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Section properties and symbols

For a rectangle with width b and depth h in the bending plane, the second moment of area is I = bh³/12. The distance from the neutral axis to the outer surface is c = h/2. In the elastic range, the maximum normal bending stress is σ = M_maxc/I. The table substitutes those section properties into the familiar beam equations.

L is the span or cantilever length, P is a point force, w is a uniformly distributed force per unit length, and E is Young’s modulus. Keep units consistent. With millimeters, newtons, and E in N/mm² (MPa), the deflection is in millimeters and the stress is in MPa.

Worked example and a useful check

For a cantilever with a 100 N end load, a 200 mm length, and a 20 mm by 40 mm section, I = 20×40³/12 = 106,666.7 mm⁴ and c = 20 mm. The maximum moment is 100×200 = 20,000 N·mm, so the elastic bending stress is M c/I = 3.75 MPa. The same load on a simply supported span, placed at its center, gives one quarter of that peak moment and a stress of 0.9375 MPa.

The section orientation matters: rotating the rectangle so that its 20 mm dimension is the bending depth changes I dramatically because depth is cubed. Confirm which dimension is vertical in the loading plane before using the result.

What the equations assume

These are Euler-Bernoulli beam results for a prismatic, linearly elastic member with small deflection and idealized supports. They do not account for shear deflection, local bearing stresses, stress concentrations, holes, notches, residual stress, yielding, buckling, fatigue, or support flexibility. A real beam’s maximum stress may occur at a detail that the simple section formula does not model.

A distributed load must be expressed as force per unit length. Do not insert a total force where w is required. Likewise, keep the point load symbol P separate from a line load and identify whether the beam is cantilevered or simply supported; changing either assumption changes the result.

Sources and limits

The load-case deflection equations are based on the beam reference tables in MIT OpenCourseWare’s Solid Mechanics course material. The rectangular section property and bending-stress relationship are covered in MIT OpenCourseWare’s introductory civil engineering design material. These general equations are not a substitute for a code-compliant structural design or qualified review.