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Load arithmetic

Shelf deflection: simple elastic model

I=bh312δ=5wL4384EIM=wL28σ=Mbh26
where
I
second moment of area (mm⁴)
b
width (mm)
h
vertical depth (mm)
δ
midspan deflection (mm)
w
uniform line load (N/mm)
L
span (mm)
E
elastic modulus (N/mm²)
M
maximum bending moment (Nmm)
σ
bending stress (N/mm²)
Explain this equation

Estimates how much a simply supported shelf or similar member sags under an even load, along with the bending moment and stress.

b is the width and h the vertical depth of the section, in millimetres. I is the second moment of area, which measures how well the shape resists bending. L is the span, E the stiffness of the material and w the load per millimetre of span. δ is the midspan sag, M the maximum bending moment and σ the bending stress.

I = bh³/12 means depth matters far more than width. Doubling h makes the shelf eight times stiffer. The 5/384 and 1/8 factors are standard results for an even load on a simply supported span. bh²/6 is the section modulus, which turns moment into stress.

A 1000 mm span of 300 × 20 mm at E = 10,000 N/mm² under 0.5 N/mm sags about 3.3 mm. Double the span and the sag rises sixteen-fold.

Watch out: this is a guide, not a design check. There is no creep, shear or safety factor.

Calculate an ideal simply supported rectangular member under a uniform load.

mm
mm
mm
N/mm²
N/mm

Enter your values to see the result.

Good to know

All lengths mm; w in N/mm (numerically equal to kN/m); E in N/mm². Ideal short-term bending only: no creep, shear, support restraint, grading, safety factors or resistance check. Not structural sizing or a safe-span verdict.

Limits

The two beam tools implement ideal short-term linear elastic bending of a simply supported rectangular section. They do not include creep, shear deflection, load combinations, material strength, buckling, service class, duration factors, connection capacity, bearing or actual restraint. They produce no safe/unsafe verdict and must not be used as structural sizing tools. Beam calculations use the stated ideal supports and loads; they do not select a safe timber size.

Sources

This is a calculation aid, not a structural design or compliance certificate. Every default is an example, not a specification for your project.