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Load arithmetic

Central point-load beam arithmetic

I=bh312δ=PL348EIM=PL4R=P2
where
I
second moment of area (mm⁴)
b
width (mm)
h
vertical depth (mm)
δ
midspan deflection (mm)
P
central point force (N)
L
span (mm)
E
elastic modulus (N/mm²)
M
maximum bending moment (Nmm)
R
reaction at each end (N)
Explain this equation

Estimates the sag, bending moment and end reactions of a simply supported member carrying a single load at midspan. It is a guide, not a design check.

b and h are the section width and vertical depth in millimetres. I is the second moment of area, the section's resistance to bending. L is the span, E the material stiffness and P the central force in newtons. δ is the midspan deflection, M the maximum moment and R the reaction at each support.

I = bh³/12 makes depth dominant. The 1/48 and 1/4 factors are standard results for a central point load on a simply supported span. The load is shared equally, so each end carries P/2.

1000 mm span, 300 × 20 mm, E = 10,000 N/mm², with 500 N at the centre: about 5.2 mm sag and 250 N at each end. Doubling the span gives eight times the sag.

Watch out: this is short-term bending only. Capacity, shear, creep and connections are not checked.

Calculate the ideal response of a simply supported rectangular member to a central point force.

mm
mm
mm
N/mm²
N

Enter your values to see the result.

Good to know

P is N, lengths mm, E N/mm². Short-term linear bending model only. No capacity, creep, shear, lateral stability or connection checks. Not a structural design result.

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

  • Geometry / arithmetic

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