Structural & Mechanical Engineering · Worked example

Beam Stress and Deflection: Simply Supported Center-Load Worked Example

Calculate reactions, bending moment, bending stress, and midspan deflection for a simply supported steel beam with a center point load.

By Neutron Studios LLC8 minute readPublished 2026-08-11Reviewed 2026-08-11

Why this calculation matters

Beam calculations are a good example of why strength and stiffness must be checked separately. A beam can remain far below material yield stress and still deflect too much for the application.

This worked example uses a simply supported rectangular steel beam with a center point load. The loading case is intentionally simple so every result can be reproduced by hand.

What you will calculate

Calculate support reactions and maximum bending moment.
Compute rectangular-section second moment of area.
Calculate elastic bending stress and midspan deflection.
Separate material-strength checks from serviceability/deflection checks.

Given values

Span L = 2.0 m
Center point load P = 1000 N
Rectangular section: width b = 50 mm, height h = 100 mm
Elastic modulus E = 200 GPa
Linear-elastic small-deflection beam theory

Governing equations

Support reactions

RA = RB = \frac{P}{2}

Symmetry gives equal reactions for a centered load.

Maximum moment

M_{\mathrm{max}} = \frac{PL}{4}

Occurs at midspan for this loading case.

Rectangular second moment of area

I = \frac{bh^{3}}{12}

The h dimension is measured in the bending direction.

Extreme-fiber bending stress

\sigma _{\mathrm{max}} = \frac{M_{\mathrm{max}}\; c}{I}

For a rectangle, c = h/2.

Midspan deflection

\delta _{\mathrm{max}} = \frac{PL^{3}}{48 EI}

Euler–Bernoulli result for a center point load on a simply supported beam.

Worked solution

1. Resolve the reactions

The load is centered, so each support carries half of the 1000 N force.

RA = RB = 500\; N

2. Calculate section stiffness

Convert the cross-section dimensions to meters before using SI units. The second moment of area is 4.167×10⁻⁶ m⁴. Rotating the same rectangle 90° would change I dramatically because height is cubed.

I = \frac{0.05 \times 0.10^{3}}{12} = 4.167 \times 10^{-6}\; m^{4}

3. Calculate moment and bending stress

The maximum moment is 500 N·m. With c = 0.05 m, the corresponding extreme-fiber bending stress is 6.0 MPa.

\begin{gathered}M_{\mathrm{max}} = \frac{1000 \times 2}{4} = 500\; N \cdot m\\\sigma _{\mathrm{max}} = \frac{500 \times 0.05}{4.167 \times 10^{-6}} \approx 6.0\; \mathrm{MPa}\end{gathered}

4. Calculate elastic deflection

The predicted center deflection is 0.00020 m, or 0.20 mm. This is a stiffness result; whether it is acceptable depends on the project-specific deflection limit and connections.

\delta _{\mathrm{max}} = \frac{1000 \times 2^{3}}{48 \times 200 \times 10^{9} \times 4.167 \times 10^{-6}} \approx 0.00020\; m = 0.20\; \mathrm{mm}

Result

Engineering interpretation

Reactions: 500 N at each support; maximum moment: 500 N·m; maximum elastic bending stress: about 6.0 MPa; center deflection: about 0.20 mm.

These results verify the mechanics for the idealized case only. A real design may also require shear, lateral-torsional buckling, local buckling, fatigue, connection, load-combination, code, and serviceability checks.

Sanity checks

The two support reactions must sum to the applied vertical load.
Stress should scale linearly with P in the elastic model.
Deflection should scale with L³, making span errors especially important.
Doubling section height greatly reduces deflection because I scales with h³.

Common mistakes

Using millimeters in the section formula while E and loads are in SI base units.
Using the weak-axis value of I accidentally.
Comparing the 6 MPa bending stress directly with a design allowable without required factors and code checks.
Assuming a low stress automatically means deflection is acceptable.

References and model boundaries

Euler–Bernoulli beam formulas from standard mechanics-of-materials and structural-analysis references.
Final structural design requires the governing load combinations, material standard, stability checks, connections, and jurisdictional code.

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