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.
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 = P/2Symmetry gives equal reactions for a centered load.
Maximum moment
Mmax = PL/4Occurs at midspan for this loading case.
Rectangular second moment of area
I = bh³/12The h dimension is measured in the bending direction.
Extreme-fiber bending stress
σmax = Mmax c / IFor a rectangle, c = h/2.
Midspan deflection
δmax = PL³ / (48EI)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 N2. 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 = 0.05 × 0.10³ / 12 = 4.167×10⁻⁶ m⁴3. 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.
Mmax = 1000×2/4 = 500 N·m; σmax = 500×0.05/(4.167×10⁻⁶) ≈ 6.0 MPa4. 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.
δmax = 1000×2³ / [48×200×10⁹×4.167×10⁻⁶] ≈ 0.00020 m = 0.20 mmEngineering 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.
For safety-critical, regulated, production, or otherwise consequential work, independently verify the result using the governing standard, current manufacturer data, and qualified engineering review. See the site methodology and engineering disclaimer.