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MechanicalMechanics of MaterialsmediumFE ExamCoursework

Maximum Bending Stress in a Simply Supported Beam

A simply supported beam spans L=6 mL = 6\ \text{m} and carries a uniformly distributed load of w=12 kN/mw = 12\ \text{kN/m} over its entire length.

The cross-section is solid rectangular, b=200 mmb = 200\ \text{mm} wide and h=400 mmh = 400\ \text{mm} deep.

Find the maximum bending stress in the beam.

Given
w12 kN/mUniformly distributed load
L6 mSpan
b200 mmSection width
h400 mmSection depth
Hint 1

Two separate steps: find the maximum bending moment, then find the section's resistance to bending.

Hint 2

For a simply supported beam under a full-span UDL, M_max = wL²/8 at mid-span.

Hint 3

Section modulus for a rectangle is S = bh²/6 — note h is squared, not cubed. That is I; S is I divided by c.

Worked solution — try the problem first

Step 1 — Maximum bending moment. For a simply supported beam under a full-span UDL, the maximum moment is at mid-span:

Mmax=wL28=12,000×628=12,000×368=54,000 N⋅mM_{max} = \frac{wL^2}{8} = \frac{12{,}000 \times 6^2}{8} = \frac{12{,}000 \times 36}{8} = 54{,}000\ \text{N·m}

Step 2 — Section modulus. For a rectangle bent about its strong axis:

S=bh26=0.200×(0.400)26=0.200×0.1606=0.03206=5.333×10−3 m3S = \frac{bh^2}{6} = \frac{0.200 \times (0.400)^2}{6} = \frac{0.200 \times 0.160}{6} = \frac{0.0320}{6} = 5.333\times10^{-3}\ \text{m}^3

Note hh is squared, not cubed — that is the section modulus, not the second moment of area. (I=bh3/12I = bh^3/12; S=I/cS = I/c where c=h/2c = h/2.)

Step 3 — Bending stress.

σmax=MmaxS=54,0005.333×10−3=1.0125×107 Pa\sigma_{max} = \frac{M_{max}}{S} = \frac{54{,}000}{5.333\times10^{-3}} = 1.0125\times10^{7}\ \text{Pa}

σmax=10.1 MPa\boxed{\sigma_{max} = 10.1\ \text{MPa}}

Sanity check: structural steel yields around 250 MPa, so a working stress near 10 MPa is comfortably elastic — a sensible result for a serviceability-governed timber or lightly loaded steel section.

Concepts:Bending stressSection modulusBending moment diagramsUnit consistency

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