Meaning and mechanism
When a beam bends, one side of its cross section stretches and the other compresses. The material’s elastic modulus and the distribution of the cross-sectional area together determine its resistance to bending.
When the height of a rectangular cross-section increases, the material is distributed further from the neutral axis. This effect is reflected in the second moment of the cross-section, increasing with the cube of the height.
Relationships and conditions
When a simply supported beam has a concentrated load at its center, the deflection is δ=PL³/(48EI)). The rectangular cross-section has a deflection of I=bh³/12, and increasing the length by a factor of two results in an eightfold increase in the same condition’s deflection.
This describes a small deformation, using linear elasticity and a defined support and load condition. For out-of-plane deformation or uniformly distributed loads, different boundary condition formulas are used.
What to observe in the simulation
For the same material and section, central-load deflection scales with span cubed.
Small-deflection Euler–Bernoulli beam theory omits shear, buckling and yielding. Large-deflection results are outside its quantitative validity. Scene shows magnification. Do not reproduce real load tests.
Key relationships and symbols
δcenter = PL³/(48EI) · Irect = bh³/12
| Symbol | Meaning | Unit |
|---|---|---|
| E | Elastic modulus | Pa |
| I | Second moment of area about the bending axis | m⁴ |
| δ | Beam deflection | m |
Worked example
Why does a deeper section resist bending?
Larger second area moment · For the same material and section, central-load deflection scales with span cubed.
Common misconceptions
- The fact that the stress will be equal regardless of shape and direction is not true if the areas are equal.