Derivation of the Flexure Formula for Beam Bending

Posted on Jul 1, 2026 in Real Estate Sciences

Derivation of the Flexure Formula

In a beam subjected to a bending moment (M), normal stresses are developed across the cross-section. The stress varies linearly from zero at the neutral axis to a maximum value at the outermost fiber.

Our objective is to derive the flexure formula:

M/I = σ/y = E/R

1. Stress Distribution in a Beam

From the theory of simple bending, plane sections remain plane after bending. The strain (ε) at a distance (y) from the neutral axis is:

ε = y/R

Where:

  • y = distance from neutral axis
  • R = radius of curvature

Using Hooke’s Law (σ = Eε), we substitute to get:

σ = E(y/R)

Since stress varies linearly with distance from the neutral axis:

σ = (σmax / c) * y

Where σmax is the maximum bending stress and c is the distance of the outermost fiber from the neutral axis.

2. Location of the Neutral Axis

The resultant force due to all bending stresses must be zero (ΣF = 0). Considering an elemental area (dA):

dF = σ dA

Integrating over the area:

∫ σ dA = 0 → (σmax / c) ∫ y dA = 0

Since (σmax / c) ≠ 0, then ∫ y dA = 0. This confirms that the neutral axis passes through the centroid of the cross-section.

3. Determination of Bending Moment

The internal moment (M) produced by the stress distribution must equal the applied bending moment:

M = ∫ y σ dA

Substituting σ = (σmax / c) * y:

M = (σmax / c) ∫ y2 dA

Since ∫ y2 dA = I (moment of inertia), we get:

σmax = (Mc) / I

4. Stress at Any Fiber

Replacing ‘c’ with any distance ‘y’, we find the bending stress at any point in the beam section:

σ = (My) / I

5. Derivation of the Bending Equation

Equating σ = E(y/R) and σ = (My)/I:

E(y/R) = (My)/I

Canceling ‘y’ yields the final flexure formula:

M/I = σ/y = E/R

Definition of Terms

  • M: Bending moment
  • I: Moment of inertia of cross-section
  • σ: Bending stress
  • y: Distance from neutral axis
  • E: Young’s modulus of elasticity
  • R: Radius of curvature of the beam

Assumptions of Simple Bending

  1. Material is homogeneous and isotropic.
  2. Beam is initially straight.
  3. Stress remains within the elastic limit.
  4. Hooke’s law is valid.
  5. Plane sections remain plane after bending.
  6. Radius of curvature is large compared to beam depth.