Euler–Bernoulli bending moment derivation

The Euler–Bernoulli beam theory relates bending moments to beam deflections. The key equation is:

1. Beam kinematics

Let the transverse deflection be $w(x)$.

For small slopes:

$$\theta(x) \approx \frac{dw}{dx}$$

Curvature is the derivative of rotation:

$$\kappa(x) = \frac{d\theta}{dx} \approx \frac{d^2 w}{dx^2}$$

Using the common structural sign convention:

$$\kappa(x) = -\frac{d^2 w}{dx^2}$$

2. Strain distribution

Assuming plane sections remain plane, the axial strain at distance $y$ from the neutral axis is:

$$\varepsilon_x(x,y) = -y\,\kappa(x)$$

3. Stress–strain relation

For linear elastic material:

$$\sigma_x(x,y) = E\,\varepsilon_x(x,y) = -E\,y\,\kappa(x)$$

4. Definition of bending moment

The internal bending moment is obtained by integrating stress times lever arm over the cross section:

$$M(x) = \int_A \sigma_x(x,y)\,y\,dA$$

Substitute $\sigma_x(x,y)$:

$$M(x) = \int_A (-E\,y\,\kappa(x))\,y\,dA$$

Factor out constants:

$$M(x) = -E\,\kappa(x)\int_A y^2\,dA$$

5. Second

Define the second moment of area:

$$I = \int_A y^2\,dA$$

Thus:

$$M(x) = -E I\,\kappa(x)$$

6. Final bending moment–deflection equation

Substitute curvature:

$$M(x) = -E I\,\frac{d^2 w}{dx^2}$$

This is the Euler–Bernoulli bending moment relation.

Short explanation

• $\frac{d^2 w}{dx^2}$ represents beam curvature for small deflections
• $I$ captures cross section geometry
• $E I$ is the flexural rigidity
• Bending moment is directly proportional to curvature