Learning topic
Deflections in Bending
Learn beam deflection w and slope θ, flexural rigidity EI, boundary conditions, stiffness checks, and the main methods used to calculate beam displacements.
Under bending moments, a beam deforms and its initially straight axis becomes an elastic curve. Stiffness calculations determine the transverse deflection $w(x)$ and the rotation $\theta(x)$ of a cross-section.
Deflection and rotation
Deflection $w$ is the transverse displacement of a point on the beam axis. For small deformations, the rotation of a cross-section is approximately the derivative of deflection: $\theta(x)\approx w'(x)$.
Flexural rigidity
In classical linear-elastic bending theory, curvature of the deflected beam axis is related to bending moment by:
$$\kappa=\frac{1}{\rho}=\frac{M}{EI},$$
where the sign of the right-hand side depends on the adopted conventions for bending moment and deflection.
For small rotations, curvature is approximately:
$$\kappa\approx w''(x).$$
Therefore, the differential equation of the elastic curve is often written as:
$$EI\,w''(x)=M(x),$$
or with a minus sign according to the chosen sign convention.
- $E$ — Young's modulus;
- $I$ — second moment of area;
- $EI$ — flexural rigidity;
- $w$ — transverse deflection;
- $\rho$ — radius of curvature.
The calculator below uses the equivalent relation $M=EI/\rho$.
The product $EI$ is called the flexural rigidity. Increasing $E$ or $I$ reduces curvature and, all else being equal, beam deflections. Therefore, the geometric distribution of material in the cross-section has a strong effect on stiffness.
Boundary conditions
| Point or support | Typical condition | Physical meaning |
|---|---|---|
| Fixed support x = a | $w(a)=0$, $\theta(a)=w'(a)=0$ | No transverse displacement or rotation |
| Pin or roller support x = a | $w(a)=0$ | Transverse displacement is restrained; rotation is allowed |
| Free end without an applied moment | $M=0$ | Boundary condition for internal bending moment |
| Free end without a transverse force | $Q=0$ | Boundary condition for internal shear force |
| Boundary between two segments without an internal hinge | $w_-=w_+$, $\theta_-=\theta_+$ | Continuity of deflection and rotation |
Boundary conditions determine the constants of integration and must represent the actual supports and connections. For a multi-segment beam, continuity of deflection and rotation is also imposed at locations without an internal hinge or discontinuity.
Methods for determining displacements
The child topics cover direct integration of the differential equation of the elastic curve and the initial-parameter method using Macaulay functions. In other parts of the course, beam displacements may also be found using energy methods.
Stiffness check
The calculated maximum deflection is compared with an allowable value prescribed by serviceability requirements or the applicable design procedure: $|w_{\max}|\le[w]$. Rotation may be limited in the same way when required.
Example of the influence of cross-section size
For a rectangular cross-section, $I=bh^3/12$. If the depth $h$ is doubled while $b$, $E$, $L$, and the loading remain unchanged, $I$ increases by $2^3=8$ times. Within the same linear model, characteristic deflections, which are inversely proportional to $EI$, therefore decrease substantially.
About this topic
Under bending moments, the beam axis deforms into an elastic curve. This section introduces linear displacements (deflections w) and angular displacements (slope angles θ). Evaluating beam deflections is necessary to verify structural stiffness and ensure peak deflection does not exceed allowable serviceability limits set by building codes.