Learning topic
Differential Equation of the Elastic Curve
Calculate beam slope and deflection from the elastic-curve differential equation EI·y″ = M(x): integrate twice, apply boundary conditions, and track signs.
The differential equation of the elastic curve relates beam loading, through the bending-moment function $M(x)$, to geometric deformation expressed by deflection and rotation.
Curvature of the elastic curve
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 exact geometric curvature of a plane curve contains derivatives of deflection. For small rotations, when $|w'|\ll1$, the denominator in the exact curvature expression is approximately one, so curvature is proportional to the second derivative of deflection.
Approximate differential equation
Depending on the adopted sign convention, the equation is written as $EIw''(x)=M(x)$ or $EIw''(x)=-M(x)$. Different sign conventions must not be mixed within one calculation.
Double integration
- Determine the support reactions and the bending-moment function $M(x)$ for each beam segment.
- Write the differential equation of the elastic curve using the sign convention adopted in the calculation: $EIw''(x)=\pm M(x)$.
- Integrate once to obtain the rotation function $\theta(x)\approx w'(x)$ together with an integration constant.
- Integrate a second time to obtain the deflection function $w(x)$ and a second integration constant.
- Determine the constants from support boundary conditions and continuity conditions between beam segments.
- Calculate the required rotations and deflections and locate extrema where necessary.
- Check units, satisfaction of boundary conditions, and the physically expected direction of deformation.
The first integration gives the rotation function, and the second gives the deflection function. For constant flexural rigidity $EI$, it may be taken outside the integral. If $E$ or $I$ changes along the beam, the variation must be handled segment by segment or directly in the differential equation.
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 |
Example
For a cantilever of length $L$ carrying a force $F$ at the free end and having constant $EI$, integration with $w(0)=0$ and $w'(0)=0$ gives the familiar magnitudes at the free end:
$$|\theta(L)|=\frac{FL^2}{2EI},\qquad |w(L)|=\frac{FL^3}{3EI}.$$
The signs depend on the force direction and the selected positive direction for deflection.
Limits of applicability
The approximate equation assumes small deflections and rotations, linear-elastic material behavior, and applicability of the classical beam model. Large displacements require a geometrically nonlinear description.
About this topic
The analytical foundation for beam deformation evaluation is the approximate differential equation of the elastic curve: E*I*y' = -M(x). This section covers its derivation from geometric strain relations and Hooke's law, double integration techniques, boundary condition application at supports, and constructing analytical displacement functions.