Learning topic
Mohr Integral and Unit-Load Method
Calculate beam, frame, and truss displacements with the Mohr integral and unit-load method: build the auxiliary unit state, integrate internal-force products, and interpret the sign.
The Mohr integral and unit-load method determine a translation or rotation by combining internal force effects from two structural states: the real loaded state and an auxiliary unit state.
The displacement at a specified point and direction can be determined by the unit-load method. For bending of a beam or frame:
$$\delta=\int_0^L\frac{M(x)\,\bar M(x)}{EI}\,dx,$$
where $M(x)$ is the bending moment from the real loading and $\bar M(x)$ is the moment from a unit force applied at the point and in the direction of the required linear displacement.
For a required rotation, apply a unit moment.
For a general member, applicable contributions may be summed:
$$\delta=\int\frac{N\bar N}{EA}\,dx+\int\frac{M\bar M}{EI}\,dx+\int\frac{T\bar T}{GJ_p}\,dx+\int\frac{Q\bar Q}{\kappa GA}\,dx,$$
- $N$, $M$, $T$, $Q$ — internal force resultants from the real loading;
- $\bar N$, $\bar M$, $\bar T$, $\bar Q$ — corresponding resultants from the unit-load state;
- $E$, $G$, $A$, $I$, $J_p$, $\kappa$ — stiffness and section parameters.
The sign of $\delta$ indicates whether the actual displacement agrees with the direction of the introduced unit load.
Procedure
- Specify the point, direction, and type of required displacement.
- Determine internal forces caused by the real loading.
- Create the unit state: apply a unit force for a translation or a unit moment for a rotation.
- Construct the corresponding unit-state internal-force diagrams.
- Multiply corresponding real and unit internal-force functions and divide by the relevant stiffness on each segment.
- Integrate and sum the contributions of all members.
Beams and frames
For slender beams and frames, the bending term $\int M\bar M/(EI)\,dx$ is often sufficient. Axial, shear, and torsional terms must be added when their deformations are significant.
Trusses
For an ideal truss with constant $N_i$, $\bar N_i$, $E_i$, and $A_i$ in each member, the integral reduces to
$$\delta=\sum_i\frac{N_i\bar N_iL_i}{E_iA_i}.$$
Sign
A positive result means displacement in the direction of the unit force or moment. A negative result means the actual displacement is opposite to the assumed unit-state direction.
About this topic
This topic explains the unit-load method and Mohr integral for structural displacements, including construction of the auxiliary unit state and integration of real and unit internal-force functions.