Learning topic
Displacements in Framed Structures
Displacements in beams, frames, and trusses using strain energy, the unit-load method, Mohr integrals, and graphical integration methods.
Displacements in framed structures are translations of points and rotations of sections caused by member deformation. Their calculation is required for serviceability checks, compatibility equations in indeterminate structures, and interpretation of the deformed shape.
Types of displacement
A linear displacement is defined for a specified point and direction; a rotation is defined for a specified section and positive sense. The sign of the calculated result is interpreted relative to that chosen direction.
Energy approach
For a linear-elastic structure, external work is related to strain energy. This relation allows displacement calculations to be expressed through internal axial force, bending moment, shear, torsion, and the corresponding member stiffnesses.
Unit-load method
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.
In beams and frames, bending deformation often provides the dominant contribution. In ideal pin-jointed trusses, axial member deformation is normally the principal contribution.
Graphical evaluation
For a segment with constant $EI$, the Mohr integral $\int M\bar M/(EI)\,dx$ can be evaluated graphically when one of the two multiplied diagrams is linear over that segment.
- Construct the real-load bending-moment diagram $M$ and the unit-load diagram $\bar M$ for the required displacement.
- Divide the beam or frame into segments where $EI$ is constant and the diagrams have convenient geometric shapes.
- For each simple part of one diagram, determine its algebraic area $\Omega$ and the coordinate of its centroid.
- On the other diagram, which must be linear over that segment, determine the ordinate $\bar M_c$ beneath the centroid of the first diagram area.
- Calculate the contribution $\Delta=\Omega\bar M_c/(EI)$, retaining the signs of the diagrams.
- Sum the contributions from all segments. A positive result acts in the direction of the introduced unit load; a negative result acts in the opposite direction.
If neither diagram is linear on a segment, the simple area-times-centroid-ordinate rule is generally not applicable without additional subdivision or direct integration.
Other sources of displacement
Displacements can also result from temperature change, temperature gradients, prescribed support movements, and settlements. In statically indeterminate structures these imposed deformations may also generate additional internal forces.
Checks
Verify dimensions, signs, the direction of the unit action, the consistency of real and unit-load diagrams, and the stiffness used on each segment. A negative result means that the actual displacement is opposite to the assumed unit-load direction.
About this topic
Energy methods for linear and angular displacements in framed structures: external and internal work, strain energy, Mohr integrals, the unit-load method, and graphical techniques for evaluating displacement integrals.