Superposition & Statically

Indeterminate Beams

Method of Superposition

Statically Indeterminate Beams

Method of Superposition

If a beam has several concentrated or distributed

loads on it, it is often easier to compute the slope

and deflection caused by each load separately.

The slope and deflection can then be determined

by applying the principle of superposition and

adding the values of the slope and deflection

corresponding to the various loads.

Method of Superposition

Assumptions

– material obeys Hooke's law

– deflections and slopes are small

– the presence of the deflections does not alter the

actions of the applied load

Statically Indeterminate Beams

Recall, Statically indeterminate beams are ones in which

the number of reactions exceeds the number of

independent equations of equilibrium

Most of the structures we encounter in everyday life,

automobile frames, buildings, aircraft, are statically

indeterminate.

4 unknowns, 3 equilibrium equations

Types of Indeterminate Beams

Usually identified by the beams support system

– Propped cantilever beam

– Fixed-end beam

– Continuous beam

The number of reactions in excess of the number of

equilibrium equations is called the Degree of Static

Indeterminacy

– A propped cantilever beam is statically indeterminate to the

first degree.

Types of Indeterminate Beams

Excess reactions are called static redundants and

must be selected in each particular case.

– In the case of a propped cantilever beam, the support at the

end may be selected as the redundant reaction

– This reaction is in excess of those needed to maintain

equilibrium, so it can be removed.

– Structure that remains when redundants are released is

called the released structure or the primary structure.

Types of Indeterminate Beams

The released structure must be stable and must be

statically determinate.

A special case: all loads action on the beam are vertical

– Horizontal reaction at A vanishes and three reactions remain

– Only two independent equations of equilibrium are available

– Beam is still statically indeterminate to the first degree.

Analysis by the deflection curve

Statically indeterminate beams may be analyzed

by solving any one of the equations of the

deflection curve

Procedure is essentially the same as for

statically determinate beams.

Illustrated by example

Method of Superposition

Fundamental in the analysis of statically indeterminate bars, trusses, beams, frames, and other structures.

First note the degree of static indeterminacy and selecting the redundant reactions

Having identified the redundants, write equations of equilibrium that relate the other unknown reactions to the redundant and the loads.

Method of Superposition

Next, assume both the original loads and the redundants act on the released structure.

– Find the deflections in the released structure by superposing the separate deflections due to the loads and redundants.

– The sum of these deflections must match the deflections in the original beam

– Since the deflections in the original beam at the restraints are 0 or a known value• We can write equations of compatibility (or equations of superposition)

Method of Superposition

The released structure is statically determinate

The relationships between loads and the

deflections of the released structure are called

Force-Displacement relations.

When these relations are substituted into the

equations of compatibility

– Unknowns are the redundants.

Method of Superposition Procedure

Study the boundary

conditions and sketch the

expected deflection curve.

Determine the degree of statical indeterminacy

Select and label redundant

forces and/or moments

Break problem into statically

determinate subproblems

– One for each load on the beam

and one for each of the

selected redundants.

Write compatibility equations– One for the deflection for each redundant force (or moment)

Write force-deflection equations

Substitute force-deflection equations into compatibility equations and solve for unknown redundants.

Write superposition equations for any additional quantities that are required by the problem statement

Complete solution (max deflection, etc.)