Superposition & Statically Indeterminate BeamsMethod of SuperpositionStatically Indeterminate Beams

Method of SuperpositionIf 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 SuperpositionAssumptions 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 BeamsRecall, Statically indeterminate beams are ones in which the number of reactions exceeds the number of independent equations of equilibriumMost of the structures we encounter in everyday life, automobile frames, buildings, aircraft, are statically indeterminate.

4 unknowns, 3 equilibrium equations

Types of Indeterminate BeamsUsually identified by the beams support systemPropped cantilever beamFixed-end beamContinuous beamThe number of reactions in excess of the number of equilibrium equations is called the Degree of Static IndeterminacyA propped cantilever beam is statically indeterminate to the first degree.

Types of Indeterminate BeamsExcess 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 reactionThis 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 BeamsThe released structure must be stable and must be statically determinate.A special case: all loads action on the beam are verticalHorizontal reaction at A vanishes and three reactions remainOnly two independent equations of equilibrium are availableBeam is still statically indeterminate to the first degree.

Analysis by the deflection curveStatically indeterminate beams may be analyzed by solving any one of the equations of the deflection curveProcedure is essentially the same as for statically determinate beams.Illustrated by example

Method of SuperpositionFundamental in the analysis of statically indeterminate bars, trusses, beams, frames, and other structures.First note the degree of static indeterminacy and selecting the redundant reactionsHaving identified the redundants, write equations of equilibrium that relate the other unknown reactions to the redundant and the loads.

Method of SuperpositionNext, 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 beamSince the deflections in the original beam at the restraints are 0 or a known valueWe can write equations of compatibility (or equations of superposition)

Method of SuperpositionThe released structure is statically determinateThe relationships between loads and the deflections of the released structure are called Force-Displacement relations.When these relations are substituted into the equations of compatibilityUnknowns are the redundants.

Method of Superposition ProcedureStudy the boundary conditions and sketch the expected deflection curve.Determine the degree of statical indeterminacySelect and label redundant forces and/or momentsBreak problem into statically determinate subproblemsOne for each load on the beam and one for each of the selected redundants.Write compatibility equationsOne for the deflection for each redundant force (or moment)Write force-deflection equationsSubstitute force-deflection equations into compatibility equations and solve for unknown redundants.Write superposition equations for any additional quantities that are required by the problem statementComplete solution (max deflection, etc.)

Could have selected the reactive moment at A as the redundant. Then when the moment restraint at A is removed, the released structure is a simple beam with a pin support at one end and a roller support at the other.