Superposition & Statically Indeterminate Beams

1. Superposition & Statically Indeterminate Beams Method of Superposition Statically Indeterminate Beams

2. 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.

3. 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

7. 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

8. 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.

9. 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.

10. 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.

11. 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

16. 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.

17. 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)

18. 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.

19. 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.) Method of Superposition Procedure