(1.13.4)--ch10-4 Method of Superposition材料力学材料力学.ppt

Chapter 10 Statically indeterminate beams Mechanics of MaterialsMethod of SuperpositionWelcome to Mechanics of Materials,this section discusses the method of superposition for analyzing statically indeterminate beams.Procedure for statically indeterminate beams:I.Free body diagram;-Report the degree of static indeterminacy,select redundants.II.Equations of equilibrium;-Relate the remaining reactions to the redundants and the load;III.Equations of compatibility;-Obtain its general solution.IV.Evaluating the unknown redundants;-Substitute Force-displacement relation into compatibility equations.V.Answer the Question;-Remaining reactions,internal forces,stresses,displacements.Method of superposition Applicable to to a wide variety of statically indeterminate structures:The method of superposition is of fundamental importance in the analysis of statically indeterminate strucutres,such as*bars,trusses,beams,frames,and many other kinds of structures.*The procedure using this method to analyze statically indeterminate beams consists of the following steps:*First step is constructing the Free body diagram,by the FBD,reporting the degree of indeterminacy and selecting redundants;then write equations of equilibrium that relate the other unknown reactions to the redundants and the loads;next estalish equations of compatibility or equations of superposition,expressing the fact that the deflections of the released structure at the points where restraints were removed are the same as the deflections in the original beam at those same points.and then*evaluate the unknownredundant reactions by substituting force-displacement relations into the equations of compatibility.Last step,Answer the Question-Typically calculate remaining reactions,shear force,bending moment,stresses,deflections.The preceding steps can be made clearer by considering a particular case,namely,a propped cantilever beam supporting a uniform load.Example 1:A propped cantilever beam AB of a length L supports a uniform load of intensity q,as shown in the figure.Analyze this beam by the method of superposition,determine all the reactions.This same beam was analyzed in the Example by solving the differential equation of the deflection curve.Now lets look at this example to see how to analyze it by the method of superposition.Solution:-3 reactions-2 equilibirum equations1st degree indeterminate 1.FBD:RB as the rdundant*Similarly,construct the free-body diagram of the entire beam and*it is indeterminate to the first degree.Then*select the reaction RB at the simple support as the redundant.2.Equilibrium:(a)*Then write the equations of equilibrium for the entire beam,*sum all the forces in the y direction and sum all the moment about point A.since the reaction RB is selected as the redundant,*the other two reactions,RA and MA are expressed in terms of RB.3.Compatibility:Splitted into two problemsHow do we get compatibility equation?Equation of compatiblity:The next step is to remove the restraint corresponding to the redundant,in this case,remove the support at end B.The released structure is a cantilever beam.*The uniform load q and*the redundant force RB are now applied as loads on the released structure,respectively.Therefore,the original indeterminate problem is splitted into two determinate problems.*Then how do we get compatibility equation?Now look at the deflection at end B.The deflection at end B of the released structure due solely to the uniform load is denoted delta B1,and the deflection at the same point due solely to the redundant is denoted delta B2.*The deflection at point B in the original structure is delta B,obtained by superposing these two deflections.and the deflection in the original beam is equal to*zero,so*the equation of compatibility is*.Force-displacement relations:Remaining reactions：Shear force;Bending moment;Internal stresses;Deflections.4.Redundants:From Table H-1 in Appendix H(see Cases 1 and 4)Compatibility equation:5.Question:Redundant:*with the aid of Table H-1 in Appendix H(see Cases 1 and 4),the force-displacement relations could be found,*which give the deflections dB1 and dB2 in terms of the uniform load q and the redundant RB,respectively.*Substitute these force-displacement relations into the equation of compatibility*solve this equation to find the*redundant RB=3ql/8.As we can see from this example,once the redundant is known,*all the remaining reactions can be found from equations of equilibrium.In effect,the structure has become statically determinate.*Therefore,shear forces and bending moments,stresses and deflections could be found by the methods described in preceding chapters.Discussion:Alternative redundant?Remarks:-The force method or the flexibility method.Force RB selected as the redundant-Applicable only to linearly elastic structures.flexibilityThis example presents the analysis of the beam by taking the reaction RB as the redundant reaction using method of superposition.Is it possible to select*MA as the redundant?The release the corresponding constraint at end A,express the*compatiblity equation in term of rotation angles at end A,and solve as before.Please try it yourself.*The method of superposition is also called the force method or the flexibility method.The name force method arises from the use of*force quantities(forces and moments)as the redundants;the flexibility method is used because in the*compatibility equation,the*coefficients of the unknown quantities terms such as L3/3EI is a flexibility.*And this method is applicable only to linearly elastic structures.EndThat is end of this video.