Analysis of Indeterminate Beams by Force Method 

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1. Analysis of Indeterminate Structures by Force Method  An Overview 2. Introduction 3. Method of Consisten Deformation 4. Indeterminate Beams 5. Indeterminate Beams with Multiple Degree of Indeterminacy 6. Truss Structures 7. Temperature Changes & Fabrication Errors 

2. IntroductionWhile analyzing indeterminate structures, it is necessary to satisfy (force) equilibrium, (displacement) compatibility and forcedisplacement relationships
Forcedisplacement requirements depend on the manner the material of the structure responds to the applied loads, which can be linear/nonlinear/viscous and elastic/inelastic; for our study the behavior is assumed to be linear and elastic Two methods are available to analyze indeterminate structures, depending on whether we satisfy force equilibrium or displacement compatibility conditions. They are: Force method and Displacement Method Force Method satisfies displacement compatibility and forcedisplacement relationships; it treats the forces as unknowns  Two methods which we will be studying are Method of Consistent Deformation and (Iterative Method of) Moment Distribution Displacement Method satisfies force equilibrium and forcedisplacement relationships; it treats the displacements as unknowns  Two available methods are Slope Deflection Method and Stiffness (Matrix) method 3. Solution Procedure:
Types of Problems to be dealt:
4.1 Propped Cantilever  Redundant vertical reaction released
Overview of Method of Consistent DeformationTo recapitulate on what we have done earlier, Structure with single degree of indeterminacy: (a) Remove the redundant to make the structure determinate (primary structure) (b) Apply unit force on the structure, in the direction of the redundant, and find the displacement Δ B0 + fBB x RB = 0 5. Indeterminate beam with Multiple Degrees of Indetermincay(a) Make the structure determinate (by releasing the supports at B, C and D) and determine the deflections at B, C and D in the direction of removed redundants, viz.,Δ BO, Δ CO and Δ DO (b) Apply unit loads at B, C and D, in a sequential manner and determine deformations at B, C and D, respectively. (c ) Establish compatibility conditions at B, C and D Δ BO + fBBRB + fBCRC + fBDRD = 0 Δ CO + fCBRB + fCCRC + fCDRD = 0 Δ DO + fDBRB + fDCRC + fDDRD = 0 Compatibility conditions at B, C and D give the following equations: Δ BO + fBBRB + fBCRC + fBDRD = Δ B Δ CO + fCBRB + fCCRC + fCDRD = Δ C Δ DO + fDBRB + fDCRC + fDDRD = Δ D 6. Truss Structures(a) Remove the redundant member (say AB) and make the structure a primary determinate structure The condition for stability and indeterminacy is: r + m > = < 2j Since, m = 6, r = 3, j = 4, (r + m =) 3 + 6 > (2j =) 2 x 4 or 9 > 8 Δ i = 1 (b) Find deformation Δ ABO along AB: Δ ABO =Δ (F0uABL)/AE F0 = Force in member of the primary structure due to applied load uAB= Forces in members due to unit force applied along AB (c) Determine deformation along AB due to unit load applied along AB: (d) Apply compatibility condition along AB: ΔABO+fAB,ABFAB=0 Hence determine FAB (e) Determine the individual member forces in a particular member CE by FCE = FCE0 + uCE FAB where FCE0 = force in CE due to applied loads on primary structure (=F0), and uCE = force in CE due to unit force applied along AB (= uAB) 7. Temperature changes affect the internal forces in a structureSimilarly fabrication errors also affect the internal forces in a structure (i) Subject the primary structure to temperature changes and fabrication errors.  Find the deformations in the redundant direction Reintroduce the removed members back and make the deformation compatible
