5 ANALYSIS OF INDETERMINATE STRUCTURES BY FORCE METHOD

5. ANALYSIS OF INDETERMINATE STRUCTURES BY FORCE METHOD 1

5. 1 ANALYSIS OF INDETERMINATE STRUCTURES BY FORCE METHOD - AN OVERVIEW • 5. 2 INTRODUCTION • 5. 3 METHOD OF CONSISTENT DEFORMATION • 5. 4 INDETERMINATE BEAMS • 5. 5 INDETRMINATE BEAMS WITH MULTIPLE DEGREES OF INDETERMINACY • 5. 6 TRUSS STRUCTURES • 5. 7 TEMPERATURE CHANGES AND FABRICATION ERRORS 2

5. 2 INTRODUCTION 5. 2 Introduction • While analyzing indeterminate structures, it is necessary to satisfy (force) equilibrium, (displacement) compatibility and force-displacement relationships • (a) Force equilibrium is satisfied when the reactive forces hold the structure in stable equilibrium, as the structure is subjected to external loads • (b) Displacement compatibility is satisfied when the various segments of the structure fit together without intentional breaks, or overlaps • (c) Force-displacement 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 3

5. 2 INTRODUCTION (Cont’d) • 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 force-displacement 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 force-displacement relationships; it treats the displacements as unknowns - Two available methods are Slope Deflection Method and Stiffness (Matrix) method 4

5. 3 METHOD OF CONSISTENT DEFORMATION • Solution Procedure: • (i) Make the structure determinate, by releasing the extra forces constraining the structure in space • (ii) Determine the displacements (or rotations) at the locations of released (constraining) forces • (iii) Apply the released (constraining) forces back on the structure (To standardize the procedure, only a unit load of the constraining force is applied in the +ve direction) to produce the same deformation(s) on the structure as in (ii) • (iv) Sum up the deformations and equate them to zero at the position(s) of the released (constraining) forces, and calculate the unknown restraining forces Types of Problems to be dealt: (a) Indeterminate beams; (b) Indeterminate trusses; and (c) Influence lines for indeterminate structures 5

5. 4 INDETERMINATE BEAMS 5. 4. 1 Propped Cantilever - Redundant vertical reaction released (i) Propped Cantilever: The structure is indeterminate to the first degree; hence has one unknown in the problem. (ii) In order to solve the problem, release the extra constraint and make the beam a determinate structure. This can be achieved in two different ways, viz. , (a) By removing the vertical support at B, and making the beam a cantilever beam (which is a determinate beam); or (b) By releasing the moment constraint at A, and making the structure a simply supported beam (which is once again, a determinate beam). 6

5. 4 INDETERMINATE BEAMS (Cont’d) (a) Release the vertical support at B: y x L/2 P P B C L/2 B B = C L B + RB BB=RB*f. BB Applied in +ve direction The governing compatibility equation obtained at B is, f. BB = displacement per unit load (applied in +ve direction) 7

5. 4 INDETERMINATE BEAM (Cont’d) 5. 4. 2 Propped cantilever - Redundant support moment released L/2 A P L = B (b) Release the moment constraint at a: P A A Primary structure B +M A A B A=MA AA Redundant MA applied 8

5. 4. 3 OVERVIEW OF METHOD OF CONSISTENT DEFORMATION To recapitulate on what we have done earlier, I. Structure with single degree of indeterminacy: P A B RB (a) Remove the redundant to make the structure determinate (primary structure) P A B Bo (b) Apply unit force on the structure, in the direction of the redundant, and find the displacement f. BB (c) Apply compatibility at the location of the removed redundant B 0 + f. BB RB = 0 9

5. 5 INDETERMINATE BEAM WITH MULTIPLE DEGREES OF INDETERMINACY w/u. l A B RB B 0 C D RC RD C 0 D 0 E (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. , BO, CO and DO 10

(b) Apply unit loads at B, C and D, in a sequential manner and determine deformations at B, C and D, respectively. A B f. BB f. CB C D E f. DB 1 A B f. BC f. CC C 1 f. DC A B f. BD f. CD C D f. DD E 1 11

(c ) Establish compatibility conditions at B, C and D BO + f. BBRB + f. BCRC + f. BDRD = 0 CO + f. CBRB + f. CCRC + f. CDRD = 0 DO + f. DBRB + f. DCRC + f. DDRD = 0 12

5. 4. 2 When support settlements occur: w / u. l. A B C B D C E D Support settlements Compatibility conditions at B, C and D give the following equations: BO + f. BBRB + f. BCRC + f. BDRD = B CO + f. CBRB + f. CCRC + f. CDRD = C DO + f. DBRB + f. DCRC + f. DDRD = D 13

5. 5 TRUSS STRUCTURES 80 k. N 60 k. N 80 k. N C 60 k. N D A C D B A 1 2 B Primary structure (a) Remove the redundant member (say AB) and make the structure (b) a primary determinate structure (c) The condition for stability and indeterminacy is: (d) (e) r+m>=<2 j, Since, m = 6, r = 3, j = 4, (r + m =) 3 + 6 > (2 j =) 2*4 or 9 > 8 i = 1 14

5. 5 Truss Structures (Cont’d) (b)Find deformation ABO along AB: ABO = (F 0 u. ABL)/AE F 0 = Force in member of the primary structure due to applied load u. AB= 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: ABO+f. AB, ABFAB=0 (d) Hence determine FAB 15

(e) Determine the individual member forces in a particular member CE by FCE = FCE 0 + u. CE FAB where FCE 0 = force in CE due to applied loads on primary structure (=F 0), and u. CE = force in CE due to unit force applied along AB (= u. AB) 16

5. 6 TEMPERATURE CHANGES AND FABRICATION ERROR • Temperature changes affect the internal forces in a structure • Similarly 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 (ii) Reintroduce the removed members back and make the deformation compatible 17