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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
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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
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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
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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
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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).
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(a) Release the vertical support at B:
The governing compatibility equation obtained at B is,
fBB = displacement per unit load (applied in +ve direction)
x
y
PP
BC
L/2L/2L
C
B= +
B B
RB
BB=RB*fBB
B + '
B B = 0
BBBB
BBBB
fR
fR
/
0)()(
F r o m e a r l i e r a n a l y s e s ,
)/)(48/5(
)16/()24/(
)2/()]2/()2/([)3/()2/(
3
33
23
EIPL
EIPLEIPL
LEILPEILPB
)3/(3 EILf BB
PEILEIPLR BB )16/5()]3/(/[)]/)(48/5([ 33
Applied in +ve direction
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G o v e r n i n g c o m p a t i b i l i t y e q u a t i o n o b t a i n e d a t A i s , )()( AAAA M , AA = r o t a t i o n p e r u n i t m o m e n t
AA
AAM
F r o m k n o w n e a r l i e r a n a l y s i s , )16(
2
EI
PLAA [ u n d e r a c e n t r a l c o n c e n t r a t e d
l o a d ])]3/()[1( EILAA
T h i s i s d u e t o t h e f a c t t h a t + v e m o m e n t c a u s e s a – v e r o t a t i o n
PL16)/(3
EI)]L/(3/[EI)]/(16PL[M 2A
5.4 INDETERMINATE BEAM (Cont’d)
5.4.2 Propped cantilever - Redundant support moment released
L
PL/2
(b) Release the moment constraint at a:
A B
A
=
A BP
Primary structure
+ BA
MA A=MAAA
Redundant MA applied
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To recapitulate on what we have done earlier,I. 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
(c) Apply compatibility at the location of the removed redundant
A BRB
A BBo
fBB
5.4.3 OVERVIEW OF METHOD OF CONSISTENT DEFORMATION
B0 + fBBRB = 0
P
P
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(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
AB C D E
RB RC RD
B0 C0D0
w/u.l
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(b) Apply unit loads at B, C and D, in a sequential manner and
determine deformations at B, C and D, respectively.
AB C D E
fBBfCB fDB
1
AB C D E
fBCfCC fDC
AB C D E
fBDfCD fDD
1
1
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(c ) Establish compatibility conditions at B, C and D
BO + fBBRB + fBCRC + fBDRD = 0
CO + fCBRB + fCCRC + fCDRD = 0
DO + fDBRB + fDCRC + fDDRD = 0
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5.4.2 When support settlements occur:
Compatibility conditions at B, C and D give the following equations:
BO + fBBRB + fBCRC + fBDRD = B
CO + fCBRB + fCCRC + fCDRD = C
DO + fDBRB + fDCRC + fDDRD = D
AB C D E
B C D Support settlements
w / u. l.
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(a) (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*4 or 9 > 8 i = 1
C
80 kN
60 kN
A B
D
C
80 kN
60 kN
A B
D
1 2
Primary structure
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5.5 Truss Structures (Cont’d)
(b)Find deformation ABO along AB:
ABO = (F0uABL)/AE
F0 = Force in member of the primary structure due to applied loaduAB= 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+fAB,ABFAB=0
(d) Hence determine FAB
AE
LABu
ABABf
2
,
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(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)
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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
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