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1. VViiddeeoo lleeccttuurreess ffoorr MMCCAA
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2. 55.. AANNAALLYYSSIISS OOFF IINNDDEETTEERRMMIINNAATTEE
SSTTRRUUCCTTUURREESS BBYY FFOORRCCEE MMEETTHHOODD

3. 5.1 ANALYSIS OOFF IINNDDEETTEERRMMIINNAATTEE SSTTRRUUCCTTUURREESS
BBYY FFOORRCCEE MMEETTHHOODD -- AANN OOVVEERRVVIIEEWW
 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

4. 55..22 IINNTTRROODDUUCCTTIIOONN
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

5. 55..22 IINNTTRROODDUUCCTTIIOONN ((CCoonntt’’dd))
 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

6. 5.3 METHOD OOFF CCOONNSSIISSTTEENNTT DDEEFFOORRMMAATTIIOONN
 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

7. 5.4 IINNDDEETTEERRMMIINNAATTEE BBEEAAMMSS
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).

8. 5.4 IINNDDEETTEERRMMIINNAATTEE BBEEAAMMSS ((CCoonntt’’dd))
(a) Release the vertical support at B:
x
y
P
P
C B
B B
DB = +
DC
L/2 L/2 L
RB
D¢BB=RB*fBB
Applied in +ve
direction
The governing compatibility equation obtained at B is,
DB
+ D'
BB = 0
R f
D+ ( ) ´ ( ) =
0
B B BB
R /
f
From earlier analyses,
=-D
B B BB
3 2
P L EI P L EI L B
D=- +- ´
( / 2) /(3 ) [ ( / 2) /(2 )] ( / 2)
3 3
PL EI PL EI
=- -
/(24 ) /(16 )
3
PL EI
=-
(5/ 48)( / )
f L3 /(3EI ) BB =
R PL EI L EI P BB =-[-(5/ 48)( 3 / )]/[ 3 /(3 )]=(5/16)
fBB = displacement per unit load (applied in +ve direction)

9. 5.4 INDETERMINATE BEAM (Cont’d)
5.4.2 Propped cantilever - Redundant support moment released
L/2 P
A B
P
Governing compatibility equation obtained at A is,
+( )´( )= A A AA q M a , AA a= rotation per unit moment
q
A
AA
A M
=-
a
From known earlier analysis, (16 )
AA q =- [under a central concentrated
2
EI
PL
load]
(1)[L /(3EI )] AA a =-
This is due to the fact that +ve moment causes a –ve rotation
M =-- [ PL2/(16EI)]/[ -
L/(3EI)]
A
=-
(3/16)PL
L
(b) Release the moment constraint at a:
qA
=
A B
Primary structure
+ A B
MA q¢A=MAaAA
Redundant MA applied

10. 5.4.3 OVERVIEW OF METHOD OF
CONSISTENT DEFORMATION
To recapitulate on what we have done earlier,
I. Structure with single degree of indeterminacy:
A B
RB
(a) Remove the redundant to make the structure determinate
(primary structure)
A B
DBo
(b) Apply unit force on the structure, in the direction of the
redundant, and find the displacement
fBB
(c) Apply compatibility at the location of the removed redundant
DB0 + fBB´RB = 0
P
P

11. 5.5 INDETERMINATE BEAM WWIITTHH MMUULLTTIIPPLLEE DDEEGGRREEEESS
OOFF IINNDDEETTEERRMMIINNAACCYY
A
w/u.l
B C D E
RB RC RD
DB0 DC0
DD0
(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., DBO, DCO and DDO

12. (b) Apply unit loads at B, C and D, in a sequential manner and
determine deformations at B, C and D, respectively.
A
B C D E
fBB
fCB fDB 1
A
B C D E
fBC
fCC fDC
A
1
B C D E
fBD
fCD fDD
1

13. (c ) Establish compatibility conditions at B, C and D
DBO + fBBRB + fBCRC + fBDRD = 0
DCO + fCBRB + fCCRC + fCDRD = 0
DDO + fDBRB + fDCRC + fDDRD = 0

14. 5.4.2 When support settlements occur:
A
B C D E
DB DC DD Support settlements
Compatibility conditions at B, C and D give the following equations:
DBO + fBBRB + fBCRC + fBDRD = DB
DCO + fCBRB + fCCRC + fCDRD = DC
DDO + fDBRB + fDCRC + fDDRD = DD
w / u. l.

15. 55..55 TTRRUUSSSS SSTTRRUUCCTTUURREESS
80 kN
C
60 kN
D
A B
80 kN
C
60 kN
D
A 1 2
B
Primary structure
(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

16. 5.5 Truss Structures (Cont’d)
(b)Find deformation Dalong AB:
ABO D=S (FuL)/AE
ABO 0ABF= Force in member of the primary structure due to applied
0 load
u= Forces in members due to unit force applied along AB
AB(c) Determine fAB uABL
=å
deformation 2
along AB due to unit load applied along
AB:
,
AB
AE
(d) Apply compatibility condition along AB:
DABO+fAB,ABFAB=0

17. (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)

18. 5.6 TEMPERATURE CCHHAANNGGEESS AANNDD FFAABBRRIICCAATTIIOONN
EERRRROORR
 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