civil engineering

1. 5. ANALYSIS OF INDETERMINATE
STRUCTURES BY FORCE METHOD

2. 5.1 ANALYSIS OF INDETERMINATE STRUCTURES
BY FORCE METHOD - AN OVERVIEW
• 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

3. 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

4. 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

5. 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

6. 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).

7. 5.4 INDETERMINATE BEAMS (Cont’d)
(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
P
P
BC
L/2L/2
L
∆C
∆B= +
B B
RB
∆′BB=RB*fBB
∆B + ∆'
BB = 0
BBBB
BBBB
fR
fR
/
0)()(
∆−=∴
=×+∆
From earlier analyses,
)/)(48/5(
)16/()24/(
)2/()]2/()2/([)3/()2/(
3
33
23
EIPL
EIPLEIPL
LEILPEILPB
−=
−−=
×−+−=∆
)3/(3
EILfBB =
PEILEIPLRBB )16/5()]3/(/[)]/)(48/5([ 33
=−−=
Applied in +ve
direction

8. Governing compatibility equation obtained at A is,
=×+ )()( AAAA M αθ , AAα= rotation per unit moment
AA
A
AM
α
θ
−=∴
From known earlier analysis, )16(
2
EI
PL
AA −=θ [under a central concentrated
load]
)]3/()[1( EILAA −=α
This is due to the fact that +ve moment causes a –ve rotation
PL16)/(3
EI)]L/(3/[EI)]/(16PL[M 2
A
−=
−−−=
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
θA
=
A B
P
Primary structure
+ BA
MA θ′A=MAαAA
Redundant MA applied

9. 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 B
RB
A B
∆Bo
fBB
5.4.3 OVERVIEW OF METHOD OF
CONSISTENT DEFORMATION
∆B0 + fBB×RB = 0
P
P

10. 5.5 INDETERMINATE BEAM WITH MULTIPLE DEGREES
OF INDETERMINACY
(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
A
B C D E
RB RC RD
∆B0 ∆C0
∆D0
w/u.l

11. (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
B C D E
fBD
fCD fDD
1
1

12. (c ) Establish compatibility conditions at B, C and D
∆BO+ fBBRB+ fBCRC+ fBDRD= 0
∆CO + fCBRB+ fCCRC+ fCDRD= 0
∆DO + fDBRB+ fDCRC+ fDDRD= 0

13. 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
A
B C D E
∆B ∆C ∆D Support settlements
w / u. l.

14. 5.5 TRUSS STRUCTURES
(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

15. 5.5 Truss Structures (Cont’d)
(b)Find deformation ∆ABO along AB:
∆ABO =Σ (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:
∆ABO+fAB,ABFAB=0
(d) Hence determine FAB
∑=
AE
L
AB
u
ABAB
f
2
,

16. (e) Determine the individual member forces in a particular member
CE by
FCE = FCE0 + uCEFAB
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)

17. 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