3. 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
school.edhole.com
4. 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
school.edhole.com
5. 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
school.edhole.com
6. 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
school.edhole.com
7. 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).
school.edhole.com
8. (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)
school.edhole.com
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
school.edhole.com
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
school.edhole.com
11. A
B C D E
RB RC RD
(a) Make the structure Ddeterminate D(Dby releasing the
B0 C0
D0
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
w/u.l
school.edhole.com
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
school.edhole.com
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
school.edhole.com
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.
school.edhole.com
15. 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
school.edhole.com
16. 5.5 Truss Structures (Cont’d)
(b)Find deformation DABO along AB:
DABO =S (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:
2
uABL
=å
AE
fAB ,
AB
(d) Apply compatibility condition along AB:
DABO+fAB,ABFAB=0
(d) Hence determine FAB
school.edhole.com
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)
school.edhole.com
18. 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
school.edhole.com