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ANALYSIS OF INDETERMINATESTRUCTURES
1.IntroductionA structure of any type is classified as statically
indeterminate when the number of unknown reaction or
internal forces exceeds the number of equilibrium
equations available for its analysis.
What is statically DETERMINATE structure? (Fig.1)
No. of unknown = 3
No. of equilibrium equations = 3
3 = 3 thus statically determinate
Fig.1
No. of unknown = 4
No. of equilibrium equations = 3
43 thus statically Indeterminate
Fig.2
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Fig.3
No. of unknown = 6
No. of equilibrium equations = 5
6 5 thus statically Indeterminate
2.Why we study indeterminate structure ? Most of the structures designed today are statically indeterminate
Reinforced concrete buildings are considered in most cases as
statically indeterminate structures since the columns & beams are
poured as continuous member through the joints & over the supports
More stable compare to determinate structure or in another wordsafer.
In many cases more economical than determinate.
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Contrast
Determinate Structure Indeterminate Structure
PL3 /48EIP
PL3 /192EIP
Considerable compared to
indeterminate
structure
DEFLECTION
PL/4
P PL/8 PSTRESS
High moment caused thicker
member & more material needed
Less moment, smaller cross
section& less material needed
TEMPERATUREP
No effect & no stress would be
Developed in the beam
Serious effect and stress would
be developed in the beam
P
3
Generally smaller than
determinate structure
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EQUILIBRIUM
Equilibrium issatisfied when the
reactive forces
hold the structure
at rest
The force -
displacement
requirement depend
upon the way the
material responds,
which assumedlinear-elastic
response
FORCE-DISPLACEMENT
COMPATIBILITY
Compatibility is
satisfied when thevarious segments
of the structure fit
together without
intentional breaks
or overlaps
Methods of Analyses
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Methods of Analysis
Two different Methods are available
Force method
known as consistent deformation, unit load method,
flexibility method, or the superposition equations method.
The primary unknowns in this way of analysis are forces
Displacement method Known as stiffness matrix method
The primary unknowns are displacements
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Force method of analysisThe deflection or slope at any point on a structure as a result of a
number of forces, including the reactions, is equal to the algebraic
sum of the deflections or slopes at this particular point as a result
of these loads acting individually
General Procedure Indeterminate to thefirst degree
1 Compatibility equation
is needed
Choosing one of the
support reaction as a redundant
The structure become statically
determinate & stable
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Downward displacementB at B calculated (load action)
BB upward deflection per unit force at
B
Compatibility equation0 =B + ByBB
Reaction By known
Now the structure is statically
determinate
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General Procedure
Indeterminate to the first
degree
1 Compatibility equation isneeded
Choosing MA at A as a
redundant
The structure become statically
determinate & stable
Rotation A at A caused by load P is
determined
AA rotation caused by unit couple
moment applied at A
Compatibility equation
0 = A + MA AA
Moment MA known
Now the structure is statically determinate 8
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General Procedure Indeterminate to the 2nd degree
2 Compatibility equations needed
Redundant reaction B & C
Displacement B &C caused by load P1 & P2 are determined
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BB & BC Deflection per unit
force at B are determined
CC &CB Deflection per unit
force at C are determined
1
Compatibility equations
0 =B + ByBB + CyBC
0 =C + ByCB + CyCC
Reactions at B & C are known
Statically determinate structure
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Maxwells Theorem
The displacement of a point B on a structure due to a unit load
acting at a point A is equal to the displacement of
point A when the unit load is acting at point B is:
fBC=fCB
The rotation of a point B on a structure due to a unit moment
acting at a point A is equal to the rotation of point A when the unitmoment is acting at point B the is
BC=CB
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Procedure for Analysis
Determine the degree of statically indeterminacy
Identify the redundants, whether its a force or a moment, that
would be treated as unknown in order to form the structurestatically determinate & stable
Calculate the displacements of the determinate structure at the
points where the redundants have been removed
Calculate the displacements at these same points in thedeterminate structure due to the unit force or moment of each
redundants individually
Workout the compatibility equation at each point where there is
a redundant & solve for the unknown redundants
Knowing the value of the redundants, use equilibrium to
determine the remaining reactions
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Determine the reaction at B
Indeterminate to the 1st degree
thus one additional equationneeded
Lets take B as a redundant
Determine the deflection at point B
in the absence of support B. Using
the momentarea method
B
= 1/EI[300x6/2{(x6)+6}]
B = 9000 kNm3/EI
Determine the deflection caused by
the unit load at point B
Example
CA
B
50kN300kNm
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BB =1/EI[12x12/2xx12]
BB = 576m3/EI Compatibility equation
0=B
+BY
BB
0= 9000 /EI + BY
576 /EI
BY
=15.6kN
The reaction at B is known now so
the structure is staticallydeterminate & equilibrium
equations can be applied to get
the rest of the unknowns
AB
1
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90kN
C
ED
BA
60kN
C
BA
CC
C ED
A B
1.671.34
0.440.56
1.0
2.67
60kN90kN
68.33 81.67
C ED BA
245230205
C
60kN 90kN
EDAB
C
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A
2.5
20kN8kN / m
B C
3 2.5
B
8kN / m20kN
A CB
B1 =RB BBA C
B
RB
Compatibility Equation
B =B1 =RBBB
RB = B / BB
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