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Structural Analysis 7th Edition in SI Units
Russell C. Hibbeler
Chapter 10:
Analysis of Statically Indeterminate Structures by the Force Method
Statically Indeterminate Structures
• Advantages & Disadvantages
• For a given loading, the max stress and deflection
of an indeterminate structure are generally smaller
than those of its statically determinate counterpart
• Statically indeterminate structure has a tendency to
redistribute its load to its redundant supports in
cases of faulty designs or overloading
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Statically Indeterminate Structures
• Advantages & Disadvantages
• Although statically indeterminate structure can
support loading with thinner members & with
increased stability compared to their statically
determinate counterpart, the cost savings in
material must be compared with the added cost to
fabricate the structure since often it becomes more
costly to construct the supports & joints of an
indeterminate structure
• Careful of differential disp of the supports as well
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Statically Indeterminate Structures
• Method of Analysis
• To satisfy equilibrium, compatibility & force-disp
requirements for the structure
• Force Method
• Displacement Method
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Force Method of Analysis: General
Procedure
• From free-body diagram, there would be 4
unknown support reactions
• 3 equilibrium eqn
• Beam is indeterminate to first degree
• Use principle of superposition & consider the
compatibility of disp at one of the supports
• Choose one of the support reactions as redundant
& temporarily removing its effect on the beam
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Force Method of Analysis: General
Procedure
• This will allow the beam to be statically
determinate & stable
• Here, we will remove the rocker at B
• As a result, the load P will cause
B to be displaced downward
• By superposition, the unknown
reaction at B causes the beam
at B to be displaced upward
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Force Method of Analysis: General
Procedure
• Assuming +ve disp act upward, we
write the necessary compatibility
eqn at the rocker as:
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
BB
y
B
BB
y
BB
BB
B
f
B
f
B
BB
f
y
B
BB















0
'
'
0
:
get
we
,
eqn
first
into
eqn
second
sub
t
coefficien
y
flexibilit
linear
B
at
reaction
unknown
B
at
disp
upward
'
Force Method of Analysis: General
Procedure
• Using methods in Chapter 8 or 9 to solve for B
and fBB, By can be found
• Reactions at wall A can then be determined from
eqn of equilibrium
• The choice of redundant is arbitrary
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Force Method of Analysis: General
Procedure
• The moment at A can be determined directly by
removing the capacity of the beam to support
moment at A, replacing fixed support by pin
support
• The rotation at A
caused by P is A
• The rotation at A
caused by the
redundant MA at
A is ’AA
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Force Method of Analysis: General
Procedure
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
AA
A
A
AA
A
AA
M
M







0
:
requires
ity
Compatibil
'
Similarly,
Maxwell’s Theorem of Reciprocal Disp:
Betti’s Law
• The disp of a point B on a structure due to a unit
load acting at point A is equal to the disp of point
A when the load is acting at point B
• Proof of this theorem is easily demonstrated using
the principle of virtual work
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
AB
BA f
f 
Maxwell’s Theorem of Reciprocal Disp:
Betti’s Law
• The theorem also applies for reciprocal rotations
• The rotation at point B on a structure due to a unit
couple moment acting at point A is equal to the
rotation at point A when the unit couple is acting
at point B
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Determine the reaction at the roller support B of the beam. EI is
constant.
Example 10.1
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Principle of superposition
By inspection, the beam is statically indeterminate to the first
degree. The redundant will be taken as By. We assume By acts
upward on the beam.
Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Compatibility equation
Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
kN
B
EI
B
EI
EI
m
f
EI
kNm
f
f
B
y
y
BB
B
BB
B
BB
y
B
6
.
15
576
9000
0
:
(1)
eqn
into
Sub
576
;
9000
table.
standard
using
obtained
easily
are
and
eqn(1)
0
)
(
3
3















Draw the shear and moment diagrams for the beam. EI is
constant. Neglect the effects of axial load.
Example 10.4
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Principle of Superposition
Since axial load is neglected,
the beam is indeterminate to the
second degree. The 2 end
moments at A & B will be
considered as the redundant.
The beam’s capacity to resist
these moments is removed by
placing a pin at A and a rocker at B.
Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Compatibility eqn
Reference to points A & B requires
The required slopes and angular flexibility coefficients can be
determined using standard tables.
Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
(2)
eqn
0
(1)
eqn
0
BB
B
BA
A
B
AB
B
AA
A
A
M
M
M
M












EI
EI
EI
EI
EI
BA
AB
BB
AA
A
A
1
;
2
;
2
1
.
118
;
9
.
151












Compatibility eqn
Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
,
1
.
28
;
9
.
61
2
1
1
.
118
0
1
2
9
.
151
0
:
gives
(2)
and
(1)
eqn
into
Sub
kNm
M
kNm
M
EI
M
EI
M
EI
EI
M
EI
M
EI
B
A
B
A
B
A


































Composite Structures
• Composite structures are composed of some
members subjected only to axial force while other
members are subjected to bending
• If the structure is statically indeterminate, the
force method can conveniently be used for its
analysis
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
The beam is supported by a pin at A & two pin-connected bars at
B. Determine the force in member BD. Take E = 200GPa & I =
300(106)mm4 for the beam and A = 1800mm2 for each bar.
Example 10.10
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Principle of superposition
The beam is indeterminate to the first degree. Force in member
BD is chosen as the redundant. This member is therefore
sectioned to eliminate its capacity to sustain a force.
Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Compatibility eqn
With reference to the relative disp of the cut ends of member BD,
we require
The method of virtual work will be used to compute BD and fBDBD
Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
(1)
0 BDBD
BD
BD f
F



Compatibility eqn
For BD we require application of the real loads and a virtual unit
load applied to the cut ends of the member BD. We will consider
only bending strain energy in the beam & axial strain energy in
the bar.
Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
mm
AE
AE
EI
dx
x
x
AE
nNL
dx
EI
Mm
o
o
L
BD
326
.
0
)
45
cos
/
8
.
1
)(
1
)(
0
(
)
30
cos
/
8
.
1
)(
816
.
0
)(
3
.
69
(
)
0
)(
333
.
3
30
(
3
0
3
0













Compatibility eqn
For fBDBD we require application of a real unit load & a virtual unit
load at the cut ends of member BD.
Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
kN
m
AE
AE
EI
dx
AE
L
n
dx
EI
m
f
o
o
L
BD
/
)
10
(
092
.
1
)
45
cos
/
8
.
1
(
)
1
(
)
30
cos
/
8
.
1
(
)
816
.
0
(
)
0
(
5
2
2
3
0
2
0
2
2











Compatibility eqn
Sub into eqn (1) yields
Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
)
(
9
.
29
)
10
)(
092
.
1
(
0003264
.
0
0
0
5
C
kN
F
F
f
F
BD
BD
BD
BD
BD
BD







Additional remarks on the force method of
analysis
• Flexibility coefficients depend on the material and
geometrical properties of the members and not on
the loading of the primary structure
• For a structure having n redundant reactions, we
can write n compatibility eqn
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
0
......
0
......
0
......
2
2
1
1
2
2
22
1
21
2
1
2
12
1
11
1



















n
nn
n
n
n
n
n
n
n
R
f
R
f
R
f
R
f
R
f
R
f
R
f
R
f
R
f
Additional remarks on the force method of
analysis
• BD are caused by both the real loads on the
primary structure and by support settlement &
dimensional changes due to temperature
differences or fabrication errors in the members
• The above eqn can be re-cast into a matrix form or
simply:
• Note that fij=fji
• Hence, the flexibility matrix will be symmetric
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method



fR
Symmetric Structures
• A structural analysis of any highly indeterminate
structure or statically determinate structure can be
simplified provided the designer can recognise
those structures that are symmetric & support
either symmetric or antisymmetric loadings
• For horizontal stability, a pin is required to support
the beam & truss.
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Symmetric Structures
• Here the horizontal reaction at the pin is zero, so both
these structures will deflect & produce the same
internal loading as their reflected counterpart
• As a result, they can
be classified as being symmetric
• Not the case if the fixed support
at A was replaced by a pin since
the deflected shape & internal
loadings would not be the same
on its left & right side
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Symmetric Structures
• A symmetric structure supports an antisymmetric
loading as shown
• Provided the structure is
symmetric & its loading is
either symmetric or
antisymmetric then a
structural analysis will only
have to be performed on half the members of the
structure since the same or opposite results will be
produced on the other half
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Symmetric Structures
• A separate structural
analysis can be performed
using the symmetrical
& antisymmetrical loading
components & the results
superimposed to obtain
the actual behaviour of
the structure
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Influence lines for Statically
Indeterminate Beams
• For statically determinate beams, the deflected
shapes will be a series of straight line segments
• For statically indeterminate beams, curve will
result
• Reaction at A
• To determine the influence line for the reaction at A
, a unit load is placed on the beam at successive
points
• At each point, the reaction at A must be computed
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Influence lines for Statically
Indeterminate Beams
• Reaction at A
• A plot of these results yields the influence line
• The reaction at A can be determined by the force
method
• The principle of superposition is applied
• The compatibility eqn for point A is:
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Influence lines for Statically
Indeterminate Beams
• Reaction at A
• A plot of these results yields the influence line
• The reaction at A can be determined by the force
method
• The compatibility eqn for point A is:
• By Maxwell’s theorem of reciprocal disp
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
AA
AD
y
AA
y
AD f
f
A
f
A
f /
0 




DA
AA
y
DA
AD f
f
A
f
f 










1
Influence lines for Statically
Indeterminate Beams
• Shear at E
• Using the force method & Maxwell’s theorem of
reciprocal disp, it can be shown that
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
DE
EE
E f
f
V 








1
Influence lines for Statically
Indeterminate Beams
• Moment at E
• The influence line for the moment at E can be
determined by placing a pin or hinge at E
• Applying a +ve unit couple moment, the beam then
deflects to the dashed position
• Using the force method & Maxwell’s theorem of
reciprocal disp, it can be shown that
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
DE
EE
E f
M 









1
Influence lines for Statically
Indeterminate Beams
• Moment at E
• The influence line for the moment at E can be
determined by placing a pin or hinge at E
• Applying a +ve unit couple moment, the beam then
deflects to the dashed position
• Using the force method & Maxwell’s theorem of
reciprocal disp, it can be shown that
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Qualitative Influence lines for Frames
• The shape of the influence line for the +ve
moment at the center I of girder FG of the frame is
shown by the dashed lines
• Uniform loads would be placed only on girders AB,
CD & FG in order to create the largest +ve
moment at I
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Draw the influence line for the vertical reaction at A for the beam.
EI is constant. Plot numerical values every 2m
Example 10.11
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
The capacity of the beam to resist reaction Ay is removed. This is
done using a vertical roller device. Applying a vertical unit load at
A yields the shape of the influence line. Using the conjugate beam
method to determine ordinates of the influence line.
Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
EI
EI
EI
M
M
M
D
D
D
B
B
67
.
34
3
2
)
2
(
2
2
1
)
2
(
18
0
0
'
'
'
,
D'
For
B'
at
beam
conjugate
on the
exists
moment
no
since
,
B'
For






















Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
EI
M
EI
EI
EI
M
M
A
A
C
C
C
72
33
.
61
3
4
)
4
(
4
2
1
)
4
(
18
0
'
'
'
,
A'
For
,
C'
For






















•Since a vertical 1kN load acting at A on the beam will cause a
vertical reaction at A of 1kN, the disp at A, A should correspond
to a numerical value of 1 for the influence line ordinate at A.
•Thus dividing the other computed disp by this factor, we obtain
Solution
© 2009 Pearson Education South Asia Pte Ltd
Structural Analysis 7th Edition
Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
x Ay
A 1
C 0.852
D 0.481
B 0

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dokumen.tips_structural-analysis-chapter-10ppt.ppt

  • 1. Structural Analysis 7th Edition in SI Units Russell C. Hibbeler Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 2. Statically Indeterminate Structures • Advantages & Disadvantages • For a given loading, the max stress and deflection of an indeterminate structure are generally smaller than those of its statically determinate counterpart • Statically indeterminate structure has a tendency to redistribute its load to its redundant supports in cases of faulty designs or overloading © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 3. Statically Indeterminate Structures • Advantages & Disadvantages • Although statically indeterminate structure can support loading with thinner members & with increased stability compared to their statically determinate counterpart, the cost savings in material must be compared with the added cost to fabricate the structure since often it becomes more costly to construct the supports & joints of an indeterminate structure • Careful of differential disp of the supports as well © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 4. Statically Indeterminate Structures • Method of Analysis • To satisfy equilibrium, compatibility & force-disp requirements for the structure • Force Method • Displacement Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 5. Force Method of Analysis: General Procedure • From free-body diagram, there would be 4 unknown support reactions • 3 equilibrium eqn • Beam is indeterminate to first degree • Use principle of superposition & consider the compatibility of disp at one of the supports • Choose one of the support reactions as redundant & temporarily removing its effect on the beam © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 6. Force Method of Analysis: General Procedure • This will allow the beam to be statically determinate & stable • Here, we will remove the rocker at B • As a result, the load P will cause B to be displaced downward • By superposition, the unknown reaction at B causes the beam at B to be displaced upward © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 7. Force Method of Analysis: General Procedure • Assuming +ve disp act upward, we write the necessary compatibility eqn at the rocker as: © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method BB y B BB y BB BB B f B f B BB f y B BB                0 ' ' 0 : get we , eqn first into eqn second sub t coefficien y flexibilit linear B at reaction unknown B at disp upward '
  • 8. Force Method of Analysis: General Procedure • Using methods in Chapter 8 or 9 to solve for B and fBB, By can be found • Reactions at wall A can then be determined from eqn of equilibrium • The choice of redundant is arbitrary © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 9. Force Method of Analysis: General Procedure • The moment at A can be determined directly by removing the capacity of the beam to support moment at A, replacing fixed support by pin support • The rotation at A caused by P is A • The rotation at A caused by the redundant MA at A is ’AA © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 10. Force Method of Analysis: General Procedure © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method AA A A AA A AA M M        0 : requires ity Compatibil ' Similarly,
  • 11. Maxwell’s Theorem of Reciprocal Disp: Betti’s Law • The disp of a point B on a structure due to a unit load acting at point A is equal to the disp of point A when the load is acting at point B • Proof of this theorem is easily demonstrated using the principle of virtual work © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method AB BA f f 
  • 12. Maxwell’s Theorem of Reciprocal Disp: Betti’s Law • The theorem also applies for reciprocal rotations • The rotation at point B on a structure due to a unit couple moment acting at point A is equal to the rotation at point A when the unit couple is acting at point B © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 13. Determine the reaction at the roller support B of the beam. EI is constant. Example 10.1 © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 14. Principle of superposition By inspection, the beam is statically indeterminate to the first degree. The redundant will be taken as By. We assume By acts upward on the beam. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 15. Compatibility equation Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method kN B EI B EI EI m f EI kNm f f B y y BB B BB B BB y B 6 . 15 576 9000 0 : (1) eqn into Sub 576 ; 9000 table. standard using obtained easily are and eqn(1) 0 ) ( 3 3               
  • 16. Draw the shear and moment diagrams for the beam. EI is constant. Neglect the effects of axial load. Example 10.4 © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 17. Principle of Superposition Since axial load is neglected, the beam is indeterminate to the second degree. The 2 end moments at A & B will be considered as the redundant. The beam’s capacity to resist these moments is removed by placing a pin at A and a rocker at B. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 18. Compatibility eqn Reference to points A & B requires The required slopes and angular flexibility coefficients can be determined using standard tables. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method (2) eqn 0 (1) eqn 0 BB B BA A B AB B AA A A M M M M             EI EI EI EI EI BA AB BB AA A A 1 ; 2 ; 2 1 . 118 ; 9 . 151            
  • 19. Compatibility eqn Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method , 1 . 28 ; 9 . 61 2 1 1 . 118 0 1 2 9 . 151 0 : gives (2) and (1) eqn into Sub kNm M kNm M EI M EI M EI EI M EI M EI B A B A B A                                  
  • 20. Composite Structures • Composite structures are composed of some members subjected only to axial force while other members are subjected to bending • If the structure is statically indeterminate, the force method can conveniently be used for its analysis © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 21. The beam is supported by a pin at A & two pin-connected bars at B. Determine the force in member BD. Take E = 200GPa & I = 300(106)mm4 for the beam and A = 1800mm2 for each bar. Example 10.10 © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 22. Principle of superposition The beam is indeterminate to the first degree. Force in member BD is chosen as the redundant. This member is therefore sectioned to eliminate its capacity to sustain a force. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 23. Compatibility eqn With reference to the relative disp of the cut ends of member BD, we require The method of virtual work will be used to compute BD and fBDBD Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method (1) 0 BDBD BD BD f F   
  • 24. Compatibility eqn For BD we require application of the real loads and a virtual unit load applied to the cut ends of the member BD. We will consider only bending strain energy in the beam & axial strain energy in the bar. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method mm AE AE EI dx x x AE nNL dx EI Mm o o L BD 326 . 0 ) 45 cos / 8 . 1 )( 1 )( 0 ( ) 30 cos / 8 . 1 )( 816 . 0 )( 3 . 69 ( ) 0 )( 333 . 3 30 ( 3 0 3 0             
  • 25. Compatibility eqn For fBDBD we require application of a real unit load & a virtual unit load at the cut ends of member BD. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method kN m AE AE EI dx AE L n dx EI m f o o L BD / ) 10 ( 092 . 1 ) 45 cos / 8 . 1 ( ) 1 ( ) 30 cos / 8 . 1 ( ) 816 . 0 ( ) 0 ( 5 2 2 3 0 2 0 2 2           
  • 26. Compatibility eqn Sub into eqn (1) yields Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method ) ( 9 . 29 ) 10 )( 092 . 1 ( 0003264 . 0 0 0 5 C kN F F f F BD BD BD BD BD BD       
  • 27. Additional remarks on the force method of analysis • Flexibility coefficients depend on the material and geometrical properties of the members and not on the loading of the primary structure • For a structure having n redundant reactions, we can write n compatibility eqn © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method 0 ...... 0 ...... 0 ...... 2 2 1 1 2 2 22 1 21 2 1 2 12 1 11 1                    n nn n n n n n n n R f R f R f R f R f R f R f R f R f
  • 28. Additional remarks on the force method of analysis • BD are caused by both the real loads on the primary structure and by support settlement & dimensional changes due to temperature differences or fabrication errors in the members • The above eqn can be re-cast into a matrix form or simply: • Note that fij=fji • Hence, the flexibility matrix will be symmetric © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method    fR
  • 29. Symmetric Structures • A structural analysis of any highly indeterminate structure or statically determinate structure can be simplified provided the designer can recognise those structures that are symmetric & support either symmetric or antisymmetric loadings • For horizontal stability, a pin is required to support the beam & truss. © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 30. Symmetric Structures • Here the horizontal reaction at the pin is zero, so both these structures will deflect & produce the same internal loading as their reflected counterpart • As a result, they can be classified as being symmetric • Not the case if the fixed support at A was replaced by a pin since the deflected shape & internal loadings would not be the same on its left & right side © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 31. Symmetric Structures • A symmetric structure supports an antisymmetric loading as shown • Provided the structure is symmetric & its loading is either symmetric or antisymmetric then a structural analysis will only have to be performed on half the members of the structure since the same or opposite results will be produced on the other half © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 32. Symmetric Structures • A separate structural analysis can be performed using the symmetrical & antisymmetrical loading components & the results superimposed to obtain the actual behaviour of the structure © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 33. Influence lines for Statically Indeterminate Beams • For statically determinate beams, the deflected shapes will be a series of straight line segments • For statically indeterminate beams, curve will result • Reaction at A • To determine the influence line for the reaction at A , a unit load is placed on the beam at successive points • At each point, the reaction at A must be computed © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 34. Influence lines for Statically Indeterminate Beams • Reaction at A • A plot of these results yields the influence line • The reaction at A can be determined by the force method • The principle of superposition is applied • The compatibility eqn for point A is: © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 35. Influence lines for Statically Indeterminate Beams • Reaction at A • A plot of these results yields the influence line • The reaction at A can be determined by the force method • The compatibility eqn for point A is: • By Maxwell’s theorem of reciprocal disp © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method AA AD y AA y AD f f A f A f / 0      DA AA y DA AD f f A f f            1
  • 36. Influence lines for Statically Indeterminate Beams • Shear at E • Using the force method & Maxwell’s theorem of reciprocal disp, it can be shown that © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method DE EE E f f V          1
  • 37. Influence lines for Statically Indeterminate Beams • Moment at E • The influence line for the moment at E can be determined by placing a pin or hinge at E • Applying a +ve unit couple moment, the beam then deflects to the dashed position • Using the force method & Maxwell’s theorem of reciprocal disp, it can be shown that © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method DE EE E f M           1
  • 38. Influence lines for Statically Indeterminate Beams • Moment at E • The influence line for the moment at E can be determined by placing a pin or hinge at E • Applying a +ve unit couple moment, the beam then deflects to the dashed position • Using the force method & Maxwell’s theorem of reciprocal disp, it can be shown that © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 39. Qualitative Influence lines for Frames • The shape of the influence line for the +ve moment at the center I of girder FG of the frame is shown by the dashed lines • Uniform loads would be placed only on girders AB, CD & FG in order to create the largest +ve moment at I © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 40. Draw the influence line for the vertical reaction at A for the beam. EI is constant. Plot numerical values every 2m Example 10.11 © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 41. The capacity of the beam to resist reaction Ay is removed. This is done using a vertical roller device. Applying a vertical unit load at A yields the shape of the influence line. Using the conjugate beam method to determine ordinates of the influence line. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
  • 42. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method EI EI EI M M M D D D B B 67 . 34 3 2 ) 2 ( 2 2 1 ) 2 ( 18 0 0 ' ' ' , D' For B' at beam conjugate on the exists moment no since , B' For                      
  • 43. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method EI M EI EI EI M M A A C C C 72 33 . 61 3 4 ) 4 ( 4 2 1 ) 4 ( 18 0 ' ' ' , A' For , C' For                      
  • 44. •Since a vertical 1kN load acting at A on the beam will cause a vertical reaction at A of 1kN, the disp at A, A should correspond to a numerical value of 1 for the influence line ordinate at A. •Thus dividing the other computed disp by this factor, we obtain Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method x Ay A 1 C 0.852 D 0.481 B 0