flexibility energy method in structural analysis
TRANSCRIPT
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BYDR. MAHDI DAMGHANI
2016-2017
Structural Design and Inspection-Energy method
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Suggested Readings
Reference 1 Reference 2
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Objective(s)
Familiarisation with force method (flexibility method)
Ability to solve indeterminate structures
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Introduction
Refer to chapter 5 of Reference 1Refer to chapter 4 of Reference 2This method is for solving indeterminate
structuresFocuses on the solution for the system
internal forces In this method the governing equations
express compatibility requirements in terms of the redundant forces and component flexibilities
Forces are determined first and then strains and displacements are found
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Procedure
This method is also called force method because: Internal forces are found
first Then displacements and
strains are calculated
1 •Determine degree of redundancy of the structure (NR)
2 •Remove NR members of the structure
3 •Represent each removed member by its associated force
4 •Calculate displacement for each force using unit load method
5 •Apply compatibility of displacements
6 •Solve a set of linear equations to calculate redundant forces
7 •Calculate displacements and strains
Let’s see this in some examples
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Example
Determine internal forces in the truss structure shown below. The cross-sectional area A and Young’s modulus E are the same for all members.
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Solution
Is the structure determinate or indeterminate?
What is the degree of redundancy? 3 members (m) 6 reaction forces (r) 4 joints=8 equilibrium
equations (2j)
NR=m+r-2j=1 At each node we have: 0,0 yx FF
1 2 3
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Solution
The redundant member is removed
The redundant member is replaced by a force R
Now the structure is determinate and can be solved under two forces P and R
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Solution
A B C
O
P
P
O
O
R
R1
2 31 32
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Solution
FA FC fA fC FAfAL/EA FCfCL/EA
P
A C
O
FCFA θ
cos2P
cos2P
cos21
cos21
3cos4EAPL
3cos4EAPL
32
1
)2(
cos2EAPL
AELfF
i ii
iiiO
1
A C
O
fCfA θ
cos2coscos
sinsinPFPFF
FFFF
ACA
CACA
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Solution
FA FB FC fA fB fC FAfAL/EA FBfBL/EA FCfCL/EA
cos2R
cos2
1
3cos4EARL
3cos4EA
RL
EARL
EARL
AELfF
i ii
iiiO
33
1
)3(
cos2
cos2R
A C
O
R
RFCFA
A C
O
1
1fCfA
1cos2
1
EARLR
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Solution
0)3()2(OO
0
cos21
cos21
33 EAPL
EARL
A B C
O
P
Rθ
3cos21
1PR
PRFFF
FFFFF
CAyO
CACAxO
coscos0
sinsin0
cos2RPFA
32
1
)2(
cos2EAPL
AELfF
i ii
iiiO
EARL
EARL
AELfF
i ii
iiiO
33
1
)3(
cos2
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Example
Calculate the forces in the members of the truss. All members have the same cross-sectional area A and Young’s modulus E.
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Solution
How many degrees of redundancy are in the structure? m=7 r=5 j=5
NR=m+r-2j=2
6
1
2
34
5 7
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Solution
2
1
4
R2
3
X1
X1
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Solution
2 4
R2
3
X1
X1
0
0)4()3()2(
)4()3()2(
CCC
ADADAD
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Solution
2
F=10
00
0
-14.14
0
11f=-0.71
0-0.71
0-0.71
1
EAAELfF
i ii
iiAD
1.276
1
)2(
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Solution
11f=-0.71
0-0.71
0-0.71
1
1
7
1
)3( 32.4 XEAAE
LfFi ii
iiAD
3
X1
X1
X1
X1
F=-0.71X1
0-0.71X1
0-0.71X1
X1
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Solution
11f=-0.71
0-0.71
0-0.71
1
2
6
1
)4( 7.2 REAAE
LfFi ii
iiAD
4
R2
R2
F=-2R2
0
R2R2
1.41R2 -1.41R2
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Solution
Therefore our first linear equation becomes;
Now we need another equation to solve two unknowns X1 and R2
This time we make sure vertical displacement at node C becomes zero
0)4()3()2(ADADAD
07.232.41.27 21 RX
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Solution
2
F=10
00
0
-14.14
0
f=-2
-1.410
1 1
1.41
EAAELfF
i ii
iiC
11.486
1
)2(
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Solution
1
7
1
)3( 7.2 XEAAE
LfFi ii
iiC
3
X1
X1
X1
X1
F=-0.71X1
0-0.71X1
0-0.71X1
X1
f=-2
-1.410
1 1
1.41
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Solution
2
6
1
)4( 62.11 REAAE
LfFi ii
iiC
R2
F=-2R2
-1.41R2
0
R2R2
1.41R2
4
R2
f=-2
-1.410
1 1
1.41
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Solution
Therefore our second linear equation becomes;
Now by solving the two linear equations simultaneously;
062.117.211.48 21 RX
0)4()3()2(CCC
062.117.211.48 21 RX
07.232.41.27 21 RXkNRkNX 15.3,28.4 21
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Example
Determine the reactions of the continuous beam in the beam structure below, using force method.
L L
w
A B C
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Solution
Is the structure determinate or indeterminate?
w
What is the degree of indeterminacy?3 equations (knowns)
A B C
0
0
0
z
y
x
M
F
F
4 reactions (unknowns)
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Solution
I intentionally choose support at B as the redundant
The released structure is then loaded by redundant member
Vertical displacement at B must be zerow
A B C
RA RC
RB=XB
0B
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Solution
To determine redundant, we superimpose deflection due to external load and a unit value of the redundant multiplied by the magnitude of redundant XB w
A C
A BC
wLwL
0B
10.5 0.5
BB
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Solution
w
A C
w
A BC
wLwL
0B
10.5 0.5
BB
00 BBBB X
EIL
EILw
BBB 4821,
38425 34
0
wLXR BB 25.1
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Solution
w
A C
w
A BC
wLwL
0B
XB1.25wL 1.25wL
BBBX
wLwLwLRA 8325.15.0 wLwLwLRC 8
325.15.0
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Solution
w
A B C
(3/8)wL(5/4)wL (3/8)wL
(3/8)wL(5/8)wL
(-5/8)wL (-3/8)wL
Shear
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Solution
w
A B C
(3/8)wL(5/4)wL (3/8)wL
(9/128)wL2
(-1/8)wL2
Moment
(9/128)wL2
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Example
How do you solve the following?
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Solution
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Example
How about this one?
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Solution
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Example
How about this one?
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Solution
01221111 RRC
02222112 RRC
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Q1
Determine member forces and reaction at supports of truss structure shown below. EA is constant for all members.
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Q2
Determine forces in the truss structure shown below. EA is constant for all members.
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Q3
A continuous beam ABC is carrying a uniformly distributed load of 1 kN/m in addition to a concentrated load of 10 kN. Draw bending moment and shear force diagram. Assume EI to be constant for all members.
A B C
1kN/m
10kN
10m5m5m
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Q4
The beam ABC shown in Figure is simply supported and stiffened by a truss whose members are capable of resisting axial forces only. The beam has a cross-sectional area of 6000mm2 and a second moment of area of 7.2×106 mm4. If the cross-sectional area of the members of the truss is 400mm2, calculate the forces in the members of the truss and the maximum value of the bending moment in the beam. Young’s modulus, E, is the same for all members.
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Q5
Draw the quantitative shear and moment diagram and the qualitative deflected curve for the Frame shown below. EI is constant.