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method of consistent deformation, structural analysis, civil engineering, construction engineering,theory of structures
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13. Method of Consistent Deformations 1
13. Method of Consistent Deformations
By James C. Maxwell in 1864
13. Method of Consistent Deformations 2
13.1 Structures with a Single Degree of Indeterminacy
AB
C
32kE = 30,000 ksi
I = 512 in4
10 10
1. Free-Body Diagram
32 k
AxAy Cy
MA
)yAyx C,M,A,(A
)0,0,0( === MFF yxUnknown variables = 4
3Equations of Equilibrium =
Statically Indeterminate Structure
Degree of Indeterminacy = 4 -3 =1
= Number of Redundancy
13. Method of Consistent Deformations 3
32 k
Ax
Ay Cy
Ma
2. Selecting Cy as a redundant
32k
Axo
Ayo
MAO Determinate Structure
Primary Structure
CO
0= xF 0=xoA= kAyo 320 y =F
kfkM AO = 32001032 =AOM0= AM
13. Method of Consistent Deformations 4
32k
32k
321 k-ft
CO
Bending Moment Diagram
A B C
-320
Determining CO using conjugate-beam method,
EI320
13. Method of Consistent Deformations 5
EI320
)101032
2110320 += (
EICO
EI67.26666=
512300001267.26666 3
=
3ftk in0.3=
= in.CO 033. Applying the redundant
Axc
MACCCf
yC1 k
Ayc fCC : Flexibility Coefficient
13. Method of Consistent Deformations 6
Axc
Ayc
MacCCf
1kyC
0=xcA0= xF= kAyc 10 y =F
kfkM AC = 200201 =+ ACM0= AM
Bending Moment Diagram
A B C
1020
yC
13. Method of Consistent Deformations 7
Determining fCC using conjugate-beam method,
EI20
)2032(
212020 =
EIfCC
EIftk 367.2666 = 51230000
1267.2666 3
= = in3.0
yCCCC Cf =
4. Compatibility Condition
0=+= yCCCOC CfyCCCO Cf=
Unknowns : Forces Force Method
Flexibility MethodFlexibility
13. Method of Consistent Deformations 8
=== kf
CCC
COy 103.0
0.3
5. Reaction Forces
Using the equilibrium equations,
0A ,0 == xxF==+= k22A0,1032 ,0 yyy AF
120 ,020101032 ,0 ftkMMM AAA ==+=
13. Method of Consistent Deformations 9
Using the principle of superposition,
32 k
32k
321 k-ft
in0.3
1k x 10
20 k-ft x 10103.0 in
32 k
Ay = 22 k
MA = 120 k-ft
Ax = 0
Cy = 10 k
13. Method of Consistent Deformations 10
6. Shear Force Diagram & Bending Moment Diagram
Using the reaction forces & the applied loads,
Using the principle of superposition,A B C
-320
A B C
20 x 1010 x 10
A
B
C
-120
100
13. Method of Consistent Deformations 11
Moment as a redundantPrimary Structure Statically Determinate & Stable, ()Redundant
AxAy Cy
MA
Selecting Ax as a redundant,
Ay Cy
MA
Unstable Structure
13. Method of Consistent Deformations 12
AxAy Cy
MA
32k
32k
Selecting MA as a redundant,
Ayo Cyo
32k
Axo
Statically Determinate & Stable Structures
13. Method of Consistent Deformations 13
2. Selecting MA as a redundant
Axo
Ayo Cyo
32k
AO
0=xoA0= xF= kCyo 160= AM= kAyo 160 y =F
EIPL
16
2
= rad0075.0=AO
13. Method of Consistent Deformations 14
3. Applying the redundant
AxA
AyACyA
ftk 1AAf
AM
0=xAA0= xF= kCyA 20
10= AM
= kAyA 201
0 y =F
EIL
3=AAf ftkrad = /0000625.0
AAAAA Mf =
13. Method of Consistent Deformations 15
4. Compatibility Condition
0=+= AAAAOA MfAAAAO Mf=
0000625.00075.0=
AA
AOA f
M = ftk =120
5. Reaction ForcesUsing the equilibrium equations,
Using the principle of superposition,
6. Shear Force Diagram & Bending Moment Diagram
Using the reaction forces & the applied loads,
Using the principle of superposition,
13. Method of Consistent Deformations 16
Example 13.1
AB
L
M
EI = constant
1. Free-Body Diagram
AxAy Cy
MAM
Unknown variables = 43Equations of Equilibrium =
Statically Indeterminate Structure
Degree of Indeterminacy = 4 -3 =1
13. Method of Consistent Deformations 17
AxAy By
MAM
2. Selecting By as a redundantMMAO
BO
Determinate Structure
Primary Structure
=BO
13. Method of Consistent Deformations 18
3. Applying the redundant
MABBBf
yB1
AyB
=BBf4. Compatibility Condition
0=+= yBBBOB Bf== BBBOy fB /
5. Reaction Forces
6. Shear Force Diagram & Bending Moment Diagram
13. Method of Consistent Deformations 19
Example 13.2
15 kN/m60 kN
AB C
D
10 m 5 m 5 m
E = 200 GPa
I = 700106 mm4
1. Free-Body Diagram
AxAy
By
15 kN/m60 kN
Dy
Unknown variables = 43Equations of Equilibrium =
Statically Indeterminate Structure
Degree of Indeterminacy = 4 -3 =1
13. Method of Consistent Deformations 20
AxAy
By
15 kN/m60 kN
Dy
2. Selecting By as a redundant
15 kN/m60 kN
AxoAyo Dyo
BO
=BO
13. Method of Consistent Deformations 21
3. Applying the redundant
1yBAxB
AyB DyB
BBf
=BBf4. Compatibility Condition
0=+= yBBBOB Bf== BBBOy fB /
5. Reaction Forces
6. Shear Force Diagram & Bending Moment Diagram
13. Method of Consistent Deformations 22
Example 13.3
25 k25 k
28 k
3 @ 20 ft
15 ft
A B C D
E F
13. Method of Consistent Deformations 23
1. Free-Body Diagram
25 k25 k
28 k
AxAy Dy
Dx
m + r Unknown variables = = 13
Equations of Equilibrium = 2j = 12
Statically Indeterminate Structure
Degree of Indeterminacy = 13 12 = 1
13. Method of Consistent Deformations 24
25 k25 k
28 k
AxAy Dy
Dx
2. Selecting Dx as a redundant
25 k25 k
28 k
Axo
Ayo Dyo
DO
13. Method of Consistent Deformations 25
= FAELFvOF
Du
== ODDO FAELu
13. Method of Consistent Deformations 26
3. Applying the redundant
1AxD
AyD DyD
DDf
xD
= FAELFv
Du
== AELuf DDD 2
13. Method of Consistent Deformations 27
4. Compatibility Condition0=+= xDDDOD Df
== DDDOx fD /
5. Reaction Forces
6. Shear Force Diagram & Bending Moment Diagram
13. Method of Consistent Deformations 28
Example 13.4
3 k/ft
30 ftA
B CEI = constant
20 ft
13. Method of Consistent Deformations 29
1. Free-Body Diagram3 k/ft
AxAy
CyCx
Unknown variables = 43Equations of Equilibrium =
Statically Indeterminate Structure
Degree of Indeterminacy = 4 -3 =1
13. Method of Consistent Deformations 30
2. Selecting Ax as a redundant3 k/ft
Ay
Cy
Cx
AO
13. Method of Consistent Deformations 31
= dxEIMM vOM
Am
== dxEIMm OAAO
13. Method of Consistent Deformations 32
3. Applying the redundant
1
AyA
CyA
CxA
AAf
= dxEIMM v
Am xA
== dxEImf AAA2
13. Method of Consistent Deformations 33
4. Compatibility Condition0=+= xAAAOA Af
== AAAOx fA /
5. Reaction Forces
6. Shear Force Diagram & Bending Moment Diagram
13. Method of Consistent Deformations 34
13.2 Internal Forces and Moments as Redundants
10 k
A B C
10 k 12 k
6 ft 8 ft 6 ft 20 ft 10 ft
EI = constant
1. Free-Body Diagram
10 k 10 k 12 k
AyBy Cy
Ax
Unknown variables = 43Equations of Equilibrium =
Statically Indeterminate Structure
Degree of Indeterminacy = 4 -3 =1
13. Method of Consistent Deformations 35
BM
B
BRBL =BL
BR Brel BLBR = 0=
13. Method of Consistent Deformations 36
2. Selecting Internal Moment (Bending Moment) MB as a redundant
10 k 10 k 12 k
BOL BOR
Determinate Structure
Primary Structure
BORBOL BOrel BOLBOR = 0
EIftk
BOR
233.533 =EI
ftkBOL
2420 =
EIftk
BOrel
233.953 =
13. Method of Consistent Deformations 37
3. Applying the redundant1 1
BBLf BBRf
BM
EIftkftk
fBBR= /10
2
EIftkftk
fBBL= /67.6
2
EIftkftk
fBBrel= /67.16
2
13. Method of Consistent Deformations 38
4. Compatibility Condition0=+= BBBrelBOrelBrel Mf
ftkfM BBrelBOrelB == 19.57/5. Reaction Forces
- Equilibrium Equations of Members & Joints
57.19 57.1910 10 57.19 1257.19
Ay = 7.14 ByAB = 12.86 Cy = 6.09By
BC = 5.9112.86 5.91
By = 18.77
6. Shear Force Diagram & Bending Moment Diagram
13. Method of Consistent Deformations 39
Internally Indeterminate StructuresSelecting a reaction as a redundant
OR Selecting an internal force or moment as a redundant
Externally determinate, but Internally indeterminate structures
Selecting a reaction as a redundant ()
Selecting an internal force or moment as a redundant
A B
CD
P
13. Method of Consistent Deformations 40
1. Static Determinacy
Externally Static Determinacy
r = 3
Statically Determinate Externally
A B
CD
P
Internally Static Determinacy
m = 6, r = 3, j = 4
m + r > 2j
Statically Indeterminate
13. Method of Consistent Deformations 41
2. Selecting Internal Force FAD as a redundant
= FAELFv
== OADADO FAELu
A B
CD
P
ADOOF
A B
CD
1ADu
1
FO : Internal Forces due to External Forces
of the Primary System
uAD : Internal Forces due to unit axial Forces
of the Primary System
ADOF 0=ADADu , 0.1=
13. Method of Consistent Deformations 42
3. Applying the redundant
A B
CD
1
1
ADADf ,
= FAELFv
ADu
ADF
== AELuf ADADAD 2,4. Compatibility Condition
0, =+= ADADADADOAD Ff== ADADADOAD fF ,/
13. Method of Consistent Deformations 43
5. Member Forces
ADADO FuFF +=
6. Reaction Forces
ADADO FrRR +=
13. Method of Consistent Deformations 44
Example 13.5
13. Method of Consistent Deformations 45
Example 13.6
13. Method of Consistent Deformations 46
13.3 Structures with Multiple Degrees of Indeterminacy
w
A B C D E
1. Free-Body Diagram
AxAy
By Cy
w
Dy Ey
Unknown variables = 63Equations of Equilibrium =
Statically Indeterminate Structure
Degree of Indeterminacy = 6 - 3 =3
13. Method of Consistent Deformations 47
AxAy
By Cy
w
Dy Ey
2. Selecting By, Cy and Dy as the redundants
w
AxOAyO EyO
BO CO DO
=BO=CO=DO
13. Method of Consistent Deformations 48
3. Applying the redundant
1yBBBfAxB
AyBEyB
CBf DBf
=BBf =CBf =DBf
1yCBCfAxC
AyCEyC
CCf DCf
=BCf =CCf =DCf
1yDBDfAxD
AyDEyD
CDf DDf
=BDf =CDf =DDf
13. Method of Consistent Deformations 49
4. Compatibility Condition
0=+++ yBDyBCyBBBO DfCfBf0=+++ yCDyCCyCBCO DfCfBf0=+++ yDDyDCyDBDO DfCfBf
No. of Unknown Redundants = 3
No. of Compatibility Equations = 3
By, Cy, Dy
13. Method of Consistent Deformations 50
5. Reaction Forces
6. Shear Force Diagram & Bending Moment Diagram
Deflections to be calculated for determining unknown reducdants
=DO=CO=BO=DBf=CBf=BBf=DCf=BCf =CCf
=BDf =CDf =DDf
12
13. Method of Consistent Deformations 51
According to the Maxwells Law of Reciprocal Deflections,
=BO =CO =DO=BBf =CBf =DBf=BCf =CCf =DCf=BDf =CDf =DDf
9Homework 4 (10 Points)Definition & Proof
- Bettis Law
- Maxwells Law of Reciprocal Deflections
13. Method of Consistent Deformations 52
Example 13.7
13. Method of Consistent Deformations 53
Example 13.8
13. Method of Consistent Deformations 54
Example 13.9
13. Method of Consistent Deformations 55
Example 13.10
13. Method of Consistent Deformations 56
Example 13.11
13. Method of Consistent Deformations 57
13.4 Support Settlements and Temperature Changes
Support Settlementsw
A B C DB C
2. Selecting By and Cy as the redundantsw
BO CO
=CO=BO
13. Method of Consistent Deformations 58
3. Applying the redundant
1
BBf CBfyB
=CBf=BBf
1
BCf CCf
=BCf =CCfyC
4. Compatibility Condition
=++ yBCyBBBO CfBf B=++ yCCyCBCO CfBf C
13. Method of Consistent Deformations 59
A B C D
AB
C D
=++ yBCyBBBO CfBf=++ yCCyCBCO CfBf
Rigid-Body Displacement
Chord of Primary BeamBR
CR
BRCR
Relative Displacement
13. Method of Consistent Deformations 60
Rigid-Body Displacement
A B C D
No Bending Deformations No Member Forces (Bending Moments) No Stresses
Support Settlements in a Externally Determinate Structure No Effects
13. Method of Consistent Deformations 61
Example 13.12
13. Method of Consistent Deformations 62
Temperature Changes
Support Settlements or Temperture Changes in a Determinate Structure
No EffectsSupport Settlements or Temperture Changes in a Indeterminate Structure
Resulting Stresses
Externally Indeterminate, and Internally Indeterminate
Reactions 0 Internal Forces 0
Externally Determinate, and Internally Indeterminate Reactions = 0 Internal Forces 0
Externally Determinate, and Internally Determinate
Reactions = 0 Internal Forces = 0
13. Method of Consistent Deformations 63
Example 13.13
A B
C D1T
2T
1. Static Determinacy
Externally Static Determinacy
r = 3
Statically Determinate Externally
Internally Static Determinacy
m = 6, r = 3, j = 4
m + r > 2j
Statically Indeterminate
13. Method of Consistent Deformations 64
2. Selecting Internal Force FAD as a redundant
A B
C D1T
2TADO
= LTFv )(
A B
C D
1
1
ADu
== LTuADADO )(
13. Method of Consistent Deformations 65
3. Applying the redundant
A B
C D
1
1ADF
ADADf ,
= FAELFv
ADu
== AELuf ADADAD 2,4. Compatibility Condition
0, =+= ADADADADOAD Ff== ADADADOAD fF ,/
13. Method of Consistent Deformations 66
5. Member Forces
ADADFu=ADADO FuFF +=
6. Reaction Forces
ADADFr=ADADO FrRR += 0=
13. Method of Consistent Deformations 67
13.5 Method of Least Workw
A B CAxAy By Cy
Selecting By as the redundantw
By
w
By
13. Method of Consistent Deformations 68
=U ),( yBwf)(wf=UAccording to the Castiglianos Second Theorem,
=
yBU
0 Compatibility Condition
For the value of the redundant that satisfies the equilibrium equation and compatibility,the strain energy of the structure is a maximum or minimum.
MINIMUM The magnitude of the redundants of a statically indeterminate structure
must be such that the strain energy (internal work) is a minimum (the least).
Principle of Least Work
13. Method of Consistent Deformations 69
For structures with n redundants,
),,,,,( 321 nRRRRwf L=U0
1
=RU 0=
nR
U02
=RU 0
3
=RU L
13. Method of Consistent Deformations 70
Example 13.1430 kN/m 80 kN
10 m 5 m 5 mB C
EI = constant
A D
AxAy
By Dy
Free-Body Diagram30 kN/m 80 kN
Unknown variables = 43Equations of Equilibrium =
Statically Indeterminate Structure
Degree of Indeterminacy = 4 - 3 = 1
13. Method of Consistent Deformations 71
Selecting By as the redundant
30 kN/m 80 kN
Ax
Ay Dy
By
yy BD 5.0135=0=xA yy BA 5.0245=
L dxEIM02
2=U
L
y
dxEIM
BM
00==
yB
U
13. Method of Consistent Deformations 72
30 kN/m 80 kN
245 0.5By 135 0.5By
ByA
B
C D
(m)Limit
scoordinate XSegment
215)5.0245( xxBy xBy )5.0135(
)5(80)5.0135( xxBy
yBM /MOriginx5.0100AABx5.0D 50 DCx5.0105CB D
=L
y
dxEIM
BM
0=
yB
U 100 dxL + 50 dxL + 105 dxL 0=)(5.242 = kNBy
=yA =yD
13. Method of Consistent Deformations 73
Example 13.15
13. Method of Consistent Deformations 74
Example 13.16
13. Method of Consistent Deformations 75
Homework 5 (10 Points)
Why cannot the method of least work be used for analyzing the effects of support settlements and temperature changes?
13. Method of Consistent Deformations 76
Homework 6 (40 Points)Solve the following problems.
13.1 ~ 13.12 : 2 problems13.13 ~ 13.25 : 2 problems13.26 ~ 13.36 : 2 problems13.37 ~ 13.45 : 2 problems13.46 ~ 13.47 : 1 problem13.48 : 1 problem13.49 ~ 13.61 : 2 problems
12 problems