flexibility method – temp. and pre-strain
TRANSCRIPT
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Lecture 10: Flexibility Method Temp. and Pre-Strain
Effects Of Temperature, Pre-strain & Support Displacement
In the previous sections we have only considered loads acting on the structure. We
would also like to consider the effects of
Temperature changes {DQT}
Prestrain of members {DQP}
These effects are taken in to account by including them in the calculation ofdisplacements (next page) in the released structure in a manner similar to {DQL} The
effects will produce displacements in the released structure, and the displacements are
associated with the redundant actions {Q}in the released structure.
The temperature displacements {DQT} in the released structure may be due to eitheruniform changes in temperature or to differential changes in temperature. A
differential change in temperature assumes that the top and the bottom of the member
changes temperature and thus will undergo a curvature along the axis of the structural
component. A uniform change in temperature will increase or decrease the length of
the structural component.
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Lecture 10: Flexibility Method Temp. and Pre-Strain
When the matrices {DQT } and {DQP}are found they can be added to the matrix {DQL }
of displacements due to loads in order to obtain the sum of all displacements in the
released structure. By superposition
As before the superposition equation is solved for the matrix of redundants {Q}.
Consider the possibility of known displacements occurring at the restraints (or
supports) of the structure. There are two possibilities to consider, depending onwhether the restraint displacements corresponds to one of the redundant actions {Q}.
If the displacement does correspond to a redundant, its effect can be taken into account
by including the displacement in the vector {DQ }.
In a more general situation there will be restraint displacements that do not correspond
to any of the selected redundants. In that event, the effects of restraint displacements
must be incorporated in the analysis of the released structure in a manner similar to
temperature displacements and prestrains. When restraint displacements occur in the
released structure a new matrix {DQR } is introduced.
QFDDDD QPQTQLQ
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Lecture 10: Flexibility Method Temp. and Pre-Strain
Thus the sum of all matrices representing displacements in the released structure will be
denoted by {DQS} and is expressed as follows
QRQPQTQLQS DDDDD
QFDD QSQ
QSQ DDFQ 1
The generalized form of the superposition equation becomes
When this expression is inverted to obtain the redundants we find that
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Lecture 10: Flexibility Method Temp. and Pre-Strain
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Lecture 10: Flexibility Method Temp. and Pre-Strain
Joint Displacements, Member End Actions And Reactions
In previous sections we have focused on finding redundants using flexibility (force) methods.
Typically redundants (Q1
, Q2
, , Qn
) specified by the structural engineer are unknown
reactions. However, we have also stipulated, and found, redundants that are internal actions,
i.e., shear forces, bending moments and/or axial forces. Redundants are determined by
imposing displacement continuity at the point in the structure where redundants are applied.
If the redundants specified are unknown reactions then after these redundants were found
other actions in the released structure could be found using equations of equilibrium.
When all actions in a structure have been determined it is possible to compute displacements
by isolating the individual subcomponents of a structure. Displacements in these
subcomponents can be calculated using concepts learned in Strength of Materials. These
concepts allow us to determine displacements anywhere in the structure but usually the
unknown displacements at the joints are of primary interest if they are non-zero.
Instead of following the procedure just outlined we will now introduce a systematic
procedure for calculating non-zero joint displacements, member end actions and reactions
directly using flexibility methods.
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Lecture 10: Flexibility Method Temp. and Pre-Strain
The principle of superposition is used to obtain the joint displacement vector {DJ}, which is
a vector of displacements that occur in the actual structure. For the structure depicted on
the previous page the rotations in the actual structure at jointsB ( =DJ1) and C( =DJ2) are
required. Initially, when the redundants Q1 and Q2 are found superposition is imposed on
the released structure requiring the displacement associated with the unknown redundantsto be equal to zero. In finding unknown, non-zero, joint displacements in the actual
structure superposition is used again and displacements in the released structure must be
equated to the displacement in the actual structure. Focusing on jointB, superposition
requires
Here
DJ1 = non-zero displacement (a rotation) at jointB in the actual structure, at
the joint associated with Q1
DJL1 = the displacement (a rotation) at jointB associated withDJ1 caused by
the external loads in the released structure.
DJQ11 = the rotation at jointB associated withDJ1 caused by a unit force at
jointB corresponding to the redundant Q1 in the released structure
DJQ12
= the rotation at jointB associated withDJ1
caused by a unit force at joint
Ccorresponding to the redundant Q2 in the released structure
21211111 QDQDDD JQJQJLJ
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Lecture 10: Flexibility Method Temp. and Pre-Strain
Thus displacements in the released structure must be further evaluated for information
beyond that required to find the redundants Q1 and Q2 . In the released structure the
displacements associated with the applied loads are designated {DJL} and are depicted
below. The displacements associated with the redundants are designated [DJQ ] and aresimilarly depicted.
In the figure to the right unit
loads are shown applied at theredundants. These unit loads
were used earlier to find
flexibility coefficients [Fij ].
These coefficients were then
used to determine Q1 and Q2 .Now the unit loads are used to
find the components of [DJQ ].
released structure
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Lecture 10: Flexibility Method Temp. and Pre-Strain
A similar expression can be derived for the rotation at C( =DJ2), i.e.,
Here
DJ2 = non-zero displacement (a rotation) at joint C in the actual structure, at
the joint associated with Q2DJL2 = the displacement (a rotation) at joint C associated withDJ2 caused by the
external loads in the released structure.
DJQ21 = the rotation at joint Cassociated withDJ2 caused by a unit force at jointB
corresponding to the redundant Q1 in the released structure
DJQ22 = the rotation at joint Cassociated withDJ2 caused by a unit force at joint C
corresponding to the redundant Q2 in the released structure
The expressionsDJ1 andDJ2 can be expressed in a matrix format as follows
where
QDDD JQJLJ
2
1
J
J
J D
D
D
2
1
JL
JL
JL D
D
D
2
1
Q
QQ
22212122 QDQDDD JQJQJLJ
2221
1211
JQJQ
JQJQ
JQ DD
DD
D
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Lecture 10: Flexibility Method Temp. and Pre-Strain
In a similar manner we can find member end actions via superposition
For the first expression
AM1 = is the shear force atB on memberAB
AML1 = is the shear force atB on memberAB caused by the external loads on
the released structure
AMQ11 = is the shear force atB on memberAB caused by a unit loadcorresponding to the redundant Q1
AMQ12 = is the shear force atB on memberAB caused by a unit load
corresponding to the redundant Q2
The other expressions follow in a similar manner.
21211111 QAQAAA MQMQMLM
22212122 QAQAAA MQMQMLM
23213133 QAQAAA MQMQMLM
24214144 QAQAAA MQMQMLM
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Lecture 10: Flexibility Method Temp. and Pre-Strain
QAAA MQMLM
2
1
Q
Q
Q
4
3
2
1
M
M
M
M
M
AA
A
A
A
4
3
2
1
ML
ML
ML
ML
ML
AA
A
A
A
4241
3231
2221
1211
MQMQ
MQMQ
MQQM
MQMQ
MQ
AAAA
AA
AA
A
The expressions on the previous slide can be expressed in a matrix format as follows
where
The sign convention for member end actions is as follows:
+ when up for translations and forces+ when counterclockwise for rotation and couples
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Lecture 10: Flexibility Method Temp. and Pre-Strain
In a similar manner we can find reactions via superposition
For the first expression
AR1 = the reaction in the actual beam at A
AR2 = the reaction in the actual beam at A
ARL1 = the reaction in the released structure due to the external loads
ARL2 = the reaction in the released structure due to the external loads
ARQ11 = the reaction at A in the released structure due to the unit actioncorresponding to the redundant Q1ARQ22 = the reaction at A in the released structure due to the unit action
corresponding to the redundant Q2ARQ12 = the reaction at A in the released structure due to the unit action
corresponding to the redundant Q2
21211111 QAQAAA RQRQRLR
22212122 QAQAAA RQRQRLR
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Lecture 10: Flexibility Method Temp. and Pre-Strain
The expressions on the previous slide can be expressed in a matrix format as
where
QAAA RQRLR
2
1
R
R
RA
AA
2
1
RL
RL
RLA
AA
2
1
Q
QQ
2221
1211
RQRQ
RQRQ
RQAA
AAA
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Lecture 10: Flexibility Method Temp. and Pre-Strain
When the effects of joint displacements, member end actions and reactions are accounted for
the only change in the superposition equation is in the first term. Take
here
where
{DJT} = joint displacement due to temperature
{DJP} = joint displacement due to prestrain
{DJR} = joint displacement corresponding to redundants
Hence there is no need to generalize the expression for {AM} and {AR} to account for
temperature effects, prestrain and displacement effects. None of these effects will produce
any actions or reactions in the statically determinate released structure. Instead the released
structure will merely change its configurations to accommodate these effects. The effects of
these influences are merely propagated into matrices {AM} and {AR} through the value of theredundants {Q}, values that account for temperature, prestrain and displacement effects.
JRJPJTJLJS DDDDD
QDDD JQJSJ
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Lecture 10: Flexibility Method Temp. and Pre-Strain
Example
PPPP
PLM
PP
3
2
1 2
Consider the two span beam to the left
where it is assumed that the objective isto calculate the various joint
displacementsDJ, member end actions
AM , and end reactionsAR. The beam has
a constant flexural rigidity EI and is acted
upon by the following loads
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Lecture 10: Flexibility Method Temp. and Pre-Strain
Consider the released
structure and the attending
moment area diagrams.
The (M/EI) diagram was
drawn by parts. Each
action and its attending
diagram is presented one at
a time in the figure starting
with actions on the far
right.
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Lecture 10: Flexibility Method Temp. and Pre-Strain
From first moment area theorem
1
2
1 12 1 1.5 0 .5
2 2
12 2
5
4
JL
P L P LD L L
E I E I
P L P L LLE I E I
P L
E I
2
2
1 2 1 3 32
2 2 2 2
1
2 2
13
8
JL
P L P L LD L
E I E I
P L PL LL
E I E I
P L
E I
13
10
8
2
EI
PL
D JL
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Lecture 10: Flexibility Method Temp. and Pre-Strain
PA
PPPA
F
RL
RL
Y
2
2
0
1
1
2
22
3
22
0
2
2
PLA
LPL
PPLL
PA
M
RL
RL
A
Using the following free body diagram of the released structure
Then from the equations of equilibrium
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Lecture 10: Flexibility Method Temp. and Pre-Strain
0
22
0
1
1
ML
ML
Y
A
PPA
F
Using a free body diagram from segment AB of the entire beam, i.e.,
then once again from the equations of equilibrium
23
222
2
0
2
2
PLA
PLPLL
PA
M
ML
ML
B
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Lecture 10: Flexibility Method Temp. and Pre-Strain
0
0
3
3
ML
ML
Y
A
PPA
F
0
4 2
4 2
MB
P LA P L
M L
P LAM L
Using a free body diagram from segmentBCof the entire beam, i.e.,
then once again from the equations of equilibrium
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Lecture 10: Flexibility Method Temp. and Pre-Strain
2
0
2
3
0
PL
PL
AML
2
2
RL
P
APL
Thus the vectorsAML andARL are as follows:
Member end actions in the released structure.
Reactions in the released structure.
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Lecture 10: Flexibility Method Temp. and Pre-Strain
Consider the released beam with a unit load at pointB
EI
L
LEI
LD JQ
2
2
1
2
11
EI
L
LEI
LD JQ
2
2
1
2
21
L
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Lecture 10: Flexibility Method Temp. and Pre-Strain
Consider the released beam with a unit load at point C
EIL
LEI
LD JQ
23
122
1
2
12
EIL
LEI
LD JQ
2
22
2
22
2
1
2L
L
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Lecture 10: Flexibility Method Temp. and Pre-Strain
2 1 3
1 42
JQ
LD
E I
L
LAMQ
0
10
0
11
1 1
2R QA
L L
Thus
In a similar fashion, applying a unit load associated with Q1 and Q2 in the previous
cantilever beam, we obtain the following matrices
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Lecture 10: Flexibility Method Temp. and Pre-Strain
6 9
6 45 6
PQ
Previously (Lecture #6)
with
2 2
2
1 0 1 3 6 9
1 3 1 4 6 48 2 5 6
1 7
51 1 2
J
P L L PD
E I E I
P L
E I
QDDD JQJLJ
then
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Lecture 10: Flexibility Method Temp. and Pre-Strain
0
1 13
0 6 92
0 0 1 6 45 6
0
2
5
2 0
6 45 6
3 6
M
P L
LPA
P L L
LP
L
Similarly, with
and knowing [AML], [AMQ] and [Q] leads to
QAAA MQMLM
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Lecture 10: Flexibility Method Temp. and Pre-Strain
7. Determine the other displacements and actions. The following are the four flexibility
matrix equations for calculating redundants member end actions, reactions and joint
displacements
where for the released structure
All matrices used in the flexibility method are summarized in the following tables
QDDD JQJSJ
QAAA MQMLM
QAAA
RQRLR
JRJPJTJLJS DDDDD
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Lecture 10: Flexibility Method Temp. and Pre-Strain
MATRIX ORDER DEFINITION
jx1 Jointdisplacementintheactualstructure(j = numberof
joint
displacement)jx1 Jointdisplacementsinthereleasedstructureduetoloads
jx1 Jointdisplacementsinthereleasedstructureduetounit
valuesof
the
redundants
jx1
Jointdisplacementsinthereleasedstructuredueto
temperature,prestrain,andrestraintdisplacements(other
thanthoseinDQ)
jx1
JLD
QL
D
JRJPJT DDD ,,
JRJPJTJLJS DDDDD
JD
JSD
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Lecture 10: Flexibility Method Temp. and Pre-Strain
MATRIX ORDER DEFINITION
mx1 Memberendactionsintheactualstructure
(m = Numberofendactions)
mx1 Member
end
actions
in
the
released
structure
due
to
loads
mxq Memberendactionsinthereleasedstructureduetounit
valuesoftheredundants
rx1 Reactions
in
the
actual
structure
(r
=
numberof
reactions)
rx1 Reactionsinthereleasedstructureduetoloads
rxqReactions
in
the
released
structure
due
to
unit
values
of
the
redundants
MLA
RA
MA
RLA
MQA
RQA