preliminaries: beam deflections virtual workacademic.csuohio.edu/duffy_s/511_06.pdf · section 6:...
TRANSCRIPT
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
1
Preliminaries: Beam Deflections – Virtual Work
There are several methods available to calculate deformations (displacements and rotations)
in beams. They include:
• Formulating moment equations and then integrating to find rotations and
displacements
• Moment area theorems for either rotations and/or displacements
• Virtual work methods
Since structural analysis based on finite element methods is usually based on a potential
energy method, we will tend to use virtual work methods to compute beam deflections.
The theory that supports calculating deflections using virtual work will be reviewed and
several examples are presented.
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
2
Consider the following arbitrarily loaded beam
Identify
actionunit a todue on acting Stress
~
actionunit a todue beam in thesection any at Moment
loads external todue beam in thesection any at Moment
dA
I
ym
m(x)m
M(x)M
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
3
The force acting on the differential area dA due to a unit action is
The stress due to external loads is
The displacement of a differential segment dA by dx along the length of the beam is
dAI
ym
dAf
~~
I
yM
dxIE
yM
dxE
dx
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
4
The work done by the force acting on the differential area dA due to a unit action as the
differential segment of the beam (dA by dx) displaces along the length of the beam by an
amount is
The work done within a differential segment (now A by dx) due to a unit action applied to
the beam is the integration of the expression above with respect to dA, i.e.,
dxdAIE
ymM
dxIE
yMdA
I
ym
fdW
2
2
~
dxEI
MmdxI
EI
Mm
dxdAyEI
MmW
dxdAIE
ymMdW
T
B
T
B
c
c
segmentaldifferneti
c
cA
2
2
2
2
2
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
5
The internal work done along the entire length of the beam due to a unit action applied to
the beam is the integration of the last expression with respect to x, i.e.,
The external work done along the entire length of the beam due to a unit action applied to
the beam is
With
or the deformation (D) of the a beam at the point of application of a unit action (force or
moment) is given by the integral on the right.
dx
EI
xmxMW
L
Internal
0
D 1ExternalW
dxEI
xmxM
dxEI
xmxM
WW
L
L
InternalExternal
D
D
0
0
1
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
Example 6.1
6
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
Example 6.2
7
Flexibility Coefficients by virtual work
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
Perspectives on the Flexibility Method
In 1864 James Clerk Maxwell published the first consistent treatment of the flexibility
method for indeterminate structures. His method was based on considering deflections, but
the presentation was rather brief and attracted little attention. Ten years later Otto Mohr
independently extended Maxwell’s theory to the present day treatment. The flexibility
method will sometimes be referred to in the literature as Maxwell-Mohr method.
With the flexibility method equations of compatibility involving displacements at each of
the redundant forces in the structure are introduced to provide the additional equations
needed for solution. This method is somewhat useful in analyzing beams, frames and
trusses that are statically indeterminate to the first or second degree. For structures with a
high degree of static indeterminacy such as multi-story buildings and large complex trusses
stiffness methods are more appropriate. Nevertheless flexibility methods provide an
understanding of the behavior of statically indeterminate structures.
8
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
The fundamental concepts that underpin the flexibility method will be illustrated by the
study of a two span beam. The procedure is as follows
1. Pick a sufficient number of redundants corresponding to the degree of
indeterminacy
2. Remove the redundants
3. Determine displacements at the redundants on released structure due to external or
imposed actions
4. Determine displacements due to unit loads at the redundants on the released
structure
5. Employ equation of compatibility, e.g., if a pin reaction is removed as a redundant
the compatibility equation could be the summation of vertical displacements in the
released structure must add to zero.
9
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
The beam to the left is statically
indeterminate to the first degree.
The reaction at the middle support
RB is chosen as the redundant.
The released beam is also shown.
Under the external loads the
released beam deflects an amount
DB.
A second beam is considered
where the released redundant is
treated as an external load and the
corresponding deflection at the
redundant is set equal to DB.
LwRB
8
5
10
Example 6.3
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
A more general approach consists in finding the displacement at B caused by a unit load in
the direction of RB. Then this displacement can be multiplied by RB to determine the total
displacement
Also in a more general approach a consistent sign convention for actions and displacements
must be adopted. The displacements in the released structure at B are positive when they are
in the direction of the action released, i.e., upwards is positive here.
The displacement at B caused by the unit action is
The displacement at B caused by RB is δB RB. The displacement caused by the uniform load
w acting on the released structure is
Thus by the compatibility equation
EI
LB
48
3
EI
LwB
384
5 4
D
LwRRB
BBBBB
DD
8
50
11
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
If a structure is statically indeterminate to more
than one degree, the approach used in the
preceeding example must be further organized
and more generalized notation is introduced.
Consider the beam to the left. The beam is
statically indeterminate to the second degree. A
statically determinate structure can be obtained
by releasing two redundant reactions. Four
possible released structures are shown.
12
Example 6.4
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
The redundants chosen are at B and C. The
redundant reactions are designated Q1 and Q2.
The released structure is shown at the left
with all external and internal redundants
shown.
DQL1 is the displacement corresponding to Q1
and caused by only external actions on the
released structure
DQL2 is the displacement corresponding to Q2
caused by only external actions on the
released structure.
Both displacements are shown in their
assumed positive direction.
13
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
We can now write the compatibility equations for this structure. The displacements
corresponding to Q1 and Q2 will be zero. These are labeled DQ1 and DQ2 respectively
In some cases DQ1 and DQ2 would be nonzero then we would write
021211111 QFQFDD QLQ
022212122 QFQFDD QLQ
21211111 QFQFDD QLQ
22212122 QFQFDD QLQ
14
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
where:
{DQ } - vector of actual displacements corresponding to the redundant
{DQL } - vector of displacements in the released structure corresponding to the
redundant action [Q] and due to the loads
[F] - flexibility matrix for the released structure corresponding to the redundant
actions [Q]
{Q} - vector of redundants
2
1
Q
Q
Q D
DD
QFDD QLQ
2
1
QL
QL
QL D
DD
2221
1211
FF
FFF
2
1
Q
The equations from the previous page can be written in matrix format as
15
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
The vector {Q} of redundants can be found by solving for them from the matrix equation
on the previous overhead.
To see how this works consider the previous beam with a constant flexural rigidity EI. If
we identify actions on the beam as
Since there are no displacements imposed on the structure corresponding to Q1 and Q2,
then
QLQ DDQF
QLQ DDFQ 1
PPPPPLMPP 321 2
0
0QD
16
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
The vector [DQL] represents the displacements in the released structure corresponding to
the redundant loads. These displacements are
The positive signs indicate that both displacements are upward. In a matrix format
EI
PLD
EI
PLD QLQL
48
97
24
13 3
2
3
1
97
26
48
3
EI
PLDQL
17
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
The flexibility matrix [F ] is obtained by subjecting the beam to unit load corresponding
to Q1 and computing the following displacements
Similarly subjecting the beam to unit load corresponding to Q2 and computing the
following displacements
EI
LF
EI
LF
6
5
3
3
21
3
11
EI
LF
EI
LF
3
8
6
5 3
22
3
12
18
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
The flexibility matrix is
The inverse of the flexibility matrix is
As a final step the redundants [Q] can be found as follows
165
52
6
3
EI
LF
25
516
7
63
1
L
EIF
64
69
56
97
26
480
0
25
516
7
6 3
3
1
2
1
P
EI
PL
L
EI
DDFQ
QQ QLQ
19
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
The redundants have been obtained. The other unknown reactions can be found from
the released structure. Displacements can be computed from the known reactions on
the released structure and imposing the compatibility equations.
Discuss the following sign conventions and how they relate to one another:
1. Shear and bending moment diagrams
2. Global coordinate axes
3. Sign conventions for actions
- Translations are positive if the follow the direction of the applied force
- Rotations are positive if they follow the direction of the applied moment
20
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
A three span beam shown at the left is
acted upon by a uniform load w and
concentrated loads P as shown. The
beam has a constant flexural rigidity EI.
Treat the supports at B and C as
redundants and compute these
redundants.
In this problem the bending moments at B
and C are chosen as redundants to
indicate how unit rotations are applied to
released structures.
Each redundant consists of two moments,
one acting in each adjoining span.
21
Example 6.5
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
The displacements corresponding to the two redundants consist of two rotations – one for
each adjoining span. The displacement DQL1 and DQL2 corresponding to Q1 and Q2.
These displacements will be caused by the loads acting on the released structure.
The displacement DQL1 is composed of two parts, the rotation of end B of member AB
and the rotation of end B of member BC
Similarly,
EI
PL
EI
wLDQL
1624
23
1
EI
PL
EI
PL
EI
PLDQL
81616
222
2
P
PwL
EI
LDQL
6
32
48
2
such that
22
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
EI
L
EI
L
EI
LF
3
2
3311
EI
LF
621
The flexibility coefficients are determined next. The flexibility coefficient F11 is the sum
of two rotations at joint B. One in span AB and the other in span BC (not shown below)
Similarly the coefficient F21 is equal to the sum of rotations at joint C. However, the
rotation in span CD is zero from a unit rotation at joint B. Thus
23
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
EI
L
EI
L
EI
LF
3
2
3322
EI
LF
612
Similarly
The flexibility matrix is
41
14
6EI
LF
The inverse of the flexibility matrix is
41
14
5
21
L
EIF
24
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
2015
2
1
PLwLQ
PwL
PwLL
P
PwL
EI
L
L
EI
DDFQ
QQ QLQ
212
68
120
6
32
480
0
41
14
5
2 2
1
2
1
40
7
60
2
2
PLwLQ
As a final step the redundants [Q] can be found as follows
and
25
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
26
Example 6.6
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
Joint Displacements, Member End Actions And Reactions
Previously we focused on finding redundants using flexibility (force) methods. Typically
redundants (Q1, Q2, … , Qn) specified by the structural engineer are unknown reactions.
Redundants are determined by imposing displacement continuity at the point in the structure
where redundants are applied, i.e., we imposed
If the redundants specified are unknown reactions then after these redundants are 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.
.27
QFDD QLQ
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
Instead of following the procedure just outlined we will now introduce a systematic
procedure for calculating non-zero joint displacements, reaction, and member end actions
directly using flexibility methods.
Consider the two span beam below where the redundants Q1 and Q2 have been computed
previously in Example 6.4. The non-zero joint displacements DJ1 and DJ2, both rotations, as
well as reactions AR1 and AR2. can be computed. We will focus on the joint displacements
DJ1 and DJ2 first. Keep in mind that when using flexibility methods translations are
associated with forces, and rotations are associated with moments.
Reactions other than redundants will be denoted {AR} and these quantities can be
determined as well. The objective here is the extension of the flexibility (force) method so
that it is more generally applied. 28
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
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 joints B ( = DJ1) and C ( = DJ2) are
required. When the redundants Q1 and Q2 were found superposition was imposed on the
released structure requiring the displacement associated with the unknown redundants to be
equal to zero. In finding joint displacements in the actual structure superposition is used
again and displacements in the released structure are equated to the displacement in the
actual structure. Focusing on joint B, superposition requires
Here
DJ1 = non-zero displacement (a rotation) at joint B in the actual structure, at
the joint associated with Q1
DJL1 = the displacement (a rotation) at joint B associated with DJ1 caused by
the external loads in the released structure.
DJQ11 = the rotation at joint B associated with DJ1 caused by a unit force at
joint B corresponding to the redundant Q1 in the released structure
DJQ12 = the rotation at joint B associated with DJ1 caused by a unit force at joint
C corresponding to the redundant Q2 in the released structure
21211111 QDQDDD JQJQJLJ
29
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
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 are
similarly depicted.
In the figure to the right unit
loads are shown applied at the
redundants. 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
30
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
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 Q2
DJL2 = the displacement (a rotation) at joint C associated with DJ2 caused by the
external loads in the released structure.
DJQ21 = the rotation at joint C associated with DJ2 caused by a unit force at joint B
corresponding to the redundant Q1 in the released structure
DJQ22 = the rotation at joint C associated with DJ2 caused by a unit force at joint C
corresponding to the redundant Q2 in the released structure
22212122 QDQDDD JQJQJLJ
31
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
The expressions DJ1 and DJ2 can be expressed in a matrix format as follows
where
and
which were determined previously
QDDD JQJLJ
2
1
J
J
JD
DD
2
1
JL
JL
JLD
DD
2
1
Q
2221
1211
JQJQ
JQJQ
JQ DD
DDD
32
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
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 action
corresponding to the redundant Q1
ARQ22 = the reaction at A in the released structure due to the unit action
corresponding to the redundant Q2
ARQ12 = the reaction at A in the released structure due to the unit action
corresponding to the redundant Q2
21211111 QAQAAA RQRQRLR
22212122 QAQAAA RQRQRLR
33
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
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
2221
1211
RQRQ
RQRQ
RQ AA
AAA
34
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
In a similar manner we can find member end actions via superposition
For the first expression
AM1 = is the shear force at B on member AB
AML1 = is the shear force at B on member AB caused by the external loads on
the released structure
AMQ11 = is the shear force at B on member AB caused by a unit load
corresponding to the redundant Q1
AMQ12 = is the shear force at B on member AB 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
35
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
The expressions on the previous slide can be expressed in a matrix format as follows
where
QAAA MQMLM
2
1
Q
4
3
2
1
M
M
M
M
M
A
A
A
A
A
4
3
2
1
ML
ML
ML
ML
ML
A
A
A
A
A
4241
3231
2221
1211
MQMQ
MQMQ
MQQM
MQMQ
MQ
AA
AA
AA
AA
A
The sign convention for member end actions is as follows:
+ when up for translations and forces
+ when counterclockwise for rotation and couples 36
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
Example 6.7
PP
PP
PLM
PP
3
2
1 2
Consider the two span beam to the left
where it is assumed that the objective is
to calculate the various joint
displacements DJ , member end actions
AM , and end reactions AR. The beam has
a constant flexural rigidity EI and is acted
upon by the following loads
37
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
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.
38
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
From first moment area theorem
1
2
1 12 1 1.5 0.5
2 2
1
2 2
5
4
JL
PL PLD L L
EI EI
PL PL LL
EI EI
PL
EI
2
2
1 2 1 3 32
2 2 2 2
1
2 2
13
8
JL
PL PL LD L
EI EI
PL PL LL
EI EI
PL
EI
13
10
8
2
EI
PLDJL
39
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
Consider the released beam with a unit load at point B
EI
L
LEI
LDJQ
2
2
1
2
11
EI
L
LEI
LDJQ
2
2
1
2
21
L
40
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
Consider the released beam with a unit load at point C
EI
L
LEI
LDJQ
2
3
122
1
2
12
EI
L
LEI
LDJQ
2
22
2
22
2
1
2L
L
41
2 1 3
1 42JQ
LD
EI
leading to
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
69
6456
PQ
Previously in Example 6.4
with
2 2
2
10 1 3 69
13 1 4 648 2 56
17
5112
J
PL L PD
EI EI
PL
EI
QDDD JQJLJ
then the displacements DJ1 and DJ2 are
42
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
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
43
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
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
2
3
222
2
0
2
2
PLA
PLPLL
PA
M
ML
ML
B
44
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
0
0
3
3
ML
ML
Y
A
PPA
F
0
4 2
4 2
MB
PLA PL
ML
PLA
ML
Using a free body diagram from segment BC of the entire beam, i.e.,
then once again from the equations of equilibrium
45
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
2
0
2
3
0
PL
PL
AML
2
2
RL
P
APL
Thus the vectors AML and ARL are as follows:
Member end actions in the released structure.
Reactions in the released structure.
46
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
Finally with
2
1 1 69
2 64562
107
3156
R
PP
A PLL L
P
L
QAAA RQRLR
then knowing [ARL], [ARQ] and [Q] leads to
47
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
L
LAMQ
0
10
0
11
1 1
2RQA
L L
In a similar fashion, applying a unit load associated with Q1 and Q2 in the previous
cantilever beam, we obtain the following matrices
48
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
0
1 13
0 692
0 0 1 6456
0
2
5
20
6456
36
M
PL
LPA
PL L
LP
L
Similarly, with
and knowing [AML], [AMQ] and [Q] leads to
QAAA MQMLM
49
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
Summary Of Flexibility Method
The analysis of a structure by the flexibility method may be described by the following steps:
1. Problem statement
2. Selection of released structure
3. Analysis of released structure under loads
4. Analysis of released structure for other causes
5. Analysis of released structure for unit values of redundant
6. Determination of redundants through the superposition equations, i.e.,
QFDD QSQ
QRQPQTQLQS DDDDD
QSQ DDFQ
1
50
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
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
51
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
MATRIX ORDER DEFINITION
q x 1 Unknown redundant actions (q = Number of redundant)
q x 1
Displacements in the actual structure Corresponding to the
redundant
q x 1
Displacements in the released structure corresponding to the
redundants and due to loads
q x q
Displacements in the released structure corresponding to the
redundants and due to unit values of the redundants
q x 1
Displacements in the released structure corresponding to the
redundants and due to temperature, prestrain, and restraint
displacements (other than those in DQ)
q x 1
QD
QLD
JQD
QRQPQT DDD ,,
QSD QRQPQTQLQS DDDDD
Q
52
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
MATRIX ORDER DEFINITION
j x 1 Joint displacement in the actual structure (j = number of joint
displacement)
j x 1 Joint displacements in the released structure due to loads
j x 1 Joint displacements in the released structure due to unit values
of the redundants
j x 1
Joint displacements in the released structure due to
temperature, prestrain, and restraint displacements (other than
those in DQ)
j x 1
q x q Matrix of flexibility coefficients
JLD
QLD
JRJPJT DDD ,,
JRJPJTJLJS DDDDD
JD
JSD
53
F
Section 6: The Flexibility Method - Beams
Washkewicz College of Engineering
MATRIX ORDER DEFINITION
m x 1 Member end actions in the actual structure
(m = Number of end-actions)
m x 1 Member end actions in the released structure due to loads
m x q Member end actions in the released structure due to unit
values of the redundants
r x 1 Reactions in the actual structure (r = number of reactions)
r x 1 Reactions in the released structure due to loads
r x q
Reactions in the released structure due to unit values of the
redundants
MLA
RA
MA
RLA
MQA
RQA
54