ce 382 l7 - deflections
TRANSCRIPT
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StructuralDisplacements
Beam Displacement
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StructuralDisplacements
P
2
Truss Displacements
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ing structures under the action ofusual design loads are known to
be small in relation to both the
overall dimensions and member
. y udeflections? Basically, the
that the predicted design loads
will not result in large deflectionsthat may lead to structural failure,
impede serviceability, or result in
distorted structure.
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Several examples which demon-
analysis include (Tartaglione,1991):
1. Wind forces on tall buildings
have been known to produce
excessive lateral deflections that
have resulted in cracked windows
,
the occupants.
. arge oor e ec ons n a
building are aesthetically
,
confidence, may crack brittlefinishes or cause other damage,
4and can be unsafe.
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3. Floor systems are often
designed to support motor-
driven machines or sensitiveequipment that will run satisfac-
undergoes limited deflections.
. arge e ec ons on a ra wayor highway structural support
,
cause passenger discomfort,
and be unsafe.
5.Deflection control and camber
behavior of re-stressed con-
crete beams during variousstages of construction and load-
5
ng are v a or a success u
design.
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6.Deflection computations serve to
dynamic characteristics ofstructures that must withstand
moving loads, vibration, and
shock environment -- inclusive of
.
Elastic Deformations structuree ec ons sappear an e
structure regains its original
the deformations are removed.
structures are referred to asinelastic or plastic deformations.
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This course will focus on linear.
deformations vary linearly with
a lied loads and the rinci le of
superposition is valid for such
structures. Furthermore, since
the deflections are expected to be
small, deflections are measured
, -
formed or reference geometry.
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Work-Energy Methods
Work-energy methods for truss,beam and frame structures are
considered. Such methods are
based on the principle of
,states that the work done by a
structure (W) equals the strain
energy stored (U) in the structure.
This statement is based on slowly
applied loads that do not produce
kinetic energy, which can bewritten as
8W = U
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A disadvantage of work-energymethods is that only one dis-
rotation can be computed with
each a lication.
Work force (moment) times
displacement (rotation) in the
Differential work of Fig. 1 can be
expresse as
dW = P (d)
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.
Displacement Curves
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ForP = F (force), equals
= F d
ForP = M (moment), equals
=
0
Eqs. (1, 2) indicate that work issimply the area under the force
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sp acemen or momen ro a-
tion) diagrams shown in Fig. 1.
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Linear Elastic Structure
W = 1 F
W =
1
M
Com lementar Work
The area above the load- .
known as complementary work,as shown in Fig 2. For aW
linear-elastic system:112
2
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Fig. 2. Complementary Work
Load complementary
work
W
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Displacement
calculations is only capable of
calculating displacements at thelocation of an applied point force
and rotations at the point of
obviously a very restrictive
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. ,
work principles are developed in
the subsequent sections.
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Virtual Work
Virtual (virtual imaginary, notreal or in essence but not in fact
work procedures can produce a
single displacement component at
any desired location on the
structure. To calculate the desired
,
load(normally of unit magnitude)
is a lied at the location and in
the direction of the desired
displacement component. Forces
associated with this virtual force
are subscripted with a V.
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Use of a virtual force in calcula-
principle of virtual forces (which
will be the focus of this cha ter :
Principle of Virtual ForcesIf a deformable structure is in
equilibrium under a virtual system
,
done by the virtual forces going
throu h the real dis lacements
equals the internal virtual work
done by the virtual stress
resultants going through the real
displacement differentials.
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Alternatively, if virtual displace-
work is defined as the principle of
virtual dis lacements:
Principle of VirtualDisplacementsIf a deformable structure is in
equilibrium while it is subject to a
virtual distortion the external
virtual work done by the external
forces acting on the structure is
equal to the internal virtual work
done by the stress resultants.
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The virtual work principles (forces
conserving the change in energy
due to the a lied virtual load or
displacement, which can be
expressed mathematically as
=V VW U
for the principle of virtual forces,
which is the focus of this chapter,
complementary energy. The realand virtual com lementar exter-
nal work is shown schematically inFig. 3.
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Pv = virtual force or moment
= real displacement orrotation P + Pv
WP
Fig. 3. Complementary Real
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an r ua or s
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Complementary Axial
Strain EnergyFor a sin le axial force
member subjected to a real
force F, the com lementarstrain energy (internal work)
isF
U2
=
F L =
2F L=
192 E A
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For a single axial force
member in equilibriumsubjected to a virtual force
FV, the virtual complemen-
tary strain energy (virtual
complementary internal
work) is
V VU F= F L
V VE A
=
-
tary strain energies for asingle member are shown
20schematically in Fig. 4.
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Fig. 4. Complementary Real and
Virtual Strain Ener ies for
a Single Truss Member
+
F
vUV
U
For a truss structure:
m mV Vi vi iU U F= =
21i 1 i 1= =
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ViU Complementary Virtual=
Strain Energy for Truss Member i
i Real Member i Deformation =
F Lii iE A
=
loaded truss member
linear coefficient of
thermal expansion
=
22T change in temperature =
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=
ferror of L in the truss
member
Non-mechanical i are positive
change in member length
consistent with tensionpositive forces in trussmembers.
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Example Deflection
Loaded Truss Structure
EA = constant
oc er
Calculate the horizontal displace-
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.
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Example Deflection
Loaded Truss Structure
=EA = constant
= constant
Calculate the horizontal displace-
ment at G if the top chord mem-
bers are subjected to a temper-
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.
Equation of condition at C!
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Complementary BendingStrain Energy
MdU d
2=
Md dx
E I =
1 MU M dx = 26
L
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V VdU M d=
V VMU M dxE I
=
M + Mv
MdUV
dU
Fig. 6. Complementary Real andd
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a Differential Bending
Segment
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= rea en ng equa on ueto the external applied loading
VM = for a virtual momentVM
for a point displacement
calculation at the desired
point in the assumed direc-
tion of the displacement
= for a virtual moment
equation for a unit virtualVM
couple for a point rotation
calculation at the desired
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direction
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For a multi-segment beam:
iV Vi
MU M dx=
iL
w
segment.
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Beam Deflection Example
and rotation for the cantilever
beam.
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For a frame structure:
jV Vj
j1
U M dxE I
=
=
where m equals the number of
frame members. Note, axial
deformation has beengnore . so, a rame
member is composed of multi-
,
over the segments must also beincluded.
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Frame Deflection Example
alculate the vertical displace-
ment and rotation at C.
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