Structural
Displacements
Beam Displacement
1
Structural
Displacements
P
T
Truss
Displacements
Di l
2
The deflections of civil engineerengineer
ing structures under the action of
usual design loads are known to
be small in relation to both the
overall dimensions and member
lengths “Why
lengths.
“Wh bother to compute
comp te
deflections?” Basically, the
design engineer must establish
that the predicted design loads
will not result in large deflections
that may lead to structural failure,
impede serviceability, or result in
an aesthetically displeasing and
distorted structure.
3
Several examples which demonstrate the value of deflection
analysis include (Tartaglione,
1991):
1. Wind forces on tall buildings
have been known to produce
excessive lateral deflections that
have resulted in cracked windows
and walls,
walls as well as discomfort to
the occupants.
2. L
2
Large flfloor d
deflections
fl ti
iin a
building are aesthetically
unattractive do not inspire
unattractive,
confidence, may crack brittle
finishes or cause other damage,
and can be unsafe.
4
3. Floor systems are often
designed to support motordriven machines or sensitive
equipment that will run satisfactorily only if the support system
undergoes limited deflections.
4 L
4.
Large d
deflections
fl ti
on a railway
il
or highway structural support
system may impair ride quality
quality,
cause passenger discomfort,
and be unsafe.
5.Deflection control and camber
pre-stressed conbehavior of p
crete beams during various
stages of construction and loadi are vital
ing
it l ffor a successful
f l
5
design.
6.Deflection computations serve to
establish the vibration and
dynamic characteristics of
structures that must withstand
moving loads, vibration, and
shock environment -- inclusive of
seismic design loads
loads.
Elastic Deformations ≡ structure
d fl ti
deflections
di
disappear and
d th
the
structure regains its original
shape when the actions causing
the deformations are removed.
Permanent deformations of
structures are referred to as
inelastic or plastic deformations.
6
This course will focus on linear
elastic deformations.
deformations Such
deformations vary linearly with
applied
app
ed loads
oads a
and
d tthe
ep
principle
cpeo
of
superposition is valid for such
structures. Furthermore, since
the deflections are expected to be
small, deflections are measured
with respect to the original
original, unde
undeformed or reference geometry.
7
Work-Energy Methods
Work-energy methods for truss,
beam and frame structures are
considered. Such methods are
based on the principle of
conservation of energy,
energy which
states that the work done by a
system of forces applied to a
structure (W) equals the strain
energy stored (U) in the structure.
This statement is based on slowly
applied
pp
loads that do not p
produce
kinetic energy, which can be
written as
W=U
8
A di
disadvantage
d
off work-energy
k
methods is that only one displacement component or
rotation can be computed with
each application.
pp
Work ≡ force (moment) times
displacement (rotation) in the
force (moment) direction
Differential work of Fig. 1 can be
expressed
d as
dW = P (dΔ)
9
Figure 1.
1 Force versus
Displacement Curves
10
For P = F (force), Δ equals
displacement δ:
δ
W=
∫
F dδ
(1)
0
For P = M (moment), Δ equals
rotation θ:
θ
W=
∫
M dθ
(2)
0
Eqs. (1, 2) indicate that work is
simply the area under the force –
di l
displacement
t ((or momentt – rotat
11
tion) diagrams shown in Fig. 1.
Linear Elastic Structure
1
W= Fδ
2
1
W= Mθ
2
Complementary
p
y Work
The area above the loaddisplacement diagrams of Fig
Fig.1
1 is
known as complementary work,
W as shown in Fig 2. For a
linear-elastic system:
1
W =W = PΔ
2
12
Load
Fig. 2. Complementary Work
complementary
work
W
Displacement
Direct use of work-energy
work energy
calculations is only capable of
calculating displacements at the
location of an applied point force
and rotations at the point of
application of a point couple;
obviously a very restrictive
condition. Consequently, virtual
work principles are developed in 13
the subsequent sections.
Virtual Work
Virtual (virtual ≡ imaginary, not
real,, 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
displacement a dummy or virtual
displacement,
load (normally of unit magnitude)
is applied
pp
at the location and in
the direction of the desired
displacement component. Forces
associated
i d with
i h this
hi virtual
i
l fforce
are subscripted with a V.
14
Use of a virtual force in calculating virtual work is defined as the
principle of virtual forces (which
will be tthe
e focus
ocus o
of tthis
s cchapter):
apte )
Principle of Virtual Forces
If a deformable structure is in
equilibrium under a virtual system
of forces,
forces then the external work
done by the virtual forces going
g the real displacements
p
through
equals the internal virtual work
done by the virtual stress
resultants going through the real
displacement differentials.
15
Alternatively, if virtual displacements are applied then the virtual
work is defined as the principle of
virtual
tua displacements:
d sp ace e ts
Principle of Virtual
Displacements
If a deformable structure is in
equilibrium and remains 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.
16
The virtual work principles (forces
and displacements) are based on
conserving the change in energy
due to tthe
e app
applied
ed virtual
tua load
oad o
or
displacement, which can be
expressed mathematically as
WV = UV
ffor the
h principle
i i l off virtual
i
l fforces,
which is the focus of this chapter,
where again the overbar signifies
complementary energy. The real
and virtual complementary
p
y external work is shown schematically in
Fig. 3.
17
Pv = virtual force or moment
Δ = real displacement δ or
rotation θ
P + Pv
P
WV
W
Δ
Fig. 3. Complementary Real
and
d Vi
Virtual
t lW
Works
k
18
Complementary Axial
Strain Energy
For a single
g axial force
member subjected to a real
force F, the complementary
p
y
strain energy (internal work)
is
F
U= δ
2
FL
δ=
EA
2
F L
⇒ U=
2EA
19
For a single axial force
member in equilibrium
subjected to a virtual force
FV, the virtual complementary strain energy (virtual
complementary internal
work) is
U V = FV δ
FL
⇒ U V = FV
EA
Real and virtual complemencomplemen
tary strain energies for a
single member are shown
schematically in Fig. 4.
20
Fig. 4. Complementary Real and
Virtual Strain Energies
g
for
a Single Truss Member
F + Fv
F
UV
U
For a truss structure:
m
m
i=1
i=1
δ
UV = ∑UVi = ∑Fvi δi
21
UVi = Complementary Virtual
Strain Energy for Truss Member i
δi = Real Member i Deformation
Fi Li
δi =
; for a mechanically
Ei Ai
loaded truss member
δi = αi Li ΔTi ; for a thermally
loaded truss member
α = linear
li
coefficient
i i
of
thermal exp ansion
ΔT = change in temperature
22
δi = ΔLf i ; for a fabrication
error of ΔLf in the truss
member
Non-mechanical δi are positive
if they produce a positive
change in member length
consistent with tension
positive forces in truss
members.
23
Example Deflection
Calculation – Mechanically
Loaded Truss Structure
EA = constant
R k
Rocker
Calculate the horizontal displacement at C
C.
24
Example Deflection
Calculation – Thermally
Loaded Truss Structure
EA = constant
EA = constant
α = constant
Calculate the horizontal displacement at G if the top chord members are subjected to a temperature increase of Δ T = 100
100.
Equation of condition at C!
25
Complementary Bending
Strain Energy
M
dU = dθ
2
M
dθ =
dx
EI
1
M
⇒ U = ∫M
dx
2
EI
L
26
dU V = M V dθ
M
⇒ UV = ∫ MV
dx
EI
L
M + Mv
M
dU V
dU
dθ
Fig. 6. Complementary Real and
Virtual Strain Energies for
a Differential Bending
Segment
27
M = reall bbending
di equation
ti d
due
to the external applied loading
M V = M δV for a virtual moment
equation due to a unit force
for a point displacement δ
calculation at the desired
point in the assumed direction of the displacement
= Mθ
V for a virtual moment
equation for a unit virtual
couple for a point rotation θ
calculation at the desired
point in the assumed rotation
28
direction
F a multi-segment
For
li
b
beam:
UV = ∑
∫
i Li
Mi
M Vi
dx
E Ii
where
here i designates beam
segment.
29
Beam Deflection Example
Calculate the tip displacement
and rotation for the cantilever
beam.
30
F a frame
For
f
structure:
m
UV = ∑
Mj
∫ MVj E I j dx
j=1 L j
where m equals the number of
frame members. Note, axial
deformation has been
i
ignored
d. Also,
Al
if a frame
f
member is composed of multisegments then a summation
segments,
over the segments must also be
included.
31
Frame Deflection Example
Calculate
C
l l
the
h vertical
i l di
displacel
ment and rotation at C.
32