BdiIII
B
en
di
ng
III
:
Deflection of Beams
MAE 314 –Solid Mechanics
Y. Zhu
Introduction
•Up to now, we have been primarily calculating normal and
shear stresses.
•In this lecture, we will learn how to formulate the deflection
curve(also known as the elastic curve) of a beam to due
transverseloading.
transverse
loading.
Slide 2
Differential Equation of Deflection
ds
d
=
θ
ρ
d
θ
tan
d
y
ds
d
=
θ
ρ
θ
1
=
d
d
dy
d
s
θ
θ
tan
=
dx
y
dx
ρ
d
s
Recall from Ch. 4 that
1/ρis the curvature of
y
dx
θ
θ
cos
=
ds
dx
θ
sin
=
摹
the beam.
θ
sin
=
ds
Slope of the deflection curve
Slide 3
Assumptions
•Assumption 1: θis small.
–
1.
⇒
≅
d
d
dd
θ
θ
≅
1
–
2
⇒
≅
d
x
d
s
dxds
ρ
≅
=
⇒
≅
=
θ
θ
tan
dy
⇒
=
2
ydd
θ
2
1yd
=
–
2
.
•
A
ssum
p
tion 2: Beam is linearl
y
elastic.
⇒
≅
θ
θ
tan
dx
⇒
=
2
dxdx
2
dx
=
ρ
M
1
py
•Thus, the differential equation for the deflection curve is:
EI
=
ρ
EI
M
dx
yd
=
2
2
Slide 4
Diff. Equations for M, V, and w
•Recall :
w
dx
dV
−=
V
dx
dM
=
•So we can write:
w
dx
yd
EI−=
4
4
V
dx
yd
EI=
3
3
M
dx
yd
EI=
2
2
•Deflection curve can be found by integrating
dx
dx
dx
–Bending moment equation (2constants of integration)
–Shearforce equation (3constants of integration)
–Load equation (4constants of integration)
•Chosen method depends on which is more convenient.
Slide 5
Boundary Conditions
•Sometimes a single equation is sufficient for the entire length of the
beam, sometimes it must be divided into sections.
Siitttithillb
t
ttfittif
•
Si
nce we
i
n
t
egra
t
e
t
w
i
ce
th
ere w
ill
b
e
t
wocons
t
an
t
s o
f
i
n
t
egra
ti
on
f
or
each section.
•These can be solved using boundary conditions.
Dflidl
–
D
e
fl
ect
i
ons an
d
s
l
opes at supports
–Known moment and shear conditions
Slide 6
Boundary Conditions cont’d
•Continuity conditions:
–Displacement continuity
Section AC: y
AC(x)Section CB: y
CB(x)
–Slope continuity
)()(C
y
C
y
CBAC
=
)
(
)
(
)
(
)
(
C
C
dy
C
C
dy
CBAC
θ
θ
=
=
=
摹
•Symmetry conditions:
)
(
)
(
)
(
)
(
C
C
dx
C
C
dx
CB
AC
θ
θ
=
=
=
0
=
dx
dy
Slide 7
Example Problem 1
The cantilever beam AB is of uniform cross section and carries a load
P at its free end A. determine the equation of the elastic curve and the
deflection and slop at A (Example 901 in Beer
’
s book (P535))
deflection and slop at A (Example 9
.
01 in Beers book (P535))
.
Slide 8
Slide 9
Example Problem 2
For the beam and loading shown, (a) express the magnitude and
location of the maximum deflection in terms of w
0, L, E, and I.
(b)Calclatethealeofthemaimmdeflectionassmingthatbeam
(b)
Calc
u
late
the
v
al
u
e
of
the
ma
x
im
u
m
deflection
,
ass
u
ming
that
beam
AB is a W18 x 50 rolled shape and that w
0
= 4.5 kips/ft, L = 18 ft, and
E = 29 x 10
6
psi.
Slide 10
Slide 11
Statically Indeterminate Beams
•When there are more reactions than can be solved using
statics, the beam is indeterminate.
Tkdtfbddititlidtit
•
T
a
k
e a
d
van
t
age o
f
b
oun
d
ary con
diti
ons
t
o so
l
ve
i
n
d
e
t
erm
i
na
t
e
problems.
Pr
ob
l
e
m:
obe
Number of reactions: 3 (M
A, Ay
, By)
Number of equations: 2 (ΣM = 0, ΣFy = 0)
x=0, y=0
x=0, θ=0
x=L, y=0
One too many reactions!
Additionally, if we solve for the deflection curve,
we will have two constants of inte
g
ration, which
g
adds two more unknowns!
Solution:Boundary conditions
Slide 12
Statically Indeterminate Beams
Problem:
Number of reactions: 4 (M
A, Ay
, MB, By)
Number of equations: 2 (ΣM = 0, ΣFy = 0)
+2constantsofintegration
x=0, y=0
x=0, θ=0
x=L, y=0
x=0, θ=0
+
2
constants
of
integration
Solution:Boundary conditions
Slide 13
Example Problem 3
•Determine the reactions at the supports for the prismatic
beam shown below
(
Exam
p
le 9.05 in Beer’s book
)
.
(p)
Slide 14
Slide 15
Example Problem 4
For the beam shown determine the reaction at the roller support when
w
0
= 65 kN/m.
0
Slide 16
Slide 17
Method of Superposition
•Deflection and slo
p
e of a beam
p
roduced b
y
multi
p
le loads
ppyp
acting simultaneously can be found by superposingthe
deflections produced by the same loads acting separately.
•Reference Appendix E in Craig’s book (Beam Deflections and
Slopes)
•Method of superposition can be applied to statically
determinate and statically indeterminate beams.
Slide 18
Superposition cont’d
•Consider the
p
roblem on the ri
g
ht.
pg
•Find reactions at A and C.
•Method 1: Choose M
C
and RC
as
redundant.
•
Method2:ChooseM
andM
asredundant
Slide 19
•
Method
2:
Choose
M
C
and
M
A
as
redundant
.
Example Problem 5
For the beam and loading shown, determine (a) the deflection at C, and
(
b
)
the slo
p
e at end A.
()p
Slide 20
Slide 21
Example Problem 6
For the beam shown, determine the reaction at B.
Slide 22
Slide 23
Example Problem 4
The overhanging steel beam ABC carries a concentrated load P at end C.
For portion AB of the beam, (a) derive the equation of the elastic curve,
(b) determine the maximum deflection (c) evaluate y
for the following
(b) determine the maximum deflection
,
(c) evaluate y
max
for the following
data:
W 14x68I = 723 in
4
E = 29x106
psi
P = 50 kipsL = 15 ft = 180 in.a = 4 ft = 48 in.
(sample problem 9.1 in Beer’s book (p. 542))
Slide 24
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