Mechanics of Deformable Bodies - Bending

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Jul 18, 2012 (5 years and 3 months ago)

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057:019:BBB Mechanics of Deformable Bodies


College of Engineering

Instructor: C.C. Swan


University of Iowa



1

Period

#
1
2
:

Spring Semester
2011

Section
s

6.1
-
6.2

of Textbook

Topic:
Bending



Objectives:

1.

Overview of Bending Behavior

2.

Sign conventions for shear and bending moments

3.

Relations between
shear and bending moments

4.

Example Problems

1. Overview of Bending Behavior


In ancient days when loads need
ed to be carried over spans,
structures that resulted in
the

material being either entirely in comp
ression, or entirely in tension were built.



With the development of modern materials, spans can now be crossed with structures
that use both tension and compression.
Such
s
tructures
have internal shear forces and bending
moments and are typically called bea
ms or girders.

Since beams are very important
in
civil,
mechanical, biomedical, and aerospace engineering
we will spend a lot of time learning to better
understand them in this course.



In Chapter 6, we will learn about
the
relationships between bending moments
and stresses in the material.



In Chapter 7, we will learn about shear forces and the internal shear stresses.



Toward the end of the course, we will learn in Chapter 12 how to calculate
deflections of beam stru
ctures.



span to be crossed

suspension structure
made of rope or vines

arch structure

made of stone or
brick masonry

057:019:BBB Mechanics of Deformable Bodies


College of Engineering

Instructor: C.C. Swan


University of Iowa



2

2. Sign Conventions for
Shear and
Bending
Moment
s


3. Relations between Shear and Bending Moments


In order to properly
analyze and
design a beam, it is important to know the variation of the
shear and
bending
moment along its axis in order to find the points where these values are a
maximum

and minimum.


a.


Relation between load and shear
















Loads acting
upward

are taken
to be positive

Internal shears on cut faces that tend to
cause counter
-
clockwise rotation are
positive.

Internal bending moments that tend to
cause
compression of the top fibers of a
beam are positive.

V(x)

V(x+dx)

dx

w(x)

)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
0
x
w
dx
dV
x
w
dx
x
V
dx
x
V
dx
x
w
x
V
dx
x
V
dx
x
V
dx
x
w
x
V
F
y












In words, the shear in a beam is obtained by
integrating the
load

function
.

Consider a s
egment of a beam of
infinitesimal length dx as shown:

057:019:BBB Mechanics of Deformable Bodies


College of Engineering

Instructor: C.C. Swan


University of Iowa



3

b.

Relation between shear and moment



















c.


Graphical Methods for Constructing Shear and Moment
Diagrams










Slope of
shear diagram at each poi
nt =
distributed


load intensity at each point









Slope of moment diagram at each point =

shear at each point





M(x)

M(x+dx)

dx

w(x)

)
(
)
(
)
(
)
(
)
(
)
(
)
(
0
)
(
)
(
)
(
0
x
V
dx
dM
x
V
dx
x
M
dx
x
M
dx
x
V
x
M
dx
x
M
dx
x
V
dx
x
M
x
M
M














In words, the
moment

in a beam is obtained by
integrating the
shear
.

V
dx
dM

)
(
x
w
dx
dV

057:019:BBB Mechanics of Deformable Bodies


College of Engineering

Instructor: C.C. Swan


University of Iowa



4






dx
x
w
V
)
(

Change in shear =
area
under distributed
loading







dx
x
V
M
)
(

Change in moment =
area under
shear
diagram


































057:019:BBB Mechanics of Deformable Bodies


College of Engineering

Instructor: C.C. Swan


University of Iowa



5

4.

Examples

Example 1

(6
-
12
)

A reinforced concrete pier is used to
support the stringers for a bridge deck.
Draw the shear and moment diagrams
for the pier when it is subjected to the
stringer loads shown. Assume the
colums at
A
and
B

exert only vertcal
reactions on the pier.









































057:019:BBB Mechanics of Deformable Bodies


College of Engineering

Instructor: C.C. Swan


University of Iowa



6

Example
2


Draw the shear and moment
diagrams for the
pipe. The end
screw is subjected to a horizontal
force of 5kN. Hint: The
reactions at the pin C must be
replaced by equivalent loadings
at point B on the axis of

the pipe.











































057:019:BBB Mechanics of Deformable Bodies


College of Engineering

Instructor: C.C. Swan


University of Iowa



7

Example 3

(6
-
23
)

Draw the shear and moment diagrams for
the beam
. It is supported by a smooth
plate at A which slides within the groove
and so it cannot support a vertical force,
although it can
support a moment and
axial load.