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MECHANICS OFFourth Edition
MECHANICS OF MATERIALS
CHAPTER
MATERIALSFerdinand P. BeerE. Russell Johnston, Jr.John T
DeWolf
Pure BendingJohn T. DeWolf
Lecture Notes:J. Walt OlerTexas Tech University
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MECHANICS OF MATERIALS
FourthEdition
Beer Johnston DeWolf
Pure BendingPure BendingOther Loading TypesSymmetric Member in
Pure Bending
Example 4.03Reinforced Concrete BeamsSample Problem 4 4Symmetric
Member in Pure Bending
Bending DeformationsStrain Due to BendingBeam Section
Properties
Sample Problem 4.4Stress ConcentrationsPlastic
DeformationsMembers Made of an Elastoplastic MaterialBeam Section
Properties
Properties of American Standard ShapesDeformations in a
Transverse Cross SectionS l P bl 4 2
Members Made of an Elastoplastic MaterialPlastic Deformations of
Members With a Single
Plane of S...Residual StressesSample Problem 4.2
Bending of Members Made of Several Materials
Example 4 03
Residual StressesExample 4.05, 4.06Eccentric Axial Loading in a
Plane of SymmetryExample 4 07Example 4.03
Reinforced Concrete BeamsSample Problem 4.4St C t ti
Example 4.07Sample Problem 4.8Unsymmetric BendingE l 4 08Stress
Concentrations
Plastic DeformationsMembers Made of an Elastoplastic
Material
Example 4.08General Case of Eccentric Axial Loading
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2
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MECHANICS OF MATERIALS
FourthEdition
Beer Johnston DeWolf
Pure Bending
Pure Bending: Prismatic members subjected to equal and opposite
couples
i i h l i di l l
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3
acting in the same longitudinal plane
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MECHANICS OF MATERIALS
FourthEdition
Beer Johnston DeWolf
Other Loading Types
Eccentric Loading: Axial loading which does not pass through
section centroid produces internal forces equivalent to an axial
force and a couple
Transverse Loading: Concentrated or distributed transverse load
producesdistributed transverse load produces internal forces
equivalent to a shear force and a couple
Principle of Superposition: The normal stress due to pure
bending may be p g ycombined with the normal stress due to axial
loading and shear stress due to shear loading to find the complete
state
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4
shear loading to find the complete state of stress.
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MECHANICS OF MATERIALS
FourthEdition
Beer Johnston DeWolf
Symmetric Member in Pure Bending Internal forces in any cross
section are equivalent
to a couple. The moment of the couple is the section bending
momentsection bending moment.
From statics, a couple M consists of two equal and opposite
forces.pp
The sum of the components of the forces in any direction is
zero.
The moment is the same about any axis perpendicular to the plane
of the couple and zero about any axis contained in the plane.
These requirements may be applied to the sums of the components
and moments of the statically i d i l i l f
y p
== ==
dAzMdAF
xy
xx
0
0indeterminate elementary internal forces.
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5
==
MdAyM xz
xy
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MECHANICS OF MATERIALS
FourthEdition
Beer Johnston DeWolf
Bending DeformationsBeam with a plane of symmetry in pure
bending: member remains symmetric
bends uniformly to form a circular arc
cross-sectional plane passes through arc centerand remains
planar
length of top decreases and length of bottom increases
a neutral surface must exist that is parallel to the upper and
lower surfaces and for which the length does not changedoes not
change
stresses and strains are negative (compressive) above the
neutral plane and positive (tension)
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above the neutral plane and positive (tension) below it
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MECHANICS OF MATERIALS
FourthEdition
Beer Johnston DeWolf
Strain Due to Bending
Consider a beam segment of length L.
Aft d f ti th l th f th t lAfter deformation, the length of the
neutral surface remains L. At other sections,
( )L ( )( )
yyyyLL
yL
===
=
li l )i( t i
m
x
cc
yyL
==
===
or
linearly)ries(strain va
mx
m
cy
=
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MECHANICS OF MATERIALS
FourthEdition
Beer Johnston DeWolf
Stress Due to Bending For a linearly elastic material,
mxx EyE ==
linearly) varies(stressm
mxx
cy
c
=
For static equilibrium,
=== dAydAF mxx 0
For static equilibrium,
= dAyc
dAc
dAF
m
mxx
0
0( ) ( )
IdAyM
dAcyydAyM
mm
mx
==
==
2
First moment with respect to neutral plane is zero. Therefore,
the neutral
f h h hSM
IMc
cdAy
cM
m ==
==
surface must pass through the section centroid.
Mycy
mx = ngSubstituti
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8
IMy
x =
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MECHANICS OF MATERIALS
FourthEdition
Beer Johnston DeWolf
Beam Section Properties The maximum normal stress due to
bending,
==SM
IMc
m
modulussection
inertia ofmoment section
==
=IS
ISI
cA beam section with a larger section modulus will have a lower
maximum stress
Consider a rectangular beam cross section,
AhbhbhIS 131
3121
==== Ahbhhc
S 662====
Between two beams with the same cross sectional area the beam
with the greater depthsectional area, the beam with the greater
depth will be more effective in resisting bending.
Structural steel beams are designed to have a
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glarge section modulus.
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MECHANICS OF MATERIALS
FourthEdition
Beer Johnston DeWolf
Properties of American Standard Shapes
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MECHANICS OF MATERIALS
FourthEdition
Beer Johnston DeWolf
Deformations in a Transverse Cross Section Deformation due to
bending moment M is
quantified by the curvature of the neutral surfaceM11
MI
McEcEcc
mm
=
===11
EI
Although cross sectional planes remain planar when subjected to
bending moments in planewhen subjected to bending moments, in-plane
deformations are nonzero,
yy xzxy ====
xzxy
Expansion above the neutral surface and contraction below it
cause an in plane curvaturecontraction below it cause an in-plane
curvature,
curvature canticlasti 1 ==
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