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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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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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acting in the same longitudinal plane
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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shear loading to find the complete state of stress.
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
xxσσ
00
indeterminate elementary internal forces.
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∫ =−=
∫
MdAyM xz
xy
σ
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
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 θ′ ( )( )
yyyyLL
yL
θδθρθθρδ
θρ−=−−=−′=
−=′
li l )i( t i
m
x
cρc
yyL
ερε
ρρθε
==
−=−==
or
linearly)ries(strain va
mx
m
cy εε
ερ
−=
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MECHANICS OF MATERIALS
FourthEdition
Beer • Johnston • DeWolf
Stress Due to Bending• For a linearly elastic material,
mxx EyE εεσ −==
linearly) varies(stressm
mxx
cy
c
σ−=
• For static equilibrium,
∫∫ −=== dAydAF mxx σσ0• For static equilibrium,
⎞⎛
∫
∫∫
−= dAyc
dAc
dAF
m
mxx
σ
σσ
0
0( ) ( )
IdAyM
dAcyydAyM
mm
mx
==
⎟⎠⎞
⎜⎝⎛−−=−=
∫
∫∫
σσ
σσ
2
First moment with respect to neutral plane is zero. Therefore, the neutral
f h h hSM
IMc
cdAy
cM
m ==
== ∫
σsurface must pass through the section centroid.
Mycy
mx −= σσ ngSubstituti
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IMy
x −=σ
MECHANICS OF MATERIALS
FourthEdition
Beer • Johnston • DeWolf
Beam Section Properties• The maximum normal stress due to bending,
==SM
IMc
mσ
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.
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 σε
ρ
EI
• Although cross sectional planes remain planar when subjected to bending moments in planewhen subjected to bending moments, in-plane deformations are nonzero,
ννεεννεε yyxzxy =−==−=
ρνεε
ρνεε 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==
′ ρν
ρ
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