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

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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

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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

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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

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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

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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

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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

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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

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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.

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Properties of American Standard Shapes

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MECHANICS OF MATERIALS

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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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