Strength of Materials

3. Stresses in Machine Elements

Lecture Number – 3.1Prof. Dr. C. S. Pathak

Department of Mechanical EngineeringSinhgad College of Engineering, Pune

Strength of Materials

Agenda

• Theory of Simple Bending • Assumptions• Derivation • 1 Illustrative Numerical • 1 Workout Example

Beam Subjected to Pure Bending

Strength of Materials

Assumptions

1. The beam is initially straight and unstressed2. The material is homogeneous and isotropic3. The beam loaded within elastic limit4. Young’s modulus is same in tension and compression5. Plane cross–section remains plane before and after bending6. Each layer of the beam is free to expand or contract7. Radius of curvature is large compared with dimension of cross sections

Strength of Materials

Bending Deformations

• bends uniformly to form a circular arc

• cross-sectional plane passes through arc center and 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 change

• stresses and strains are negative (compressive) above the neutral plane and positive (tension) below it

• member remains symmetric

Strength of Materials

Derivation

• Constant BM is applied• Beam will bend @ O, R• tension, compression

Ref – Strength of Materials by Dr. R. K. Bansal, Laxmi Publications

Strength of Materials

Strain variation along the depth

• Increase in the length of the layer EF =

• Strain in layer EF = • As R is constant, strain in a layer is proportional to its

distance from the neutral axis

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Stress variation

• Let,

• Rewriting

• Variation of strain and stress is linear

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Moment of resistance

• Figure shows cross section of a beam• Small layer at distance y• dA = area of layer• Force on the layer = stress on layer x area of layer = = • Total force on the beam section is obtained by integrating it

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Derivation continued….

• Moment of the force on the layer about N.A.

• Total moment of the forces on the section of the beam

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Bending equation

• Let M = external moment applied on the beam section• For equilibrium moment of resistance = M• M = • The term represents moment of inertia @ NA

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Moment of Inertia

• Moment of inertia is • a measure of the resistance of the section to – applied moment or – load that tends to bend it.

• Moment of inertia depends on shape and not material• It is a derived property

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Loading Pattern

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𝜋

𝜋

𝜋

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Section Modulus

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Illustrative Example

A cast-iron machine part is acted upon by a 3 kN-m couple. Knowing E = 165 GPa and neglecting the effects of fillets, determine

(a) the maximum tensile and compressive stresses,

(b) the radius of curvature.

Ref:- Mechanics of Materials by Beer and Johnston

Strength of Materials

SOLUTION

• Based on the cross section geometry, calculate the location of the section centroid and moment of inertia.

2dAIIA

AyY x

• Apply the elastic flexural formula to find the maximum tensile and compressive stresses.

I

Mcm

• Calculate the curvature

EI

M

R

1

Strength of Materials

SOLUTION

Based on the cross section geometry, calculate the location of the section centroid and moment of inertia.

mm 383000

10114 3

A

AyY

3

3

3

32

101143000

104220120030402

109050180090201

mm ,mm ,mm Area,

AyA

Ayy

49-3

2312123

121

231212

m10868 mm10868

18120040301218002090

I

dAbhdAIIx

Strength of Materials

SOLUTION

49

49

mm10868

m038.0mkN 3mm10868

m022.0mkN 3

I

cM

I

cMI

Mc

BB

AA

m

MPa 0.76A

MPa 3.131B

• Calculate the curvature

49- m10868GPa 165

mkN 3

1

EI

M

R

m 7.47

m1095.201 1-3

RR

• Apply the elastic flexural formula to find the maximum tensile and compressive stresses.