Lecture Mechanics of materials (Third edition) - Chapter 8: Principle stresses under a given loading

7, you learned how the components of stress are transformed by a rotation of the coordinate axes and how to determine the principal planes, principal stresses, and maximum shearing str[r]

MECHANICS OF MATERIALS

Ferdinand P Beer

E Russell Johnston, Jr.

John T DeWolf

Lecture Notes:

J Walt Oler Texas Tech University

CHAPTER

Principle Stresses Under a Given

Loading

Introduction Principle Stresses in a Beam Sample Problem 8.1

Sample Problem 8.2 Design of a Transmission Shaft Sample Problem 8.3

Stresses Under Combined Loadings Sample Problem 8.5

• In Chaps 1 and 2, you learned how to determine the normal stress due

to centric loads

In Chap 3, you analyzed the distribution of shearing stresses in a

circular member due to a twisting couple

In Chap 4, you determined the normal stresses caused by bending

couples

In Chaps 5 and 6, you evaluated the shearing stresses due to transverse

loads

In Chap 7, you learned how the components of stress are transformed

by a rotation of the coordinate axes and how to determine the principal planes, principal stresses, and maximum shearing stress

at a point

• Prismatic beam subjected to transverse loading

It

VQ It

Mc I

My

m xy

m x

=

−

=

=

−

=

τ τ

σ σ

• Principal stresses determined from methods

of Chapter 7

• Can the maximum normal stress within the cross-section be larger than

I

Mc

m =

σ

• Cross-section shape results in large values of τxy

near the surface where σx is also large

• σmax may be greater than σm

• Determine shear and bending

moment in Section A-A’

• Calculate the normal stress at top surface and at flange-web junction

A 160-kN force is applied at the end

of a W200x52 rolled-steel beam

Neglecting the effects of fillets and

of stress concentrations, determine

• Evaluate the shear stress at flange-web junction

• Calculate the principal stress at flange-web junction

• Determine shear and bending moment in

Section A-A’

kN 160

m -kN 60 m

375 0 kN 160

=

=

=

A

A

V M

• Calculate the normal stress at top surface and at flange-web junction

MPa 9

102

mm 103

mm 4 90 MPa 2

117

MPa 2

117

m 10 512

m kN 60

3 6

=

=

=

=

×

⋅

=

= −

c

y σ

S M

b a b

A a

σ σ

• Evaluate shear stress at flange-web junction.

MPa 5

95

m 0079

0 m 10 7 52

m 10 6 248 kN

160

m 10 6 248

mm 10

6 248 7

96 6 12 204

4 6

3 6

3 6

3 3

=

×

×

=

=

×

=

×

=

×

=

−

−

−

It

Q V

Q

A b

τ

• Calculate the principal stress at flange-web junction

( )

2

2 2

2

1 2

1 max

⎞

⎛

+ +

= σb σb τb σ

The overhanging beam supports a

uniformly distributed load and a

concentrated load Knowing that for

the grade of steel to used σall = 24 ksi

and τall = 14.5 ksi, select the

wide-flange beam which should be used

SOLUTION:

• Determine reactions at A and D.

• Find maximum shearing stress

• Find maximum normal stress

• Calculate required section modulus and select appropriate beam section

• Determine maximum shear and bending moment from shear and bending moment diagrams