Lecture Mechanics of materials (Third edition) - Chapter 5: Analysis and design of beams for bending

Lecture Mechanics of materials (Third edition) - Chapter 5: Analysis and design of beams for bending. The following will be discussed in this chapter: Introduction; shear and bending moment diagrams; relations among load, shear, and bending moment; design of prismatic beams for bending.

MECHANICS OF MATERIALS

CHAPTER

Analysis and Design

of Beams for Bending

IntroductionShear and Bending Moment DiagramsSample Problem 5.1

Sample Problem 5.2Relations Among Load, Shear, and Bending MomentSample Problem 5.3

Sample Problem 5.5Design of Prismatic Beams for BendingSample Problem 5.8

• Beams - structural members supporting loads at

various points along the member

• Objective - Analysis and design of beams

• Transverse loadings of beams are classified as

concentrated loads or distributed loads

• Applied loads result in internal forces consisting

of a shear force (from the shear stress distribution) and a bending couple (from the normal stress distribution)

• Normal stress is often the critical design criteria

S

M I

c M I

Classification of Beam Supports

• Determination of maximum normal and shearing stresses requires identification of maximum internal shear force and bending couple.

• Shear force and bending couple at a point are determined by passing a section through the beam and applying an equilibrium analysis on the beam portions on either side of the

section

• Sign conventions for shear forces V and V’

and bending couples M and M’

For the timber beam and loading

shown, draw the shear and

bend-moment diagrams and determine the

maximum normal stress due to

• Section the beam at points near supports and load application points Apply equilibrium analyses on

resulting free-bodies to determine internal shear forces and bending couples

• Apply the elastic flexure formulas to determine the corresponding

maximum normal stress

kN 20 0

kN 20 0

1 1

1

1 1

=

= +

M

V V

F y

(20 kN)(2 5 m) 0 50 kN m 0

kN 20 0

kN 20 0

2 2

2

2 2

⋅

−

=

= +

M

V V

F y

0 kN

14

m kN 28 kN

14

m kN 28 kN

26

m kN 50 kN

26

6 6

5 5

4 4

3 3

=

−

=

⋅ +

=

−

=

⋅ +

= +

=

⋅

−

= +

=

M V

M V

M V

M V

• Identify the maximum shear and moment from plots of their distributions.

bending-m kN 50 kN

3 6

2 6

1 2 6 1

m 10 33 833

m N 10 50

m 10 33 833

m 250 0 m 080 0

h b S

B m

σ

Pa 10 0

60 × 6

=

m

σ

The structure shown is constructed of a

W10x112 rolled-steel beam (a) Draw

the shear and bending-moment diagrams

for the beam and the given loading (b)

determine normal stress in sections just

to the right and left of point D.

SOLUTION:

• Replace the 10 kip load with an

equivalent force-couple system at D Find the reactions at B by considering

the beam as a rigid body

• Section the beam at points near the support and load application points Apply equilibrium analyses on

resulting free-bodies to determine internal shear forces and bending couples

• Apply the elastic flexure formulas to determine the maximum normal

stress to the left and right of point D.

• Replace the 10 kip load with equivalent

force-couple system at D Find reactions at B.

• Section the beam and apply equilibrium analyses on resulting free-bodies

( )3 ( ) 0 1 5 kip ft 0

kips 3

0 3

0

:

2 2

M x

x M

x V

V x F

C to A From y

( 4) 0 (96 24 )kip ft 24

0

kips 24 0

24 0

M x

M

V V

F

D to C From y

(226 34 )kip ft kips

• Apply the elastic flexure formulas to determine the maximum normal stress to

the left and right of point D.

From Appendix C for a W10x112 rolled

steel shape, S = 126 in3 about the X-X axis.

3

3

in 126

in kip 1776

:

in 126

in kip 2016 :

D of right the

To

S M

D of left the To

=

m

σ

( )

x w V

x w V

V V

C

V

w dx

dV

• Relationship between load and shear:

( )

( )2 2

1

0 2 :

0

x w x

V M

x x w x V M M

∆

−

−

∆ +

• Taking the entire beam as a free body,

determine the reactions at A and D.

• Apply the relationship between shear and load to develop the shear diagram

Draw the shear and bending

moment diagrams for the beam

and loading shown

• Apply the relationship between bending moment and shear to develop the bending moment diagram

kips 12 kips 26 kips 12 kips 20 0

0 F

kips 26

ft 28 kips 12 ft

14 kips 12 ft

6 kips 20 ft

24 0

0

y

=

− +

A

A

D D M

• Apply the relationship between shear and load to develop the shear diagram

dx w dV

w dx

- zero slope between concentrated loads

- linear variation over uniform load segment

• Apply the relationship between bending moment and shear to develop the bending moment diagram.

dx V dM V

dx

- bending moment at A and E is zero

- total of all bending moment changes across the beam should be zero

- net change in bending moment is equal to areas under shear distribution segments

- bending moment variation between D

and E is quadratic

- bending moment variation between A, B,

C and D is linear

• Taking the entire beam as a free body,

determine the reactions at C.

• Apply the relationship between shear and load to develop the shear diagram

Draw the shear and bending moment

diagrams for the beam and loading

shown

• Apply the relationship between bending moment and shear to develop the bending moment diagram

• Taking the entire beam as a free body,

determine the reactions at C.

0

0

0 2

1 0

2 1

0 2

1 0

2 1

a L a w M

M

a L a w M

a w R

R a w F

C C

C

C C

y

Results from integration of the load and shear distributions should be equivalent

• Apply the relationship between shear and load

to develop the shear diagram

(area under load curve)

a w V

a

x x w dx

a

x w

V V

B

a a

A B

0

2 0

0

0

2 1

- No change in shear between B and C.

- Compatible with free body analysis

• Apply the relationship between bending moment and shear to develop the bending moment

diagram

2 0 3 1

0

3 2

0 0

2 0

6 2 2

a w M

a

x x

w dx

a

x x w M

M

B

a a

A B

0 6 1

0 2

1 0

2 1

a L w a a L a w M

a L a w dx

a w M

M

C

L a

C B

Results at C are compatible with free-body

analysis

• The largest normal stress is found at the surface where the maximum bending moment occurs.

S

M I

c M

m = max = max

σ

• A safe design requires that the maximum normal stress be less than the allowable stress for the material used This criteria leads to the determination of the minimum

acceptable section modulus

all

all m

M S

σ

σ σ

max min =

≤

• Among beam section choices which have an acceptable section modulus, the one with the smallest weight per unit length or cross sectional area will be the least expensive and the best choice

A simply supported steel beam is to

carry the distributed and concentrated

loads shown Knowing that the

allowable normal stress for the grade

of steel to be used is 160 MPa, select

the wide-flange shape that should be

used

SOLUTION:

• Considering the entire beam as a

free-body, determine the reactions at A and

D.

• Develop the shear diagram for the beam and load distribution From the diagram, determine the maximum bending moment

• Determine the minimum acceptable beam section modulus Choose the best standard section which meets this criteria

• Considering the entire beam as a free-body,

determine the reactions at A and D.

( ) ( )( ) ( )( )

kN 0 52

kN 50 kN 60 kN 0 58 0

kN 0 58

m 4 kN 50 m

5 1 kN 60 m

5 0

=

−

− +

A

A

A F

D

D M

• Develop the shear diagram and determine the maximum bending moment

( )

kN 8

kN 60

kN 0 52

y A

V

curve load

under area

V V

A V

• Maximum bending moment occurs at

V = 0 or x = 2.6 m.

kN 6 67

• Determine the minimum acceptable beam section modulus

3 3

3 6

max min

mm 10

5 422 m

10 5 422

MPa 160

m kN 6 67

σ

• Choose the best standard section which meets this criteria

448 1

46 W200

535 8

44 W250

549 7

38 W310

474 9

32 W360

637 38.8