Normal stress varies linearly with distance from neutral axis

Elastic flexure formula for maximum normal stress,

I = moment of inertia of cross section

Normal stress at any distance y distance from neutral axis

S = elastic section modulus

section rectanglefor 3

121 bhI

section rectanglefor 2

61 bhI

curvature of the member

S= elastic section modulus

4 - 3

Sample Problem 4.2SOLUTION:

Based on the cross section geometry, calculate

the location of the section centroid ( 𝑌) and

moment of inertia (I).

mm 383000

10114 3

A

AyY

3

3

3

32

101143000

104220120030402

109050180090201

mm ,mm ,mm Area,

AyA

Ayy

49-3

2312123

121

23

1212

m10868 mm10868

18120040301218002090

I

dAbhdAIIx

Find I and 𝑌1. (Ans: I = 301.3 × 10−6 𝑚4 , Y = 170mm 2. Ans: I = 79.7 × 10−6 𝑚4 , Y = 75mm

3. Ans: I = 13.8694 × 10−6 𝑚4 , X = 44.25mm

Y

X

• A 56N.m coupled is applied to the steel bar shown. (a) Assuming that the couple is applied about the z-axis as shown, determine the maximum stress and the radius of curvature of the bar. Use E= 200GPa.[ans: (a) 94.3 MPa, 19.1m]

Analysis and Design of Beam for Bending

• Sign conventions for shear forces V and V’

and bending couples M and M’

• Sign conventions for shear

forces V and V’ and bending

couples M and M’

Sample Problem 5.3

5 - 9

Draw the shear and bending-

moment diagrams for the beam

and loading shown.

SOLUTION:

• 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.

• Apply the relationship between bending

moment and shear to develop the bending-

moment diagram.

Sample Problem 5.3

5 - 10

SOLUTION:

• 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.

dxwdVwdx

dV

zero slope between concentrated loads

linear variation over uniform load segment

Sample Problem 5.3

5 - 11

• Apply the relationship between bending

moment and shear to develop the bending-

moment diagram.

dxVdMVdx

dM

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

• Normal stress is often the critical design criteria

S

M

I

cM

I

Mymx

Requires determination of the location and

magnitude of largest bending moment.

Sample Problem 5.1

5 - 13

• Identify the maximum shear and bending

moment from plots of their distributions.

mkN50kN26 Bmm MMV

• Apply the elastic flexure formulas to

determine the corresponding

maximum normal stress.

36

3

36

2

612

61

m1033.833

mN1050

m1033.833

m250.0m080.0

S

M

hbS

Bm

Pa100.60 6m