BTE 1013 ENGINEERING SCIENCEs

BTE 1013ENGINEERING SCIENCES

8. SHEAR FORCE AND BENDING MOMENT

NAZARIN B. [email protected]

IntroductionClassification 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’

Shear and Bending Moment Diagrams

Sample Problem 8.1

For the timber beam and loading shown, draw the shear and bend-moment diagrams and determine the maximum normal stress due to bending.

SOLUTION:• Treating the entire beam as a rigid

body, determine the reaction forces

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

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

• 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

Sample Problem 8.1SOLUTION:• Treating the entire beam as a rigid body, determine

the reaction forces kN14kN46:0 from DBBy RRMF

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

00m0kN200

kN200kN200

111

11

MMM

VVFy

mkN500m5.2kN200

kN200kN200

222

22

MMM

VVFy

0kN14

mkN28kN14

mkN28kN26

mkN50kN26

66

55

44

33

MV

MV

MV

MV

Sample Problem 8.1• 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

2612

61

m1033.833mN1050

m1033.833

m250.0m080.0

SM

hbS

Bm

Pa100.60 6m

Sample Problem 8.2

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.

Sample Problem 8.2SOLUTION:• 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.

ftkip5.1030

kips3030:

221

1

xMMxxM

xVVxFCtoAFrom

y

ftkip249604240

kips240240:

2

xMMxM

VVFDtoCFrom

y

ftkip34226kips34:

xMVBtoDFrom

Sample Problem 8.2• 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

in126inkip1776

:in126

inkip2016

:

SM

DofrighttheTo

SM

DoflefttheTo

m

m

ksi0.16m

ksi1.14m

5 - 11

Sample Problem 8.3

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.

5 - 12

Sample Problem 8.3SOLUTION:• Taking the entire beam as a free body, determine the

reactions at A and D.

kips18

kips12kips26kips12kips200

0Fkips26

ft28kips12ft14kips12ft6kips20ft2400

y

y

y

A

A

A

DD

M

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

dxwdVwdxdV

- zero slope between concentrated loads- linear variation over uniform load segment

Sample Problem 8.3• 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

Sample Problem 8.5

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 C.

• 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 8.5SOLUTION:• Taking the entire beam as a free body,

determine the reactions at C.

330

0

021

021

021

021

aLawMMaLawM

awRRawF

CCC

CCy

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

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

curveloadunderareaawV

axxwdx

axwVV

B

aa

AB

021

0

20

00 2

1

- No change in shear between B and C.- Compatible with free body analysis

Sample Problem 8.5• Apply the relationship between bending moment

and shear to develop the bending moment diagram.

203

10

320

0

20 622

awM

axxwdx

axxwMM

B

aa

AB

323 0

061

021

021

aLwaaLawM

aLawdxawMM

C

L

aCB

Results at C are compatible with free-body analysis

5 - 17

Sample Problem 8.8

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.

5 - 18

Sample Problem 8.8• Considering the entire beam as a free-body,

determine the reactions at A and D.

kN0.52

kN50kN60kN0.580kN0.58

m4kN50m5.1kN60m50

y

yy

A

A

AFD

DM

• Develop the shear diagram and determine the maximum bending moment.

kN8

kN60

kN0.52

B

AB

yA

VcurveloadunderareaVV

AV

• Maximum bending moment occurs at V = 0 or x = 2.6 m.

kN6.67

,max

EtoAcurveshearunderareaM

Sample Problem 8.8• Determine the minimum acceptable beam

section modulus.

3336

maxmin

mm105.422m105.422

MPa160mkN6.67

all

MS

• Choose the best standard section which meets this criteria.

4481.46W2005358.44W2505497.38W3104749.32W36063738.8W410

mm10 33

SShape

9.32360W

• Shear force diagram (SFD): Graph of shear force V vs x

• Bending moment diagram (BMD): Graph of bending moment M vs x

Example 8.1: Draw the shear force and bending moment dagrams for the beam shown in Fig. 1.

SOLUTIONS

QUESTION 1

If the beam carries loads at the positions shown in figure, what are the reactive forces at the supports? The weight of the beam may be neglected.

QUESTION 2

If the beam carries loads at the positions shown in figure, what are the reactive forces at the beam? The weight of the beam may be neglected.

QUESTION 3

Determine the shear force and bending moment at points 3.5m and 8.0m from the right-hand end of the beam. (neglect the weight of the beam)

QUESTION 4A beam of length 5.0m and neglect the weight rests on supports at each end and a concentrated load of 255N is applied at its midpoint. Determine the shear force and bending moment at distances from the right-hand end of the beam of a) 1.5mb) 2.4mc) Draw the shear force and bending moment diagram.

QUESTION 5A cantilever has a length of 2m and a concentrated load of 8kN is applied to its free end. Determine the shear force and bending moment at distances of a) 0.5m b) 1.0mc) Draw the shear force and bending moment diagram.(neglect the weight of the beam.

QUESTION 6A beam of length 5.5m supports at each end and a concentrated load of 135N is applied at 2.5m from the left hand end. Determine the shear force and bending moment at distances of;a) 0.8m b) 1.2mc) Draw the shear force and bending moment diagram.(neglect the weight of the beam)

QUIZ

A beam of length 1m supports at each end and a concentrated load of 1.5N is applied at the centre. Determine the shear force and bending moment at distances of;a) 0.25m b) 0.65mc) Draw the shear force and bending moment diagram.(neglect the weight of the beam)

THANK YOU