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Page 1: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

Chapter Objectives

To develop methods for designing beams to resist both bending and shear loads.

To develop methods for determining the deflection of beams subject to bending and shear loads.

To use beam deflection methods to solve statically indeterminate equilibrium problems.

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Page 2: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

1. Prismatic beam design

2. Fully stressed beam design

In-class Activities

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Page 3: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

PRISMATIC BEAM DESIGN

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• Basis of beam design• Strength concern (i.e. provide safety margin to

normal/shear stress limit)• Serviceability concern (i.e. deflection limit – See

Chapter 12)

• Section strength requirement

allowdreq

MS

max

'

Page 4: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

PRISMATIC BEAM DESIGN (cont)

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• Choices of section:• Steel sections e.g. AISC standard

W 460 X 68

(height = 459 ≈ 460 mm

weight = 0.68 kN/m)

Page 5: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

PRISMATIC BEAM DESIGN (cont)

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• Wood sections

Nominal dimensions (in multiple of 25mm) e.g. 50 (mm) x 100 (mm) actual or “dressed” dimensions are smaller, e.g. 50 x 100 is 38 x 89.

• Built-up sections

Page 6: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

PRISMATIC BEAM DESIGN (cont)

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Procedures:• Shear and Moment Diagram

• Determine the maximum shear and moment in the beam. Often this is done by constructing the beam’s shear and moment diagrams.

• For built-up beams, shear and moment diagrams are useful for identifying regions where the shear and moment are excessively large and may require additional structural reinforcement or fasteners.

Page 7: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

PRISMATIC BEAM DESIGN (cont)

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• Normal Stress– If the beam is relatively long, it is designed by

finding its section modulus using the flexure formula, Sreq’d = Mmax/σallow.

– Once Sreq’d is determined, the cross-sectional dimensions for simple shapes can then be computed, using Sreq’d = I/c.

– If rolled-steel sections are to be used, several possible values of S may be selected from the tables in Appendix B. Of these, choose the one having the smallest cross-sectional area, since this beam has the least weight and is therefore the most economical.

Page 8: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

PRISMATIC BEAM DESIGN (cont)

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• Normal Stress (cont)– Make sure that the selected section modulus, S, is

slightly greater than Sreq’d, so that the additional moment created by the beam’s weight is considered.

• Shear Stress– Normally beams that are short and carry large

loads, especially those made of wood, are first designed to resist shear and then later checked against the allowable-bending-stress requirements.

– Using the shear formula, check to see that the allowable shear stress is not exceeded; that is, use τallow ≥ Vmax Q/It.

Page 9: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

PRISMATIC BEAM DESIGN (cont)

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• Shear Stress (cont)– If the beam has a solid rectangular cross section,

the shear formula becomes τallow ≥ 1.5(Vmax/A), Eq.7-5, and if the cross section is a wide flange, it is generally appropriate to assume that the shear stress is constant over the cross-sectional area of the beam’s web so that τallow ≥ Vmax/Aweb, where Aweb is determined from the product of the beam’s depth and the web’s thickness.

Page 10: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

PRISMATIC BEAM DESIGN (cont)

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• Adequacy of Fasteners– The adequacy of fasteners used on built-up beams

depends upon the shear stress the fasteners can resist. Specifically, the required spacing of nails or bolts of a particular size is determined from the allowable shear flow, qallow = VQ/I, calculated at points on the cross section where the fasteners are located.

Page 11: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 1

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A beam is to be made of steel that has an allowable bending stress of σallow = 170 MPa and an allowable shear stress of τallow = 100 MPa. Select an appropriate W shape that will carry the loading shown in Fig. 11–7a.

Page 12: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 1 (cont)

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• The required section modulus for the beam is determined from the flexure formula,

Solution

33

33

33

33

33

33

mm 10987 100200

mm 10984 80250

mm 101060 74310

mm 101030 64360

mm 101200 67410

mm 101120 60460

SW

SW

SW

SW

SW

SW

Page 13: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 1 (cont)

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• The beam having the least weight per foot is chosen, W460 x 60

• The beam’s weight is

• From Appendix B, for a W460 x 60, d = 455 mm and tw = 8 mm.

• Thus

• Use a W460 x 60.

Solution

kN 55.32.3552681.935.60 W

(OK) MPa 100 MPa 7.24

8455

10900 3max w

avg A

V

Page 14: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 2

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The laminated wooden beam shown in Fig. 11–8a supports a uniform distributed loading of 12 kN/m. If the beam is to have a height-to-width ratio of 1.5, determine its smallest width. The allowable bending stress is 9 MPa and the allowable shear stress is 0.6 MPa. Neglect the weight of the beam.

Page 15: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 2 (cont)

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• Applying the flexure formula,

• Assuming that the width is a, the height is 1.5a.

Solution

36

3max m 00119.0

109

1067.10

allowreq

MS

m 147.0

m 003160.0

75.0

5.100119.00

33

3121

a

a

a

aa

c

ISreq

Page 16: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 2 (cont)

Copyright © 2011 Pearson Education South Asia Pte Ltd

• Applying the shear formula for rectangular sections,

• Since the design fails the shear criterion, the beam must be redesigned on the basis of shear.

• This larger section will also adequately resist the normal stress.

Solution

MPa 6.0929.0

147.05.1147.0

10205.15.1

3max

max A

V

(Ans) mm 183 m 183.0

5.1

10205.1600

5.1

3

max

a

aa

A

Vallow

Page 17: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 3

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The wooden T-beam shown in Fig. 11–9a is made from two 200mm x 30mm boards. If the allowable bending stress is 12 MPa and the allowable shear stress is 0.8 MPa, determine if the beam can safely support the loading shown. Also, specify the maximum spacing of nails needed to hold the two boards together if each nail can safely resist 1.50 kN in shear.

Page 18: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 3 (cont)

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• The reactions on the beam are shown, and the shear and moment diagrams are drawn in Fig. 11–9b.

• The neutral axis (centroid) will be located from the bottom of the beam.

Solution

m 1575.0

2.003.02.003.0

2.003.0251.02.003.01.0

A

Ayy

Page 19: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 3 (cont)

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• Thus

• Since c = 0.1575 m,

Solution

46

23

23

m 10125.60

1575.0215.02.003.003.02.012

1

1.01575.02.003.02.003.012

1

I

(OK) Pa 1024.510125.60

1575.01021012 6

6

39

max

I

cMallow

Page 20: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 3 (cont)

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• Maximum shear stress in the beam depends upon the magnitude of Q and t.

• We will use the rectangular area below the neutral axis to calculate Q, rather than a two-part composite area above this axis, Fig. 11–9c

• So.

Solution

33 m 10372.003.01575.02

1575.0'

AyQ

(OK) Pa 10309

03.010125.60

10372.0105.110800 3

6

333

max

It

QVallow

Page 21: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 3 (cont)

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• From the shear diagram it is seen that the shear varies over the entire span.

• Since the nails join the flange to the web, Fig. 11–9d, we have

• The shear flow for each region is therefore

Solution

3-3 m 100.34503.00.2015.00725.0' AyQ

kN/m 74.510125.60

10345.0101

kN/m 61.810125.60

10345.0105.1

6

33

6

33

I

QVq

I

QVq

CDCD

BCBC

Page 22: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 3 (cont)

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• One nail can resist 1.50 kN in shear, so the maximum spacing becomes

• For ease of measuring, use

Solutions

m 261.074.5

5.1

m 174.061.8

5.1

CD

BC

s

s

(Ans) mm 250

(Ans) mm 150

CD

BC

s

s

Page 23: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

FULLY STRESSED BEAMS

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• The strength requirements are not the same for every cross sections of the beam.

• To optimize the design so as to reduce the beam’s weight, non- prismatic beam is a choice.

• A beam designed in this manner is called a fully stressed beam; e.g.

Page 24: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 4

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Determine the shape of a fully stressed, simply supported beam that supports a concentrated force at its center, Fig. 11-11a.The beam has a rectangular cross section of constant width b, and the allowable stress is σallow.

Page 25: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 4 (cont)

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• The internal moment in the beam, Fig. 11–11b, expressed as a function of position,

• The required section modulus is

• For a cross-sectional area h by b we have

Solution

xP

M2

xPM

Sallowallow 2

xb

Ph

xP

h

bh

c

I

allow

allow

3

22/

2

3121

Page 26: Chapter Objectives To develop methods for designing beams to resist both bending and shear loads. To develop methods for determining the deflection of

EXAMPLE 4 (cont)

Copyright © 2011 Pearson Education South Asia Pte Ltd

• If

• By inspection, the depth h must therefore vary in a parabolic manner with the distance x.

Solution

L/2,at x 0 hh

(Ans) 2

2

3

202

20

xL

hh

b

PLh

allow


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