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MECHANICS OF MATERIALS
Fifth SI Edition
Ferdinand P. BeerE. Russell Johnston, Jr.John T. DeWolfDavid F. Mazurek
Lecture Notes:J. Walt OlerTexas Tech University
CHAPTER
© 2009 The McGraw-Hill Companies, Inc. All rights reserved.
6Shearing Stresses in
Beams and Thin-Walled Members
© 2009 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSFifthEdition
Beer • Johnston • DeWolf • Mazurek
6- 2
Contents
IntroductionShear on the Horizontal Face of a Beam ElementExample 6.01Determination of the Shearing Stress in a BeamShearing Stresses txy in Common Types of BeamsFurther Discussion of the Distribution of Stresses in a ...Sample Problem 6.2Longitudinal Shear on a Beam Element of Arbitrary ShapeExample 6.04Example 6.04Shearing Stresses in Thin-Walled MembersPlastic DeformationsSample Problem 6.3Unsymmetric Loading of Thin-Walled MembersExample 6.05Example 6.06
© 2009 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSFifthEdition
Beer • Johnston • DeWolf • Mazurek
6- 3
Introduction
MyMdAFdAzMVdAF
dAzyMdAF
xzxzz
xyxyy
xyxzxxx
stst
tts
00
00
• Distribution of normal and shearing stresses satisfies
• Transverse loading applied to a beam results in normal and shearing stresses in transverse sections.
• Longitudinal shearing stresses must exist in any member subjected to transverse loading.
• When shearing stresses are exerted on the vertical faces of an element, equal stresses must be exerted on the horizontal faces
© 2009 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSFifthEdition
Beer • Johnston • DeWolf • Mazurek
6- 4
Shear on the Horizontal Face of a Beam Element
• Consider prismatic beam
• For equilibrium of beam element
A
CD
ACDx
dAyIMMH
dAHF
ss0
xVxdxdMMM
dAyQ
CD
A
• Note,
flowshearI
VQxHq
xI
VQH
• Substituting,
© 2009 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSFifthEdition
Beer • Johnston • DeWolf • Mazurek
6- 5
Shear on the Horizontal Face of a Beam Element
• where
section cross full ofmoment second
above area ofmoment first
'
21
AA
A
dAyI
y
dAyQ
• Same result found for lower area
HH
qIQV
xHq
axis neutral torespect h moment witfirst
0
flowshearI
VQxHq
• Shear flow,
© 2009 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSFifthEdition
Beer • Johnston • DeWolf • Mazurek
6- 6
Example 6.01
A beam is made of three planks, nailed together. Knowing that the spacing between nails is 25 mm and that the vertical shear in the beam is V = 500 N, determine the shear force in each nail.
SOLUTION:
• Determine the horizontal force per unit length or shear flow q on the lower surface of the upper plank.
• Calculate the corresponding shear force in each nail.
© 2009 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSFifthEdition
Beer • Johnston • DeWolf • Mazurek
6- 7
Example 6.01
46
2
3121
3121
36
m1020.16
]m060.0m100.0m020.0
m020.0m100.0[2
m100.0m020.0
m10120
m060.0m100.0m020.0
I
yAQ
SOLUTION:
• Determine the horizontal force per unit length or shear flow q on the lower surface of the upper plank.
mN3704
m1016.20)m10120)(N500(
46-
36
IVQq
• Calculate the corresponding shear force in each nail for a nail spacing of 25 mm.
mNqF 3704)(m025.0()m025.0(
N6.92F
© 2009 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSFifthEdition
Beer • Johnston • DeWolf • Mazurek
6- 8
Determination of the Shearing Stress in a Beam
• The average shearing stress on the horizontal face of the element is obtained by dividing the shearing force on the element by the area of the face.
ItVQ
xtx
IVQ
Axq
AH
ave
t
• If the width of the beam is comparable or large relative to its depth, the shearing stresses at D1 and D2 are significantly higher than at D.
• On the upper and lower surfaces of the beam, tyx= 0. It follows that txy= 0 on the upper and lower edges of the transverse sections.
© 2009 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSFifthEdition
Beer • Johnston • DeWolf • Mazurek
6- 9
Shearing Stresses txy in Common Types of Beams
• For a narrow rectangular beam,
AV
cy
AV
IbVQ
xy
23
123
max
2
2
t
t
• For American Standard (S-beam) and wide-flange (W-beam) beams
web
ave
AV
ItVQ
maxt
t
© 2009 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSFifthEdition
Beer • Johnston • DeWolf • Mazurek
6- 15
Sample Problem 6.2
A timber beam is to support the three concentrated loads shown. Knowing that for the grade of timber used,
MPa8.0MPa12 allall ts
determine the minimum required depth d of the beam.
SOLUTION:
• Develop shear and bending moment diagrams. Identify the maximums.
• Determine the beam depth based on allowable normal stress.
• Determine the beam depth based on allowable shear stress.
• Required beam depth is equal to the larger of the two depths found.
© 2009 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSFifthEdition
Beer • Johnston • DeWolf • Mazurek
6- 16
Sample Problem 6.2
SOLUTION:
Develop shear and bending moment diagrams. Identify the maximums.
kNm95.10kN5.14
max
max
MV
© 2009 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSFifthEdition
Beer • Johnston • DeWolf • Mazurek
6- 17
Sample Problem 6.2
2
261
261
3121
m015.0
m09.0
d
d
dbcIS
dbI
• Determine the beam depth based on allowable normal stress.
mm246m246.0
m015.0Nm1095.10Pa 1012 2
36
max
dd
SM
alls
• Determine the beam depth based on allowable shear stress.
mm322m322.0
m0.0914500
23Pa108.0
23
6
max
dd
AV
allt
• Required beam depth is equal to the larger of the two. mm322d
