1586_C006

6

Shear Stresses in Beams

6.1 Shear Stresses in Rectangular Beams

6.2 Shear Stresses in Circular Beams

6.3 Shear Stresses in the Webs of Beams with Flanges

The loads acting on a beam [Figure 6.1(a)] usually produce both bending moments

M

and shear forces

V

at cross-sections such as

ab

[Figure 6.1(b)]. The longitudinal normal stresses

s

x

associated with thebending moments can be calculated from the flexure formula (see Chapter 5). The transverse shearstresses

t

associated with the shear forces are described in this chapter.Since the formulas for shear stresses are derived from the flexure formula, they are subject to the same

limitations:

1. The beam is symmetric about the

xy

plane and all loads act in this plane (the

plane of bending

).2. The beam is constructed of a linearly elastic material.3. The stress distribution is not disrupted by abrupt changes in the shape of the beam or by discon-

tinuities in loading (

stress concentrations

).

6.1 Shear Stresses in Rectangular Beams

A segment of a beam of rectangular cross-section (width

b

and height

h

) subjected to a vertical shearforce

V

is shown in Figure 6.2(a). We assume that the shear stresses

t

acting on the cross-section areparallel to the sides of the beam and uniformly distributed across the width (although they vary as wemove up or down on the cross-section). A small element of the beam cut out between two adjacent cross-sections and between two planes that are parallel to the neutral surface is shown in Figure 6.2(a) aselement

m n.

Shear stresses acting on one face of an element are always accompanied by complementaryshear stresses of equal magnitude acting on perpendicular faces of the element, as shown in Figure 6.2(b)and Figure 6.2(c). Thus, there are horizontal shear stresses acting between horizontal layers of the beamas well as transverse shear stresses acting on the vertical cross-sections.

The equality of the horizontal and vertical shear stresses acting on element

m n

leads to an interestingconclusion regarding the shear stresses at the top and bottom of the beam. If we imagine that the element

m n

is located at either the top or the bottom, we see that the horizontal shear stresses vanish becausethere are no stresses on the outer surfaces of the beam. It follows that the vertical shear stresses alsovanish at those locations; thus,

t

=

0 where

y

=

h

/2. (Note that the origin of coordinates is at the centroidof the cross-section and the

z

axis is the neutral axis.)

James M. Gere

Stanford University

1586_book.fm Page 1 Friday, May 7, 2004 3:56 PM

The magnitude of the shear stresses can be determined by a lengthy derivation that involves only theflexure formula and static equilibrium (see References). The result is the following

1

formula for the shearstress:

(6.1)

in which

V

is the shear force acting on the cross-section,

I

is the moment of inertia of the cross-sectionalarea about the neutral axis, and

b

is the width of the beam. The integral in Equation (6.1) is the firstmoment of the part of the cross-sectional area below (or above) the level at which the stress is beingevaluated. Denoting this first moment by

Q

, that is,

(6.2)

we can write Equation (6.1) in the simpler form

FIGURE 6.1

Beam with bending moment

M

and shear force

V

acting at cross-section

ab.

FIGURE 6.2

Shear stresses in a beam of rectangular cross-section.

1

Selected material (text and figures) from Chapter 5 of Gere, J. M. and Timoshenko, S. P. 1990.

Mechanics ofMaterials

, 3rd ed. PWS, Boston. With permission.

O O Ma

(a) (b)

b

a

b

y y

x

V

x

m

n

n

z

x

y

h

h

(a)

(b) (c)

m

V

O

t =

V

Iby dA

Q y dA=


(6.3)

This equation, known as the

shear formula

,

can be used to determine the shear stress

t

at any point inthe cross-section of a rectangular beam. Note that for a specific cross-section, the shear force

V

, momentof inertia

I

, and width

b

are constants. However, the first moment

Q

(and hence the shear stress

t

) variesdepending upon where the stress is to be found.

To evaluate the shear stress at distance

y

1

below the neutral axis (Figure 6.3), we must determine thefirst moment

Q

of the area in the cross-section below the level

y

=

y

1

. We can obtain this first momentby multiplying the partial area

A

1

by the distance from its centroid to the neutral axis:

(6.4)

Of course, this same result can be obtained by integration using Equation (6.2):

(6.5)

Substituting this expression for

Q

into the shear formula [Equation (6.3)], we get

(6.6)

This equation shows that the shear stresses in a rectangular beam vary quadratically with the distance

y

1

from the neutral axis. Thus, when plotted over the height of the beam,

t

varies in the manner shown bythe parabolic diagram of Figure 6.3(c). Note that the shear stresses are zero when

y

1

=

h

/2.The maximum value of the shear stress occurs at the neutral axis, where the first moment

Q

has itsmaximum value. Substituting

y

1

=

0 into Equation (6.6), we get

(6.7)

FIGURE 6.3

Distribution of shear stresses in a beam of rectangular cross-section. (a) Side view of beam showingthe shear force

V

and bending moment

M

acting at a cross-section. (b) Cross-section of beam showing shear stresses

t

acting at distance

y

1

from the neutral axis. (c) Diagram showing the parabolic distribution of shear stresses.

h/2h/2

h/2h/2

OM

y y

y1V

x z

max

O

b

(a) (b) (c)

t =

VQ

Ib

y1

Q A y bh

y yh y b h

y= = -

+-

= -

1 1 1 11

2

12

2

2

2 2 4

/

Q y d A yb dyb h

yy

h

= = = -

2 4

2

12

2

1

/

t = -

V

I

hy

2 4

2

12

t max = =Vh

I

V

A

2

8

3

2


in which

A

=

bh

is the cross-sectional area. Thus, the maximum shear stress is 50% larger than the averageshear stress (equal to

V/A

). Note that the preceding equations for the shear stresses can be used to calculateeither vertical shear stresses acting on a cross-section or horizontal shear stresses acting between hori-zontal layers of the beam.

The shear formula is valid for rectangular beams of ordinary proportions; it is exact for very narrowbeams (width

b

much less than height

h

) but less accurate as

b

increases relative to

h

. For instance, when

b

=

h

, the true maximum shear stress is about 13% larger than the value given by Equation (6.7).A common error is to apply the shear formula to cross-sectional shapes, such as a triangle, for which

it is not applicable. The reasons it does not apply to a triangle are:

1. We assumed the cross-section had sides parallel to the

y

axis (so that the shear stresses actedparallel to the

y

axis).2. We assumed that the shear stresses were uniform across the width of the cross-section.

These assumptions hold only in particular cases, including beams of narrow rectangular cross-section.

6.2 Shear Stresses in Circular Beams

When a beam has a circular cross-section (Figure 6.4), we can no longer assume that all of the shearstresses act parallel to the

y

axis. For instance, we can easily demonstrate that at a point on the boundaryof the cross-section, such as point

m

, the shear stress

t

acts tangent to the boundary. This conclusionfollows from the fact that the outer surface of the beam is free of stress, and therefore the shear stressacting on the cross-section can have no component in the radial direction (because shear stresses actingon perpendicular planes must be equal in magnitude).

Although there is no simple way to find the shear stresses throughout the entire cross-section, we canreadily determine the stresses at the neutral axis (where the stresses are the largest) by making somereasonable assumptions about the stress distribution. We assume that the stresses act parallel to the

y

axisand have constant intensity across the width of the beam (from point

p

to point

q

in Figure 6.4). Inasmuchas these assumptions are the same as those used in deriving the shear formula [Equation (6.3)], we canuse that formula to calculate the shear stresses at the neutral axis. For a cross section of radius

r

, we obtain

(6.8)

in which

Q

is the first moment of a semicircle. Substituting these expressions for

I

,

b

, and

Q

in the shearformula, we obtain

FIGURE 6.4

Shear stresses in a beam of circular cross section.

p

y

qz

r

O

m

Ir

b r

Q A yr r r

= =

= =

=

p

p

p

4

1 1

2 34

2

2

4

3

2

3


(6.9)

in which

A

is the area of the cross-section. This equation shows that the maximum shear stress in acircular beam is equal to 4/3 times the average shear stress

V/A

.Although the preceding theory for the maximum shear stress in a circular beam is approximate, it

gives results that differ by only a few percent from those obtained by more exact theories.If a beam has a

hollow circular cross-section

(Figure 6.5), we may again assume with good accuracythat the shear stresses along the neutral axis are parallel to the

y

axis and uniformly distributed. Then,as before, we may use the shear formula to find the maximum shear stress. The properties of the hollowsection are

(6.10)

and the maximum stress is

(6.11)

in which is the area of the cross-section. Note that if

r

1

=

0, this equation reduces toEquation (6.9) for a solid circular beam.

6.3 Shear Stresses in the Webs of Beams with Flanges

When a beam of wide-flange shape [Figure 6.6(a)] is subjected to a vertical shear force, the distributionof shear stresses is more complicated than in the case of a rectangular beam. For instance, in the flangesof the beam, shear stresses act in both the vertical and horizontal directions (the y and z directions).Fortunately, the largest shear stresses occur in the web, and we can determine those stresses using thesame techniques we used for rectangular beams.

Consider the shear stresses at level ef in the web of the beam [Figure 6.6(a)]. We assume that the shearstresses act parallel to the y axis and are uniformly distributed across the thickness of the web. Then theshear formula will still apply. However, the width b is now the thickness t of the web, and the area usedin calculating the first moment Q is the area between ef and the bottom edge of the cross-section [thatis, the shaded area of Figure 6.6(a)]. This area consists of two rectangles the area of the flange (thatis, the area below the line abcd) and the area efcb (note that we disregard the effects of the small fillets

FIGURE 6.5 Shear stresses in a beam of hollow circular cross-section.

Or1

r2

z

y

t

p p

max

( / )

( / ) ( )= = = =

VQ

Ib

V r

r r

V

r

V

A

2 3

4 2

4

3

4

3

3

4 2

I r r b r r Q r r= - = - = -p

42

2

324

14

2 1 23

13( ) ( ) ( )

t max = =+ +

+

VQ

Ib

V

A

r r r r

r r

4

322

2 1 12

22

12

A= r rp( )22

12

-


at the juncture of the web and flange). After evaluating the first moments of these areas and substitutinginto the shear formula, we get the following formula for the shear stress in the web of the beam at distancey1 from the neutral axis:

(6.12)

in which I is the moment of inertia of the entire cross section, t is the thickness of the web, b is the flangewidth, h is the height, and h1 is the distance between the insides of the flanges. The expression for themoment of inertia is

(6.13)

Equation (6.12) is plotted in Figure 6.6(b), and we see that t varies quadratically throughout the heightof the web (from y1 = 0 to y1 = h1/2).

The maximum shear stress in the beam occurs in the web at the neutral axis (y1 = 0), and the minimumshear stress in the web occurs where the web meets the flanges (y1 = h1/2). Thus, we find

(6.14)

For wide-flange beams having typical cross-sectional dimensions, the maximum stress is 10 to 60%greater than the minimum stress. Also, the shear stresses in the web typically account for 90 to 98% ofthe total shear force; the remainder is carried by shear in the flanges.

When designing wide-flange beams, it is common practice to calculate an approximation of themaximum shear stress by dividing the total shear force by the area of the web. The result is an averageshear stress in the web:

(6.15)

For typical beams, the average stress is within 10% (plus or minus) of the actual maximum shear stress.

FIGURE 6.6 Shear stresses in the web of a wide-flange beam. (a) Cross-section of beam. (b) Graph showingdistribution of vertical shear stresses in the web.

h1/2

h1/2

h1/2

h1/2

minmax

b

h z

e

bafc d

y

y1

t

(a) (b)

O

t = = - + -VQ

It

V

Itb h h t h y

842 1

212

12[ ( ) ( )]

Ibh b t h

bh bh th= --

= - +3

13

313

13

12 12

1

12

( )( )

t tmax min( ) ( )= - + = -V

Itbh bh th

Vb

Ith h

8 82

12

12 2

12

t ave =

V

th1


The elementary theory presented in the preceding paragraphs is quite satisfactory for determiningshear stresses in the web. However, when investigating shear stresses in the flanges, we can no longerassume that the shear stresses are constant across the width of the section, that is, across the width b ofthe flanges [Figure 6.6(a)]. For instance, at the junction of the web and lower flange (y1 = h1/2), the widthof the section changes abruptly from t to b. The shear stress at the free surfaces ab and cd [Figure 6.6(a)]must be zero, whereas across the web at bc the stress is tmin. These observations indicate that at thejunction of the web and either flange the distribution of shear stresses is more complex and cannot beinvestigated by an elementary analysis. The stress analysis is further complicated by the use of fillets atthe reentrant corners, such as corners b and c. Without fillets, the stresses would become dangerouslylarge. Thus, we conclude that the shear formula cannot be used to determine the vertical shear stressesin the flanges. (Further discussion of shear stresses in thin-walled beams can be found in the references.)

The method used above to find the shear stresses in the webs of wide-flange beams can also be usedfor certain other sections having thin webs, such as T-beams.

Example

A beam having a T-shaped cross section (Figure 6.7) is subjected to a vertical shear force V = 10,000lb. The cross-sectional dimensions are b = 4 in., t = 1 in., h = 8 in., and h1 = 7 in. Determine theshear stress t1 at the top of the web (level nn) and the maximum shear stress tmax. (Disregard theareas of the fillets.)

Solution

The neutral axis is located by calculating the distance c from the top of the beam to the centroidof the cross-section. The result is

c = 3.045 in.

The moment of inertia I of the cross-sectional area about the neutral axis (calculated with the aidof the parallel-axis theorem) is

I = 69.66 in.4

To find the shear stress at the top of the web we need the first moment Q1 of the area above levelnn. Thus, Q1 is equal to the area of the flange times the distance from the neutral axis to the centroidof the flange:

FIGURE 6.7 Example.

b = 4 in.

h = 8 in.h1 = 7 in.

n n c

z

y

t = 1 in.

O


Substituting into the shear formula, we find

Like all shear stresses in beams, this stress exists both as a vertical shear stress and as a horizontalshear stress. The vertical stress acts on the cross section at level nn and the horizontal stress acts onthe horizontal plane between the flange and the web.

The maximum shear stress occurs in the web at the neutral axis. The first moment Q2 of the areabelow the neutral axis is

Substituting into the shear formula, we obtain

which is the maximum shear stress in the T-beam.

Defining Terms

Shear formula The formula t = VQ/Ib giving the shear stresses in a rectangular beam of linearly elasticmaterial [Equation (6.3)].

(See also Defining Terms for Chapter 5.)

References

Beer, F. P., Johnston, E. R., and DeWolf, J. T. 2001. Mechanics of Materials, 3rd Ed. McGraw-Hill, Inc.,New York.

Gere, J. M. 2001. Mechanics of Materials, 5th Ed. Brooks/Cole, Pacific Grove, CA. Hibbeler, R. C. 2000. Mechanics of Materials, 4th Ed., Prentice Hall, Inc., Upper Saddle River, NJ.Lardner, T. J. and Archer, R. R. 1994. Mechanics of Solids, McGraw-Hill, Inc., New York.Popov, E. P. and Balan, T. A. 1999. Engineering Mechanics of Solids, 2nd Ed., Prentice Hall, Inc., Upper

Saddle River, NJ.

Further Information

Extensive discussions of bending with derivations, examples, and problems can be found intextbooks on mechanics of materials, such as those listed in the References. These books also cover manyadditional topics pertaining to shear stresses in beams. For instance, built-up beams, nonprismatic beams,shear centers, and beams of thin-walled open cross-section are discussed in Gere [2001].

Q A y c1 1 1 4 1 0 5 10 18= = - =( .) ( .) ( . .) . . in in in in3

t11

3

4

10 000 10 18

69 66 11460= = =

VQ

It

( ) ( . . )

( . . ) ( .)

lb in

in in psi

Q A y cc

2 2 2 1 88

212 28= = -

-

=( ) ( ) . in. in. in.

in.3

t max

( . )

( . ) (= = =

VQ

It2 10 000 12 28

69 66 11760

lb) ( in.

in. in.) psi

3

4


ContentsChapter 6Shear Stresses in Beams6.1 Shear Stresses in Rectangular Beams6.2 Shear Stresses in Circular Beams6.3 Shear Stresses in the Webs of Beams with FlangesDefining TermsReferencesFurther Information

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