Bracket Design

(1)

21

Brackets

21.

21.1 1 INTINTRODRODUCTUCTIONION

Brackets have manifold applications in structures and machinery. They serve as pipe supports,

motor mounts, connecting joints, fasteners, and seats of various types. [They involve rolled shapes,

plate components, and prefabricated structural elements to meet requirements like strength, rigidity,

appearance, and low manufacturing cost.]

Due to their many applications, brackets have a wide variety of geometry and loading con

Due to their many applications, brackets have a wide variety of geometry and loading con��g-

g-urations. The loads may be dynamic with changing directions. Optimal design is thus elusive.

urations. The loads may be dynamic with changing directions. Optimal design is thus elusive.

Nevert

Neverthelessheless, , in this in this chaptechapter, we r, we offer a offer a methomethodologdology y and design philosophand design philosophy y for safe, for safe, reliabreliable, andle, and

economical designs.

As with the design of all structural components, bracket design criteria must also be based upon

the fundamentals of strength of materials, elasticity theory, and elastic stability. For brackets, the

design

designation of ation of criticcritical dimensions may al dimensions may also be also be govergoverned by ned by the elastoplastthe elastoplastic response and ic response and the localthe local

buckling resistance. Thus, experimental stress analysis results may be useful. All these

consider-ati

ations ons may may be be impimportortant ant in in evaevaluatluating ing brabrackeckets ts and and in in prepredicdictinting g thetheir ir strstructuctural ural safsafety ety andand

perfo

performancrmance. e. There is, There is, howevhowever, relatively little informatier, relatively little information available on on available on brackbracket et design in design in thethe

open literature. One of the reasons for this, as noted earlier, is the inherent diversity of con

open literature. One of the reasons for this, as noted earlier, is the inherent diversity of con��gur-

gur-ations, loading conditions, and safety of individual bracket applications.

ations, loading conditions, and safety of individual bracket applications.

21.

21.2 2 TYPTYPES OF COES OF COMMOMMON BRAN BRACKECKET DESIT DESIGNGN

We focus upon generic designs that are commonly employed in structural systems. Specialized

brackets for a given application are expected to be similar to those described here.

Brackets in general have a bearing plate to distribute a load together with an edge-loaded plate,

or plates, to act as a stiffener or gusset. The bearing and stiffening plates are usually attached or

bonded by welding.

Figures 21.1 through 21.7 show a number of typical bracket designs, as previously documented

by Blake [1]. These designs do not exhaust all possibilities but they illustrate some of the more

imp

importortant ant strstructucturaural l feafeaturtures es thathat t affaffect ect the the desdesign ign chochoice ice and and metmethodhods s of of strstress ess anaanalyslysis. is. TheThe

examples selected indicate welded con

examples selected indicate welded con��gurations, which, with modern fabricating techniques aregurations, which, with modern fabricating techniques are

likely to be reliable and economic. However, this statement is not intended to imply that welding

processes never cause problems. Despite signi

processes never cause problems. Despite signi��cant progress during the past years, strict qualitycant progress during the past years, strict quality

control of welding should be maintained at all times. Fracture-safe design, for instance, can easily be

compromised by a change in material properties in the head-affected zone due to welding,

compromised by a change in material properties in the head-affected zone due to welding, ��ameame

cutting, or other operation.

The mechanical characteristics of the various support brackets can be summarized as follows.

The short bracket shown in Figure 21.1 is made of a standard angle with equal legs. This component

can be designed on the basis of bending and transverse shear. When loading arm

can be designed on the basis of bending and transverse shear. When loading arm d d is relatively is relatively

short, the structural element is rigid and the effect of bending may be neglected.

A box-type support bracket (Figure 21.2) can be made out of two channels using butt-welding

tec

technihniquesques. . The The strstrengength th checheck ck herhere e is is perperforformed med usiusing ng a a simpsimple le beam model beam model undunder er benbendindingg

and shear.

Rug

RuggedgedbrabrackeckettconconstrstructuctionionisisillillustustratratededininFigFigureure21.21.3,3,whewherereheaheavyvyloaloadsdshavetohavetobebesupsupporportedted..

Bec

Becausauseeofofthethefraframe-me-typtypeeappappearearancanceeandandmecmechanihanicscsofofthithisstyptypeeofofaasupsupporport,t,extexternernalalloaloadindinggcancanbebe

337

(2)

B H d W T T

FIGURE 21.1 _{Shear-type bracket.}

B H T T W d

FIGURE 21.2 _{Box-type support bracket.}

B

W

T T

H f

FIGURE 21.3 _{Heavy-duty plate bracket.}

T H W e a x L

(3)

resolved into tensile and compressive forces for design purposes. In this design, the cross sections of the tensile and compression members are large enough to carry substantial loads.

A simple and light construction is illustrated in Figure 21.4. When the plate is relatively long, the bracket must be designed to resist bending, shear, and local buckling loads.

A more conventional type of bracket design is shown in Figure 21.5. This bracket can be made

either by �ame cutting and welding separate plate members or by cutting standard rolled shapes

such as I or T beams.

For larger loads, a double-T con�guration bracket design, shown in Figure 21.6, may be

recommended. The design should be checked, however, for bending effects, shear strength, and stability of the free edges due to the compressive stresses.

Yet another version, shown in Figure 21.7, can be �ame-cut from a standard channel and

welded to the base plate to form a solid unit. The design analysis in this case is similar to that

employed for the con�guration given in Figure 21.6.

21.3 WELD STRESSES

With welding being a bonding agent between the plates forming brackets, it is essential that stresses in the welds be considered in overall stress analyses of brackets. In this section, we present a review

of formulas for calculating welding stresses. The major �ndings are documented by the American

Welding Society. For additional details, refer to welding handbooks, publications of the Welding Research Council, and of texts on materials science (see, for example, Ref. [2]).

T B

W_{= Resultant load} Top

plate Loaded edge L Vertical support edge Free edge of triangular plate H a

FIGURE 21.5 _{T-section bracket.}

B B

a

W

(4)

In reviewing the designs illustrated by Figures 21.1 through 21.7, we see that we have to consider both transverse and parallel welds subjected to bending moments. To examine the

principles involved, consider the case shown in Figure 21.8: for the �llet weld shown, the size of

the weld leg is h. The overall linear dimensions of the weld are B and H for the transverse and

longitudinal welds respectively. The bending moment M 1 on the transverse welds can be imagined

to be a couple consisting of two equal forces F acting at the center of the weld legs, as shown. Since it is standard practice to calculate the stresses on the basis of a weld-throat section, the area on which the component force F is acting must be approximately equal to Bh

=

p

ffiffiffi

2 . This is somewhat conservative because of the additional weld material found at the corner, which is not accounted for in calculating the weld area. Thus, we have

M 1

¼

F ( H

þ

h) (21

:

1)

and the tensile stress across the throat section is

s 1

¼

F

p

ffiffiffi

2

Bh (21

:

2)

Combining these expressions gives

s 1

¼

ffiffiffi

2

p

_M 1 Bh( H

_þ

h) (21

:

3) W a B

FIGURE 21.7 _{Channel-type heavy-duty bracket.}

W h_/2 h_/2 A B Transverse P a r a l l e l l o n g i t u d i n a l d h _h H F F

(5)

The effect of the external load W on the parallel welds can be treated with the help of simple beam theory. The section modulus z of the parallel weld throat is approximately equal to

z

_¼

bH

2

6

p

ffiffiffi

2 (21

:

4)

Since both longitudinal sections are involved in resisting M 2, we have

s 2

¼

3

p

ffiffiffi

2M 2

hH 2 (21

:

5)

As noted earlier, the stress at a common point must be the same for both the transverse and longitudinal welds. That is,

s 1

¼

s 2 (21

:

6)

Then, from Equations 21.1 and 21.5, we obtain a relation between the bending moments as M 1

¼

3 B( H

_þ

h) M 2

H 2 (21

:

7)

Since M is M 1

þ

M 2, this equation provides an expression for M as

M

_¼

M 2 1

þ

3 B( H

_þ

h) H 2





(21

:

8) Finally, from Equations 21.5 and 21.7 and by observing further in Figure 21.8 that M is Wd , we obtain the bending stress s in terms of the load W as

s

_¼

3

ffiffiffi

2

p

_Wd

h H 2

þ

3 B( H

_þ

h)

½



(21

:

9)

Correspondingly, the average shear stress t due to the load W is

t

_¼

ffiffiffi

2

p

_W

2h( H

_þ

h) (21

:

10)

Consider the bracket loaded in tension as in Figure 21.9. At the plate junction (or ‘‘throat ’’), the

nominal stress S on the �llet weld is simply

S

_¼

P

=

2 Bh (21

:

11)

where B is the weld length. For design purposes, it is customary to use the more conservative stress estimate:

S

_¼

p

ffiffiffi

2P

=

2 Bh (21

:

12)

where the

p

ffiffiffi

2factor is included since in actual welding practice, the effective weld area is generally

(6)

To look into this further, consider the double �llet weld represented in Figure 21.10 where a sketch of the probable outline of an actual weld is given. Thus, if the effective area is reduced due to the welding, the thickness of the weld is probably greater than that used in the mathematical model. To explore the theoretical model in more detail, consider an arbitrary section of the weld designated

by the angle u as in Figure 21.10. Let F n and F s be the normal and shear forces on the section

respectively and let t be the thickness of the section as shown. From Figure 21.10, we see that t may be expressed in terms of the weld height h and angle u as

t

_¼

h

=

(sin u

_þ

cos u) (21

:

13)

To see this, consider an enlarged view of, say, the upper weld pro�le of Figure 21.10 as shown in

Figure 21.11. By focusing upon triangle ABC and by using the law of sines we have t

sin p

=

4

¼

h

sin f (21

:

14)

By noting that angle f is (p

=

4

_þ

u) we see that sin f is

sin f

_¼

sin p

_½

_

(p

=

4

_þ

u)

_{ ¼}

sin(p

=

4

_þ

u)

P P

Standard throat section

h

4 5 

FIGURE 21.9 _Symmetrical _�_{llet weld in tension.}

F _s F _n h t h P q q

Probable outline of the actual weld Assumed outline for the

mathematical model

h

(7)

or

sin f

_¼

sin p

=

4 cos u

_þ

cos p

=

4 sin u

_¼



p

ffiffiffi

2

=

2



(cos u

_þ

sin u) (21

:

15) Then, by substituting into Equation 21.14 we have the result of Equation 21.13. That is,

t

_¼

h sin(p

=

4)

=

sin f

_¼

h

=

(sin u

_þ

cos u) (21

:

16)

Next, referring again to Figure 21.10, consider a free-body diagram of the shaded portion of the lower weld of the bracket as in Figure 21.12.

By adding forces horizontally and vertically we have

FIGURE 21.11 _{Bracket weld pro}_�_le.

Cutting plane t q h F _s F _n P _/2

cos u)

2 Bh (21

:

21)

and

s n

¼

(P cos u)(sin u

_þ

cos u)

2 Bh (21

:

22)

An examination of Equations 21.21 and 21.22 shows that at no point of the weld do these theoretical stress values exceed the value estimated by Equation 21.12. Therefore, Equation 21.12 may be viewed as a safe upper bound on the stresses, and that the actual weld stresses are likely to be considerably smaller.

21.4 STRESS FORMULAS FOR VARIOUS SIMPLE BRACKET DESIGNS

Consider the simple shear bracket of Figure 21.1 and shown again in Figure 21.13. This bracket is simple both in design and manufacture. If the line of action of the load is a distance d from the main

plate, the bending (s b) and shear (s s) stresses on the bracket plate may be computed as

3 BH

_þ

3 Bh) (21

:

25) W d T T B H

(9)

and

s s

¼

0

:

_þ

2T ) (21

:

28)

The corresponding weld stresses are

s b

¼

4

:

24Wd h H ( H

_½

plate members in use, Equations 21.31 and 21.32 suf �ce for sizing calculations. However, it

should be appreciated that the compressive member of the bracket can become elastically unstable if its thickness is drastically reduced. Since in the angle brace of Figure 21.4, the two edges of the plate are free to deform, we have the case of buckling of a relatively wide beam subjected to axial compression. Denoting the width and length of this beam by B and H , respectively, and assuming end �xity due to welding, the following expression for the critical buckling stress can be used

S cr

¼

3

:

62 ET 2

H 2 (21

:

35)

This formula is limited to elastic behavior, and therefore the yield strength of the material S y can be

used to determine the maximum allowable value of H

=

T to avoid failure by buckling. The

corresponding critical ratio is

H T

¼

1

:

9 E S y

 

1=2 (21

:

36)

The term E

=

S y may be called the inverse strain parameter because it follows directly from Hooke’s

law. For the conventional metallic materials, the ratio E

=

S y varies between 100 and 1000 for

high-strength and low-high-strength materials, respectively. B T H T f W

(11)

21.5 STRESS AND STABILITY ANALYSES FOR WEB-BRACKET DESIGNS

Consider the tapered plate bracket of Figure 21.4 shown again in Figure 21.16. This design, in effect, is a cantilever plate loaded on edge. The normal stresses on a section, say at x , must vary from

tension to compression. The maximum bending stress s b depends upon the taper. It can be

calculated using the expression

s b

¼

6WL 2( x

_

e)

T aL

_½

_þ

x ( H

_

a)

_

2 (21

:

37)

The distance x at which the highest bending stresses develop can be found form the condition

ds b

=

d x

¼

0, calculated from Equation 21.37. This yields

x

_¼

e

_þ

(e2

_þ

c)1=2 (21

:

38)

where c is

c

_¼

aL 23( H

½



a)

þ

aL



( H

_

a)2 (21

:

39)

The procedure is to compute x from Equations 21.38 and 21.39, and to substitute this value into Equation 21.37 to obtain the maximum stress value. With the usual proportion of brackets found in practice, the aspect ratio H

=

L can be used to make a rough estimate of the relevant buckling coef �cient K b from Figure 21.17. This coef �cient is then used in calculating the critical elastic

stress of the free edge of the bracket using the expression

s Cr

¼

K b E

T H

 

2

(21

:

40) The plate buckling coef �cient K b given in Figure 21.17 can be determined experimentally for

each case of plate proportions, boundary conditions, and type of stress distribution. It represents the tendency of a free edge of the plate element to move toward local instability when the

T H W e a x L

(12)

compressive stresses reach a certain critical value. The consequence of local buckling may then be interpreted in two ways:

1. Overall collapse by rendering the plate element less effective in the postbuckling region of structural response

2. Detrimental stress redistribution in�uencing the load-carrying capacity of the system

The design values given in Figure 21.17 depend largely on the type of stress distribution in compression. Although K b values are sensitive to the type of stress distribution and vary in a

nonlinear fashion, their dependence on the aspect ratio H

=

L is only moderate.

When the actual compressive stress given by Equation 21.37 exceeds that given by Equation 21.40, it is customary to assume that the free edge of the bracket is susceptible to local elastic buckling. To make a conservative allowance for the critical buckling stress in the plastic range, the following set of design formulas may be used

S Cr

¼

K b E h T H

 

2 (21

:

41) where h is h

_¼

E t E

 

1=2 (21

:

42) and E t is E t

¼

dS d« (21

:

43) B u c k l i n g c o e f f i c i e n t , k b 25 1. 2. 3. L H 20 15 10 1. 2. 3. 5 0.6 0.8 1.0 1.2 Tension Compr. Compr. Compr.

Plate aspect ratio, H _/ L

(13)

In Equation 21.43, the terms S and « denote the normal stress and uniaxial strain, respectively.

Therefore, Equation 21.43 de�nes the tangent modulus of the stress

_–

strain characteristics of the

material at a speci�ed level of stress.

The strength of the weld in bending is estimated as follows:

s b

¼

4

:

24W ( L

_

e) h( H 2

þ

3 HT

_þ

3hT ) (21

:

44)

The numerical value of shear stress for this case can be obtained from Equation 21.26.

Figures 21.5 through 21.7 reproduced here as Figures 21.18 through 21.20 show various common designs of tapered plate brackets. In spite of their common use, comparatively few stress formulas for these brackets are available.

The design shown in Figure 21.18 contains two basic elements of structural support: the top support plate which helps to distribute the load; and the triangular plate loaded on edge and designed

to carry the major portion of the load. The two plates acting together form a relatively rigid ‘‘tee’’

con�guration.

Experience indicates that the free edge carries the maximum compressive stress X max, which

depends on the aspect ratio L

=

H . Practical design situations give aspect ratios somewhere between

B

T

Free edge of triangular plate

Vertical support edge

W _{= Resultant}

load _plateTop

Loaded edge

H L

a

FIGURE 21.18 _{T-section bracket.}

W

a

B B

(14)

0.5 and 2.0 [3]. For this particular case, the maximum allowable total load W near the center of the upper plate can be estimated as

W

_¼

S max(0

:

60 H



0

:

21 L )

TL

H (21

:

45)

When the working load W is speci�ed, the maximum corresponding stress S max can be calculated

from Equation 21.45. It is then customary to make S max

<

S y, where S y denotes the yield

compres-sive strength of the material. The design condition for the critical values of L

=

:

47) Equations 21.45 through 21.47 are valid when the resultant load W is located reasonably close to

the center of the top plate and when the yield strength S y is expressed in ksi. However, when this

load moves out toward the edge of the plate, the analytical method described above loses its degree of conservatism and an alternative approach based on the concept of increased eccentricity should be designed. The strength of a welded connection in this design may be checked from Equations 21.33 and 21.34.

Some of the speci�c features of the triangular-plate bracket can be analyzed with reference to

Figure 21.21. The maximum stress at the free edge of the triangular part may be calculated on the

basis of elementary beam theory, by combining the stresses due to the bending moment W

_

e and

the compressive load equal to W

=

cos a. This gives

S max

¼

W ( L

_þ

6e)

TL 2_cos2_a (21

:

 

_HL S 1y=2 (21

:

49)

Here, the yield strength S y is expressed in ksi [3].

The maximum permissible load on the bracket under fully plastic conditions can be calculated from the expression:

W pl

¼

TS y cos2a (L

h

2

þ

4e2)1=2



2e

i

(21

:

50)

The cross-sectional area of the top plate should be designed for the horizontal component of the external load

A

_¼

W pl

tan a S y

(21

:

51) The design formulas given by Equations 21.23 through 21.51 are applicable to various practical situations wherever a particular structure can be modeled as a support bracket similar to one of the

con�gurations illustrated in Figure 21.13 through 21.20. By checking the weld strength, beam

strength, and stability the structural integrity of a bracket can be assured, provided that material and fabrication controls are not compromised.

SYMBOLS

A Area of cross section

Ar Depth of tapered rib

At Total cross-sectional area

a Moment arm; depth of tapered plate

W_tana W _/cosa W L_/2 e L_cosa /4 Top plate H L a Centrally located triangular plate a

(16)

a0 Mean length of �ange section

B Width of bracket

Br Depth of rib

c Tapered plate parameter

d Distance to loaded point

d 1 Mean pipe diameter

d b Bolt hole diameter

E Elastic modulus

E 0 Reduced modulus of elasticity

E t Tangent modulus

e Eccentricity of load application

F Load on weld seam

F n Normal force component

F s Shear force component

F t Width of �ange cross section

f Load-sharing ratio

H Depth of standard�ange; maximum depth of bracket

H e Equivalent depth of �ange

h Thickness of �ange ring; size of weld leg

I b Second moment of area

I g Moment of area of a component section

I m Moment of area of wall element of unit width

I n Moment of area of a rib cross section

I 0 Moment of area of main�ange section

I x Moment of area about central axis

J First moment of area

K Modulus of elastic foundation

K b Buckling coef �cient

k

_¼

Ro

=

Ri Flange ring ratio

L Length of rib of constant depth; length of bracket

L g Length of tapered rib

‘

1 Moment arm

M General symbol for bending moment

M 1, M 2 Bending moment components

M c Bending moment per one rib spacing

M F Toroidal moment on �ange ring

M 0 Discontinuity bending moment

M R Bending moment on rib

M y Bending moment about longer edge of plate

m

_¼

Ri

=

T Ratio of inner radius to wall thickness

N Number of ribs

n

_¼

H

=

T Ratio of �ange to pipe thickness

P Tensile load on bracket; edge force on plate

Q0 Discontinuity shearing force

q Radial load intensity

R Radius to bolt circle; mean�ange radius

Ri Inner radius of pipe

Ro Outer radius of �ange

r Mean radius of pipe

S General symbol for stress

(17)

S bR Rib bending stress

S c Compressive stress

S Cr Critical compressive stress

S F Flange dishing stress

S n Normal stress

S p Plastic stress

S R Radial stress in�ange ring

S s Shear stress

S t Tensile stress; toroidal stress in�ange

S TR Total stress in rib

S u Ultimate strength

S y Yield strength

S max Maximum principal stress

s0 Wall thickness

s1 Depth of section at failure

T Thickness of pipe; thickness of plate

t Distance from bolt circle to outer pipe surface; thickness of �llet weld

T r Thickness of rib

T 0 Thickness of backup ring

V Moment factor in plate analysis

W Total bolt load; external load on bracket

W i Load per inch of pipe circumference

W 0 Load per inch of bolt circle

W R Tensile load on rib

W pl Plastic load on bracket

x Arbitrary distance

Y De�ection of beam on elastic foundation

Y p Plate edge de�ection under concentrated load

Y q Plate edge de�ection under uniform load

y Coordinate; ring de�ection

Z Section modulus

a Angle of fractured part; bracket angle, rad

b Elastic foundation parameter

bs Shell parameter

d Slope of stress

–

strain curve, rad

« Strain

«y Uniaxial strain at yield

h Modulus ratio

u Angle of twist; angle in weld analysis, rad

uF Angle of twist of main�ange ring, rad

up Rib half-angle, rad

uR Bending slope at end of rib, rad

m Poisson’s ratio

s , S Stress

s b, Sb Bending stress

t Shear stress component

t max Principal shear stress

f Plate angle, rad

f0 Flange factor

f ( b, L) Auxiliary function for a beam on elastic foundation

(18)

REFERENCES

1. A. Blake, Design of welded brackets, Machine Design, 1975.

2. D. R. Askeland, The Science and Engineering of Materials , 3rd ed., PWS Publishing Co., Boston, MA, 1989.