Bracket Design

21 21.1 21. 1

  Brackets

INTROD INT RODUCT UCTION ION

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 �gurations. The loads may be dynamic with changing directions. Optimal design is thus elusive. Nevertheless Nevert heless,, in this chapte chapter, r, we offer a metho methodolog dology y and design philosophy philosophy for safe, reliab reliable, le, 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 designation design ation of critic critical al dimensions may also be gover governed ned by the elastoplastic elastoplastic response and the local buckling resistance. Thus, experimental stress analysis results may be useful. All these considerations ati ons may be imp import ortant ant in eva evaluat luating ing bra bracke ckets ts and in pre predic dictin ting g the their ir str struct uctural ural saf safety ety and performanc perfo rmance. e. There is, howev however, er, relatively little information information available on brack bracket et design in the open literature. One of the reasons for this, as noted earlier, is the inherent diversity of con�gurations, loading conditions, and safety of individual bracket applications. 21.2 21. 2

TYPES TYP ES OF COMMO COMMON N BRACKE BRACKET T DESIGN DESIGN

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 import imp ortant ant str struct uctura urall fea featur tures es tha thatt aff affect ect the des design ign cho choice ice and met method hodss of str stress ess ana analys lysis. is. The examples selected indicate welded con�gurations, 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�cant 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, �ame 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  d  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 techni tec hniques ques.. The str streng ength th che check ck her heree is per perfor formed med usi using ng a simp simple le beam model und under er ben bendin ding g and shear. Rugged Rug ged bra bracke ckett con constr struct uction ion is ill illust ustrat rated ed in Fig Figure ure 21. 21.3, 3, whe where re hea heavy vy loa loads ds haveto be sup suppor ported ted.. Becaus Bec ausee of the fra frameme-typ typee app appear earanc ancee and mec mechani hanics cs of thi thiss typ typee of a sup suppor port, t, ext extern ernal al loa loadin ding g can be 337

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338

W 

d  T 

 H 

T 

 B

FIGURE 21.1

Shear-type bracket.

W 

T 

d  T 

 H 

 B

FIGURE 21.2

Box-type support bracket.

W 

   T

T 

 B

 H   

FIGURE 21.3

Heavy-duty plate bracket.

W  e

a

T 

H   x

 L

FIGURE 21.4

Tapered-plate bracket.

f 

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339

W=

Top plate

Resultant load

 B

 L

T 

Loaded edge  H 

a 

Free edge of triangular plate FIGURE 21.5

Vertical support edge

T-section bracket.

W 

 B

B

a 

FIGURE 21.6

Double-T section bracket.

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]).

Practical Stress Analysis in Engineering Design

340

W 

 B  a

FIGURE 21.7

Channel-type heavy-duty bracket.

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= 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

p  ffiffi

(21:1)

 ¼ F ( H  þ h)

 M 1

and the tensile stress across the throat section is

 ¼

s 1

p  ffiffi

F  2

(21:2)

 Bh

Combining these expressions gives

p  ffi2ffi M  s   ¼  Bh( H  þ h) 1

(21:3)

1

W  d   F 

h/2

 A h

Transverse h

 H   B  F  h/2

FIGURE 21.8

Example of  �llet weld in bending.

    l   a   n    i     d   u    t    i   g   n   o     l     l   e     l     l   a   r   a    P

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341

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 2

 z

p  ffi2ffi ¼ 6bH 

(21:4)

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

 ¼

s 2

p  ffiffi

3 2M 2

(21:5)

hH 2

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

s 1

(21:6)

 ¼ s 

2

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

Since  M  is  M 1

 ¼ 3 B( H  H þ h) M 

2

(21:7)

2

þ M  , this equation provides an expression for  M  as  3 B( H  þ h)  M  ¼  M  1 þ 2

2

(21:8)

 H 2

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

p  ffiffi s  ¼ ½ þ þ h) 3 2Wd  h H 2 3 B( H 

(21:9)

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

p  ffi2ffi W  t  ¼ 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 

p  ffiffi

¼

p  ffiffi

2P=2 Bh

(21:12)

where the 2 factor is included since in actual welding practice, the effective weld area is generally smaller than 2 Bh.

Practical Stress Analysis in Engineering Design

342

Standard throat section        4  5

h  P 

h

FIGURE 21.9

P 

h 4  5           

  Symmetrical �llet weld in tension.

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

 ¼ h=(sin u þ cos u)

(21:13)

t 

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 

h ¼ sin p =4 sin f

(21:14)

By noting that angle  f  is (p =4

þ u) we see that sin f  is sin f ¼ sin½ p   (p =4 þ u) ¼ sin(p =4 þ u) Assumed outline for the mathematical model

q    

h

Probable outline of the actual weld h  P 

t  

 F s

h  F n  q

FIGURE 21.10

Free-body diagram of  � llet weld.

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343

C 

p /4

h  B

f  t 

q 

p /4

 A

FIGURE 21.11

h

Bracket weld pro�le.

or  sin f

¼ sin p =4 cos u þ cos p =4 sin u ¼

p  ffiffi = 

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,

 ¼ h sin(p =4)= sin f ¼ h=(sin u þ cos u)

t 

(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 P=2

 F   sin u  F   cos u ¼ 0 s

n

(21:17)

and F s cos u

 F   sin u ¼ 0 n

(21:18)

Then, by solving these expressions for  F s  and  F n, we have

 ¼ (P=2) sin u

F s

and

 ¼ (P=2) cos u

F n

 P /2 h  F s

t 

q   F n

Cutting plane FIGURE 21.12

Free-body diagram of a weld segment.

(21:19)

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344

Finally, for a weld length B, the shear and normal stresses on the cutting plane surface of Figure 21.12 are

 ¼ F  =2 Bt 

s s

and

s

 ¼ F  =2 Bt 

s n

(21:20)

n

By substituting for  t ,  F s, and  F n  from Equations 21.16 and 21.19, the stresses become

u þ cos u)  ¼ (P sin u)(sin 2 Bh

(21:21)

u þ cos u)  ¼ (P cos u)(sin 2 Bh

(21:22)

s s and

s n

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 2

 ¼ 6W (d   T )= BT 

(21:23)

 ¼ W = BT 

(21:24)

Wd   ¼ h( H  þ4:324 BH  þ 3 Bh)

(21:25)

s b and

s s The corresponding weld stresses are

s b

2

W  d  T 

 H 

 B

FIGURE 21.13

Shear-type bracket.

T 

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345

W 

T 

d  T 

 H 

 B

FIGURE 21.14

Box-type support bracket.

and 7071W   ¼ 0h:( H  þ h)

s s

(21:26)

Next, consider the box-type bracket of Figure 21.2 and shown again in Figure 21.14. Here, the bending shear stresses are

 ¼  HT ( H  þ3Wd  2 B þ 4T )

(21:27)

 ¼ 2( H W  þ 2T )

(21:28)

 ¼ h½H ( H  þ 4T ) 4þ:243( BWd þ 2T )( H  þ h)

(21:29)

W   ¼ h( H 0:þ7071 2T  þ h)

(21:30)

s b and

s s The corresponding weld stresses are

s b and

s s

For the heavy-duty plate bracket of Figure 21.3 and shown again in Figure 21.15, we can perform a  stress analysis by assuming that the load W  may be resolved into two components acting along the central planes of the plates. The stress is then tensile in the horizontal member and compressive in the inclined member. These stresses are W  sin f  ¼  BT   cos f

(21:31)

 ¼  BT  W  cos f

(21:32)

s t and

s c

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346

W 

   T  B

 H   

FIGURE 21.15

T 

f 

Heavy-duty plate bracket.

The corresponding weld stresses are approximately

 ¼ 0:7071 BhW  tan f

(21:33)

 ¼ 0:5W  Bh tan f

(21:34)

s b and

s s

It is not practical to use large angles f  because the corresponding plate forces become relatively high, as can be seen from the foregoing expressions. In addition, the bracket having high f  loses its frame character of structural behavior and tends to become a cantilevered member for which even small transverse loads can cause substantial bending stresses. A bracket angle  f  of 45  is often selected in practical design. With the typical proportions of  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 8

2

 ¼ 3:62 H  ET 

S cr 

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 highstrength and low-strength materials, respectively. ’

Brackets

347

W  e

a

H 

T 

 x

 L

FIGURE 21.16

21.5

Tapered-plate bracket.

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 2

6WL  ( x   e)  ¼ T ½aL  þ x ( H   a)

s b

2

(21:37)

The distance x  at which the highest bending stresses develop can be found form the condition ds b=d x  0, calculated from Equation 21.37. This yields

¼

¼ e þ (e þ c) =

(21:38)

 H   a) þ aL  ¼ aL ½ 23(( H   a)

(21:39)

 x 

2

1 2

where  c  is

c

2

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

 T 

 ¼ K  E   H 

s Cr 

b

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

Practical Stress Analysis in Engineering Design

348

Compr.

Compr. 1.

2.

3.

Tension 25

Compr.

 H   L

1.

    b

        k

 ,    t   n   e    i   c    i     f     f   e   o   c   g   n    i     l     k   c   u    B

20 15 10 5

2. 3.

0.6 0.8 1.0 1.2 Plate aspect ratio, H / L FIGURE 21.17

Buckling coef �cients for simply supported plates under nonuniform longitudinal stresses.

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



 ¼ K  E h

S Cr 

b

T 

 H 

2

(21:41)

where  h  is

h

 E t

¼

1=2

 E 

(21:42)

and  E t  is

 ¼ ddS «

 E t

(21:43)

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349

W = Resultant

Top plate

  load  B

 L

T 

Loaded edge  H  a 

Free edge of  triangular plate FIGURE 21.18

Vertical support edge

T-section bracket.

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: W ( L   e)  ¼ h( H 4:24 þ 3 HT  þ 3hT )

s b

(21:44)

2

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 ‘‘

W 

 B

 B

a 

FIGURE 21.19

Double-T section bracket.

’’

Practical Stress Analysis in Engineering Design

350

W 

 B a 

FIGURE 21.20

Channel-type heavy-duty bracket.

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 max (0:60 H 

 ¼ S 

W 

TL   0:21 L )  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 compressive strength of the material. The design condition for the critical values of  L =T  can be represented by the following criteria. For 0.5  L = H  1.0,





 180  = T 

 L 

(21:46)

1 2

S y

For 1.0

  L = H   2.0, L   H 

 120 þ  = T  60

 L 

1 2

S y

(21:47)

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



( L  þ 6e)  ¼ W  TL  cos a

S max

2

2

(21:48)

A conservative check on free-edge stability can be made by assuming that the shaded portion of the plate acts as a column with a cross section equal to (TL  cos a)=4 and length equal to  H =cos  a .

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351

W tan a   L/2

W 

e

W /cos a 

Top plate a 

Centrally  located triangular plate

 H   L cos

/4

a 

a 

 L

FIGURE 21.21

Approximate model for a triangular plate bracket.

When a relatively small value of  L =T   must be used, there is little danger of elastic instability and the bracket can be designed to undergo a certain amount of local yielding. For 0.5  L = H  2.0, the recommended  L =T   ratio is



L   H 

 þ 24  = T  48

 L 



(21:49)

1 2

S y

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:

 ¼ TS   cos

W pl

y

2

h

2

a (L

2 1=2

þ 4e )  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  Ar   At  a

Area of cross section Depth of tapered rib Total cross-sectional area  Moment arm; depth of tapered plate

352

a0  B  Br  c d  d 1 d b  E   E 0  E t  e F  F n F s F t   f   H   H e h  I b  I g  I m   I n  I 0  I  x   J  K  K b k   Ro= Ri  L   L g

¼

‘1  M   M 1,  M 2  M c  M F  M 0  M R  M  y m  Ri=T   N  n  H =T  P Q0 q  R  Ri  Ro r  S  S 1, S 2

¼ ¼

Practical Stress Analysis in Engineering Design Mean length of  �ange section Width of bracket  Depth of rib Tapered plate parameter  Distance to loaded point  Mean pipe diameter  Bolt hole diameter  Elastic modulus Reduced modulus of elasticity Tangent modulus Eccentricity of load application Load on weld seam  Normal force component  Shear force component  Width of  �ange cross section Load-sharing ratio Depth of standard �ange; maximum depth of bracket  Equivalent depth of  �ange Thickness of  �ange ring; size of weld leg Second moment of area  Moment of area of a component section Moment of area of wall element of unit width Moment of area of a rib cross section Moment of area of main �ange section Moment of area about central axis First moment of area  Modulus of elastic foundation Buckling coef �cient  Flange ring ratio Length of rib of constant depth; length of bracket  Length of tapered rib Moment arm  General symbol for bending moment  Bending moment components Bending moment per one rib spacing Toroidal moment on �ange ring Discontinuity bending moment  Bending moment on rib Bending moment about longer edge of plate Ratio of inner radius to wall thickness Number of ribs Ratio of  �ange to pipe thickness Tensile load on bracket; edge force on plate Discontinuity shearing force Radial load intensity Radius to bolt circle; mean �ange radius Inner radius of pipe Outer radius of  �ange Mean radius of pipe General symbol for stress Weld stress components

Brackets S bR S c S Cr  S F S n S p S R S s S t  S TR S u S y S max s0 s1 T  t  T r  T 0 V  W  W i W 0 W R W pl  x  Y  Y p Y q  y  Z  a  b  bs d «   «y h u uF up uR m   s , S s b, Sb t  t max f f0 f  ( b, L) v

353

Rib bending stress Compressive stress Critical compressive stress Flange dishing stress Normal stress Plastic stress Radial stress in �ange ring Shear stress Tensile stress; toroidal stress in �ange Total stress in rib Ultimate strength Yield strength Maximum principal stress Wall thickness Depth of section at failure Thickness of pipe; thickness of plate Distance from bolt circle to outer pipe surface; thickness of �llet weld Thickness of rib Thickness of backup ring Moment factor in plate analysis Total bolt load; external load on bracket  Load per inch of pipe circumference Load per inch of bolt circle Tensile load on rib Plastic load on bracket  Arbitrary distance De�ection of beam on elastic foundation Plate edge de�ection under concentrated load Plate edge de�ection under uniform load Coordinate; ring de�ection Section modulus Angle of fractured part; bracket angle, rad Elastic foundation parameter  Shell parameter  Slope of stress–strain curve, rad Strain Uniaxial strain at yield Modulus ratio Angle of twist; angle in weld analysis, rad Angle of twist of main �ange ring, rad Rib half-angle, rad Bending slope at end of rib, rad Poisson s ratio Stress Bending stress Shear stress component  Principal shear stress Plate angle, rad Flange factor  Auxiliary function for a beam on elastic foundation Ratio of rib to �ange moment  ’

354

Practical Stress Analysis in Engineering Design

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. 3. L. Tall, L. S. Beedle, and T. V. Galambos,   Structural Steel Design , Ronald Press, New York, 1964.