IMPROVED ANALYSIS OF BOLTED SHEAR CONNECTION UNDER ECCENTRIC LOADS

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1 Jornal of Marine Science and Technology, Vol. 5, No. 4, pp (17) 373 DOI: /JMST IMPROVED ANALYSIS OF BOLTED SHEAR ONNETION UNDER EENTRI LOADS Dng-Mya Le 1, heng-yen Liao, hien-hien hang, and Wei-Ting Hs 3 Key words: instantaneos center of rotation, eccentric load, bolted shear connections, ltimate analysis. ABSTRAT Almost all bolted connections are eccentrically loaded. The American Institte of Steel onstrction (AIS) permits the se of elastic and instantaneos center of rotation (I) methods to analyze eccentric bolted connections. The elastic method normally yields relatively conservative designs and the I method, which provides more realistic analyses, is rather complex and tedios. The crrent AIS manal provides tables for determining coefficient, which is sed to obtain the design strength of bolt grop patterns. However, the tables provide vales for only six angles of inclination (, 15, 3, 45, 6, and 75). For other angles, a direct analysis sing the I method mst be condcted. The straight-line interpolation between vales for loads at different angles may be non-conservative and it is not recommended by the AIS. This work develops an iterative algorithm for implementing the tedios I method in the general analysis or design of eccentric bolted connections. To eliminate the tediosness of the I method, a method is proposed to provide a reasonable reslt for all angles (between and 9) inclding those not being considered in the crrent AIS design tables. The proposed method is easy to implement bt reasonably accrate, and replaces both the straight-line interpolation between vales for loads at varios angles and direct analysis. This work eliminates the crrent limitations on AIS design. It provides a qick and reliable tool for preliminary design of eccentric bolted connection. I. INTRODUTION Steel strctres generally have eccentrically loaded joints. Paper sbmitted 6/8/16; revised 1/7/16; accepted /3/17. Athor for correspondence: Wei-Ting Hs ( wths@cyt.ed.tw). 1 Professor of ivil Engineering, National hng Hsing University, Taichng, Taiwan, R.O.. Gradate Stdent, Department of ivil Engineering, National hng Hsing University, Taichng, Taiwan, R.O.. 3 Associate Professor, Department of onstrction Engineering, haoyang University. of Technology, Taichng, Taiwan, R.O.. s G g e x P s G (a) < 75 cases (b) 75 < 9 cases Fig. 1. Loading cases covered and not covered in the AIS manals. Some common examples of sch joints are bracket-type connections and web splices in beams and girders. For a bolted connection, as shown in Fig. 1(a), both the eccentric load and the indced torsion contribte to bolt shear. The effect of the combined load is eqivalent to a rotation of the bolts abot a particlar point, which is called the instantaneos center of rotation (I). The exact position of I is important in the analysis and design of eccentrically loaded joints. The location of an I depends on the pattern of the grop of bolts and the location and direction of the loading. Theoretically, the force eqilibrim eqations are satisfied at the tre I. The American Institte of Steel onstrction (AIS) design manals (1986, 1989, 1993, 1999, 5, and 1) provide two practical methods for evalating the design strength of an eccentrically bolted connection. The first method is essentially an elastic method and is regarded as conservative; the second is based on the concept of the instantaneos center of rotation (I), and is a strength-based method that provides more realistic reslts. In the method of the AIS, the design strength of an eccentrically loaded bolt connection is evalated sing a tablated coefficient, which is proportional to the reqired strength of the bolt grop. The AIS allowable stress design (ASD) manal (1989) contains the coefficients for only vertical eccentric loads. Iwankiw (1987) proposed an approximate method to handle bolted connections nder eccentric and inclined loading. The AIS load and resistance factor design (LRFD) manals (1986, 1993, 1999, 5, and 1) provide the coefficients for only six inclination angles of the load (, 15, 3, 45, 6, and 75), which were evalated sing the I method. Design engineers tend to interpolate linearly the coefficient for a nonspecific vale. However, doing so is not entirely jstified. Additionally, the direct implementation of the I method is dif- g e x P

2 374 Jornal of Marine Science and Technology, Vol. 5, No. 4 (17) ficlt becase it involves a tedios trial-and-error process. Moreover, no design table or tool is available for > 75 (Fig. 1(b)), even thogh this angle exists in practice. A more effective and efficient method for analyzing an eccentrically bolted connection withot the above limitations and shortcomings is therefore needed. This work develops an iterative algorithm to locate the instantaneos center of rotation of an eccentrically bolted connection, and proposes a simple rational method for approximating the coefficient for loads at any angle withot direct analysis. The reslts ths obtained are compared with those obtained sing other available methods, inclding the elastic method, the I method, the AIS manal-based interpolation method, and the approximation method of Iwankiw (1987). II. PROPOSED METHOD FOR ANALYZING EENTRI BOLTED ONNETIONS Iwankiw (1987) presented a comptationally simple bt rather conservative method for approximating the coefficients for bolt patterns nder inclined loads. This work proposes an improved method that is based on the work of Iwankiw (1987) and yields sfficiently accrate reslts for loads at varios angles between and 9 withot the complex iteration of the I method. According to the 1 AIS design manal, the tablated non-dimensional coefficient,, represents the nmber of effective bolts that resist the eccentric force. is proportional to the available strength (R n ) of the eccentrically loaded bolts. The coefficient represents for the resistance at the six specified angles (, 15, 3, 45, 6, and 75), as indicated in the AIS manals. Any inclined load is conventionally split into vertical and horizontal components. To be consistent with the tablated coefficients in AIS design manals, an applied inclined load (P ) can be divided into two components (P' and P'15), where P' and P'15 are parts of the connection capacities of P and P15. P and P15 are proportional to and 15, which are listed in the manal in increments of 15, as displayed in Fig.. Since the algebraic addition of two components is always greater than the vector addition thereof, these two components are added algebraically to provide a conservative estimate of the strength of bolted connections. To pt this proposed approach into mathematical form which is compatible with the AIS Manal tables, part of the connection capacity () resists P', and the remainder resists P'15, based on the algebraic addition simplification. The magnitdes of P' and P'15 are assmed to be as follows. P r P r r n and n 15 n where r n is the nominal shear strength per bolt; and 15 are the coefficients at the six specified angles (, 15, 3, 45, 6, and 75), as tablated in the 1 AIS manal, and ' is (1) P' γ +15 γ + 15 γ P' γ P γ γ Fig.. P' γ P' γ 1 P' γ +15 P 1 P' γ +15 P ' γ ' γ γ γ ' γ +15 ' γ +15 γ +15 γ +15 Derivation of proposed method from trigonometric relationship (P' γ and P' γ+15 ). a derived eccentricity coefficient to resist a part of P. From the simple trigonometric relationship between P' and P'15, we have P P 15 sin( 15 ) sin( ) Sbstittion of Eq. (1) into Eq. () gives sin( 15 ) r n r sin( ) 15 n Frther simplification of Eq. (3), the eqation then becomes 15 sin( 15 ) 15 sin( ) sin( 15 ) 15 sin( ) sin( 15 ) 15 Hence the derived eccentricity coefficient ' can be given as sin( ) sin( 15 ) if where is the AIS-tablated coefficient that corresponds to the angle γ(, 15, 3, 45, 6, or 75) and γ < < γ + 15 in degrees. From the law of cosines, we have Sbstitting r n, we obtain () (3) (4) (5) P P P PP cos165 (6) sin( ) P rn and P 15 P sin( 15 )

3 D. M. Le et al.: Bolted onnection nder Eccentric Loads 375 r (kips) I d i G r i r o e (a) I method P r = r lt (1 e -1Δ ) Deformation Δ (in.) (b) Load-deformation relationship of one ASTM A35 bolt nder single shear Fig. 3. Instantaneos center of rotation method. cos165 1 cos165 Then the proposed method can be conclded as follows: (1) alclate ', which is based on Eq. (5). () ompte, which is calclated based on Eq. (7) and is the approximate coefficient for any angle () of inclination between and 9. (3) Obtain P, which is the maximm eccentric load of the bolt grop. (7) P ( r ) (8) n To evalate the accracy of the proposed method, the exact reslt obtained by the I method is developed as follows. III. PROEDURE FOR IMPLEMENTING THE I METHOD The combined effect of rotation and translation nder the eccentric load is eqivalent to a rotation abot a particlar location, which is called the instantaneos center of rotation (I), as shown in Fig. 3(a). The exact location of the I depends on the bolt patterns and the location and direction of the applied load. Fig. 3(b) plots the load-deformation relationship (rawford and Klak, 1971) of a high-strength bolt, which is given by r r e 1.55 lt (1 ) (9) where r denotes the nominal bolt shear strength (kips) (1 kip kn); r lt is the ltimate bolt shear strength (kips), and is the combined bolt shear, bearing, and bending deformation (inches) (1 inch.54 cm). Eq. (9) was obtained from an experiment on an ASTM A35 bolt with a diameter of ¾ inch nder single shear with r lt 74 kips and max.34 in. The nominal shear strength (r) of the farthest bolt from the I is obtained by applying a maximm deformation max to the bolt. The vales of r of the other bolts are obtained by considering a deformation that varies linearly with distance from the I between zero (at the I) and max (at the farthest bolt). The nominal shear strength of the bolt grop is, then, the sm of the individal bolt strengths. The AIS design manals (1986, 1989, 1993, 1999, 5, and 1) sed the I method to analyze eccentric bolted connections by specifically evalating. Brandt (198) presented a techniqe for locating the I. However, none of the aforementioned design manals provided detailed steps for the implementation of the tedios I method. The present work develops a different algorithm for locating the I, as follows. In the I method, three force eqilibrim eqations (F x, F y, and M ) are reqired to locate the instantaneos center of rotation. The exact soltion to the problem is almost impossible to obtain directly, so an iterative algorithm that is based on the concept of gradient descent for line search (Nocedal and Stephen, 1999; S and Si, 7) is developed to solve it. Given M, the reqired trial strength (P ) of the eccentrically loaded bolt grop is P M /r o (r i d i )/r o, whose corresponding load components are P x and Py. At static eqilibrim (F x, Fy ), the resltants are expressed as F F P r and F F P r (1) x h x x y y y y where F h and F v are called nbalanced forces if F x or Fy. Let F(x, y ) be the magnitde of the resltant of the eccentrically loaded bolt grop at the first trial point of the I: F F F( x, y) i j x y ( x, y) ( x, y) ( x, y) ( x, y) F if jf if j where F F F x y h v h v (11) The magnitde of resltant F(x, y) increases rapidly in the direction of positive gradient (F), and falls rapidly in the direction of negative gradient (-F). The negative gradient (-F) specifies the direction of descent of the resltant F(x, y). In the iterative algorithm, this negative gradient is applied to redce the x- and y-components of nbalanced forces, F h and Fv, in each direction of descent. Gradient is perpendiclar to the force

4 376 Jornal of Marine Science and Technology, Vol. 5, No. 4 (17) F v s i ( F v) s i ( F h) y I x 1. Determine the sectional properties and geometry of the bolt grop and se the center of gravity (G) of the bolt grop as the first trial location of I.. With reference to Fig. 4(b), find the normal form of the eqation (Sisam and Atchison, 1955) along the load application line (l). The perpendiclar distance (e) from the G to l is expressed as e ( x x )cos ( y y )sin (13) p cg p cg (a) Direction of descent with linear search y o r o N I (x ic, y ic ) A e l β α P (x p, y p ) G(x cg, y cg ) (b) Perpendiclar distance (r o ) from I to a line (l) Fig. 4. Iterative algorithm to find the location of I. vector. Accordingly, the direction of descent opposes the normal to the force vector, so Fv declines in the positive x direction and F h falls in the negative y direction. Then, step length, s i Fv or s i F h, is adjsted as a shift along each direction of descent with reference to Fig. 4(a). The positive s i is the steplength parameter, which can be set for each iterative process. Therefore, the iterated coordinates are given by x x s ( F ) x F and i1 i i y i y y y s ( F ) y F i1 i i y i h x (1) The algorithm reqires that the initially gessed I position is the centroid of the section; the iterative process generates the next point by moving one step length in the direction of negative gradient from the preceding I point. The comptational procedre is described in detail herein sing an illstrative example. The iterative process terminates when the nbalanced forces Fv and F h have been obtained to the desired accracy, which approaches zero. The reqired design force (P ) and available bolt strength (r n ) are then set, yielding the coefficient = P / (r n ). The detailed procedre for implementing the I method is smmarized as follows: and the perpendiclar distance (r o ) from the I to l is calclated as r ( x x )cos ( y y )sin e o ic cg ic cg ( x x )cos ( y y )sin ic p ic p (14) where is the angle between the line that is normal to P and the horizontal axis. 3. alclate the deformation ( i ) and resistance (r i ) from the load-deformation relationship that is given in the AIS manals for each bolt: ( d / d ).34( d / d ) and i i max max i max r r (1 e ) i lt 1 i.55 (15) 4. alclate the resltant moment (M ) and force components (r x and ry) for the bolt grop. r ( ry )/ d, r ( rx )/ d,and M x i i i y i i i ( rd ) i i (16) where (x i, y i ) are the coordinates of each bolt, r i is the resistance of each bolt, and d i is the radial distance from G to the center of each bolt. 5. alclate the corresponding applied load (P ) and its components (P x and Py) by considering the static eqilibrim, P M / r ( rd ) / r, P P cos, and P o i i o x y P sin (17) where is the angle of inclination of P with respect to the horizontal line and r o is the load eccentricity, which is obtained sing Eq. (14). 6. onfirm the force eqilibrim. F F P r and F F P r (18) x h x x y y y y 7. If an eqilibrim condition is violated (F h or F v ),

5 D. M. Le et al.: Bolted onnection nder Eccentric Loads 377 Table 1. omparisons of vales of obtained sing proposed iterative algorithm. Angle () omptational reslt () AIS manal () (Based on 1 AIS Tables 7 and 8 with g 5.5 in., s, e x 16 in., and n 6; 1 in..54 cm) then alter the location of I to redce the difference between the applied load (P ) and the resltant force (Σr ) of the bolt grop. The next I coordinates are adjsted to x x s ( F ) x F and i1 i i y i y y y s ( F ) y F i1 i i y i h (19) where positive s i is the step-length parameter for F h and F v. 8. Repeat Steps to 7 ntil the convergence vale of.1 percent for nbalanced forces F h and Fv is reached. A design example is presented below to illstrate the above procedre, whose reslts are compared with those of the proposed method. A compter program was developed to execte the above iterative process. The otpts (coefficients ) of the program were verified against the tables in the 1 AIS design manal. Table 1, which presents a set of otpts, shows that the calclated coefficients in the example compare favorably with those in the AIS design tables. IV. ILLUSTRATIVE EXAMPLE Fig. 5(a) shows a bolted connection nder an applied load (P ). The connection is spported by the bracket. Both the colmn and the bracket are made of steel with Fy 36 ksi. A35-N bolts with a diameter of ⅞ in. are sed in standard holes. Assme the colmn flange and the bracket plate have adeqate strength. The objective is to evalate the maximm load (P ) sing both the available methods and the proposed method. 1. Elastic Method With reference to Fig. 5(a), the center of gravity (G) of the given bolt grop is located at x ( x A ) / A.75 in. and y ( y A ) / A 7.5 in., cg i b b cg i b b where A b is the cross-sectional area of each bolt. By calclation, d in. with respect to the G of the bolt grop. The components of direct bolt shear are r r sin P sin / n P sin 8 / n.81 P ( ) px p r r cos P cos / n P cos8 / n.145 P ( ) py p olmn s y y e x = = 8 P e Bracket x G(x cg, y cg) 5.5 in. (a) Bolted connection r o s x I d i y e G α (b) Perpendiclar distance (r o) from trial I to line along which loadis applied The torqe at the G is e x r i β P Fig. 5. Illstrative example. M P cos ( P 16).7784 P (clockwise) cg The components of torsion-indced bolt shear are MGdy.7784P 7.5 r mx P d MGdx.7784P.75 r my P d Hence, the reqired strength per bolt, r, is px mx py my r r r r r. 81P. 5136P. 145P. 1883P.1375 P x () () According to the 1 AIS LRFD, r r n. With r n 4.3 kips, the obtained maximm P vale is (P ) max kips From P R n e r n, e (P ) max /(r n ) /

6 378 Jornal of Marine Science and Technology, Vol. 5, No. 4 (17). I Method As stated above, the G of the bolt grop is located at xcg.75 in. and ycg 7.5 in. With reference to Fig. 5(b), the line along which the load is applied can be expressed in normal form. The angle between the horizontal line and the normal to the load direction is The perpendiclar distance from the G to the line along which the load is applied is where e( x x )cos ( y y )sin p cg p cg 16cos1sin1.778 in. ( x, y ) (18.75 in., 7.5 in.) and p p ( x, y ) (.75 in., 7.5 in.) cg cg The initial gess of I at G is (x, y ) (.75 in., 7.5 in.). The perpendiclar distance from the initially gessed I to the line along which the load is applied is r ( x x )cos ( y y )sin e o cg cg ( x.75)cos1 ( y 7.5)sin in. The AIS load-deformation relationship for one bolt is r r e d d 1.55 lt (1 ),.34( i / max ), r r( d / d),and r r( d / d) x y y x From the developed program otpts, (r d) = k-in, r x, and ry Therefore, P M /r o (r d)/ / kips The eqilibrim of the nbalanced forces in the horizontal and vertical directions is checked as follows. The angle between the line along which the load is applied and the horizontal axis is Fh Fx Px rx cos kips ( ) Fy Fy Py ry sin kips ( ) NG. NG. In this example, the step-length parameter (s i ) is set to 5% in each iteration. The iterative algorithm generates the next trial location of I as x x s F x.75.5( ) 1.94 in. i1 i i v 1 y y s F y 7.5.5( ) in. i1 i i h 1 The step length and direction of descent in this iteration are given by in. in the negative x direction and 6.69 in. in the positive y direction, respectively. The iteration algorithm generates the next location (x 1, y 1 ) (-1.94 in., in.) from the crrent point (x, y ) (.75 in., 7.5 in.). Repeating the above steps, as described above, yields a seqence of locations of I. At the correct I (x ic, y ic ), the force eqilibrim eqations are satisfied, (ΣF x, ΣFy and ΣM ), and the final P vale is ths determined. In this example, the correct I coordinates, (x ic, y ic ) ( in., in.) are obtained after 5 iterations. The distance from the I to the G, S 3.78 in. (with S x 1.4 leftward and Sy in. pward). Table presents in detail the calclations that are associated with, and the reslts that are obtained sing, the iterative algorithm that is based on the I method. P /[( ) cos1 ( )sin1.778] 18.5 kips The eqilibrim of the horizontal forces is confirmed as follows. P P cos 18.5cos kips () x Fx Px Rx kips OK The eqilibrim of the vertical forces is confirmed as follows. P P cos 18.5sin kips () y Fy Py Ry kips OK r n 4.3 kips as per the 1 AIS LRFD, and P /(r n ) 18.5/ Straight-Line Interpolation Tables 7-6 to 7-13 in the 1 AIS manal provide the vales of for the six specified load inclination angles (, 15, 3, 45, 6, and 75). For a non-tablated vale, straight-line interpolation between vales for loads at different angles may be non-conservative, so the AIS recommends direct analysis. In this case, linear interpolation yields 9.7 for 8 where 7.9 for 75 and 1 for 9.

7 D. M. Le et al.: Bolted onnection nder Eccentric Loads 379 Table. Reslts in illstrative example ( 8) obtained sing the iterative algorithm. Bolt No. d x (in.) d y (in.) d (in.) (in.) r (kip) r x (kip) r y (kip) R d (k-in.) (1 kip kn; 1 in..54 cm) Table 3. Differences among vales of in illstrative example ( 8). Elastic method (A) I method (B) AIS Manal interpolated () Method of Iwankiw (D) Proposed method (E) oefficient P (kips) Difference (%) (A B)/B. (B B)/B 3.5 ( B)/B (D B)/B (E B)/B The reqired strength of the bolt grop is P (r n ) kips. As expected, this reslt is an overestimate, relative to that obtained by the I method. 4. Method of Iwankiw max 1 (total nmber of bolts) and o 3.55 (AIS s tablated vale for ). max 1 A o The approximate eccentricity coefficient for the inclined load ( a ) is given by a o A 3.38 sin Acos sin cos The reqired strength (P ) is then calclated as P ( r ) ( ) (4.3) kips a n 5. Proposed Method For 75, , as obtained from the AIS table. Here, 9 1, so when 8, sin( 75 ) sin(875 ).519 sin(9 ) sin(98 ) cos Therefore, P r n kips V. DISUSSION Table 3 presents the reslts that are obtained sing the varios methods for 8. The vales of for other vales can be compted similarly, and are shown in Table 4 and Fig. 6. Fig. 7 compares the vales of that are obtained sing the varios methods. The following observations are made. (1) For the six specified vales of (, 15, 3, 45, 6 & 75), the coefficients that are calclated sing the proposed iterative algorithm eqal those that are tablated in the 1 AIS manal, which are presented in Table 1 and

8 38 Jornal of Marine Science and Technology, Vol. 5, No. 4 (17) Angle (degree) Table 4. omparisons of vales of obtained by varios methods. elastic [1] I method [] interpolated [3] Iwankiw [4] proposed [5] (%) [1] [] [] (%) [3] [] [] (%) [4] [] [] (%) [5] [] [] oefficient in. P -elastic -I method -interpolated -Iwankiw -proposed Fig. 6. oefficients compted sing varios methods. Fig. 6, verifying the very high accracy of the proposed iterative algorithm. () As indicated in Table 4 and Fig. 7 in this example, all methods except the linear interpolation method nderestimate, as determined by comparison with those obtained sing the more exact I method. The linear interpolation method overestimates the strength by -3%. For -75, the Difference (%) P 5.5 in. -elastic -I method -interpolated -Iwankiw -proposed Fig. 7. omparisons of vales obtained sing varios methods. nderestimations by the elastic method, the approximate method of Iwankiw (1987), and the proposed method are 7%, 18%, and.9%, respectively. For 75-9, not considered in any AIS manal, the maximm degrees of conservatism are 3%, 17%, and 1.91%, respectively. Apparently, the proposed method yields fairly accrate reslts.

9 D. M. Le et al.: Bolted onnection nder Eccentric Loads 381 Table 5. Bolt grop geometry in 1 AIS design tables. AIS Tables olmn No. Row No. Horizontal eccentricity (in.) Horizontal spacings (in.) Vertical spacings (in.) , , , , , , , , 6 (1 in..54 cm) oefficient P -I method -proposed oefficient P -I method -proposed oefficient P -I method -proposed oefficient P -I method -proposed Fig. 8. Accracy of the proposed method for other bolt grops in 1 design manal. (3) For all other bolt patterns in the 1 AIS manal, shown in Table 5, the proposed approach yields reslts that are very close to those obtained sing the I method for -75, shown in Fig. 8. For 75-9, not covered in any AIS manal, the proposed approach yields sfficiently accrate reslts. (4) The proposed approach yields reslts with greater accracy than straight-line interpolation between vales for loads at the specified angles; straight-line interpolation may be non-conservative and is not recommended by the AIS. (5) The derived eccentricity coefficient ' is obtained by a nonlinear interpolation between and 15, which are listed in the AIS design tables. ' decreases as the angle of inclination increases in range of each 15 interval for all bolt patterns in the 1 AIS manal. This finding reflects that the assmption, as represented by Eq. (1), is a rational one. The proposed method is derived from a simple trigonometric relationship among the applied load and two components of force. Accordingly, the method can be reasonably applied to estimate the strength of an eccentrically loaded inplane connection withot any restriction on materials or a tedios iterative process. VI. ONLUSIONS This work presents a rational procedre for determining the instantaneos center of rotation and strength of an eccentric grop of bolts. The procedre is simple and reliable. This work

10 38 Jornal of Marine Science and Technology, Vol. 5, No. 4 (17) overcomes the limitations in crrent AIS design manals. The reslts in this work spport the following conclsions. 1. The proposed iterative algorithm constittes a general procedre for implementing the tedios I method to find the exact location of an I. The iterative procedre yields identical eccentricity coefficients at the six specified load inclination angles (, 15, 3, 45, 6, and 75) being tablated in the AIS design manals, frther demonstrating the accracy and reliability of the iterative algorithm.. The proposed method yields fairly accrate reslts withot the need for a tedios trial-and-error procedre for all angles of inclination ( 9) as shown in Fig. 6. The accracy is also applied to all other bolt patterns listed in the 1 AIS design manal. Some examples of the accracy are displayed as shown in Fig The proposed method is a rational and reliable tool for approximating the eccentricity coefficients for loads at angles between and 9 instead of straight-line interpolation, which is not recommended by the AIS. The proposed model can get rid of engaging on a direct analysis or the vales for the next lower angle increment in the tables as recommended by the 1 AIS design manal. The accracy of the proposed method sbstantially exceeds the reqirements of engineering. This work overcomes the design limitations (for 75 only) which are evident in crrent AIS design manals. AKNOWLEDGEMENTS The athors wold like to thank the Ministry of Science and Technology of Taiwan for financially spporting this research. REFERENES AIS (1986). Manal of steel constrction: load and resistance factor design, 1 st ed. American Institte of Steel onstrction: hicago, IL., USA. AIS (1989). Manal of steel constrction: allowable stress design, 9 th ed. American Institte of Steel onstrction: hicago, IL., USA. AIS (1993). Manal of steel constrction: load and resistance factor design, nd ed. American Institte of Steel onstrction: hicago, IL., USA. AIS (1999). Manal of steel constrction: load and resistance factor design, 3 rd ed. American Institte of Steel onstrction: hicago, IL., USA. AIS (5). Manal of steel constrction: load and resistance factor design, 13 th ed. American Institte of Steel onstrction: hicago, IL., USA. AIS (1). Manal of steel constrction: load and resistance factor design, 14 th ed. American Institte of Steel onstrction: hicago, IL., USA. Brandt, G. D. (198). Rapid determination of ltimate strength of eccentrically loaded bolt grops. Engineering Jornal, AIS, 19(), rawford, S. H. and G. L. Klak (1971). Eccentrically Loaded Bolted onnections. Jornal of the Strctral Division, ASE 97(3), Iwankiw, N. R. (1987). Design for Eccentric and Inclined Loads on Bolt and Weld Grops. Engineering Jornal, AIS 4(4), Nocedal, J. and J. W. Stephen (1999). Nmerical Optimization, 1 st ed. Springer: New York, NY, USA, 1-3. Sisam,. H. and W. F. Atchison (1955). Analytic Geometry, 3 rd ed. Holt, Rinehart and Winston, Inc.: New York, NY, USA, S, R. K. L. and W. H. Si (7). Nonlinear response of bolt grops nder inplane loading. Engineering Strctres 9,