A UNIFIED APPROACH TO DESIGN FOR BLOCK...

A UNIFIED APPROACH TO DESIGN FOR BLOCK SHEAR

R.G. Driver, G.Y. Grondin, and G.L. Kulak Dept. of Civil & Environmental Engineering, University of Alberta, Canada

ABSTRACT

Design equations in Canadian, American, and European standards for determining the block shear capacity of structural members vary significantly. A detailed survey of test results published in the literature, combined with the experimental results of two recent research projects at the University of Alberta, demonstrates that these equations do not always provide an acceptable level of safety. For coped beams, in particular, it was found that the North American standards do not provide an acceptable level of safety and that predictions of the strength of such connections with two lines of bolts are often on the unsafe side by a considerable margin. Reliability methods are used to provide a simple, unified approach to design for block shear failure of gusset plates, angles, tees, and coped beams. This method provides both an adequate and a consistent level of safety and predicts capacities that compare well with test results over a broad array of connection configurations. Of particular importance, the proposed approach also reflects the failure modes that have been observed in tests, which is an aspect that is lacking in some current design equations.

INTRODUCTION Block shear failure is characterized by a combination of rupture on the tension plane and yielding on the shear plane(s) of a block of material. Although this mode of failure can occur in either welded or bolted connections, it is more common in the latter because of the reduced area that results from the bolt holes. This paper deals only with bolted connections. Design rules in various codes and standards base block shear capacity on a combination of yield and rupture strength of the net or gross areas in shear and tension on the failure planes. North American design provisions are examined in order to assess the level of safety provided by each design standard for gusset plates, angles, tees, and coped beams. The Eurocode 3 design equations are also examined as one alternative to current North American practice. Finally, a unified approach that provides a uniform level of safety for all of these cases is proposed. DESIGN PROVISIONS CSA–S16–01 In the current Canadian design standard, CSA–S16–01 (1), the block shear capacity of gusset plates and angles is taken as the lesser of the following two equations:

u nt u gv yP A F 0.6 A F= + (1)

u nt u nv uP A F 0.6 A F= + (2)

Connections in Steel Structures V - Amsterdam - June 3-4, 2004 323

The capacity of coped beams is provided by similar equations except that the contribution of the tension area to the connection capacity is reduced by one-half to reflect the non-uniform stress distribution on the tension plane. Thus, for coped beams:

u nt u gv yP 0.5 A F 0.6 A F= + (3)

u nt u nv uP 0.5 A F 0.6 A F= + (4)

where yF and uF are the yield and the tensile strength of the material, respectively, ntA and nvA are the net tension and shear areas, respectively, and gtA and gvA are the gross

tension and shear areas, respectively. Equations (1) and (3) are based on the observation that rupture on the tension plane occurs before rupture on the shear planes. This is supported by several test programs, including the University of Alberta test observations presented below. Equations (2) and (4) are provided in order to limit the capacity of the shear planes to the rupture strength of the net shear area.

AISC 1999

The block shear provisions in the AISC LRFD 1999 Specification (2) make use of two equations that depend on the relative strength of the net tension and shear areas of the connection:

when ≥u nt u nvF A 0.6 F A , = + ≤ +u u nt y gv u nt u nvP F A 0.6 F A F A 0.6 F A (5)

when <u nt u nvF A 0.6 F A , = + ≤ +u y gt u nv u nt u nvP F A 0.6 F A F A 0.6 F A (6)

where all the variables are as defined above. The block shear capacity combines either rupture on the net tension area, ntA , with yielding on the gross shear area,

gvA , or yielding

on the gross tension area, gtA , with rupture on the net shear area, nv

A . Both equations have an upper bound corresponding to the combination of rupture on both the net tension and net shear areas. Although equation (5) represents a plausible failure mode, the validity of equation (6) is questionable (3). Although not yet adopted, the current proposal for the 2005 AISC Specification is to use the S16-01 equations (equations (1) through (4)), except that the 0.5 coefficient for the tension plane is used for angles as well as for coped beams.

Eurocode

Eurocode 3 (4) combines shear yield acting over the gross shear area with rupture on the net tension plane. Thus, the block shear capacity of a gusset plate or tension member can be obtained as follows:

( )u u nt y gvP F A F 3 A= + (7)

(Eurocode uses the von Mises shear yield stress equal to yF 3 , whereas the North American standards simply take this as 0.60 yF .)

Although equation (7) is not given specifically in Eurocode, it can be deduced from the description of the mode of failure. A more elaborate procedure is presented for block shear failure in beam webs, where a series of equations, when combined, gives the following result:

= − +u gt h u gv1 1

P w (L k d ) F A Fy3 3 (8)

where w is the web thickness, gtL is the gross tension length, k is a tension area coefficient (0.5 for one line of bolts or 2.5 for two), and hd is the bolt hole diameter.

324 Connections in Steel Structures V - Amsterdam - June 3-4, 2004

GUSSET PLATE TESTS There are a large number of gusset plate tests reported in the literature for which block shear is the failure mode. Table 1 presents a summary of the results for 133 gusset plate tests from eight different sources.

All gusset plate tests show that the ultimate load is reached when the tensile ductility of the gusset plate material at the first (i.e., inner) transverse row of bolts is exhausted. This is true even in cases where oversize holes were used and in cases where the connection was short (i.e., little shear area available). Recent tests conducted at the University of Alberta (5) show clearly that fracture on the net tension plane takes place before shear fracture on the shear planes, as illustrated in Figure 1 on the left. The displacement of a block of material is seen only when the test is continued until the parts separate (Figure 1, right), and this occurs well past the ultimate load capacity of the connection. Figure 1 also illustrates that shear rupture does not take place at the net area, as assumed in some design equations.

Figure 1. Block shear failure in gusset plates (5).

Although many tests have been conducted on gusset plates, the range of geometries tested in the past is fairly limited. Most tests have been conducted on gusset plates with two lines of bolts with from two to four bolts in each line. Huns et al. (5) have extended the number of bolts in a line to eight using finite element analysis. The 10 tests reported in Table 1 from reference (5) include five results from a finite element model that has been validated against existing test results.

The test-to-predicted ratios for all 133 gusset plates presented in Table 1 average 1.18 for CSA–S16–01, 1.19 for AISC LRFD, and 1.10 for Eurocode 3. Thus, all design standards provide a conservative estimate of the block shear capacity of gusset plates.

ANGLE AND TEE TESTS

Experience and test results show that block shear is potentially a failure mode for angles and tees, particularly when the connection is short. Tests of 41 such members (including nine structural tees from Orbison et al. (16) connected at the stem that are considered to give the same failure mode as angles) are shown in Table 1. Although Epstein (14) reported the results of 144 tests, only the three specimens that failed in block shear with the tension face perpendicular to the load direction are included in Table 1. Other test programs have also

Tension fracture


investigated block shear in single angle connections and structural tees (14, 15, 16) with one line of bolts. Block shear failure of angles and tees can be affected by both out-of-plane and in-plane eccentricity. Tests on structural tees have been used to assess the effect of out-of-plane eccentricity (16). Although Orbison et al. (16) found that out-of-plane eccentricity was not a significant factor, some tests (15, 16) have shown that the block shear capacity decreases with an increase in eccentricity. In-plane eccentricity is present in all of the angle and tee tests reported in Table 1. Table 1 indicates that the block shear capacity is predicted reasonably well by the North American and European design standards.

Table 1. Block shear test results.

Source No. Tests Test

S16 01-

Test

AISC

Test

EURO

Test

Eq. (13)

Gusset Plates

Hardash and Bjorhovde (6) 28 1.20 (0.07)

1.22 (0.08)

1.19 (0.06)

1.05 (0.06)

Rabinovitch and Cheng (7) 5 1.22 (0.06)

1.22 (0.06)

1.05 (0.05)

0.96 (0.05)

Udagawa and Yamada (8) 73 1.18 (0.05)

1.18 (0.05)

1.06 (0.09)

0.96 (0.06)

Nast et al. (9) 3 1.35 (0.02)

1.35 (0.02)

1.07 (0.01)

1.00 (0.01)

Aalberg and Larsen (10) 8 1.21 (0.04)

1.21 (0.04)

1.05 (0.18)

0.95 (0.10)

Swanson and Leon (11) 1 1.18 1.18 0.99 0.88

Huns et al. (5) 10 1.14 (0.17)

1.13 (0.19)

1.10 (0.14)

0.98 (0.13)

Mullin (12) 5 1.13 (0.04)

1.15 (0.05)

1.16 (0.03)

1.02 (0.05)

Angles and Tees

Barthel et al. (13) 13 1.00 (0.04)

1.01 (0.05)

0.89 (0.04)

0.87 (0.04)

Epstein (14) [only mean reported] 3 0.98 0.98 1.00 1.02

Gross et al. (15) 13 0.97 (0.05)

0.96 (0.05)

0.93 (0.07)

0.92 (0.06)

Orbison et al. (16) [includes 9 tees] 12 1.09 (0.05)

1.12 (0.07)

1.09 (0.05)

1.08 (0.05)

Coped Beams Birkemoe and Gilmor (17) 1 1.17 0.95 1.18 0.90

Yura et al. (18) 3 1.22 (0.16)

1.03 (0.13)

1.25 (0.16)

0.96 (0.13)

Ricles and Yura (19) 7 1.02 (0.10)

0.70 (0.08)

1.13 (0.06)

1.00 (0.12)

Aalberg and Larsen (10) 8 1.33 (0.12)

1.13 (0.08)

1.21 (0.23)

1.01 (0.11)

Franchuk et al. (20, 21) 17 1.20 (0.10)

1.07 (0.13)

1.21 (0.09)

1.02 (0.12)

Note : The number in parentheses is the standard deviation.


COPED BEAM TESTS In contrast to the number of gusset plate test results available, there are relatively few tests of coped beams. Of the 36 tests shown in Table 1, ten had connections with two lines of bolts and 26 had a single line. A recent test program at the University of Alberta (20, 21) provided nearly one-half of the available test results and expanded the number of test parameters by including several variables related to connection geometry and loading conditions. Seventeen full scale tests were conducted on coped beams and included the effect of beam end rotation, gross shear area, bolt end and edge distances, number of bolt lines and bolt rows, bolt diameter, section depth, connection depth, and double copes vs. single cope. Analysis of the load vs. deformation results showed that none of these parameters affected connection capacity significantly, apart from the associated changes in tension and shear areas. It was concluded that only the tension and shear areas need be considered in a design equation to reasonably represent the failure mode. The test results indicated that rupture on the tension plane occurs before shear rupture, as shown in Figure 2 (left). It was also evident from the tests that shear rupture does not take place on the net shear area. As shown in Figure 2 (right), shear failure takes place on a plane that intercepts the holes very near to their edges.

Figure 2. Block shear failure in coped beams (20, 21). The ratio of test ultimate load to the AISC predicted ultimate load is significantly non-conservative for the tests by Ricles and Yura (19), which were all tests on coped beams with two lines of bolts. The average AISC test-to-predicted ratio for the three test specimens of Franchuk et al. (20, 21) with two lines of bolts is 0.90, compared to 1.10 for the test specimens with one line. Both the Canadian and European standards seem to predict the test results conservatively, although a large standard deviation is observed in some cases. RELIABILITY ANALYSIS Although the test-to-predicted ratios presented in Table 1 provide an indication of the ability of each equation to predict block shear capacity, they do not by themselves quantify the level of safety being provided. Therefore, the suitability of the various design equations was assessed using reliability methods. Since the equations presented above do not contain a performance factor, the following analysis derives the performance factors required to achieve pre-selected levels of safety. Because the desired level of safety is a matter of

Tension fracture


continuing debate, the calculations are performed for three different safety levels.

In the development of a limit states design equation, a performance factor, φ, is determined that is a function of the mean values and variabilities—expressed as the coefficients of variation—of the relevant material and geometric properties, as well as the ability of the equation itself to predict capacity. The performance factor is calculated using the following equations from Ravindra and Galambos (22):

( )ρ α βφ = −R R Rexp V (9)

where β is the safety index, which is directly related to the probability of failure during the life of the structure. It is desirable to have a higher safety index (lower probability of failure) for connections than for structural members, such as beams, that tend to be more ductile. As such, members are usually assigned a safety index of about 3.0, while connections have typically been assigned a value of approximately 4.5 (22). The bias coefficient for resistance,

Rρ , is given as:

R M G Pρ = ρ ρ ρ (10)

where Mρ is the ratio of the mean measured to nominal material strength, Gρ is the ratio of mean measured to nominal connection geometric properties, and Pρ is the professional factor, or mean ratio of measured to predicted resistance, which reflects the ability of the equation to predict the capacity. The coefficient of variation for the resistance, RV , is given as:

2 2 2R M G PV V V V= + + (11)

where MV , GV , and PV are the coefficients of variation associated with Mρ , Gρ , and Pρ , respectively. Ravindra and Galambos (22) recommend that the separation variable, αR , in equation (9) be taken as 0.55.

There is an interdependence between φ and the load factors, shown by Fisher et al. (23). This means that the use of a safety index other than 3.0 in equation (9) requires that a modification factor be applied. As shown in (20, 24), equation (9) must be modified to:

( )⎡ ⎤φ = β − β + ρ −α β⎣ ⎦2

R R R0.0062 0.131 1.338 exp V (12)

Material Factor The material factor reflects the difference between the nominal and measured material strengths (yield or ultimate). The bias coefficient and coefficient of variation for the material properties were obtained using data from Schmidt and Bartlett (25) for plates and rolled wide flange sections. The material factors selected for gusset plates and angles and tees are presented in Table 2. The values selected for coped beams are presented in Table 3. Geometry Factor The geometry factor accounts for the difference between the nominal and measured plate thickness and hole layout dimensions. Since insufficient data exist to evaluate the geometry factors for gusset plates, angles and tees, the bias coefficient and coefficient of variation for the geometry factor proposed by Hardash and Bjorhovde (6) was used. The values selected for coped beams were obtained from measurements on the 17 specimens tested by Franchuk et al. (20, 21, 24) and data reported by Kennedy and Gad Aly (26). The values of

Gρ and GV adopted are presented in Tables 2 and 3.


Professional Factor The professional factor (the ratio of the measured capacity, obtained either by laboratory testing or from a validated finite element analysis, to the capacity predicted by the equation using measured material properties and geometry) represents the ability of a model to predict the block shear capacity. The mean and coefficient of variation of the test-to-predicted ratios for the individual test programs presented in Table 1 have been consolidated for gusset plates, angles and tees, and coped beams and are summarized in Tables 2 and 3. Although the North American standards require the inclusion of a 2 mm allowance when punched holes are used, it was not included in the professional factors presented in this work. This is because in many cases it is not known how the holes were made. The effect of this approximation was assessed for coped beams by Franchuk et al. (24) and found to be relatively small. The performance factor, φ , was calculated for three levels of safety index, namely, 3.5, 4.0, and 4.5.

Table 2. Reliability parameters for gusset plates and angles and tees.

Table 3. Reliability parameters for coped beams.

Using the performance factor adopted by each standard (0.9 or 0.75 for the Canadian and American standards, respectively) the current safety indices for each type of connection can be determined. Use of the Canadian standard gives a safety index of 4.5 for gusset plates, 3.4 for angles and tees, and 4.1 and 3.0 for one- and two-line coped beam connections. Use


of the AISC specification gives a safety index of 5.8 for gusset plates, 4.6 for angles and tees, and 4.4 and 1.9 for one- and two-line coped beam connections. Both standards provide an inconsistent level of safety. The level of safety in the Canadian standard is too low for angles and tees and for two-line coped beam connections. The level of safety provided by the AISC specification is too low for two-line coped beam connections. Both North American standards provide an unacceptably low level of safety for the two-line connections. The Eurocode approach, however, provides a consistent level of safety for one- and two-line connections, although there is inconsistency when including gusset plates, angles, and tees, as can be deduced from Tables 2 and 3. For comparison, the equations currently proposed for the 2005 AISC Specification, with the associated performance factor of 0.75, give a safety index of 5.9 for gusset plates, 5.5 for angles and tees, and 5.3 and 4.1 for one- and two-line coped beam connections. Had the punched hole allowance been included, the safety indices could be as much as 7% higher than the values presented above (24). PROPOSED UNIFIED EQUATION In order to address the shortcomings and inconsistencies of existing design equations, a new equation for the design of gusset plates, angles, tees, and coped beams is proposed. It uses laboratory observations of the failure mode as its basis. A single equation is considered appropriate because the failure mode observed over a large variety of tests consists of rupture on the tension plane after yielding has occurred on the shear plane, but prior to shear rupture. Although most of the existing design equations predict the test capacity for angles reasonably well, the capacity of gusset plates is generally under-predicted by a considerable margin, and none of the equations adequately predict the block shear capacity of all coped beams. It is also desirable to have a single equation rather than multiple equations, as is currently used in the design standards. Although the stress on the shear plane is lower than the rupture stress when rupture on the tension plane occurs, it is likely significantly greater than the yield stress. Simply taking the mean of the yield and ultimate shear stresses provides excellent correlation with test results. Another issue that must be taken into account is that shear failure has been observed to occur on a plane that intercepts the holes very near to their edges (see Figures 1 and 2). Therefore, the gross area is deemed to be more appropriate for determining the strength of the shear components of the block.

The proposed unified equation combines effective stresses on both the net tension area and the gross shear area:

u v

F Fy uP R A F R At nt u gv 2 3

+⎛ ⎞⎜ ⎟= +⎜ ⎟⎝ ⎠

(13)

where tR and vR are the tension area and shear area stress correction factors, respectively, as given in Table 4. These factors account for the non-uniform stress distributions that occur in some cases on the tension and shear planes of the block shear failure surface. These factors are semi-empirical in that they have been selected to optimize the consistency of the safety indices, although they have support from finite element studies (20).

The proposed model is simplified over those of the existing North American and European standards in that only one equation need be checked for a particular connection type. Nevertheless, it not only provides an accurate prediction of connection capacity in an average sense, with mean professional factors varying from 0.96 to 1.07 for all types of connections considered, it also results in coefficients of variation that are among the lowest of all procedures investigated. These values are tabulated in Tables 2 and 3. The excellent correlation is attributed to the consistency of the structure of the proposed unified equation with experimental observations.

Tables 2 and 3 give the performance factors, φ, required to achieve safety indices of 3.5, 4.0,


and 4.5. The tables indicate that a consistent performance factor is achieved with the unified equation for gusset plates, angles, tees, and coped beams with one or two lines of bolts. The performance factor varies from about 0.8 to 0.7 to achieve safety indices from 3.5 to 4.5 for all connection types considered. As a point of reference, the traditional target safety index for connections of 4.5 is achieved for all cases by adopting a performance factor of 0.67, although it can legitimately be argued that a somewhat lower safety index would be sufficient for this type of failure. Table 4. Stress correction factors. tR vR

Gusset plates 1.0 1.0 Angles and tees 0.9 0.9

One-line connections 0.9 1.0 Coped Beams

Two-line connections 0.3 1.0 SUMMARY AND CONCLUSIONS A comprehensive research program was carried out to evaluate the level of safety currently being provided by design standards for determining the block shear capacity of gusset plates, angles, tees, and coped beams. A reliability analysis indicates that the current North American standards do not provide an adequate level of safety for all coped beam connections, as compared to the traditional target safety index for connections of 4.5. Both standards provide a sufficient level of safety for gusset plates and coped beams with one line of bolts. For angles and tees and for coped beam connections with two lines of bolts, the level of safety in the Canadian standard is unacceptable. The safety index provided by the 1999 AISC Specification for coped beams with two lines of bolts is far below the target value of 4.5 for connections and even lies well below the target for ductile member failure of 3.0. On the other hand, the equations currently proposed for the 2005 AISC Specification result in safety indices very much higher than 4.5 for most common connections, while the value for coped beams with two lines of bolts is considerably lower. The Eurocode equation provides an inconsistent level of safety when considering various connection types. The proposed unified design model (Equation 13) is recommended for the prediction of block shear capacity. It is simpler than existing procedures and reflects the failure mode observed consistently in a wide variety of tests. Moreover, with a single performance factor the model provides a consistent and adequate level of safety for all types of connections investigated in this study. ACKNOWLEDGEMENT The research conducted at the University of Alberta was funded by the the Steel Structures Education Foundation, the Natural Sciences and Engineering Research Council of Canada, and the C.W. Carry Chair in Steel Structures Research. REFERENCES 1. Canadian Standards Association (CSA). (2001). “Limit States Design of Steel

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