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1、Prying Actiona General Treatment* WILLIAM A. THORNTON INTRODUCTION The 8th Edition AISC Manual1 uses a model (Fig. 1) for predicting the prying force which was recommended in the book by Fisher and Struik.2 Unlike the approach taken in the 7th Edition Manual, this method is not restricted to specifi
2、c bolt-plate combinations, since all major parameters which influence the prying action are included in the model. The Q denotes the prying force per bolt and is assumed to act as a line load at the edge of the flange. Test results have shown this to be a reasonable assumption for conditions near ul
3、timate, as long as the edge distance a is within certain limits. The tensile load in the fastener is Bc, and the corresponding applied load per bolt is equal to T. The bending moment at the interface between the web and the flange is taken as Mc, and the moment at the bolt line due to prying force Q
4、 is taken equal to Mc where is equal to the ratio of the net area (at the bolt line section bb) and the gross area (at the web face section aa) of the flange. The represents the ratio between the moment per unit width at the centerline of the bolt line and the flange moment at the web face. When = 0
5、, it corresponds to the case of single curvature bending, i.e., no prying action, and = 1 corresponds to double curvature bending and maximum prying action. Note that, from physical considerations, 0 1. GENERAL DEVELOPMENT Considering equilibrium of the portions of the flange shown in Figs. 1c and 1
6、d, the following independent equilibrium equations result: Mc Tb + Qa = 0 T + Q Bc = 0 Qa Mc = 0 William A. Thornton, PhD, is Chief Engineer, Cives Corporation, Atlanta, Georgia. Fig. 1. Prying action analytical model * This paper is part of the authors paper “Details in Bolted Steel Construction,“
7、presented at the AISC National Engineering Conference, Tampa, Florida, March, 1984. It was also presented in the BSCE/ASCE Structural Group Lecture Series, at MIT, October 1983 SECOND QUARTER / 198567 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any
8、 part thereof must not be reproduced in any form without the written permission of the publisher. If T is taken as a known applied load, Mc, Q, Bc and are unknowns. The problem is statically indeterminate and no elastic solution is possible without recourse to compatibility and constitutive relation
9、ships. Alternately, limit analysis can be used. This is the approach taken in Ref. 2. Reference 2 also proposed an adjustment in the position of the bolt force as shown in Fig. 2 to bring the theoretical and experimental results closer together. Replacing b with b = b d/2 and a with a = a + d/2, the
10、 equilibrium equations can be rearranged into the following two equations: Tb Mc + = 1 TBc1 1 + + = Fig. 2. Influence of flange deformations on location of resultant bolt force where p = b/a. These are the basic equations for prying analysis. A third equation which provides an explicit result for Q
11、is: QT= + 1 Next, introducing the limit state conditions: Mc M where Mpt Fy= 1 8 2 and Bc B where B = specified allowable bolt tension, any solution to the following two inequalities: Tb pt Fy + 1 1 8 2 TB1 1 + + is a valid solution to the prying action problem. The solution space for these inequali
12、ties is shown in Fig. 3 in dimensionless form by introducing the parameter tc, where: t Bb pF c y = 8 In terms of T/B and t/tc, the above two inequalities can be rewritten as: T B t tc + ()1 2 (1) T B + + 1 11 ) ( (2) The family of curves labeled a in Fig. 3 is obtained from the Inequality 1 above w
13、ith the inequality sign replaced by the equality sign. The family of curves labeled b in Fig. 3 is obtained from the prying force equation: Q B T B = + 1 with: T B t tc =+ ()1 2 (3) 68ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION 2003 by American Institute of Steel Construction, Inc
14、. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher. Fig. 3. Solution space for prying action analysis Thus, curves b are given by: Q B t tc = 2 (4) The curve c in Fig. 3 is the locus of points for which: T B
15、Q B +=1 and is given by: T B t tc = + + 1 1 1 2 The boundary between the region of solutions to Inequalities 1 and 2, Region A and the remainder of the solution space, Region B, is denoted by the cross-hatched curve ROABP of Fig. 3. It will be apparent from Fig. 3 that there is no unique solution to
16、 the prying action problem. For instance, if the applied tension is given, T/B is known and any value of t/tc from curve OAB to t/tc is a solution. Likewise, if t is given, t/tc is known, and any value of T/B from 0 to curve OABP is a solution. Obviously, efficient solutions are those that lie on cu
17、rve OAB. Points on this curve give the least required material thickness t for a given applied tension T, or the largest allowable applied load T for a given material thickness. Thus, methods for achieving points which are on or close to curve OAB will be developed. METHODS OF SOLUTION Method 1. Thi
18、s method solves the problem: Given: T, a, b, p, Fy, and B Find:the smallest value of t Such that:Inequalities 1 and 2 are satisfied It can be verified that the solution to this problem is given by the following algorithm: 1.Check T B; if so proceed, if not use more or stronger bolts 2.Then calculate
19、 = 1 1 B T SECOND QUARTER / 198569 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher. 3.If 1 set = 1 4.If 0 treqd, the design point will not lie on curve
20、OAB, but will be to the right of OAB. Thus, the actual value of will be less than the value calculated above. This reduced value of , say act, can be calculated from Eq. 3 above, as: act actc T B tt = 1 1 2 / (/ ) ifact treqd = .651, the W1860 is o.k. To calculate the prying force: tc= = 81941417 45
21、36 11651 . . act= = 1 819 11 194 695 11651 17246 2 . /. (./ .) . Qact= 19481972466582 695 11651 2 . . = 2.696 kips Example 2 This example is also drawn from the Manual. It is Ex. 2 on p. 4-92, and involves the same situation as Ex. 1, but with a fatigue loading of more than 20,000 but less than 500,
22、000 cycles. Section B3 of the AISC Specification has a provision for reducing the allowable bolt tension B to .6 B, exclusive of prying force if Q/T .10. In this example, Q/T = 2.696/11 = .2451 .10, thus B = .6 19.4 = 11.64. Since 11.64 11.0, the connection is satisfactory. Example 3 This is Ex. 4,
23、pp. 4-92 and 4-93 of the Manual, and it serves to demonstrate the solution to problems in which the bolts are subjected to both tension and shear. The bolts are A325N 3/4-dia. Skipping the preliminary selection routine which is performed here exactly as it was in Ex. 1, as a 5/8-angle is chosen and
24、the geometric data are a = 1.875, b = 1.50, = .819, p = .8000, T = 8.95, and the shear per bolt V = 26.8/6 = 4.47. Interaction enters this problem as (see Table 1): B = .4418 55 1.8 4.47 = 16.253 kips Since 16.253 8.95, we proceed to calculate: = = 1 8 16253 895 1102 . . . .and = 1 Table 1. Interact
25、ion Expressions for Bearing Connections Value of B Bolt Type Threads IncludedThreads Excluded A32555Ab 1.8V 44Ab55 1.4V 44Ab A49068Ab 1.8V 54Ab68Ab 1.4V 54Ab A30726Ab 1.8V 20Ab 70ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION 2003 by American Institute of Steel Construction, Inc. All
26、 rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher. Thus treqd= = Br, more or larger bolts would have to be used to proceed. Continuing: = = 1 8 194 895 114595 . . . . Since 1, set = 1, and: treqd= = 1, set = 1 (m
27、aterial thickness controls), and T pt F b allow y = + 2 8 1 () 5.If 0 1 (bolts and material thickness both control), and T B allow= + + () () 1 11 or T pt F b allow y = + 2 8 1 () The two values given for Tallow for the latter case will always be equal. The designer can choose which one he prefers t
28、o calculate. As in Method 1, an initial choice of section is made to get t, a and b. The initial choice can be based, as before, on: t Tb pFy prelim= 2 Once a section is chosen, Tallow is calculated. If Tallow T, the choice is adequate. If Tallow 1, set = 1, and Tallow= = 4569536 81417 18191256 2 .
29、kips Since 12.56 kips 11.0 kips, the W1860 tee and -dia. bolts are o.k. To calculate the prying force: tc= = 81941417 4536 11651 . . act= = 1 819 11 194 695 11651 17246 2 . /. (./ .) . Qact= 19481972466582 695 11651 2 . . = 2.696 kips As expected (and obvious from the equations used), this is the sa
30、me result obtained by Method 1. Example 6 This is the same as Ex. 3 of Method 1. Given the data of Ex. 3, T = 8.95 1, set = 1, and Tallow= = 4562536 8 15 1819959 2 . kips Since 9.59 kips 8.95 kips, the -angle and -dia. A325N bolts are o.k. As will be obvious from previous calculations, act = .8513 a
31、nd Qact = 2.942 kips. Example 7 This is the same as Ex. 4 of Method 1. The first (basic) method given in Ex. 4 proceeds as follows: = 14.542 T = 8.95, so calculate as: = = 1 819 18 8 14524 15 45 62536 11192 2 .( . ) . Since = 1.1922 1, set = 1, and: Tallow = 9.59 kips 8.95 kips o.k. From the calcula
32、tions of Ex. 3, act = .8514 and Qact = 2.942 kips. The second (alternate) method given in Ex. 4 proceeds as follows: Br = 14.542 kips 8.95 kips, so calculate as: = = 1 819 18 8 194 15 4562536 118171 2 .( . ) . Since = 1.8171 1, set = 1, then Tallow = 9.59 kips 8.95 kipso.k. From the calculations of
33、Ex. 3, act = .8514. Qact = 2.942 kips. As mentioned earlier, the alternate method for friction connections can yield significantly lighter (cheaper) connections than the first method, but the above examples, which are taken from the AISC Manual, do not show this. Consider then the following example:
34、 Example 8 The framed connection shown in Fig. 4 is subjected to 65 kips of shear. The shop and field bolts are A325 -dia. A friction-type connection is required and the surface class is A clean mill scale. Standard holes are used, so Fv = 17.5 ksi. Determine the maximum tension this connection can
35、carry. The fundamental parameters can be calculated from the given information. Thus: b = 3 .625 = 2.3750 b= 2.3750 .3750 = 2.0 a = = 1, set = 1, and Tallow= += 362536 8 20 172924559 2 . . (.). Remembering that the applied tension cannot exceed Br: Tallow = min 4.559, Br = 4.4554 kips and the total
36、allowable applied tension is: Ttotal = 4.4554 10 = 44.55 kips which is 32% greater than the previously obtained value of 33.78 kips. It can be seen that the alternate method is the significantly more economical method of the two. It must be kept in mind there are other checks, involving the shop bol
37、ts and beam web, that must be made to assess the capacity of this joint. Thus, 44.55 kips calculated above may not be the tensile capacity of the joint. The reader can verify that the maximum allowable tension, at 65 kips shear, is 41.84 kips, based on resultant shear in the shop bolts. Methods 1 an
38、d 2 for the solution to the prying action problem provide optimal solutions from the point of view of least material thickness or maximum capacity, respectively. Any other method of solution which achieves a point (t/tc, T/B) in region A of Fig. 3, is an acceptable method. Method 3, which follows, i
39、s just such a method. It is an organized version of the method given on pp. 4-89 and 4-90 of the Manual. Method 3. An initial choice of thickness t is required for this method. Note that in Methods 1 and 2, an initial t was not required except that it was needed to estimate a and b. After choosing n
40、umber, type and arrangement of bolts, proceed as follows: Choose t = tact, calculate a, b, p, T, V, B, . Then 1.Check T B; if not increase number of bolts or use larger or stronger bolts. 2.Calculate = 1 1 1 1 2 T B tt Tb M actc / (/ ) (AISC Eq. 3) 3.If 1, set = 1 t Tb pF t reqd y act = + 8 1() n.g.
41、 Choose a new tact treqd, or increase number of bolts; go to Step 1. 4.If 0, set = 0 t Tb pF t reqd y act = 8 o.k. 74ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must no
42、t be reproduced in any form without the written permission of the publisher. Choice of bolts and tact is satisfactory, no further calculations are required; go to Step 6. 5. If 0 1: t Tb pF t reqd y act = + = 8 1() . o.k Check BT p B c= + + 11 1 () (AISC Eq. 4) If o.k., go to Step 6, otherwise choos
43、e more or stronger bolts, or increase tact, and go to Step 1. 6. Solution is complete. Prying force, if required, is calculated from: QB t t act act c = 2 In the above algorithm, when shear is present, B is determined from the interaction equation for bearing connections or friction connections. In
44、the alternative method for friction connections B is replaced by the reduced B = Br only in Step 1. Everywhere else B appears it is the unreduced tension value. The above algorithm will seem to differ from the Manual procedure in that AISC Eq. 5 (Manual p. 4-89) does not appear. Actually, AISC Eq. 5
45、 can be written as: reqd t Tb pF f y = + 8 1() (AISC Eq. 5) which can be verified by direct substitution of AISC Eq. 4 into AISC Eq. 5, thereby eliminating the appearance of B and simplifying the expression. In this simplified form, it can be seen that AISC Eq. 5 does indeed appear in Steps 3, 4 and
46、 5 of the above algorithm. As a final comment on this method, it will be noticed in Steps 3, 4 and 5 that the result of the comparison of treqd with tact is known in advance. This occurs because the same equation is being used to calculate in Step 2 and treqd in Steps 3, 4 and 5. Thus, there is no n
47、eed actually to calculate treqd in Steps 4 and 5. In Step 3, treqd is calculated to provide a new guess for tact if a thicker angle or tee is decided upon rather than more bolts. REFERENCES 1.American Institute of Steel Construction, Inc. Manual of Steel Construction 8th Ed., 1980, Chicago, Ill., pp. 4-88 through 4- 93. 2.Fisher, J. W. and J. H. A. Struik Guide to Design Criteria for Bolted and Riveted Joints Wiley-Interscience, New York, N.Y., 1974, pp. 270-279. SECOND QUARTER / 198575 2003 by American Institute of Steel Co