A More GENERALIZED solution to ANALYZe BOLTED AND WELDED CONNECTIons
Dung M. LUE[1], Ching H. LIN[2], and Wei T. HSU2
Abstract: Two general approaches for the analysis of eccentrically loaded connections, which bolted or welded, have been developed through the years. The first one is a very conservative approach. The second one, the ultimate strength method, provides the most realistic values as compared with the test results but is extremely tedious to apply. In AISC-ASD design, a tabulated solution is intended for eccentric loads that are vertical and is not applicable directly to eccentric loads that are inclined at an angle from vertical. In AISC-LRFD design, a tabulated solution for eccentric loads that are inclined at specified angles from vertical is presented. In AISC-ASD design, an approximate method is also furnished for the use in dealing with eccentric loads that are not vertical. However, the approximate approach was appreciably underestimated and unreasonable predictions were detected when the inclined angle from vertical was approaching 90 degrees. This study provides a more generalized solution to overcome the problems as specified above. Firstly, the methods to evaluate the strength of eccentrically loaded connections based on the elastic and ultimate analysis with detailed steps are presented. A numerical procedure for the analysis of eccentric loads that are inclined at any angles from vertical is furnished. Secondly, a written program is developed to perform the calculations involved. This is because of the long and tedious procedure for the strength evaluation is not a common practice for a designer. Finally, the completion of this study will remove the deficiencies of the current AISC-ASD and AISC-LRFD design manuals. This will also accomplish a better solution for practicing engineers in their routine design.
KEYWORDS: AISC, Bolted and welded connections, Eccentric load, Instantaneous center of rotation
1. Introduction
Two approaches for the analysis of eccentrically loaded connections are being specified in the AISC Specifications [1-3]. The first method is the very conservative elastic method in which friction or slip resistance between the connected parts is neglected. In addition these connected parts are assumed to be perfectly rigid. The second method, which is also called the ultimate strength method, provides the most realistic values as compared with test results but is extremely tedious to apply, at least with hand-held calculators. The tables in the AISC design manuals for eccentrically loaded connections are based on the ultimate strength method and enable us to solve most of these types of problems quite easily as long as the bolted or welded patterns are those listed in the AISC design manuals. Although the ultimate method and the use of design manuals have been known and developed in the AISC design manual (2005), the results for the inclined loads, with the angles somewhere between 75 and 90 degrees, may result in a value much larger than the one by theory. This is obviously not acceptable and a better way to treat this is anticipated. In the AISC Specification [3], ultimate methods are introduced and the values for inclined load at some particular angles but not any angles are furnished. A further investigation on the inclined loads at any angles is anticipated and this has been done in this research.
2. norminal shear strength of bolted and welded connections
2-1 Bolted connections
The combined effect of rotation and translation is equivalent to a rotation about a point defined as the instantaneous center of rotation (IC) as illustrated in Figure 1. The location of the IC depends upon the geometry of the bolt group as well as the direction and point of application of the load.
(a) Instantaneous center of rotation (IC) / (b) Forces on bolts in groupFigure 1. Instantaneous center of rotation method / bolted connections
The load-deformation relationship [4] for one bolt is illustrated in Figure 2.
(2)
where
R = nominal shear strength of one bolt at a deformation D, kips. (1 kip = 4.448 kN)
Rult = ultimate shear strength of one bolt, kips
D = total deformation, including shear, bearing and bending deformation in the bolt and bearing deformation of the connection elements, in. (1 in. = 2.54 cm)
e = 2.718…., base of the natural logarithm.
The nominal shear strength of the bolt most remote from the IC can be determined by applying a maximum deformation to the bolt. The load-deformation relationship is based upon data obtained experimentally for 3/4-in. diameter ASTM A325 bolts, where kips, and in. Determined the bolt deformation that varies linearly with distance from the IC. The nominal shear strength of the bolt group is, then, the sum of the individual strengths of all bolts.
Figure 2. Load-deformation relationship for one 3/4-in. ASTM A325 bolt
2-2 Weld connections
The location of the IC depends upon the geometry of the weld group as well as the direction and point of application of the load, illustrated in Figure 3.
(a) Instantaneous center of rotation (IC) / (b) Forces on weld elementsFigure 3. Instantaneous center of rotation method / welded connections
Figure 4. Fillet-weld strength as a function of load angle,
The load-deformation relationship [5] for a unit-length segment of the weld, as illustrated in Figure 4,
(3)
where
= nominal shear strength of the weld segment at a deformation , kips.
= weld electrode strength, ksi.
= load angle measured relative to the weld longitudinal axis, degrees.
= ratio of element deformation to its deformation at maximum stress.
Unlike the load-deformation relationship for bolt, the strength and deformation of welds are dependent upon the angle that the resultant elemental force makes with the axis of the weld element. The nominal shear strength of the weld group is governed by of the weld segment that first reaches its limits,
where
(4)
w = weld leg size, in.
The nominal shear strength of the weld group is the sum of the individual strengths of all weld segments. Because of the non-linear nature of the requisite iterative solution, for sufficient accuracy, a minimum of twenty weld elements for the longest line segment is generally recommended.
3. Analysis for eccentrically loaded connections
In the analysis for eccentrically loaded connections, the bolts or welds must be designed to resist the combined effect of direct shear and the additional shear from the induced moment. Eccentricity produces both a rotation and a translation of one connection element with respect to the other. Analytical methods used to evaluate the strength of bolted or welded connections are given as follows:
3-1 Elastic method
The elastic method may be used to design or analyze eccentrically loaded connection not conforming to the AISC manual tables. Each unit element is assumed to support:
a. An equal share of the vertical component of the load.
b. An equal share of the horizontal component of the load.
c. A proportional share (dependent on the fastener’s or element’s distance from the centroid of the group) of the eccentric moment portion of the load.
For a force applied as illustrated in Figure 5, the eccentric force, P, is resolved into a force, P, acting through the center of gravity (CG ) of the group and a moment,
(a) Bolt group / (b) Weld groupFigure 5. Illustration for elastic method
The shear due to the concentric force, , is determined as or (5)
The shear due to the moment, , is determined as (6)
Thus, the required strength,, is determined as (7)
where c is the radial distance from CG to point in bolt or weld group most remote from CG. Other notations used above are being specified or defined in the current AISC design manual.
The maximum load is determined from the vectorial resolution of these stresses at the fastener or element most remote from the group’s centroid. The elastic method is simplified, but may be excessively conservative because it neglects the ductility of the bolt or weld group and the potential for load redistribution and does not render a consistent factor of safety.
3-2 Instantaneous center of rotation method / ultimate strength method
When fastener or weld groups are loaded in shear by an external load that does not act through the center of gravity of the group, the load is eccentric and will tend to cause a relative rotation and translation of the connected material. This condition is equivalent to that of pure rotation about a single point. This point is called the instantaneous center of rotation.
The instantaneous center of rotation method is more accurate, but generally requires the use of tabulated values or an iterative solution. If the correct IC location has been selected, the three equations of in-plane static equilibrium (SFx = 0, SFy = 0, SM = 0) will be satisfied.
3-3 Method by using tables in AISC design manual [3]
Tabulated values are obtained based on the instantaneous center of rotation method. The tables are applicable only for inclined angles (θ) at 0°,15°, 30°, 45°, 60°, and 75°. The design strength of the eccentrically loaded, , is determined as
For bolt group (8)
For weld group (9)
where
C = coefficient for eccentrically loaded bolt and weld groups
C1 = electrode coefficient for relative strength of electrodes
D = number of sixteenths-of-an-inch in the weld size
l = length of weld, in.
rn = nominal strength per bolt from LRFD Specification
4. ILLustrated examples for bolted and welded connections
Three methods are used in the following examples and they are specified as follows:
Method 1: Elastic method / LRFD is the elastic analysis as specified in the 2005 AISC design manual. The calculated Pu is obtained through the use of Eq. (7).
Method 2: Ultimate method / LRFD is the instantaneous center of rotation method as specified in the 2005AISC design manual. The calculated Pu is obtained through the use of LRFD design equation as shown below.
Method 3: AISC / LRFD is the method by employing the tabulated values from the tables listed in the 2005 AISC design manual. This is the same method as the ultimate method. The only difference is that the AISC / LRFD method assumes that the external load (Pu) is applied on the same lever as the center gravity of the bolt group. However, this is not the general cases in common practice.
4-1 Bolted connections
Determine the largest eccentric force Pu for which the design shear strength of the bolts in the connection is adequate using the elastic/LRFD, the ultimate/LRFD, and the AISC/LRFD methods as specified above and in 2005 AISC design manual if the inclined angles (q ) are in the range of 0 to 360 degrees and h = 10 in., which is the vertical distance from the level of center of gravity of the bolt group. Use 7/8-in. A325-N bolts. The e is the eccentricity location of the applied load (Pu) from the center of gravity of the bolt group as shown in Figure 6. Three methods as specified here are employed and the results are compared. When the inclined angles from the vertical (0 £ q £ 360°) are applied, the obtained loads (Pu ) are plot as shown in Figure 7.
Figure 6. Bolted connection / Figure 7. Load-angle relationship (Pu-q )The comparison of Pu-values for various eccentricities (e = 2, 6, 12, 16, 24, and 36 in.) with h = 10 in.
and 0≦q ≦360° are given as shown in Table 1 and Figure 8. The Pu-values are obtained from the AISC design manuals.
Comments on the results:
The results for the inclined loads, with the angles somewhere between 75 and 90 degrees, may result in a value (AISC/LRFD method) much larger than the one by theory (ultimate method). This is obviously not acceptable in practice.
Table 1 Comparison of Pu-values for various eccentricities (e)
with h = 10 in. and 0 ≦q ≦360°.
e / 2 in. / 6 in. / 12 in. / 16 in. / 24 in. / 36 in.Angle, θ
(Degree) / Difference in percentage (%) = (Pu – Pu theory ) / Pu theory
0 / 180 / 0 / 0 / 0 / 0 / 0 / 0
15 / 195 / -8 / -23 / -17 / -14 / -10 / -7
30 / 210 / 16 / -35 / -34 / -29 / -21 / -15
45 / 225 / 52 / -13 / -48 / -44 / -35 / -25
50 / 230 / 64 / 0 / -50 / -48 / -40 / -29
60 / 240 / 87 / 28 / -27 / -49 / -50 / -40
70 / 250 / 107 / 62 / 11 / -13 / -47 / -52
75 / 255 / 117 / 81 / 37 / 13 / -22 / -55
80 / 260 / 126 / 101 / 66 / 46 / 15 / -21
90 / 270 / 144 / 144 / 144 / 144 / 144 / 144
100 / 280 / 137 / 132 / 123 / 117 / 107 / 92
140 / 320 / 83 / 70 / 51 / 61 / 31 / 22
160 / 340 / 39 / 35 / 24 / 19 / 14 / 10
180 / 360 / 0 / 0 / 0 / 0 / 0 / 0
Min. % / -11 / -36 / -51 / -54 / -57 / -58
Max. % / 144 / 144 / 144 / 144 / 144 / 144
Figure 8. Comparison of Pu-values for various eccentricities (e) and angles (q )
4-2 Welded connections
Determine the largest eccentric force Pu for which the design shear strength of the weld elements in the connection is adequate using the elastic method/LRFD, the ultimate method/LRFD, and the AISC/LRFD method as specified above and in 2005 AISC design manual if the inclined angles (q ) are in the range of 0 to 360 degrees and the values of h vary with weld patterns, which are listed in the AISC design manuals. Three methods as specified here are employed and the results are compared. When the inclined angles from the vertical (0 £ q £ 360°) are applied, the obtained loads (Pu ) are plot as shown in Figures 9 and 10. Three methods used here are the elastic method/LRFD, the ultimate method/LRFD, and the AISC/LRFD methods, respectively. They have been specified in previous section.
Comments on the results:
From Figure 9, the followings were found.
The values of Pu are exactly the same in either the channel weld or the box weld case when the value of h is 0 in. When the inclined angle is getting higher, the values of Pu of AISC (vertical welds or horizontal welds) are higher than the ones based on the ultimate method, which is also called instantaneous center of rotation method.