Eccentric bolt group

A group of bolts under an in-plane load whose line of action misses the centroid, by the instantaneous center of rotation (IC) method or the elastic method of the AISC Manual, 16th Edition, Part 7, with the per-bolt strengths of AISC 360-22 Section J3. The result is the coefficient C, the available strength C rn against your load, the force in every bolt and the critical bolt. Use it for a bracket, a shear tab with a large eccentricity or any pattern the Manual does not tabulate.

Inputs of the eccentric bolt group #

• Load: service loads by type (each ASCE 7-22 2.3.1 or 2.4.1 combination of the types present is evaluated, all along the same load line; W and E act both ways) or the required strength directly. See Loads and combinations.
• Load line: the angle θ from the vertical (0 is straight down; a positive angle turns the load clockwise, so the arrow points down and to the left, as in the Manual tables) and the point the load passes through, ex to the right of and ey above the centroid. The Manual's ex is measured at the level of the centroid: ex − ey tan θ, as in Design Example II.A-24 Solution B. The calculation shows it.
• Bolt pattern: bolts per vertical row n, number of vertical rows, pitch s and gage g; or custom coordinates (any origin, the centroid is computed), up to 200 bolts.
• Bolts: diameter, grade (Group 120, 150, 200 or A307), threads in or out of the shear plane, single or double shear, hole type, joint type (snug-tightened, pretensioned or slip-critical). A slip-critical joint also takes the faying surface class, the pretension multiplier Du (AISC 360-22 J3.9 gives 1.13, another value only as approved by you as the engineer of record) and whether two or more fillers are present (hf = 0.85). An A307 bolt takes the grip.
• Connected ply: thickness of the thinnest connected part, material and edge distance le along the load, used for bearing, tearout and the spacing and edge limits. A thickness of 0 checks bolt shear alone and says so in the results, as Design Examples II.A-24 and II.A-25 assume. Whether hole deformation at service load is a design consideration is your judgment (J3.11a: 2.4 d t Fu and 1.2 lc t Fu when it is, prefilled, or 3.0 and 1.5 when it is not). Unpainted weathering steel sets the smaller maximum spacing of J3.6.
• Code options: the strength method, and whether the connection is end-loaded. Table J3.2 note [c] reduces the shear stress to 83.3 % when an end-loaded fastener pattern is longer than 38 in. Whether your bolts are end-loaded is your judgment; on is the prefilled, conservative choice, and it changes nothing below 38 in.

Auto-design varies the bolt diameter, the bolts per vertical row (1 to 16) and the number of vertical rows (1 to 4), fewest bolts first; the pitch, gage, load line, ply and joint stay as you entered them and the grade is locked unless you unlock it. A custom pattern keeps its coordinates and varies the diameter and grade only.

Checks performed #

IC method. Each bolt's force follows the load-deformation curve R = Rult (1 − e^(−10Δ))^0.55, with the deformation proportional to the distance from the instantaneous center and 0.34 in. at the farthest bolt (Manual Eqs. 7-9 to 7-14). The calculation finds the center by equilibrium and reports C = P / Rult. The Manual's iterative tables use a 1 % convergence, so the printed values can differ from the exact solution by about that much.

Elastic method. Direct shear P / n plus the moment shear r = M l / Ip at each bolt, added as vectors (Manual Eqs. 7-2 to 7-8). C is P over the largest resultant. It ignores the ductility of the bolts and is more conservative.

Tearout. The clear distance lc is the smaller of le − h/2 and s − h, and that value is used for every bolt: the simplification of Design Example II.A-22, which is conservative. The hole dimension is the one along the nearer of the vertical and horizontal axes to the load, and le is entered, not derived from an outline, because the connected parts are not modelled.

Manual table note. Between tabulated angles the Manual warns that straight-line interpolation may be wrong and says to use a direct analysis or the next lower angle: the note reads the next lower angle, and the check uses the direct analysis at your angle. The note appears for s = 3 or 6 in., up to 12 bolts per row and up to four rows. For a gage the Manual does not tabulate it says so, and gives the IC value at the table's grid points. With Group 150 or Group 200 bolts a note repeats the Manual's statement that the tables, built on the Group 120 curve, are conservative.

The drawings show the bolt pattern with the pitch, gage, edge and load point dimensioned, and the bolt forces of the strength method in proportion, with the instantaneous center and the critical bolt marked.

Limitations #

• In-plane loads on one group. Out-of-plane loads (bolts in shear and tension, Manual Part 7 Case II) and a pure moment with no force are not covered: enter a load at the eccentricity you want. The printed C' of the tables is verified but not offered as a result.
• The connected parts are not designed: plate yielding, rupture, block shear and buckling are separate checks.
• One thinnest ply stands for the group's bearing and tearout. Different plies per bolt, and a separate interior bolt strength (as Examples II.A-30 and II.A-17 use), are not modelled; the result is on the conservative side.
• An inclined load uses the hole dimension and the slot direction of the nearer axis (vertical or horizontal).
• ASTM F3148 (Group 144) is not in the grade list: its Table J3.2 strengths are not modelled.
• Manual Table 7-14 is a table of fastener dimensions and is not used; nor are the clearance tables.

Verification examples #

Verification status: 98 values compared in 37 examples (37 published, 0 hand calculations from the cited equations): 97 within tolerance, 1 recorded deviations. Code basis: the provisions it uses were read in the target edition.

AISC Design Examples v16.0, Vol. 1, against the published values (more values, LRFD and ASD, are on the Verification page):

AISC Manual, 16th Edition, tables, read from the page images:

The other 818 cells of Tables 7-9 to 7-13 were not compared: two machine readings of the page disagreed and they were not read by hand. The 12,142 cells were read by machine, and those that missed the method or could not be read (209) were read by eye; the 16 sample cells and Tables 7-1 and 7-3 by eye. Table 7-14 is not a coefficient table (it gives fastener dimensions) and is not part of this calculation.

The one difference: at 75 degrees, s = 6 in. and ex = 2 in., the larger groups (n = 9 to 12, and the row for Tables 7-6 to 7-8 likewise) are printed up to about 2.3 % above the IC method, for example Table 7-10, n = 9: 26.4 against 25.85. The page images confirm the printed values, and several are printed the same for two different gages. The calculation uses the method, which is on the safe side; this is a listed deviation.

Slip through a group, and bearing or tearout through a group, have no published example with an eccentric load: the per-bolt strengths are verified as above and the group strength is C rn, as Manual Part 7 states.