000_[Jonathan_Ochshorn]_Structural_Elements_for_Architecture_400

218 CHAPTER 9 Connections

FIGURE 9.23
Double-shear bolted connection with multiple fasteners and steel side plates for Example 9.7
(same as Figure 9.16 for Example 9.4)

Douglas Fir-Larch (North) No. 1; the side plates are ASTM A36 steel; and the bolts are fabri-
cated from ordinary, low-strength, A307 steel, as is typical for wood connections. Assume live
and dead loads only, dry fabrication and service conditions, and spacing as shown in Figure
9.23. Use yield limit and group action equations, rather than tabular values (see Example 9.4
for solution using tabular values).

Solution overview
1. Find the capacity for a single fastener, Z, using yield limit equations.
2. For lag screws and nails, check that penetration into the main member is at least 4D (for

lag screws) or 6D (for nails), and reduce capacity, Z, if necessary.
3. Adjust for duration of load, moisture, and geometry.
4. Adjust for group action using group action factor equations, and then multiply the adjusted

single-fastener capacity by the number of fasteners in the connection.
5. Check that the element itself is designed in a manner that accounts for the presence of

bolt or lag screw holes (not included in this example).

Problem solution
1. To find the lateral design value, Z, for a single fastener using yield limit equations, follow

the step-by-step method outlined in Table A-9.15. The main member is oriented so that
the load is parallel to the direction of grain, as defined in Figure 9.8. The orientation of the
steel side plates to the direction of load is not relevant, since there is no “grain” in the steel
plates that influences its strength.

From Table A-3.11 (specific gravity), G ϭ 0.49 for Douglas Fir-Larch (North).
D ϭ 5⁄8 in. ϭ 0.625 in.

Wood 219

Main member (D Ͼ 0.25 in., wood, loaded parallel to grain): Fem ϭ 11,200G ϭ 11,200
(0.49) ϭ 5488 psi. Side member (A36 steel): Fes ϭ 87,000 psi. It is common to round
these values to the nearest 50 psi, so we will use Fem ϭ 5500 psi.

Fyb ϭ 45,000 psi for bolts.
Dowel bearing lengths are lm ϭ 5.5 in. and ls ϭ 0.25 in.
Compute the terms Re ϭ Fem /Fes ϭ 5500/87,000 ϭ 0.06322; and Rt ϭ lm /ls ϭ
5.5/0.25 ϭ 22.0.

Rd ϭ 4Kθ ϭ 4(1.0) ϭ 4 (for Yield Modes Im and Is); Rd ϭ 3.6Kθ ϭ 3.6(1.0) ϭ 3.6 (for
Yield Mode II); and Rd ϭ 3.2Kθ ϭ 3.2(1.0) ϭ 3.2 (for Yield Modes IIIm. IIIs, IV). In these
equations, Kθ ϭ 1 ϩ 0.25(θ/90) ϭ 1.0, since θ ϭ 0°.
Compute the following coefficients:

k1 ϭ Re ϩ 2Re2(1 ϩ Rt ϩ Rt 2) ϩ Rt 2Re3 Ϫ Re (1 ϩ Rt )
(1 ϩ Re )

ϭ 0.06322 ϩ 2(0.06322)2(1 ϩ 22 ϩ 222) ϩ 2220.063223 Ϫ 0.06322(1 ϩ 22)
(1 ϩ 0.06322)

ϭ 0.5687

k2 ϭ Ϫ1 ϩ 2(1 ϩ Re ) ϩ 2Fyb (1 ϩ 2Re )D 2
3Femlm2

ϭ Ϫ1 ϩ 2(1 ϩ 0.06322) ϩ 2(45,000)(1 ϩ 2x 0.06322)(0.625)2
3(5500)(5.5)2

ϭ 0.4852

k3 ϭ Ϫ1 ϩ 2(1 ϩ Re ) ϩ 2Fyb (2 ϩ Re )D 2
Re
3Feml 2
s

ϭ Ϫ1 ϩ 2(1 ϩ 0.06322) ϩ 2(45,000)(2 ϩ 0.06322)(0.625)2
0.06322 3(5500)(0.25)2

ϭ 9.1967

Compute Z for all applicable yield modes (four applicable modes for double shear):

For Yield Mode Im, Z ϭ DlmFem /Rd ϭ 0.625(5.5)(5500)/4 ϭ 4726.6 lb.
For Yield Mode Is, Z ϭ 2DlsFes/Rd ϭ 2(0.625)(0.25)(87,000)/4 ϭ 6796.9 lb for double shear.
Yield Mode II does not apply to double-shear connections.

Yield Mode IIIm does not apply to double-shear connections.

For Yield Mode IIIs, Z ϭ 2k3DlsFem ϭ 2(9.1967)(0.625)(0.25)(5500) ϭ 2394.1 lb.
(2 ϩ Re )Rd (2 ϩ 0.06322)(3.2)

For Yield Mode IV, Z ϭ 2D2 2FemFyb ϭ (2)(0.625)2 2(5500)(45,000) ϭ 3041.4 lb.
Rd 3(1 ϩ Re ) 3.2 3(1 + 0.06322)

220 CHAPTER 9 Connections

The smallest of the various yield mode values is then selected: Z ϭ 2394.1 lb based on
Yield Mode IIIs.
2. Penetration is only an issue with lag screws and nails, since bolts must always fully pen-
etrate the members being connected. Therefore, no reduction of the lateral design value is
necessary, and it remains equal to Z ϭ 2394.1 lb.
3. Adjustments are as follows (same as for Example 9.4):
CD for typical values of live and dead load is 1.0 (Table A-9.4).
CM for members fabricated and used dry is 1.0 (Table A-9.5).
CΔ is found by testing four separate criteria (Table A-9.7): spacing between fasteners in
a row, spacing between rows of fasteners, end distance, and edge distance. It is some-
times useful to sketch the members separately, showing dimensions for the relevant geom-
etry factor parameters (Figure 9.24). Only the wood main member is considered here; the
tension capacity and bolt spacing in the steel plate must be considered separately (see
Chapter 6 for discussion of tension and the steel section of this chapter for discussion of
bolt spacing).

In the calculations that follow, D is the fastener diameter of 5⁄8 in. ϭ 0.625 in.
Spacing criteria: Adjustment criteria for spacing appear in Table A-9.7, parts A and B. For
spacing between fasteners in a row, where the loading direction is parallel to grain, the mini-
mum spacing for full value is 4D ϭ 4(0.625) ϭ 2.5 in. Since the actual spacing is 2.5 in.,
the full value applies, and CΔ ϭ 1.0. For spacing between rows of fasteners, again with the
loading direction parallel to grain, the minimum required spacing is 1.5D ϭ 1.5(0.625) ϭ
0.9375 in. Since the actual spacing (between rows) of 2.5 in. exceeds this value and is no
greater than 5 in. (the maximum distance allowed between the outer rows of fasteners), the
geometry factor is CΔ ϭ 1.0.
End distance: Adjustment criteria for end distance appear in Table A-9.7, part C. For the
main member, the loading direction is parallel to grain. Where the fasteners are bearing
toward the member end (in tension) and for softwood, the minimum end distance for full
value (i.e., for CΔ ϭ 1.0) is 7D ϭ 7(0.625) ϭ 4.375 in. The actual end distance of 5 in. is
greater than this, so the geometry factor is CΔ ϭ 1.0.
Edge distance: Adjustment criteria for edge distance appear in Table A-9.7, part D. For
the main member, the loading direction is parallel to grain, so the minimum edge distance

FIGURE 9.24
Geometry factor parameters for Example 9.7 (same as Figure 9.17 for Example 9.4)

Wood 221

is determined from the so-called slenderness ratio of the fastener, l/D. The fastener length,

l, within the main member is 5½ in., so l/D ϭ 5.5/0.625 ϭ 8.8. Since this value is greater

than 6, the minimum edge distance is either 1.5D ϭ 1.5(0.625) ϭ 0.9375 in., or one-half

of the spacing between rows ϭ 0.5(2.5) ϭ 1.25 in., whichever is greater: the minimum

edge distance is therefore 1.25 in., which the actual edge distance of 1.5 in. exceeds. The

geometry factor, therefore, is CΔ ϭ 1.0.
The geometry factor for the entire connection is found by using the smallest of the geom-

etry factors found for any of the four conditions just tested (end, edge, and the two spacing

conditions where applicable); therefore, we use CΔ ϭ 1.0.
The adjusted lateral design value for a single bolt in the connection is found by multiply-

ing the lateral design value from step 2 by the various adjustment factors determined in

step 3: CD, CM, and CΔ: Z Ј ϭ Z(CD)(CM)(CΔ) ϭ 2394.1(1.0)(1.0)(1.0) ϭ 2394.1 lb.
4. The group action factor, Cg, can be found based on the method described in Note 3 of

Table A-9.6:

D ϭ 0.625 in.
γ ϭ 270,000(D1.5) ϭ 270,000(0.6251.5) ϭ 133,409.

s ϭ 2.5 in.

Em ϭ 1,600,000 psi (Table A-3.9); Es ϭ 29,000,000 psi (Table A-3.12, Note 1).
Am ϭ 30.25 in2; As ϭ 2(0.25 ϫ 5.5) ϭ 2.75 in2 (Table A-4.1).

u ϭ 1 ϩ (133,409) 2.5 ⎡⎣⎢⎢ (1,600,0010)(30.25) ϩ (29,000,0100)(2.75)⎥⎤⎥⎦ ϭ 1.0055; and
2

m ϭ 1.0055 Ϫ 1.00552 Ϫ 1 ϭ 0.900.

REA ϭ [(1,600,000)(30.25)]/[(29,000,000)(2.75)] ϭ 0.607.
n ϭ 2.

( )Cg ⎣⎡⎢⎢⎢⎢ 2 ⎥⎤⎥⎥⎦⎥
ϭ ⎣⎢⎡(1 0.900 1 Ϫ 0.9002(2) 0.9002(2) ⎤⎦⎥ ⎣⎢⎢⎡11 ϩ 0.607 ⎥⎤⎦⎥ ϭ 0.999.
ϫ 0.9002)(1 ϩ 0.900) Ϫ Ϫ 0.900
ϩ 0.607 1 ϩ

Adjusting for group action and multiplying the single-fastener value for Z Ј found in step 3
by the number of fasteners in the connection, we get a total adjusted connection capacity
equal to 2394.1(0.999)(4) ϭ 9567 lb.
5. We are not considering the design of the structural elements themselves in this example.
Tension, row, and group tear-out are considered in Chapter 6, Example 6.2.
6. Conclusion: The total capacity of the connection (consisting of six ½-in.-diameter bolts) is
9567 lb.

Withdrawal

Where a fastener is itself stressed in tension, it is considered to be loaded in “with-
drawal,” as a failure of the connection would cause it to “withdraw”—pull out—
from the member into which it was inserted. For lag screws and nails, selected
withdrawal design values, designated W to distinguish them from lateral design