Bearing at an angle and under washers: braces and bolt heads

Two bearing questions came up in earlier chapters and were left open. The fire tower braces (Hibbeler Chapter 12) push into the legs at an angle, not straight along or across the grain. And the roof's hold-down bolt (Breyer Chapter 6) pulls on a washer that bears on the wood, which that chapter did not check. Wood is far stronger along the grain than across it, so both depend on the geometry.

Bearing at an angle to the grain

When the load makes an angle θ with the grain, the NDS (3.10.3, Equation 3.10-1) gives the allowed pressure as Fθ′ = Fc* × Fc⊥′ ÷ (Fc* × sin²θ + Fc⊥′ × cos²θ). Fc* is the along-the-grain compression value with every adjustment except column stability, so it includes the load duration factor. Fc⊥′ is the across-the-grain value, which gets no load duration increase because it is based on how far the wood squashes. At 0° the formula returns Fc*; at 90° it returns Fc⊥′; in between it falls fast.

• A No. 2 Southern Pine 4×4 (Fc 1,450 psi, Fc⊥ 565 psi, no load duration increase): 1,450 psi along the grain, 1,312 at 15°, 1,042 at 30°, 813 at 45°, 667 at 60°, 589 at 75°, and 565 across the grain. A 45° push is allowed 56% of the along-the-grain pressure.
• A fire tower brace, a 2×4 end (1.5 × 3.5 in) bearing at 45° on a No. 2 timber leg (Fc 525, Fc⊥ 375 psi): about 437 psi, or about 2,300 lb. With the wind duration factor of 1.6 on the along-the-grain term it is about 519 psi, or about 2,720 lb.
• Chapter 12 has no brace force yet (the tower wind load is still an open item), so this chapter gives the capacity to compare with it, not a verdict.

Under a washer

A bolt in tension pulls its head or nut against a washer, and the washer bears on the wood across the grain. The NDS asks for a standard cut washer, or a plate or strap at least as large, under both the head and the nut (12.1.3.3). The bearing area is the washer's face minus the bolt hole (the net bearing area, 3.10.2). For a round washer the bearing length used for the area factor is its diameter (3.10.4), so a small washer away from the member's end gets a bonus of (diameter + 0.375) ÷ diameter. At the end of a member, or within 3 in of it, there is no bonus.

• Standard cut washers (NDS Appendix Table L7): 1/2 in bolt 1.375 in across, 5/8 in bolt 1.75, 3/4 in bolt 2.0, 7/8 in bolt 2.25, 1 in bolt 2.5. The calculator takes the hole as the largest the NDS allows, 1/16 in over the bolt.
• On a No. 2 Southern Pine 4×4 (Fc⊥ 565 psi) the wood under the washer carries about: 1/2 in bolt, 700 lb near the end of the post and 890 lb away from it; 5/8 in, 1,150 and 1,400 lb; 3/4 in, 1,480 and 1,760 lb; 7/8 in, 1,860 and 2,170 lb; 1 in, 2,270 and 2,610 lb.
• Timber is weaker across the grain: a No. 2 timber (Fc⊥ 375 psi) with a 1/2 in washer away from the end carries about 590 lb.

The roof hold-down

Chapter 6 found the roof's net uplift is about 450 lb. A 1/2 in bolt in a 4×4 post, with its standard washer, can only transfer about 700 lb through that washer when it sits within 3 in of the end of the post, and about 890 lb elsewhere. Wet wood cuts that to about 470 and 600 lb. The 450 lb uplift is within reach of the washer, but only just, wet and near the end. The wind load duration factor does not help here, because bearing across the grain gets none. A larger plate washer, or a bolt in a position where the wood is stronger, buys the margin back.

Answer

Both checks pass for the loads we have, but both are tighter than the along-the-grain numbers suggest. A brace at 45° bears at roughly 55 to 85% of the along-the-grain pressure, depending on the wood and the load duration factor, and the standard washer on a 1/2 in hold-down bolt is about 1.6 times the net uplift when dry and near the post's end, and close to it when wet. For a tie-down that matters, use a plate washer (a square or round steel plate several times the standard washer's area) and keep the bolt out of the 3 in zone at the post's end.

Where to be cautious

• The angle is between the load and the grain, not between the brace and the horizontal. A brace that is 45° from horizontal pushing into a vertical leg is 45° from the grain; one pushing into the leg's side face squarely is 90°.
• Do not apply the load duration factor to the across-the-grain part: it applies only to the along-the-grain value inside the formula.
• The washer area bonus is for a bearing away from the end of the member and under 6 in across, and it does not apply near the end. Wet service lowers the across-the-grain value by a third for dimension lumber; timber values already cover wet or dry service.
• None of this checks the steel: the bolt's tension strength, the washer or plate itself, or the nut's thread. Those are separate.
• Everything here is a worked example. A tower people climb or a roof over people is for a structural engineer to design and stamp.

Open items for a later pass

The real brace force from the tower wind load. The shear check for a bolted connection close to a beam's end (NDS 3.4.3.3): its equations were read, but the effective-depth definition depends on a figure that could not be read here, so it is not built. A rectangular plate or strap as the bearing (use its real length and width in the angle calculator), end-grain crushing of a post under a vertical load, and the bolt's tension strength.

Sources

• American Wood Council, NDS 2018 with Commentary (free at awc.org): 3.10.2 (net bearing area), 3.10.3 and Equation 3.10-1 (bearing at an angle), 3.10.4 and Table 3.10.4 (area factor Cb, round washers use the diameter), 12.1.3.3 (washers), Table 2.3.2 note 1 (no load duration factor on Fc⊥), and Appendix Table L7 (standard cut washers).
• Southern Forest Products Association, Southern Pine Use Guide, 2013: Table 1 (4× dimension lumber), Table 3 (timbers) and Table A-2 (wet service).
• No published numeric example was found for these checks; the equations are read from the NDS text and checked at their limits (0° returns Fc*, 90° returns Fc⊥′). Written from public documents, not from the Breyer textbook itself. If the book disagrees, the book and the current NDS win.