Where the beam meets the post: bearing and notches

The roof beam from Hibbeler Chapter 1 ends on a post, and the Ogopogo joinery in Chapter 14 cuts notches in beam ends. Two checks go with that: does the wood crush where the beam sits, and does a notch leave enough wood to carry the end reaction without splitting. Both are about the ends of a beam, where bending is zero and shear is at its highest.

Bearing: the wood crushing across its grain

A beam sits with its grain running along the beam and the post pushes up across it, so the allowed pressure is the compression-perpendicular-to-grain value, Fc⊥, which for No. 2 Southern Pine is 565 psi (SFPA Southern Pine Use Guide, Table 1). The check is simple: the reaction divided by the bearing area must not exceed Fc⊥′. The NDS bases Fc⊥ on how far the wood squashes, not on breaking, so the load duration factor does not apply to it: snow does not buy extra bearing the way it buys extra bending. Wet wood is weaker (×0.67).

• The roof beam end on a 4×4 post: 1.5 in wide × 3.5 in long = 5.25 in², so 675 ÷ 5.25 ≈ 129 psi against 565 psi, about 23% used. The beam needs only 0.8 in of bearing for this reaction. Bearing almost never governs a light beam on a post.
• Wet: the allowed pressure drops to about 379 psi. Still comfortable here.
• Bearing gets a bonus only in one case (NDS 3.10.4): a bearing under 6 in long and at least 3 in from the end of the member, such as a post under the middle of a beam or a washer or plate. Then Fc⊥ is multiplied by Cb = (bearing length + 0.375) ÷ bearing length, which is 1.25 for a 1.5 in bearing and about 1.1 for a 3.5 in post. At the end of a beam, the usual place, the table value applies as it stands: no bonus.

Notches: what a cut in the bottom of the beam end costs

Cutting a notch in the bottom of a beam end so it sits flush on a ledger or post removes the bottom of the board right where shear is highest, and it leaves a sharp inside corner where the wood wants to split. The NDS (3.4.3.2) takes the notch seriously in three ways. The shear capacity of the notched end falls as the square of the ratio of the remaining depth to the full depth: V′ = (2/3) × Fv′ × b × dn × (dn ÷ d)², where dn is the depth left and d the full depth. It uses the full end reaction, with no allowance for load near the support that an unnotched beam gets. And it limits the notch itself (4.4.3): an end notch may not be deeper than one quarter of the beam depth; an interior notch only in the outer thirds of the span and no deeper than one sixth; and no interior notch on the tension side of a board 4 in thick or more. A notch no deeper than d/6 and no longer than d/3 leaves the beam's stiffness practically unchanged (3.2.3.2), and a gradual taper in place of a square corner reduces the stress concentration.

• The roof beam, a single 2×10 (d = 9.25 in) under the snow load duration factor (Fv′ = 175 × 1.15 ≈ 201 psi), no notch: capacity ⅔ × 201 × 1.5 × 9.25 ≈ 1,860 lb against the 675 lb reaction, about 36% used.
• A 1.5 in notch (dn = 7.75 in): capacity about 1,095 lb, 62% used.
• The deepest notch the NDS allows, d/4 = 2.31 in (dn = 6.94 in): capacity about 785 lb, 86% used. Without the snow increase (CD = 1.0) the same notch has about 683 lb, 99% used: no margin.
• A 4 in notch (dn = 5.25 in, not permitted): capacity about 340 lb against 675 lb, about 198% used. It fails by a factor of two.

A notch cut in the top instead

If the cut is in the top (compression) face of the beam end, the NDS (3.4.3.1(e), Equation 3.4-5) treats it much more gently, because the sharp inside corner sits where the wood is being squeezed, not pulled apart: V′ = (2/3) × Fv′ × b × [d − ((d − dn) ÷ dn) × e], where e is how far the notch reaches in from the inner edge of the support (at most dn). At e = 0, a notch that stops at the support, there is no loss at all; capacity falls as the notch reaches farther in. The same end-notch limit of d/4 applies.

• Same roof beam, snow duration factor, a 2.31 in (d/4) notch in the top: 1,862 lb if it stops at the support, about 1,760 lb reaching 1.5 in, and about 1,396 lb at its full reach (48% used). The same depth cut in the bottom gives 785 lb (86% used).
• A 1.5 in top notch: about 1,800 lb reaching 1.5 in and about 1,560 lb at full reach, against 1,095 lb for the same notch in the bottom.

Answer

For the roof beam on a post, bearing is not a concern and a full-depth end is far from its shear limit. A notch changes that fast: even the largest notch the NDS permits uses 86% of the shear capacity with the snow increase and essentially all of it without, and a deep notch fails outright. If an Ogopogo joint needs the beam end to sit in a notch, prefer cutting it in the top, which costs far less shear than the bottom (1,396 lb against 785 lb at the largest allowed depth); if it must go in the bottom, keep it shallow (well under d/4), taper the corner, and check the shear with the real reaction; if it needs more than that, let the beam's full depth carry the load on a hanger, a ledger or a bolted cleat instead of cutting into it.

Where to be cautious

• Do not use the area bonus Cb at the end of a beam, and do not use the load duration factor on bearing: both are common mistakes that overstate the bearing capacity.
• The bottom-face and top-face notches use different equations (3.4-3 and 3.4-5). Check which face the cut is on, and for the top face, how far it reaches in from the support.
• Never notch the middle third of a span or the tension side of a thick board (4.4.3.2): that is where bending is highest.
• A notched end uses the full reaction. The allowance for loads close to the support does not apply to a notched beam.
• Crushing of the end grain of the post itself, bearing on a steel hanger or a plate, and bearing at an angle to the grain are separate checks that are not covered here.
• Everything here is a worked example. A joint that carries a roof over people is for a structural engineer to design and stamp.

Open items for a later pass

The real Ogopogo moon-eye and melon-head geometry, which decides the notch depth and length. Bearing at an angle to the grain and under washers are in Breyer Chapter 7. Still open: the post's own end-grain compression, the shear check for a bolted connection near a beam end (NDS 3.4.3.3, whose effective-depth figure could not be read here), and the Dense and Non-Dense grades of Southern Pine.

Sources

• American Wood Council, NDS 2018 with Commentary (free at awc.org): Chapter 3, sections 3.10.2 and 3.10.4 and Table 3.10.4 (bearing and Cb), 3.2.3 (notches), 3.4.3.1 including (e) with Equation 3.4-5, and 3.4.3.2 with Equation 3.4-3 (shear at notches); Chapter 4, section 4.4.3 (notch limits for sawn lumber); Table 2.3.2 note 1 (no load duration factor on Fc⊥).
• Southern Forest Products Association, Southern Pine Use Guide, 2013: Table 1 (No. 2 Southern Pine Fc⊥ 565 psi, Fv 175 psi) and Table A-2 (wet service: Fc⊥ ×0.67, Fv ×0.97).
• Written from these public documents, not from the Breyer textbook itself. If the book disagrees, the book and the current NDS win.