Coped Beam Design — Block Shear, Local Buckling, and Flexural Checks

A coped beam has a portion of one or both flanges cut away (coped) at the beam end to allow it to frame into a supporting member. Coping is common in beam-to-girder connections where the beam top flange would otherwise interfere with the girder top flange. Coping introduces additional limit states that must be checked beyond the standard connection design.

Why beams are coped

In steel framing, beams often connect to the webs of girders. If the beam top flange is at the same elevation as the girder top flange (as required for a flat floor), the beam flange must be cut back (coped) to clear the girder flange. Without the cope, the beam cannot physically be erected into position.

Cope types: Top cope only (most common), top and bottom cope (double cope, used when the beam is deeper than the available clearance), and bottom cope (rare).

Limit states for coped beams

1. Block shear rupture (AISC 360-22 Section J4.3)

Block shear is frequently the governing limit state for coped beams. The failure mode involves shear tearing along the bolt line and tension rupture along the bottom of the cope.

phiRn = 0.75 * (0.60*Fu*Anv + Ubs*Fu*Ant)
     <= 0.75 * (0.60*Fy*Agv + Ubs*Fu*Ant)    [Eq. J4-5]

Where Anv = net shear area along the vertical bolt line, Ant = net tension area along the horizontal cope line, Ubs = 1.0 for uniform tension stress distribution.

For a coped beam with n bolts at spacing s and cope depth dc:

Agv = tw * ((n-1)*s + Lev) where Lev = vertical edge distance
Anv = Agv - (n-0.5)dhtw
Agt = tw * Leh (horizontal edge distance from bolt to cope)
Ant = Agt - 0.5dhtw

2. Flexural yielding at the cope (AISC Manual Part 9)

The cope removes the top flange, creating a tee-shaped cross-section at the critical section. This reduced section must resist the moment caused by the beam reaction acting at the eccentricity from the support to the critical section.

phiMn = 0.90 * Fy * Snet

Where Snet = net section modulus of the coped section. For a top cope: the neutral axis shifts downward, and the section modulus is based on the remaining web and bottom flange.

3. Lateral-torsional buckling of the coped section (AISC Manual Part 9)

Removing the top flange eliminates the torsional restraint at the beam end. The coped region can buckle laterally, especially for deep copes:

Fcr = 0.62 * pi * E * tw * c / (ho^2)      [for c/ho <= 1.0]
Fcr = 1.65 * (tw/ho)^2 * E * (c/dc)        [for c/ho > 1.0]

Where c = cope length, ho = reduced beam depth (d - dc), tw = web thickness, dc = cope depth. The available moment is phiMn = 0.90 _ Fcr _ Snet.

4. Local web buckling at the cope

For deep copes (dc/d > 0.2) or long copes (c/d > 2), the web plate above the cope can buckle locally. Check per AISC Manual Part 9 using the plate buckling equation for the web panel.

Cope geometry limits

Parameter	Recommended Limit	Reason
Cope depth dc	dc <= d/2	Avoid excessive section loss
Cope length c	c <= 2*d	Avoid LTB and local buckling
Minimum cope depth	Top flange thickness + k distance	Clear the supporting flange
Cope re-entrant corner	1/2" minimum radius	Prevent stress concentration and fatigue cracking

Critical detail: The re-entrant corner of the cope must have a smooth radius (minimum 1/2", preferred 3/4"). A sharp corner creates a stress concentration that can initiate fatigue cracks, especially under cyclic loading.

Worked example — W16x40 with top cope

Given: W16x40, top cope dc = 2.5 in, cope length c = 9 in, 3 bolts at 3" spacing, 3/4" A325-N, Fy = 50 ksi, Fu = 65 ksi.

Properties: d = 16.0 in, tw = 0.305 in, bf = 7.0 in, tf = 0.505 in. Reduced depth ho = 16.0 - 2.5 = 13.5 in. Vertical edge distance Lev = 1.5 in. Horizontal edge distance Leh = 1.75 in. Bolt hole diameter dh = 13/16" = 0.8125 in.

Step 1 — Block shear: Agv = 0.305 * (2*3 + 1.5) = 0.305 _ 7.5 = 2.29 in^2. Anv = 2.29 - 2.5 _ 0.8125 _ 0.305 = 2.29 - 0.619 = 1.67 in^2. Agt = 0.305 _ 1.75 = 0.534 in^2. Ant = 0.534 - 0.5 _ 0.8125 _ 0.305 = 0.534 - 0.124 = 0.410 in^2. phiRn = 0.75 _ (0.60 _ 65 _ 1.67 + 1.0 _ 65 _ 0.410) = 0.75 _ (65.1 + 26.7) = 0.75 * 91.8 = 68.9 kips.

Step 2 — Flexural yielding at cope: The coped section is a tee (web + bottom flange). For W16x40 with dc = 2.5 in: ho = 13.5 in, Snet approximately 16.0 in^3 (from AISC Manual Table 9-2 or direct calculation of the tee section modulus). phiMn = 0.90 _ 50 _ 16.0 = 720 kip-in. Applied moment at the cope = R * e (where e = distance from bolt line to face of support, typically 3-5 in). For R = 68.9 kips and e = 4 in: M = 275.6 kip-in < 720 kip-in. OK.

Step 3 — LTB of coped section: c/ho = 9/13.5 = 0.667 < 1.0. Fcr = 0.62 _ pi _ 29000 _ 0.305 _ 9 / (13.5^2) = 0.62 _ 3.1416 _ 29000 _ 2.745 / 182.25 = 0.62 _ 250,536 / 182.25 = 852 ksi >> Fy. LTB does not govern for this short cope.

Result: Block shear governs at 68.9 kips allowable beam reaction.

Multi-code comparison

AISC 360-22 (USA): Coped beam design per Manual Part 9. Block shear per Section J4.3 (Eq. J4-5). Flexural yielding and LTB of the coped section per Manual Part 9 equations. phi = 0.75 for block shear, 0.90 for flexure and LTB. The AISC Manual provides Tables 9-2 through 9-4 for coped beam capacities for standard W-shapes.

AS 4100-2020 (Australia): Clause 9.1.10 covers block shear (called "block shear tearout"). The capacity formula is similar: phiVbs = phi(0.6fyAgv + fu*Ant). phi = 0.75. No explicit tabulated coped beam capacities. LTB of coped sections is treated as a plate buckling problem per Section 5.6, using the elastic buckling stress of the reduced section.

EN 1993-1-8 (Europe): Block tearing per Clause 3.10.2: Veff,1,Rd = fuAnt/gamma_M2 + fyAnv/(sqrt(3)*gamma_M0). gamma_M0 = 1.00, gamma_M2 = 1.25. The approach differs from AISC by applying the 1/sqrt(3) reduction to shear yielding rather than using 0.60. EN 1993 does not provide specific coped beam LTB equations -- designers are expected to check the tee section using the general LTB provisions of EN 1993-1-1 Section 6.3.2.

CSA S16-19 (Canada): Block shear per Clause 13.11: Tr + Vr = phi_uAnFu + 0.6phiAgv*Fy (or the alternative using net shear area with Fu). phi_u = 0.75, phi = 0.90. CSA S16 Clause 14.3.5.1 addresses reduced sections and requires that the stability of the coped region be checked. The approach is generally consistent with AISC Manual Part 9 methodology.

Common mistakes

Not checking block shear at the cope. Block shear at coped beams frequently governs over bolt shear. The failure plane includes the vertical shear path along the bolt line and the horizontal tension path along the cope bottom. This check is mandatory per AISC 360 J4.3.
Ignoring lateral-torsional buckling of the coped section. Removing the top flange eliminates torsional restraint. For long or deep copes (c/ho > 1.0), the LTB critical stress can drop well below Fy, sometimes to 20-30 ksi. Always check LTB when c > d/2.
Sharp re-entrant corners. The corner of the cope must have a smooth radius (minimum 1/2") to avoid fatigue cracking. Flame-cut copes without grinding are particularly susceptible. AWS D1.1 Section 5.15 covers thermal cutting quality requirements.
Not checking the reduced section for flexure. The tee-section remaining after coping has a section modulus that may be only 30-50% of the original beam. The eccentricity between the beam reaction and the cope face creates a moment that this reduced section must resist.
Coping too deep or too long without reinforcement. Keep dc <= d/2 and c <= 2d. When these limits are exceeded, consider cope reinforcement: a plate welded horizontally across the top of the cope to restore the flange area and improve both flexural capacity and LTB resistance.

Frequently asked questions

What is a coped beam? A beam with a portion of one or both flanges cut away at the end to allow it to fit into a supporting member. The cope creates clearance for the supporting member's flange.

When is cope reinforcement needed? When the coped section capacity (block shear, flexure, or LTB) is less than the required beam reaction. Reinforcement is a plate welded across the cope to restore capacity.

Does double coping reduce capacity more? Yes, significantly. A double-coped beam (top and bottom) loses both flanges at the end, creating a rectangular web plate with much less flexural and torsional resistance. Double-coped beams almost always require a reinforcement plate.

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Disclaimer

This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against AISC 360-22 and the AISC Manual Part 9. The site operator disclaims liability for any loss arising from the use of this information.