Why are steel beams coped and what limit states does coping introduce?

Beams are coped when their top flange would interfere with the supporting girder's top flange — a top cope cuts back the beam flange and part of the web so the beam can fit into the girder web at the same top-of-steel elevation. This is essential for flat-floor framing. However, coping removes the top flange, creating a tee-shaped cross-section and introducing four additional limit states beyond standard connection design: (1) block shear rupture along the bolt line and cope line per AISC 360 J4.3 — often the governing limit state because the cope removes material adjacent to the bolt group; (2) flexural yielding at the cope, where the reduced section modulus Snet of the tee section must resist the eccentric moment from the beam reaction; (3) lateral-torsional buckling of the coped region, where removal of the top flange eliminates torsional restraint and the coped web can buckle laterally; (4) local web buckling at the cope corner if the cope is deep relative to the beam depth. AISC Manual Part 9 provides the complete design procedure. Bottom copes and double copes introduce additional complications because both flanges are removed.

How is block shear calculated for a coped beam connection?

Block shear in a coped beam follows AISC 360-22 Equation J4-5: φRn = 0.75 × (0.60Fu × Anv + Ubs × Fu × Ant) ≤ 0.75 × (0.60Fy × Agv + Ubs × Fu × Ant). The shear area Agv = tw × [(n−1)×s + Lev] where n is the number of bolts, s is bolt spacing, and Lev is the vertical edge distance from the top bolt to the cope. The net shear area Anv = Agv − (n−0.5) × dh × tw, subtracting bolt holes. The tension area Agt = tw × Leh where Leh is the horizontal edge distance from the bolt line to the cope. The net tension area Ant = Agt − 0.5 × dh × tw (only half a hole is deducted from the tension plane because only the bolt closest to the cope affects it). Ubs = 1.0 for uniform tension stress distribution. Example: coped W16x26 beam (tw=0.25 in) with 4 bolts at 3 in. spacing, Lev=1.5 in, cope depth dc=1.5 in, Leh=1.25 in, Fy=50 ksi, Fu=65 ksi: Agv=0.25×(3×3+1.5)=2.625 in², Anv=2.625−(4−0.5)×0.8125×0.25=1.914 in², Agt=0.25×1.25=0.313 in², Ant=0.313−0.5×0.8125×0.25=0.211 in². φRn=0.75×(0.60×65×1.914+1.0×65×0.211)=0.75×(74.66+13.72)=66.3 kips. Compare to upper limit: 0.75×(0.60×50×2.625+1.0×65×0.211)=0.75×(78.75+13.72)=69.4 kips. Block shear capacity = 66.3 kips.

How is lateral-torsional buckling checked in a coped beam?

Lateral-torsional buckling of the coped region is governed by AISC Manual Part 9 equations derived from Cheng (1993). The critical stress Fcr depends on the cope geometry: for c/ho ≤ 1.0 (short cope), Fcr = 0.62π × E × tw × c/ho²; for c/ho > 1.0 (long cope), Fcr = 1.65 × (tw/ho)² × E × (c/dc). Here c is the cope length, ho = d − dc is the reduced beam depth (full depth minus cope depth), tw is the web thickness, and dc is the cope depth. The available moment φMn = 0.90 × Fcr × Snet where Snet is the elastic section modulus of the net coped section. Example: W18x35 coped 2 in. deep × 6 in. long, d=17.7 in, ho=15.7 in, tw=0.30 in, dc=2.0 in, Snet=27.8 in³ (tee section). c/ho = 6/15.7 = 0.38 ≤ 1.0 → short cope: Fcr = 0.62π × 29,000 × 0.30 × 6/15.7² = 164,500/246.5 = 667 ksi, but limited to Fy=50 ksi. φMn = 0.90 × 50 × 27.8 = 1,251 kip-in = 104.3 kip-ft. For a long cope (c=18 in, c/ho=1.15): Fcr = 1.65 × (0.30/15.7)² × 29,000 × (18/2.0) = 1.65 × 0.000365 × 29,000 × 9 = 157.3 ksi, capped at 50 ksi. Cope LTB rarely governs for typical shear connection cope dimensions (short cope, shallow depth) but can control for deep copes and long cope lengths.

When is cope reinforcement required and what options exist?

Cope reinforcement is required when the coped beam fails any of the four cope limit states (block shear, flexural yielding, LTB, or local web buckling) at the design reaction. Reinforcement options in order of preference: (1) Double clip angles or shear end plate extended beyond the cope — increases the bolt group and moves the critical section further from the cope, increasing both block shear and flexural capacity. (2) Horizontal stiffener (web plate) welded to the web in the coped region — a triangular or rectangular plate fillet-welded to the web and bottom flange resists the flexural tension at the cope, effectively restoring some of the lost section modulus. (3) Full-depth web stiffener pairs at the cope — provides torsional restraint and increases the warping resistance at the critical section. (4) Reduce the cope depth and length — the simplest fix if the geometry allows: shallower cope increases Snet, shorter cope improves LTB resistance. (5) Increase the beam size — sometimes the most economical if multiple beams are affected. For seismic applications (SMF beams), copes in the protected zone are prohibited per AISC 341 — the beam must be sized to fit without coping at the moment connection, or a deeper girder used to avoid the interference entirely.

What is the difference between top cope, bottom cope, and double cope?

Top cope is the most common — the beam's top flange and web are cut back to clear the girder top flange when both share the same top-of-steel elevation. Only the top flange is removed; the bottom flange remains intact, providing a compression stress block for flexure and anchoring the coped region against LTB. Bottom cope is rare — used when the beam bottom flange interferes with a stiffener, bracket, or lower-level obstruction. It removes the bottom flange, which is more detrimental because the bottom flange is typically in tension under gravity load, and the remaining top flange is the compression element that may buckle. Bottom copes should be avoided when possible. Double cope removes both top and bottom flanges, leaving only the web at the beam end — the most critical condition because both flanges are gone, the section modulus is drastically reduced (only the web remains), and there is no flange to restrain either lateral-torsional buckling or local buckling. Double-coped beams almost always require reinforcement. For double-coped beams where both flanges are removed for a length exceeding 2d, the beam behaves essentially as a shear-only element and must be checked for shear buckling of the web panel — a stiffener pair at the cope end is typically required.

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.

Cope Types — Detailed Classification

Top Cope (Most Common)

A top cope removes a portion of the top flange and a section of the web, creating an L-shaped notch at the beam end. This is the standard condition when a beam frames into a girder at the same top-of-steel elevation. The minimum cope depth must clear the girder flange (cope depth = girder tf + k distance of the supported beam). The cope length must extend past the girder web to allow the beam end to be inserted during erection.

Bottom Cope (Uncommon)

A bottom cope is used when the beam must frame under an obstruction (e.g., a mechanical duct or a shallower beam above). It creates a notch at the bottom flange. Bottom copes are structurally less efficient than top copes because the bottom flange typically carries more compression in positive bending.

Double Cope (Top and Bottom)

A double cope removes both flanges at the beam end, leaving only a rectangular web stub. This is required when the beam is deeper than the available clearance at both flanges. Double-coped beams have significantly reduced capacity and almost always require reinforcement. The remaining web plate has minimal flexural and torsional resistance.

Cope Type	Capacity Reduction vs. Uncoped	When Reinforcement Required
Top cope only	30-50% of original section	When dc > d/3 or c > d
Bottom cope only	40-60% of original section	When dc > d/4
Double cope	70-90% reduction	Almost always required

AISC Manual Cope Design Procedure (Part 9)

The AISC Steel Construction Manual Part 9 provides a systematic procedure for coped beam design. The checks proceed in order of likelihood to govern:

Step 1: Determine cope dimensions (dc and c) from the connection geometry and required erection clearance.

Step 2: Check block shear rupture at the cope per Section J4.3. This is the most commonly governing limit state.

Step 3: Calculate reduced section properties of the coped cross-section. For a top cope, the remaining section is a tee shape comprising the beam web and bottom flange. Compute the new neutral axis location, Snet (elastic section modulus), and Znet (plastic section modulus) of the tee section.

Step 4: Check flexural yielding at the cope: phiMn = 0.90 x Fy x Snet. The applied moment equals the beam reaction times the eccentricity from the bolt line to the face of support.

Step 5: Check lateral-torsional buckling of the coped section using the AISC Manual Part 9 equations. LTB is critical for long or deep copes where c/ho > 0.5.

Step 6: Check local web buckling at the cope for deep copes (dc/d > 0.2) or long copes (c/d > 2).

Step 7: If any limit state fails, either reduce the cope dimensions, add a reinforcement plate, or increase the beam size.

Reduced Section Properties Calculation

For a W-shape with a top cope of depth dc, the remaining tee-section properties are calculated as follows:

Neutral axis location (measured from the bottom of the beam): y_bar = (tw x ho^2/2 + bf x tf x (tf/2)) / (tw x ho + bf x tf), where ho = d - dc.
Moment of inertia: Ix = tw x ho^3/12 + bf x tf^3/12 + tw x ho x (ho/2 - y_bar)^2 + bf x tf x (y_bar - tf/2)^2
Elastic section modulus: Snet = Ix / (ho + tf - y_bar) (governs at the top of the web, the most stressed fiber)
Plastic section modulus: Znet = computed from the equal-area axis of the tee section

For common W-shapes with moderate top copes, Snet is typically 25-40% of the original Sx. This dramatic reduction means the coped section must be checked even for relatively modest beam reactions.

Reduced Section Properties for Common Sections (Top Cope)

Section	Cope dc (in)	ho (in)	Snet (in^3)	Snet / Sx	Znet (in^3)
W16x40	2.0	14.0	17.8	0.48	20.1
W16x40	2.5	13.5	16.0	0.43	18.2
W16x40	3.5	12.5	12.8	0.35	14.8
W18x46	2.0	16.0	23.4	0.48	26.5
W18x46	3.0	15.0	20.2	0.42	23.0
W21x44	2.0	19.0	25.3	0.45	28.8
W24x55	2.0	22.0	35.1	0.44	39.7
W24x55	3.0	21.0	31.2	0.39	35.4

These values illustrate why even a moderate 2" cope on a W16x40 reduces the section modulus to less than half the original capacity.

Local Web Buckling Check

Local web buckling occurs when the unstiffened web plate above the cope buckles under the combination of bending and shear. AISC Manual Part 9 provides the following check:

For a cope with length c and reduced depth ho, the web plate buckling stress is:

Fcr_web = 26.2 x (tw/ho)^1.5 x E^0.5 / sqrt(c)     [for dc/ho <= 0.75]

The available moment considering local web buckling: phiMnw = 0.90 x Fcr_web x Snet. This limit state typically governs for deep copes (dc > d/3) or long copes (c > 2d). When it governs, a vertical stiffener or doubler plate at the cope eliminates the buckling concern.

Reinforcement Options for Coped Beams

When the coped section capacity is insufficient, several reinforcement strategies are available:

Doubler Plate (Most Common)

A horizontal plate is welded across the top of the cope, effectively restoring the top flange. The plate is fillet-welded to the beam web on both sides and must extend beyond the cope to develop the force in the plate.

Plate force = M_cope / (ho - t_plate/2)
Plate width = beam bf - 2 x weld_return (typically 1" less than bf on each side)
Plate thickness = Plate force / (0.90 x Fy x plate_width)

Vertical Stiffener

A vertical plate welded to the beam web at the face of the cope prevents local web buckling. This is used when LTB or local buckling governs but flexural yielding is adequate. The stiffener should be at least as thick as the beam web and extend the full depth of the cope.

Extended Cope with Reduced Eccentricity

Sometimes the cope dimensions can be reduced by using a different connection type. For example, switching from a single-plate shear connection to a double-angle connection may reduce the required cope length by 1-2 inches, improving the coped section capacity.

Cope Limitations by Beam Size

Not all beams can be coped without significant capacity loss. Lighter sections with thinner webs are more severely affected:

Beam Depth	Weight (lb/ft)	tw (in)	Max Recommended dc	Max Recommended c	Comments
W16	26-40	0.25-0.31	d/3	1.5d	Web very thin; check local buckling
W18	35-50	0.30-0.36	d/3	2d	Moderate web; cope usually manageable
W21	44-57	0.32-0.36	d/2.5	2d	Good web thickness; cope rarely governs
W24	55-68	0.34-0.40	d/2.5	2d	Deepest common beams; cope capacity good
W12	40-65	0.30-0.39	d/3	1.5d	Shorter beams; cope length often limited

For beams lighter than those shown (e.g., W16x26, W18x35), the thin web makes even moderate copes problematic. Consider using a deeper beam to avoid coping, or specify a double-angle connection that requires less cope length.

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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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.

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