Coped Beam — Cope Dimensions, Stress Concentration, and Block Shear
A coped beam has its flange and a portion of its web cut away at the end so it can frame into a supporting girder or column without the beam flange colliding with the girder flange. Coping is universal in steel framing — virtually every floor beam framing into a girder web requires a top cope, and many require bottom copes as well. The cope trades geometric clearance for reduced capacity: every square inch of material removed weakens the beam at its most critical location — the connection.
PRELIMINARY — NOT FOR CONSTRUCTION. All content is for educational and reference use only. Must be independently verified by a licensed Professional Engineer (PE) or Structural Engineer (SE) before use in any project.
Cope Configurations
Three basic cope configurations cover the vast majority of framing situations:
Top cope only. The top flange and a portion of the web are removed above the top of the girder. This is the most common configuration — the beam frames into the girder web, and the top flange must clear the girder's top flange for erection. The bottom flange remains intact, so the tension flange is uncoped.
Bottom cope only. The bottom flange and a portion of the web are removed to clear the bottom girder flange. The beam is bearing on the girder from below (less common in building construction, frequent in industrial platforms where beams sit on top of girders).
Double cope (top and bottom). Both flanges are removed. The beam frames within the depth of the girder, clear of both flanges. Rare in simple framing because top cope alone usually suffices — double coping signals a beam that is deeper than the girder, which is structurally inadvisable (the supported member should not be deeper than the supporting member).
Cope Geometry and Dimensional Limits
The cope is defined by three dimensions:
- c = cope depth (vertical cut into the web, measured from the flange)
- L = cope length (horizontal distance from the beam end to the start of the full section)
- r = cope radius (at the re-entrant corner where the vertical and horizontal cuts meet)
Dimensional Limits per AISC Design Guide 2
| Parameter | Top Cope Limit | Bottom Cope Limit | Reason |
|---|---|---|---|
| c/d | ≤ 0.50 (d/2) | ≤ 0.20 (d/5) | Bottom cope removes tension flange — more critical |
| L/d | ≤ 1.0 (d) | ≤ 2.0 (2d) | Longer cope = longer unbraced length |
| r (minimum) | 3/4 in | 3/4 in | Sharp corners concentrate stress; radiused corners reduce Kt |
The bottom cope limits are more restrictive because the bottom flange carries tension under positive bending. Removing the tension flange eliminates the primary flexural resistance path — the coped region must transfer the tension force from the reduced section into the remaining web through shear, creating a stress concentration at the cope corner. Tensile fracture initiating at the cope corner is the dominant failure mode for bottom-coped beams.
Stress Concentration at the Cope Radius
The re-entrant corner at the cope — where the vertical cut meets the horizontal cut — is a classic stress concentration. For a sharp 90-degree corner, the theoretical elastic stress concentration factor Kt ≈ 3.0 to 4.0, meaning the localized stress is 3-4 times the nominal web stress. Adding a radius r reduces Kt:
- r = 0.25 in (sharp): Kt ≈ 3.5
- r = 0.50 in: Kt ≈ 2.5
- r = 0.75 in: Kt ≈ 1.8
- r = 1.00 in: Kt ≈ 1.5
The 3/4-inch minimum radius is the point of diminishing returns — additional radius reduces Kt only marginally while increasing the amount of web removed. Flame-cut copes naturally produce a radius because the cutting torch cannot turn a sharp inside corner; the minimum radius is the kerf width of the cutting tip.
IMPORTANT: The cope corner must be ground smooth after flame cutting. The heat-affected zone from flame cutting produces a hard, brittle martensitic surface layer with residual tensile stresses. Grinding removes this layer and eliminates surface notches from the cutting process. If grinding is not specified, the fatigue life at the cope corner may be reduced by an order of magnitude.
Failure Modes at Beam Copes
Block Shear (J4.3) — The Governing Check
Block shear is the most critical failure mode for coped beam ends. The failure block is defined by:
- The cope face (tension plane)
- The bolt line(s) (shear plane(s))
Block shear rupture strength:
Rn = 0.60 × Fu × Anv + Ubs × Fu × Ant ≤ 0.60 × Fy × Agv + Ubs × Fu × Ant
φ = 0.75 (LRFD)
Where:
Anv = net shear area along bolt line(s)
Ant = net tension area at cope face
Ubs = 1.0 when tension stress is uniform (copes)
For a beam coped top and bottom with bolts along the web, the shear area runs the vertical length of the bolt group, and the tension area is the cope face above (or below) the bolts. Block shear frequently governs over bolt shear and bearing — it's the "check many engineers forget" at beam copes.
Flexural Yielding at the Reduced Section
The cope reduces the section modulus from S_full to S_net. For a W16×26 with a 2-inch top cope, S_net may be only 40-50% of S_full. The moment at the cope face must be checked:
Mu (at cope face) = Vu × L
φMn = φ × Fy × S_net (φ = 0.90)
If φMn < Mu → reinforce or reduce cope length
Reducing the cope length L is the simplest remediation — shorter cope = smaller moment arm for the shear reaction = smaller Mu at the critical section.
Lateral-Torsional Buckling of the Coped Region
When the top flange is coped, the coped T-shaped portion loses lateral support from the compression flange. The unbraced length is the cope length L (typically 6-12 inches). For short copes (L/d ≤ 1.0), LTB is rarely critical because L is too short for the lateral buckling mode to develop. For long copes (L/d > 1.0), LTB must be checked using AISC 360 F2 with the reduced section properties.
Web Local Buckling at the Cope
The cope removes the flange that provides rotational restraint to the web, converting it from a flange-supported web (stiff) to a free-edge web (flexible). Under compression from the flexural stresses at the coped section, the web may buckle laterally. AISC Design Guide 2 provides an empirical buckling check based on the cope depth-to-web thickness ratio c/tw.
Cope Reinforcement Options
When the unreinforced cope fails any of the four checks above, reinforcement is required:
Horizontal stiffener. A flat plate (typically 3/8 or 1/2 inch thick) is welded to the web at the cope radius, extending into the beam. The stiffener restores the tension flange path lost to the cope and reduces the web slenderness at the cope corner. Most common for bottom copes in tension.
Doubler plate. An additional plate is welded to the web at the coped region, effectively thickening the web and increasing block shear resistance. Used when block shear governs and increasing bolt count is not possible.
Extended shear tab. Instead of coping the beam, a deeper shear tab extends past the beam flanges and bolts directly to the beam web. This eliminates the cope entirely — the beam flanges remain intact — but requires a custom shear plate that may be uneconomical for standard framing.
Frequently Asked Questions
Can copes be field-cut instead of shop-cut?
Yes, but with consequences. Field cutting (oxyacetylene torch) cannot achieve the dimensional accuracy of shop cutting (CNC plasma or saw). Field-cut copes are rougher, with larger heat-affected zones and less consistent radii. AWS D1.1 permits field cutting if the cut surfaces are ground smooth, but the cost of field grinding often exceeds the shop fabrication cost. Where possible, copes should be shop-fabricated to controlled dimensions and inspected before the beam leaves the fabricator.
Does the cope affect the beam's overall deflection?
The cope affects only the local stress state — it has negligible effect on overall beam deflection. The cope length L is typically 6-12 inches, or 2-5% of the beam span. The reduction in moment of inertia over this short length produces a negligible increase in midspan deflection (<1%). The cope is a local strength issue, not a global stiffness issue.
Can I cope an HSS (hollow structural section)?
No. Coping an HSS removes a portion of its closed perimeter, converting the HSS from a closed torsionally-stiff section to an open C-section at the connection — with dramatically reduced torsional stiffness and strength. Connection to HSS uses through-plates, external gussets, or direct welding to the HSS face — never coping. The shear tab for an HSS beam is welded directly to the HSS sidewall, and the beam runs past the HSS rather than into it.
International Code References
- AISC 360-22: Section J4.3 — Block shear strength. Section F2 — Flexural strength (for coped section checks).
- AISC Design Guide 2 (Steel and Composite Beams with Openings): Appendix A — Coped beam design, including all four failure mode checks and worked examples.
- AS 4100: Section 9.1.9 — Block shear. Section 5.2 — Flexural capacity of coped sections.
- EN 1993-1-8: Section 3.10.2 — Block tearing (block shear). EN 1993-1-1 Section 6.2.6 provides shear buckling checks for coped web panels.
Educational reference only. Coped beam design must be checked for block shear per AISC 360 J4.3, flexural yielding, LTB, and web buckling per AISC Design Guide 2 by a licensed Professional Engineer. Cope corners must be ground smooth after thermal cutting per AWS D1.1.
Disclaimer: This content is for educational purposes only. Results must be verified by a licensed professional engineer. Steel Calculator provides preliminary design tools — NOT a substitute for professional engineering judgment.