Engineering story

A century of steel — from concept to skyline

Engineer reviewing blueprints against a steel-frame construction site at sunrise

Shear does not care about Zx — it lives in the web.

For most rolled W-shapes, phi_v*Vn = 0.60*Fy*Aw. Everything else is web-slenderness bookkeeping.

Chapter 7 covers AISC 360-22 Chapter G: web shear yielding for non-slender webs (G2.1 with phi_v = 1.00 or 0.90), shear buckling for slender webs, tension-field action for plate girders, plus the AISC J10 concentrated-force checks (web yielding, web crippling, and side-sway).

Iconic steel structures built on engineering excellence

Basler tension-field theory
1961
AISC LRFD shear provisions
1986
AISC 360-22 Chapter G (current)
2022

Load pathBeam reactionWeb shearBearing plateSupportFoundation

02

Learning objectives

What you will be able to do after finishing Chapter 1 — and why each objective matters in practice

Objective 01

Compute Vn = 0.6FyAw*Cv1 (G2.1)

For rolled I-shapes with h/tw <= 2.24*sqrt(E/Fy), Cv1 = 1.0 and phi_v = 1.00.

Why it matters: Most rolled W-shapes fall in this pocket — the shear check is one line.
Where it is used: Every gravity beam and girder shear check.
Connects to: AISC G2.1; Manual Table 3-2 phi_v*Vn column.

Objective 02

Classify web slenderness h/tw

h/tw versus 2.24*sqrt(E/Fy) (~53.9 for A992). Below: Cv1 = 1.0 with phi_v = 1.00. Above: Cv1 reduces and phi_v = 0.90.

Why it matters: The pocket determines phi and the buckling knock-down.
Where it is used: Deep W-shapes, plate girders, custom sections.
Connects to: AISC G2.1; Table B4.1b web limits.

Objective 03

Handle shear buckling / tension-field

For h/tw > 260 and stiffened webs, tension-field action (G2.2/G3) recovers strength beyond elastic buckling.

Why it matters: Plate girders would be uneconomic without tension-field capacity.
Where it is used: Plate girders in bridges and long-span roofs.
Connects to: AISC G2.2, G3; Basler theory.

Objective 04

Run J10 concentrated-force checks

Web local yielding (J10.2), web crippling (J10.3), sidesway web buckling (J10.4).

Why it matters: Reactions at supports and point loads under columns govern local web failures.
Where it is used: Every beam bearing on a column or wall; every point-loaded girder.
Connects to: AISC J10.

Objective 05

Read Manual Table 3-2 (phi_v*Vn)

Table 3-2 tabulates phi_v*Vn for every W in one column.

Why it matters: Fastest FE-exam shortcut.
Where it is used: Every design office; every FE shear question.
Connects to: AISC Manual Part 3.

Objective 06

Verify Aw = d*tw

For a W-shape, Aw is taken as the full depth times the web thickness.

Why it matters: Aw is not the clear web height — a common student error.
Where it is used: Every G2.1 calc.
Connects to: AISC G2.1(b).

03

Engineering motivation

What each part of a steel-frame building actually does — and why it exists

Before you design any single member, you have to see the whole system. A steel-frame building is not a collection of independent shapes bolted together — it is a deliberate load path, engineered so that every kilonewton of gravity, wind, or seismic demand has a continuous route from where it starts to the ground where the earth can resist it.

The photograph below shows a typical steel framing detail. Drag each labelled chip onto the structural element it names — the drop is only accepted when it lands inside the correct element's outlined region. A correct answer locks in with a green outline and a short explanation; a wrong answer flashes the region red, tells you what you actually hit, and returns the chip so you can try again. Press Reveal expected placements to see the reference solution (that attempt is then marked as assisted).

Structural steel framing — identify each element by dragging the labels — Fig. 1.3 · Drop a chip inside the outlined element it names. Green = correct and locked; red flash = wrong element, chip returns.

04

Failure mechanisms

Why we design the way we do — six ways steel structures have failed, and what each disaster taught the profession

Every provision in AISC 360 is a scar. Behind each equation, load factor, and detailing rule is a bridge, a walkway, or a tower whose failure cost lives and rewrote the profession. The six case studies below trace the mechanisms that motivate the code you are about to learn.

Read each one as an engineer, not a spectator: identify the load, the limit state, the missing check, and the specific clause that exists today because that check was missed. When you meet those clauses again in Chapters 5–17, they will read as answers, not rules.

Failure mechanism

Web shear buckling

A slender web crinkles diagonally under shear before yielding.

Root cause

h/tw > 2.24*sqrt(E/Fy) triggers Cv1 < 1.0.

Lesson learned

Use G2.1 with Cv1 reduction, or add stiffeners and mobilise tension-field.

§AISC G2.1 / G2.2

Failure mechanism

Web crippling

A concentrated load buckles the web locally at the bearing.

Root cause

N (bearing length) too short and web too thin; J10-4 or J10-5a.

Lesson learned

Use bearing plates or bearing stiffeners; check J10.

§AISC J10.3

Failure mechanism

Web local yielding

Web yields directly under a point load or reaction over a short bearing length.

Root cause

Rn = (2.5*k + N)*Fy*tw not satisfied.

Lesson learned

Increase N or add a bearing stiffener.

§AISC J10.2

How failure propagates

The five-stage failure progression

1Applied load

2Elastic shear

3Diagonal principal stress

4Web buckle

5Collapse

Design codes intervene at the transition from yield to instability. Everything before yield is elastic and reversible; everything after instability is a race to collapse. LRFD keeps the demand well below the first transition.

05

Lecture notes

The full textbook chapter — figures, equations, and engineering narrative

Reflection · Think before you read

For a typical rolled W-shape, flexure almost always governs — yet for short, deep beams and plate girders, shear can control. What geometric ratio flips that balance, and can you think of a real structure where it matters?

Chapter 7 — Beam Shear Design (AISC 360-22 Chapter G)

Chapter focus. Beam shear is almost always carried by the web alone (flanges resist moment, web resists V). For most rolled shapes h/tw is stocky enough that φVn = 1.0·(0.6·Fy·Aw) — one line of arithmetic. Where webs are slender (deep girders), a shear-buckling coefficient Cv1 or Cv2 reduces it. This chapter also covers where to add stiffeners.

1. Behavior

In an I-shaped beam, the parabolic shear stress in the flanges is small and reverses direction; the web carries essentially all of the transverse shear V. AISC uses a simplified uniform stress on the web area:

Idealized shear-stress distribution — web governs

(1)

A_{w} = d \cdot t_{w} (f u l l d e pt h \cdot w e b t hi c k n ess)

2. Nominal Shear Strength (§G2)

(2)

V_{n} = 0.6 \cdot F_{y} \cdot A_{w} \cdot C_{v 1}

C_v1 is the web-shear coefficient:

If h/t_w ≤ 2.24 √(E/F_y): C_v1 = 1.0 and φ_v = 1.00 (most rolled W-shapes at F_y = 50 ksi).
Otherwise C_v1 reduces per Eq. G2-3/G2-4; φ_v = 0.90.

(3)

D es i g n : ϕ_{v} V_{n} \geq V_{u}

3. Shear Buckling and Stiffeners (§G2.2)

For plate girders with slender webs, transverse stiffeners spaced at ratio a/h allow post-buckling tension-field action per §G2.2. Rolled W-shapes rarely need stiffeners; plate girders always require analysis.

4. Block Shear at Beam End (Coped Beams, §J4.3)

Where the top flange is coped for a shear tab connection, the web can tear out along the bolt lines. Same formula as Chapter D block shear applied to the coped beam web.

5. Design Procedure

Compute V_u at each critical section (support face, concentrated loads).
Check web slenderness h/t_w vs 2.24 √(E/F_y); classify.
Compute A_w = d·t_w, then V_n.
Verify φ_v V_n ≥ V_u.
Check block shear at coped ends and bearing on webs at concentrated loads (§J10).

⚠ Common mistakes

Using cross-sectional area A_g instead of A_w = d·t_w.
Using φ_v = 0.90 when 1.00 applies for h/t_w ≤ 2.24√(E/F_y).
Ignoring coped-end block-shear on shear-tab connections.
Applying tension-field action rules to non-stiffened webs.

Worked Example 7.1 — Web Shear Check, W18×50 Floor Beam

Given: The floor beam sized in Ch. 6 (W18×50, A992, span 30 ft, w_u = 3.06 k/ft). Check web shear per AISC G2. Section: d = 17.99 in, t_w = 0.355 in, h/t_w = 45.2.

Figure 7.1a — Elevation, cross-section, and coped-end block-shear pattern

Step 1 — Factored shear at support

(1)

V_{u} = w_{u} L /2

V_u = 3.06 · 30 / 2 = 45.9 kips

Step 2 — Web slenderness

§G2.1 stocky-web limit

(2)

h / t_{w} \leq 2.24 (E / F_{y})

C_{v 1} = 1.0, ϕ_{v} = 1.00

Limit = 2.24 √(29000/50) = 53.95

h/t_w = 45.2 < 53.95 → C_v1 = 1.00, φ_v = 1.00

Step 3 — Web area

(3)

A_{w} = d \cdot t_{w}

A_w = 17.99 · 0.355 = 6.39 in²

Step 4 — Nominal & design shear

Eq. G2-1

(4)

V_{n} = 0.6 F_{y} A_{w} C_{v 1}

ϕ_{v} V_{n} = 1.00 \cdot V_{n}

V_n = 0.6 · 50 · 6.39 · 1.0 = 191.7 k
φ_v V_n = 191.7 k

Step 5 — Verify

Demand/Capacity = 45.9 / 191.7 = 0.24 ✓ (comfortable margin)

Step 6 — Coped-end block-shear check (J4.3)

Assume shear tab with three 3/4″ A325 bolts at 3″ pitch; top flange coped 2″ × 5″.

Formulas

(5)

A_{g v} = t_{w} \cdot [L_{e v} + (n - 1) s]

A_{n v} = A_{g v} - (n - 0.5) \cdot d_{h} \cdot t_{w}

A_{n t} = t_{w} \cdot [L_{e h} - 0.5 \cdot d_{h}]

R_{n} = min (0.60 F_{u} A_{n v}, 0.60 F_{y} A_{g v}) + U_{b s} F_{u} A_{n t}

L_ev = 1.5″, L_eh = 1.5″, t_w = 0.355 in, d_h = 7/8″
A_gv = t_w(2·3 + 1.5) = 0.355·7.5 = 2.66 in²; A_nv = 2.66 − 2.5(0.875)(0.355) = 1.88 in²
A_nt = t_w(1.5 − 0.5·0.875) = 0.355·1.06 = 0.377 in²; U_bs = 1.0
Fracture: 0.60·65·1.88 + 1.0·65·0.377 = 73.3 + 24.5 = 97.8 k
Yield: 0.60·50·2.66 + 24.5 = 79.8 + 24.5 = 104.3 k → fracture governs
φR_n = 0.75 · 97.8 = 73.4 k ≥ 45.9 k ✓

Result: W18×50 web shear φV_n = 192 k ≫ V_u = 45.9 k ✓; end block-shear φR_n = 73.4 k ≥ 45.9 k ✓. Beam adequate.

Engineering tips

Elastic vs post-buckling shear. Use Aw = d*tw for W-shapes.
Cv1 and the phi_v split. Memorise 2.24*sqrt(E/Fy) ~ 53.9 for A992.
Tension-field action. Not permitted in end panels or where tension-field development is blocked.
J10 concentrated-force checks. Interior loads use larger bearing coefficients than end reactions.

Common mistakes

Elastic vs post-buckling shear. Averaging V/A over the entire section — that gives the flange too much credit.
Cv1 and the phi_v split. Using phi_v = 0.90 when the shape is in the phi_v = 1.00 pocket.
Tension-field action. Applying tension-field to a rolled W-shape (usually h/tw is too small).
J10 concentrated-force checks. Skipping web crippling because 'the beam has enough Mp'.

Real-world use

Elastic vs post-buckling shear. Every rolled W-shape beam shear check.
Cv1 and the phi_v split. All AISC 360 shear checks.
Tension-field action. Plate girders.
J10 concentrated-force checks. Beam supports, girder point loads.

06

Professional practice, safety & ethics

Shear design practice

Professional practice

Show all web penetrations, copes, and blocks on the framing plans — each one reduces shear area.
Where tension-field action is used, note the requirement for adequate end panels and stiffeners on the drawings.
Coordinate stiffener locations with connection and MEP clearances early.

Safety in design & construction

High shear regions are at supports, where copes and bolt holes also occur — the worst detail sits at the worst force.
Web crippling and local yielding at concentrated loads are separate, frequently missed limit states.
Never allow an unreinforced hole to be burned in a beam web near a support.

Engineering ethics

Field-cut copes deeper than shown are a construction defect: require documentation and re-analysis, not a verbal approval.
Advise the contractor in writing when a proposed penetration is unacceptable.
Do not sign off on shop drawings you have not actually checked.

Ironworkers bolting a steel beam connection while tied off at height — Erection safety: OSHA Subpart R fall protection and stable temporary bracing.

Engineers reviewing sealed structural drawings across a conference table — Design review: documenting assumptions before the drawings are sealed.

ABET / licensure link. These points map to ABET Student Outcomes 2 and 4 — engineering design within realistic constraints, and recognition of ethical and professional responsibilities. Expect NCEES FE and PE exam questions on the NSPE Code of Ethics, OSHA construction requirements, and the engineer's standard of care.

07

Cost analysis

Web reinforcement economics

Approach

Rolled beams rarely need stiffeners; adding them is usually more expensive than a thicker web or heavier section.
Each pair of fitted stiffeners is a cut, fit, and weld operation — price it as shop labor, not steel.
Coping and reinforcing a coped web at high shear can cost more than the beam itself.

Worked cost example — Stiffeners vs heavier beam at a high-shear support

Basis: Vu = 210 kip

Line item	Qty	Rate	Cost
Option A — add bearing stiffener pair	2 pairs	$320	$640
Option B — upsize beam, +8 lb/ft × 32 ft	0.128 ton installed	$2,700	$346
Shop welding + inspection (Option A)	3 hr	$95	$285
Estimated total			$1,271

Takeaway. The heavier beam (≈$346) undercuts the stiffener detail (≈$925) — avoid shop labor whenever steel can do the job.

Unit rates are representative US averages for teaching purposes. On a real project, price with current local rates (RSMeans, fabricator quotes, or contractor pricing) and state the estimate date.

08

Animated concepts

Key mechanics visualised — watch the strain profile, stress block, or buckled shape evolve

Web shear buckling & tension field

Slender webs buckle diagonally before yielding; a post-buckling tension field (G3) recovers strength between stiffeners.

09

Engineering figures

Full-page reference diagrams — the visual vocabulary you will use for the rest of the course

§7.6.1

Girder in service

Fig. 7.1Steel plate girder with stiffeners
Deep plate girder — the web is doing all the shear work.

§7.6.2

Web buckling

Fig. 7.2Diagonal wrinkle in a slender web
Slender-web shear buckling — Cv1 < 1.0, then post-buckling tension-field if stiffeners exist.

§7.6.3

Intermediate stiffeners

Fig. 7.3Bolted transverse stiffener
Transverse stiffeners divide the web into shear panels of aspect ratio a/h.

§7.6.4

Web crippling

Fig. 7.4Web crippled under bearing
Local web failure at a short-bearing reaction — the J10 check.

10

Worked examples

Full textbook solutions — problem, theory, step-by-step, verification, interpretation

Example 7.1

W18x50 shear check — phi_v*Vn per G2.1

The W18x50 (A992) filler beam of Ex. 6.1 has factored end reaction Vu = 90 kips. Verify web shear per AISC 360-22 G2.1 and cross-check with Manual Table 3-2.

Problem statement

A W18x50 (A992) beam has end reaction Vu = 90 kips. Verify phi_v*Vn per AISC 360-22 G2.1 and confirm with Manual Table 3-2.

Steel plate girder web — FIG. 7.1 — Web shear is the entire story of Chapter G.

DIMW18x50 web carries the shear — Aw = d·tw, Vn = 0.6·Fy·Aw·Cv1.

Given

W18x50: d = 18.0 in, tw = 0.355 in, h/tw = 45.2
A992 steel: Fy = 50 ksi, E = 29,000 ksi
Vu = 90 kips (factored)

Find

phi_v*Vn per G2.1
Utilisation Vu/phi_v*Vn

Assumptions

Rolled I-shape (not built-up girder)
Bearing details verified separately (J10)

Code references

AISC 360-22 G2.1(a) — phi_v pocket
AISC 360-22 G2.1(b) — Aw definition
AISC Manual Table 3-2

Theory & approach

For rolled I-shapes with h/tw <= 2.24*sqrt(E/Fy), Cv1 = 1.00 and phi_v = 1.00. Vn = 0.60*Fy*Aw*Cv1.

Step-by-step solution

1
Web-slenderness pocket
FormulaAISC G2.1(a)
h / t_w <= 2.24 * sqrt(E / F_y)
2.24 * sqrt(29,000 / 50) = 2.24 * 24.08 = 53.9
h/t_w = 45.2 < 53.9 → C_v1 = 1.00, phi_v = 1.00
2
Web area Aw
FormulaG2.1(b)
A_w = d * t_w
A_w = 18.0 * 0.355 = 6.39 in^2
3
Design shear phi_v*Vn
FormulaG2.1
phi_v*V_n = phi_v * 0.60 * F_y * A_w * C_v1
phi_v*V_n = 1.00 * 0.60 * 50 * 6.39 * 1.00
phi_v*V_n = 192 kips
4
Manual cross-check
AISC Manual Table 3-2 for W18x50: phi_v*Vn = 192 kips ✓
5
Utilisation
V_u / phi_v*V_n = 90 / 192 = 0.47 ⇒ 47%

Verification

The pocket check, Aw definition, and product all reconcile with Manual Table 3-2 exactly.

Final answer

phi_v*Vn = 192 kips; utilisation = 47%. Beam is comfortable in shear.

Design interpretation

For most rolled W-shapes under gravity loading, shear utilisation runs 20-50%. Shear controls only for very short spans or heavy point loads.

Common mistakes

Using Aw = (d − 2tf)*tw — that under-counts the web area.
Using phi_v = 0.90 when h/tw is in the phi_v = 1.00 pocket.
Forgetting to run J10 at the reaction — G2.1 alone is not enough for bearing.

Engineering insight

The 2.24*sqrt(E/Fy) = 53.9 threshold covers 90%+ of rolled W-shapes. If h/tw > 53.9 you likely have a plate girder or a slender custom web.

References

· AISC 360-22 Chapter G
· AISC 360-22 J10
· AISC Manual 16th ed., Table 3-2

11

Guided practice

Compute the governing variables — hints unlock as you need them

A W14×22 (A992) has d = 13.7 in, t_w = 0.230 in, h/t_w = 53.3. Compute the design shear strength φ_v·V_n per AISC 360 §G2.1, state whether C_v1 = 1.0 applies, and interpret the result relative to a Vu = 60 kip end reaction.

Your turn

A_w =in^2phi_{v*V_n} =kip

Hints

1.A_w = d·t_w = 13.7·0.230 = 3.151 in² (full web depth × thickness, §G2.1b).

12

Independent practice

Solve the chapter's design task — compute each governing variable

Design task

A W12×26 (A992) beam is checked for shear per AISC G2. Compute the web area Aw, verify the web slenderness so Cv1 = 1.0, and report the design shear strength φv Vn.

Given

Fy = 50 ksi, E = 29,000 ksi
W12×26: d = 12.22 in, tw = 0.230 in, h/tw = 47.2
Rolled I-shape: φv = 1.00 when h/tw ≤ 2.24·√(E/Fy) = 53.95

Approach

Aw = d·tw (full-depth web for rolled shapes).
Since h/tw = 47.2 < 53.95, Cv1 = 1.0 and φv = 1.00 (Section G2.1(a)).
Vn = 0.6·Fy·Aw·Cv1; φv Vn = 1.00·Vn.

Submit your answer

A_w =in^2phi_{v*V_n} =kip

13

Mini design challenge

Select the option that satisfies every code and serviceability requirement in the brief

Brief

A 24-ft simply-supported W16 A992 beam carries w_u = 4.0 klf. Screen candidates so that (1) shear φ_v·V_n ≥ V_u = w_u·L/2, (2) flexure φ_b·M_n ≥ M_u = w_u·L²/8, AND (3) the h/t_w compact-web limit is checked (Exception §G2.1a). Pick the lightest section.

Requirements

V_u = w_u·L/2 = 4.0·24/2 = 48 kip
M_u = w_u·L²/8 = 4.0·24²/8 = 288 kip·ft
Verify h/t_w < 2.24·√(E/F_y) = 53.95 → φ_v = 1.00, C_v1 = 1.00
Check flexure φ_b·M_n ≥ M_u (AISC Table 3-2, Lb ≤ Lp)
A992, W16 family; pick the lightest passing BOTH shear and flexure

Section

Wt (lb/ft)

Δ (in)

Ru/Rn

Cost

Pick

14

Chapter summary

A mind map of how every concept connects

Graded Chapter Quiz(13 FE-style questions · AISC Manual required)

These questions reference AISC Steel Construction Manual (16th ed.) — sections, equations, and tables are cited explicitly. Use a calculator. Each question offers a clue you may reveal before answering. Submissions are recorded to your account once signed in.

Upload your worked solution (PDF)

Attach your handwritten or typed step-by-step solution for this chapter's graded quiz. The instructor can download every submission. PDF only, up to 25 MB.

How your upload will be graded

Your file — PDF, Word document, scanned handwriting or a photo — is read page by page like an experienced structural engineering instructor would. The scan is validated first, then your reasoning, structural model, calculations, diagrams, code basis and final answers are graded on process, not just the final number. Design work is additionally reviewed against AISC 360-22 and ACI 318-19. Partial credit applies, and one early mistake carried correctly forward is only penalized once.

Before you attach the file

Include every page, in order and right way up — a missing page cannot earn credit.
Keep margins in frame: nothing cropped at the edges, especially boxed final answers.
Scan or photograph in good, even light — no shadows, glare or blur; 300 dpi or a steady phone scan.
Write in dark pen; faint pencil is the most common 'UNREADABLE — INSTRUCTOR REVIEW REQUIRED' flag.
Include all diagrams, FBDs, shear/moment diagrams and section sketches — label them.
Number each question the same way the assignment does, and note anything you skipped.
Show units on every line and box your final answers.
Combine everything into ONE file (PDF preferred; Word, JPG or PNG accepted) under 20 MB.

I have read the grading rubric and the required output format, my scan is complete and readable, and this is my own work. Required before submitting.

Sign in to upload your worked solution.

16

FE exam preparation

NCEES-style practice with timer, equation sheet, and mastery tracking

Exam mode

30:00 Calculator

Question 1 / 9

A W18x50 has d = 18.0 in, tw = 0.355 in. Aw is closest to:

◆ EasyAISC G2.1(b)

A. 4.8 in^2
B. 6.4 in^2
C. 8.0 in^2
D. 10 in^2

Beam Shear Design

Compute Vn = 0.6*Fy*Aw*Cv1 (G2.1)

Classify web slenderness h/tw

Handle shear buckling / tension-field

Run J10 concentrated-force checks

Read Manual Table 3-2 (phi_v*Vn)

Verify Aw = d*tw

Web shear buckling

Web crippling

Web local yielding

Chapter 7 — Beam Shear Design (AISC 360-22 Chapter G)

1. Behavior

2. Nominal Shear Strength (§G2)

3. Shear Buckling and Stiffeners (§G2.2)

4. Block Shear at Beam End (Coped Beams, §J4.3)

5. Design Procedure

⚠ Common mistakes

Worked Example 7.1 — Web Shear Check, W18×50 Floor Beam

Step 1 — Factored shear at support

Step 2 — Web slenderness

Step 3 — Web area

Step 4 — Nominal & design shear

Step 5 — Verify

Step 6 — Coped-end block-shear check (J4.3)

Girder in service

Web buckling

Intermediate stiffeners

Web crippling

W18x50 shear check — phi_v*Vn per G2.1

Web-slenderness pocket

Web area Aw

Design shear phi_v*Vn

Manual cross-check

Utilisation

Graded Chapter Quiz(13 FE-style questions · AISC Manual required)

Upload your worked solution (PDF)

Compute Vn = 0.6FyAw*Cv1 (G2.1)