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. Every beam that carries transverse load must transfer that load to its supports as shear, and every connection at a beam end is really a shear connection. Chapter G answers one deceptively simple question: how much shear can the web alone carry before it yields or buckles? For the vast majority of rolled W-shapes at F_y = 50 ksi the answer is almost trivial — φ_v = 1.00 and V_n = 0.6F_yA_w — but as webs get thinner and deeper (plate girders, built-up sections, channels, HSS), shear buckling, stiffener spacing, and coped-end block shear all become governing checks that a careless designer will miss.

1. Why the Web (Almost) Alone?

In an I-shaped beam under transverse load, elastic shear-flow theory (VQ/I_xt) shows the shear stress is parabolic through the depth and is concentrated almost entirely in the thin web; the flanges, being far from the neutral axis and oriented perpendicular to V, carry a negligible fraction (typically < 5%). AISC exploits this by ignoring the flange contribution altogether and using a single, conservative, uniform stress on a simplified web area A_w, rather than requiring a full elastic shear-flow calculation for every beam.

Figure 7.1 — Idealized shear-stress distribution in a W-shape: web carries essentially all of V; AISC replaces the true parabola with a uniform average stress over A_w.

2. The Web Shear Area, A_w

AISC's shear area is deliberately simple — the full section depth times the web thickness, not the clear distance between flanges (h) and not the gross area A_g. Using the full depth is a conservative simplification that already builds in margin for the small shear carried by the fillets/flange webs.

AISC 360-22 §G2.1(b)

(1)

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

d = overall (nominal) depth of the section, in.
t_w = web thickness, in.

Physically: a deeper, thicker web gives more A_w and more shear capacity — this is why deep, heavy beams rarely have a shear problem even when their moment demand is very high.

Figure 7.2 — A_w uses the full outside depth d, not the clear web height h. Do not confuse the two: h is only used to compute the slenderness ratio h/t_w.

3. Nominal Shear Strength — Web Yielding vs. Web Shear Buckling (§G2.1)

A slender plate loaded in pure shear behaves just like a plate loaded in compression: if it is stocky it yields in shear; if it is thin it buckles diagonally before the shear yield stress 0.6F_y is reached. AISC captures both behaviors with a single equation and a web shear coefficient C_v1 that ranges from 1.0 (full yielding) down toward zero (very slender, elastic buckling).

Spec. Eq. G2-1

(2)

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

C_v1 is found from the web slenderness ratio h/t_w compared against two limits that depend on the plate-buckling coefficient k_v (k_v = 5.34 for unstiffened webs of most rolled shapes):

Spec. Eq. G2-2 — stocky web (web shear yielding)

(3)

I f h / t_{w} \leq 1.10 (k_{v} E / F_{y}) : C_{v 1} = 1.0

Spec. Eq. G2-3 — inelastic buckling

(4)

I f h / t_{w} > 1.10 (k_{v} E / F_{y}) : C_{v 1} = 1.10 (k_{v} E / F_{y}) / (h / t_{w})

User Note shortcut (§G2.1(a)). For all standard rolled I-shapes with F_y ≤ 50 ksi that satisfy h/t_w ≤ 2.24 √(E/F_y) (≈ 53.95 at F_y = 50 ksi), AISC simply sets C_v1 = 1.0 and gives the higher resistance factor φ_v = 1.00 (Ω_v = 1.50) instead of the usual φ_v = 0.90 (Ω_v = 1.67). Almost every rolled W-shape in the Manual at 50 ksi qualifies — this is why beam shear is so rarely a governing check in ordinary buildings.

Figure 7.3 — C_v1 as a function of web slenderness. Region 1: full shear yielding, φ_v = 1.00. Region 2: inelastic buckling, φ_v drops to 0.90. Region 3: elastic buckling (Eq. G2-4, not shown numerically here), C_v1 ∝ 1/(h/t_w)².

4. Design (Available) Shear Strength

(5)

L R F D : ϕ_{v} V_{n} \geq V_{u} ∣ A S D : V_{n} / Ω_{v} \geq V_{a}

Condition	φ_v (LRFD)	Ω_v (ASD)	Applies to
h/t_w ≤ 2.24√(E/F_y), rolled I-shape, F_y ≤ 50 ksi	1.00	1.50	Nearly all rolled W, S, HP shapes (§G2.1(a))
All other doubly-symmetric shapes / built-up girders	0.90	1.67	Plate girders, slender webs, non-qualifying shapes (footnote [v] in Table 1-1)
Channels (all cases)	0.90	1.67	§G1(a) — the §G2.1(a) exception never applies to channels
Single angles	0.90	1.67	§G4
Round HSS	0.90	1.67	§G5, uses F_cr from torsional/shear buckling
Rectangular HSS / box sections	0.90	1.67	§G4/G2.1, A_w = 2ht (two webs)
Doubly-symmetric shapes, minor-axis shear	0.90	1.67	§G6, flanges act as the "web"

Common trap: the φ_v = 1.00 / Ω_v = 1.50 pocket is a narrow exception, not the default. Channels, built-up girders, and minor-axis shear always use φ_v = 0.90 even when their h/t_w is small.

5. Transverse Stiffeners and Tension-Field Action (§G2.2, §G3)

When a web is too slender to rely on C_v1 alone, transverse stiffeners can be added. Stiffeners do two things: (1) they raise the plate-buckling coefficient k_v, effectively subdividing the web into shorter, stockier panels; and (2) for stiffened interior panels, AISC permits tension-field action (TFA) — post-buckling reserve strength in which the buckled web acts like the diagonal member of a Pratt truss, anchored by the stiffeners and flanges, developing shear far beyond the elastic buckling load. TFA is not permitted in end panels (no anchorage on one side) or when a/h > 3 or a/h > [260/(h/t_w)]².

Figure 7.4 — Transverse stiffeners subdivide a slender web into panels of width a. Interior panels can mobilize a diagonal tension field; end panels (adjacent to a free edge/reaction) cannot.

Almost all AISC Manual rolled W-shapes never need stiffeners for ordinary gravity beams — this topic matters mainly for plate girders and deep built-up sections, developed further with worked stiffener-spacing design in Chapter 10.

6. Shear in Channels (§G1, §G2.1)

A channel has only one web, so A_w = d·t_w exactly as for a W-shape, and the nominal strength equation (Eq. G2-1) is identical. The only difference is the resistance factor: per §G1(a), the φ_v = 1.00 / Ω_v = 1.50 shortcut of §G2.1(a) applies only to I-shaped rolled sections — never to channels. Channels always use φ_v = 0.90, Ω_v = 1.67, even though many rolled channels satisfy the same stocky-web slenderness limit.

7. Minor-Axis (Weak-Axis) Shear — §G6

Occasionally a member must carry shear parallel to its weak axis (e.g., a beam framing at 90° into a girder web, or wind load on a column oriented to bend about y). In that orientation, the two flanges are the "webs" resisting shear, each behaving like a rectangular plate in shear bending about its own weak axis; the actual web of the shape contributes negligibly.

Spec. Eq. G6-1

(6)

V_{n} = 0.6 \cdot F_{y} \cdot b_{f} \cdot t_{f} \cdot C_{v 2} (p er f l an g e

m u l t i pl y b y 2 f or b o t h f l an g es)

C_v2 is computed the same way as C_v1 but using h/t_w → b_f/(2t_f) and k_v = 1.2 (per §G2.2, referenced from G6). Per the §G6 User Note, C_v2 = 1.0 for essentially all rolled W-, S-, M-, and HP-shapes at F_y ≤ 70 ksi.

Figure 7.5 — (a) Channel major-axis shear: single web area A_w = d·t_w. (b) Minor-axis shear in a doubly-symmetric shape: both flanges act as parallel "webs" per §G6.

8. Coped Beam Ends: Block Shear and Local Section Checks (§J4.3, Manual Part 9)

When the top (or top and bottom) flange of a beam is trimmed away ("coped") to fit around a supporting girder flange, two new limit states appear that have nothing to do with Chapter G's uniform web-shear check: (1) block shear rupture of the bolted or welded connection region (a combined shear-yield/shear-fracture plane plus a perpendicular tension plane, per §J4.3 — the identical mechanics used for gusset plates and tension members), and (2) local buckling or flexural yielding of the reduced (coped) web section, checked using AISC Manual Part 9 procedures because the coped region is a short, deep cantilever stub with its own effective depth and slenderness.

Spec. Eq. J4-5 (block shear)

(7)

R_{n} = 0.6 F_{u} A_{n v} + U_{b s} F_{u} A_{n t} \leq 0.6 F_{y} A_{g v} + U_{b s} F_{u} A_{n t}

A_gv, A_nv = gross/net area on the shear plane (vertical, through the bolt line).
A_nt = net area on the tension plane (horizontal, from the last bolt to the coped edge).
U_bs = 1.0 when the tension stress is uniform (typical for coped beam shear tabs).

φ = 0.75 (LRFD), Ω = 2.00 (ASD) — the standard rupture factors, not the shear-yielding factors from Chapter G.

Figure 7.6 — Coped beam end. The shear plane A_nv runs vertically through the bolt line; the tension plane A_nt runs horizontally to the coped edge. The reduced depth d_c must also be checked for local flexural yielding/buckling.

9. Combined Shear and Flexure — When Do They Interact?

For ordinary rolled shapes, AISC Chapter G shear and Chapter F flexure are checked independently — there is no shear-moment interaction equation in the main Specification body, because the web area that resists shear and the flanges that resist moment are (to a good approximation) structurally separate load paths. The one place interaction reappears is exactly at a coped or connection region: locally, the reduced net section must resist shear and the eccentric moment from the bolt group simultaneously (AISC Manual Part 9 "eccentrically loaded bolt/weld group" tables), and in plate-girder design at high-shear/high-moment panels near interior supports where webs can be governed by a combined check in some design guides (Design Guide 25).

Rule of thumb: for a standard simply-supported gravity beam, check V_u at the support (max shear, near-zero moment) and M_u at midspan (max moment, near-zero shear) — they rarely coincide, so no interaction equation is normally needed outside of coped or heavily loaded connection regions.

10. Design Workflow

Determine the factored (or ASD) required shear V_u (or V_a) at each critical section — usually at the face of the support and at any concentrated load.
Pull d, t_w, h/t_w from the Manual (Table 1-1/1-5) for the trial shape.
Compute the stocky-web limit 2.24√(E/F_y) (rolled I-shapes only) and compare to h/t_w to select C_v1 and the correct φ_v/Ω_v pair.
Compute A_w = d·t_w, then V_n = 0.6F_yA_wC_v1.
Verify φ_vV_n ≥ V_u (or V_n/Ω_v ≥ V_a) — or simply pull φ_vV_n straight from AISC Manual Table 3-2/3-6/3-8.
If the beam is a channel or is loaded about its minor axis, re-select φ_v/Ω_v and the correct A_w (or b_ft_f for G6) per §G1/§G6.
If the end is coped, run the block-shear check (§J4.3) and the coped-section local check (Manual Part 9) — these govern far more often than the uniform web check.
For deep built-up girders, check whether stiffeners are required and whether tension-field action may be used (Chapter 10 develops this fully).

⚠ Common mistakes

Using the gross area A_g or the clear web height h instead of A_w = d·t_w (full outside depth).
Applying φ_v = 1.00/Ω_v = 1.50 to a channel, a built-up girder, or a minor-axis shear check — that pocket is for rolled I-shapes in major-axis shear only.
Forgetting to check block shear (and the local coped-section check) at coped beam ends — the uniform Chapter G shear check does not capture this limit state.
Mixing up the tension and shear planes (A_nt vs. A_nv) in the block-shear equation, or using the wrong φ (block shear uses φ = 0.75, not φ_v = 0.90 or 1.00).
Assuming tension-field action is always available once stiffeners are present — it is barred in end panels and when a/h exceeds the Spec limits.
Ignoring web local yielding/crippling and bearing at concentrated loads (§J10) — a separate, related limit state from Chapter G shear, covered later.

Exam/FE tip. Memorize A_w = d·t_w, the stocky-web limit 2.24√(E/F_y) ≈ 54 at F_y = 50 ksi, and the φ_v/Ω_v pairs (1.00/1.50 vs. 0.90/1.67). On the FE/PE exam, if the problem says "channel" or "minor-axis," reflexively use φ_v = 0.90 even if the numbers look stocky.

Worked Example 7.1 — W-shape in Major-Axis Shear, via Manual Tables

Adapted from AISC Design Examples V16.0, Example G.1A

Shape: W24×62, ASTM A992/A992M, F_y = 50 ksi
End shear from dead load: V_D = 48 kips
End shear from live load: V_L = 145 kips

Figure 7.1a — W24×62 beam with dead- and live-load end shears

Step 1 — Required shear strength (ASCE/SEI 7 Ch. 2).

LRFD Combo 2

(1)

V_{u} = 1.2 V_{D} + 1.6 V_{L}

LRFD: V_u = 1.2(48) + 1.6(145) = 57.6 + 232 = 290 kips
ASD: V_a = 48 + 145 = 193 kips

Step 2 — Available shear strength from AISC Manual Table 3-2. LRFD: φ_vV_nx = 306 kips ≥ 290 kips ✓
ASD: V_nx/Ω_v = 204 kips ≥ 193 kips ✓

Result: W24×62 is adequate for shear — φ_vV_n = 306 k ≥ V_u = 290 k (D/C = 0.95). Table lookup is fast but Example 7.2 shows the underlying hand calculation.

Engineering interpretation: the utilization is close to 1.0 (95%), so this beam is shear-critical — unusual for a rolled W-shape, and a signal that the design was likely also governed or nearly governed by shear rather than moment.

Common mistakes:

Reading Table 3-2 values for the wrong grade of steel (Manual tables are F_y-specific).
Forgetting that the tabulated φ_vV_n already embeds the correct φ_v (1.00 here) — don't re-multiply.

Worked Example 7.2 — W-shape in Major-Axis Shear, by Direct Specification Calculation

Adapted from AISC Design Examples V16.0, Example G.1B

Same beam as Example 7.1: W24×62, A992, F_y = 50 ksi
Section properties: d = 23.7 in., t_w = 0.430 in.
V_u = 290 kips (LRFD), V_a = 193 kips (ASD) — from Example 7.1

Step 1 — Check §G2.1(a) applicability and web slenderness.

§G2.1(a) User Note

(2)

N e a r l y a l l r o l l e d I - s ha p es a t F_{y} \leq 50 k s i s a t i s f y h / t_{w} \leq 2.24 (E / F_{y})

C_{v 1} = 1.0, ϕ_{v} = 1.00

W24×62 is not footnoted [v] in Table 1-1, so §G2.1(a) applies directly.

Spec. Eq. G2-2

(3)

C_{v 1} = 1.0

Step 2 — Web shear area.

§G2.1(b)

(4)

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

A_w = 23.7 × 0.430 = 10.2 in²

Step 3 — Nominal shear strength.

Spec. Eq. G2-1

(5)

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

V_n = 0.6(50)(10.2)(1.0) = 306 kips

Step 4 — Available shear strength. LRFD: φ_v = 1.00 → φ_vV_n = 1.00(306) = 306 kips
ASD: Ω_v = 1.50 → V_n/Ω_v = 306/1.50 = 204 kips

Quantity	LRFD	ASD
Required, V_u/V_a	290 k	193 k
Available, φV_n / V_n/Ω	306 k	204 k
D/C ratio	0.948 ✓	0.946 ✓

Result: hand calculation exactly matches the Manual Table 3-2 tabulated value (φ_vV_n = 306 kips) because the tables are generated from these same equations.

Engineering interpretation: this example demonstrates that Manual tables are shortcuts, not different rules — knowing the hand method lets you check non-tabulated or built-up sections the same way.

Common mistakes:

Using d = clear web height h instead of full depth d in A_w.
Applying φ_v = 0.90 out of habit when §G2.1(a) actually grants 1.00 for this shape.

Worked Example 7.3 — Channel in Major-Axis Shear, via Manual Tables

Adapted from AISC Design Examples V16.0, Example G.2A

Shape: C15×33.9, ASTM A992/A992M, F_y = 50 ksi
End shear from dead load: V_D = 25 kips
End shear from live load: V_L = 75 kips

Figure 7.3a — C15×33.9 channel cross-section, single web resisting shear

Step 1 — Required shear strength. LRFD: V_u = 1.2(25) + 1.6(75) = 30 + 120 = 150 kips
ASD: V_a = 25 + 75 = 100 kips

Step 2 — Available shear strength from AISC Manual Table 3-8. LRFD: φ_vV_n = 162 kips ≥ 150 kips ✓
ASD: V_n/Ω_v = 108 kips ≥ 100 kips ✓

Result: C15×33.9 is adequate for shear — φ_vV_n = 162 k ≥ V_u = 150 k. Note φ_vV_n here corresponds to φ_v = 0.90, not 1.00, even though this is a stocky rolled section — Example 7.4 shows why.

Engineering interpretation: channels are frequently used as horizontal bracing struts or light beams; their shear tables already reflect φ_v = 0.90 baked in, so no separate reduction is needed by the user.

Common mistakes:

Assuming a channel gets φ_v = 1.00 just because its h/t_w is small — it never does.

Worked Example 7.4 — Channel in Major-Axis Shear, by Direct Specification Calculation

Adapted from AISC Design Examples V16.0, Example G.2B

Same channel as Example 7.3: C15×33.9, A992, F_y = 50 ksi
Section properties: d = 15.0 in., t_w = 0.400 in.
V_u = 150 kips (LRFD), V_a = 100 kips (ASD)

Step 1 — Web slenderness qualifies for C_v1 = 1.0.

Spec. Eq. G2-3, per Manual User Note

(6)

A l l A 992 r o l l e d c hann e l ss a t i s f y h / t_{w} \leq 1.10 (k_{v} E / F_{y})

C_{v 1} = 1.0

Step 2 — Web area.

(7)

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

A_w = 15.0 × 0.400 = 6.00 in²

Step 3 — Nominal shear strength.

Spec. Eq. G2-1

(8)

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

V_n = 0.6(50)(6.00)(1.0) = 180 kips

Step 4 — Available shear strength — note the φ_v switch.

§G1(a)

(9)

B ec a u se § G 2.1 (a) n e v er a ppl i es t oc hann e l s, u se ϕ_{v} = 0.90 an d Ω_{v} = 1.67 (n o tt h e 1.00/1.50 I - s ha p e p oc k e t)

LRFD: φ_vV_n = 0.90(180) = 162 kips
ASD: V_n/Ω_v = 180/1.67 = 108 kips

Quantity	LRFD	ASD
V_n	180 k
φ_v or Ω_v	0.90	1.67
Available strength	162 k	108 k
Required	150 k	100 k

Result: φ_vV_n = 162 kips ≥ 150 kips ✓ — matches Table 3-8 exactly, confirming the C_v1 = 1.0 but φ_v = 0.90 (not 1.00) treatment for channels.

Engineering interpretation: the nominal strength V_n is identical whether computed for an equivalent-slenderness W-shape or a channel — it is only the resistance factor that differs, purely because of §G1(a)'s scope restriction to I-shapes.

Common mistakes:

Copy-pasting the W-shape φ_v = 1.00 pocket onto a channel calculation.
Using k_v = 1.2 (the G6 minor-axis value) instead of 5.34 for this major-axis channel check.

Worked Example 7.5 — Doubly Symmetric Shape in Minor-Axis Shear

Adapted from AISC Design Examples V16.0, Example G.6

Shape: W21×48, ASTM A992/A992M, F_y = 50 ksi
Section properties: b_f = 8.14 in., t_f = 0.430 in.
End shear from dead load (weak-axis direction): V_D = 20.0 kips
End shear from live load (weak-axis direction): V_L = 60.0 kips

Figure 7.5a — W21×48 with shear applied about the weak (minor) axis; flanges act as the shear-resisting elements per §G6

Step 1 — Required shear strength. LRFD: V_u = 1.2(20.0) + 1.6(60.0) = 24.0 + 96.0 = 120 kips
ASD: V_a = 20.0 + 60.0 = 80.0 kips

Step 2 — Flange slenderness for minor-axis shear (§G2.2, per §G6).

(10)

h / t_{w} (f l an g e ana l o g) = b_{f} / (2 t_{f})

u se k_{v} = 1.2

b_f/(2t_f) = 8.14 / (2 × 0.430) = 9.47
Limit: 1.10√(k_vE/F_y) = 1.10√(1.2 × 29,000/50) = 1.10(26.4) = 29.0
9.47 < 29.0 → stocky

Step 3 — Web shear coefficient for minor axis.

Spec. Eq. G2-9

(11)

C_{v 2} = 1.0

Per the User Note of §G6, C_v2 = 1.0 for all rolled A6/A6M W-, S-, M-, HP-shapes at F_y ≤ 70 ksi — this is automatic here.

Step 4 — Nominal minor-axis shear strength.

Spec. Eq. G6-1

(12)

V_{n} = 0.6 \cdot F_{y} \cdot b_{f} \cdot t_{f} \cdot C_{v 2} \cdot 2 (b o t h f l an g es)

V_n = 0.6(50)(8.14)(0.430)(1.0)(2) = 210 kips

Step 5 — Available strength and check.

§G1

(13)

ϕ_{v} = 0.90 (L R F D), Ω_{v} = 1.67 (A S D) — min or - a x i ss h e a r n e v er g e t s t h e § G 2.1 (a) p oc k e t

LRFD: φ_vV_n = 0.90(210) = 189 kips ≥ 120 kips ✓
ASD: V_n/Ω_v = 210/1.67 = 126 kips ≥ 80.0 kips ✓

Limit state	φV_n (LRFD)	V_n/Ω (ASD)	Governs?
Minor-axis flange shear (§G6)	189 k	126 k	Yes — only applicable limit state here

Result: W21×48 has adequate minor-axis shear strength — φ_vV_n = 189 kips ≥ V_u = 120 kips ✓ (D/C = 0.63).

Engineering interpretation: minor-axis shear demands are usually small in ordinary framing, but this check matters for members loaded transversely in their weak direction (e.g., girts, wind columns, or beams framing perpendicular to the primary bending plane).

Common mistakes:

Using A_w = d·t_w (major-axis area) for a minor-axis shear check — §G6 uses the flange area 2b_ft_f, not the web.
Forgetting the ×2 multiplier for two flanges in Eq. G6-1.
Applying φ_v = 1.00 to a minor-axis shear check — always 0.90 per §G1.

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.

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Include every page, in order and right way up — a missing page cannot earn credit.
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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.
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Show units on every line and box your final answers.
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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. Why the Web (Almost) Alone?

2. The Web Shear Area, Aw

3. Nominal Shear Strength — Web Yielding vs. Web Shear Buckling (§G2.1)

4. Design (Available) Shear Strength

5. Transverse Stiffeners and Tension-Field Action (§G2.2, §G3)

6. Shear in Channels (§G1, §G2.1)

7. Minor-Axis (Weak-Axis) Shear — §G6

8. Coped Beam Ends: Block Shear and Local Section Checks (§J4.3, Manual Part 9)

9. Combined Shear and Flexure — When Do They Interact?

10. Design Workflow

⚠ Common mistakes

Worked Example 7.1 — W-shape in Major-Axis Shear, via Manual Tables

Worked Example 7.2 — W-shape in Major-Axis Shear, by Direct Specification Calculation

Worked Example 7.3 — Channel in Major-Axis Shear, via Manual Tables

Worked Example 7.4 — Channel in Major-Axis Shear, by Direct Specification Calculation

Worked Example 7.5 — Doubly Symmetric Shape in Minor-Axis Shear

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)

2. The Web Shear Area, A_w