| name | shear-stress |
| description | Shear stress — transverse shear (VQ/Ib), shear flow in thin-walled sections, shear center, horizontal shear in composite beams, direct shear, punching shear. |
| metadata | {"priority":7,"promptSignals":{"phrases":["shear stress","transverse shear","shear flow","shear center","VQ/Ib","punching shear","horizontal shear"],"minScore":3}} |
Shear Stress — Complete Skill
Transverse Shear Stress (VQ/Ib)
τ = VQ/(Ib)
Where:
- V = transverse shear force [N]
- Q = first moment of area above (or below) point of interest: Q = ȳ'× A' [m³]
- I = moment of inertia of full section about NA [m⁴]
- b = width of section at point of interest [m]
Assumptions (Euler-Bernoulli beam):
- Prismatic beam, homogeneous, linear elastic
- τ uniform across width b (valid for b << d)
- Valid for solid sections and thin-walled open sections
Rectangular Section (b×d):
Q at neutral axis = b×(d/2)×(d/4) = bd²/8
τ_max = VQ/(Ib) = V(bd²/8)/(bd³/12×b) = 1.5V/(bd) = 1.5V/A
(max at NA, zero at top/bottom)
W-Shape (I-beam):
τ_web ≈ V/A_web = V/(d×t_w) [approximately uniform over web]
τ_flange = small (wide flange → large b, small τ)
More precise: Q_at_web = Q_flange + Q_web-above = A_f×ȳ_f + t_w×(c-y)×(c+y)/2
Circular Section (diameter d):
τ_max = 4V/(3A) = 4V/(3π r²) at neutral axis
Hollow Circular Section (r_o, r_i):
τ_max = 4V(r_o² + r_o r_i + r_i²)/(3π(r_o⁴ - r_i⁴))
Shear Flow (q = VQ/I)
For thin-walled sections: τ = q/t where q = VQ/I [N/m]
q varies continuously around the section
Open thin-walled (channel, Z, T):
Start at free end (q=0), integrate Q around perimeter:
q(s) = V/I × Q(s) = V/I × ∫₀ˢ t(s')ȳ(s')ds'
Closed thin-walled (box, tube): statically indeterminate
q(s) = q₀ + V/I × Q_open(s)
q₀ = constant determined by twist compatibility: ∮ q ds/(Gt) = 0 for single closed cell
Shear Center
For unsymmetric cross-sections: shear center ≠ centroid
Load must pass through shear center to avoid twisting
Channel section (flange b_f, web h, flange thickness t_f, web thickness t_w):
e = b_f² h t_f / (4I_x) [distance from web centerline to shear center]
I_x = (t_w h³/12) + 2(b_f t_f h²/4)... [standard formula]
I-section (doubly symmetric): shear center = centroid
Z-section: shear center at centroid (doubly point-symmetric)
Angle section: shear center at vertex intersection
Horizontal Shear in Composite Beams
At interface between materials (e.g., steel+concrete, flange+web):
q_interface = VQ/I [shear flow, N/m or lb/in]
Connectors (shear studs) spacing:
s = Q_stud × I / (V × Q) where Q_stud = capacity per stud [N]
Or: s ≤ φQ_n × n / (V_u × Q/I) [AISC, LRFD]
Glued wood beam: τ_glue = VQ/(Ib), b = glue width at interface
Punching Shear (Concrete Slabs — ACI 318)
At column-slab connection:
Critical perimeter: b₀ = 4(c + d) [for square column size c, effective depth d]
Nominal shear strength: V_c = (2 + 4/β_c)√f'c × b₀d ≥ (α_s × d/b₀ + 2)√f'c × b₀d ≥ 4√f'c × b₀d
[f'c in psi] — use minimum value
Design: φV_c ≥ V_u (φ = 0.75)
If fails → add shear reinforcement (headed studs, shear caps) or increase slab thickness
Direct Shear (Fasteners, Pins, Keyways)
Single shear: τ = F/A
Double shear: τ = F/(2A)
Shear area A = π d²/4 for round pins, bolts
Bearing stress (pin on plate):
σ_b = F/(d × t) [projected area]
Allow: σ_b ≤ 1.9F_y (AISC), or σ_b ≤ 0.9Sy (machine design)
Shear Lag (Partial Connection)
Net section effective for tension but reduced for eccentricity:
A_e = U × A_n
U = 1 - x̄/L (AISC Eq. D3-1)
x̄ = distance from connection plane to centroid of connected element
L = connection length (parallel to load)
Output
Provide: τ_max [MPa] location (NA, interface, etc.), Q [mm³], shear flow q [N/mm], shear center location (e from reference), stud spacing s [mm].
