----------------- | ------ | ------------- | ---------------------------------------------------------- | | Design code | -- | -- | ACI 318-19, EN 1992-1-1, or AS 3600 | | Column dimension x | c1 | in / mm | Column width in the primary direction | | Column dimension y | c2 | in / mm | Column depth in the perpendicular direction | | Slab thickness | h | in / mm | Total slab depth | | Effective depth | d | in / mm | Distance from compression face to tension steel centroid | | Column location | -- | -- | Interior, edge, or corner | | Factored shear force | Vu | kip / kN | Factored shear at the critical section | | Unbalanced moment | Mu | kip-ft / kN-m | Moment to be transferred at the connection | | Concrete strength | fc' | psi / MPa | Specified compressive strength at 28 days | | Stud yield strength | fyt | ksi / MPa | Yield strength of shear reinforcement (max 60 ksi per ACI) | | Openings near column | -- | -- | Size and location of slab openings within 4d |

Design methodology

Critical Section Determination (ACI 318-19)

Per ACI 318-19 22.6.4, the critical section for two-way shear is located at d/2 from the column face. The critical perimeter bo is measured along the perimeter of this critical section:

Rectangular columns:  bo = 2(c1 + d) + 2(c2 + d)
Circular columns:     bo = pi x (D + d)

For edge columns, the exterior edge of the critical section is taken at the free edge. For corner columns, the perimeter follows the two free edges.

Concrete Shear Strength (ACI 318-19)

Per ACI 318-19 22.6.5.1, the nominal two-way shear stress capacity is the minimum of:

vc = min(
  4 x lambda x sqrt(fc'),                          (a)
  (2 + 4/beta_c) x lambda x sqrt(fc'),             (b)
  (alpha_s x d/bo + 2) x lambda x sqrt(fc')        (c)
)

where:
  beta_c = c1/c2 (ratio of long to short column side)
  alpha_s = 40 for interior, 30 for edge, 20 for corner columns
  lambda = 1.0 for normal-weight concrete
  sqrt(fc') limited to 100 psi for fc' > 10,000 psi

The total shear strength is Vc = vc x bo x d and the design strength is phi x Vc with phi = 0.75 per ACI 318-19 Table 21.2.1.

Unbalanced Moment Transfer

When column moments are present, the shear stress is non-uniform around the critical perimeter:

vu = Vu/(bo x d) + gamma_v x Mu x c_AB / Jc

where:
  gamma_v = 1 - 1/(1 + (2/3) x sqrt(b1/b2))
  b1 = c1 + d (critical section width parallel to moment)
  b2 = c2 + d (critical section width perpendicular to moment)
  c_AB = distance from centroid of critical section to point of interest
  Jc = polar moment of inertia of the critical section about its centroid

Shear Reinforcement Design (ACI 318-19)

When vu > phi x vc, shear reinforcement is required. Per ACI 318-19 22.6.6:

vs = (vu - phi x vc) / phi  (required steel contribution)

For headed shear studs:
  Av/s = vs x bo / (fyt)  per peripheral line
  Maximum spacing s_max = d/2 along radial lines
  First stud line at d/2 from column face
  Extend to where vu <= phi x 2 x sqrt(fc')

Maximum shear stress including reinforcement: vn = vc + vs <= 8 x sqrt(fc') (psi units). Stud rails typically provide 50-80% capacity increase over unreinforced punching strength.

EN 1992-1-1 Provisions

Eurocode 2 Section 6.4 uses the basic control perimeter at 2.0d from the column face:

vRd,c = CRd,c x k x (100 x rho_l x fck)^(1/3) + k1 x sigma_cp

where:
  CRd,c = 0.18 / gamma_c (gamma_c = 1.5)
  k = 1 + sqrt(200/d) <= 2.0 (d in mm)
  rho_l = sqrt(rho_ly x rho_lz) <= 0.02
  k1 = 0.15

The Eurocode approach differs from ACI in using a single formula rather than the three-term minimum, and in locating the control perimeter at 2.0d rather than d/2 from the column face.

AS 3600 Provisions

Australian Standard AS 3600 Section 9.3:

Vuc = beta1 x beta2 x beta3 x bv x do x fcv^(1/3)

where:
  beta1 accounts for slab aspect ratio
  beta2 accounts for loading area shape
  beta3 accounts for concentrated loads near supports
  bv = critical perimeter at do/2 from the column face
  do = effective depth to tension reinforcement

Common pitfalls

Frequently Asked Questions

How is the critical section for punching shear determined per ACI 318? Per ACI 318-19 22.6.4, the critical section is located at d/2 from the column face, where d is the effective slab depth. For rectangular columns, the critical perimeter bo = 2(c1 + d) + 2(c2 + d). For circular columns, bo = pi(D + d). The shear stress is vu = Vu/(bo x d). The concrete strength vc is the minimum of three expressions: 4 x sqrt(fc'), (2 + 4/beta_c) x sqrt(fc'), and (alpha_s x d/bo + 2) x sqrt(fc'), each with the lambda lightweight concrete factor. Edge and corner columns have reduced perimeters because the critical section is interrupted at free edges.

When is punching shear reinforcement required? Per ACI 318-19 22.6.5 and 22.6.6, shear reinforcement is required when vu exceeds phi x vc (phi = 0.75). Headed shear studs are the most effective reinforcement type, placed radially from the column face. The first stud row is at d/2 from the column face, with subsequent rows spaced at no more than d/2. Maximum shear stress with reinforcement is limited to phi x 8 x sqrt(fc'). Stud rails typically increase capacity by 50-80%. Reinforcement must extend to where vu <= phi x 2 x sqrt(fc').

How do different design codes handle punching shear? ACI 318-19 uses the three-term minimum approach with critical section at d/2 and phi = 0.75. EN 1992-1-1 places the basic control perimeter at 2.0d from the column face and uses vRd,c = CRd,c x k x (100 x rho_l x fck)^(1/3) with gamma_c = 1.5. AS 3600 uses Vuc = beta1 x beta2 x beta3 x bv x do x fcv^(1/3). The Eurocode generally produces lower unreinforced capacities for lightly reinforced slabs but higher capacities for heavily reinforced slabs due to the reinforcement ratio term. Always design to the code governing your jurisdiction.

How is unbalanced moment transferred at slab-column connections? Unbalanced moment is transferred through a combination of flexure and eccentric shear. Per ACI 318-19 22.6.4.2, gamma_f = 1/(1 + (2/3)sqrt(b1/b2)) of the moment goes to flexure within a slab width of c2 + 3h. The remainder gamma_v = 1 - gamma_f goes to eccentric shear, creating non-uniform shear stress: vu = Vu/(bo x d) + gamma_v x Mu x y / Jc. This combined stress must be checked around the full perimeter, with the maximum occurring at the face where the moment adds to the direct shear.

How do slab openings affect punching shear capacity? Per ACI 318-19 22.6.4.3, openings within 4d of the column face reduce the effective critical perimeter. The portion of bo falling within the projection of the opening is considered ineffective. Maintain at least 4d clear from the column face to any opening to avoid perimeter reduction. When openings cannot be relocated, increase slab thickness or add shear reinforcement to compensate for the reduced perimeter length.

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