Punching Shear Design per ACI 31819 with ideCAD
How does ideCAD calculate the punching shear required and design strength according to ACI 31819?
Slab punching shear is checked automatically in accordance with ACI 318  Chapter 8.
Punching shear reinforcement is defined by user.
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Notation
A_{g }= gross area of the concrete section, in^{2}
A_{s}_{ }= area of non prestressed longitudinal tension reinforcement, in^{2}
b_{o} = perimeter of critical section for twoway shear in slabs and footings, in.
b_{slab} = effective slab width, in.
b_{1 } = dimension of the critical section b_{o} measured in the direction of the span for which moments are determined, in.
b_{2 } = dimension of the critical section b_{o} measured in the direction perpendicular to b_{1}, in.
d = distance from extreme compression fiber to centroid of longitudinal tension reinforcement, in.
E_{c } = modulus of elasticity of concrete, psi
E_{s } = modulus of elasticity of reinforcement, psi
√f_{c}‘ = square root of specified compressive strength of concrete, psi
f_{c}^{'}_{ } = specified compressive strength of concrete, psi
f_{y } = specified yield strength for nonprestressed reinforcement, psi
J_{c } = property of assumed critical section analogous to polar moment of inertia
M_{sc} = factored slab moment that is resisted by the column at a joint, in.lb
v_{c }= stress corresponding to nominal twoway shear strength provided by concrete, psi
v_{uv }= factored shear stress on the slab critical section for twoway action, from the controlling load combination, without moment transfer, psi
α_{s }= constant used to calculate V_{c} in slabs and footings
ϕ = strength reduction factor
λ = modification factor to reflect the reduced mechanical properties of lightweight concrete relative to normalweight concrete of the same compressive strength
λ_{s} = factor used to modify shear strength based on the effects of member depth, commonly referred to as the size effect factor
ε_{t }= net tensile strain in extreme layer of longitudinal tension reinforcement at nominal strength, excluding strains due to effective prestress, creep, shrinkage, and temperature
ε_{ty }= value of net tensile strain in the extreme layer of longitudinal tension reinforcement used to define a compressioncontrolled section
γ_{f} = factor used to determine the fraction of M_{sc} transferred by slab flexure at slabcolumn connections
γ_{v} = factor used to determine the fraction of M_{sc} transferred by eccentricity of shear at slabcolumn connections
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Factored Slab Moment Resisted by the Columns
According to ACI 8.4.2.2.1, if gravity, wind, earthquake, or other loads cause a moment at the slabcolumn joint, a fraction of M_{sc} , the factored slab moment resisted by the column at a joint, should be transferred by flexure in accordance with ACI 8.4.2.2.2 through ACI 8.4.2.2.5.
According to ACI 8.4.2.2.2, the fraction of unbalanced factored slab moment resisted by the column, γ_{f}M_{sc} , is assumed to be transferred by flexure. γ_{f} is calculated in accordance with ACI Eq.(8.4.2.2.2);
The fraction of unbalanced moment transferred by the eccentricity of shear is taken to be γ_{v}M_{sc} and the factor γ_{v} equals 1 γ_{f}._{ }(γ_{v}=1 γ_{f })
According to ACI 8.4.2.2.4, for nonprestressed slab systems without beams, where the limitations on v_{uv} and ε_{t }in ACI Table 8.4.2.2.4 are satisfied, γ_{f} can be increased to maximum modified values provided in ACI Table 8.4.2.2.4. The calculation of twoway shear strength provided by concrete without shear reinforcement v_{c }is explained in TwoWay Shear Strength per ACI 31819 with ideCAD title.
Column location  Span direction  v_{uv}  ε_{t}  Max. γf 

Corner colum  Either direction 

 1.0 
Edge column  Perpendicular to the edge 

 1.0 
Parallel to the edge 


 
Interior column  Either direction 



Critical Section
According to ACI 22.6.4.1, for twoway shear, critical sections should be located so that perimeter b_{o} is a minimum but need not be closer than d/2 to the two cases given below;
Edges or corners of columns, concentrated loads, or reaction areas
Changes in slab or footing thickness, such as edges of capitals, drop panels, or shear caps
According to ACI 22.6.4.1.1, concentrated loads, or reaction areas, critical sections for twoway shear in accordance with ACI 22.6.4.1 (case 1 and case 2) can be defined assuming straight sides for square or rectangular columns.
According to ACI 22.6.4.1.2, critical sections for twoway shear in accordance with ACI 22.6.4.1 (case 1 and case 2) can be defined assuming a square column of equivalent area for a circular or regular polygonshaped column.
According to ACI 22.6.4.2, for twoway members with single (or multi) leg stirrup or headed stud shear reinforcement, one more critical section with perimeter b_{o} located d/2 beyond the point where shear reinforcement is discontinued. The shape of this critical section should be a polygon selected to minimize b_{o}.
The value of b_{o} is shown in the figure below.
According to ACI 22.6.4.3, If there is an opening located closer than 4h from the periphery of a column, concentrated load, or reaction area, the portion of b_{o} enclosed by straight lines projecting from the centroid of the column, concentrated load or reaction area and tangent to the boundaries of the opening should be considered ineffective.
Factored Twoway Shear Stress Due to Shear and Factored Slab Moment Resisted by the Column
According to ACI 8.4.4.2.1, for punching shear with factored slab moment resisted by the column, factored shear stress v_{u} should be calculated at critical sections. Factored shear stress v_{u} corresponds to combinations of v_{uv} and γ_{v}M_{sc}.
According to ACI 8.4.4.2.2, γ_{v}M_{sc} is the fraction of unbalanced moment transferred by eccentricity of shear. The fraction of M_{sc} should be applied at the centroid of the critical section. γ_{v} is taken as specified in ACI Eq. (8.4.4.2.2);
According to ACI 8.4.4.2.3, the factored shear stress caused by γ_{v}M_{sc} is assumed to vary linearly about the centroid of the critical section.
The stress distribution is assumed for the interior and edge columns, as illustrated in ACI Fig. R8.4.4.2.3. For the interior column, the perimeter of the critical section equals 2(c_{1}+d+c_{2}+d). For the edge column, the perimeter of the critical section equals 2(c_{1}+d) + (c_{2}+d).
The factored shear stress v_{uv} and factored slab moment resisted by the column M_{sc} are determined at the centroidal axis cc of the critical section. According to the stress distribution shown in ACI Fig. R8.4.4.2.3. The maximum factored punching shear stress is calculated from;
For an interior column, the property of the assumed critical section analogous to the polar moment of inertia, J_{c,} is calculated by:
Punching shear stress with biaxial bending can similarly be applied for two perpendicular moment directions. For the interior column, punching shear stresses v_{u,1} and v_{u,2} is calculated as follows;
For the interior column, polar moment of inertia values, J_{c,1} and J_{c,2} are calculated along the respective axes c_{1} and c_{2}. v_{uv} means factored shear stress on the slab critical section for twoway action, from the controlling load combination, without moment transfer. v_{uv} calculated by dividing the axial force in the column by the critical section area [ v_{uv} = V / 2d(c_{1}+d+c_{2}+d) ]
Minimum flexural reinforcement in nonprestressed slabs
A minimum area of flexural reinforcement is satisfied near the slab's tension face in the span's direction.
A_{s,min}=0.0018A_{g }
According to ACI 8.6.1.2, if v_{uv} > 2ϕλ_{s}λ√f_{c}‘ on the critical section for twoway shear surrounding a column, concentrated load, or reaction area, A_{s,min}, provided over the width b_{slab}, shall satisfy ACI Eq. (8.6.1.2)
α_{s} and λ_{s} is given in TwoWay Shear Strength per ACI 31819 with ideCAD title.
Shear Reinforcement  Stirrups
Singleleg, simpleU, multipleU, and closed stirrups are used as shear reinforcement. Stirrup anchorage and geometry are determined with ACI 25.7.1. Detailed information: Stirrups per ACI 31819 with ideCAD.
If stirrups are provided, location and spacing are determined with ACI Table 8.7.6.3.
The shear reinforcement engages longitudinal reinforcement at both the top and bottom of the slab, as shown for typical details in ACI Fig. R8.7.6(a) to (c). The shear reinforcement should be symmetrical
about the centroid of the critical section shown in ACI Fig. R8.7.6(d). Spacing limits defined in ACI 8.7.6.3 are shown below.
Shear Reinforcement  Headed Studs
According to ACI 8.7.7.1.1, the overall height of the shear stud assembly should be at least the thickness of the slab minus the sum of all given values below;
concrete cover on the top flexural reinforcement
concrete cover on the base rail
Onehalf the bar diameter of the flexural tension reinforcement
According to ACI 8.7.7.1.2, Headed shear stud reinforcement location and spacing should be in accordance with ACI Table 8.7.7.1.2.