# Punching Shear Design

Slab punching shear is checked automatically in accordance with

**ACI 318 - Chapter 8.**

**Notation**

* A_{g }= *gross area of concrete section, in

^{2}

**A**_{s}**area of nonprestressed longitudinal tension reinforcement, in**

_{ }=^{2}

*perimeter of critical section for two-way shear in slabs and footings, in.*

**b**_{o}=*effective slab width, in.*

**b**_{slab}=*dimension of the critical section*

**b**_{1 }=*measured in the direction of the span for which moments are determined, in.*

**b**_{o}*dimension of the critical section*

**b**_{2 }=*measured in the direction perpendicular to*

**b**_{o}*, in.*

**b**_{1}*distance from extreme compression fiber to centroid of longitudinal tension reinforcement, in.*

**d =***modulus of elasticity of concrete, psi*

**E**_{c }=*modulus of elasticity of reinforcement, psi*

**E**_{s }=

**√f**_{c}‘*square root of specified compressive strength of concrete, psi*

**=***specified compressive strength of concrete, psi*

**f**_{c}^{'}_{ }=*specified yield strength for nonprestressed reinforcement, psi*

**f**_{y }=*property of assumed critical section analogous to polar moment of inertia*

**J**_{c }=*factored slab moment that is resisted by the column at a joint, in.-lb*

**M**_{sc}=*stress corresponding to nominal two-way shear strength provided by concrete, psi*

**v**_{c }=*factored shear stress on the slab critical section for two-way action, from the controlling load combination, without moment transfer, psi*

**v**_{uv }=*constant used to calculate*

**α**_{s }=*in slabs and footings*

**V**_{c}*strength reduction factor*

**ϕ =***= modification factor to reflect the reduced mechanical properties of lightweight concrete relative to normalweight concrete of the same compressive strength*

**λ***= factor used to modify shear strength based on the effects of member depth, commonly reffered to as the size effect factor*

**λ**_{s}*net tensile strain in extreme layer of longitudinal tension reinforcement at nominal strength, excluding strains due to effective prestress, creep, shrinkage, and temperature*

**ε**_{t }=*value of net tensile strain in the extreme layer of longitudinal tension reinforcement used to define a compression-controlled section*

**ε**_{ty }=*= factor used to determine the fraction of*

**γ**_{f}*transferred by slab flexure at slab-column connections*

**M**_{sc}*= factor used to determine the fraction of*

**γ**_{v}*transferred by eccentricity of shear at slab-column connections*

**M**_{sc}**Factored Slab Moment Resisted by the Columns**

According to **ACI 8.4.2.2.1; **if gravity, wind, earthquake, or other loads cause moment at the slab-column 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.

*is calculated in accordance with*

**γ**_{f}**ACI Eq.(8.4.2.2.2);**

The fraction of unbalanced moment transferred by eccentricity of shear is taken to be * γ_{v}M_{sc}* and the factor

*equals to*

**γ**_{v}

**1-***.*

**γ**_{f}

_{ }(γ_{v}=1-

**γ**_{f })According to **ACI 8.4.2.2.3; **the effective slab width * b_{slab}* for resisting

*should be the width of column or capital plus a distance on each side in accordance with*

**γ**_{f}M_{sc }**ACI Table 8.4.2.2.3**.

According to **ACI 8.4.2.2.4; **for nonprestressed slab systems without beams, where the limitations on * v_{uv}* and

*in*

**ε**_{t }**ACI Table 8.4.2.2.4**are satisfied,

*can be increased to maximum modified values provided in*

**γ**_{f}**ACI Table 8.4.2.2.4**. the calculation of two-way shear strength provided by concrete without shear reinforcement

*is explained in Two-Way Shear Strength title.*

**v**_{c }**Critical Section**

According to **ACI 22.6.4.1, **for two-way shear, critical sections should be located so that perimeter * b_{o}* is a minimum but need not be closer than

*to given two cases given below;*

**d/2**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, **concerntrated loads, or reaction areas, critical sections for two-way shear in accordance with **ACI 22.6.4.1 (**case 1 and case2**) **can be defined assuming straight sides for square or rectengular columns.

According to **ACI 22.6.4.1.2,** critical sections for two-way shear in accordance with **ACI 22.6.4.1 (**case 1 and case2**) **can be defined assuming a square column of equivalent area for a circular or regular polygon-shaped column.

According to **ACI 22.6.4.2,** for two-way members with single (or multi) leg stirrup or headed stud shear reinforcement, one more critical section with perimeter * b_{o}* located

*beyond the point where shear reinforcement is dicontinued. The shape of this critical section should be a polygon selected to minimize*

**d/2***.*

**b**_{o}Value of * b_{o}* is shown in

**ACI Fig. R22.6.4.2a, b**and

**c.**

The critical cross section types of columns are shown in the picture below.

**Factored Two-way 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

*corresponds to combinations of*

**v**_{u}*and*

**v**_{uv}*.*

**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

*should be applied at the centroid of the critical section.*

**M**_{sc}*is taken as specified in*

**γ**_{v}**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.

For interior column and edge column, the stress distribution is assumed as illustrated in** ACI Fig. R8.4.4.2.3**. For interior column, the perimeter of the critical section equals to * 2(c_{1}+d+c_{2}+d). *for edge column, the perimeter of the critical section equals to

*.*

**2(c**_{1}+d) + (c_{2}+d)The factored shear stress * v_{uv}* and factored slab moment resisted by the column

*are determined at the centroidal axis*

**M**_{sc}*of the critical section. According to the stress distribution shown in*

**c-c****ACI Fig. R8.4.4.2.3**. The maximum factored punching shear stress is calculated from;

For an interior column, property of assumed critical section analogous to 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 interior colum, punching shear stresses * v_{u,1}* and

*is calculated as follows;*

**v**_{u,2}For interior column, polar moment of inertia values, * J_{c,1}* and

*are calculated along the respective axes c*

**J**_{c,2}_{1}and c

_{2}.

*means factored shear stress on the slab critical section for two-way 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**_{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 tension face of the slab in the direction of the span.

A_{s, min}=0.0018A_{g }

According to **ACI 8.6.1.2, **if * v_{uv}* >

*on the critical section for two-way shear surropnding a column, concentrated load, or reaction area,*

**2ϕλ**_{s}λ√f_{c}‘*, provided over the width*

**A**_{s,min}*, shall satisfy*

**b**_{slab}**ACI Eq. (8.6.1.2)**

* α_{s}* and

*is given in Two-Way Shear Strength title.*

**λ**_{s}**Shear Reinforcement - Stirrups**

Single-leg, simple-U, multiple-U, and closed stirrups are used as shear reinforcement. Stirrup anchorage and geometry is determined with **ACI 25.7.1.** Detailed information: Stirrups.

If stirrups are provided, location and spacing is determined with **ACI Table 8.7.6.3.**

The shear reinforcement engage 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 in **ACI Fig** **R8.7.6(d) **to** (e).**

**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 thinkness of the slab minus the sum of given all values below;

concrete cover on the top flexural reinforcement

concrete cover on the base rail

One-half 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**.