Calculation of punching resistance for the unreinforced section

Control Perimeter Section Properties and Equations for the Calculation of Actual Stresses

Before any calculations are performed, the following manipulations are carried out on the reactions at the column center:

The column reactions are transformed to the control perimeter elastic centroid.
k factors are calculated using ratios about the column principal axes.
The reactions are rotated to the column principal axes and multiplied by appropriate k factors.
The reactions are rotated to the control perimeter elastic principal axes.

EN 1992-2004 control perimeter

The remainder of the calculations are carried out about the elastic principal axes of the control perimeter. Since a plastic stress distribution is used, if the "punching" area of the control perimeter on each side of the elastic neutral axes is not equal, the magnitudes must vary to maintain vertical equilibrium. This is handled by using multiplication factors representing the ratio of stress on one side of the principal axis over the stress on the other side. These factors are represented in the following form:

α_{x} = \frac{{A r e a}_{a}}{{A r e a}_{b}}

α_{y} = \frac{{A r e a}_{c}}{{A r e a}_{d}}

The stresses in each quadrant (considering bending about each axis separately) can then be represented as:

σ_b = -α_x σ_a

Equation 1

σ_d = -α_y σ_c

Equation 2

Two simultaneous equations can then be set up and solved for the state of stress around the critical section:

M_ox = unbalanced moment about the principal x-axis of the critical section (after adjustment by k)

M _oy = unbalanced moment about the principal y-axis of the critical section (after adjustment by k)

d = effective depth at location in critical section

M_{o x} = σ_{a} \int_{a} d y ⅆ l + σ_{b} \int_{b} d y ⅆ l + σ_{c} \int_{c} d y ⅆ l + σ_{d} \int_{d} d y ⅆ l

Substituting equations 1 and 2 and collecting terms,

M_{o x} = σ_{a} [\int_{a} d y ⅆ l - \int_{b} α_{x} d y ⅆ l] + σ_{c} [\int_{c} d y ⅆ l - \int_{d} α_{y} d y ⅆ l]

Equation 3

M_{o x} = σ_{a} [\int_{a} d x ⅆ l - \int_{b} α_{x} d x ⅆ l] + σ_{c} [\int_{c} d x ⅆ l - \int_{d} α_{x} d y ⅆ l]

Equation 4

We refer to the following terms:

$\int_{a} d y ⅆ l - \int_{b} α_{x} d y ⅆ l$ as Z_xx

$\int_{c} d y ⅆ l - \int_{d} α_{y} d y ⅆ l$ as Z_xy

$\int_{a} d x ⅆ l - \int_{b} α_{x} d x ⅆ l$ as Z_yx

$\int_{c} d x ⅆ l - \int_{d} α_{y} d x ⅆ l$ as Z_yy

These terms can be envisioned as plastic section moduli and each term has units of cubic length. Due to the interaction of a in the above equations and the equations below, these values are only valid for the axes about which they are calculated.

Equation 3 and 4 then become:

M_ox = σ_aZ_xx + σ_cZ_xy

M_oy = σ_aZ_yx + σ_cZ_yy

Solving the simultaneous equations for σ_a and σ_c, we get:

σ_{a} = \frac{M_{o x} Z_{y y} + M_{o y} Z_{x y}}{Z_{x x} Z_{y y} - Z_{x y} Z_{y x}}

σ_{c} = \frac{- M_{o y} Z_{x x} - M_{o x} Z_{y x}}{Z_{x x} Z_{y y} - Z_{x y} Z_{y x}}

For symmetric sections, Z_xy and Z_yx are equal to zero and the equations above reduce to:

σ_{a} = \frac{M_{o x}}{Z_{x x}}

σ_{a} = \frac{M_{o y}}{Z_{y y}}

We can then use equations 1 and 2 to solve for σ_b and σ_d .

The stress in any given "quadrant" of the critical section is then solved for as:

ν_{E d} = \frac{V_{E d}}{u d} + σ_{a / b} + σ_{c / d}

Calculation of Allowable Stress

The punching resistance for an unreinforced section is calculated as follows:

v_{R d, c} = C_{R d, c} k {(100 ρ_{1} f_{c k})}^{1 / 3} + k_{1} σ_{c p} \leq v_{\min} + k_{1} σ_{c p}

where

C_Rd,c	=	0.18/γ_c
k	=	$1 + \sqrt{\frac{200}{d}} \leq 2.0 d$ , d in mm
ρ₁	=	$\sqrt{ρ_{1 y} + ρ_{1 z}} \leq 0.02$ relates to bonded tension steel in y and z directions
f_ck	=	characteristic compressive cylinder strength of concrete at 28 days
k₁	=	0.1
σ_cp	=	(σ_cy + σ_cz )/2 average compression in the punching check region

When ρ₁ is automatically calculated for use in equation 6.47, the following apply :

Reinforcement ratios are calculated about each punch check axis individually.
A distance 3d beyond the column width is used.
Only user reinforcement is considered (program reinforcement is not used). Once the flexural design is completed the program reinforcement can be changed to user reinforcement to perform the auto calculation.
Untensioned reinforcement on the specified face (top or bottom) only is considered (bonded tendons are excluded).
Vector components of user reinforcement are used.
Development lengths are considered (only the developed portion of the bar is used).
A ratio at each face of the column is calculated, with the results of each opposing face averaged. If one opposing face has no reinforcement it is excluded from the calculation (this will be common in edge and corner conditions where reinforcement will cross only a single column face).
The averaged results in each direction are combined using the equation above.
If no user reinforcement is detected in a given axis, a warning will be issued and a reinforcement ratio of 0 (zero) will be used.

Calculation of Punching Resistance with SSR

Where SSR is used the punching resistance is calculated as follows:

v_Rd,cs = 0.75v_Rd,c + [1.5(d_sw/s_r)A_swf_ywd]/u_id_i

(6.52)

where

d_sw	=	average effective depth of slab containing shear reinforcement
s_r	=	radial spacing of shear reinforcement
A_sw	=	area of one peripheral line of shear reinforcement
d_i	=	the average effective depth of the perimeter under consideration
f_ywd	=	effective design strength of the shear reinforcement = f_yk/γ_s
d_i	=	the average effective depth of the perimeter under consideration

Note: Because the head sizes of SSR are typically selected to ensure 100% development of the stem, the yield strength of the SSR reinforcement is used without adjustment for effective depth, d. If the Engineer needs to make reductions to the effective yield strength of the studs due to depth issues these modifications can be made by specifying a reduced yield stress in the "SSR Systems" on the "Materials" page.

Limitation of Punching Stress at the Perimeter of the Column or Loaded Area

At the perimeter of the column face the maximum shear stress is limited to:

v_Ed = v_Rd,max

(6.53)

where

v_Ed	=	βV_Ed/(u₀d)
v_Rd,max	=	0.5νf_cd
β	=	maximum beta from calculated control perimeters, u₁
v	=	0.6[1 - f_ck/250] f_ck in N/mm²
f_cd	=	design value of concrete compressive strength
u₀	=	length of column or loaded area periphery. For edge columns, u₀ ≤ 6d For corner columns, u₀ ≤ 3d

Note: This simplification for edge columns was necessary due to difficulty in calculating the code equation for irregular situations. This provision is not in strict compliance with the code and should be reviewed by the Engineer as necessary.

Miscellaneous Provisions

The control perimeter at which shear reinforcement is not required is calculated using eq. 6.47. The outermost perimeter of shear reinforcement is placed not greater than 1.5d within this perimeter.

The spacing to the first stud is calculated as 0.5 d.

The maximum typical stud spacing is 0.75 d.

The maximum transverse rail spacing is 1.5 d within the first control perimeter and 2.0 d outside the first control perimeter.

A minimum quantity of SSR reinforcement is provided in accordance with EC2 equation 9.11:

A_{s w, \min} \geq (0.08 \sqrt{f_{c k}} s_{r} s_{t}) / 1.5 f_{y k}

where s_t is assumed to be ≤ 2d (Final rail layout should be confirmed/adjusted to be in agreement with this assumption)

Note: EC2 has special provisions for column bases. These provisions are not implemented in Concept (the provisions above are applied to all punching checks). For slabs without prestress, this will always be conservative. For slabs with prestress, the Engineer will need to evaluate the validity of the results.