Punching shear design with control perimeters subjected to asymmetrical bending

Abstract A numerical procedure is proposed for asymmetrical plastic shear diagrams in punching control perimeters. Asymmetrical diagrams occur for edge and corner columns and for internal columns with biaxial unbalanced moments. The procedure intends to support the use of NBR 6118, which covers asymmetrical shear distributions due to internal moments of edge and corner columns. The study of columns in different positions of the slab proves the robustness and numerical efficiency of the proposal. The practical application of the procedure is tested against Model Code 1990, Eurocode 2, NBR 6118, and with combinations of criteria from these codes. The estimated capacities are compared with experimental data from the literature. Eurocode 2 initially presents better results, but this code does not consider moments with internal eccentricities in edge and corner columns. The Eurocode 2 evaluations are significantly improved by the inclusion of NBR 6118 criteria that partially apply these moments, whose asymmetrical shear diagrams can be determined by the proposed procedure.


INTRODUCTION
The design of flat slabs is often controlled by the punching shear strength of the slab-column connections.Slabcolumn connections are usually subjected to unbalanced moments that yield additional shear stresses and reduce the punching shear capacity.

PUNCHING SHEAR DESIGN WITH UNBALANCED MOMENTS
Any evaluation of shear forces along a control perimeter must consider that the bending and torsional moments resist part of the unbalanced moment.The  factor is defined as the fraction of the unbalanced moment that is resisted by shear forces.
The moments transferred to the slab by bending and shear are experimentally investigated by Hanson and Hanson [12] on square columns.The ratios of unbalanced moments resisted by shear are analytically discussed by Mast [13] for internal columns (Figure 1).The corresponding  factor is estimated by the elastic solution of a plate subjected to a concentrated moment.The plate is simply supported in the main direction and is infinite in the other direction (Girkmann [14]).The  factor can be defined as a function of the control perimeter shape in the region close to the column.
Design  values indicated in MC90 are compared with Mast's [13] approximate elastic solution (Figure 1).The analytical values assume that lengths  and  vary between /20 and /5, where  is the span between the columns.Figure 1 also shows the elastic distribution of shear forces along a square perimeter, where  =  = /10.Shear force distribution in reinforced concrete flat slabs has also been studied by nonlinear finite element analyses.Shu et al. [15] investigate internal columns without unbalanced moments in flat slabs without shear reinforcement.The shear force distributions along the control perimeters are determined by shell and solid nonlinear finite element models.The results show that reinforcement arrangement, cracking and nonlinear material behavior influence the shear force distribution.Laguta [16] uses a concrete damaged plasticity material model to describe the nonlinear behavior of the concrete and present a typical shear stress distribution at a control perimeter under combined vertical load and unbalanced moment.
Normal force and bending moment are considered separately in the MC90 design model and produce distinct plastic shear force diagrams per unit length (Figure 2).The plastic shear forces related to normal force  and bending moment  are respectively denoted by   and   .   is here defined as the effective force that is calculated on Perimeter .The following expressions apply: where    is the plastic modulus and   is the developed length of Perimeter .Reduced lengths  *  are defined for edge and corner columns.Reduced lengths  *  correspond to developed lengths   in internal columns.The combined shear force per unit length    is expressed by The effective force    on Perimeter  is given by

PLASTIC MODULUS FOR ASYMMETRICAL BENDING
A numerical procedure is used to determine the plastic shear diagram and the plastic flexural modulus of an arbitrary perimeter, which is subjected to a bending moment  about an oblique axis.
and   are the vector components of bending moment  about the  − and  −axes, respectively (Figure 3).

Moments in the principal coordinate system
Figure 3 presents the principal coordinate system ̄̄.The system is rotated from the  coordinate system by an angle , which is defined by Considering ̄=    +    and ̄= −   +   , the equilibrium conditions of the shear forces per unit length  along the perimeter  yield where  ̄ and  ̄ are the moments about the ̄− and ̄−axes (Figure 3).

Distribution of shear forces per unit length
Shear forces  = −  and  = +  (Figure 2) are considered uniformly distributed along perimeter lengths  − and  + , respectively.The equilibrium conditions in the  −direction and about the  − and  −axes lead to the following equations: ̄= ∫ (−  )̄ + ∫ (+  )̄ Collecting like terms in Equations 7 to 9 yields where the plastic flexural moduli  ̄ and  ̄ are A numerical algorithm establishes perimeter lengths  − and  + .

Discretization and parametrization of the control perimeter
The perimeter is divided into linear and arc segments for the application of the numerical procedure.
The developed length of the control perimeter is defined as .The numerical procedure demands that all segments have lengths less than the semi-perimeter /2.This condition is satisfied by the division into segments shown in Figure 4.The points that divide the segments are numbered from 1 to .The control perimeter is parameterized according to the developed length , where  1 = 0 at start point 1 and   =  at endpoint  (Figure 5).Equation 10 shows that the developed lengths of the perimeter lengths  − and  + are both equal to the semiperimeter /2.

Plastic flexural modulus
The parameters  of points  and  are defined as   and   , respectively.Equation 10 is automatically respected by adopting the following A parameter   yields   by Equation 14. Perimeter lengths  − and  + are defined in Figure 5.The plastic flexural moduli  ̄(   ) and  ̄(   )are determined by Equations 11 and 12, respectively.
The algorithm searches for a parameter   * that yields  ̄(   * ) ≃ 0. The solution   * yields a unit diagram proportional to the shear force diagram that satisfies Equations 5 and 6.
The plastic flexural modulus   and the shear force per unit length   are given by where   is always positive.
If ̄ coordinates are always negative in  − and positive in  + , or always positive in  − and negative in  + , Equations 11 and 15 yield Equation 17 cannot be used in the general case, but it is valid for symmetrical perimeters about the ̄− axis.It is applicable in specific cases, such as edge columns subjected to moments  ̄=   (Figure 4).

Partial integration of a same-sign length of a segment
Changes in the sign of unit shear forces can occur in linear and arc segments.Equations 11 and 12 yield the plastic flexural moduli  ̄ and  ̄ by integrating lengths with shear forces of the same sign.
A segment's start and end points are defined as  and , respectively (Figure 6).Points  and  determine a length with positive shear forces.
In the case of linear segments,  is defined as the barycenter of .The parameter   and coordinates ̄ and ̄ are equal to where  P  and  P  are the contributions of  to the plastic moduli  ̄ and  ̄.In linear segments, they are expressed by where  is the length between  and .
In arc segments, the following variables are also considered: angles   and   at points  and , coordinates   and   of the center, and radius .Angles   and   are interpolated by The following expressions yield the contributions of  to the plastic moduli  ̄ and  ̄ in arc segments: The developed length  between  and  is

Full segment integration
Unit shear forces on lengths  (Figure 6) can be positive or negative.Figure 7 discusses the positive and negative shear forces that should be considered during the complete integration of a segment .
Each segment cannot simultaneously contain points  and , since all segments have a developed length less than the semi-perimeter /2.The discretization into segments shown in Figure 4 meets this requirement.
Parameters   and   define segments   and   , which respectively contain points  and .Table 1 defines the coordinates   and   of each segment  integration step, according to its location.Equations 21 and 24 are established for a positive unit shear force.The effective signs of unit shear forces will be considered as indicated in the table.

IMPLEMENTATION AND EXAMPLES
The previous steps define an iteration that yields the plastic flexural moduli  ̄(   ) and  ̄(   ) as a parameter function   .The parameter   * associated with  ̄(   * ) ≅ 0 gives the principal plastic modulus   = � ̄(   * )�, which depends on the direction of the applied bending moment .The solution   * is searched for in the interval 0 ≤   <  2 .
As the computational cost of the procedure is low, the solution can be investigated by sequentially examining many values in the range.This process can be optimized by dividing the original interval into  subintervals.The solution subinterval is identified by the change in the sign of  ̄ at its endpoints, but the endpoints themselves should be previously verified as possible solutions.The solution subinterval is iteratively divided into  subintervals until the required tolerance is reached.This work uses  = 20.The sections of all the columns are 0.60 x 0.30 m, and all the slabs have 0.15 m effective depth.The signs of the shear forces change in the arc and linear segments.The shear force diagrams depend on the direction of the acting bending moment and do not show symmetry.

COMPARISON WITH EXPERIMENTS IN THE LITERATURE
Experimental results in the literature are analyzed using MC90, EC2, and NBR6118 together with the proposed procedure.
Asymmetrical bending results can be compared with values usually accepted by these codes as the dataset also includes tests that yield symmetrical diagrams in bending.
The formulas from the codes are used with   =   =   = 1, where   is the partial factor for actions.Parameters   and   are the partial factors for concrete and reinforcing steel, respectively.
Shear reinforcement arrangements are always distributed uniformly in the literature tests discussed in this work.Perimeter  may be discontinuous since perimeter lengths with distances greater than  to the nearest transverse reinforcement will not be considered.uniformly distributed shear reinforcement, the effective shear force  , is estimated by the following equation: The combined shear force   for continuous perimeters is determined according to each code.The average spacing between transverse bars in the outer reinforcement contour is   .The maximum spacing   is defined as 2.
Although the procedures in EC2 and NBR6118 are based on MC90, the differences between them significantly affect the results.The following code criteria are discussed in this work: a. Size effect () In MC90, EC2, and NBR6118, design shear stress   depends on the size effect parameter  = 1 + � 0.2m  .EC2 also assumes  ≤ 2.

b. Reinforcement ratio for longitudinal reinforcement (𝜌𝜌)
Design shear stress   is also a function of the longitudinal reinforcement ratio  = �     in MC90, EC2, and NBR6118.Parameters   and   are the ratios in the  − and  −directions, respectively.EC2 also assumes  ≤ 0.02.c.Effective design yield stress of shear reinforcement ( , ) All the codes limit the effective design yield stress of shear reinforcement  , .MC90 adopts  , ≤ 300MPa.In EC2, the maximum value depends on the effective depth  of the slab.NBR6118 defines different limits for connectors and stirrups depending on total depth ℎ.In this work, the NBR6118 formulation is adapted for effective depth  = ℎ − 0.03m.d.Distance between Perimeter n and the outer transverse reinforcement contour () MC90 and NBR6118 establish the distance  = 2 between Perimeter  and the outer transverse reinforcement contour.EC2 assumes  = 1.5.e. Edge and corner column moments with internal eccentricity ( * ) MC90 and EC2 do not consider moments with internal eccentricity in edge and corner columns.NBR6118 partially includes internal moments that are larger than  * , which are the moments that can be resisted by eccentricities between columns and reduced control perimeters.f.Perimeter 0 length on edge and corner columns ( 0 * ) In edge and corner column connections, MC90 and EC2 assume a reduced length for Perimeter 0, which is here denoted as  0 * .NBR6118 does not assume a reduced length for Perimeter 0. Ten combinations (C1 to C10) of eight criteria from the codes are presented in Table 2. Combinations C1 to C3 correspond to MC90, EC2, and NBR6118, respectively.C4 to C10 investigate the response of different criteria a to h in EC2 and NBR6118.Ninety-four experiments of slab-column connections subjected to punching shear were compiled from the literature and their experimental capacity was compared with the theoretical capacity given by the combinations in Table 2.The dataset contains internal, reentrant corner, edge, and corner columns subjected to normal forces, with and without unbalanced moments.The tests include slabs without shear reinforcement.Transverse reinforcement is provided by shear studs.

Figure 1 .
Figure 1.Unbalanced moment transfer in slab-column connections

Figure 2 .
Figure 2. Coordinate system, applied forces and moments, and distributed shear forces

Figure 3 .
Figure 3. Rotated coordinate system associated with the principal moments

Figure 4 .
Figure 4. Discretization of control perimeters into segments

Figure 5 .
Figure 5. Parameter  and unit shear force distribution along the control perimeter

Figure 5
also presents a flat diagram of unit shear forces per unit length.The longitudinal axis indicates parameter , which is the developed length from the origin ( 1 = 0).The unit shear forces change signs at points  and .The perimeter length of positive shear forces  + is defined between points  and .The perimeter length of negative shear forces  − is defined in intervals 1 −  and  − .

Figure 6 .
Figure 6.Partial integration between points  and

Figure 7 .
Figure 7. Unit shear force along the control perimeter

Figure 8
Figure8shows examples of internal, reentrant corner, edge, and corner columns.All columns are subjected to unbalanced moments about an oblique ̄−axis, which is rotated 60 degrees from the  −axis, in the counterclockwise direction.Perimeter 1 unit shear diagrams are shown.

Figure 8 .
Figure 8. Examples: asymmetrical diagrams of unit shear forces

Table 1 .
Integration of segment g. Effective normal force due to unbalanced moments (  )    is defined as the effective normal force that is calculated on a control Perimeter .MC90 uses   1 and    on Perimeters 1 and , respectively, and reuses   1 on Perimeter 0. EC2 only calculates   1 , which is used as the effective normal force on all perimeters.Effective forces are not discussed in NBR6118.h.Concrete strength reduction factor in diagonal compression (  )The concrete stress in diagonal compression is limited to       at Perimeter 0, where   is the concrete design strength and   = �1 − The reduction factor   is given as 0.30, 0.24, and 0.27 in MC90, EC2, and NBR6118, respectively.

Table 2 .
Combinations of criteria from Model Code 90, Eurocode 2, and NBR 6118