1. General ACI Requirements for Reinforced Concrete Flat Plate Design
The proper design and construction of buildings of structural concrete is dictated by “Building Code Requirements for Structural Concrete and Commentary (ACI 318-02)” [2], published by the American Concrete Institute. According to ACI 318-02, structures and structural members must be designed with a load capacity greater than the required strength as determined by a suitable analysis.
The required strength, U, is computed based on combinations of the loadings required in the general building code. These loads are increased by load factors, depending on the type of load and the specific loading combination. Loads which must be considered include dead loads, live loads, temperature loads, earthquake loads, and loads due to snow, rain, and wind. Loadings considered to be determined less accurately and variable are assigned a higher load factor. ACI 318 Section 9.2 presents nine loading combinations that must be considered for the design of a reinforced concrete system [2].
The design strength at a location in the system is computed as the nominal capacity, based on mechanics and code assumptions, multiplied by a strength reduction factor, φ. φ-factors are always less than 1.0 and account for statistical variations in material properties and inaccuracies in design equations. φ-factors vary based on the specific response quantity being designed. For flexure, φ ranges from 0.65 to 0.9 depending on the strain condition, while for shear and torsion, φ is 0.75 [2]. The basic requirements for strength design are
expressed as
expressed as
Design Strength ≥ Required Strength
or
φ(Nominal Strength) ≥ U
1.1 Analysis Requirements
ACI 318 Section 13.5.1 states:
A slab system shall be designed by any procedure satisfying conditions of equilibrium and geometric compatibility, if shown that the design strength at every section is at least equal to the required strength and that all serviceability conditions, including limits on deflections, are met[2].
Generally a linear elastic analysis is performed to provide the basis for an initial design, although use of a linear elastic analysis is not required. In fact, if inadequate reinforcement is provided in a particular location, the reinforcement will yield locally and the moment will be redistributed internally due to the redundancy of the slab system. Although such behavior may satisfy strength requirements, it can lead to poor serviceability of the slab, evidenced by undesirable cracking. If the slab is reinforced based on the distribution of linear elastic moments, however, serviceability conditions will generally be satisfied [18].
ACI 318 presents two simplified linear elastic analysis techniques which are permitted, providing the structure satisfies various requirements. These two methods are the ACI direct design technique and the ACI equivalent frame technique. Both of these methods are based on analytical studies of moment distributions using elastic theory, strength requirements from yield line theory, experimental testing of physical models, and previous experience of slabs constructed in the field [27].
Practice has shown that it is acceptable to proportion reinforcement for ultimate limit states as well based on the results of a linear elastic structural analysis. At this limit state, the linear elastic moments do not closely predict the actual moment distribution in the slab, but are acceptable for design provided that equilibrium and compatibility are satisfied. The primary advantage of distributing reinforcement for the ultimate limit state based on a linear elastic analysis is that deflection and cracking requirements are likely to be satisfied, though this is not guaranteed [18].
The actual behavior of the slab at the ultimate limit state is plastic, and it is permitted to apply limit analysis to the design of slabs. Using such a method, the ultimate load is determined, and then the distribution of moments and shears at the ultimate level are determined. Either a lower bound method or upper bound method may be employed.
A lower bound method assumes a distribution of moments at the ultimate condition such that equilibrium is satisfied at all locations in the system, the yield criterion defining the strength of the slab is not exceeded, and all boundary conditions are satisfied. Equilibrium is applied, and with the assumed distribution of moments, the ultimate load can be determined.
The lower bound method converges to the correct ultimate load from below the actual value, and never overestimates the ultimate load [27]. The strip design technique is a lower bound method. An upper bound method assumes a collapse mechanism at the ultimate condition. In this collapse mechanism, the moments at the plastic hinges are assumed to be less than or equal to the ultimate moment capacity at the hinges, and the collapse mechanism must satisfy the imposed boundary conditions of the structure. The upper bound method converges to the actual ultimate load from above the actual value, and never underestimates this value. As such, it is possible to compute an ultimate load that is too high if an incorrect collapse mechanism is assumed [27]. The yield line technique is an upper bound method.
 |
| Figure 1: General Flexure: (A) Cross-Section, (B) Strain Distribution, (C) Stress Distribution |
ACI 318 Section 13.5 sets out several other important principles related to the analysis of slabs. For instance, for lateral load analysis, the model of the supporting frame must account for the stiffness effects of cracking and reinforcement for frame members. Also, ACI 318 permits the results of gravity load analyses to be superimposed with results of lateral load analyses per Section 13.5.1.3, even though different models may be used to compute each of these structural responses. Displacements under both gravity and lateral loads must remain small for this to remain applicable.
1.2 Flexural Strength Requirements
One of the significant design limit states considered for flat plate systems is the design of reinforcement to resist flexure. The general requirements for flexural design are set forth in ACI Chapter 10 [2]. A cross-section in a flat plate system is designed using the same procedure as that in reinforced concrete beam design. The provided ultimate strength of the member, φMn, must exceed the ultimate factored moment, Mu, for all locations in a structure.
1.2.1 Ultimate Flexural Capacity
Figure 1-A shows the cross-section of a general reinforced concrete member reinforced for flexure. This figure will be used to develop the equations for the ultimate moment capacity of the section. The concrete compressive strength is f c. The distance d is the distance from the compression face of the member to the centroid of the layer of reinforcing steel. The width of the beam is given by bw. The area of steel reinforcement is given as As, and the steel has a yield strength equal to fy.
Figure 1-A shows the cross-section of a general reinforced concrete member reinforced for flexure. This figure will be used to develop the equations for the ultimate moment capacity of the section. The concrete compressive strength is f c. The distance d is the distance from the compression face of the member to the centroid of the layer of reinforcing steel. The width of the beam is given by bw. The area of steel reinforcement is given as As, and the steel has a yield strength equal to fy.
When the reinforced concrete member is subjected to a loading and a moment is produced, an assumed linear distribution of strain develops through the cross-section, as shown in Figure 1-B. In this figure, c is the strain at the extreme compression fiber of the concrete, and s is the tensile strain in the steel reinforcement at a distance d from the compression face of the member. ACI 318 Section 10.2.5 states that the tensile strength of concrete is to be neglected in the calculation of required flexural reinforcement, such that the only tensile component in the system is the steel reinforcement. The neutral axis of the member is located at a distance c from the compression face.
ACI 318 states that the crushing strain of the concrete, cu, shall not be taken greater than 0.003 [2]. Though experimental results show concrete can achieve compressive strains significantly higher than this value, concrete is assumed to crush when the strain reaches this lower bound for design purposes [49]. In flexure, there are three ways in which the section will fail. First, a section is said to fail at a balanced strain condition when the tension reinforcement strain, s, reaches the strain corresponding to the specified yield stress, fy, at the same time that the extreme compression fiber in the concrete reaches the assumed ultimate strain, 0.003. Second, a section is said to be compression-controlled if the extreme concrete fiber reaches the ultimate compressive strain prior to the tensile strain in the reinforcement reaching y. (For Grade 60 steel, this value may be taken as 0.002.)
In a compression-controlled section, the concrete will crush before any significant yielding of reinforcement occurs, leading to a brittle failure mode with little or no visible warning. Third, a section is said to be tension-controlled if the tensile strain in the steel is equal to or higher than 0.005 when the extreme compression fiber reaches its assumed strain limit of 0.003 [2]. In a tension-controlled failure, the steel will undergo large strains, and the concrete will experience significant cracking before the compression zone crushes, providing warning of impending failure. Because of this, flexural members are designed to fail in a tension-controlled manner when possible.
Per ACI 318 Section 9.3, the strength reduction factor φ is taken to be 0.9 for tension controlled sections and 0.65 for compression-controlled sections. At failure of the section, if the strain in the steel falls between the limits for compression-controlled and tension controlled strains, the section is said to fail in the transition zone, and the strength reduction factor is to be linearly interpolated between 0.65 and 0.9 based on the value of the strain in the steel [2]. Initially in design, φ is assumed to equal 0.9, and after the section is designed, the strain is computed at the ultimate state and verified to be greater than 0.005 in the steel.
Based on the limiting criteria for the strains in both the steel and concrete, the ultimate capacity of the cross-section can be computed. Since a tension-controlled failure is assumed, the strain in the steel at failure is greater than the yield strain. The stress in the steel, fs, is therefore set equal to the yield stress, fy, producing a tension force, T, in the section, given by Equation 1.
T = As. fy [Equation 1]
T acts at a distance d from the compression face of the member and is shown in Figure 1-C. The tension force in the section must be resisted by an equal and opposite compression force. Although the distribution of compressive stress in the concrete is nonlinear at the ultimate level, ACI 318 permits the compression zone to be represented by a rectangular distribution of stress of magnitude 0.85f c. This uniform stress distribution is also shown in Figure 1-C, and extends a depth a into the section from the compression face. The depth of the rectangular stress block is related to the depth of the neutral axis, c, by the expression, a = β1c, where β1 is 0.85 for concrete strengths less than 4000 psi, 0.65 for concrete strengths greater than 8000, and linearly interpolated between 0.65 and 0.85 for concrete strengths between 4000 and 8000 psi. As such, the compressive force within the member, C, is given in Equation 2.
C = 0.85f'c.a.bw [Equation 2]
Equating the tension and compression forces in the cross-section, the depth of the rectangular compression block, a, can be computed using Equation 3.
a = As.fy/0.85f'c.bw [Equation 3]
Now that the depth of the compression zone can be explicitly computed, the strain in the steel should be checked to verify that the section is indeed tension-controlled as assumed. The strain in the steel is determined in several steps. First of all, the distance to the neutral axis, c, is computed as c = a/β1. Then by assuming ∈cu = 0.003, s is computed by applying similar triangles to the strain distribution, shown in Equation 4.
∈ = ∈cu(d-c)/c [Equation 4]
∈ = ∈cu(d-c)/c [Equation 4]
If s is less than 0.005, the value of φ must be accordingly computed. Then, the moment capacity of the section, φMn, is determined by summing the internal moments from the compression stress in the concrete and tension force in the steel, as shown in Equation 5.
φMn = φAs.fy (d-a/2) [Equation 5]
1.2.2 Computation of Required Area of Steel
Now that the ultimate strength of the member has been formulated, the computation can be modified for design application. Equation 5 computes the ultimate strength when a known steel reinforcement area is provided. In design, however, the ultimate strength required is known, but not the area of steel. By setting Mu = φMn and substituting Equation 3 into Equation 5, a quadratic expression for As as a function of Mu can be formulated (Eqn. 6).
Mu = φAs.fy [d-(As.fy / 1.7f'c.bw)] [Equation 6]
After several algebraic manipulations, Equation 2.6 can be rewritten using the quadratic
equation to explicitly compute the area of steel required
[Equation 7]The importance of Equation 7 is that all of the arguments are independent, the only exception being the strength reduction factor, φ. Thus, once As is computed using Equation 7, the strain in the steel must be computed by Equation 4, indicating the appropriate φ-factor. It may be necessary to iterate on φ using these equations until As converges.
Although the case of a rectangular cross-section reinforced only on the tension side is one of the more simple cases for the design of reinforced concrete beams, it provides a sound basis for the design of flat plate sections for several reasons. First of all, flat plate sections are generally rectangular, so the formulation need not be specialized for more complicated geometries. Second, although flat plates may contain reinforcement on both faces, the width of design cross-sections (large with respect to the thickness of the slab) usually produces an a-depth smaller than the cover between the compression-side bars and the extreme compression face. Thus, compression reinforcement in flat slabs is usually located near the neutral axis and can be neglected since the strain will be very small in comparison to the strain in the tension-face reinforcement [43].
1.2.3 Minimum Flexural Reinforcement Requirements
Section 10.5 of ACI 318 dictates that a minimum area of steel is required at all sections of a beam where analysis demonstrates a need for tensile reinforcement. If the amount of reinforcement computed using Equation 2.7 is very small, the computed ultimate moment strength based on a reinforced concrete cracked section analysis can be less than the computed ultimate moment strength of the same unreinforced concrete section computed based on the modulus of rupture [2]. To prevent this, minimum flexural reinforcement must be provided in beams according to Equation 8.
[Equation 8]1.4 Serviceability Requirements - Will be published later
1.5 Detailing of Reinforcement - Will be published later)
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