Menu



Review of Reinforced Concrete Flat Plate Design as per ACI 318-02
The objective of this article is to provide an overview of the current state of practice of the design of reinforced concrete flat plate systems. General ACI design requirements are presented, along with a summary of the ACI direct design, ACI equivalent frame, yield line, and strip design techniques. Each procedure as well as its limitations are discussed. The following discussion is limited to flat plate systems. That is, the design methodologies presented below pertain only to slabs of constant thickness without drop panels, column capitals, or edge beams. In addition, prestressed concrete is not considered.

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

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
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.
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]

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]
(1.3 Shear Strength Requirements - Will be published later

1.4 Serviceability Requirements - Will be published later
1.5 Detailing of Reinforcement - Will be published later)

References

[1] Adini, A.  and  Clough,  R. W., “Analysis  of Plate  Bending  By the  Finite  Element ,”  1960. G7337 Report to the National  Science Foundation.

[2] American Concrete  Institute, ACI 318 Building Code Requirements for Structure Concrete and  Commentary, 2002.

[3] Anderheggen, E.,   e. a., “Reinforced-Concrete  Dimensioning  Based  on  Element Nodal   Forces,”  ASCE Journal of the Structural  Division,  vol.  102,  pp.  1718–1731, June  1994.

[4] Armer, G. S. T., “Discussion of Reference [72],” Concrete, vol. 2, pp. 319–320, August 1968.

[5] Barzegar, F. and  Maddipudi, S.,  “Generating Reinforcement  in FE  Modeling of Concrete  Structures,” ASCE Journal of the Structural Division,  vol. 120, pp.  1656–1662, May 1994.

[6] Bashur, F. K. and Darwin,  D., “Nonlinear  Model for Reinforced Concrete  Slabs,” ASCE Journal of the Structural Division, vol. 104, pp. 157–170, January 1978.

[7] Bica´nic,  N., de   Borst, R.,   Mang,  H., and  Meschke,  G.,   eds.,  COMPUTATIONAL   MODELING  OF CONCRETE STRUCTURES.    Lisse,  The  Netherlands: A.A. Balkema  Publishers, 2003.

[8] Borkowski,  A.,   “Optimization  of Slab  Reinforcement  by  Linear  Programming,” Computer Methods in Applied Mechanics and Engineering, vol. 12, pp. 1–17, September 1977.

[9] Brotchie, J. F. and Russell, J. J., “Flat  Plate  Structures:  Elastic-Plastic Analysis,”  Journal of the American Concrete Institute, vol. 61, pp. 959–996, August  1964.

[10] Cheung, Y. K., King, I. P., and  Zienkiewicz, O.  C.,  “Slab  Bridges  with  Arbitrary   Shape and Support Conditions:   A General  Method  of Analysis Based on Finite Elements,”   Institution of Civil Engineers - Proceedings, vol. 40, pp. 9–36, May 1968.

[11] Clough, R., “The Finite  Element Method After Twenty-Five Years,” Computers and Structures, vol. 12, pp. 361–370, 1980.

[12] Computer Aided Structural Engineering  Center, Georgia Institute of Technology,  Atlanta,  GA, GT STRUDL User Reference Manual Abstract of Commands, 1999. Revi- sion R.

[13] Computer Aided Structural Engineering  Center, Georgia Institute of Technology,  Atlanta,  GA, GT STRUDL User Reference Manual Volume 3: Finite Element Analysis, Nonlinear Analysis, and Dynamic Analysis Commands, 2002. Revision R.

[14] Computer Aided Structural Engineering  Center, Georgia Institute of Technology,  Atlanta,  GA, GT STRUDL User Reference Manual Volume 4: Reinforced Concrete De- sign, 2002. Revision R.

[15] Computer Aided Structural Engineering  Center, Georgia Institute of Technology,  Atlanta,  GA, GT STRUDL User Guide: Getting Started, 2003. Revision 7.

[16] Computers  and  Structures,  Incorporated,  Berkeley,  California,   SAFE Integrated
Analysis and Design of Slabs by The Finite  Element Method - User’s Manual, 1998. Version 6.0.

[17] Cook, R. D. e. a., CONCEPTS AND  APPLICATIONS OF FINITE ELEMENT ANALYSIS.  New York:  John
Wiley & Sons, Inc., fourth  ed., 2002.

[18] Cope, R. and  Clark, L.,  CONCRETE SLABS: Analysis and Design. London:  Elsevier Applied
Science Publishers, 1984.

[19] Corley,  W. G. and  Jirsa, J. O.,  “Equivalent  Frame  Analysis  for Slab  Design,” Journal of the American Concrete Institute, vol. 67, pp. 875–884, November 1970.

[20] Cusens, A.  R.  and  Besser, I. I., “Elastic  Stress  Parameters in Simply Supported Skew Plates  in Flexure,”  Journal of Strain Analysis, vol. 15, no. 2, pp. 103–111, 1980.

[21] Davies, J. D.,  “Analysis  of Corner  Supported Rectangular Slabs,”  The Structural Engineer, vol. 48, pp. 75–82, February 1970.

[22] Davies, J. D.,  Ma, S. P. K., and  Cheung,  Y. K., “Analysis  of Corner  Supported Skew Slabs,” Building Science, vol. 3, pp. 81–92, November 1968.

[23] de  Figueiredo,  A. J. M., da Fonseca,  A.  M.  A.,  and  Azevedo, A.  F. M., “Analysis  and   Optimization of Reinforced  Concrete  Slabs,”  in COMPUTER AIDED OPTIMUM  DESIGN OF STRUCTURES VI  (Hernandez,  S.,  Kassab, A.  J., and Brebbia, C.  A.,  eds.), (Southhampton, Boston,   Massachusetts), WIT  Press,  1999.

[24] Douglas, M.  R.  and Lunniss, R. C., “An Application  of the Finite  Element Tech- nique in  Bridge Design,” Concrete, vol. 4, pp. 197–200, May 1970.

[25] Famiyesin,  O.   O.   R.  and  Hossain,  K.  M.  A.,  “Optimized  Design  Charts  for Fully  Restrained Slabs by FE Predictions,” ASCE Journal of the Structural Division, vol. 124, pp.  560–569, May 1998.

[26] French, S., Kabaila, A.  P., and Pulmano, V.  A., “Single Element Panel  for Flat Plate   Structures,” ASCE Journal of the Structural Division, vol. 101, pp. 1801–1812, September  1975.

[27] Gamble,  W. L. and  Park,  R., REINFORCED  CONCRETE SLABS.  New York: John  Wiley & Sons, Inc., second ed., 2000.

[28] Gamble, W. L., Measured and Theoretical Bending Moments in Reinforced Concrete Floor Slabs.  PhD  thesis, University  of Illinois, Urbana,  Illinois, June  1962.

[29] Gentry,  T. R., “The  Use of Elastic  Finite  Elements  in the Design  of Reinforced Concrete  Flat  Plates,” Master’s thesis, Georgia Institute of Technology,  1986.


[30] GTICES Systems Laboratory, Atlanta,  GA,  GTICES Programmers Reference Guide, 1991.

[31]  Gupta, A.  K.  and Sen, S.,  “Design  of Flexural Reinforcement in Concrete Slabs,” ASCE Journal of the Structural Division,  vol. 103,  pp. 793–804, April 1977.

[32] Hernandez, S.,  Kassab, A.  J., and Brebbia, C.  A.,  eds., COMPUTER  AIDED OPTIMUM DESIGN OF  STRUCTURES VI. Southhampton, Boston, Massachusetts: WIT Press, 1999.

[33] Hillerborg, A., STRIP DESIGN METHOD HANDBOOK.  London, Great Britain: E & FN Spon, 1996.

[34] Hrabok, M. M., Stiffened Plate Analysis by the Hybrid Stress Finite Element Method. PhD thesis, University of Alberta, Edmonton, Alberta, Canada, 1981.

[35] Hrabok, M.  M.  and Hrudey, T. M., “Finite Element Analysis in Design of Floor Systems,” ASCE Journal of the Structural Division,  vol. 109, pp. 909–929, April 1982.

[36] Huang, Y. H. and Wang, S. T., “Finite-Element Analysis of Concrete Slabs and its Implications for Rigid Pavement Design,” Highway Research Record, no. 466, pp. 55–69, 1973.

[37] Hudson, W. R. and Stelzer, C. F., “A Direct Computer Solution for Slabs on Foundation,” Journal of the American Concrete Institute, vol. 65, pp. 188–201,  March 1968.

[38] Hughes, T. J. R. and Cohen, M., “The Heterosis Finite Element for Plate Bending,” Computers and Structures, vol. 9, pp. 445–450, 1978.

[39] Hughes, T. J. R., Taylor, R. L., and Kanoknukulchai,  W., “A Simple and Effi- cient Finite  Element for Plate Bending,” International Journal for Numerical Methods in Engineering, vol. 11,  pp. 1529–1543, 1977.

[40] Isenberg, J., ed., FINITE ELEMENT ANALYSIS OF REINFORCED CONCRETE STRUCTURES II, (New York,  New York), Committee on Finite Element Analysis of Reinforced Concrete Structures, American Society of Civil Engineers, June 1991.

[41] Jirsa, J. O.,  Sozen, M.  A.,  and Siess, C.  P., “Pattern  Loadings on Reinforced Concrete FLoor Slabs,”  ASCE Journal of the Structural  Division, vol. 95, pp. 1117– 1137, June 1969.

[42] Jofriet, J. C.  and McNeice, G.  M.,  “Finite Element Analysis of Reinforced Con- crete  Slabs,”  ASCE Journal of the Structural  Division, vol. 97, pp. 785–806, March 1971.

[43] Kahn,  L., “Reinforced  Concrete Slab Systems.”  Course Notes, Georgia Institute of Technology, 2004.

[44] Law, F. M.,  “Design of Irregular Shaped Two-Way Slabs,”  Journal of the American Concrete Institute, vol. 68, pp. 844–847, November 1971.


[45] Lewi´nski, P. M. and Wojew´odzki, W., “Integrated Finite Element Model for Reinforced Concrete Slabs,” ASCE Journal of the Structural Division, vol. 117, pp. 1017–1038, April 1991.

[46] Liepins, A. A. and Bell, G. R., “How To Avoid Common Pitfalls in Structural Analysis by Computer.” ASCE International Convention Exposition, 1992.

[47] Logan, D. L., A FIRST COURSE IN THE FINITE ELEMENT METHOD. Pacific
Grove, CA: Brooks/Cole, third ed., 2002.

[48] Lourenc¸o, P. B. and Figueiras, J. A., “Solution for the Design of Reinforced Concrete Plates and Shells,” ASCE Journal of the Structural Division, vol. 121, pp. 815– 823, May 1995.

[49] MacGregor, J. G., REINFORCED CONCRETE: Mechanics and Design. Upper Saddle River, New Jersey: Prentice Hall, third ed., 1997.

[50] Malhotra, V. M., ed., IMPACT OF COMPUTERS ON THE PRACTICE OF STRUCTURAL ENGINEERING IN CONCRETE. Detroit, Michigan: American Concrete Institute, 1970. SP-33.

[51] McCormac, J. C., DESIGN OF REINFORCED CONCRETE. New York: John Wiley & Sons, Inc., fifth ed., 2001.

[52] Melosh, R. J., “Basis of Derivation of Matrices for the Direct Stiffness Method,” Journal of the American Institute of Aeronautics and Astronautics, vol. 1, pp. 1631– 1637, 1963.

[53] Meyer, C. and Okamura, H., eds., FINITE ELEMENT ANALYSIS OF REINFORCED CONCRETE STRUCTURES, (New York, New York), Committee on Finite Element Analysis of Reinforced Concrete Structures, American Society of Civil Engineers, May 1985.

[54] Mohr, G. A., “Elastic and Plastic Predictions of Slab Reinforcement Requirements,” Institute of Engineers, Australia, Civil Engineering Transactions, vol. CE 21, no. 1, pp. 16–20, 1979.

[55] Nilson, A. H., ed., FINITE ELEMENT ANALYSIS OF REINFORCED CONCRETE. New York, New York: American Society of Civil Engineers, 1982. Committee on Finite Element Analysis of Reinforced Concrete Structures.

[56] Pfaffinger, D. D., “Column-Plate Interaction in Flat Slab Structures,” ASCE Journal of the Structural Division, vol. 98, pp. 307–326, January 1972.

[57] Phuvoravan, K. and Sotelino, E. D., “Nonlinear Finite Element for Reinforced Concrete Slabs,” ASCE Journal of the Structural Division, vol. 131, pp. 643–649, April 2005.

[58] Portland Cement Association, Shokie, Illinois, Concrete Structural Floor Systems and More..., 2000. CD-ROM Version 1.

[59] Price, W. A., ed., COMPUTER APPLICATIONS IN CONCRETE TECHNOLOGY, (Farmington Hills, Michigan), ACI Committee 118 - Use of Computers, American Concrete Institute, 1987. SP-98.



[60] Raisanen, David, e. a., “A  CASE  Project   Study  of Finite  Element  Analysis  of Concrete   Flat  Slabs,”  1987.   U.S.  Army  Engineer  Waterways Experiment  Station  - Technical  Report ITL-87-2.

[61] RAM International, Carlsbad,  CA, RAM Conce pt,  2003. Version 1.0.

[62] Research  Engineers  International, STAAD.Pro  2004 Technical  Reference, 2004.

[63] Saleh, L.  R.  R.,  “Finite   Element  Analysis  of Shear  in  Reinforced  Concrete  Flat Plates,” Master’s thesis, Georgia Institute of Technology,  1987.

[64] Sawko, F. and  Cope, R. J., “The  Analysis  of Skew Bridge  Decks - a New Finite Element Approach,” The Structural Engineer, vol. 47, pp. 215–224, June  1969.

[65] Severn, R. T. and  Taylor,  P. R., “The  Finite  Element  Method  for Flexure  of Slabs when  Stress Distributions are Assumed,”  Proceedings of the Institution of Civil Engineers, vol. 34, pp.  153–170, 1966.

[66] Smith, E.  T. and Faulkes, K.  A., “Flexural  Properties of Flat  Plate  Floors,”  Institute  of Engineers, Australia, Civil Engineering Transactions, vol. CE 18, no. 2, pp. 140– 145, 1976.

[67] Timoshenko,   S. and  Woinowsky-Krieger,  S.,   THEORY  OF  PLATES  AND SHELLS. New York:   McGraw-Hill  Book Company,  second ed., 1959.

[68] Tracy, Fred T., e. a., “User’s Guide for Revised Computer Program to Calculate Shear,  Moment  and Thrust (CSMT),” 1988. U.S. Army Engineer  Waterways Experi- ment Station  - Technical  Report   ITL-88-4.

[69] Vanderbilt,  D. M., “Equivalent Frame  Analysis for Lateral  Loads,”  ASCE Journal of the  Structural Division, vol. 105, pp. 1981–1998, October  1979.

[70] Will, Kenneth M.,  e. a., “Procedure for Static  Analysis of Gravity  Dams Using the Finite   Element Method - Phase 1a,” 1987. U.S. Army Engineer Waterways Experiment Station  - Technical   Report  ITL-87-8.

[71] Willam, K.  and  Tanabe,  T., eds.,  FINITE ELEMENT  ANALYSIS  OF REIN- FORCED CONCRETE  STRUCTURES. Farmington Hills, Michigan:  American Con- crete Institute, 2001. SP-205.

[72] Wood,  R. H., “The  Reinforcement  of Slabs  in Accordance  with  a Pre-Determined Field of Moments,” Concrete, vol. 2, pp. 69–76, February 1968.

[73] Zienkiewicz, O.  C.  and Cheung, Y. K., “The  Finite  Element Method  for Analysis of Elastic  Isotropic and Orthotropic Slabs,” Institution of Civil Engineers - Proceedings, vol. 28, pp.  471–488, August  1964.

[74] Zwillinger,  D. e. a., ed.,  STANDARD  MATHEMATICAL TABLES  AND FOR- MULAE.  Washington, D.C. Chapman & Hall / CRC Press LLC, 31 ed., 2000.









0 comments:

Post a Comment

Share your views or discuss.

 
Top