Optimal Domains for Flexural and Axial Loading by ProQuest


More Info
									 ACI STRUCTURAL JOURNAL                                                                                  TECHNICAL PAPER
Title no. 105-S66

Optimal Domains for Flexural and Axial Loading
by Mark Aschheim, Enrique Hernández-Montes, and Luisa María Gil-Martín

Domains associated with the optimal design of rectangular reinforced
concrete sections are established according to the provisions of
ACI 318 with respect to P-M coordinates using results obtained
with reinforcement sizing diagrams (RSDs). Characteristics of the
optimal solutions are identified for each domain, analytical
expressions for the domain boundaries are established, and a two-
step solution procedure is presented for optimally determining the
top and bottom reinforcement required to resist an arbitrary
combination of axial load and moment. Examples illustrate the
application of this computationally efficient procedure to the design of
columns, singly reinforced beams, and doubly reinforced beams.
    The optimal domains provide new insight into reinforcement             Fig. 1—Sketch defining terms for ultimate strength analysis
requirements for beam-column sections. A design approach is                per ACI 318.
presented that is applicable to both beams and columns alike. This
contrasts with the variety of piecemeal approaches currently used to
determine non-optimal reinforcement solutions for beam and column                         RESEARCH SIGNIFICANCE
sections, which can involve manual iterations, charts, and trial-and-         The characteristics of optimal (minimum) reinforcement
error procedures. The savings in reinforcement can be significant for      solutions over the entire P-M space are described. Analytical
sections subjected to nonsymmetric loading and provide an opportunity      expressions defining the domain boundaries and optimal
to improve the sustainability of reinforced concrete construction.         reinforcement quantities are derived. A novel two-step design
                                                                           approach is articulated, in which optimal reinforcement is
Keywords: beams; column; cross-section design.                             determined to provide adequate flexural and axial strength
                                                                           according to the assumptions of ACI 318.1 The conceptual
                         INTRODUCTION                                      advances provide a new holistic perspective on reinforcement
   A variety of methods are currently used to design the                   requirements and enable a unified treatment of the design of
reinforcement in beam and column sections. Beams typically                 beams and columns subjected to nonsymmetric combinations
are designed with nonsymmetric distribution of reinforcement.              of axial load and moment. Benefits include potential reductions
Whereas mathematically exact solutions are typically used                  in the required amount of reinforcement and increases in
for the design of singly reinforced beams, approximate and                 curvature ductility, as well a deeper understanding of the
sometimes iterative solutions are more commonly used for the               beam-column design problem.
design of doubly-reinforced beams. Columns, on the other hand,
are routinely designed assuming a predetermined pattern of                                 PROBLEM FORMULATION
symmetric reinforcement, often using P-M interaction charts to                A rectangular cross section having two layers of reinforce-
represent the effect of axial load on the flexural strength of             ment is considered, as shown in Fig. 1(a). The section resists
the section. Whereas this practice is well-suited to members               a combination of axial load P and moment M. The ultimate
subjected to symmetric loading, it can be very uneconomical                strength of the cross section may be determined as a function
for the design of those elements that are subjected to                     of the reinforcement and cross section geometry for a range
nonsymmetric loading. The optimal domains provide new                      of eccentricities, in accordance with conventional design
insight into the beam-column design problem, revealing for                 assumptions. Axial load is positive in compression. Moment,
the first time patterns in the distribution of optimal reinforcement       if present, is positive if the top fiber is compressed relative to
for sections subjected to nonsymmetric loading.                            the bottom fiber. These terms and sign conventions are
   This study presents the relatively complex domains that                 consistent with the conventional analysis of a cross section
characterize the optimal distributions of reinforcement for                in flexure (for which tensile strain develops in the bottom
the special case of uniaxial bending for a nonsymmetric P-M                layer of reinforcement), and they are maintained even as the
loading. The results are directly applicable to the design of              analyses are extended to include conditions in which
special elements such as the walls of box culverts and the                 compressive or tensile axial loads dominate the behavior of
columns of bridge C-bents, which are subjected to                          the cross section.
nonsymmetric loading. The design approach is significantly                    Following the provisions of ACI 318,1 the ultimate
more computationally efficient than its predecessor, which                 strength is determined assuming plane sections remain plane
requires brute force calculations for a large number of                    for an extreme fiber compressive strain of 0.003. It is
neutral axis depths. The results are useful for more complex
scenarios involving multiple uniaxial load combinations and                  ACI Structural Journal, V. 105, No. 6, November-December 2008.
                                                                             MS No. S-2007-144 received April 25, 2007, and reviewed under Institute publication
provide a key baseline for making sense of results obtained                policies. Copyright © 2008, American Concrete Institute. All rights reserved, including the
for more complex optimizations involving sections                          making of copies unless permission is obtained from the copyright proprietors. Pertinent
                                                                           discussion including author’s closure, if any, will be published in the September-October
subjected to biaxial loading.                                              2009 ACI Structural Journal if the discussion is received by May 1, 2009.

720                                                                               ACI Structural Journal/November-December 2008
                                                                                             where As is the cross-sectional area of steel located at a
Mark Aschheim, FACI, is a Professor of civil engineering at Santa Clara University,
Santa Clara, CA, and is a member of ACI Committees 341, Earthquake Resistant                 distance d from the top of the section and
Concrete Bridges; 374, Performance-Based Seismic Design, and Joint ACI-ASCE
Committee 445, Shear and Torsion. His research interests include the earthquake-resistant
design of reinforced concrete structures.                                                                            f s∗ = f s + 0.85f c ′ if β 1 c > d
Enrique Hernández-Montes is a Professor of structural mechanics at the University                                                 = f s otherwise
of Granada, Granada, Spain. His research interests include structural concrete and
earthquake engineering.
                                                                                                The stress fs is positive in tension and is negative in the
Luisa María Gil-Martín is an Associate Professor of structural mechanics at the University
of Granada. Her research interests include structural steel and structural concrete.         event that β1c > d.
                                                                                                The internal stress resultants Cc, Cs, and Ts equilibrate the
                                                                                             applied load P and moment M. A conventional solution
common, though not required, to represent the concrete                                       approach would determine nominal strengths Pn and Mn for
contribution in compression using the Whitney stress block,2                                 a given neutral axis depth, material properties, and reinforcement
for which a stress of 0.85fc′ is present over the width of the                                                  ′
                                                                                             areas As and As , by applying equilibrium equations to the
section and for a depth of β1c, where fc′ is the specified                                   free body diagram of Fig. 1(d).
compressive strength of concrete; β1 is a factor relating the
depth of equivalent rectangular compressive stress block 
To top