ADVANCED NONLINEAR FORMULATION FOR ...

ADVANCED NONLINEAR FORMULATION FOR REINFORCED CONCRETE BEAM-COLUMNS By B. A. Izzuddin, 1 C. G. Karayannis, 2 and A. S. Elnashai 3 ABSTRACT: This is the first of two papers addressing the efficient and simulta-

neouslyaccurate analysisof reinforcedconcrete frames through the use of adaptive nonlinear analysis techniques. The present paper discusses the requirements of adaptive analysisand highlightsthe essentialneed for a beam-columnformulation that accurately models the behavior of reinforced concrete members in the range where the response can be consideredelastic and fully recoverable. A new elastic formulationis therefore proposed for reinforcedconcretebeam-columns;it assumes a quartic shape functionfor the transverse displacementsand differs from conventional finite-elementformulationsin the use of the constant-axial-forceCriterion. The quartic formulationis capable of modelinga typicalreinforced concrete member of arbitrary cross-sectionalshape and reinforcementlayout with only one element, includingthe effects of concrete tensile cracking,the nonlinear compressive response of concrete, and the beam-columnaction. Verificationexamples using the nonlinear analysisprogram ADAPTIC demonstrate the accuracy of the proposed formulationand its ability to model an entire reinforced concrete member using one element. INTRODUCTION

The past few decades have witnessed significant advances in methods of nonlinear analysis for structures of multifarious forms and constituent materials. While these advances were accompanied by great technological achievements in the field of digital computers, the n o n l i n e a r analysis of real structures still poses huge demands on the most powerful of available computers. The advent and continuous development of the finite-element method has allowed several aspects of reinforced concrete behavior to be modeled to various levels of accuracy and detail. At one extreme, two- and threedimensional elements, enhanced with advanced material constitutive laws (Beshara and Virdi 1991), were used to investigate the n o n l i n e a r response of reinforced concrete members. However, this approach requires a large number of elements per m e m b e r for accurate modeling, hence posing huge computational and modeling demands that render it inappropriate to the nonlinear analysis of real structures. A t the other extreme, simplified assumptions are often made, such as adopting a stick model for a multistory reinforced concrete frame (Liou and Kang 1992), so that nonlinear analysis can be performed on available computers within a reasonable time scale. This approach, while suitable for a preliminary investigation, cannot model 1Lect. in Engrg. Computing, Civ. Engrg. Dept., Imperial College, London SW7 2BU, U.K. 2Asst. Prof., University of Thrace, Xanthi 67100, Greece; formerly, Academic Visitor, Civ. Engrg. Dept., Imperial College, London SW7 2BU, U.K. 3Reader in Earthquake Engrg., Civ. Engrg. Dept., Imperial College, London SW7 2BU, U.K. Note. Discussion open until March 1, 1995. Separate discussions should be submitted for the individual papers in this symposium. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on August 19, 1993. This paper is part of the Journal of Structural Engineering, Vol. 120, No. 10, October, 1994. 9 ISSN 0733-9445/94/0010-2913/$2.00 + $.25 per page. Paper No. 6750. 2913

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the most salient characteristics of the behavior of reinforced concrete members with an accuracy comparable to that of the previous approach. Concerted research has been devoted to the development of nonlinear analysis tools for reinforced concrete frames that provide a desirable balance between accuracy and efficiency. In this context, one-dimensional elements have been developed for modeling reinforced concrete beams and columns, although the issue of accuracy versus efficiency still persisted. It has been observed by Kim and Lee (1992) that the layered approach, which is a reasonably accurate method for the formulation of cross-sectional response characteristics from material response models, is computationally demanding in comparison with the "modified', stiffness approach in which the crosssectional response is obtained directly on the generalized forces level. While the latter approach has been followed widely for modeling reinforced concrete members (Otani 1974; Darvall and Mendis 1985; Sfakianakis and Fardis 1991), it cannot accurately model the various salient characteristics of the cross-sectional response. This includes characteristics such as the interaction between the axial force and crack opening/closure as well as the hysteretic response of concrete in the compressive range. The present work uses the accuracy of the layered approach and addresses its inefficiency through 'the application of nonlinear adaptive analysis techniques (Izzuddin and Elnashai 1993). This recognizes the necessity of applying the layered approach only in those zones of the structure that develop material inelasticity, while the elastic parts of the structure can be modeled using a less computationally intensive approach. In this regard, adaptive analysis relies on the existence of a powerful elastic formulation capable of representing a whole member with one element; hence, the nonlinear analysis is always started using one element per member. During analysis, various parts of the structure are checked for the development of material inelasticity, and elements based on the layered approach are inserted in the inelastic zones. Apart from modeling advantages, the proposed methodology leads to considerable computational savings, since expensive elements based on the layered approach are inserted only when and where necessary, during analysis, and within the structure, respectively. In the present paper the first component of adaptive analysis is presented, namely, an efficient and simultaneously accurate elastic formulation for reinforced concrete beam-columns. The formulation details and underlying assumptions are provided hereafter. This is followed by verification ex, amples, using the nonlinear analysis program ADAPTIC (Izzuddin and Elnashai 1989), which demonstrate the accuracy of the proposed formulation and its ability to model a whole elastic reinforced concrete beam-column using only one element. The framework of adaptive analysis and the use of the proposed formulation within the inelastic large displacement analysis of reinforced concrete frames is laid out in the companion paper (Karayannis et al. 1994). MAIN CONSIDERATIONS

There are several complexities associated with the accurate simulanon of the behavior of reinforced concrete members, even when the simulation domain is restricted to low strains. Apart from the nonuniformity of concrete materials that usually imposes limitations on the accuracy of analytical models, the cracking phenomenon exhibited by concrete adds further complications to the formulation of accurate models. 2914

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In the context of reinforced concrete beam-columns, the following assumptions are often adopted in the formulation of realistic analytical models:

1. Plane sections remain plane after flexural deformation. 2. Full strain compatibility exists between concrete and steel reinforcement. 3. Reinforcement steel bars cannot buckle under compression. 4. Concrete has a uniform composition. 5. Mechanical properties of concrete may vary according to confinement levels. The first two assumptions allow the formulation of analytical models to be transferred from the three-dimensional spatial domain to a combination of two-dimensional cross-sectional domain and one-dimensional longitudinal domain. This allows the formulation details to be considered on two distinct levels, namely, the cross-sectional level and the member longitudinal axis level. The interaction between the formulation characteristics at these two levels, which determines the overall properties of the formulation, can be established through the translation of the given assumptions into compatibility constraints, as discussed in later sections. Cross-Sectional Level Reinforced concrete members use a variety of cross-sectional geometric shapes, as illustrated, for example, in Fig. 1, and also use a wide range of reinforcement bar sizes and numbers. The mechanical properties of concrete, evaluated on the direct stress/strain level, may vary according to confinement conditions, with concrete in the core associated with higher confinement. Typically, concrete exhibits a nonlinear compressive stressstrain relationship and cracks under low levels of tensile strains. A simplifying assumption is made herein that the response of concrete is elastic and fully recoverable under tensile and low compressive strains. Longitudinal Axis Level The primary consideration herein is the deformed shape of a reference line along the member length. The main departure from the prediction of classical beam theory, even in the elastic range and for loads applied only at the member ends, is due to the following considerations:

1 Unconfined Concrete 2 Confined Concrete

R,orcmCnts.

'

):

I"

" 2

."

t

4

FIG. 1. Typical Cross Sections for Reinforced Concrete Members 2915

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1. The effects of concrete cracking and nonlinear mechanical properties on the cross-sectional level lead to a nonlinear relationship between the bending moment and curvature, which significantly depends on the magnitude of the axial force. Since bending moments vary along the member length, complex deformed shape may result that cannot be accurately represented by the conventional cubic polynomial function. 2. Reinforcement bars may vary in size and location along the member, leading to a coupling between the axial and bending actions, since the chosen longitudinal reference line would not coincide with the cross-sectional centroid at all points along the member. This again leads to complexities in the member deformed shape. Hereafter, the main requirements discussed previously are considered in a new formulation for modeling elastic reinforced concrete members, with the requirements for adaptive elastoplastic analysis laid out in Karayannis et al. (1994). The properties of the formulation are first established on the level of the cross section, after which the overall characteristics are determined on an element level suitable for use within a general purpose nonlinear analysis environment. CROSS-SECTION MODELING

With the assumptions made in the previous section, the properties of the present formulation on the section level depend on the chosen material models for steel and concrete, as well as the adopted representation for the cross-sectional shape, including confinement effects. These properties are expressed in terms of a relationship between generalized stresses and strains as well as a corresponding generalized section stiffness, which are directly used by the formulation on the element level. Material Models

As pointed out earlier, this formulation is intended to represent the nonlinear elastic response of reinforced concrete members. With this in mind, elastic material models are adopted on the direct stress-strain level for both steel and concrete materials. For steel, the stress-strain relationship is simply expressed ,~s =

(a)

E,~s

where E, is Young's modulus. Whereas for concrete, the following stress-strain relationship is used: ~rc =

- kfc;

r162

r2 ec + L

~c = 0;

e c O

]

;

-ec0-