ENGINEERING APPROACH TO NONLINEAR ANALYSIS

OF CONCRETE STRUCTURES

By Paolo Riva1 and M. Z. Cohn,2 Fellow, ASCE

ABSTRACT: The objective of this paper is to explore the potential of the lumped-plasticity program STRUPL-1C and to illustrate its use as a practical engineering tool for the nonlinear analysis of concrete structures. This result is achieved through the development of a moment-rotation constitutive law that uses realistic material laws and is valid for any reinforced, prestressed and partially prestressed concrete planar frame at all behavioral states along the loading history. The effects of the main parameters that influence the moment-rotation relationship are investigated and discussed. Finally, the developed constitutive law is applied to the nonlinear analysis of a few reinforced and prestressed concrete continuous beams. Compar­ison to both experimental tests and more-complex nonlinear analysis techniques proves the accuracy of solutions obtained by the STRUPL-1C computer program. The examples reported also prove that the proposed approach enables considerable simplification of the structural model required for a historical nonlinear analysis of concrete structures and demonstrate the high potential of the approach with re­spect to its use in engineering practice.

INTRODUCTION

The object of nonlinear analysis (NLA) of concrete structures is to predict structural response at any state, under a given loading history. From an en­gineering point of view, the difficulties of NLA stem from the need to rec­oncile the conflicting requirements of accuracy and simplicity.

This study describes a NLA method that combines a highly accurate ma­terial constitutive model with a relatively simple lumped-plasticity structural model. The result is an approach that tends to balance the need for realistic and practical analysis procedures, and extends some earlier investigations on flexural concrete constitutive laws (Cohn and Gosh 1973; Cohn and Bartlett 1982; Cohn and Riva 1987) and computer-based NLA (Cohn and Franchi 1979; Cohn and Krzywiecki 1987). As a result, an approach to NLA that combines a moment-rotation (M — 8) constitutive law program (MOCURO) and a NLA routine (STRUPL-1C) is developed. This is referred to as an "engineering" approach, as it has the following features.

1. Simple: the structural description is similar to that for elastic analysis. 2. General: it is applicable to RC, PC, and PPC hyperstatic structures. 3. Realistic: it has reliable stress-strain relations for concrete and steel. 4. Accurate: its analysis is consistent with the required degree of sophistica­

tion for a specified problem. 5. Versatile: it is able to handle various loading types.

The engineering value of the proposed method is validated by comparing the accuracy and ease of arriving at the results to solutions based on more complex FEM methods and by investigating the sensitivity of the results to various parameters considered in the analysis.

'Post-Doctoral Fellow, Dept. of Civ. Engrg., Univ. of Waterloo, Ontario, Canada, N2L 3G1.

2Prof., Dept. of Civ. Engrg., Univ. of Waterloo, Ontario, Canada. Note. Discussion open until January 1, 1991. 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 June 19, 1989. This paper is part of the Journal of Structural Engineering, Vol. 116, No. 8, Au­gust, 1990. ©ASCE, ISSN 0733-9445/90/0008-2162/$1.00 + $.15 per page. Paper No. 24975.

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NONLINEAR ANALYSIS METHODS

Available computer nonlinear analysis methods for structural concrete plane frames may be classified according to their increasing degree of complexity as indicated in Fig. 1. Differences between various approaches within each category arise from the structural discretization, the adopted constitutive law, and the prestressing modeling. An ample discussion of the hypotheses, lim-

TYPE AUTHOR

3

2

CO

2

s

STRUCTURAL DISCRETIZATION

Macchi, 1972

Maier, De Donato, Corradi, 1973

Cohnand Franchi, 1979 Cohn and Krzywiecki, 1987

3 Hung, 1984

Cauvin, 1983

Appleton, Carnara, Almeida, 1983

CONSTITUTIVE LAW

M

Joint

'4>-d

J j+1

Sub-el.

j , j+l: Critical sec.

J j+l

Suh-el. dx

Plastic sec. I

Sub-element

Sub-element \EI

PRESTRESSING MODELLING

NOT

CONSIDERED

• Equivalent load and in definitioi

Joint of M - d> law

Imposed distortion, Effect var.

with EI

Initial distortion considered

PI. Sec. through M-i>

Q-dx

O

8

Aparicio, Arenas, Alonso, 1983

10 equally spaced elements per span

Levi, Mancini, 10 Sub-el. T Munari, 1983 I

3 Cedolin, — BU Dei Poli, g Malerba, 1977 S ui

o

Kangand Scordelis, 1980

Element

Initial distortion considered

through M - <|>

External load

Introduced

through of • ep

e Introduced through ar - tp

FIG. 1. Computer Nonlinear Analysis Methods for Structural Concrete

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its, and potential applications of these approaches is found elsewhere (Riva 1988).

The microscopic finite element approach (Cedolin et al. 1977; Kang and Scordelis 1980) appears to be the most accurate of the surveyed approaches: structural behavior may be studied in detail at a local level, and phenomena such as cracking, dowel action, bond, and aggregate interlocking may be modeled. The time-dependent behavior of concrete, geometric nonlineari-ties, and any loading history may be considered. However, for engineering purposes, the degree of sophistication involved is excessive in most practical cases. Furthermore, the complexity of formulation and the number of vari­ables involved are such that the few reported examples of application are devoted more to checking the validity of the method on simple beams than to the nonlinear analysis of actual structures. The method is valuable in re­search, as it enables a better understanding of the mechanics of concrete structures and may serve as a yardstick for comparison to simplified meth­ods.

Although simpler than the former, the macroscopic finite element ap­proach (Aparicio et al. 1983; Appleton et al. 1983; Cauvin 1983; Levi et al. 1983) has a limited potential for engineering applications. In fact, be­cause of the assumption of constant curvature in an element, the structure must be discretized in short elements, limiting the class of practically solv­able problems to continuous beams or frames with a limited number of ele­ments.

In the lumped-plasticity approach, the structure is discretized by linear-elastic elements, with the behavioral nonlinearity lumped at rigid-plastic joints, for which a piecewise linearized rigid-plastic moment-rotation law is as­sumed. Nonlinear analysis methods of this kind (Cohn and Franchi 1979; Cohn and Krzywiecki 1987; Hung 1984; Macchi 1972; Maier et al. 1973) have characteristics similar to those of the elastic frame analysis, and, if based on realistic moment-rotation constitutive laws, have a great potential in structural engineering practice.

MOMENT-ROTATION CONSTITUTIVE LAW

Moment and curvature distributions in any structure are determined by satisfying the relevant equilibrium, compatibility, and constitutive relations. The compatibility at joint j of a continuous beam may be expressed as

e7. = ej + e; <i) where 6j and 6} = the rotations on the left and right sides of joint j , re­spectively. Splitting the rotation into elastic and plastic components, we can write

e, = (e„ + %)] + (e„ + %P)rj (2) The elastic component may be defined as the elastic rotation generated by the loads and by an inelastic distortion Qpj so as to satisfy the compatibility and constitutive laws.

On each side of the joint, the plastic rotation is determined as

H>(x) - <M*)]<& (3)

where z = the abscissa of the contraflexure point (Fig. 2). In attempting to establish a moment-rotation (M-8) constitutive law for a

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—* \ lp \+-

(a) (b)

FIG. 2. Plastic Rotation at Joint;: (a) Curvature 4>(x) Variation; and (b) Equivalent Plastic Rotation

lumped-plasticity model, we are interested in developing an M-8P relation­ship for the rigid-plastic joint, with the beam element considered perfectly elastic (Fig. 3). If an equivalent "plastic hinge length lp" on one side of joint j is defined as

hj = (4> — f

_ <&el)j Jo

H>(*) - 4>e,(x)]dx (4)

the plastic rotation may be expressed as a function of the plastic curvature at the same section

%j = ( < ) > - fyedjlpj = fypjlpj (5)

where the elastic and plastic curvatures (<$>ei and 4>p, respectively) are deter­mined from the moment-curvature relationship at section j (Fig. 4). Note that the plastic rotation is calculated from the onset of inelastic behavior (i.e., concrete cracking).

The relationship between the plastic rotation 6P and curvature <j> expressed by Eqs. 3 and 5 is shown in Fig. 2. Adopting Eq. 5 for the plastic rotation 9P, the development of an M-Qp relation is reduced to the study of the mo­ment-curvature (Af-<(>) constitutive law and of the plastic hinge length lp expression. All the following material, geometric, and loading parameters

M rigid-plastic flexural spring

elastic element /

M,

(a)

-erf j+i

M,

Any number of hardening branches

(b) e„.

FIG. 3. Lumped-Plasticity Model: (a) Elastic Element i; and (b) Rigid-Plastic Flex­ural Springy

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FIG. 4. Typical Moment-Curvature Relationship for Structural Concrete Sections

that influence the M-(j> relation and plastic hinge length lp should be consid­ered in the development of this M-Qp relation.

Material Parameters The a-e law for concrete in compression; the a-e law for concrete in ten­

sion; the a-e law for reinforcing steel; the a-e law for prestressing steel; and the bond-slip law for reinforcing and prestressing steel.

Geometric Parameters The shape of the section; the mechanical percentage of tension steel q; the

mechanical percentage of compression steel q'\ the mixed reinforcing index 7; the degree of prestressing ft; and the stirrups percentage p".

Loading Parameters The duration of loading; the axial loading; the loading repetition; and the

loading reversal. The parameters that govern lp refer to the loading description (load inten­

sity, distribution, and history) and structure [structural layout, support (boundary) conditions, and prestressing cable layout]. In existing lumped-plasticity approaches (Fig. 1), the adopted moment-rotation laws are based on a tri-linear constitutive law (Macchi 1972; Maier et al. 1973) defined by the cracking, yielding, and ultimate limit states, or by a piecewise linear constitutive law (Cohn and Franchi 1979; Cohn and Krzywiecki 1987). For the former, cracking and yielding limit states are determined according to the classical elastic theory of concrete sections. Rotations at these limit states are determined through integration of curvature distributions along the beam, adopting either the uncracked or cracked section stiffnesses. The ultimate limit state is defined by the ultimate resistant moment of the critical section, and by an ultimate rotation determined empirically from results of several tests on simply supported RC beams subjected to a concentrated load at mid-span. The latter type of moment-rotation constitutive law is based on Eq. 5, where the plastic hinge length lp is assumed constant throughout the loading history, and equal to the effective depth of the section.

These constitutive relationships appear to be unsatisfactory for historical

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nonlinear analysis of concrete structures, because the influence of such im­portant parameters as the section shape, loading arrangement, variation of lp with the loading history, and presence of prestressing are not properly taken into consideration. This study attempts to remove these limitations and to develop an M-Qp relationship that accounts for all relevant parameters.

MOMENT-CURVATURE CONSTITUTIVE LAW

The moment-rotation constitutive law to be used with a lumped-plasticity nonlinear analysis program is developed from the moment-curvature rela­tions presented elsewhere (Cohn and Riva 1987; Riva 1988). The main as­sumptions and results concerning the M-(j> law are: (1) Quasi-static (mono-tonic, nonrepeated, nonreversible) loading; (2) negligible shear effect; (3) linear strain distribution; (4) known uniaxial material stress-strain relation­ships (analytical, experimental, etc.) valid for section analysis; and (5) only bonded tendons are considered.

The M-(j) law is determined from the study of an element whose length equals the crack spacing lc, by assuming the moment is constant in the ele­ment, and expressing the compatibility and equilibrium conditions at the cracked section B in Fig. 5. The curvature <}> is defined as the ratio between the relative rotation of sections A and C and crack spacing lc, Fig. 5. The material constitutive laws adopted for the analysis of sections, along with relevant notation and equations, are summarized in Figs. 6-8.

A general computer program, MOCURO (moment curvature rotation) has been developed to handle automatically the governing conditions of section response at all loading states. Any symmetrical concrete section with up to 15 layers of mild and/or prestressing steel, under either negative or positive moment, using any experimental, analytical, or assumed point-by-point ma­terial constitutive law, may be analyzed.

A parametric study has been performed to assess the influence of the ma­terial and geometric parameters previously identified on the moment-cur­vature law. To account for the actual material behavior and to ensure the continuity of response of reinforced, partially prestressed, and prestressed concrete sections (RC, PPC, PC, respectively), much attention has been paid to the definition of yielding and ultimate limit states, as well as to the gov­erning parameters q (mechanical steel percentage) and 7 (mixed reinforce­ment index).

The yielding curvature §y is defined as the curvature at which the ordinary reinforcing steel reaches its yielding strain ey. For PC sections with only prestressing steel, <$>y is conventionally assumed to correspond to a strain increment in the steel (from its effective prestressing value) equal to the ordinary steel yielding strain (i.e., Aep = ep e, = 0.2%).

N- i * j j !-«4-i

A B C

(b)

- { , 1

5-T -i

h44J^U^H A B C

. r» *e - |

FIG. 5. Typical Element for which Moment-Curvature Law Is Studied

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(«cr.fe))-V • '

XE^

fct

V ^ / ^ C " C, w/tc+C2

w w/f .

FIG. 6. Constitutive Relations for Concrete: (a) Uniaxial Compression (Sargin 1971); and (b) Uniaxial Tension (Giuriani and Rosati 1986)

FIG. 7. Constitutive Relations for Reinforcing and Prestressing Steels (Sargin 1971)

To

bPA

-IDEALIZED BEHAVIOUR

-EXP. RESULTS

Su

(a) (b) "Tp

FIG. 8. Bond Stress-Slip Relations for: (a) Reinforcing Steel (Giuriani 1981); and (b) Bonded Prestressing Steel (Giuriani and Riva 1985)

For sections failing by concrete crushing (highly reinforced rectangular sections) the ultimate limit state is assumed to correspond to the maximum moment capacity (Cohn and Bartlett 1982; Naaman et al. 1986). For other sections, the ultimate limit state is identified by prestressing steel failure (if present) or by reinforcing steel reaching a limiting strain, for RC sections.

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0.0 D.5 1.0 1.5 2.0

<f,d (%)

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

U (%) (h)

FIG. 9. Typical Moment-Curvature Diagrams: Effects of Reinforcing Index q and Degree of Prestressing -y: (a) Rectangular Section; and (b) l-Section

This limiting strain is assumed approximately equal to the strain increment in prestressing steel from its effective prestressing value to its ultimate value (i.e., es/ = Aep = ep„ - epe = 3%).

Since the final design and section response are not known a priori, the definition of the mechanical steel percentage should be independent of the actual steel stresses at the ultimate limit state and should reflect only material characteristics at this limit state and the section geometry. Correspondingly, the following definition of the mechanical steel percentage q (equivalent/ replacing the reinforcement index co) is adopted:

<7 = Kdf'c

(6)

where fpu = the ultimate prestressing steel stress; fst = the reinforcing steel stress at the limiting strain e.sl; f'c = the concrete compressive strength; d = the effective depth of the steel; bw = the section web width; and As and Ap = the reinforcing and prestressing steel area, respectively.

The mixed reinforcement index 7 reflects the proportion of prestressing to total steel in a section, and is defined as

7 = A-pJph

Apfpu + Asfsi (7)

This definition implies that for sections with a given q value, whose ultimate limit state is characterized by the steel reaching its limiting strain, the ul­timate moment is independent of 7.

Typical moment-curvature diagrams for a rectangular and an I-section, with gamma = 0 or 1 and q = 0.10, 0.20, and 0.30, are illustrated in Fig. 9. A complete discussion of the influence of all relevant parameters is given elsewhere (Cohn and Riva 1987; Riva 1988).

PLASTIC HINGE LENGTH

The definition of lp Eq. 4 is general and valid throughout the loading his­tory, provided the distance z of the contraflexure point to the critical section and the curvature distribution 4>(x) are known at any instant. Since both z and 4>0t) are functions of the loading history and structural configuration, they cannot be determined a priori.

Accordingly, an approximate expression of the plastic hinge length lp, based on a parametric investigation of the relationship between the plastic rotation and curvature of a critical section, is derived. To obtain results of

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general validity, the relationship between the nondimensional quantities lp and 4>P/(()OT is studied, where 4>p and §py are the plastic curvature and the plastic curvature at yielding of a critical section, respectively (Fig. 4).

The parametric study involves the nonlinear analysis of 56 simply sup­ported and 32 cantilevered reinforced and prestressed concrete beams with different sections, subjected to either uniformly distributed or point loads (Fig. 10). The influence of the following governing parameters on the lp-$p/$py relationship is studied: section shape, mechanical percentage of ten­sion steel q, mixed reinforcing index 7 (partial prestressing ratio), load dis­tribution, structural configuration and support conditions (boundary condi­tions), and prestressing cable layout.

The influence of material parameters is not investigated, as they were found to have a limited influence on the moment-curvature law: concrete and steel reinforcement properties selected for the parametric analysis are given in Table 1.

Statically determinate rather than continuous beams are studied, because for the former the distance z is constant throughout the loading history, and equals either the half-span, or the entire span length, for simply supported or cantilevered beams, respectively.

Nonlinear analyses of the studied beams have been carried out using the lumped-plasticity program STRUPL-1C (Cohn and Krzywiecki 1987) with elements of length smaller than the adopted section height. STRUPL-1C as­sumes linear elastic elements, with the material nonlinearity lumped at the element ends, where a piecewise linear moment-rotation law is assumed (Fig. 3). Accordingly, the constitutive laws adopted for the parametric study are derived from the developed M-§ law assuming an elastic, uncracked behav­ior of the elements, and a piecewise M-Qp law for the rigid plastic joints, where the plastic rotation is obtained considering a constant plastic curvature distribution along the half-element (i.e., 9P = <^PJle/2, where le is the element length).

Values of the plastic rotations at critical sections are found to be quite sensitive to the adopted element length, because of the assumption of con­stant curvature for an element. To eliminate this effect, elements of length equal to approximately z/100 are used around the critical sections. Elements of progressively increasing length, up to a maximum of 200 mm (assumed stirrup spacing), are used at increasing distance from the critical sections. To ensure a close approximation at the plastic beam-element ends, the M-4> law is piecewise linearized using nine linear planes in all cases investi­gated.

It is noted that bending moment and curvature distributions of the beams investigated are representative of moment and curvature distributions be­tween a contraflexure point and a neighboring critical section of any con­tinuous beam subjected to concentrated or distributed loads. For hyperstatic beams, however, the value of z is not constant throughout the loading his­tory; it can either increase or decrease with the loading, depending on sec­tion, support conditions, and the reinforcement layout of the beam.

Figs. 11—13 summarize the main results of the parametric study, while the complete set of results is given elsewhere (Riva 1988). Results show that the lp/z ratio is not sensibly affected by the variation of z/h ratio, and is influenced mostly by the bending moment distribution considered (Fig. 11), mechanical steel percentage q, compression and tension flange to web width ratios (Fig. 12), and partial prestressing index 7 (Fig. 13). Significantly, they

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TABLE 1. Material Properties for Parametric Study

Material

0) Concrete compression

Concrete tension

Mild steel

Mild steel bond

Prestressing steel

Prestressing steel bond

Figure number (2)

6(a)

6(b)

7

8(a)

7

8(b)

Numerical constant (3)

Ec = 29,930 MPa fc, = 0.8 / ; = 40 MPa e0 = 0.00264 A = 2.5 D = 0.362 / „ = 4.5 MPa Cjlc = 12,000 C2 = 2.0 E, = 200,000 MPa EsH = 6,500 MPa e„ = 0.070 f„ = 400 MPa «*, = 0.010 / „ = 600 MPa T0 = 3.0 MPa TS„ = 10.0 MPa su = 0.5 mm Ep = 190,000 MPa e„ = 0.010 ipu = 0.035 /p0,0 = 1,300 MPa fpl = 1,500 MPa fpu = 1,860 MPa T„ = 4.0 MPa

also indicate that the relationship between lp /z and <$>p /<t>py is characterized by the following three distinct phases.

1. Post-cracking state (§p/§py s 1.0), in which the plastic hinge length in­creases monotonically from 0 to a maximum value corresponding to the yielding limit state (except for y approaching 1.0, where the maximum value of lp/z corresponds to curvature values higher than the conventional yielding limit state). This behavior is related to the progressive crack development along the beam under increasing loads.

FIG. 11. Variation of lp with Loading: Effect of B.M. Distribution

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SECTION A

1 D fl = 0.10

ZL A <? = 0.15

- M V <? = 0.20

O -7 = 0.25

21.0

21.0

21.0

FIG. 12. Variation of /„ with Loading: Effect of: (a) Reinforcement Index q; (b) Compression Flange to Web Width; and (c) Tension Flange to Web Width

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FJG. 13. Variation of /„ with Loading: Effect of Degree of Prestressing 7

2. Post-yielding state, up to the reinforcing steel strain hardening (1.0 < 4>p/ §py =£ =7.0), where the plastic hinge length is drastically reduced. This is due to the almost perfectly plastic behavior of critical sections in this range of <j>p/ <\>py ratio (Fig. 9). This leads to a progressive concentration of deformation at these sections, followed by a drastic reduction of the plastic hinge length lp. An exception to this behavior is observed for prestressed sections, where the plastic hinge length continues to increase for a while after the yielding limit state (Fig. 13). This is a consequence of the conventional definition assumed for the yield­ing limit state, which does not correspond to a sensible slope change in the moment-curvature law.

3. Post strain-hardening state (<$>p/§py > 7.0), in which the plastic hinge length increases up to the ultimate limit state value.

PROPOSED MOMENT-ROTATION CONSTITUTIVE LAW

The results of the parametric analysis described in the previous section allow the development of an equivalent plastic hinge length expression. For this purpose, a nonlinear regression analysis, considering various possible formulations of the lp /z-$p /$py relation, has been performed on the data set obtained from the numerical simulation program.

A simple formulation of the lp/z-$p/$py relationship may be obtained by separately considering the three identified behavioral states. Further simpli­fication may be obtained by considering at first only the reinforced concrete rectangular beam case (7 = 0, b/bw = 1, and b' /b„ = 1) for varying q values. A set of approximate expressions, one for each identified state of behavior, relating lp/z ratios to $p/$py values, has been developed for this basic case. These expressions have later been modified to include the effects of different 7, b/bw, and b' /bw values. As a result of this investigation, the proposed lp /z expressions for reinforced and prestressed concrete beams are the following.

From Cracking to Yielding Limit State (4>„/4>w =£ 1.0)

-r{A-T^)k) k) ™ (8)

From Yielding to Reinforcement Strain-Hardening (1.0 s $p/$w s 7.0)

z \ \mq)\tj \bj J™

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BEAM MODEL A B C D E F G f(y)

I I I I 1 I I 1 ITTT1 0.58 3.0 3.5 3.0 5.0 6.5 0.5 ( 1 - 0 . 5 0 y 3 " ) W'0?

1 JST * »

0.39 7.0 6.5 5.0 5.4 0.0 0.75 ( 1 - 0 .75Y 3 n ) " ,fti

1 -•! 1-1-! !. !-!- i-1- I JF 0.25 7,0 8.0 6.0 2.8 0.0 0.8 l - 0 . 8 0 y

FIG. 14. Numerical Constants for 1,,/z, Eqs. 8, 9, and 10

J>U <9)

Ultimate Limit State (only if 4?pu/$py > 7.0)

^ = ( A + _ ^ ^ ) ( A ) G (10)

2 Vioo l.ooo^/Viw/ where Zp in Eq. 9 is the value of lp/z from Eq. 8 when <$>P/§Py - 1, and the constants A, 2?, C, D, E, F, G, f(y) depend on the bending moment distribution considered and are given in Fig. 14. These values are approx­imations of the estimates determined by the nonlinear regression analysis. Finally, for values of §p/§py between 7.0 and <$>pu/§py, a linear interpolation between lp/z values computed from Eq. 9 with §p/$py = 7.0 and the value at the ultimate limit state (Eq. 10) may be used.

With lp values from Eqs. 8, 9, or 10, the moment-rotation constitutive law can be derived from the moment-curvature relationship by expressing the plastic rotation as a function of the plastic curvature at the critical section, i.e.

6„ = V P (ID This expression of the plastic rotation requires the evaluation of the vari­

able z (distance of the critical section from the contraflexure point). For statically determinate beams z is constant throughout the loading history and its value is a function of only the loading distribution and support conditions. However, for hyperstatic beams its value varies throughout the loading his­tory and is also a function of the reinforcement distribution in the beam. Hence, an a priori definition of z is an approximation only. For historical nonlinear analysis an average of the elastic and plastic z values may be used. The steps required for the determination of the M-Qp law with the proposed formulation are outlined in the flow chart, Fig. 15.

It is noted that the concept of an equivalent plastic hinge length does not apply when only one crack is present at the critical section (e.g., after the first crack formation), because in this case lp = 0 and the plastic rotation 6P approximately equals the crack opening to crack penetration ratio. However, errors resulting from the application of the proposed formulation also at the crack formation state may not affect nonlinear analyses solutions at later states, because the amount of plastic deformation at the end of the transition phase from the first (uncracked) to the second (cracked) state is small relative to the total plastic deformation.

Comparison of plastic hinge length values predicted with Eqs. 8 ,9 , and

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0 GIVEN: Section Shape, q and 7 values

1 :

C

SELECT # of points for M - 0 law

I DETERMINE M, <f>, <£„ and 4>Pl4>n at each point

: i ENTER </>,/<frn in Eqs. 9, 10, 11 and determine Jp/z at each point

0 X

DEFINE 2 for the section considered

f OBTAIN M - 9P with 9P = <f,p('-f) z; M and <t> from Af - </, law V

FIG. 15. Flow Diagram for Derivation of M-8P Law

10 and theoretical values determined in the parametric study shows that the absolute value of the error is lower than 10% in most of the cases (Riva 1988). Exceptions are found close to the cracking limit state (where the plas­tic hinge length concept is not applicable), between the onset of reinforce­ment strain-hardening and the ultimate limit state (due to the assumed lin­earity of the lp/z-$p/$py relationship in this phase), and for prestressed sections at the beginning of the post-yielding state (due to the adopted definition of the yielding limit state for these sections). However, the errors implied in the prediction of the lp/z value with Eqs. 8, 9, and 10 are acceptable also for PC sections, at any behavior state other than the vicinity of the yielding limit state. This result suggests that the proposed formulation can be adopted in practice also for 7 values approaching 1.

NONLINEAR ANALYSIS EXAMPLES

The three examples of nonlinear analysis of reinforced and prestressed concrete structures in Fig. 16 are presented to illustrate the potential and limitations of the lumped-plasticity program STRUPL-1C when the devel-

* E X ™ O T / A N A L ™ S " J ^ STRUCTURE LAYOUT SECTION

P ^ 1 Dilger, 1966 Aparicio a al„ R.C | |

1983

2 Lin, 1955 Kang and Scordelis, 1980 " = u - w ' J j

P.P.C. (Y = 0.90)

K

7

3 Moucessian, Moucessian, "-P-C. 1986 1986 (Y=0.92)

FIG. 16. Numerical Analysis Examples

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, BEAM H4 (DILGER, 1966), I P X 3=E

JGT 0 I—

13.3 H—h

125 "2L -H2

2<|>12 2*12

2*20

2*18 k

1

All measures in cm, bar diameters in mm. Stimips <t> 10 mm in both beams.

APARICIQ MODEL (Aparicio, Arenas, Alonso, 1983)

jxr

STRUPL-1C + M - 6

JT

Typical Section

7 5 l k-30-H

T T R T P T 35.2 1 132.0:

—iiok- T T8 5

ft' = 266 kg/cm2

Ec = 200 000 kg/cm-

STRUPL-1C + M

1

1 9»

i

',

-f

1 1

K-

7 17

O j

0000 3500 3640 4170 5460

es

0.00000 0.00173 0.00190 0.00406 0.00710

CONSTITUTIVE LAWS

M

l/r— End nodes of element i

FIG. 17. Analysis Example 1: Beam #4 Specimen (Dilger 1966): Loading, Ge­ometry, Modeling, and Material Properties

oped moment-rotation constitutive law is adopted to describe the behavior of the nonlinear element joints (Fig. 3). Further numerical examples are dis­cussed elsewhere (Riva 1988).

The present version of STRUPL-1C is based on the following assump­tions.

1. Static loading. 2. Small displacements. 3. First-order analysis.

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BEAM B (LIN, 1955)

192" • 103.2" T '-I k !

58.8'

1

rt ' ' • • STEEL CHARACTERISTICS

MILD STEEL:

PREST. STEEL:

f, = 45.5 ksi E, m 28 400 ksi

fpoo - 200 ksi Ep = 29 000 ksi

13

e,h = 0.0135

fp, - 228 ksi e „ = 0.0099

!ZD fc = 6.9 ksi i A, = 0.476 sq in ! Ap - 0.963 sq in

19J fp,-120.0 ksi

f„ = 59 ksi e.,-0.16

fpu - 256 ksi Ep„ = 0.05

CONSTITUTIVE LAWS

PCFRAME MODEL fKang and Scordelis, 1980)

1 6

Elements divided 0

inio 10 layers

£ s -2@46.8"

STRUPL-lC + M-*t»

-54"—>|^ 2@22.2"

»+«—4@ 19.68" 24.48'

23.2"

STRUPL-1C + M-6

1

£ I

-192"- • 103.2" H

FIG. 18. Analysis Example 1: Comparison of Results: (a) Load-Deflection Dia­grams; and (b) Moment-Load Diagrams

4. Negligible shear effects. 5. Known M-Qp constitutive laws at all rigid plastic joints, valid throughout

the loading history, and for any RC, PPC, or PC structure. 6. Reversible nonlinear material behavior. 7. Adoption of realistic (unfactored) material constitutive laws. 8. Ultimate limit state corresponding to the load for which the first section

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BEAM S IB (MOUCESSIAN, 19861

2D3

St. D2@80 st. D2@80 l-< 1600 — • + • 1600

STEEL CHARACTERISTICS

MILD STEEL: o

3 st.D2@160 4 | 1. -#4* »4«i »4-a*-l T"

12?

/

/ t ,

7

l<t> 13mm

2D3

st. D2@80 st. D2@80 1600 «+. 1600 H 150

f c = 36.6 MPa Ap = 98.71 mm:

ft* • var.

000 310 428

0.0 0.00217 0.0380

PREST. STEEL: fpOo= 1350 MPa E. - 190 000 MPa

fpr= 1715 MPa £p, = 0.011

fp„= 1911 MPa e,.-0.035 CONSTITUTIVE LAWS

a...

PCFRAME MODEL (Moucessian. 1986)

1 10

A ' ' '1' ' ' 19 w^^ •:

Concrete lavers

2@457 2@254 355 355 2@254 2@457 E ements

432 2@381 2@229 2@381 432 iwolofafers

P.S.

M.S. Steel layers

STRUPL-lC + M - e

1 2 3

JJ7 1

-1600-

* 2

-1600 e+

3

-1600-

i r -1600 H

FIG. 19. Analysis Example 2: Beam B Specimen (Lin 1955): Loading, Geometry, Modeling, and Material Properties

exceeds its plastic rotation capacity or to the plastic collapse load, whichever occurs first.

The mathematical formulation of the lumped-plasticity approach developed for STRUPL-1C, as well as a detailed description of the program capability, may be found in other papers (Cohn and Franchi 1979; Cohn and Krzywiecki 1987).

The nonlinear analyses performed allow the comparison of the response predicted by using STRUPL-1C and the developed M-Qp law versus both experimental results and more refined macroscopic and microscopic finite element models. The experimental specimens and the different nonlinear analysis models considered are shown in Fig. 17 (beam H4, Dilger 1966), Fig. 18 (beam B, Lin 1955), and Fig. 19 (beam S I B , Moucessian 1986). The experimental and numerical load-mid-span deflection (P-A) and mo­ment-load (M-P) curves for examples 1 and 2 are plotted in Figs. 20 and

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EXPERIMENTAL rP M l

••••••••• APAMCIO

8 I- « - —A— STRUPL-lC + M-4>

— a - STRUPL-IC + M • ef

0 100 200 300

P (kN) (b)

FIG. 20. Analysis Example 2: Comparison of Results: (a) Load-Deflection Dia­grams; and (b) Moment-Load Diagrams

21, respectively. The P-A and M-i? plots for example 3 are shown in Fig. 22, where the deflection A is measured under the load P, and /?! and R2 are the reactions at the interior supports. In all cases, a historical analysis under proportional loads is performed.

Prestressing effects in example 2 are introduced in STRUPL-1C describing the cable layout as piecewise linear. As a concordant cable profile is used for this beam, no secondary moments are present. In example 3, the sec­ondary moments are introduced in the STRUPL-1C analysis through an equivalent load approach, and are assumed constant throughout the loading history (Cohn and Frostig 1983; Levi et al. 1983).

The adoption of the proposed M-Qp law with STRUPL-1C enables a struc­ture to be described by a number of nodes equal to the number of potential critical sections. This leads to considerable simplification and time saving in the data preparation, result interpretation, and CPU time.

Example 1 (Fig. 20) analysis results indicate the following. For the second state, numerical data yield deflections up to 40% smaller

than the experimental measurements. These large discrepancies between the numerical and experimental results most likely reflect the neglect of shear effects in the theoretical models. As pointed out in the experimental test

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0 10 20 30 40 P (kips) (b)

FIG. 21. Analysis Example 3: Beam S1B Specimen (Moucessian 1986): Loading, Geometry, Modeling, and Material Properties

report (Dilger 1966), shear effects are particularly relevant for this beam, characterized by a z/h ratio lower than 2.

Except for the deflection at the ultimate limit state, no appreciable dif­ference is noted between the numerical results obtained with STRUPL-1C using two elements or 16 elements. Differences in ultimate deflections in­dicate that the moment-rotation constitutive law underestimates the rotation capacity of critical sections at the ultimate limit state.

The ultimate load and deflection predicted by Aparicio's model are only 90% and 45%, respectively, of their experimental values. On the other hand, the ultimate load and deflection computed by STRUPL-1C are 95% and 71% (94% when using 16 elements) of their respective experimental values.

Study of Fig. 21 allows the following remarks on the results of exam­ple 2.

The analysis based on STRUPL-1C with 18 elements and an M-§ law gives the most accurate prediction of the experimental behavior throughout the loading history, with the exception of the ultimate limit state, for which PCFRAME gives results closer to the experimental tests.

The use of the developed M-6P law with STRUPL-1C results in an ov­erestimate of the beam stiffness and an underestimate of the ultimate strength

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8

o g 1 • ' ' •"

0 IS 30 15 60

Aj(mm) (a)

Si 1

0 25 SO 75 100

P INN) (b)

FIG. 22. Analysis Example 3: Comparison of Results: (a) Load-Deflection Dia­grams; and (b) Moment-Load Diagrams

of the beams. This is a consequence of the approximations introduced in the definition of the relationship between the conventional plastic hinge length and the plastic curvature at critical sections of prestressed beams.

The analysis of the results relative to example 3 (Fig. 22) allows the fol­lowing conclusions.

Both STRUPL-1C and PCFRAME overestimate the stiffness of the beam at early states of behavior. However, a close agreement between the two numerical approaches is noted.

Strengths and ultimate deflections of the experimental beams are under­estimated by any of the numerical analysis methods examined. This may be due to a difference between the theoretical and actual cable profile, in which no sharp angles are possible, because of obvious physical limitations.

The similarity of the M-P and M-R curves obtained with STRUPL-1C and those obtained in the experimental tests and with PCFRAME, where no a priori assumptions are made on the hyperstatic effects of prestressing, tends to confirm the assumption that hyperstatic prestressing effects are constant throughout the loading history (Cohn and Frostig 1983; Levi et al. 1983).

These examples show unequivocally that the differences between results obtained with sophisticated and complex nonlinear analysis techniques and

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those derived with a simple engineering-oriented approach (such as STRUPL-1C combined with an appropriate M-Qp constitutive law) are so small that there is no need for complex analytical techniques. This is even more evident when comparing the time required for data preparation and solution of the problems.

CONCLUSIONS

The studies presented in this paper demonstrate the broad potential of a lumped-plasticity approach (STRUPL-1C computer program) as a rational, accurate, and practical engineering tool for the nonlinear analysis of concrete structures. This result is achieved through the development of a realistic moment-rotation constitutive law, valid for any RC, PPC, and PC planar frame at all behavioral states.

It is shown that a moment-plastic rotation constitutive law can be derived from a moment-curvature relationship by multiplying the plastic curvature and an equivalent plastic hinge length lp. Accordingly, a moment-curvature constitutive law, based on realistic material laws, considering any RC, PPC, or PC section is developed. Finally, to obtain a plastic hinge length expres­sion valid throughout the loading history for any type of structural concrete section, a parametric study of the relationship between the plastic curvature and lp at a critical section is performed using a refined macroscopic finite element approach.

The major features of the proposed formulation of plastic rotations 0P at critical sections are the validity at any state of behavior throughout the load­ing history, and the explicit consideration of the bending moment distribu­tion, section shape, and partial prestressing index effects on the inelastic response.

The study of the plastic hinge length evolution throughout the loading history shows that separate consideration must be given to different behavior states. As a result, Eqs. 8, 9, and 10 are developed to express the relation­ship between the plastic hinge length and the plastic curvature at three dif­ferent states of behavior (i.e., post-cracking, post-yielding, and post strain-hardening).

The nonlinear analysis examples demonstrate that the adoption of a re­alistic moment rotation constitutive law with a lumped-plasticity program such as STRUPL-1C is a viable approach in structural concrete engineering practice. Analysis results of the structures investigated, and comparison of the method to some more-refined finite element approaches, and to experi­mental tests, lead to the following conclusions.

STRUPL-1C enables considerable simplification of the structural model required for a historical analysis of concrete structures. In all cases, the num­ber of nodes necessary to describe the behavior of the structure can be re­duced to the number of potential critical sections. This leads to a structural discretization equivalent to that used for the elastic analysis of the same structure, whereas more sophisticated methods require much finer structural meshes.

The developed model allows a detailed study of the nonlinear flexural response of any reinforced, partially prestressed, and prestressed planar frame throughout all behavioral states up to the structural collapse.

Comparison of analysis results based on STRUPL-1C to experimental tests and more-refined microscopic and macroscopic finite element methods shows that more-sophisticated methods lead only to marginal improvements of pre-

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dieted structural response. However, STRUPL-1C enables considerable time and effort savings in data preparation and computer time required for the solution of a given nonlinear analysis problem. Thus, adequate accuracy and application efficiency tend to confirm the validity of STRUPL-1C as a prac­tical engineering tool.

Present limits of the model are related to the fact that only first-order, flexural, static actions are considered. Further studies are needed to inves­tigate the influence of shear on the structural deformability, moment-axial force interaction, geometric nonlinearities, and dynamic response.

ACKNOWLEDGMENTS

Results presented in this paper are part of a thesis prepared by the first writer under the direction of the second writer in partial fulfillment of the requirements for a Ph.D. degree in the Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada. The financial support of the National Sciences and Research Council (NSERC) of Canada under Grant A-4789, is gratefully acknowledged.

APPENDIX I. REFERENCES

Aparicio, A. C , Arenas, J. J., and Alonso, C. (1983). "Examples of moment re­distribution in continuous P.P.C. bridges." Int. Symp. on Nonlinearity and Con­tinuity in Prestressed Concr., Preliminary Publication, 2, Univ. of Waterloo, Wa­terloo, Ontario, Canada, 185-204.

Appleton, J., Camara, J., and Almeida, J. F. (1983). "Elastoplastic analysis and design of partially prestressed concrete beams." Int. Symp. on Nonlinearity and Continuity in Prestressed Concr., Preliminary Publication, 2, Univ. of Waterloo, Waterloo, Ontario, Canada, 77-106.

Cauvin, A. (1983). "Nonlinear analysis of P.C. continuous beams and frames." Int. Symp. on Nonlinearity and Continuity in Prestressed Concr., Preliminary Publi­cation, 2, Univ. of Waterloo, Waterloo, Ontario, Canada, 107-124.

Cedolin, L., Dei Poli, S., and Malerba, P. G. (1977). "Finite element analysis of prestressed concrete beams." Costruzioni in Cemento Armato, Studi e Rendiconti, Corso di Perfezionamento per le Costruzioni in Cemento Armato F.lli Pesenti, Politecnico di Milano, Milan, Italy, 15-30.

Cohn, M. Z., and Ghosh, S. K. (1979). "Ductility of reinforced concrete sections." IABSE Publications, 32(2), 51-81.

Cohn, M. Z., and Franchi, A. (1979). "STRUPL: A computer system for structural plasticity." J. Struct. Div., ASCE, 105(4), 789-804.

Cohn, M. Z., and Bartlett, M. (1982). "Nonlinear flexural response of partially pre­stressed concrete sections." J. Struct. Div., ASCE, 108(12), 2747-2765.

Cohn, M. Z., and Frostig, Y. (1983). "Inelastic behavior of PC beams." J. Struct. Engrg., ASCE, 109(10), 2292-2309.

Cohn, M. Z., and Krzywiecki, W. (1987). "Nonlinear analysis system for concrete structures: STRUPL-1C." Engrg. Struct., 9(Apr.), 104-123.

Cohn, M. Z., and Riva, P. (1987). "A comprehensive study of the flexural behavior of structural concrete elements." Studi e Ricerche, Corso di Perfezionamento per le Costruzioni in Cemento Armato F.lli Pesenti, Politecnico di Milano, Milan, Italy, 9, 365-414.

Dilger, W. (1966). "Veranderlichkeit (ter biege- und schubsteifigkeit bei stahlbeton-tragwerken und ihr einfluB auf schnittkraftverteilung und traglast bei statisch un-bestimmter lagerung." Deutscher Auschuss fur Stahlbeton, 179, W. Ernst Sohn, Berlin, W. Germany (in German).

Giuriani, E. (1981). "Experimental investigation on the bond-slip law of deformed bars in concrete." Proc. IABSE Colloquium on Advanced Mech. of Reinforced Concr., Delft, The Netherlands, Dec, 121-142.

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Giuriani, E., and Riva, P. (1985). "Effects of cracking on the moment-curvature relationship in partially prestressed concrete beams." Studi e Richerche, Corso di Perfezionamento per le Costruzioni in Cemento Armato F.lli Pesenti, Politecnico di Milano, Milan, Italy, 7, 189-220 (in Italian).

Giuriani, E., and Rosati, G. (1986). "Behavior of concrete elements under tension after cracking." Studi e Ricerche, Corso di Perfezionamento per le Costruzioni in Cemento Armato F.lli Pesenti, Politecnico di Milano, Milan, Italy, 8, 65-82 (in Italian).

Hung, N. D. (1984). "CEPAO—An automatic program for rigid-plastic and elastic-plastic analysis and optimization of frame structures." Engrg. Struct., 6(Jan.), 33 -51.

Kang, Y. J., Scordelis, A. C. (1980). "Nonlinear analysis of prestressed concrete frames." J. Struct. Div., ASCE, 106(2), 445-462.

Levi, F., Mancini, G., and Munari, D. (1983). "Hyperstatic effects of prestressing between serviceability and ultimate limit states." Int. Symp. on Nonlinearity and Continuity in Prestressed Concr., Preliminary Publication, 2, Univ. of Waterloo, Waterloo, Ontario, Canada, 1-23.

Lin, T. Y. (1955). "Strength of continuous prestressed concrete beams under static and repeated loads." ACIJ., 26(10), 1037-1059.

Macchi, G. (1972). "Analysis of hyperstatics structures with the imposed rotations methods." C.E.B., Annexes aux Raccomandations Internationales pour le Calcul et I'Execution des Ouvrages en Beton, AITEC, Rome, Italy, 313-368 (in French).

Maier, G., De Donato, O., and Corradi, L. (1973). "Inelastic analysis of reinforced concrete frames by quadratic programming." Symp. on Inelasticity and Nonlineari­ty in Struct. Concr., SM Study 8, Univ. of Waterloo, Waterloo, Ontario, Canada, University of Waterloo Press, 265-288.

Moucessian, A. (1986). "Nonlinearity and continuity in prestressed concrete beams," thesis presented to Queen's University, at Kingston, Ontario, Canada, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Naaman, A. E., Harajli, M. H., and Wight, J. K. (1986). "Analysis of ductility in partially prestressed concrete members." PCI J., 31(3), 64-87.

Riva, P. (1988). "Engineering approaches to nonlinear analysis of concrete struc­tures," thesis presented to the University of Waterloo, at Waterloo, Ontario, Can­ada, in partial fulfillment of the requirements for the degree of Doctor of Philos­ophy.

Sargin, M. (1971). "Stress-strain relationships for concrete and the analysis of struc­tural concrete sections." SM Study No. 4, Solid Mech. Div., Univ. of Waterloo, Waterloo, Ontario, Canada.

APPENDIX II. NOTATION

The following symbols are used in this paper:

lp = plastic hinge length; M = bending moment referred to centroid of gross cross section;

Mu = ultimate moment; My = yielding moment;

q = Apfpu — Asfs,/bwdf'c = mechanical percentage of tension steel;

q' = A'sfs,/bwdf'c = mechanical percentage of compression steel; z = distance of contraflexure point to critical section; 7 = Apfpu/Apfpu + Asfst = mixed reinforcement index; 6 = rotation;

8e, = elastic rotation; 9P = plastic rotation;

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K = fpe If pa = degree of prestressing; <|> = curvature;

<}>«.; = elastic curvature; <t>p = plastic curvature; <|>„ = ultimate curvature; and i?y = yielding curvature.

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