designing rc continuous beams

F O U N D A T I O N S O F C I V I L A N D E N V I R O N M E N T A L E N G I N E E R I N G

No. 6 2005

© Publishing House of Poznan University of Technology, Pozna 2005 ISSN 1642-9303

Andrzej GARSTECKI, Jacek CIGA O,Poznan University of Technology, ul. Piotrowo 5, PL-60-965 Pozna , Poland

DESIGNING RC CONTINUOUS BEAMS ACCOUNTING

FOR PLASTIC REDISTRIBUTION OF MOMENTS

The Application of classic methods of the theory of plasticity and the linear method with redistribution of moments have been briefly discussed here. The major part of the paper is devoted to methods of static analysis when the moment-curvature relation is assumed in non-linear form accounting for tension stiffening effect. Two different definitions of plastic hinge have been presented, namely, the idealised hinge concentrated in a cross-section and the hinge smeared over a limited region. Numerical examples illustrate practical application of both definitions and demonstrate quantitative difference. Following the concept of the smeared hinge, the angles of mutual plastic rotation of two cross-sections at the distance of 1.2×depth of beam were computed using three different moment-curvature relations. Potential savings in structural material resultant from the application of plastic redistribution of bendings moments are discussed as well.

Key words: rotation capacity, redistribution of moments,

1. INTRODUCTION

Designing of engineering structures accounting for plastic properties of material has a long tradition. However, the knowledge and experience concerning RC structures are not as extensive as those concerning steel structures. Conservative application of the theory of plasticity in the static analysis of RC structures is justified. The basic reason is that concrete cannot be treated as ideally plastic material. Besides, the appearance of the plastic hinge in the beam or slab cross-section is preceded by cracks in the tensile zone. In this situation the rotational capacity in the plastic hinge is very limited. After

Andrzej Garstecki, Jacek ciga o22

exceeding a certain limit value of the plastic rotation sudden failure may occur with the brittle fracture features. Despite these unfavourable characteristics, foreign standards and the international code EC2 [4] allow to use static analysis methods based on the elastic-plastic model or rigid-ideally-plastic model, provided that the structure is able to safely withstand plastic deformations. Several international research programmes were opened to examine this problem. As a result, the detailed provisions in the sequential editions of EC2 concerning plastic hinges were drastically changed. The problem has not yet been fully solved and is still the field for the future interesting research. The possibility of using the plastic capacity reserve in static analysis and design of RC structures expands the range of designing. In this paper we have tried to show the trends observed in the literature and in the EC2 in respect of employing plastic models in static analysis. The application of classic theory of plasticity and the linear method with redistribution are discussed. Then, the non-linear analysis allowing plastic hinges is presented, where different methods of assessment of the rotation capability in the plastic hinge are shown. It is illustrated by a numerical example of the analysis of a continuous beam. The quantitative differences between various methods of evaluation of the plastic rotation angle are shown as well.

2. PRINCIPLES OF STATIC ANALYSIS ACCOUNTING FOR

THE PLASTIC PROPERTIES OF REINFORCED CONCRETE

2.1. Classic methods of theory of plasticity Classic methods exploit the rigid-ideally-plastic material model. Using the kinematically admissible states one obtains the upper bound of limit load, i.e. the hazardous one. The static approach leads to the lower bound which is better accepted by designers. In the case of continuous beams and, to some extent, the continuous slabs, one can easily find the real kinematic mechanism and then the upper bound coincides with the true limit load. However, the applied material model does not allow to find the displacements and strains, hence one cannot check if the hinge rotations are admissible. To this end, one must use more sophisticated model, i.e. the elastic-plastic model. It is obvious that the classic methods of the plastic analysis can provide only introductory assessment of the load bearing capacity of beams or plates. However, these methods provide important information about the failure mechanisms. In some special types of structures, e.g. corbels, the classic methods of theory of plasticity are widely used.

Designing RC continuous beams accounting for plastic redistribution of moments. 23

2.2. Linear method with plastic redistribution of moments The initial distribution of internal forces is computed using the linear elastic analysis. Then, the plastic hinges are allowed. They result in the plastic redistribution of internal forces (moments). In this method the rotations in the plastic hinges are not checked, hence only a relatively small redistribution can be allowed. The so called coefficient of redistribution is defined as the ratio between the plastic moment after redistribution and the initial elastic moment. The limit value of coefficient depends on the relative depth of compression zone of concrete cross-section under bending and on the plastic properties of reinforcing steel and concrete. The application of this method is very easy but one cannot expect significant profits. The results are close to the ones obtained from the linear elastic analysis. Note that in fact the method is non-linear, hence the superposition rule cannot be used. The theoretical background, current experience and the evolution of code regulations concerning this method are given in [3, 6]. The following examples illustrate the idea.

2.3. Non-linear method with plastic redistribution of moments Briefly speaking, one uses non-linear physical law, which, in the best

possible way, describes the elastic-plastic properties of concrete and steel on the level of point and cross-section. One must take into account cracking, tension stiffening effect and creep. The limit load level usually results from the condition of the admissible plastic deformations. In beams and slabs this condition can be expressed as a constraint on the angle of rotation in the plastic hinge. The admissible value of the angle depends on the relative depth of the compression zone, plastic properties of steel and concrete and the contribution of shear or normal force. The above described method has commonly been accepted for at least 10 years. However, recently the interpretation of the term plastic hinge has evolved substantially and the rules of introduction of partial (material) safety coefficients in the calculation of the angle of rotation have been changed.

Following the classic definition, the plastic hinge in a beam is a point on the longitudinal beam axis, where the value of the bending moment is equal to the plastic moment and where the discontinuity of first derivative of the deflection line exists. In other words it is the idealised hinge, where the plastic limit moment Mpl exists. The angle of rotation in the hinge is equal to the angle

between two planes adjacent to the hinge point (Fig. 1). A similar definition is used for the hinge line in the slab and the respective angle of rotation. Until now the angles of rotation defined in this way were subject to limitations. Experiments carried out on a large number of beams [5] proved that the classic, idealised definition of a plastic hinge has limited applicability. The experiments clearly demonstrated that the plastic hinge in the


RC beam has a form of the curvature location observed on a small section of the beam length. Following this idea, EC2 suggests to assume that this section is a = 1.2h, where h is the beam depth. The angle of rotation in the hinge is defined as the relative angle between two planes located at the distance of 0.5a = 0.6h

from the ideal hinge (Fig. 1). Note that the angle of rotation in thus defined smeared hinge is greater than the angle defined for the ideal hinge. Therefore, the constraint on the angle in the smeared hinge is more rigorous than it was in the previous editions of EC2, where the angle in the ideal hinge was constrained. The method of calculation of both angles will be illustrated in an example and the quantitative differences will be shown. The question of the length of the section of the generalized hinge is still open. One may suppose that it should depend not only on the beam depth but also on other features of the structure, e.g. on the width of the support. The problem is ‘of delicate nature’ since, as a matter of fact, the value of the plastic moment exists only in the cross-section of an ideal point support. In the cross-sections located at the 0.6h from the ideal support, moments are much lower than the plastic moment. It may happen that the slopes of the bending moment diagram at both sides of the support differ significantly. Then, the assumption of the plastic section equal to a = 1.2h is dubious. Note that when the plastic hinge emerges in a span under a distributed load, then the bending moment in this zone remains nearly constant, whereas the moment over a support or in a span under a point load changes significantly. Is it correct to assume the same length a = 1.2h in all these cases? The problem requires more research since the definition of the mutual angle of rotation in the plastic hinge influences strongly the results of the analysis.

Fig. 1. Plastic angle of rotation .

A major problem encountered in the computation of the angle of rotation in a plastic hinge is the application of proper relation between generalised stress and strain, in case of beams the relation between bending moment and curvature. Only advanced computer programs for the static analysis of RC structures allow the introduction of a physical law at the point level. Usually a non-linear


physical law is formulated at the level of cross-section. The crucial issue is the determination of the mean strain smr in the reinforcement under tension accounting for cracking of the concrete, tension stiffening effect and creep. The following relation may be used:

1E

2

s

sr21

s

ssmrsm (2.1)

where: smr - is the steel strain calculated on the basis of an uncracked section under the cracking load,

sr - in the steel stress calculated on the basis of a cracked section under the cracking load,

s - is the steel stress calculated on the basis of a cracked section under the loading considered,

1 - is a coefficient which takes account of the bond properties of the reinforcement ( 1 = 1 for deformed bars and 0.5 for plain bars),

2 - is a coefficient which takes account of the duration and nature of the loading ( 2 = 1 for short term loading and 0.5 for long term or frequently repeated loading).

Knowing sm one can compute the mean curvature 1/ from the Bernoulli rule

d

1 csm

m

(2.2)

where: sm - is the average steel strain calculated taking the tension stiffening into account,

c - is the strain at the most compressed fibre (negative for compression) calculated ignoring tension stiffening.

This method relatively well describes the dependence of curvature 1/ on the value of the bending moment M. A slightly different method of the curvature calculation uses the linear interpolation between the crack-free state and the fully cracked state, namely

,1

111

IIIm

(2.3)


where (1/ )I and (1/ )II denote the curvatures calculated for the cross-section before and after cracking, respectively. The interpolation coefficient can be obtained from

2

cr21

M

M1 (2.4)

where: Mcr - the bending moment producing a crack of the cross-section.

In a next method the linearization of the dependence of curvature 1/ on the bending moment M can be used. In the co-ordinate system (1/ ) - M the straight line starts from the origin. The second point is the limit point, which follows from the formulae (1) and (2). In (1) the limit bending moment inducing the stress in the reinforcement, equal to the plasticity limit, must be used. Then, for all cross-sections, where intermediate values of the bending moment M

appear the curvature can be calculated from the linear relationship

,M

M11

ydydm

(2.5)

where: M - the actual value of the bending moment, Myd - the bending moment corresponding to the attainment of fyd at the hinge considered, where fyd is design yield strength of reinforcement.

In the presented example the angle of rotation in the idealised hinge and the relative rotation of cross-sections at the distance a = 0.6h were calculated. The curvatures were determined using the presented methods. This allowed to compare the results quantitatively. Alternative approach to the limitation of plastic strains was proposed in [2] and [1] in the form of explicit constraints on curvatures. The allowable curvatures followed from the limit allowable values of strains in concrete and steel. The actual strains were computed using layered beam finite element with non-linear stress-strain relations for concrete and steel.

3. NUMERICAL EXAMPLE

Let us consider a two-span continuous beam shown in Fig. 2. The rotation capability of critical cross-section at the support B will be checked. The beam is made of the concrete C30, reinforced with ribbed steel bars, class A-II, 18G2A. The ductility of steel is high ( uk > 5%). The bending moments and shear forces

have been calculated for different coefficients elB

plB M/M of redistribution of


bending moment at the support B. They are illustrated in Fig. 2 and presented in Table 1. The diagram for = 1 corresponds to the linear elastic method (without redistribution). The internal forces for = 0.93 correspond to the ultimate, admissible redistribution according to [4] using the linear method with redistribution. Note that this means only 7% reduction of the support bending moment. Higher redistribution requires a verification of the plastic angle of rotation. Table 2 presents the results of calculation of the curvature 1/connected with the redistribution coefficient = 0.8.

Fig. 2. The beam, static scheme, bending moment diagrams for various values of the redistribution coefficient .

Three methods described in Chapter 2 were used: method a – according to the formulae (1) and (2), method b – according to the formulae (3) and (4), method c – linearized according to the formula (5).


Table 1. Internal forces for various values of the support bending moment reduction

MB VBL=VBP

x

where: 0)x(M

x

where: max

ABM)x(Mmax

ABM

[KN m] [kN] [m] [m] [kN m] 1.00 280.00 175.00 6.00 3.00 157.50 0.93 260.40 172.55 6.14 3.07 164.94 0.80 224.00 168.00 6.40 3.20 179.20 0.70 196.00 164.50 6.60 3.30 190.58

Table 2. Calculation of curvature 1/ according to various methods for the internal forces at = 0.80

ix )x(M iCurvature (1/ ) 103 m-1, method:

[m] [kN m] a b c

0.000 0.00 0.00 0.00 0.00 1.067 99.58 2.189 2.451 2.204 2.134 159.31 3.511 3.930 3.526 3.200 179.20 3.951 4.422 3.966 4.267 159.31 3.511 3.930 3.526 5.334 99.58 2.189 2.451 2.204 6.400 0.00 0.00 0.00 0.00 7.200 100.80 2.388 2.666 2.472 7.700 175.18 4.226 4.640 4.295 8.000 224.00 5.394 5.935 5.493

We can observe that the method b provides the highest values of curvatures, whereas the curvatures computed using methods a and c are similar. The relative angle of rotation in the plastic hinge was calculated for two values of the coefficient and the results are presented in Tables 3 and 4. Table 3 refers to the case of the ideal hinge, while Table 4 - to the case of the plastic hinge distributed at the length 1.2h. The second set of columns presents the admissible values of the plastic rotation pl,d according to EC2.

Table 3. Values of the angle of plastic rotation B with ideal plastic hinge assumed

pl,d Angle of rotation B, method: [m] [rad 10-3] a b c

0.70 8.657 9.442 10.610 9.332 0.80 7.628 6.016 6.802 5.862

Comparing the rotation angle in the ideal hinge (Table 3) with the rotation in the distributed hinge of the length 1.2 h (Table 4), we can observe a high increase of over 20%. The comparison of results presented in columns a, b and cdemonstrates the difference due to the application of various methods of


curvature calculation. Finally, comparing the actual angles of rotation pl,d

(Table 4, column 3) with the admissible values (Table 4, column 2) we observe that the constraint was not satisfied. It appears that the coefficient of redistribution of the bending moment must be increased to the value = 0.85 to satisfy the rotational capacity requirements.

Table 4. Values of the angle of plastic rotation B with the plastic hinge distributed at the length a = 1,2h.

pl,d Angle of rotation B, method: [m] [rad 10-3] a b c

0.70 8.657 11.857 13.282 11.810 0.80 7.628 8.332 9.384 8.254

The example demonstrates that the constraint on the rotation angle according to EC2 is very conservative. This may be explained by the fact that EC2 permits to ignore the effect of previous application of loading and does not require checking the shake down. The further research will definitely lead to less conservative but still safe method allowing for the plastic redistribution of stress.

4. FINAL REMARKS

The concept of the plastic hinge distributed on a certain length of the beam as the generalisation of the ideal plastic hinge has been discussed in the paper. It was shown on an example of a two-span beam that the relative angle of rotation of cross-sections spaced at the distance 1.2h in the distributed plastic hinge was 20% larger than the one calculated for the model of the ideal hinge. Three different methods of curvature calculation were used in the evaluation of the plastic rotation and the results were presented. The possibility of application of the static analysis accounting for the plastic redistribution of moments enriches the range of methods for a structural designer. However, we cannot expect too big advantages from these methods in the process of design of new structures. Nevertheless, they may be useful in assessment of the load capacity reserve in an existing structure. These methods may also be useful in the process of reinforcement of the existing structures, e.g. with the glued polymer bands. The reinforcement of the span is usually easier than the reinforcement of the support zone. Therefore, reinforcing of the span and allowing the plastic hinges to appear at intermediate supports may, in some cases, prove to be the best way of increasing the load bearing capacity of continuous beams.


BIBLIOGRAPHY

1. Czkwianinc A.: Furma czyk S., Analiza konstrukcji w: Komentarz naukowy do PN-B-03264:2002 Konstrukcje betonowe i spr one, ed. B. Lewicki, Warszawa, ITB 2003.

2. Czkwianianc A.: Kami ska M. Metoda nieliniowej analizy elbetowych elementów pr towych, Warszawa, PAN, KILiW, IPPT 1993.

3. D browski K.: Kilka uwag na temat metody plastycznego wyrównania momentów stosowanej w konstrukcjach elbetowych, In ynieria i Budownictwo 8 (2002).

4. Eurocode 2: Design of concrete structure, EN-2002, CEN European Committee for Standardization Brussels, April 2002.

5. Henriksen M.S.: Deformation capacity and cracks of reinforced concrete beams, Research Project S. Henriksen, R. Brincker. et all. Aalborg Univ. Conf. Proc. Summer Course on Mechanics of Concrete, Janowice-Kraków 1996.

6. J drzejczak M.: Knauff M., Redystrybucja momentów zginaj cych w elbetowych belkach ci g ych – zasady polskiej normy na tle Eurocodu.

In ynieria i Budownictwo 8 (2002).

A. Garstecki, J. ciga o

PROJEKTOWANIE ELBETOWYCH BELEK CI G YCHZ UWZGL DNIENIEM PLASTYCZNEJ REDYSTRYBUCJI MOMENTÓW

S t r e s z c z e n i e

W pracy podj to problem projektowania elbetowych belek z uwzgl dnieniem plastycznej redystrybucji momentów. Na wst pie omówiono klasyczne metody teorii plastyczno ci oraz metod liniow z plastyczn redystrybucj momentów. G ówn czpracy po wi cono metodom analizy statycznej uwzgl dniaj cym nieliniow zale nomoment-krzywizna. Przedstawiono dwie ró ne definicje przegubu plastycznego:idealnego zlokalizowanego w przekroju krytycznym oraz rozmytego, zlokalizowanego na pewnym ograniczonym obszarze. W przyk adzie numerycznym przestawiono praktyczne zastosowanie obu definicji przegubu plastycznego. Omówiono korzy ciwynikaj ce z wykorzystania plastycznej redystrybucji momentów przy projektowaniu konstrukcji.

Financed by university sources DS-11-557/2004 Received, 05.08.2005.

designing rc continuous beams

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