Modelling of reinforced concrete structures and composite structures with concrete strength degradation taken into consideration

ARCHIVES OF CIVIL AND MECHANICAL ENGINEERING Vol. XI

2011

No. 3

Modelling of reinforced concrete structures and composite structures with concrete strength degradation taken into consideration P. KMIECIK, M. KAMISKI Wrocaw University of Technology, Wybrzee Wyspiaskiego 25, 50-370 Wrocaw, Poland.

Because of the properties of the material (concrete), computer simulations in the field of reinforced concrete structures are pose a challenge. As opposed to steel, concrete when subjected to compression exhibits nonlinearity right from the start. Moreover, it much quicker undergoes degradation under tension. All this poses difficulties for numerical analyses. Parameters needed to correctly model concrete under compound stress are described in this paper. The parameters are illustrated using the Concrete Damaged Plasticity model included in the ABAQUS software. Keywords: numerical modelling, concrete degradation, stress-strain relation, reinforced concrete structures, composite structures, Abaqus, concrete damaged plasticity

1. Introduction The two main concrete failure mechanisms are cracking under tension and crushing under compression. However, concrete strength determined in simple states of stress (uniaxial compression or tension) radically differs from the one determined in complex states of stress. For example, the same concrete under biaxial compression reaches strength of between ten and twenty per cent higher than in the uniaxial state while in the hydro-static state (uniform triaxial compression) its strength is theoretically unlimited. In order to describe strength with the equation for triaxial stress, its plane should be presented in a threedimensional stress space (since concrete is considered to be an isotropic material in a wide range of stress). The states of stress corresponding to material failure are on this surface while the states of safe behaviour are inside. Also the so-called plastic potential surface is located inside this space. After the plasticity surface is crossed, two situations arise [9]: x an increase in strain with no change in stress (ideal plasticity), x material weakening (rupture).

2. Strength hypothesis and its parameters One of the strength hypotheses most often applied to concrete is the Drucker– Prager hypothesis (1952). According to it, failure is determined by non-dilatational strain energy and the boundary surface itself in the stress space assumes the shape of a

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energy and the boundary surface itself in the stress space assumes the shape of a cone. The advantage of the use of this criterion is surface smoothness and thereby no complications in numerical applications. The drawback is that it is not fully consistent with the actual behaviour of concrete 0.

Fig. 1. Drucker–Prager boundary surface 0: a) view, b) deviatoric cross section

The CDP (Concrete Damaged Plasticity) model used in the ABAQUS software is a modification of the Drucker–Prager strength hypothesis. In recent years the latter has been further modified by Lubliner 0, Lee and Fenves 0. According to the modifications, the failure surface in the deviatoric cross section needs not to be a circle and it is governed by parameter K c.

Fig. 2. Deviatoric cross section of failure surface in CDP model 0

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Physically, parameter Kc is interpreted as a ratio of the distances between the hydrostatic axis and respectively the compression meridian and the tension meridian in the deviatoric cross section. This ratio is always higher than 0.5 and when it assumes the value of 1, the deviatoric cross section of the failure surface becomes a circle (as in the classic Drucker–Prager strength hypothesis). Majewski reports that according to experimental results this value for mean normal stress equal to zero amounts to 0.6 and slowly increases with decreasing mean stress. The CDP model recommends to assume Kc = 2/3. This shape is similar to the strength criterion (a combination of three mutually tangent ellipses) formulated by William and Warnke (1975). It is a theoretical-experimental criterion based on triaxial stress test results. Similarly, the shape of the plane’s meridians in the stress space changes. Experimental results indicate that the meridians are curves. In the CDP model the plastic potential surface in the meridional plane assumes the form of a hyperbola. The shape is adjusted through eccentricity (plastic potential eccentricity). It is a small positive value which expresses the rate of approach of the plastic potential hyperbola to its asymptote. In other words, it is the length (measured along the hydrostatic axis) of the segment between the vertex of the hyperbola and the intersection of the asymptotes of this hyperbola (the centre of the hyperbola). Parameter eccentricity can be calculated as a ratio of tensile strength to compressive strength [4]. The CDP model recommends to assume  = 0.1. When  = 0, the surface in the meridional plane becomes a straight line (the classic Drucker-Prager hypothesis).

Fig. 3. Hyperbolic surface of plastic potential in meridional plane 0

Another parameter describing the state of the material is the point in which the concrete undergoes failure under biaxial compression. b0/c0 ( fb0 / fc0) is a ratio of the strength in the biaxial state to the strength in the uniaxial state. The most reliable in this regard are the experimental results reported by Kupler (1969). After their approximation with the elliptic equation, uniform biaxial compression strength fcc is equal to 1.16248 fc 0. The ABAQUS user’s manual specifies default b0/c0 = 1.16. The last parameter characterizing the performance of concrete under compound stress is dilation angle, i.e. the angle of inclination of the failure surface towards the

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hydrostatic axis, measured in the meridional plane. Physically, dilation angle  is interpreted as a concrete internal friction angle. In simulations usually  = 36° 0,0 or  = 40° 0 is assumed.

Fig. 4. Strength of concrete under biaxial stress in CDP model 0 Table 1. Default parameters of CDP model under compound stress Parameter name Value Dilatation angle 36 Eccentricity 0.1 fbo /fco 1.16  0.667 Viscosity parameter 0

The unquestionable advantage of the CDP model is the fact that it is based on parameters having an explicit physical interpretation. The exact role of the above parameters and the mathematical methods used to describe the development of the boundary surface in the three-dimensional space of stresses are explained in the ABAQUS user’s manual. The other parameters describing the performance of concrete are determined for uniaxial stress. Table 1 shows the model’s parameters characterizing its performance under compound stress.

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3. Stress-strain curve for uniaxial compression The stress-strain relation for a given concrete can be most accurately described on the basis of uniaxial compression tests carried out on it. Having obtained a graph from laboratory tests one should transform the variables. Inelastic strains H~cin are used in the CDP model. In order to determine them one should deduct the elastic part (corresponding to the undamaged material) from the total strains registered in the uniaxial compression test:

H~cin H 0elc

H c  H 0elc , Vc E0

(1)

.

(2)

Fig. 5. Definition of inelastic strains 0

When transforming strains, one should consider from what moment the material should be defined as nonlinearly elastic. Although uniaxial tests show that such behaviour occurs almost from the beginning of the compression process, for most numerical analyses it can be neglected in the initial stage. According to Majewski, a linear elasticity limit should increase with concrete strength and it should be rather assumed than experimentally determined. He calculated it as a percentage of stress to concrete strength from this formula: § f · elim 1  exp¨ c ¸. © 80 ¹

(3)

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This ceiling can be simply arbitrarily assumed as 0.4 fcm. Eurocode 2 specifies the modulus of elasticity for concrete to be secant in a range of 0–0.4 fcm. Since the basic definition of the material already covers the shear modulus and the longitudinal modulus of concrete, at this stage it is good to assume such an inelastic phase threshold that the initial value of Young’s modulus and the secant value determined according to the standard will be convergent. In most numerical analyses it is rather not the initial behaviour of the material, but the stage in which it reaches its yield strength which is investigated. Thanks to the level of 0.4·fcm there are fewer problems with solution convergence. Having defined the yield stress-inelastic strain pair of variables, one needs to define now degradation variable dc. It ranges from zero for an undamaged material to one for the total loss of load-bearing capacity. These values can also be obtained from uniaxial compression tests, by calculating the ratio of the stress for the declining part of the curve to the compressive strength of the concrete. Thanks to the above definition the CDP model allows one to calculate plastic strain from the formula:

H~cpl

H~cin 

dc V c , 1  d c E0

(4)

where E0 stands for the initial modulus of elasticity for the undamaged material. Knowing the plastic strain and having determined the flow and failure surface area one can calculate stress c for uniaxial compression and its effective stress V c .

Vc Vc

1  d c E0 H c  H~cpl , Vc

1  d c



(5)



E0 H c  H~cpl .

(6)

3.1. Plotting stress-strain curve without detailed laboratory test results

On the basis of uniaxial compression test results one can accurately determine the way in which the material behaved. However, a problem arises when the person running such a numerical simulation has no such test results or when the analysis is performed for a new structure. Then often the only available quantity is the average compressive strength ( fcm) of the concrete. Another quantity which must be known in order to begin an analysis of the stress-strain curve is the longitudinal modulus of elasticity (Ecm) of the concrete. Its value can be calculated using the relations available in the literature 0:

Ecm

220.1 f cm , 0.3

(7)

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where: fcm [MPa], Ecm [GPa]. Other values defining the location of characteristic points on the graph are strain c1 at average compressive strength and ultimate strain cu 0:

H c1 0.7 f cm 0.31 ,

(8)

cu = 3.5 ‰.

(9)

The formulas (8–9) are applicable to concretes of grade C50/60 at the most. On the basis of experimental results Majewski proposed the following (quite accurate) approximating formulas:

H c1

0.0014 >2  exp  0.024 f cm  exp  0.140 f cm @,

(10)

H cu

0.004  0.0011>1  exp  0.0215 f cm @.

(11)

Knowing the values of the above one can determine the points which the graph should intersect.

Fig. 6. Stress-strain diagram for analysis of structures, according to Eurocode 2

The curve can be also plotted on the basis of the literature 0, 0, 0,0,0. The most popular formulas are presented in Table 2, but the original symbols have been replaced with the uniform denotations used in Eurocode 2.

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Choosing a proper formula form to describe relation c – c one should note whether the longitudinal modulus of elasticity represents initial value Ec (at stress c = 0) or that of secant modulus Ecm. Most of the formulas use initial modulus Ec which is neither experimentally determined nor taken from the standards. Another important factor is the functional dependence itself. Even though the Madrid parabola has been recognized as a good relation by CEB (Comité Euro-International du Béton), this function is not flexible enough to correctly describe the performance of concrete. Table 2. Stress-strain relation for nonlinear behaviour of structure Formula name/ source

Formula form ª 1 § H ·º EcH c «1  ¨¨ c ¸¸ » ¬« 2 © H c1 ¹ ¼»

Vc

Madrid parabola

Vc

§H · 1  ¨¨ c ¸¸ © H c1 ¹

f cm

EN 1992-1-1

k 1.05Ecm

Vc

Vc

f Ec , H c1

Vc

f Ec , H c1

EcH c

Vc

Desay & Krishnan formula

Variables

2

kK  K 2 1  k  2 K

H c1 f cm

Hc H c1

, K

Vc

f Ecm , f cm , H c1

Vc

f Ec , f cm , H c1

EcH c if V c d elim f cm

½ elim  2 2 §¨ H c ·¸  ° ¨ ¸ 4elim  1 © H c1 ¹ ° ¾ 2 2  ° elim  2 § H c · elim ¨ ¸  f cm  f cm ¨ ¸ 2elim  1 © H c1 ¹ 4elim  1 °¿ 2

Vc Majewski formula

f cm

Ec

f cm

Hc

2  elim ,

if V c ! elim f cm elim in formula (3)

Wang & Hsu formula

Vc

Sáenz formula

ª § H · § H ·2 º Hc c ¸ ¨  ¨ c ¸¸ » if ]H d 1 ¸ c1 «¬ © ]H c1 ¹ © ]H c1 ¹ »¼ Hc ª § H / ]H  1 · 2 º c1 ¸¸ » if ]H ! 1 ] f cm «1  ¨¨ c c1 2 / ] 1 ¹ » ¬« © ¼

V c ] f cm «2¨¨

Vc

Vc

Hc

A  BH c  CH c  DH c symbols in formula (12) 2

3

Vc

f  f cm , H c1

§ Ec , f cm , f cu , · ¸¸ f ¨¨ © H c1 , H cu1 ¹

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Fig. 7. Property of 2nd order parabola

The 2nd order parabola has this property that the tangent of the angle of a tangent passing through a point on its branch, measured relative to the horizontal axis passing through this point, is always double that of the angle measured as the inclination of the secant passing through the same point and the extremum of the parabola, relative to the same horizontal axis.

Fig. 8. Relation c– c for Madrid parabola depending on longitudinal modulus of elasticity

The consequence of this property of the parabola is either the exceedance of the concrete’s strength for a correct initial modulus value or the necessity to lower the value in order to reach a specific stress value in the extreme. Figure 8 shows relation c – c for the Madrid parabola for grade C16/20 concrete. The following batch denotations were assumed: x Ecm – Ec = Ecm = 28608 MPa was assumed as the initial modulus, calculated extremum fcm = 26.81 MPa; x Ec/Ecu = 2 – the doubled tangent of the angle of the secant passing through point (c1, fcm), amounting to Ec = 25602 MPa, calculated extremum fcm = 24.00 MPa (correct);

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x 0.4 fcm – the value of initial modulus Ec = 31808 MPa matched so that the curve intersects point (c, 0.4·fcm), calculated extremum fcm = 29.81 MPa; x Ec = Ecm – a straight line describing the elastic behaviour of the concrete up to (c, 0.4·fcm). As one can see, when initial modulus Ec is assumed to amount to Ecm, the strength of the concrete is much overrated despite the fact that the initial modulus is still underrated (numerically Ecm is not the highest value). In the case of parabolic relations one should artificially lower modulus Ec in order for the graph to intersect the correct value fcm. A sufficiently detailed description of relation c – c has been proposed by Sáenz. The function with a 3rd order polynomial in the denominator (Table 2) depends on the variables:

A C P1 P3

P3  P4  2 P3 f cm

½ ° ° P4  1 ° 2 P4  1 , D  P3 f cmH c1 P3 f cmH c1 °° ¾. f cm H cu ° , P2 ° f cu H c1 ° EcH c1 P3 P2  1 1 ° , P4  f cm P1  1 2 P1 °¿ 1 , B Ec

Fig. 9. Comparison of curves c-c based on table 2 relations for grade C16/20 concrete

(12)

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The above notation allows one to shape the function graph so that it intersects points: (c1, fcm) and (cu, fcu). The relation proposed by Wang and Hsu is an interesting notation. These are two functions describing the curve’s ascending and descending part. They also include coefficient  representing the reduction in compressive stress of concrete resulting from locating reinforcing bars in the compressed zone. In figure 9  = 1.0 (no reinforcement taken into account). It is worth noticing that Wang and Hsu relation, the Majewski relation and the Madrid parabola almost coincide. The same applies to the Desay and Krishanan relation and the Sáenz relation, but in the latter case the same point (cu, fcu) which followed from the Desay and Krishanan formula was assumed since a lower value of function fcu would result in an improper shape of the curve. The standard relation yields intermediate results.

4. Stress-strain curve for uniaxial tension The tensile strength of concrete under uniaxial stress is seldom determined through a direct tension test because of the difficulties involved in its execution and the large scatter of the results. Indirect methods, such as sample splitting or beam bending, tend to be used [2]: f ctm

0.30 f ck2 / 3 .

(13)

Fig. 10. Definition of strain after cracking – tension stiffening 0

The term cracking strain H~tck is used in CDP model numerical analyses. The aim is to take into account the phenomenon called tension stiffening. Concrete under tension is not regarded as a brittle-elastic body and such phenomena as aggregate interlocking

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in a crack and concrete-to-steel adhesion between cracks are taken into account. This assumption is valid when the pattern of cracks is fuzzy. Then stress in the tensioned zone does not decrease sharply but gradually. The strain after cracking is defined as the difference between the total strain and the elastic strain for the undamaged material:

H~tck H 0elt

H t  H 0elt ,

(14)

Ht

(15)

Ec

.

Plastic strain H~t pl is calculated similarly as in the case of compression after defining degradation parameter dt. In order to plot curve t – t one should define the form of the weakening function. According to the ABAQUS user’s manual, stress can be linearly reduced to zero, starting from the moment of reaching the tensile strength for the total strain ten times higher than at the moment of reaching fctm. But to accurately describe this function the model needs to be calibrated with the results predicted for a specific analyzed case.

Fig. 11. Modified Wang & Hsu formula for weakening function at tension stiffening for concrete C16/20

The proper relation was proposed by, among others, Wang and Hsu [11]:

Vt Vt

EcH t if H t d H cr §H f cm ¨¨ cr © Ht

· ¸¸ ¹

0.4

½ ° ¾, if H t ! H cr ° ¿

(16)

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where cr stands for strain at concrete cracking. Since tension stiffening may considerably affect the results of the analysis and the relation needs calibrating for a given simulation, it is proposed to use the modified Wang & Hsu formula for the weakening function:

Vt

§H f cm ¨¨ cr © Ht

· ¸¸ ¹

n

if H t ! H cr ,

(17)

where n represents the rate of weakening.

5. Conclusion Problems with solution convergence may arise when full nonlinearity of the material (concrete) with its gradual degradation under increasing (mainly tensile) stress is assumed. Simple FE techniques, consisting in reducing the size of stress increment or increasing the maximum number of steps when solving the problem by means of the Newton-Raphson approach, may prove to be insufficient. Therefore the CDP model uses viscosity parameter  which allows one to slightly exceed the plastic potential surface area in certain sufficiently small problem steps (in order to regularize the constitutive equations). Viscoplastic adjustment consists in choosing such  > 0 that the ratio of the problem time step to  approaches infinity. This means that it is necessary to try to match the value of  a few times in order to find out how big an influence it has on the problem solution result in ABAQUS/Standard and to choose a proper minimum value of this parameter. The CDP model makes it possible to define concrete for all kinds of structures. It is mainly intended for the analysis of reinforced concrete structures and concrete-concrete and steel-concrete composite structures. However, it is recommended that before an analysis of the structure one should test the behaviour of the material itself, e.g. by carrying out a numerical analysis of cylindrical samples under compression, in order to compare it with the given stress-strain relation. Because of the character of concrete failure, some quantities can be rather assumed than determined in laboratory tests. Therefore the assumptions should be verified by comparing other parameters, e.g. the deflection of the modelled structural component. This means that the model parameters often need to be calibrated several times in the course of the numerical analysis. Wrocaw Centre for Networking and Supercomputing holds a licence for the Abaqus software (http://www.wcss.wroc.pl), grant No. 56. References [1] ABAQUS: Abaqus analysis user's manual, Version 6.9, 2009, Dassault Systèmes. [2] Eurocode 2: Design of concrete structures. Part 1-1: general rules and rules for buildings, Brussels, 2004.

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[3] Godycki-wirko T.: The mechanics of concrete, Arkady, Warsaw, 1982. [4] Jankowiak I., Kkol W., Madaj A.: Identification of a continuous composite beam numerical model, based on experimental tests, 7th Conference on Composite Structures, Zielona Góra, 2005, pp. 163–178. [5] Jankowiak I., Madaj A.: Numerical modelling of the composite concrete-steel beam interlayer bond, 8th Conference on Composite Structures, Zielona Góra, 2008, pp. 131–148. [6] Kmita A., Kubiak J.: Investigation of concrete structures. Guide to laboratory classes, Wrocaw University of Technology Publishing House, Wrocaw, 1993. [7] Lee J., Fenves G.L.: Plastic-damage model for cyclic loading of concrete structures, Journal of Engineering Mechanics, Vol. 124, No. 8, 1998, pp. 892–900. [8] Lubliner J., Oliver J., Oller S, Oñate E.: A plastic-damage model for concrete, International Journal of Solids and Structures, Vol. 25, 1989, pp. 299–329. [9] Majewski S.: The mechanics of structural concrete in terms of elasto-plasticity, Silesian Polytechnic Publishing House, Gliwice, 2003. [10] Szumigaa A.: Composite steel-concrete beam and frame structures under momentary load, Dissertation No. 408, Pozna Polytechnic Publishing House, Pozna, 2007. [11] Wang T., Hsu T.T.C.: Nonlinear finite element analysis of concrete structures using new constitutive models, Computers and Structures, Vol. 79, Iss. 32, 2001, pp. 2781–2791. Modelowanie konstrukcji elbetowych oraz zespolonych z uwzgldnieniem degradacji wytrzymaociowej betonu Symulacje komputerowe w dziedzinie konstrukcji elbetowych s wyzwaniem z uwagi na waciwoci materiau, jakim jest beton. W przeciwiestwie do stali, jest to materia, który podczas ciskania wykazuje nieliniowo ju od samego pocztku swojej pracy. Ponadto podczas rozcigania ulega znacznie szybszej degradacji, co skutkuje problemami natury numerycznej. W niniejszej pracy opisano parametry niezb dne do prawidowego zamodelowania betonu w zoonym stanie napr enia. Parametry te przedstawiono na przykadzie modelu „Concrete Damaged Plasticity” zawartego w programie ABAQUS.