discrete crack analysis of rc structure...

11th World Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM V)

6th European Conference on Computational Fluid Dynamics (ECFD VI) E. Oñate, J. Oliver and A. Huerta (Eds)

DISCRETE CRACK ANALYSIS OF RC STRUCTURE USING HYBRID- TYPE PENALTY METHOD WITH DELAUNAY TRIANGULATION

ATSUSHI KAMBAYASHI 1, YOSHIHIRO FUJIWARA2,A, NORIO TAKEUCHI 2,B AND TADAHIKO SHIOMI 3

1 Research & Development Institute, Takenaka Corporation 1-5-1 Ohtsuka, Inzai, Chiba 270-1395, Japan e-mail: [email protected]

2 Graduate School of Engineering and Design, Hosei University

2-33 Ichigaya Tamachi, Shinjuku, Tokyo 162-0843, Japan A email: [email protected]

B email: [email protected]

3 Engineering Director, Mind Inc. 7-17-19 Maebara-nishi, Funabashi, Chiba 274-0825, Japan

email: [email protected] Key Words: Hybrid-type Penalty Method, Discrete Crack, Reinforced Concrete.

Abstract. The hybrid-type penalty method (HPM) is suitable for representing failure phenomena that occur during the transition from continua to discontinua in materials such as concrete. The initiation and propagation of dominant cracks and the branching of cracks can easily be modeled as discrete cracks. The HPM represents a discrete crack by eliminating the penalty that represents the separation of elements at the intersection boundary. This treatment is easy because no change is required in the degrees of freedom for the discrete crack. We present a numerical method of a constitutive model of concrete for application of the hybrid-type penalty method (HPM), and describe the simulation of an anchor bolt pullout test and deep beam test using fine meshes and Delaunay triangulation to validate of the method for evaluating progressive failure.

1 INTRODUCTION

Recent large earthquakes caused significant damage to concrete structures. Therefore, understanding the failure mechanism of concrete structures is important. A dominant crack is initiated in a concrete structure because of tensile stress. The crack subsequently grows, propagates, and branches until the structure finally collapses. To predict the progressive failure of a concrete structure, accurate computation of a discrete crack is essential. Computer simulations can predict the crack growth, propagation, and branching that lead to failure of the concrete structure.

The rigid body spring model (RBSM) developed by Kawai [1] is a good method for

Atsushi Kambayashi, Yoshihiro Fujiwara, Norio Takeuchi and Tadahiko Shiomi.

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modelling a discrete crack. The advantage of this method is in its simplicity; there is no need to track the crack path. Initially, the model obtained good results when solving the problem of the strong nonlinearity of steel. It was then applied to discrete limit analysis of soils and concrete structures [2-5]. Unfortunately, the elastic deformation in continua obtained by the RBSM is not accurate because it models a continuum that connects the spring elements between the edges of rigid bodies.

The hybrid-type penalty method (HPM) developed by Takeuchi et al. [6-8] refined the RBSM method for computing the elastic deformations of elements using linear or higher displacement field; a Lagrange multiplier was also introduced to satisfy the subsidiary condition of continuous displacement in the hybrid-type virtual work formulation. The HPM is suitable for analyzing the progressive failure of concrete structures; this method offers the following features: Accurate deformation can be calculated before crack initiation because an elastic element

(called a subdomain in the HPM) is used. Even after crack initiation, accurately calculating the deformation within the elastic area between cracks is important [9,10].

The HPM models a discrete crack by eliminating the penalty caused by separation of the elements at the intersection boundary. This treatment is easy because no change is required in the degrees of freedom for a discrete crack.

The concentrated stress at the crack tip can be calculated without the use of the J-integral, which was originally developed by Rice [11]. The HPM can easily and accurately obtain concentrated stresses around the crack by directly using the correct relationship between the tensile stress and displacement of the crack mouth opening. It can calculate not only the growth of the existing cracks but also the formation of new cracks.

In the present paper, we introduce the basic formulation of the HPM and describe an

implemented constitutive model of reinforced concrete. We validated the accuracy of the constitutive model by simulating an anchor bolt pullout test and a deep beam test.

2 THEORY OF HPM

2.1 Governing equation The basic equations of the elastic problem are as follows:

, (1) , (2)

, (3)

where is the Cauchy stress tensor; is the body force per unit volume; is the infinitesimal strain tensor; is the constitutive tensor; is the differential operator; is the symmetric part of ; and is the displacement field in , where is the reference configuration of the continuum body with a smooth boundary . Here, is the geometric boundary, and is the stress boundary. At the boundaries, the following conditions are satisfied:

, (4)


3

, (5)

where is the traction and is the field normal to the boundary . Let consist of M subdomains with the closed boundary , as shown in

Figure 1, that is,

where . (6)

Figure 1: Subdomain and its boundary . Figure 2: Common boundary between subdomains and .

We denote as the common boundary between two subdomains and , which are adjoined, as shown in Figure 2; is defined as follows:

. (7)

The relation for the displacement on , which is the intersection boundary between and , is as follows:

. (8)

Equation (8) introduces a subsidiary condition into the framework of the virtual work equation with the Lagrange multiplier as follows:

, (9)

where represents the variation in . From Equations (1) and (9), the following hybrid-type virtual work equation is obtained :

., (10)

where N represents the number of common boundaries of the subdomain, and represents the virtual displacement. The superscripts and represent the subdomains and

, respectively, related to the common boundary . The physical interpretation of the Lagrange multiplier is that of a surface force at the

boundary . In this paper, the Lagrange multiplier on the boundary is defined


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as follows: . (11)

where represents the relative displacement on the boundary , and is the penalty function.

2.2 Discretized equation in matrix form The independent linear displacement field in each subdomain is assumed to be as

follows:

. (12)

where is the rigid displacement and rigid rotation at point , and is a constant strain in the subdomain .

In the case of a two-dimensional problem, the coefficients in Equation (12) are as follows:

, ,

, (13)

.

where and represent rigid displacements at point in a subdomain; represents rigid rotation; and , , and represent the constant strains in the subdomain.

We obtain the following discretized equation: , (14)

where

, . (15)

The discretized equation of the HPM is thus transformed into the simultaneous linear equation of Equation (14) [12]. The coefficient matrix of on the left-hand side can be obtained to assemble each stiffness matrix of the subdomain and the subsidiary condition on the boundary . Discontinuous phenomena such as opening can be expressed without changing the degrees of freedom by setting the right-hand side of Equation (11) to zero.

3 IMPLEMENTATION OF CONSTITUTIVE MODEL OF REINFORCED CONCRETE

3.1 Constitutive law for compressive stress of concrete material A typical compressive stress–strain relationship for concrete is shown by the dashed line in

Figure 3. The stiffness gradually degrades with increasing stress up to the compressive strength fc. After the stress exceeds fc, softening occurs. The solid line in Figure 3 represents a


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trilinear approximation for the skeleton curve, which was also adopted in the RBSM by Takeuchi et al. [4].

Figure 3: Skeleton curve of compressive stress.

3.2 Constitutive law for tensile stress of concrete material The HPM can separate two subdomains by simply eliminating the penalty. This feature is

suitable for representing a discrete crack in concrete. When the surface force at boundary reaches tensile strength , the penalty can then be eliminated, and the discrete crack can

be computed as shown in Figure 4.

Figure 4: Surface force and tensile strength. Figure 5: Tension-softening curve for concrete.

The stress in the concrete gradually decreases as the crack opening displacement increases once the tensile failure is exceeded. This behavior is called tension softening. Hillerborg et al. [13] introduced the fracture energy for this tensile softening behavior in a fictitious (or cohesive) crack model. The fracture energy is the area enclosed by the tension-softening curve, and it has a unique value that represents the strength of a concrete material with tensile strength .

We used an empirical expression of the tension-softening curve proposed by Hordijk et al. (Equation (16))[14], which can be directly related to the fracture energy (Figure 5).

, (16)

where is the crack opening displacement (mm), and is the limit virtual crack opening displacement (mm) when the tensile stress is zero; this is given by

, (17)

where is the tensile strength (MPa) and is the fracture energy (N/mm).

Trilinear approximationExperiment

Reached

Crack: Normal stress: Crack opening

displacement

Eliminating the penaltyReleasedaccording to

: Tensile strength: Fracture energy


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Thus, stresses and displacements that occur after crack initiation can be calculated accurately because the fracture energy is determined directly.

3.3 Constitutive law for reinforcing bar The reinforcing bar was implemented using layered elements. Figure 6 shows a schematic

image of the layered element for reinforced concrete. The element consists of a concrete layer and arbitrary reinforced bar layers. The layer of a reinforced bar was modeled using a spring element identical to that used in the RBSM. The stiffness matrix for the penalty at the intersection boundary is obtained as follows:

, (18)

where is the penalty value, is the stiffness matrix of the concrete material, is the stiffness matrix of the i-th layer of the reinforced bar, and n is the number of reinforced layers. is obtained from the following relationship between traction and relative displacement at the intersection boundary:

, (19)

where and are normal stress and tangential stress, respectively, at the surface ; and are the relative displacements at the ; is the Young’s modulus of the steel; is the coefficient of the Dowel effect; is the length between two adjacent subdomains; and is the angle to the normal direction of the surface from the axial reinforcement steel (Figure 7).

A bilinear model was used to solve for the nonlinearity of the reinforcing bar.

First reinforced bar layer

Second reinforced bar layer

Concrete layer

Reinforced bar

Figure 6: Modeling of reinforced concrete. Figure 7: Direction of the steel.

4 NUMERICAL EXAMPLES To verify the validity of the HPM with a newly implemented reinforced concrete

constitutive model, we simulate an anchor bolt pullout test and a deep beam test which show typical progressive failure.

4.1 Simulation of anchor bolt pullout test The experimental model is shown in Figure 8. Two specimens were tested (Models 1 and

2). The details of the experiment are available in a publication of the Japan Concrete Institute

a

bReinforcement steel


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(JCI) [15]. Analytical models are shown in Figures 9 and 10. Figure 9 shows the half-size model used

to simulate symmetrical deformation. Figure 10 shows the full model used to simulate asymmetrical deformation.

The material properties of the concrete and anchor bolt are listed in Table 1. The values are based on the JCI report [15].

a b d l Model 1 300 100 150 900 Model 2 60 80 60 350

(unit: mm)

Figure 8: Experimental model.

Figure 9: Symmetrical analytical model.

Figure 10: Asymmetrical analytical model.

Figure 11 shows the relationship between the normalized force F/ and the crack displacement in Model 1, where F is the applied load, , and is the compressive strength (MPa). The thick blue line represents the experimental results and the red line represents the numerical results. The numerical results are in good agreement with the experimental results.

Figure 12 shows the transition of the contour of the maximum principal stress σ1 in Model 1. Panels (a)−(c) correspond to the respective yellow circle markers in Figure 11. The numerical results are in good agreement with the experimental results before the crack (a), after crack initiation (b), and during the propagation of the dominant crack (c).

Table 1. Material properties.

Concrete Anchor bolt Model 1 Model 2

Compressive strength fc (MPa) 32.6 34.3 － Tensile strength ft (MPa) 2.8 3.4 －

Young’s modulus E (GPa) 24.7 29.4 205.9 Poisson’s ratio ν 0.167 0.167 0.3

Fracture energy Gf (N/mm) 0.06 0.08 －

Figure 11: Relationship between normalized load and vertical displacement.

d

a a

b

l

l

F

3d 10

d 10

d 10

Pull-up force Anchor bolt

Concrete block

Lubricant coating

Two attachedsupport bars

00.20.40.60.8

11.21.41.61.8

22.2

0 0.1 0.2 0.3 0.4

F/bd

f t

Displacement (mm)

(c)

(a)

(b)

Analytical results Experimental results


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σ1 (MPa) 2.8 1.9 0.9 0.0

(a) F/ = 0.303 (b) F/ = 0.748 (c) F/ = 1.154

Figure 12: Deformation with contour of maximum principal stress σ1 (Model 1). Figures 13 and 15 respectively show the symmetrical and asymmetrical propagations of the

dominant crack in Model 2 with experiments. Figures 14 and 16 show the deformations with the contour of the maximum principal stress

σ1. Generally it is difficult to simulate branching of the cracks such as Figure 13, but the crack patterns are in good agreement with the experimental results. The propagation of the progressive crack was well simulated. The red part of the σ1 contour corresponds to the tensile strength in the area where the crack eventually appeared. The tensile stress concentration is related to fracture energy Gf. In the HPM, the crack displacement can be used directly for the tension softening curve which is corresponding to the fracture energy Gf. Therefore, we can simulate accurate crack propagation and stress concentration.

σ1 (MPa) 3.4 2.3 1.1 0.0

Figure 13: Symmetrical crack pattern with experiment (Model 2).

Figure 14: Deformation with contour of σ1 (Model 2 symmetry).

σ1 (MPa)

3.4 2.3 1.1 0.0

Figure 15: Asymmetrical crack pattern with experiment

(Model 2).

Figure 16: Deformation with contour of σ1 (Model 2 asymmetry).

Anchor bolt Support points

start


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4.2 Simulation of deep beam test The experimental model is shown in Figure 17. The deep beam was placed on steel plates,

and a vertical load was applied to the loading plates. The specimen was reinforced with six bars. Details on this experiment are available in the publication of the JCI [15].

Figure 17: Schematic of test model. Figure 18: Simulation model.

The simulation model is shown in Figure 18. The supported point is indicated by a triangular marker. Only the vertical direction was fixed because the concrete block could be rotated during the test. A static load was applied to the top of the plate.

The material properties of the concrete, reinforced layer and steel plates are listed in Table 2. These values were set according to the JCI report [15].

Table 2: Material properties.

(a) Concrete Parameter Value

Compressive strength fc (MPa) 54.4 Tensile strength ft (MPa) 3.3 Young’s modulus E (GPa) 33.3 Poisson’s ratio ν 0.167 Thickness (mm) 100.0 Fracture energy Gf (N/mm) 0.13

(b) Reinforcement Parameter Value

Young’s modulus E (GPa) 210.0 Thickness (mm) 29.79 Angle to the horizontal (°) 0.0 Tensile strength ft (MPa) 375.3 Coefficient of Dowel effect β 0.005

(c) Steel plate

Parameter Value Young’s modulus E (GPa) 210.0 Thickness (mm) 100.0 Poisson’s ratio ν 0.3

Figure 19 shows the relationship between the shear force and vertical displacement. The

dashed line represents the experimental results, and the solid line represents the numerical results. The numerical results matched well with the experimental results. In the numerical results, a large vertical displacement occurred after the concrete compressive stress under the loading plates reached the compressive strength.

Figure 20 shows the numerical deformation (amplification factor 40.0) with a contour of the minimum principal stress . Multiple progressive cracks propagated from the bottom support steel plate toward the upper loading steel plate in the concrete.

160460900

320

400

100

30

100

ForceCL

(mm)

Reinforcement


10

Figure 19: Relationship between shear force and vertical displacement.

σ3 -544.0

0.0

MPa

Figure 20: Deformation and contour of minimum principal stress .

5 CONCLUSIONS We proposed the numerical method to simulate progressive concrete failure and verified

the validity of the method by using experimental test results. The presented method is an extension of the HPM, in which a constitutive model (discrete crack model) for reinforced concrete is implemented. The discrete crack was evaluated at the intersection boundary between subdomains of the HPM. Simulating the tensile stress behavior and crack displacement is easy; these were directly related to the fracture energy through empirical expressions that were introduced by Hordijk et al. [14]. The nonlinearity of the compressive behavior is considered in the compressive stress–strain components of the subdomain of the HPM on the basis of the trilinear approximation function of the empirical stress–strain relationship.

Reinforcing bars were implemented using a layered spring element that is similar to that used in the RBSM. A bilinear model was used to solve for the nonlinearity of the reinforcing bar.

To confirm the validity of the new HPM, we carried out a simulation of an anchor bolt pullout test and a deep beam test. The qualitative and quantitative agreement between the simulation and experimental results proved the validity of the proposed numerical method.

050

100150200250300350400450500

0 0.2 0.4 0.6 0.8 1

Shea

r for

ce (k

N)

Displacement (mm)

Experimental resultsNumerical results


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ACKNOWLEDGMENTS The experimental results in Figures 11 and 19 were published in the Committee Report of

Japan Concrete Institute (JCI) [15]. We extend special thanks to Dr. Ueda for sharing his knowledge of discrete limit analysis

and the constitutive model of reinforced concrete.

REFERENCES [1] Kawai, T. [1977], “New element models in discrete structural analysis,” Journal of the

Society of Naval Architects of Japan, 141, pp. 187–193. [2] Kawai, T. and Takeuchi, N. [1981], “New discrete models and their application to rock

mechanics“, Proc. of the International Symposium on Weak Rock, 21-24 September, pp.725-730

[3] Ueda, M., Takeuchi, N., Higuchi, H. and Kawai, T. [1984]. “A discrete limit analysis of reinforced concrete structures”, Proc. of the International Conference on Computer Aided Analysis and Design of Concrete Structures, pp.1369-1384

[4] Takeuchi, N., Ueda, M., Kambayashi, A. and Kito, H. [2005], Discrete limit analysis of reinforced concrete structure (Maruzen, Tokyo (in Japanese)).

[5] Gedik, Y. H., Nakamura, H., Yamamoto, Y. and Kunieda, M. [2011], “Evaluation of three-dimensional effects in short deep beams using a rigid-body-spring-model,” Cement and Concrete Composites, 33(9–10), pp. 978–91.

[6] Takeuchi, N. [2007], " Elasto-plastic analysis in soil and rock mechanics by using hybrid-type penalty method, " Proc. of International Conference dedicated to the 95th Anniversary of Academician Nagush Kh. Arutyunyan,, Institute of Mechanics National Academy of Sciences of Armenia, Yerevan-2007, pp.492-496.

[7] Takeuchi, N., Tajiri, Y. and Hamasaki, E. [2009], “Development of modified RBSM for rock mechanics using principle of hybrid-type virtual work,” Analysis of Discontinuous Deformation: New Developments and Applications, (Ma, G. and Zhou, Y.). pp. 395–403, Research Publishing Service, Singapore.

[8] Vardanyan, A. and Takeuchi, N. [2008], " Discrete analysis for plate bending problems by using hybrid-type penalty method," Bulletin of Research Center for Computing and Multimedia Studies, Hosei University, 21, pp. 131-141 (URL: http://hdl.handle.net/10114/1532).

[9] Ohki, H. and Takeuchi, N. [2005], “Upper and lower bound solution with hybrid-type penalty method,” Proc. of 7th International Conference on the Analysis of Discontinuous Deformation, December 10-12,2005, Honolulu, Hawaii, pp.199-210.

[10] Y. Fujiwara, N. Takeuchi, T. Shiomi and A. Kambayashi: Discrete crack modeling of RC structure using hybrid-type penalty method, International Journal of Aerospace and Lightweight Structures, Vol. 3 (2), pp. 263–274, 2013.

[11] Rice, J. R. [1968], “Path independent integral and the approximate analysis of strain consideration by notches and cracks,” Journal of Applied Mechanics, 35, pp. 379–386.

[12] Mihara, R. and Takeuchi, N. [2007], "Nonlinear analysis of Riedel shearing test by using mesh dividing method in HPM," Proc. of the 8th International Conference on Analysis of Discontinuous Deformation : Fundamentals & Applications to Mining & Civil


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Engineering, , pp.201-206. [13] Hillerborg, A., Modéer, M. and Petersson, P.-E., [1976], “Analysis of crack formation

and crack growth in concrete by means of fracture mechanics and finite elements,” Cement and Concrete Research, 6, pp. 773–782.

[14] Cornelissen, H. A. W., Hordijk, D. A. and Reinhardt, H. W. [1986], “Experiments and theory for the application of fracture mechanics to normal and lightweight concrete,” International Conference on Fracture Mechanics of Concrete. (Edited by Wittmann, F. H.), Elsevier, Amsterdam.

[15] JCI [1993] Japan Concrete Institute, Application of fracture mechanics to concrete structures, Japan.

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