- Email: [email protected]
Peridynamic modelling of reinforced concrete structures
Engineering Failure Analysis 103 (2019) 266–274
Contents lists available at ScienceDirect
Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Peridynamic modelling of reinforced concrete structures ⁎
Nicolas Sau , Jose Medina-Mendoza, Ana C. Borbon-Almada
T
Civil Engineering and Mines Department, University of Sonora, Mexico
A R T IC LE I N F O
ABS TRA CT
Keywords: Fracture Cracks Stress analysis
Reinforced concrete structures subjected to loads that cause the formation and propagation of cracks exhibit a behavior where continuum mechanics is no longer applicable, because most conventional models do not consider the discontinuities that appear and propagate because of concrete tension failure. In an effort to correct the shortcomings of continuum models, the peridynamic bond-based model was proposed in 2000 and the peridynamic state-based model in 2007. The micropolar peridynamic model improves the bond-based peridynamic model by allowing different values of the Poisson's ratio. A numerical scheme for the calculation of the micropolar peridynamic stress was developed where results show convergence with classical linear elastic solutions for small values of the material horizon. With this approach, the stress tensor can be analyzed at points in failure zones. Fractures patterns using physical experimentation are obtained and compared with the model.
1. Introduction Quasibrittle structures that are subjected to loads that cause the formation and propagation of cracks exhibit highly nonlinear behavior, where models based on continuum mechanics are no longer applicable. With the development of the finite element method, models based upon. Fracture Mechanics concepts were formulated, where the main disadvantage of these models is that they are impractical, since multiple cracks need to be represented in the material. The use of smeared crack models using continuum damage theories appeared to be a more suitable technique to model damage in concrete structures. Nevertheless, spurious mesh sensitivity was one of the many problems associated with the smeared crack approach [1]. In an effort to correct the shortcomings of continuum models, the peridynamic model was proposed in 2000 by Silling. Governing equations in the peridynamic model do not assume spatial differentiability of the displacement field and permit discontinuities to arise as part of the solution [2]. In addition, the peridynamic theory is non-local, where locality is recovered as a special case [3]. The micropolar peridynamic model, which is a bond-based model, improves the peridynamic model proposed by Silling in 2000, by allowing moment densities in addition to force densities to interact among particles inside the material horizon [4]. The micropolar peridynamic model is able to model materials with Poisson's ratios different from 1/4 in three-dimensions and 1/3 for twodimensional plane stress problems; in addition, particles are capable of having rotational as well as translational degrees of freedom [4]. Strength capacity of reinforced concrete beams has not been easy to predict with the use of non-empirical methods, since most conventional models do not consider the discontinuities that appear and propagate as a result of concrete tension failure. With this
⁎
Corresponding author. E-mail address: [email protected] (N. Sau).
https://doi.org/10.1016/j.engfailanal.2019.05.004 Received 22 November 2018; Received in revised form 11 April 2019; Accepted 2 May 2019 Available online 10 May 2019 1350-6307/ © 2019 Elsevier Ltd. All rights reserved.
Engineering Failure Analysis 103 (2019) 266–274
N. Sau, et al.
approach, the stress tensor is analyzed at different points in failure zones. Crack patterns are obtained with the model and compared with four point-loaded plain and reinforced concrete beam tests. 2. Peridynamic model The peridynamic model postulates a vector function in units of force per unit volume squared fi j that represents the interaction between particles i and j inside a material horizon δ [2]. The sum of all internal forces per unit volume acting on a particle i is expressed as:
∫V
fij dVj + bi = ρi
J
∂ 2u i , ∂t 2
(1) ∂2ui ∂t 2
and ρi are the acceleration and density of particle i respectively. This integral is in which bi is the body force acting on particle i; performed on all particles j within δ. In the micropolar peridynamic model a moment equation must be added [4]:
∫V
mij dVj + mi = Ii
J
∂2θi , ∂t 2
(2)
where the moment per unit volume squared is mij and miis a moment density function [4]. These moments and forces can be functions of the relative positions ξij, the relative displacements ηij, and the rotations of particles θi andθj, where forces fij and moments mij between particles can be expressed using the following matrix at the linear microelastic level as
⎡ f ⎤ ⎡ uî ⎤ ⎢ xi ⎥ − c /L 0 0 0 0 ⎤⎢ ⎥ ⎢ f ⎥ ⎡ c/L vi ̂ − 12d/ L3 6d/ L2 ⎥ ⎢ i ⎥ 12d/ L3 6d/ L2 0 ⎢ y⎥ ⎢ 0 i⎥ ⎢ ⎥ ⎢ ⎢m θ 2 2 z z⎥ − 6d/ L 0 6d/ L 4d/ L 0 2d/ L ⎥⎢ ⎥ ⎢ ⎥=⎢ j u − c /L c /L 0 0 0 0 ⎥ ⎢ ĵ ⎥ ⎢ f ⎥ ⎢ 3 − 6d / L2 3 ⎢ x⎥ ⎢ 0 − − d L d L d/ L2 ⎥ ⎢ vj ̂ ⎥ 12 / 0 12 / 6 j ⎥⎢ ⎥ ⎢ 2 2 fy⎥ ⎢ − d L d L d L d 0 6 / 2 / 0 6 / 4 /L ⎦ ⎢ j ⎥ ⎣ ⎢ ⎥ ⎣θ z ⎦ j ⎢m ⎥ ⎣ z⎦ i
′
(3)
The parameters c and d are given by c = E ′ A and d = E I′ respectively, where A is the cross-sectional area of the link between particles, E′ is the modulus of elasticity, I′ is the moment of inertia and L is its length (Fig. 1). The relative displacements in the x and the y direction are ηijx = u î − u ̂j and ηijy = v î − v ̂j respectively. Similarly, links in a tree-dimensional model have a circular cross-
Fig. 1. Peridynamic link in the 2-D linear microelastic model in local coordinates. 267
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sectional area with a moment of inertia I′, polar moment of inertia 2I′ and an area A′, all these links have a finite length with an infinite number of particles j, attached to each particle i [5]. If a pair of particles have different material properties, such as steel and concrete, then values for c and d are averaged for the link connecting these particles. Relationships between peridynamic constants (c, d) and conventional linear elastic constants (E, ν) can be obtained by calculating the strain energy density of all the peridynamic links attached to a particle inside a material horizon δ. These relationships are given by
c=
6E E (1 − 3ν ) ,d= , πδ 3t (1 − ν ) 6πδt (1 − ν 2)
(4)
for two-dimensional plane stress problems and
c=
6E E (1 − 4ν ) ,d= , πδ 4 (1 − 2ν ) 4πδ 2 (1 − 2ν )(1 + ν )
(5)
for three-dimensional models [5]. Using the definition given by Lehoucq [6], the peridynamic stress tensor in cylindrical coordinates is given by
σkl =
1 2
2π
δ
∫0 ∫0 ∫0
δ−y
(y + z ) fk ml dz dy dθ
(6)
where σkl is the stress tensor, fk is the pair wise force or moment function per unit volume squared,y, z are the distances between particles i and j, and ml is the unit vector in the direction of the normal of the plane where the stress is measured. For concrete structures a microelastic damage model was proposed, where the stiffness of a peridynamic link depends upon the axial stretch s of the link between particles i and j, and also upon the maximum stretch st, of all other peridynamic links connected to particle i. The link shown remains linearly elastic as long as its stretch s is smaller than a tensile limit, stens and larger than a compressive limit, scomp. If s exceeds the tensile limitstens, the tensile peridynamic force remains constant until s exceeds another specific limit, αtensstens, after which the force goes down to zero. On the other hand if s is less than a compressive limit,scomp, there are two possibilities. If the maximum transverse stretch sc is smaller than scomp, the link remains linear elastic, while if the stretch sc is greater than scomp, the compressive force remains constant until s exceeds another limit, αcompscomp, after which it plunges to zero. The plateau in this model is able to simulate dissipation of energy due to aggregate contact and friction. In addition, cement paste crushing is simulated with the limitαcompscomp. Thus, this model has a reasonable physical explanation. The total number of parameters of this micropolar peridynamic damage model is seven c, d, δ, stens, scomp, αtens, and αcomp (Fig. 2). 3. Application to concrete structures Using the Micropolar Peridynamic Model a computer model was developed and applied to Concrete Structures, two types of concrete structures were analyzed, simply supported reinforced concrete deep beams and four point loaded reinforced concrete beams. Results were compared with the American Concrete Institute (ACI) code and physical experimentation.
Fig. 2. Microelastic damage model for concrete. 268
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Fig. 3. Example of a simply supported plane stress reinforced concrete deep beam.
According to ACI, deep beam action must be considered when the ratio of length versus depth is less than five [7]. Tests performed by Schlaich and Weischede on simply supported shear deep beams indicated that crushing zones form close where the load is applied, and that inclined cracks initiate at the bottom of the beam [7]. Ashour performed tests on continuum reinforced concrete deep beams, and results show that inclined cracks develop between the loading plates and the supports [8]. An example of a reinforced deep beam with a length-depth ratio of one was analyzed using the Peridynamic Model. The beam is simply supported and model dimensions are shown in Fig. 3. A reinforced bar with a cross sectional area of 1 in2 is located at the bottom of the beam, and concrete properties are E= 3605 ksi, ν= 0.22, δ=3″, c= 319 kip/in6, d= 319 kip/in4, stens = 1.387 × 10−4, αtens = 2.0, scomp = 1.1 × 10−3 and αcomp = 5.0. Steel properties are E= 29,000 ksi, ν= 0.30, δ=3″, c= 2930 kip/in6, d= 56.3 kip/in4, stens = 1.0 × 10−2, αtens = 2 × 103, scomp = 2 × 103 and αcomp = 5.0. Fig. 4 show the damage sequence of the deep beam described in Fig. 3, where Fig. 4a illustrates multiple crack initiation at the bottom of the beam. At an applied load of 33 kips a major crack is shown in Fig. 4b. In addition, this Figure shows the growth of several secondary cracks, and an area of concrete crushing close to the zone where the load is applied. At a failure load of 35 kips, the
Fig. 4. Simulation of a simply supported reinforced concrete deep beam with a concentrated load at the middle. 269
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Fig. 5. Plot of a load versus deflection of a simply supported reinforced concrete deep beam.
major crack grows almost vertically connecting the crushing region at the left edge of the applied load. Fig. 5 shows a plot of load versus deflection from this example. The ACI 318 code for deep beams establishes that the shear strength is given by the following formula [7]:
M Vd Vc = ⎛3.5 − 2.5 u ⎞ ⎛1.9 f ′c + 2500ρw u ⎞ bw d Vu d ⎠ ⎝ Mu ⎠ ⎝ ⎜
⎟ ⎜
⎟
where Mu and Vu are the maximum factored moment and shear respectively, Vu is the concrete compressive strength, bw and d are the beam width and depth respectively, and ρw is the steel proportion ratio (As/bwd). The value of Vc in the previous equation should not be larger than Vc = 6 f ′c bw d and the value of (3.5 − 2.5Mu/Vud) should be smaller than 2.5 [7]. For a simply supported beam with a point load at the middle the values Mu of and Vu can be computed with classical beam theory as Mu = PL/4 and Vu = P/2. By using a ratio ρw of 0.01, and by substituting these equations, the ACI formula reduces to:
L d Vc = ⎛3.5 − 1.25 ⎞ ⎛1.9 f ′c + 50 ⎞ bw d L⎠ d⎠ ⎝ ⎝ From this example problem L = 100", f′c = 4000 psi, b= 1”, and d= 97″, and by substituting these values, the shear strength becomes Vc= 36.1 kips, which is close to the value of Vc= 35 kips obtained in the simulations.
Fig. 6. Experimental setup and model description of the four point bending test. 270
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Fig. 7. Plain and reinforced concrete beam tests configuration.
Four point bending plain and reinforced concrete beams were simulated using the Peridynamic Model and compared with physical tests. Model dimensions are shown in Fig. 6. One of the beams is unreinforced, another one has only longitudinal reinforcement and the last beam has longitudinal and transverse reinforcement (Fig. 7). Fig. 8 shows simulation results using the Peridynamic Model and Fig. 9 shows photos of the test, where is seen that the crack path is very similar with the crack shown in the simulation. Flexural stress distribution is shown in Fig. 8 and was obtained using Eq. (6). In order to obtain good integration results, the size of the material horizon was three times the grid spacing. Grid spacing is 0.125″ × 0.125″, so that the material horizon is 3/8″. Material properties for concrete are E= 2260 ksi, ν= 0.2, stens = 1.33 × 10−4, αtens = 1.4, scomp = 1.1 × 10−3 and αcomp = 5.0, and for steel E= 29,000 ksi, ν= 0.30,stens = 1.2 × 10−2, αtens = 2 × 103, scomp = 2 × 103 and αcomp = 5.0. Values of c and d are obtained with Eq. (4) using these model properties. Simulations and tests were also performed on reinforced concrete beams. For the four point loaded beam with only longitudinal reinforcement, peridynamic results show major inclined shear cracks starting at the beam supports (Fig. 10), whereas the physical test shows four instead two shear cracks inclined at a different angle (Fig. 11). Average crack width size is about ¾” in the model. Size of longitudinal reinforcement may play a role. For the four point loaded concrete beam with longitudinal and transverse reinforcement peridynamic simulation results show several cracks almost parallel to the shear reinforcement (Fig. 12), in which crack propagation stopped at the transverse reinforcement since it was a two-dimensional plane stress simulation. The same beam using physical experimentation showed cracks at the support with a sudden brittle failure at the maximum load. The reason for this sudden failure is that the beam was over reinforced (Fig. 13). 4. Discussion and conclusions By comparing the failure paths and fracture, propagation results with the model and the physical tests showed similar results, where crack size width average is about 3/4″. In addition, strength results compared with the ACI code showed equivalent results. Nevertheless, for the four point loaded concrete beam with longitudinal without transverse reinforcement, tests results showed two instead of a one major inclined crack. In addition, literature shows that cracks for reinforced concrete beams with transverse reinforcement do not stop at the stirrups, as the numerical model suggests. One possible cause of this is that a two-dimensional plane-
Fig. 8. Simulation and flexural stress distribution results of the plain concrete four point loaded beam using the Peridynamic Model. 271
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Fig. 9. Failure of the four point loaded plain concrete beam test.
Fig. 10. Simulation results of the four point loaded beam with longitudinal reinforcement.
stress Peridynamic model was used, instead a three-dimensional model. In order to obtain better results, mesh convergence studies are required, where more degrees of freedom are included. In addition, three-dimensional computer simulation studies must be performed. In addition, more reinforced and plain concrete beam physical tests should be implemented. Nevertheless, the present work showed that better and safe failure prediction results could be obtained with the implementation of the Peridynamic Model.
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Fig. 11. Four point loaded beam with longitudinal reinforcement test.
Fig. 12. Simulation results of the four point loaded beam with longitudinal and transverse reinforcement.
Fig. 13. Four point loaded beam with longitudinal and transverse reinforcement test.
Acknowledgements We would like to thank the University of Sonora Experimental Laboratory, Samuel Castro, Yesenia A. Valenzuela and Alan F. Ramirez. References [1] ACI, American Concrete Institute, Finite Element Analysis of Fracture in Concrete Structures, Report ACI 446.3 R-97, (1997). [2] S.A. Silling, Reformulation of elasticity theory and long-range forces, J. Mech. Phys. Solids 48 (2000) 175–209.
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[3] S.A. Silling, R.B. Lehoucq, Convergence of peridynamics to classical elasticity theory, J. Elast. 93 (2008) 13–37. [4] W. Gerstle, N. Sau, S. Silling, Peridynamic modeling of concrete structures, nuclear engineering and design, 237 (2007), pp. 1250–1258 Gerstle, W., Sau N., Silling S. (2007). [5] N. Sau, Peridynamic Modeling of Quasibrittle Structures, Doctoral Dissertation University of New Mexico, 2008. [6] R.B. Lehoucq, S.A. Silling, Force flux and the peridynamic stress tensor, J. Mech. Phys. Solids 56 (4) (2008) 1566–1577. [7] J.C. MacGregor, Reinforced Concrete, Mechanics and Design, Second edition, Prentice Hall Ed, 2002. [8] A. Ashour, Tests of reinforced continuous deep beams, ACI Struct. J. 94 (6) (1997) 3–12.
274
Contents lists available at ScienceDirect
Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Peridynamic modelling of reinforced concrete structures ⁎
Nicolas Sau , Jose Medina-Mendoza, Ana C. Borbon-Almada
T
Civil Engineering and Mines Department, University of Sonora, Mexico
A R T IC LE I N F O
ABS TRA CT
Keywords: Fracture Cracks Stress analysis
Reinforced concrete structures subjected to loads that cause the formation and propagation of cracks exhibit a behavior where continuum mechanics is no longer applicable, because most conventional models do not consider the discontinuities that appear and propagate because of concrete tension failure. In an effort to correct the shortcomings of continuum models, the peridynamic bond-based model was proposed in 2000 and the peridynamic state-based model in 2007. The micropolar peridynamic model improves the bond-based peridynamic model by allowing different values of the Poisson's ratio. A numerical scheme for the calculation of the micropolar peridynamic stress was developed where results show convergence with classical linear elastic solutions for small values of the material horizon. With this approach, the stress tensor can be analyzed at points in failure zones. Fractures patterns using physical experimentation are obtained and compared with the model.
1. Introduction Quasibrittle structures that are subjected to loads that cause the formation and propagation of cracks exhibit highly nonlinear behavior, where models based on continuum mechanics are no longer applicable. With the development of the finite element method, models based upon. Fracture Mechanics concepts were formulated, where the main disadvantage of these models is that they are impractical, since multiple cracks need to be represented in the material. The use of smeared crack models using continuum damage theories appeared to be a more suitable technique to model damage in concrete structures. Nevertheless, spurious mesh sensitivity was one of the many problems associated with the smeared crack approach [1]. In an effort to correct the shortcomings of continuum models, the peridynamic model was proposed in 2000 by Silling. Governing equations in the peridynamic model do not assume spatial differentiability of the displacement field and permit discontinuities to arise as part of the solution [2]. In addition, the peridynamic theory is non-local, where locality is recovered as a special case [3]. The micropolar peridynamic model, which is a bond-based model, improves the peridynamic model proposed by Silling in 2000, by allowing moment densities in addition to force densities to interact among particles inside the material horizon [4]. The micropolar peridynamic model is able to model materials with Poisson's ratios different from 1/4 in three-dimensions and 1/3 for twodimensional plane stress problems; in addition, particles are capable of having rotational as well as translational degrees of freedom [4]. Strength capacity of reinforced concrete beams has not been easy to predict with the use of non-empirical methods, since most conventional models do not consider the discontinuities that appear and propagate as a result of concrete tension failure. With this
⁎
Corresponding author. E-mail address: [email protected] (N. Sau).
https://doi.org/10.1016/j.engfailanal.2019.05.004 Received 22 November 2018; Received in revised form 11 April 2019; Accepted 2 May 2019 Available online 10 May 2019 1350-6307/ © 2019 Elsevier Ltd. All rights reserved.
Engineering Failure Analysis 103 (2019) 266–274
N. Sau, et al.
approach, the stress tensor is analyzed at different points in failure zones. Crack patterns are obtained with the model and compared with four point-loaded plain and reinforced concrete beam tests. 2. Peridynamic model The peridynamic model postulates a vector function in units of force per unit volume squared fi j that represents the interaction between particles i and j inside a material horizon δ [2]. The sum of all internal forces per unit volume acting on a particle i is expressed as:
∫V
fij dVj + bi = ρi
J
∂ 2u i , ∂t 2
(1) ∂2ui ∂t 2
and ρi are the acceleration and density of particle i respectively. This integral is in which bi is the body force acting on particle i; performed on all particles j within δ. In the micropolar peridynamic model a moment equation must be added [4]:
∫V
mij dVj + mi = Ii
J
∂2θi , ∂t 2
(2)
where the moment per unit volume squared is mij and miis a moment density function [4]. These moments and forces can be functions of the relative positions ξij, the relative displacements ηij, and the rotations of particles θi andθj, where forces fij and moments mij between particles can be expressed using the following matrix at the linear microelastic level as
⎡ f ⎤ ⎡ uî ⎤ ⎢ xi ⎥ − c /L 0 0 0 0 ⎤⎢ ⎥ ⎢ f ⎥ ⎡ c/L vi ̂ − 12d/ L3 6d/ L2 ⎥ ⎢ i ⎥ 12d/ L3 6d/ L2 0 ⎢ y⎥ ⎢ 0 i⎥ ⎢ ⎥ ⎢ ⎢m θ 2 2 z z⎥ − 6d/ L 0 6d/ L 4d/ L 0 2d/ L ⎥⎢ ⎥ ⎢ ⎥=⎢ j u − c /L c /L 0 0 0 0 ⎥ ⎢ ĵ ⎥ ⎢ f ⎥ ⎢ 3 − 6d / L2 3 ⎢ x⎥ ⎢ 0 − − d L d L d/ L2 ⎥ ⎢ vj ̂ ⎥ 12 / 0 12 / 6 j ⎥⎢ ⎥ ⎢ 2 2 fy⎥ ⎢ − d L d L d L d 0 6 / 2 / 0 6 / 4 /L ⎦ ⎢ j ⎥ ⎣ ⎢ ⎥ ⎣θ z ⎦ j ⎢m ⎥ ⎣ z⎦ i
′
(3)
The parameters c and d are given by c = E ′ A and d = E I′ respectively, where A is the cross-sectional area of the link between particles, E′ is the modulus of elasticity, I′ is the moment of inertia and L is its length (Fig. 1). The relative displacements in the x and the y direction are ηijx = u î − u ̂j and ηijy = v î − v ̂j respectively. Similarly, links in a tree-dimensional model have a circular cross-
Fig. 1. Peridynamic link in the 2-D linear microelastic model in local coordinates. 267
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sectional area with a moment of inertia I′, polar moment of inertia 2I′ and an area A′, all these links have a finite length with an infinite number of particles j, attached to each particle i [5]. If a pair of particles have different material properties, such as steel and concrete, then values for c and d are averaged for the link connecting these particles. Relationships between peridynamic constants (c, d) and conventional linear elastic constants (E, ν) can be obtained by calculating the strain energy density of all the peridynamic links attached to a particle inside a material horizon δ. These relationships are given by
c=
6E E (1 − 3ν ) ,d= , πδ 3t (1 − ν ) 6πδt (1 − ν 2)
(4)
for two-dimensional plane stress problems and
c=
6E E (1 − 4ν ) ,d= , πδ 4 (1 − 2ν ) 4πδ 2 (1 − 2ν )(1 + ν )
(5)
for three-dimensional models [5]. Using the definition given by Lehoucq [6], the peridynamic stress tensor in cylindrical coordinates is given by
σkl =
1 2
2π
δ
∫0 ∫0 ∫0
δ−y
(y + z ) fk ml dz dy dθ
(6)
where σkl is the stress tensor, fk is the pair wise force or moment function per unit volume squared,y, z are the distances between particles i and j, and ml is the unit vector in the direction of the normal of the plane where the stress is measured. For concrete structures a microelastic damage model was proposed, where the stiffness of a peridynamic link depends upon the axial stretch s of the link between particles i and j, and also upon the maximum stretch st, of all other peridynamic links connected to particle i. The link shown remains linearly elastic as long as its stretch s is smaller than a tensile limit, stens and larger than a compressive limit, scomp. If s exceeds the tensile limitstens, the tensile peridynamic force remains constant until s exceeds another specific limit, αtensstens, after which the force goes down to zero. On the other hand if s is less than a compressive limit,scomp, there are two possibilities. If the maximum transverse stretch sc is smaller than scomp, the link remains linear elastic, while if the stretch sc is greater than scomp, the compressive force remains constant until s exceeds another limit, αcompscomp, after which it plunges to zero. The plateau in this model is able to simulate dissipation of energy due to aggregate contact and friction. In addition, cement paste crushing is simulated with the limitαcompscomp. Thus, this model has a reasonable physical explanation. The total number of parameters of this micropolar peridynamic damage model is seven c, d, δ, stens, scomp, αtens, and αcomp (Fig. 2). 3. Application to concrete structures Using the Micropolar Peridynamic Model a computer model was developed and applied to Concrete Structures, two types of concrete structures were analyzed, simply supported reinforced concrete deep beams and four point loaded reinforced concrete beams. Results were compared with the American Concrete Institute (ACI) code and physical experimentation.
Fig. 2. Microelastic damage model for concrete. 268
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Fig. 3. Example of a simply supported plane stress reinforced concrete deep beam.
According to ACI, deep beam action must be considered when the ratio of length versus depth is less than five [7]. Tests performed by Schlaich and Weischede on simply supported shear deep beams indicated that crushing zones form close where the load is applied, and that inclined cracks initiate at the bottom of the beam [7]. Ashour performed tests on continuum reinforced concrete deep beams, and results show that inclined cracks develop between the loading plates and the supports [8]. An example of a reinforced deep beam with a length-depth ratio of one was analyzed using the Peridynamic Model. The beam is simply supported and model dimensions are shown in Fig. 3. A reinforced bar with a cross sectional area of 1 in2 is located at the bottom of the beam, and concrete properties are E= 3605 ksi, ν= 0.22, δ=3″, c= 319 kip/in6, d= 319 kip/in4, stens = 1.387 × 10−4, αtens = 2.0, scomp = 1.1 × 10−3 and αcomp = 5.0. Steel properties are E= 29,000 ksi, ν= 0.30, δ=3″, c= 2930 kip/in6, d= 56.3 kip/in4, stens = 1.0 × 10−2, αtens = 2 × 103, scomp = 2 × 103 and αcomp = 5.0. Fig. 4 show the damage sequence of the deep beam described in Fig. 3, where Fig. 4a illustrates multiple crack initiation at the bottom of the beam. At an applied load of 33 kips a major crack is shown in Fig. 4b. In addition, this Figure shows the growth of several secondary cracks, and an area of concrete crushing close to the zone where the load is applied. At a failure load of 35 kips, the
Fig. 4. Simulation of a simply supported reinforced concrete deep beam with a concentrated load at the middle. 269
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Fig. 5. Plot of a load versus deflection of a simply supported reinforced concrete deep beam.
major crack grows almost vertically connecting the crushing region at the left edge of the applied load. Fig. 5 shows a plot of load versus deflection from this example. The ACI 318 code for deep beams establishes that the shear strength is given by the following formula [7]:
M Vd Vc = ⎛3.5 − 2.5 u ⎞ ⎛1.9 f ′c + 2500ρw u ⎞ bw d Vu d ⎠ ⎝ Mu ⎠ ⎝ ⎜
⎟ ⎜
⎟
where Mu and Vu are the maximum factored moment and shear respectively, Vu is the concrete compressive strength, bw and d are the beam width and depth respectively, and ρw is the steel proportion ratio (As/bwd). The value of Vc in the previous equation should not be larger than Vc = 6 f ′c bw d and the value of (3.5 − 2.5Mu/Vud) should be smaller than 2.5 [7]. For a simply supported beam with a point load at the middle the values Mu of and Vu can be computed with classical beam theory as Mu = PL/4 and Vu = P/2. By using a ratio ρw of 0.01, and by substituting these equations, the ACI formula reduces to:
L d Vc = ⎛3.5 − 1.25 ⎞ ⎛1.9 f ′c + 50 ⎞ bw d L⎠ d⎠ ⎝ ⎝ From this example problem L = 100", f′c = 4000 psi, b= 1”, and d= 97″, and by substituting these values, the shear strength becomes Vc= 36.1 kips, which is close to the value of Vc= 35 kips obtained in the simulations.
Fig. 6. Experimental setup and model description of the four point bending test. 270
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Fig. 7. Plain and reinforced concrete beam tests configuration.
Four point bending plain and reinforced concrete beams were simulated using the Peridynamic Model and compared with physical tests. Model dimensions are shown in Fig. 6. One of the beams is unreinforced, another one has only longitudinal reinforcement and the last beam has longitudinal and transverse reinforcement (Fig. 7). Fig. 8 shows simulation results using the Peridynamic Model and Fig. 9 shows photos of the test, where is seen that the crack path is very similar with the crack shown in the simulation. Flexural stress distribution is shown in Fig. 8 and was obtained using Eq. (6). In order to obtain good integration results, the size of the material horizon was three times the grid spacing. Grid spacing is 0.125″ × 0.125″, so that the material horizon is 3/8″. Material properties for concrete are E= 2260 ksi, ν= 0.2, stens = 1.33 × 10−4, αtens = 1.4, scomp = 1.1 × 10−3 and αcomp = 5.0, and for steel E= 29,000 ksi, ν= 0.30,stens = 1.2 × 10−2, αtens = 2 × 103, scomp = 2 × 103 and αcomp = 5.0. Values of c and d are obtained with Eq. (4) using these model properties. Simulations and tests were also performed on reinforced concrete beams. For the four point loaded beam with only longitudinal reinforcement, peridynamic results show major inclined shear cracks starting at the beam supports (Fig. 10), whereas the physical test shows four instead two shear cracks inclined at a different angle (Fig. 11). Average crack width size is about ¾” in the model. Size of longitudinal reinforcement may play a role. For the four point loaded concrete beam with longitudinal and transverse reinforcement peridynamic simulation results show several cracks almost parallel to the shear reinforcement (Fig. 12), in which crack propagation stopped at the transverse reinforcement since it was a two-dimensional plane stress simulation. The same beam using physical experimentation showed cracks at the support with a sudden brittle failure at the maximum load. The reason for this sudden failure is that the beam was over reinforced (Fig. 13). 4. Discussion and conclusions By comparing the failure paths and fracture, propagation results with the model and the physical tests showed similar results, where crack size width average is about 3/4″. In addition, strength results compared with the ACI code showed equivalent results. Nevertheless, for the four point loaded concrete beam with longitudinal without transverse reinforcement, tests results showed two instead of a one major inclined crack. In addition, literature shows that cracks for reinforced concrete beams with transverse reinforcement do not stop at the stirrups, as the numerical model suggests. One possible cause of this is that a two-dimensional plane-
Fig. 8. Simulation and flexural stress distribution results of the plain concrete four point loaded beam using the Peridynamic Model. 271
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Fig. 9. Failure of the four point loaded plain concrete beam test.
Fig. 10. Simulation results of the four point loaded beam with longitudinal reinforcement.
stress Peridynamic model was used, instead a three-dimensional model. In order to obtain better results, mesh convergence studies are required, where more degrees of freedom are included. In addition, three-dimensional computer simulation studies must be performed. In addition, more reinforced and plain concrete beam physical tests should be implemented. Nevertheless, the present work showed that better and safe failure prediction results could be obtained with the implementation of the Peridynamic Model.
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Fig. 11. Four point loaded beam with longitudinal reinforcement test.
Fig. 12. Simulation results of the four point loaded beam with longitudinal and transverse reinforcement.
Fig. 13. Four point loaded beam with longitudinal and transverse reinforcement test.
Acknowledgements We would like to thank the University of Sonora Experimental Laboratory, Samuel Castro, Yesenia A. Valenzuela and Alan F. Ramirez. References [1] ACI, American Concrete Institute, Finite Element Analysis of Fracture in Concrete Structures, Report ACI 446.3 R-97, (1997). [2] S.A. Silling, Reformulation of elasticity theory and long-range forces, J. Mech. Phys. Solids 48 (2000) 175–209.
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[3] S.A. Silling, R.B. Lehoucq, Convergence of peridynamics to classical elasticity theory, J. Elast. 93 (2008) 13–37. [4] W. Gerstle, N. Sau, S. Silling, Peridynamic modeling of concrete structures, nuclear engineering and design, 237 (2007), pp. 1250–1258 Gerstle, W., Sau N., Silling S. (2007). [5] N. Sau, Peridynamic Modeling of Quasibrittle Structures, Doctoral Dissertation University of New Mexico, 2008. [6] R.B. Lehoucq, S.A. Silling, Force flux and the peridynamic stress tensor, J. Mech. Phys. Solids 56 (4) (2008) 1566–1577. [7] J.C. MacGregor, Reinforced Concrete, Mechanics and Design, Second edition, Prentice Hall Ed, 2002. [8] A. Ashour, Tests of reinforced continuous deep beams, ACI Struct. J. 94 (6) (1997) 3–12.
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