FEM-modeling of reinforced concrete and verification of the concrete material models available in ABAQUS Daniel Eriksson and Tobias Gasch 2010-12-17
Abstract Reinforced concrete is a very complex material and is widely used as a construction material; for example in buildings, bridges and nuclear containment vessels. When design codes are not accurate enough and experiments are not possible to perform, other analyze methods are needed. Today’s most common analyze method, for such cases, is the finite element method in combination with nonlinear concrete material models. These concrete models are very intricate to use and it is hard to find instructions on how to use them. Therefore, this study includes a brief instruction on how to define and use the concrete smeared cracking model and the concrete damaged plasticity model available in ABAQUS. The material models are also verified against experimental results from the literature with focus on the torsional behavior of concrete.
Royal Institute of Technology Department of Mechanics SE-100 44 Stockholm, Sweden
1
Contents 1 Introduction
3
1.1
Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2 Constitutive material models 2.1
2.2
2.3
4
Concrete smeared cracking . . . . . . . . . . . . . . . . . . . . . .
4
2.1.1
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1.2
Defining the material model . . . . . . . . . . . . . . . . .
7
Concrete damaged plasticity . . . . . . . . . . . . . . . . . . . . .
10
2.2.1
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.2.2
Defining the material model . . . . . . . . . . . . . . . . .
13
Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3 FEM-modeling of concrete with ABAQUS
15
3.1
Modeling aspects . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.2
Convergence problems . . . . . . . . . . . . . . . . . . . . . . . .
16
4 Verification of material models
17
4.1
Cube test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4.2
Unreinforced concrete beam . . . . . . . . . . . . . . . . . . . . .
19
4.3
Unreinforced concrete axle subjected to torsion . . . . . . . . . .
21
4.4
Reinforced concrete beam . . . . . . . . . . . . . . . . . . . . . .
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5 Conclusions 5.1
26
Further research . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
27
1
Introduction
When designing concrete structures according to design codes such as ACI 2008, BBK04 and Eurocode 2 the calculations often simplify the behavior of concrete. With the built–in safety margins in the codes the equations are often accurate enough for the design of reinforced concrete structures. But to capture the actual behavior of the structure one must use other methods that are able to account for the anisotropic nature of concrete, including cracking due to tensile stresses. The best method to describe the behavior is of course to do experiments on the actual structure, but unfortunately experiments are expensive and often not possible to perform. Another method that is able to account for some of these effects is the Finite Element Method (FEM) which is less expensive and easier to perform on complicated structures. Although it may seem easy to use commercial FEM software including concrete material models the non linearity of concrete causes many difficulties for the user. The fact, that FEM is an approximate solution method in combination with the non linearity, results in an uncertainty whether the analysis is relevant or not. Some of the difficulties are: • Discretization of the real structure • Choosing and defining the material model • Obtaining a converging solution • Interpreting the results To overcome all these difficulties the user must have an extensive knowledge of FEM-theory and a good understanding about reinforced concrete as a structural material.
1.1
Purpose
The purpose of this study is partially to give the reader and the authors an introduction to the concrete models available in the FEM software ABAQUS. And partially to verify the material models and to examine whether these material models are able to analyze concrete members subjected to torsional loading. This study is made as a pre–study for future investigations of concrete structures.
1.2
Structure
In the second section, two of the material models for analyzing concrete structures available in ABAQUS will be introduced. First the basic theory of the material models is given and then the input parameters needed to define the material are described. Suggestions are also given on how to define the input parameters.
3
In the third section, some modeling aspects involving ABAQUS and concrete analysis will be discussed. A few difficulties often encountered are mentioned and suggestions are given on how to overcome these problems. In the fourth section, the material models are verified by using different examples from the literature. In the fifth, section the conclusions of this study are presented and some suggestions of further research are given.
2
Constitutive material models
In commercial FEM software there are many different material models for modeling crack propagation in concrete. Most of them are based on either nonlinear fracture mechanics, with a discrete or smeared approach, or plasticity theory. In this thesis, the focus will be on two material models in the FEM software ABAQUS, these are concrete smeared cracking and concrete damaged plasticity. They are based on nonlinear fracture mechanics and a coupled damaged plasticity theory, respectively. In this section, the basic theory of these material models will be given and then it will be explained how to define them in a more practical manner. A brief introduction on how to model and define the reinforcement in concrete structures will also be given in this section.
2.1
Concrete smeared cracking
The concrete smeared cracking model can be used to model all types of concrete structures; both unreinforced and reinforced. The smeared cracking model is limited to static problems with no load reversals, i.e. structures subjected to monotonic loads. In tension the model is based on nonlinear fracture mechanics while its compressive behavior is based on a simple elastic–plastic theory (Hibbit et al., 2009). 2.1.1
Theory
Cracking The smeared crack approach assumes that a crack in the concrete is composed by a number of micro cracks instead of a large discrete macro crack. The discrete approach demands a new mesh for each step of the calculation when a crack starts to develop and as it propagates, this approach requires a lot of computational power. In a smeared crack the material is softened perpendicular to the crack direction, which means that the original mesh can be kept and instead the material stiffness is reduced. Because of the reduced stiffness, the material transfers less stress, as the crack develops further the material stiffness continues to degrade until it reaches zero. At this point no stress can be transferred and the crack propagates further without the stress (Hibbit et al., 2009).
4
A crack initiates when the principal stress reaches the ultimate tensile strength of the concrete. The crack growth direction is determined by the principal stress direction at crack initiation. As the crack grows further this direction is kept even though the principal stress direction may change, as a result shear stresses are induced on the crack surface. This approach is called a fixed crack model (Cervenka et al., 2010). ABAQUS uses a crack detection surface to determine when the ultimate tensile or compressive strength is reached. Due to the reduction of the material stiffness when a crack opens, the material will also get a reduced shear stiffness. The ability of an open crack to transfer shear stresses is due to the interlocking of aggregate. When the crack width exceeds the average aggregate size, the crack is considered to no longer transfer any shear stress. The material model deals with this effect by introducing a post crack shear behavior, called shear retention. This behavior is a linear function of the crack strain, which reduces the shear modulus from its fully elastic value to zero (Hibbit et al., 2009). Uniaxial behavior The uniaxial behavior can be divided into two parts, tensile and compressive behavior. See Fig. 1 for the uniaxial behavior. Stress Failure point in compression (peak stress)
Start of inelastic failure
Unload/reload response Idealized (elastic) unload/reload response Strain Cracking failure Softening
Figure 1: Uniaxial behavior (Hibbit et al., 2009). The tensile behavior is linear elastic up to the point of failure which is defined by the concrete tensile strength; this is the first stage of the tensile behavior. The next stage is called tension softening and describes the transformation of micro cracks to macro cracks. During the formation of micro cracks the material stiffness will gradually degrade until it reaches zero and a macro crack is formed, as mentioned above. In the concrete smeared cracking model, the tension softening can be described by a stress–strain or stress–displacement relationship. 5
These relationships describe the amount of fracture energy required to open a stress free unit area of a crack. The third and last stage of the tensile behavior is reached when the crack becomes stress free. In this stage the crack continues to open without any stress. If the load changes from tensile to compressive, no compression strength is lost even though a crack has opened, i.e. the crack closes completely. The concrete exhibits a linear elastic behavior in compression until it reaches its yield stress. At that point some plastic straining occurs and the material behavior becomes nonlinear. This nonlinear behavior is due to bond failure between the aggregate and the cement paste which starts to occur at stress levels of 70–75 % of the ultimate concrete strength (Malm, 2006). After the ultimate strength is reached, the strain continues to increase while the strength is reduced until the concrete cannot carry any more stress. In reality the unloading response of the concrete is softer beyond the inelastic point than the elastic response. In the model this effect is neglected and the normal elastic response is used during unloading. Multiaxial behavior The multiaxial behavior of the concrete is described by a simplified yield surface fitted to experimental data. The experimental data is fitted with an exponential associated flow rule in the deviatoric plane, see Fig. 2, which results in an inaccurate surface. Normally the inelastic volume strain is over predicted by the associated flow assumption. To overcome this drawback the third stress invariant would be needed i.e. a yield volume. The model contains even more assumptions but no attempts have been made to overcome them because the accuracy of the model is considered to be enough. Another reason is to keep the computational efficiency. The yield surface can be seen in Fig. 3 (Hibbit et al., 2009).
q
-pt
p
Figure 2: Exponential plastic flow rule (Hibbit et al., 2009).
6
σ2 Uniaxial tension
Uniaxial compression
Biaxial tension
σ1
Biaxial compression
Figure 3: Yield surface in the deviatoric plane (Hibbit et al., 2009).
2.1.2
Defining the material model
To define the model a number of input parameters have to be assigned in ABAQUS; some of these are optional. Below follows a description of the different parameters and suggestions are given on how to calculate them according to the CEB-FIP Model Code 90. The only input parameter needed is the cube strength fcu . Another way, and often a better, is to use experimental data of the current concrete to define the input parameters. Parameter Initial elastic modulus Poisson’s ratio Compressive cylinder strength Strain at fc Plastic softening compression Tensile strength Fracture energy Gf Crack opening displacement
Equation √ Ec = (600 − 15.5fcu ) fcu υ = 0.2 fc = 0.85fcu εc0 = 2fc /Ec wd = 5 · 10−4 2/3 ft = 0.24fcu Gf = 25 · 10−6 · ft wc = 5.14Gf /ft
Unit MPa – MPa – m MPa Nm/m2 m
Table 1: Equations for input parameters (Malm, 2006).
Elastic behavior The elastic behavior of the concrete is defined by the linear elastic model in ABAQUS. This model is separate from the concrete smeared
7
cracking but is mandatory in order to get a working material model. Input data is Young’s modulus Ec and the Poisson’s ratio ν according to above. Concrete smeared cracking The compressive behavior is defined by the yield stress as a function of the plastic strain and specified in ABAQUS as tabular data. The first row of the table is defined by the point where the concrete starts to exhibit a nonlinear behavior (plastic straining). According to Malm (2006), the stress at this point can be estimated as 30% of the ultimate compressive strength fc . Note that the plastic strain at this point must be zero in ABAQUS. The remaining points up to fc are given by Eq. (1), where the user selects the number of points to describe the compressive behavior depending on the desired accuracy.
σc = fc
kx − x2 1 + (k − 2)x
x=
ε εc0
(1)
(1a)
where σc is the concrete compressive stress, fc is the concrete compressive strength, x is the normalized strain, ε is the strain, εc0 is the strain at peak stress and k is a shape parameter which depends on initial elastic modulus and the secant elastic modulus at peak stress. For k = 2 the shape is parabolic and for k = 1 the shape is linear. To calculate the plastic strain εpl at the chosen points Eq. (2) is used: εpl = ε −
σc Ec
(2)
After the ultimate strength, the behavior can be considered linear to a point where the stress is zero and the strain εc calculated according to Eq. (3). εc = εc0 +
wd L
(3)
where wd is the plastic softening compression and L is the finite element length. As seen in Eq. (3), εc is dependent on the element size. In other words, if the mesh is changed, a new value for εc has to be calculated. A satisfactory approximation for εc is 0.01 for normal concrete. The tensile behavior is defined by the tension stiffening suboption in ABAQUS. The tension stiffening can either be defined as stress–strain or stress–displacement. Up to the ultimate tensile strength ft the tensile behavior is defined by the elastic behavior of the concrete and no input for ft has to be given.
8
If the stress–strain alternative is chosen the input for tension stiffening can be defined for multiple points. The input needed for each point in ABAQUS is the ratio between current stress and ft and the absolute value of the difference between the current strain and the strain at cracking. The tension stiffening can be calculated by an exponential crack opening law according to Cornelissen et al. (1986). Note that the crack opening displacement has to be divided by the finite element length to obtain the strain which is the input parameter in ABAQUS: w σ = f (w) − f (wc ) ft wc
(4)
where f (w) is a displacement function given by f (w) =
1+
c1 w wc
3 !
c2 w exp − wc
(4a)
where w is the crack opening displacement, wc is the crack opening displacement at which stress no longer can be transfered, c1 is a material constant (c1 = 3.0 for normal density concrete) and c2 is a material constant (c2 = 6.93 for normal density concrete). If the stress–displacement alternative is chosen, only one point can be defined for the tension stiffening, in that case a linear crack opening law is used. The only input is the crack opening displacement at which stress no longer can be transferred and is calculated as: wc =
2Gf ft
(5)
It is recommended to use the stress–displacement alternative to avoid mesh sensitivity in regions that lack reinforcement. Another reason to use this approach is that the tensile behavior does not need to be redefined if the mesh is changed, which is the case for the stress–strain alternative (Malm, 2006). The aforementioned yield surface is defined by the four input parameters, described in the list below, in the failure ratio suboption in ABAQUS. If these parameters are not specified, ABAQUS will use default values (Hibbit et al., 2009). • Failure ratio 1 is the ratio between the ultimate biaxial compressive strength and the ultimate uniaxial strength. Default value is 1.16. • Failure ratio 2 is the ratio between the uniaxial tensile strength and the ultimate uniaxial compressive strength, i.e. ft /fc . Default value is 0.09. • Failure ratio 3 is the ratio between the principal plastic strain at ultimate biaxial compression to the plastic strain at ultimate uniaxial compression. Default value is 1.28.
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• Failure ratio 4 is the ratio between the tensile principal stress at cracking in plane stress and the tensile cracking stress under uniaxial tension. Default value is 1/3. Shear retention is an optional input that describes how the shear stiffness is affected by cracking. The shear modulus is assumed to decrease linearly to zero as the crack opens. Two input variables needs to be defined in this suboption. The first is ρclose which describes the amount of shear retention when a crack closes. This value is often set to 1.0 which means no loss in shear capacity for a closed crack. The other parameter is εmax which describes the strain when all shear capacity is lost, a reasonable value is 10−3 according to Hibbit et al. (2009). If the shear retention is not defined, ABAQUS assumes full shear retention, i.e. no shear capacity is lost when the concrete cracks. An omission of the shear retention may, according to Malm (2006), lead to convergence difficulties as well as unrealistic and distorted crack patterns.
2.2
Concrete damaged plasticity
The concrete damage plasticity model can be used to model all types of concrete structures, both unreinforced and reinforced. It can also be used to model other quasi-brittle materials such as soils. The model can be used to analyze structures subjected to monotonic, cyclic and dynamic loading. For both tension and compression, the model is based on a coupled damage plasticity theory. 2.2.1
Theory
Uniaxial behavior As for the smeared crack model, described in section 2.1, the uniaxial behavior for the concrete damage plasticity can be divided into a tensile and a compressive part. The tensile behavior is linear elastic up to the ultimate tensile strength of the concrete. As the strain increases the model uses a tension softening response to describe how the formation of micro cracks affects the concrete at a macroscopical level. This response can be described by a stress–strain or stress– displacement function or by simply defining the fracture energy Gf needed to open a unit area of a crack. After the tension softening the crack continues to open without transferring any stress (Hibbit et al., 2009). In compression the behavior of the concrete damage plasticity model resembles that of the smeared cracking model. First, the behavior is linear elastic after which the material starts to lose its stiffness and the behavior becomes nonlinear. The loss of stiffness is due to bond failure between the aggregate and the cement paste. After the ultimate compressive strength the strain continues to increase while the stress decreases; this phase is called softening (Hibbit et al., 2009). Multiaxial behavior The yield surface used in the concrete damage plasticity model was developed by Lubliner et al. (1989) and later modified by Lee and Fenves (1998), see Fig. 4.
10
σ2
‘Crack detection’ surface
Uniaxial tension Biaxial tension
σ1
Uniaxial compression
‘Compression’ surface
Biaxial compression
Figure 4: Yield surface (Hibbit et al., 2009). The concrete damage plasticity model uses a Drucker–Prager hyperbolic plastic potential function, which is a non-associated flow rule. This means that the plastic flow potential and the yield surface use separate functions and that the plastic flow develops in normal direction to the plastic flow potential. The plastic flow potential can be seen in Fig. 5. Hardening
dε p
ψ
q
Hyperbolic Drucker-Prager flow potenial
p
Figure 5: Hyperbolic plastic flow rule (Hibbit et al., 2009).
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The evolution of the yield surface is governed by an isotropic hardening model which depends on the plastic flow potential through the plastic flow. Cyclic/dynamic loading As mentioned above, the concrete damage plasticity model can also be used to analyze structures subjected to cyclic and dynamic loads. This ability is incorpareted into the model by using scalar damage parameters which reduces the stiffness of the concrete in either compression or tension. To account for plastic strains that concrete exhibits during unloading, due to the micro cracks, the damage has to be coupled with the plasticity of the concrete. This is done as, (Hibbit et al., 2009): σt = (1 − dt ) Ec εt − εpl t
(6)
σc = (1 − dc ) Ec εc − εpl c
(7)
where dt is the tension damage parameter, dc is the compression damage parameter and εpl is the equivalent plastic strains. A typical load cycle is depicted in Fig. 6. If the load changes from tension to compression, the concrete compressive stiffness is fully recovered as the crack closes. In the opposite case, compression to tension, the tensile stiffness will not be recovered if crushing micro cracks has developed. This is the default behavior of the material model, and is shown in Fig. 6, but it can be modified through the use of stiffness recovery factors (Hibbit et al., 2009). σt
ft 0
E0
(1 − dt ) E0 (1 − d c ) E0
ε
(1 − dt )(1 −d c ) E0
E0
Figure 6: Uniaxial load cycle (Hibbit et al., 2009).
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2.2.2
Defining the material model
As for the smeared crack model, the material model is defined through a number of parameters in ABAQUS. The proposed formulas and values in Tab. 1 on page 7 are recommended to use when defining this material model. Elastic behavior The elastic behavior of the concrete is defined by the linear elastic model in ABAQUS. This model is separate from the concrete damage plasticity but is mandatory in order to get a working material model. Input data is Young’s modulus Ec and the Poisson’s ratio ν according to Tab. 1. Density To perform a dynamic analysis or use the ABAQUS/Explicit solver, the density of the concrete has to be defined. As for the elastic behavior, the density is defined in a separate material model. A typical value for concrete is 2400 kg/m3 . Concrete damaged plasticity The dilation angle ψ is used, in ABAQUS, to calculate the plastic flow potential. A dilation angle close to the materials friction angle, for concrete ψmax = 56.3◦ which also is the upper limit, results in a ductile behavior. On the other hand a low value, close to 0◦ , results in a very brittle behavior. The effect of the dilation angle for values between the upper and the lower limit (10◦ ≤ ψ ≤ 40◦ ) is almost negligible (Malm, 2006). For normal concrete structures a value around 30◦ gives results with a sufficient accuracy. According Lovén and Rosengren (2009) the dilation angle should be set to the material friction angle when analyzing deep beams in plane stress as a lower angle will underestimate the strength of the deep beam. The flow potential eccentricity defines the rate at which the plastic potential function approaches the asymptote, see Fig. 5. If no experimental data is available it should be set to the ABAQUS default value 0.1. The ratio fb0 /fc0 is the ratio between the initial biaxial compressive yield stress and the initial uniaxial compressive yield stress. The default value in ABAQUS is 1.16. The multiaxial behavior of the material model is defined by the input parameterKc , which has to be set to a value in the range of 0.5 < Kc ≤ 1. The default value in ABAQUS is 2/3 and is the recommended value to use. The viscosity parameter µ is used to help to accomplish a good convergence in ABAQUS/Standard analyses. If no input is given, the viscoplastic regularization is ignored. The value should be small compared to the characteristic time increment. A reasonable value is µ = 10−7 which is used in analyses by Malm (2006). The compressive behavior is defined in the same manner as for the concrete smeared cracking model. But in the concrete damage plasticity model the inelastic strains are used instead of the plastic strains. The inelastic strain is the same as the plastic strain in the smeared crack model and therefore the same behavior can be used, Eq. (1). The plastic strain in the damage plasticity model
13
is used to describe the permanent strain after unloading and is dependent on the damage parameter. In the absence of damage the plastic and inelastic strains are equal. The compressive damage parameter dc can be defined linearly dependent on the inelastic strain in the range 0 < dc ≤ 0.99. A recommended max value is 0.9 to account for aggregate interlocking and similar effects. If no cyclic or dynamic loads are expected, the damage suboption can be omitted. The tension stiffness recovery factor describes how much of the tensile strength that remains after crushing, when the load changes to tension. The default value is 0 which agrees with the actual behavior of concrete. The tensile behavior can be defined as either stress–strain, stress–displacement or stress–fracture energy. For plain concrete, the tensile behavior can be described by tension softening and the stress–displacement alternative is recommended. The behavior can be defined by Eq. (4) developed by Cornelissen et al. (1986). With a sufficiently fine mesh the tension softening can just as well be defined linearly, Eq. (4). For a coarse mesh, the crack opening is much better described by using the equation by Cornelissen et al.. The tension damage parameter dt must be in the range 0 < dt ≤ 0.99 but as for the compressive damage a maximum value of 0.9 is recommended. It should be defined with the same shape as the tension softening but mirrored, i.e. the maximum damage ought to occur when a stress free crack is obtained. If cyclic or dynamic loads are not present, the tension damage parameter is not necessary to define. However, the tension damage parameter is a good way to visualize the crack pattern and propagation. Therefore, it is recommended to define this parameter in a static analysis. The compression stiffness recovery factor describes how much of the compressive strength that is recovered after crack closure. The default value is 1.0, which corresponds well with the actual behavior of concrete.
2.3
Reinforcement
There are many different ways of modeling reinforcement in ABAQUS; for example as smeared reinforcement in the concrete, as cohesive elements, with a built–in rebar layer which is available in certain element types and as discrete truss or beam elements with the embedded region constraint. All the analyses in this study will use the embedded region alternative and therefore this section will only explain how to model reinforcement in this manner. The material is defined as an ideal elasto–plastic material, i.e. no hardening behavior. This is done by using the elastic and plastic material models in ABAQUS. The input parameters should be set according to the reinforcement used in the structure. The required input parameters are Young’s modulus, Poisson’s ratio and the yield strength. Normally it is sufficient to model the reinforcement with truss elements, for which the only input parameter needed is the cross–sectional area of the reinforcement. The benefit of beam elements is that they can account for the dowel effect, which only gives a small increase of the load capacity in the structure. As the beam elements require more input parameters and also more computational power this alternative is not recommended. 14
This way of modeling the reinforcement does not include the bond slip effect between the concrete and the reinforcement. Instead these effects are included in the tension stiffening behavior of the concrete material models described in sections 2.1 and 2.2 (Hibbit et al., 2009). When modeling the embedded region in ABAQUS with normal reinforcement bars the default constraint values has been used in this study.
3
FEM-modeling of concrete with ABAQUS
FEM modeling of concrete is associated with many difficulties as the analysis has to be nonlinear. Hence, the user should be familiar with finite element analysis theory and the program to be used for the actual analysis. If the user is new to ABAQUS the recommendation is to go through the “Getting Started with ABAQUS: Interactive Edition” in the ABAQUS documentation (Hibbit et al., 2009) which introduces the user to the basics of ABAQUS. This section will cover some of the modeling difficulties associated with concrete analysis in ABAQUS and guidance on how to overcome them.
3.1
Modeling aspects
The concrete material models are defined in section 2 and should be read carefully before starting the analysis. How to model the interaction between concrete and reinforcement is also described in the aforementioned section. More information about the material models is given in the ABAQUS documentation (Hibbit et al., 2009) which also describes a third concrete material model included in ABAQUS, concrete brittle cracking. The loading can be defined as either a force or a forced displacement. All the analyses in this study are performed with the displacement method of applying loads. The reason is that this method allows the analyst to observe the post failure behavior of the structure. When using the force approach the analysis will be aborted when the structure reaches its collapse load. To reduce stress concentrations in the concrete at the load application points, steel load plates can be used. ABAQUS offers the user many different solution algorithms, called step functions. The most commonly used nonlinear solution schemes for concrete modeling are the Static/General and Static/Riks. The Static/General can either be defined as a load controlled or displacement controlled solution method depending on how the load is defined. Static/Riks is an arc length method suitable for problems including “snap back” behavior. The recommendation is to use as simple elements as possible. In this article all the presented analyses are performed with linear elements.
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3.2
Convergence problems
When modeling and analyzing reinforced concrete, convergence problems will often occur in ABAQUS during the solution. There are many different ways to handle these problems and some of them are presented below. The time increment parameters should always, when analyzing concrete with the Static/General time step, be set to lower values than the default values in ABAQUS. As an effect of this, the maximum number of time increments has to be increased. The reason to this is the need to allow a more accurate solution at points when cracking occurs. Convergence problems can arise if the reinforcement nodes coincide with the concrete element nodes. These convergence problems occur because the high stresses in the reinforcement heavily distort the less stiff material elements. To avoid this, reinforcement should be placed where it does not coincide with any other material element node. Another way of handle convergence problems is to introduce automatic stabilization in the time steps. According to results in Malm and Ansell (2008) this stabilization does not interfere with the concrete behavior and is therefore an effective way of handling convergence problems. The automatic stabilization can be turned on in the “Edit step” window in ABAQUS. In this study the “Specify dissipated energy fraction” option with default values is used. The default value for the automatic stabilization is 0.0002 and for the adaptive stabilization with maximum ratio of stabilization to strain energy is 0.05. If convergence problems still are present, the number of iterations and the tolerance in the general solution control for the load time step can be changed (Malm, 2006). The recommended parameters to change are: Rnα
Default value is 0.005. Use a higher value to avoid convergence problems. In this study values up to 0.1 have been used.
I0
Default value is 4 and can be increased to 3–4 times the default value.
IR
Default value is 8 and can be increased to 3–4 times the default value.
IA
Default value is 4 and can be increased to 3–4 times the default value.
If the aforementioned measures to avoid convergence problems do not help, there are still some material model specific measures that can be implemented. The tension stiffening/softening can affect the convergence. A too low fracture energy can cause unstable behavior which abort the solution premature. By setting a higher fracture energy than the actual one in the concrete, the solution may converge. However, defining a higher fracture energy than the actually one alters the quality of the concrete and therefore the solution may not be relevant. In the concrete smeared cracking model an incorrect shear retention factor may result in distorted and unrealistic crack patterns. A more accurate definition of 16
the shear retention factor may therefore help the solution to converge. A major drawback with the smeared cracking model is the fact that the shear retention may only be defined as linearly dependent on the strains. A more accurate way to describe the shear retention, is to use an exponential function as the one used in the SBETA material model in ATENA, Cervenka et al. (2010). According to section 2.2, the viscosity parameter in the concrete damaged plasticity model can be used to help the solution to converge. If none of the above mentioned measures result in a converged solution, the concrete damaged plasticity model can be used with the ABAQUS/Explicit solver. ABAQUS/Explicit is a dynamic solver which uses a central differences scheme to solve the problem. Because the solver account for dynamic effects the load has to be applied with low speed to make the dynamic effects negligible for a static problem.
4
Verification of material models
When using the material models for complex structures it is difficult to determine whether the behavior of the models can describe the actual behavior of concrete in a satisfactory way. Therefore it is recommended to verify the material models against examples, for which analytical solutions easily can be obtained or for which experimental data is known. In this section the behavior of the material models described in the previous section will be examined and verified through a couple of examples that analyze different aspects of the material behavior.
4.1
Cube test
In this example, the biaxial compressive behavior of the material models will be examined. This will be made by subjecting a three–dimensional cube to a uniform biaxial compressive pressure. The dimensions of the cube are shown in Fig. 7 and a concrete of quality C25/30 will be used. The material models are defined according to section 2.
17
150
[mm]
t = 150
150 Figure 7: Cube A uniform biaxial pressure is obtained by constraining three of the sides against translation in their normal directions and applying a forced displacement onto the fourth side as shown in the Fig. 7. The two other sides are free to deform. For both material models the biaxial compressive strength is defined as fb = 1.18fc which corresponds to the cube strength, 30 MPa, of the chosen concrete. The load deformation curve from the FEM analysis is shown in Fig. 8. For the concrete smeared cracking model the compressive failure occurred at a load of 31.8 MPa. When the same analysis was carried out with the concrete damaged plasticity model the compressive failure occurred for a load of 30.8 MPa.
18
[MPa] 35.
30.
Pressure
25.
20.
15.
10.
5.
0. 0.00
0.05
0.10
0.15
Displacement
0.20
0.25 [mm]
Concrete Damaged Plasticity Concrete Smeared Cracking
Figure 8: Load deformation curve The calculated biaxial compressive strength from both material models is close enough to the expected results. The two material models show a slightly different behavior even though the compressive material behavior is defined from the same equation. An explanation to this difference is probably the different biaxial failure surfaces of the two material models shown in section 2 since tensile stresses will occur in the unconstrained direction.
4.2
Unreinforced concrete beam
To examine the tensile behavior of the concrete material models, analyses of a unreinforced beam are made. The beam has dimensions according to Fig. 9 and will be subjected to a point load at its mid span. A concrete of quality C25/30 was used in the analyses. To be able to observe the post failure behavior, the point load will be represented by a forced displacement. The crack patterns obtained from the analyses can be seen in Fig 10.
P [mm] 100
t = 150
50
350 800
Figure 9: Unreinforced beam 19
Figure 10: Crack pattern from all analyses The same crack pattern was obtained with all material models; concrete smeared cracking with linear tension stiffening and concrete damaged plasticity with both linear and exponential tension stiffening. As can be seen in Fig. 10, a distinct flexural crack was received from the analyses, which corresponds well with the expected result. The load–displacement curves from the analyses can be seen in Fig. 11. [kN] 3.5
3.0
Force
2.5
2.0 First crack appears 1.5
1.0
0.5
0.0 0.0
0.05
0.10
0.15
0.20
Displacement
0.25
0.30 [mm]
CDP Linear Tension Stiffening CDP Cornelisen Tension Stiffening CSC Linear Tension Stiffening
Figure 11: Load–displacement curves The first crack shown in Fig. 11 is calculated with Navier’s formula and is considered to appear as the first part of the beam reaches its ultimate tensile strength. As can be seen, the analyses show an overcapacity compared to the calculated first crack. This is probably due to the defined tension softening of the material models. Which implies that the concrete still has a tensile capacity after the ultimate tensile strength is reached.
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4.3
Unreinforced concrete axle subjected to torsion
To observe whether the material models can account for the crack pattern created by a torsional load an unreinforced concrete axle will be analyzed. The axle is made of concrete with quality C25/30, has a radius of 0.5 m and is 1 m long. Both end surfaces of the axle are constrained in all degrees of freedom except the rotation around the longitudinal direction of the axle. These degrees of freedom are rotated in opposite direction to represent the torsional load. The aim of this analysis is not to observe the point of failure but rather to see if the material models can handle the crack pattern due to torsion. The expected crack pattern for a structure subjected to a torsional load is cracks that propagate in an angle of 45◦ to the rotational axis. The torsional moment is carried through shear stresses, which for a circular member reaches its maximum at the surface of the member. If the torsional moment is the only loading, all other stresses, except the shear stress, are zero. This results in a principal tensile stress and a principal compressive stress of equal magnitude. The equal principal stresses result in a principal direction of 45◦ to the rotational axis. When the principal tensile stress eventually reaches the ultimate tensile strength of the concrete, cracks will initiate and spiral around the axle in the principal direction (Wight and MacGregor, 2009).
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(a) Concrete smeared cracking with linear tension stiffening
(b) Concrete damaged plasticity with linear tension stiffening
(c) Concrete damaged plasticity with tension stiffening according to Cornelissen et al. 1986 Figure 12: Crack patterns of axle subjected to torsion 22
The crack pattern from the three analyses can be seen in Fig. 12. The crack pattern obtained with the concrete smeared cracking model is in good agreement with the expected result, as the crack pattern spirals around the axle at an angle of approximately 45◦ . The two analyses made with the concrete damaged plasticity model do not agree with the expected result; although the cracks seem to spiral around the axle, the angle to the rotational axis is far from 45◦ . For a comparison of the load response of the different material models, see Fig. 13. [kNm] 80.
Moment
60.
40.
20.
0. 0.0
0.5
1.0
1.5
2.0
[mRadians]
Angle
CDP Cornelisen Tension Stiffening CDP Linear Tension Stiffening CSC Linear Tension Stiffening
Figure 13: Moment-angle curves
4.4
Reinforced concrete beam
To study and verify the interaction between the concrete material models and reinforcement in ABAQUS, a reinforced concrete beam will be analyzed. To keep the focus on the torsional behavior of concrete, a beam that fails because of shear cracks was chosen for this verification analysis. A number of beams exhibiting such a behavior were tested by Leonhardt and Walter (1962) to determine their maximum shear stress capacity. The beam referred to as beam 5 in Leonhardt and Walter was chosen for this verification example. The beam has earlier been examined by both Malm (2006) and Cervenka (2001) among others and hence there are a lot of results to compare with. The dimensions of the beam are shown in Fig. 14. The concrete material parameters were calculated from a cube strength of 33.5 MPa but with a tensile strength of 1.64 MPa according to Malm (2006) and Cervenka (2001). The reinforcement is composed of two Ø26 bars with a Young’s modulus of 200 GPa, Possion’s ratio of 0.3 and a yield stress of 500 MPa. The material behavior of the steel is assumed to be ideal elastic–plastic.
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P
P [mm]
As = 1066 mm2
320
t = 190
50 300
810
330
810
300
Figure 14: Leonhardt’s beam 5. The result from the experiments in Leonhardt and Walter (1962) show an ultimate load of 58.9 kN with a mid deflection of 2.57 mm. The results from the present finite element analysis are shown in Fig. 15. [kN] 60.
50.
Force
40.
30.
20.
10.
0. 0.0
0.5
1.0
1.5
2.0
Displacement
2.5
3.0
[mm]
CDP Cornelisen Tension Stiffening CDP Linear Tension Stiffening CSC Linear Tension Stiffening
Figure 15: Load–displacement curves. The concrete smeared cracking model gives an ultimate load of 65.5 kN and a mid deflection of 2.4 mm. An ultimate load of 59.5 kN is reached with the concrete damaged plasticity model using linear tension stiffening and at the point of failure the mid deflection is 2.5 mm. The same material model but with tension stiffening according to Cornelissen et al. fails at 45 kN with a mid deflection of 1.8 mm. The concrete smeared cracking model shows good agreement with the experimental results, but it tends to be a bit too stiff. When comparing with the results from Leonhardt and Walter (1962), the concrete damaged plasticity model with linear tension stiffening corresponds best of the material models and agrees very well the experimental results. The analysis made with tension stiffening according to Cornelissen et al. reaches its point of failure to early and the response is too soft, which also was found in previous verification examples.
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(a) Concrete smeared cracking with linear tension stiffening
(b) Concrete damaged plasticity with linear tension stiffening
(c) Concrete damaged plasticity with tension stiffening according to Cornelissen et al. 1986 Figure 16: Crack patterns for the reinforced concrete beam. The crack patterns from the analyses can be seen in Fig. 16. According to the experiments the beam should fail due to shear stresses and the main crack should propagate towards to the load point in an angle of approximately 45◦ . As can be seen in Fig. 16(a) the concrete smeared cracking model cannot capture this crack and this is probably the reason to the too stiff behavior of the model. Both concrete damaged plasticity models capture this crack very well. The underestimation by the concrete damaged plasticity model with Cornelissen et al. tension stiffening, might be explained by Fig. 15(c). Were it can be seen that the shear crack occur without presence of any flexural cracks, as in Fig. 15(b). This means that less strain engery has been consumed as fracture engery when the beam fails, i.e. an underestimation of the result.
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5
Conclusions
Modeling reinforced concrete using the finite element method can be very difficult. It is important that the analyst not only have good knowledge about FEM, but also about concrete as a material to be able to define the material models in the FEM software. The input parameters used to define the material models often have no theoretical origin, but are rather results from experiments. It is therefore necessary to gather the required knowledge about the parameters and to understand the effect they have on the material model before using it in an analysis. Even the smallest erroneousness in one input parameter, can develop into large errors in the analysis due to the complex interaction between the parameters. Difficulties do not only arise when defining the material models, but also due to some technicalities in the FEM itself. The most important difficulties that have been encountered in this study are presented in section 3. Some of these are associated with the concrete material models in ABAQUS, and some of them can be encountered when using the FEM in general. It is important to be observant when one starts to alter parameters to overcome convergence problems. Otherwise it is easy to obtain a solution which is no longer relevant for the given problem. When performing analyses with the smeared cracking model in ABAQUS, it was difficult to obtain converging solutions; especially for complex reinforced concrete structures. The convergence and the result of the analyses were very dependent on the type and size of the element used. The only element that showed good convergence with the smeared cracking model was the four node element with reduced integration (CPS4R) and enhanced hourglass control turned on. Even with this element, the computational time was almost 10 times that of the analyses made with the concrete damaged plasticity model. The material model also shows a too stiff response compared with experimental results. This behavior is often observed for concrete material models based on fracture mechanics and is of course problematic in engineering applications. Also the enhanced hourglass control may contribute to the too stiff behavior. A further drawback is that the material model cannot capture shear cracks; this is probably due to the fixed crack approach and the crack detection method. Further, the material model cannot observe the post failure behavior for reinforced concrete structures; instead it stops shortly after the point of failure. The strength of the concrete smeared cracking model is the “easy to understand” input parameters, which are closely linked to normally used concrete parameters. In conclusion, the smeared cracking model is easy to define but it is hard to get a relevant and converging solution. Therefore, it is recommended to use this material model only in analyses of simple concrete structures. Analyses made with the concrete damaged plasticity material model in ABAQUS, in most cases, shows good and fast convergence. The analyses also show results that correspond well with experimental data. The major drawback of the material model is that some of the input parameters have no counterparts in the normal material science of concrete; for example the dilation angle and the eccentricity used to define the plastic flow. But an advantage is that most of the material parameters can be chosen relatively free, in comparison with the smeared cracking model. This study showed that if the tensile behavior was defined according to Cornelissen et al. (1986), the structure reached the point 26
of failure to early, i.e. a too low failure load. In other studies, for example Malm (2006), it has been shown that this tensile behavior is the best and most accurate way to describe the tension softening in the concrete. Instead this study found that a linear crack opening law was most accurate. The concrete damaged plasticity model has no trouble to observe the post failure behavior and to visualize the numerous crack patterns that can occur in a concrete structure. In summary, the concrete damaged plasticity model is a very powerful analysis tool. It can handle many different load cases, including cyclic loading, and complex structures. It is also possible to use the ABAQUS/Explicit solver with this material model, which enables that a solution always can be obtained. Hence, the concrete damaged plasticity model is the recommended model to use for analysis of all type of concrete structures. Even though the concrete smeared cracking model captures the torsional crack pattern better than the concrete damaged plasticity model; the authors still recommend to use the concrete damaged plasticity model for structures subjected to torsional loading. This is because of the severe convergence problems associated with the concrete smeared cracking model. Although, the analyses made with the concrete damaged plasticity model in the scope of this study were not able to capture the correct crack pattern. The model succeeds in capturing shear cracks and should therefore also be able to describe torsional cracks.
5.1
Further research
Although the concrete damaged plasticity shows good agreement with experimental results there are still a lot of effects, such as the bonding interaction between the concrete and the reinforcement, that can be improved. Therefore it is necessary to investigate other concrete material models and other ways to model the reinforcement if very accurate results are required. There are very few papers about torsion of reinforced concrete members and it is therefore a subject that needs further research.
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References CEB-FIP, 1993. Ceb-fip model code 1990. Thomas Telford Ltd. Cervenka, J., May 2001. Atena program documentation part 4-1 Tutorial for progaram ATENA 2D. Prague. Cervenka, V., Jendele, L., Cervenka, J., 2010. ATENA Program Documentation - Part 1 Theory. Prague. Cornelissen, H., Hodijk, D., Reinhardt, H., 1986. Experimental determination of crack softening characteristics of normal weigth and lightweigth conrete. Heron 31 (2), –. Hibbit, H., Karlsson, B., Sorensen, E., 2009. Abaqus 6.9 Online Documentation. Dassault Systèmes. Lee, J., Fenves, G., 1998. Plastic-damage model for cyclic loading of concrete structures. Journal of engineering mechanics, ASCE 124 (8), 892–900. Leonhardt, F., Walter, R., 1962. Schubversuche an einfeldrigen stahlbetonbalken mit und ohne schubbewehrung. Deutscher suschuss für stahlbeton 151, –. Lovén, S. L., Rosengren, P., 2009. Design of deep concrete beams using strutand-tie, stress field and finite element methods. Master’s thesis, Royal institute of technology, Stockholm, Sweden. Lubliner, J., Oliver, J., Oller, S., Onate, E., 1989. A plastic-damage model for concrete. International journal of solids and structures 25 (3), 299–326. Malm, R., 2006. Shear cracks in concrete structures subjected to in-plane stresses. Ph.D. thesis, Royla institute of technology, Stockholm, Sweden. Malm, R., Ansell, A., July 2008. Nonlinear analysis of thermally induced cracking of a concrete dam. ACI Structual Journal -, –. Wight, J. K., MacGregor, J. G., 2009. Reinforced concrete mechanics and design, fifth edition Edition. Pearson education.
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