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245 lines
21 KiB
Markdown
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# 23.6 Concrete
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• “Concrete smeared cracking,” Section 23.6.1
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• “Cracking model for concrete,” Section 23.6.2
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• “Concrete damaged plasticity,” Section 23.6.3
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# 23.6.1 CONCRETE SMEARED CRACKING
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Products: Abaqus/Standard Abaqus/CAE
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# References
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• “Material library: overview,” Section 21.1.1
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• “Inelastic behavior,” Section 23.1.1
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• \*CONCRETE
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• \*TENSION STIFFENING
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• \*SHEAR RETENTION
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• \*FAILURE RATIOS
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• “Defining concrete smeared cracking” in “Defining plasticity,” Section 12.9.2 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
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# Overview
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The smeared crack concrete model in Abaqus/Standard:
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• provides a general capability for modeling concrete in all types of structures, including beams, trusses, shells, and solids;
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• can be used for plain concrete, even though it is intended primarily for the analysis of reinforced concrete structures;
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• can be used with rebar to model concrete reinforcement;
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• is designed for applications in which the concrete is subjected to essentially monotonic straining at low confining pressures;
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• consists of an isotropically hardening yield surface that is active when the stress is dominantly compressive and an independent “crack detection surface” that determines if a point fails by cracking;
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• uses oriented damaged elasticity concepts (smeared cracking) to describe the reversible part of the material’s response after cracking failure;
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• requires that the linear elastic material model (see “Linear elastic behavior,” Section 22.2.1) be used to define elastic properties; and
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• cannot be used with local orientations (see “Orientations,” Section 2.2.5).
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See “Inelastic behavior,” Section 23.1.1, for a discussion of the concrete models available in Abaqus.
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# Reinforcement
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Reinforcement in concrete structures is typically provided by means of rebars, which are one-dimensional strain theory elements (rods) that can be defined singly or embedded in oriented surfaces. Rebars
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are typically used with metal plasticity models to describe the behavior of the rebar material and are superposed on a mesh of standard element types used to model the concrete.
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With this modeling approach, the concrete behavior is considered independently of the rebar. Effects associated with the rebar/concrete interface, such as bond slip and dowel action, are modeled approximately by introducing some “tension stiffening” into the concrete modeling to simulate load transfer across cracks through the rebar. Details regarding tension stiffening are provided below.
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Defining the rebar can be tedious in complex problems, but it is important that this be done accurately since it may cause an analysis to fail due to lack of reinforcement in key regions of a model. See “Defining reinforcement,” Section 2.2.3, for more information regarding rebars.
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# Cracking
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The model is intended as a model of concrete behavior for relatively monotonic loadings under fairly low confining pressures (less than four to five times the magnitude of the largest stress that can be carried by the concrete in uniaxial compression).
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# Crack detection
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Cracking is assumed to be the most important aspect of the behavior, and representation of cracking and of postcracking behavior dominates the modeling. Cracking is assumed to occur when the stress reaches a failure surface that is called the “crack detection surface.” This failure surface is a linear relationship between the equivalent pressure stress, p, and the Mises equivalent deviatoric stress, q, and is illustrated in Figure 23.6.1–5. When a crack has been detected, its orientation is stored for subsequent calculations. Subsequent cracking at the same point is restricted to being orthogonal to this direction since stress components associated with an open crack are not included in the definition of the failure surface used for detecting the additional cracks.
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Cracks are irrecoverable: they remain for the rest of the calculation (but may open and close). No more than three cracks can occur at any point (two in a plane stress case, one in a uniaxial stress case). Following crack detection, the crack affects the calculations because a damaged elasticity model is used. Oriented, damaged elasticity is discussed in more detail in “An inelastic constitutive model for concrete,” Section 4.5.1 of the Abaqus Theory Guide.
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# Smeared cracking
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The concrete model is a smeared crack model in the sense that it does not track individual “macro” cracks. Constitutive calculations are performed independently at each integration point of the finite element model. The presence of cracks enters into these calculations by the way in which the cracks affect the stress and material stiffness associated with the integration point.
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# Tension stiffening
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The postfailure behavior for direct straining across cracks is modeled with tension stiffening, which allows you to define the strain-softening behavior for cracked concrete. This behavior also allows for the effects of the reinforcement interaction with concrete to be simulated in a simple manner. Tension
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stiffening is required in the concrete smeared cracking model. You can specify tension stiffening by means of a postfailure stress-strain relation or by applying a fracture energy cracking criterion.
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# Postfailure stress-strain relation
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Specification of strain softening behavior in reinforced concrete generally means specifying the postfailure stress as a function of strain across the crack. In cases with little or no reinforcement this specification often introduces mesh sensitivity in the analysis results in the sense that the finite element predictions do not converge to a unique solution as the mesh is refined because mesh refinement leads to narrower crack bands. This problem typically occurs if only a few discrete cracks form in the structure, and mesh refinement does not result in formation of additional cracks. If cracks are evenly distributed (either due to the effect of rebar or due to the presence of stabilizing elastic material, as in the case of plate bending), mesh sensitivity is less of a concern.
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In practical calculations for reinforced concrete, the mesh is usually such that each element contains rebars. The interaction between the rebars and the concrete tends to reduce the mesh sensitivity, provided that a reasonable amount of tension stiffening is introduced in the concrete model to simulate this interaction (Figure 23.6.1–1).
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<details>
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<summary>line</summary>
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| Strain, ε | Stress, σ |
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| --------- | --------- |
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| 0 | 0 |
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| εₜᵘ = σₜᵘ/E | σₜᵘ |
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| εₜ | 0 |
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</details>
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Figure 23.6.1–1 “Tension stiffening” model.
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The tension stiffening effect must be estimated; it depends on such factors as the density of reinforcement, the quality of the bond between the rebar and the concrete, the relative size of the concrete aggregate compared to the rebar diameter, and the mesh. A reasonable starting point for relatively heavily reinforced concrete modeled with a fairly detailed mesh is to assume that the strain softening after failure reduces the stress linearly to zero at a total strain of about 10 times the strain at failure. The strain at failure in standard concretes is typically 10−4 , which suggests that tension stiffening that reduces the stress to zero at a total strain of about $1 0 ^ { - 3 }$ is reasonable. This parameter should be calibrated to a particular case.
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The choice of tension stiffening parameters is important in Abaqus/Standard since, generally, more tension stiffening makes it easier to obtain numerical solutions. Too little tension stiffening will cause the local cracking failure in the concrete to introduce temporarily unstable behavior in the overall response of the model. Few practical designs exhibit such behavior, so that the presence of this type of response in the analysis model usually indicates that the tension stiffening is unreasonably low.
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Input File Usage: Use both of the following options:
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\*CONCRETE
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\*TENSION STIFFENING, TYPE=STRAIN (default)
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Concrete
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Smeared Cracking: Suboptions→Tension Stiffening: Type: Strain
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# Fracture energy cracking criterion
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As discussed earlier, when there is no reinforcement in significant regions of a concrete model, the strain softening approach for defining tension stiffening may introduce unreasonable mesh sensitivity into the results. Crisfield (1986) discusses this issue and concludes that Hillerborg’s (1976) proposal is adequate to allay the concern for many practical purposes. Hillerborg defines the energy required to open a unit area of crack as a material parameter, using brittle fracture concepts. With this approach the concrete’s brittle behavior is characterized by a stress-displacement response rather than a stress-strain response. Under tension a concrete specimen will crack across some section. After it has been pulled apart sufficiently for most of the stress to be removed (so that the elastic strain is small), its length will be determined primarily by the opening at the crack. The opening does not depend on the specimen’s length (Figure 23.6.1–2).
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# Implementation
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The implementation of this stress-displacement concept in a finite element model requires the definition of a characteristic length associated with an integration point. The characteristic crack length is based on the element geometry and formulation: it is a typical length of a line across an element for a first-order element; it is half of the same typical length for a second-order element. For beams and trusses it is a characteristic length along the element axis. For membranes and shells it is a characteristic length in the reference surface. For axisymmetric elements it is a characteristic length in the r–z plane only. For cohesive elements it is equal to the constitutive thickness. This definition of the characteristic crack length is used because the direction in which cracks will occur is not known in advance. Therefore, elements with large aspect ratios will have rather different behavior depending on the direction in which
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<details>
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<summary>line</summary>
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| u, displacement | Stress, σ |
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| --------------- | --------- |
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| 0 | 0 |
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| u₀ | σₜᵘ |
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</details>
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Figure 23.6.1–2 Fracture energy cracking model.
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they crack: some mesh sensitivity remains because of this effect, and elements that are as close to square as possible are recommended.
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This approach to modeling the concrete’s brittle response requires the specification of the displacement $u _ { 0 }$ at which a linear approximation to the postfailure strain softening gives zero stress (see Figure 23.6.1–2).
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The failure stress, $\sigma _ { t } ^ { u }$ , occurs at a failure strain (defined by the failure stress divided by the Young’s modulus); however, the stress goes to zero at an ultimate displacement, $u _ { 0 }$ , that is independent of the specimen length. The implication is that a displacement-loaded specimen can remain in static equilibrium after failure only if the specimen is short enough so that the strain at failure, $\varepsilon _ { t } ^ { u }$ , is less than the strain at this value of the displacement:
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$$
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\varepsilon_ {t} ^ {u} < u _ {0} / L,
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$$
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where L is the length of the specimen.
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Input File Usage: Use both of the following options:
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\*CONCRETE
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\*TENSION STIFFENING, TYPE=DISPLACEMENT
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Concrete Smeared Cracking: Suboptions→Tension Stiffening: Type: Displacement
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# Obtaining the ultimate displacement
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The ultimate displacement, $u _ { 0 } .$ , can be estimated from the fracture energy per unit area, $G _ { f }$ , as $u _ { 0 } =$ $2 G _ { f } / \sigma _ { t } ^ { u }$ , where $\sigma _ { t } ^ { u }$ is the maximum tensile stress that the concrete can carry. Typical values for $u _ { 0 }$ are 0.05 mm $( 2 \times 1 0 ^ { - 3 }$ in) for a normal concrete to 0.08 mm $( 3 \times 1 0 ^ { - 3 } \ \mathrm { i n } )$ for a high strength concrete. A typical value for $\varepsilon _ { t } ^ { u }$ is about $1 0 ^ { - 4 }$ , so that the requirement is that $L < 5 0 0$ mm (20 in).
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# Critical length
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If the specimen is longer than the critical length, L, more strain energy is stored in the specimen than can be dissipated by the cracking process when it cracks under fixed displacement. Some of the strain energy must, therefore, be converted into kinetic energy, and the failure event must be dynamic even under prescribed displacement loading. This implies that, when this approach is used in finite elements, characteristic element dimensions must be less than this critical length, or additional (dynamic) considerations must be included. The analysis input file processor checks the characteristic length of each element using this concrete model and will not allow any element to have a characteristic length that exceeds $u _ { 0 } / \varepsilon _ { t } ^ { u }$ . You must remesh with smaller elements where necessary or use the stress-strain definition of tension stiffening. Since the fracture energy approach is generally used only for plain concrete, this rarely places any limit on the meshing.
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# Cracked shear retention
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As the concrete cracks, its shear stiffness is diminished. This effect is defined by specifying the reduction in the shear modulus as a function of the opening strain across the crack. You can also specify a reduced shear modulus for closed cracks. This reduced shear modulus will also have an effect when the normal stress across a crack becomes compressive. The new shear stiffness will have been degraded by the presence of the crack.
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The modulus for shearing of cracks is defined as $\varrho G .$ , where G is the elastic shear modulus of the uncracked concrete and is a multiplying factor. The shear retention model assumes that the shear stiffness of open cracks reduces linearly to zero as the crack opening increases:
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$$
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\varrho = \left(1 - \varepsilon / \varepsilon^ {\max}\right) \quad \text {for} \quad \varepsilon < \varepsilon^ {\max}, \qquad \varrho = 0 \quad \text {for} \quad \varepsilon \geq \varepsilon^ {\max},
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$$
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where is the direct strain across the crack and $\varepsilon ^ { \mathrm { m a x } }$ is a user-specified value. The model also assumes that cracks that subsequently close have a reduced shear modulus:
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$$
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\varrho = \varrho^ {\text { close }} \quad \text { for } \quad \varepsilon < 0,
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$$
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where you specify $\varrho ^ { \mathrm { c l o s e } }$ .
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$\varrho ^ { \mathrm { c l o s e } }$ and $\varepsilon ^ { \mathrm { m a x } }$ can be defined with an optional dependency on temperature and/or predefined field variables. If shear retention is not included in the material definition for the concrete smeared cracking model, Abaqus/Standard will automatically invoke the default behavior for shear retention such that the shear response is unaffected by cracking (full shear retention). This assumption is often reasonable: in many cases, the overall response is not strongly dependent on the amount of shear retention.
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Input File Usage: Use both of the following options:
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$$
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\begin{array}{l} * \text { CONCRETE } \\ * \text {SHEAR RETENTION} \\ \end{array}
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$$
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Concrete Smeared Cracking: Suboptions→Shear Retention
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# Compressive behavior
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When the principal stress components are dominantly compressive, the response of the concrete is modeled by an elastic-plastic theory using a simple form of yield surface written in terms of the equivalent pressure stress, p, and the Mises equivalent deviatoric stress, q; this surface is illustrated in Figure 23.6.1–5. Associated flow and isotropic hardening are used. This model significantly simplifies the actual behavior. The associated flow assumption generally over-predicts the inelastic volume strain. The yield surface cannot be matched accurately to data in triaxial tension and triaxial compression tests because of the omission of third stress invariant dependence. When the concrete is strained beyond the ultimate stress point, the assumption that the elastic response is not affected by the inelastic deformation is not realistic. In addition, when concrete is subjected to very high pressure stress, it exhibits inelastic response: no attempt has been made to build this behavior into the model.
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The simplifications associated with compressive behavior are introduced for the sake of computational efficiency. In particular, while the assumption of associated flow is not justified by experimental data, it can provide results that are acceptably close to measurements, provided that the range of pressure stress in the problem is not large. From a computational viewpoint, the associated flow assumption leads to enough symmetry in the Jacobian matrix of the integrated constitutive model (the “material stiffness matrix”) such that the overall equilibrium equation solution usually does not require unsymmetric equation solution. All of these limitations could be removed at some sacrifice in computational cost.
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You can define the stress-strain behavior of plain concrete in uniaxial compression outside the elastic range. Compressive stress data are provided as a tabular function of plastic strain and, if desired, temperature and field variables. Positive (absolute) values should be given for the compressive stress and strain. The stress-strain curve can be defined beyond the ultimate stress, into the strain-softening regime.
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Input File Usage: \*CONCRETE
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Concrete Smeared Cracking
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# Uniaxial and multiaxial behavior
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The cracking and compressive responses of concrete that are incorporated in the concrete model are illustrated by the uniaxial response of a specimen shown in Figure 23.6.1–3.
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When concrete is loaded in compression, it initially exhibits elastic response. As the stress is increased, some nonrecoverable (inelastic) straining occurs and the response of the material softens. An ultimate stress is reached, after which the material loses strength until it can no longer carry any stress. If the load is removed at some point after inelastic straining has occurred, the unloading response is softer than the initial elastic response: the elasticity has been damaged. This effect is ignored in the model, since we assume that the applications involve primarily monotonic straining, with only occasional, minor unloadings. When a uniaxial concrete specimen is loaded in tension, it responds elastically until, at a stress that is typically 7%–10% of the ultimate compressive stress, cracks form. Cracks form so quickly that, even in the stiffest testing machines available, it is very difficult to observe the actual behavior. The
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<!-- source-page: 450 -->
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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A["Start of inelastic behavior"] --> B["Cracking failure"]
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B --> C["Softening"]
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C --> D["Failure point in compression (peak stress)"]
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D --> E["Implied (elastic) unload/reload response"]
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E --> F["Unload/reload response"]
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F --> G["Idealized (elastic) unload/reload response"]
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G --> H["Stress"]
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```
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</details>
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Figure 23.6.1–3 Uniaxial behavior of plain concrete.
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model assumes that cracking causes damage, in the sense that open cracks can be represented by a loss of elastic stiffness. It is also assumed that there is no permanent strain associated with cracking. This will allow cracks to close completely if the stress across them becomes compressive.
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In multiaxial stress states these observations are generalized through the concept of surfaces of failure and flow in stress space. These surfaces are fitted to experimental data. The surfaces used are shown in Figure 23.6.1–4 and Figure 23.6.1–5.
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# Failure surface
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You can specify failure ratios to define the shape of the failure surface (possibly as a function of temperature and predefined field variables). Four failure ratios can be specified:
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• The ratio of the ultimate biaxial compressive stress to the ultimate uniaxial compressive stress.
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• The absolute value of the ratio of the uniaxial tensile stress at failure to the ultimate uniaxial compressive stress.
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• The ratio of the magnitude of a principal component of plastic strain at ultimate stress in biaxial compression to the plastic strain at ultimate stress in uniaxial compression.
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