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## Multi-fiber beam model with embedded strain discontinuity in fibers

In this section we present shortly about the multi fibers beam model, which provides the basis for identifying the parameters of stress resultant model. This multi-fiber model is helpful to analyse and handle a beam-column with multi materials embedded in one crosssection in general or for reinforced concrete in particular. In this method, we also present the steps for embedding strain discontinuity in each fibers, including concrete fibers and steel reinforced fibers. The constitutive laws of concrete and steel are introduced as the corresponding Elasto-Plastic material models with softening.

### Multi-fibers beam model

The beam in Fig. 1.7 is introduced as the fine micro scale beam model used to explore the details of inelastic constitutive behavior of material such as the reinforced concrete. The analysis of this kind is quite equivalent to nonlinear homogenization,which will provide the best possible definition of (’macro’) stress resultant beam model. The analysis is carried out on the built-in beam with the length L, divided in a number of elements with length le. The beam rectangular cross-section with the width b and the depth h is divided into a number of concrete fibers and reinforced fibers. The coordinate yi denotes the distance from the neutral line to the centre of a given fiber. This beam is further submitted to a loading program with a constant value of axial force N and an increasing value of moment M applied at free end.

#### Embedding strain discontinuity in fiber

Each fiber of this beam element can be considered as a truss-bar element, for which the stress-strain behavior is selected as either concrete or steel material. For any fiber placed at any point in the cross section, we will consider that the strain discontinuity can be embedded (see [1] and [2]). The fiber axial strain can be split into the regular part and singular part ei x = ei x+aidi xc.

**Constitutive laws of the concrete and the steel**

In this part we present two constitutive models for concrete and steel materials referred to as Elasto-Plastic-Softening (EPSM). Either model is developed to include the strain discontinuity introduced in the previous section 3.2, ([2]). In the constitutive law of concrete, the compression phase and the tension phase are different. For the concrete material, the Young modulus is denoted as Eb, the ultimate stress and the plastic stress in the compression phase are denoted as f′c and fyc, the ultimate stress in the tension phase is denoted as ft . All these values of stress are correlated with the different strains eu, ey and et , which can be obtained from the simple tests of the concrete sample. For clarity we picture the concrete model with the stress/strain relation described in Fig.3.4. We denote with K1 the hardening tangent value of the compression phase, and with K2 andK3 we denote for the softening phases in compression and tension. In the computation, they are used as the negative values ( see [2]).

All values of the Young modulus, strength, tangent modulus for both hardening and softening phases are also compared to the other model. In the tension phase, no hardening part is considered. Therefore, the stress reaching the ultimate value ft , will trigger the softening response with the tangent modulus K3 < 0.

**Reinforced concrete simple beam**

The beam used for test is of rectangular cross-section b×d = 20×50cm and the length L = 5m. Two reinforcement longitudinal bars of diameter f = 8mm are placed at the top side, and two with diameter f = 32mm are placed at the bottom side of the cross section. The concrete material parameters used in this example are the same as those in [3], (see Fig.3.4). Young’s modulus Eb = 37272MPa, fracture process zone threshold f′y = 30.6MPa, compressive strength f′c = 38.3MPa, tensile strength fct = 3.727MPa, hardening modulus in compression K1 = 9090MPa, softening modulus in compression K2 = −18165MPa and softening modulus in tension K3 = −30000MPa. The steel reinforcement material parameters chosen for this computation are: Young’s modulus Es = 200000MPa, yield stress fys = 400MPa and t = Et/Es = 0.0164. The stirrups with the diameter f = 8mm and the distance a = 100mm are placed along the beam. Detailed plan of the reinforcement is presented in Fig.1.9.

**Two-storey frame ultimate load computation**

In the second numerical example, we consider a reinforced-concrete frame with two floors and one span. The dimensions of the frame are detailed in the fig.1.12. The crosssection of both column and beam is b×d = 30×40(cm), . In both beam and column, 4f20mm of the longitudinal bar are placed at each side, and the stirrups f10mm at the distance a = 125mm are used along to the length of span and the height of two-storey. This example is based on the experiment presented in [3]. Two fixed vertical forces P=700KN are applied at two nodes on the top of the frame representing the effect of the dead load. The lateral force is imposed on one side at the top node with the values increasing from zero to the time of the complete collapse of the frame.

The finite element model used in the numerical computations is as follows: each column with the height h = 2m is divided into 8 elements with Le = 0.25m and each beam with the length L = 3.5m is divided into 14 elements with Le = 0.25m. The concrete has compressive strength f′c = 30MPa, tensile strength fct = 1.8MPa, modulus of elasticity Eb = 28,600MPa. All the details on material parameters and geometry for the test can be found in [3].

For obtaining the transversal force-displacement diagram by the multi-fiber model computation, the cross-section is divided into 20 layers of concrete. In the cross-section are embedded 8 fibers of reinforcement steel, 4 on each side. The dimensions of cross section can be seen in Fig.1.12. The result of this computation is obtained by FEAP program, and the relation of lateral load versus deflection at the top storey of frame can be described by the hidden line in the Fig.1.16 and be compared to the dash-dot line of the Stress-resultant models for optimal design of reinforced-concrete frames result of experiment.

**Table of contents :**

Contents

List of Figures

List of Tables

Introduction

**1 Stress resultant and multi-fiber beam model for bending failure **

1 Introduction

2 Embedded rotation discontinuity for Timoshenko beam

3 Multi-fiber beam model with embedded strain discontinuity in fibers

3.1 Multi-fibers beam model

3.2 Embedding strain discontinuity in fiber

3.3 Assembly of all fibers in a beam element

3.4 Constitutive laws of the concrete and the steel

4 Numerical examples

4.1 Reinforced concrete simple beam

4.2 Two-storey frame ultimate load computation

5 Conclusions

**2 Stress resultant and multi-fiber beam model for combined axial force and moment **

1 Introduction

2 Prediction for the relations of ultimate and yield moments with axial force

2.1 Material properties and basic assumptions

2.2 Formulas for doubly reinforced-concrete rectangular section

2.3 Prediction for ultimate moment and axial force relation

3 Parameter identifications and macro model for stress-resultant failure criteria in frame

3.1 Stress-resultant macro model for reinforced concrete frames and chosen material behavior of concrete and reinforcement

3.2 Parameter identification for doubly reinforced concrete rectangular section

3.3 Function identification for moment-axial force relation

3.4 Function identification for curvature-axial force relations

3.5 Moment and curvature yield functions for stress-resultant macro model at the time t

4 Numerical applications

4.1 Single-element console computation

4.2 Two-storey reinforced concrete frame computation

5 Conclusion

**3 Stress resultant and multi-fiber beam model with shear failure and crack spacing **

1 Introduction

2 Reinforced concrete model and its finite element implementation

2.1 Concrete, bond-slip and steel models

2.2 Numerical formulation of the reinforced concrete element

2.3 Numerical examples

3 Stress-resultant macro model: embedded displacement discontinuities

3.1 Theoretical formulation and finite element representation

3.2 Stress-resultant constitutive laws

4 Numerical examples

5 Conclusions

Conclusion

**Bibliography **