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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.

#### 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.

In the computation using the stress resultant macro model, we also use two small secondary computations on the console very much the same as in the first example, except for the cross section b×d = 30×40(cm) and the length of element Le = 0.25m for the frame case (see Fig.1.13). The first computation is carried out to find the relation of moment-curvature of the beam macro elements, when only moment with no axial force is applied, M 6= 0 and N = 0. The second computation is performed to obtain the relation of moment-curvature of the column macro element, where there are combined effects of moment and of axial force, M 6= 0 and N 6= 0. In this particular case, the column is first loaded by a constant axial force N = 700KN followed by imposed moment. The results for moment-curvature relationship is given in Fig.1.14 for both cases.

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

In this section we first review the basic assumptions for design computation in reinforcedconcrete frame components, some limitations of concrete and steel materials proposed EC2 as introduced in [18] and [19]. With these formulations in hand, we can easily predict the figure of yield and ultimate moments with respect to the axial force in rectangular cross section submitted to the axial force and bending moment.

**Material properties and basic assumptions**

We can have many kinds of simplified material constitutive models for both steel reinforcement and concrete as suggested by EC2, and the other using codes. Hence, in this work, it is first necessary to give some guidelines and conditions for material used herein.

Generally, we use some basic assumptions in the design computation for beam or column: one is that plane sections remain plane once loaded, and the strain in reinforcement is the same as the strain in the concrete at the same level (no bond slip). These assumptions are universally accepted for the design of members containing bonded ordinary reinforcement. With perfect bond, the change in strain in the steel is assumed to be the same as the change in strain in the concrete. In computation of the resistance for a reinforced-concrete column, many factors in material properties of both steel and concrete affect to the load carrying capacity of elements. However, in a simplified version for the consideration to follow, we just concentrate on main factors, such as the yield and ultimate strengths, strain limits and moduli. At first we introduce steel material model proposed in EC2. Absolute limit is not given for the maximum tensile strain in the reinforcement, but there is clearly a limiting strain that defines the failure of any particular type of reinforcement. In Figure.2.1 we introduce the stress-strain diagram in steel material used for design, where the curve B is expressed for design while the curve A is considered as the idealized curve. Strain limit eu for normal ductility steel is equal to 0.025 and for high ductility steel it is equal to 0.05. More discussions and informations about steel material can be found in [18].

**Prediction for ultimate moment and axial force relation**

From two formulas in (2.1) and (2.2), we can readily obtain the corresponding relationship between the ultimate moment and axial force (see Fig.2.5). There are two cases related to load compression for a particular rectangular cross-section. The first case is called the large-eccentricity load, where in the couple of (M, N), the value of M is very big compared to the value of N, so that we obtain a large value e = M/N. In this case, in comparison to the limit-eccentricity elimit of the section, we shall have e > elimit . This limiteccentricity is considered as the frontier between large-eccentricity and small-eccentricity (see [18]). In case of the large-eccentricity, cross-section is divided clearly into two parts with compression and traction zone, with the neutral axis that always remains within the section. The failure process in the cross-section will derive from the yielding process of steel in the traction zone. We use an increasing relation of ultimate moment and axial force in the case of large-eccentricity, which means when axial force increases we shall also obtain a increasing ultimate moment.

In the second case, we consider the small-eccentricity, where the eccentric value e=M/N is smaller than the limit-eccentricity elimit . The failure process will derive mostly from the failure process of concrete in compressive zone. We shall obtain a descending relation of ultimate moment and axial force. This further implies that in case of smalleccentricity, the axial force increase will also be accompanied by a descending values of ultimate moment.

We propose these relations as shown in Fig.2.5, as the functions of yield and ultimate moments with respect to the axial force. For either of those relations, we note three particular points: the first point is when axial force is equal to zero; the second point is when the value of ultimate moment of the section reaches maximum value Mmax, corresponding to axial force N∗; the third point is when axial force reaches the ultimate value, that corresponds to the moment be equal to zero M = 0. For each rectangular cross-sections of reinforced concrete column, and each value of axial force N, we can thus easily determine the correspondence of yield and ultimate moment (My and Mu).

**Parameter identifications and macro model for stressresultant failure criteria in frame**

In this section, we elaborate upon the tasks of identifying the limit values of moment (Mc, My, Mu) and hardening/softening moduli (K1, K2, K3) for stress-resultant macro model. We also deal with the tasks of creating the functions of moment-axial force and/or curvature-axial force relations. The relations identified in this manner are embedded into threshold functions for combined action of bending moments and axial force, which can be used for computing the complete failure of a reinforced concrete beam-column element with the rectangular cross-section. We present in Fig.2.6 the multi-fiber beam used for such an identification procedure. The computation is carried out under free-end rotation and axial load, which is also indicated in Fig.2.6.

**Parameter identification for doubly reinforced concrete rectangular section**

Here we will consider the case of rectangular cross-section for reinforced concrete beamcolumn element. The section dimensions are denoted as (b×h), with two layers of reinforcements with the areas (A′s, As) that are embedded symmetrically in both compressive and tension zone of section. The identification procedure is carried out on the built-in beam. The length of the beam L is equal to the length of single element Le, that is used for this computation. At free end of the console, we impose an axial load and rotational displacement.

We present a representative for the built-in beam-column, which is used to describe for all needed steps in the process of parameter identification. The dimensions of rectangular cross-section of the beam are b×h equal to 30×40(cm). The length of the beam is as the same as the length of element, for this built-in beam-column we choose L = Le equal to 0.25m. Inside the cross-section, 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 console.

The concrete has compressive strength f′c = 30MPa, tensile strength fct = 1.8MPa, modulus of elasticity Eb = 28,600MPa. By using the multi-fiber model for parameter Stress-resultant models for optimal design of reinforced-concrete frames identification, we use the divided rectangular cross-section of the beam with 20 concrete layers and 8 discrete fibers for describing the steel reinforcements. The geometry and detail of the beam-column can be seen in Fig.2.10.

In the first step, we compute the limit values of axial force for this beam without the effort of the imposed rotation displacement, which purposes to obtain the third point, Nu, in the relationship of ultimate moment and axial force. By imposing only the axial displacement we can obtain the yield and ultimate axial force (Ny, Nu) as presented in chosen value of axial load N specified by the ratio n = N/Ny we can consider the steps n := 0 : 0.1 : 1.05, and thus obtain the corresponding diagram for moment-curvature for 12 different cases presented in Fig.2.12. We can separate these diagrams into two main groups: the first one for axial force N changes of n from zero to 0.4 of yield axial force Ny and the second for axial force with changes of n from 0.4 to 1.05. The first group corresponds to the large-eccentricity cases, where the failure in the reinforced-concrete beam is induced by the yielding of reinforcement in tension zone. With the increasing values of axial force we can obtain the corresponding increase in values of cracking, yielding and ultimate moment (Mc, My, Mu), where the values of My and Mu will differ clearly. We also have fairly long part with hardening response diffuse of plastic phase. The second group corresponds to the small-eccentricity case, where the failure process is governed by the compressive failure of concrete in compression zone. With the increase of axial force, we obtain a small difference between the values of yield and ultimate moments, and the value of ultimate moment can even become smaller than the yield moment (Mu 6My).

**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 Stress-resultant models for optimal design of reinforced-concrete frames

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 **