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## Softening plastic hinge in bending

Behavior of the softening plastic hinge is described by a plastic softening law, presented in Fig. 2.11. It associates the moment in the hinge t to the jump in rotation . When the hinge forms, the rotational jump is zero and the moment is equal to the ultimate moment Mu of the cross-section. If the imposed nodal displacements of the finite element are increased, the carrying capacity of the hinge reduces. The moment t decreases, while the rotational jump increases. This is referred to as plastic softening. If the imposed nodal displacements of the finite element are reduced, the rotational jump remains the same, representing plastic deformation. The moment in the hinge decreases in such a way to remain in equilibrium with the moment in the bulk, as demanded by equation (2.38).

This process is called elastic unloading. When t reaches the admissible value again (in absolute value), the plastic softening continues and changes accordingly to the sign of the moment. The diagram in Fig. 2.11 depends on the present axial force N, and may be different for positive and negative bending moment, if the cross-section is not symmetric. In this chapter, we limit ourselves to symmetrical behavior for both load signs. Mathematical representation of the described behavior is condensed in the following equations, which can be derived by the principle of maximum plastic dissipation, see e.g. [54].

### Bending response in the softening phase

This section describes the computational procedure for an element in the softening phase. It is applied if the current value of the discontinuity flag crack(e) n+1 = true, which happens if the discontinuity already existed in the previous step, or if the carrying capacity of the element was exceeded in this iteration, see equation (2.73). In any case, the hardening internal variables take the values from the previous step, which are the last converged results. ¯(k) p,n+1 = ¯ p,n, ¯(k) n+1 = ¯n (2.74) We start by assuming a trial solution, keeping the softening internal variables at the values from the previous step.

The moment in the bulk and the moment at the discontinuity are computed according to equations (2.42) and (2.38), respectively.

#### Computation of nodal degrees of freedom

In this section we will describe the computations of phase (B) of k-th iteration, mentioned in the introduction ofsection 2.3. In this phase, a linearized form of equilibrium equations (2.58) is solved to provide the k-th update of the nodal displacements/rotations at pseudo-time point n+1. The computation is performed with known values of internal variables ¯(e),(k) p,n+1 , ¯(e),(k) n+1 , (e),(k) n+1 and ¯¯ (e),(k) n+1 for each finite element, freshly updated in preceding phase (A) of the same iteration. Since the nodal degrees of freedom are generally common to several finite elements, the equations of phase (B) must be handled on structural (global) level. Hence, they are also referred to as global equations.

The first of equations (2.58) would be sufficient for calculating the new values of generalized displacements d (e),(k) n+1 , if all rotational jumps (e),(k) n+1 were fixed at the values, computed in phase (A). To improve convergence, however, it is useful to update the rotational jumps as well. For that purpose, the second of equations (2.58) are engaged. Actually, they have once already been satisfied by using expression (2.38) for the moment at the discontinuity, but that equality held for the displacements from the previous iteration d (e),(k−1) n+1 . Updating the displacements would disrupt the equilibrium between the moment at the discontinuity and the moment in the bulk of the element, unless the rotational jumps are updated as well. Solving the whole system of equations (2.58) therefore promises a more accurate solution.

**Linearization of equilibrium equations**

The first of equations (2.58) ensures the equilibrium of the structure, i.e. of its each and every node. It is linearized around the current values of nodal degrees of freedom of the structure dstr,(k−1) n+1 . @f int,str,(k) n+1 @dstr,(k−1) | n{+z1 } K str,(k) n+1 Ddstr,(k) +1 = f ext,str n+1 −f int,str,(k) n+1 , @f ext,str n+1 @dstr,(k−1) n+1 = 0.

**Failure of a cantilever beam**

We consider a cantilever beam of rectangular cross-section for three different load cases, presented in Fig. 2.13.

In the first load case, the beam is loaded with moment at the free end. In order to perform the analysis up to the total collapse, the load is controlled with imposed rotation of the free end. The following geometrical and material properties are chosen: length of the beam L = 2.5m, elastic bending stiffness EI = 77650kNm2, elastic axial stiffness EA = 3727200kN, moment at elasticity limit Mc = 37.9kNm, yield moment My = 268kNm, ultimate moment Mu = 274kNm, hardening moduli H1 = 29400kNm2 and H2 = 272kNm2, and softening modulus K = −18000kNm. Response of the structure is computed for meshes of 1, 2, 5 and 10 finite elements.

**Table of contents :**

BIBLIOGRAPHIC-DOCUMENTALISTIC INFORMATION AND ABSTRACT

BIBLIOGRAFSKO-DOKUMENTACIJSKA STRAN IN IZVLECˇ EK

INFORMATION BIBLIOGRAPHIQUE-DOCUMENTAIRE ET RESUME

**1 INTRODUCTION **

1.1 Motivation

1.2 Theoretical background

1.3 Goals and outline of the thesis

**2 STRESS RESULTANT EULER-BERNOULLI BEAM FINITE ELEMENT WITH EMBEDDED DISCONTINUITY IN ROTATION **

2.1 Introduction

2.2 Finite element formulation

2.2.1 Kinematics

2.2.2 Derivation of operator G

2.2.3 Relations between global and local quantities

2.2.4 Virtual work equation

2.2.5 Constitutive models

2.3 Computational procedure

2.3.1 Computation of internal variables

2.3.2 Computation of nodal degrees of freedom

2.4 Numerical examples

2.4.1 Failure of a cantilever beam

2.4.2 Failure of simply supported and clamped beams

2.4.3 Four point bending test of a simply supported beam

2.4.4 Two story reinforced concrete frame

2.5 Concluding remarks

**3 MULTI-LAYER EULER-BERNOULLI BEAM FINITE ELEMENT WITH LAYER-WISE EMBEDDED DISCONTINUITIES IN AXIAL DISPLACEMENT **

3.1 Introduction

3.2 Finite element formulation

3.2.1 Kinematics

3.2.2 Relations between global and local quantities

3.2.3 Virtual work equation

3.2.4 Derivation of operators GR and GV

3.2.5 Constitutive models

3.3 Computational procedure

3.3.1 Computation of internal variables

3.3.2 Computation of nodal degrees of freedom

3.4 Numerical examples

3.4.1 One element tension and compression tests

3.4.2 Cantilever beam under end moment

3.4.3 Cantilever beam under end transversal force

3.4.4 Two story reinforced concrete frame

3.5 Concluding remarks

**4 MULTI-LAYER TIMOSHENKO BEAM FINITE ELEMENT WITH LAYER-WISE EMBEDDED DISCONTINUITIES IN AXIAL DISPLACEMENT **

4.1 Introduction

4.2 Finite element formulation

4.2.1 Kinematics

4.2.2 Derivation of operator G

4.2.3 Relations between global and local quantities

4.2.4 Virtual work equation

4.2.5 Constitutive models

4.3 Computational procedure

4.3.1 Computation of internal variables

4.3.2 Computation of nodal degrees of freedom

4.4 Numerical examples

4.4.1 One element tension and compression tests

4.4.2 Cantilever beam under end moment

4.4.3 Cantilever beam under end transversal force

4.4.4 Simply supported beam

4.4.5 Reinforced concrete portal frame

4.4.6 Two story reinforced concrete frame

4.5 Concluding remarks

**5 VISCOUS REGULARIZATION OF SOFTENING RESPONSE FOR MULTI-LAYER TIMOSHENKO BEAM FINITE ELEMENT **

5.1 Introduction

5.2 Virtual work equation

5.3 Computation of internal variables

5.3.1 Discontinuity in concrete layer

5.3.2 Discontinuity in reinforcement layer

5.4 Computation of nodal degrees of freedom

5.5 Numerical examples

5.5.1 One element tension and compression tests

5.5.2 Tension and compression tests on a mesh of several elements

5.5.3 Cantilever beam under end moment

5.6 Concluding remarks

**6 CONCLUSIONS **

RAZˇSIRJENI POVZETEK

**BIBLIOGRAPHY **