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5.4 Linear elastic analysis

5.5 Linear analysis with limited redistribution

### C5.4 Linear elastic analysis

For the determination of load effects, linear analysis may be used assuming: •uncracked cross sections, •linear stress-strain relationships and •mean value of the modulus of elasticity. With these assumptions, stresses are proportional to loads and therefore the superposition principle applies. For thermal deformation, settlement and shrinkage effects at the ultimate limit state (ULS), a reduced stiffness corresponding to the cracked sections, neglecting tension stiffening but including the effects of creep, may be assumed in accordance with [5.4(3)-EC2]. For the serviceability limit state (SLS) a gradual evolution of cracking should be considered.

### Comments.

- In the previous explanation the expression For the determination of load effects' was used, whereas [5.4(3)-EC2] admits "For the determination of action effects", therefore of all actions, including thermal deformation, settlement and shrinkage for which [5.4(3)-EC2] admits different assumptions, without which the effects of impressed deformations would be devastating and quantitatively incorrect.

- the fact that no limits were set to (xu/d) for the application of the linear analysis method at the ultimate limit states, does not mean that any value of (xu/d) may be used in design: it's opportune to observe a limit consistent with the method of linear elastic analysis with limited redistribution, for which xu/d < 0,45. It must be remembered that increasing values of xu/d the model uncertainty also increases and higher safety factors should be assumed for precaution.

C5.5 Linear elastic analysis with limited redistribution [5.5 - EC2]

At ultimate limit state plastic rotations occur at the most stressed sections. These rotations transfer to other zones the effect of further load increase, thus allowing to take, for the design of reinforcement, a reduced bending moment 5M, smaller than the moment M resulting from elastic linear design, provided that in the other parts of the structure the corresponding variations of load effects (viz. shear), necessary to ensure equilibrium, are considered.

Despite being named "Linear analysis with limited redistribution", this is a design method.

In clause [5.5(4)-EC2], in relation with continuous beams and slabs with ratio between adjacent spans in the range [0,5 - 2] expressions are given for the redistribution factor 5 in function of the concrete class, the type of steel and the xu/d ratio after redistribution. For instance, for concrete up to C50 and reinforcing steel of type B and C, respectively of average and high ductility, the expression is:

where:

5 is the ratio of the redistributed moment to the elastic bending moment Xu is the depth of the neutral axis at the ultimate limit state after redistribution d is the effective depth of the section

The limits of this formula are: 5 = 0,70 per (xu/d) = 0,208 and 5 = 1 per (xu/d) = 0,45

It must be considered that a redistribution carried out in observance of the ductility rules only ensures equilibrium at the ultimate limit state. Specific verifications are needed for the serviceability limit states. Very high redistributions, which may be of advantage at the ultimate limit states, very often must be lowered in order to meet the requirements of serviceability limit states. For the design of columns the elastic moments from frame action should be used without any redistribution.

5.6 Plastic analysis C56 Plastic analysis

Plastic analysis should be based either on the lower bound (static) method or on the upper bound

(kinematic) method for the check at ULS only.

### 5.6.1 Static method

It is based on the static theorem of the theory of plasticity, which states: "whichever load Q, to which a statically admissible tension field corresponds, is lower or equal to the ultimate load Qu". The expression "statically admissible" indicates a field that meets both the conditions of equilibrium and the boundary condition without exceeding the plastic resistance.

An important application of this method is the strut-and-tie scheme [5.6.4 - EC2]. Other applications are the management of shear by the method of varying 0 and the analysis of slabs by the equivalent frame analysis method [Annex I - EC2].

### 5.6.2 Kinematic method

In this method, the structure at ultimate limit states becomes a mechanism of rigid elements connected by yield hinges. The method is based on the kinematic theorem, which states: "every load Q, to which corresponds a kinematically admissible mechanism of collapse, is higher or equal to the ultimate load Qu".

The method is applied for continuous beams, frames and slabs (in this last case with the theory of yield lines.

For beams, clause [5.6.2)-EC2] states that the formation of plastic hinges is guaranteed provided that the following are fulfilled:

i) the area of tensile reinforcement is limited such that, at any section

(xu /d) < 0,25 for concrete strength classes < C50/60 (xu /d) < 0,15 for concrete strength classes > C55/67

ii) reinforcing steel is either Class B or C

iii) the ratio of the moments at intermediate supports to the moments in the span shall be between 0,5 and 2.

If not all the conditions above are fulfilled, the rotation capacity must be verified, by checking the required rotations against those allowed in accordance with [Fig. 5.6N-EC2].

It should be remembered that the plastic analysis methods shall only be used for checking ultimate limit states. Serviceability limit states requirements should be checked by specific verifications.

### 5.7 Non-linear analysis C57 Non-linear analysis

Non-linear analysis is a procedure for calculation of action effects, based on idealisations of the non-linear behaviour of materials [non-linear constitutive laws: for concrete cf. Eurocode 2, 3.1.5(1) expression (3.14) and Fig. 3.2; for steel 3.2.7(1) Fig. 3.8], of the elements and of the structure (cracking, second order effects), suitable for the nature of the structure and for the ultimate limit state under consideration.

It requires that the section geometry and reinforcement are defined, because it is a process of analysis. Resulting stresses are not proportional to the applied actions.

The process is developed by computer-aided calculations, by verifying equilibrium and compatibility at every load increase. Compatibility conditions are normally expressed by assigning to each section its moment - curvature law, and integrating the curvatures along the axis of the elements. Inelastic rotations are generally concentrated in the critical sections. Deformations due to shear are generally neglected, those in relation with axial load are taken into account only in case have significant influence on the solution. As the superposition principle does not apply because of the non-linearity, the calculations must be developed for each load condition: for each one it is conventionally assumed that the ultimate limit state is reached through a single proportional increase of the applied load.

In the case of elements mainly subjected to bending, trilinear idealizations of the moment / rotations law of each critical section can be adopted as in Fig. 4.6, representing the three following states:

- first state (elastic and linear): characterized by EI rigidity of the entirely reacting sections; it ends when the tensional strength of concrete is reached (cracking moment)

- second state (cracked): from the cracking moment to the moment corresponding to steel yielding, moment increases are related to the curvature increases on the basis of rigidity

Es As z(d-x), where As is the cross section of the tensioned reinforcement, z the lever arm, x the depth of the neutral axis. The rigidity can be increased by taking into account the contribution of concrete in tension between cracks ("tension stiffening"), but with caution in case of load cycles .

- third state (plastic): a third linear line can be idealized from the steel yielding clause to the point of failure moment. The line corresponds to a 0pl plastic rotation at the critical section, with a value that can be deducted from the diagram in Fig 5.6N of Eurocode 2 in function of the relative depth of the neutral axis. Following the evolution of response to actions, it is possible to verify the conditions for the serviceability limit states and for the ultimate limit state.

5.8 Analysis of second order C5.8 Second order effects with axial load effects with axial load C5.8.1. Definitions

5.8.1 Definitions Definitions specific to chapter 5.8 are listed in 5.8.1. Some comments are given below.

### Braced - bracing

The distinction braced - bracing is simple: units or systems that are assumed to contribute to the stabilization of the structure are bracing elements, the others are braced. Bracing units/systems should be designed so that they, all together, have the necessary stiffness and resistance to develop stabilization forces. The braced ones, by definition, do not need to resist such forces.

### Buckling

The word buckling has been reserved for the "pure", hypothetical buckling of an initially straight member or structure, without load eccentricities or transverse loading. It is pointed out in a note that pure buckling is not a relevant limit state in real structures, due to the presence of imperfections, eccentricities and/or transverse loads. This is also a reason why the word "buckling" is avoided in the title of 5.8. In the text, buckling is mentioned only when a nominal buckling load is used as a parameter in certain calculation methods.

### First order effects

First order effects are defined to include the effect of imperfections, interpreted as physical deviations in the form of inclinations or eccentricities. The ENV is ambiguous in this respect; see also clause C5.2.

### Nominal second order moment

The nominal second order moment is used in certain simplified methods, to obtain a total moment used for design of cross sections to their ultimate moment resistance. It can be defined as the difference between the ultimate moment resistance and the first order moment, see 6.3. If the ultimate load is governed by instability before reaching the cross section resistance, then the nominal second order moment is greater than the true one; this is the reason for using the word "nominal".

### Sway - non-sway and global second order effects

The terms sway - non-sway have been omitted in the final draft, after many comments for or against. The words in themselves are misleading, since all structures are more or less "sway"; a

5.8.2 General structure that would be classified as "sway" could be just as stiff as one classified as "non-sway". These terms are now replaced by unbraced - braced.

In the ENV the concept of sway - non-sway was linked to the criterion for neglecting global second order effects in structures. The classification of structures from this point of view remains in the EN, but without using the "sway" - "non-sway" terminology.

A stiffness criterion like the one in ENV-A.3.2 was avoided in earlier drafts of the EN, since it was considered as too crude, and in some cases misleading. However, during the conversion process there were many requests to include some simple criterion for evaluating the significance of global second order effects, without the need for calculating them. This led to the present rules in 5.8.3.3 and Informative Annex H, which are more general than those in ENV-A.3.2. The details are given in clause 3.3.

C5.8.2. Basic criteria for neglecting second order effects

Two basic criteria for ignoring second order effects have been discussed during the conversion process, namely:

1) < 10 % increase of the corresponding first order effect,

2) < 10 % reduction of the load capacity, assuming a constant eccentricity of the axial force.

The first criterion is the one stated in 5.8.2 (6), and in the ENV, 4.3.5.1 (5). The second one has been claimed by some to be the "true", hidden criterion behind the ENV-rules.

Figure 5.6 illustrates the consequences of these two criteria in an interaction diagram for axial force and bending moment. Their effects on the slenderness limit are discussed in chapter 3.

In a column or a structure it is the bending moment that is influenced by second order effects. The axial force is governed by vertical loads, and is not significantly affected by second order effects. Most design methods are based on calculating a bending moment, including a second order moment if it is significant. From this point of view, criterion 1 is the most logical and natural one.

The basic criterion is further discussed in chapter 3 in connection with slenderness limits.

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