5. Models for physical behaviour¶
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5.1. General¶
Calculation models shall describe the structure and its behaviour up to the limit state under consideration, accounting for relevant actions and environmental influences. Models should
generally be regarded as simplifications which take account of decisive factors and neglect the less important ones.
One can often distinguish between:
action models
structural models which give action effects (internal forces, moments etc.)
resistance models which give resistances corresponding to the action effects, and are based on.
material models and geometry models .
However, in some cases it is not possible or convenient to make this distinction, for example, if the instability or loss of equilibrium of an entire structural system is studied or if interactions between loads and structural response are of interest.
5.2. Action models¶
A complete action model should describe several properties of the action such as its magnitude, position, direction, duration etc. In some cases there is an interaction between the different properties and also between these properties and the response of the structure. Such interactions should be taken into account.
The magnitude \(F\) of an action may often be described by two different types of variables so that
where \(\varphi\) is an appropriate function and
\(F_0\) is a basic action variable, often with time and space dependent variations (random or non-random) and is generally independent of the structure
\(W\) is a random or non-random variable or a random field which may depend on the structural properties and which transformes \(F_0\) to \(F\).
Eq. (5.1) should be regarded as a symbolic expression where \(F_0\) and \(W\) may represent several variables.
One example may be snow load where \(F_0\) is the time dependent snow load on ground and \(W\) is the conversion factor for snow load on ground to snow load on roof which normally is assumed to to be time independent.
Further information on action models is provided in part 2. It is noted that action models may include material properties (earthquake action depends for example on material damping).
5.3. Geometrical models¶
A structure can generally be described by a model consisting of one-dimensional elements (beams, columns, cables, arches, etc), two-dimensional elements (slabs, walls, shells, etc) and three-dimensional elements.
The geometrical quantities which are included in the model generally refer to nominal values,
i.e. the values given in drawings, descriptions etc. Normally, the geometrical quantities of a real structure differ from their nominal values, i.e. the structure has geometrical imperfections. If the structural behaviour is sensitive to such imperfections, these shall be inlcuded in the model.
In many cases the deformation of a structure causes significant deviations from nominal values of geometrical quantities. If such deformations are of importance for the structural behaviour, they have to be considered in the design in principally the same way as imperfections. The effects of such deformations are generally denoted geometrically nonlinear or second order effects and should be accounted for.
5.4. Material models¶
When strength or stiffness is considered the material model normally consists of relations between forces or stresses and deformations i.e costitutive relationships. The parameters of such relations are modulus of elasticity, yield limit, ultimate strength etc. which generally are considered as random variables, Sometimes they are time dependent or space dependent. There is often an correlation between the parameters e.g. the modulus of elasticity and the ultimate strength of concrete.
Other material properties, e.g. resistance against material deterioration may often be treated in a similar way. However the principles are strongly dependent on type of material and the property considered.
Further information related to models of several material types is given in part 3.
5.5. Mechanical models¶
The following mechanical models may be classified
models describing static response
models decribing dynamic response
models for fatigue
a) models describing static response
In almost all design calculations some assumptions concerning the relation between forces or moments and deformations (or deformation rates) are necessary. These assumptions can vary and depend on the purpose and type of calculation. The most general relationship regarding structural response is considered to be elastic) developing into plastic behaviour in certain parts of the structure at high action effects. In other parts of the structure intermediate stages occur. Such relationships may be used generally. However the use of any theory taking into account in-elastic or post-critical behaviour may have to take into account repetitions of variable actions that are free. Such actions may cause great variations of the action effects, repeated yielding and exhaustion of the deformation capacity.
The theory of elasticity may be regarded as a simplification of a more general theory and may generally be used provided that forces and moments are limited to those values, for which the behaviour of the structure is still considered as elastic. However, the theory of elasticity may also be used in other cases if it is applied as a conservative approximation.
Theories in which fully developed plasticity is assumed to occur in certain zones of the structure (plastic hinges in beams, yield lines in slabs, etc) may also be used, provided that the deformations which are needed to ensure plastic behaviour, occur before the ultimate limit state is reached. Thus theory of plasticity should be used with care to determine the load carrying capacity of a structure, if this capacity is limited by:
brittle failure
failure due to instability
b) models for dynamic response
In most cases dynamic response of a structure is caused by a rapid variation of the magnitude, position or direction of an action However, a sudden change of the stiffness or resistance of a structural element may also cause dynamic behaviour.
The models for dynamic response consist in general of:
a stiffness model
a damping model
an inertia model
c) models for fatigue
Fatigue models are used for the description of fatigue failures caused by fluctuating actions. Two types of models are distinguished:
a) S-N model based on experiments
b) fracture mechanics model
It is further noted here, that other types of degradation such as chemical attack or fire can modify the parameters entering the aforementioned models or the models themselves.
5.6. Model uncertainties¶
A calculation model is a physically based or empirical relation between relevant variables, which are in general random variables:
\(Y\) = model output
\(f( )\) = model function
\(X_i\) = basic variables
The model \(f(\dots)\) may be complete and exact, so that, if the values of \(X_i\) are known in a particular experiment (from measurements), the outcome \(Y\) can be predicted without error. This, however, is not normally the situation. In most cases the model will be incomplete and inexact. This may be the result of lack of knowledge, or a deliberate simplification of the model, for the convenience of the designer. The difference between the model prediction and the real outcome of the experiment can be written down as:
\(\theta_i\) are referred to as parameters which contain the model uncertainties and are treated as random variables. Their statistical properties can in most cases be derived from experiments or observations. The mean of these parameters should be determined in such a way that, on average, the calculation model correctly predicts the test results.