Calculation Model

Calculations with the Ultimate Load Module (see D 2.3) are based on the FE model of the Basic Module and the reinforcement model (see D 2.2), which are automatically generated from the user input of structure geometry and reinforcement arrangement. Thereby the following basic assumptions are made:

S constant or linear support conditions

In this way the nonlinear modelling is restricted to the nonlinear material law.

Nonlinear Moment-Curvature Relationship

As described in Chapter D 2.1, the nonlinear behaviour of a slab subjected to bending actions can be described by an elastoplastic moment−curvature relationship. The dia gram for a section of unit width used in the Ultimate Load Module is shown in D 2.1.1 c) and is defined by four characteristic values: The cracked stiffness of the cross section EI_II, the plastic moment m_plR, the hardening parameter m and the limiting curvature xu. CEDRUS5 determines the values for EIII, mplR and xu automatically on the basis

of the concrete cross section and the actual reinforcement (see D 2.1.4). The calculation is performed following the method of the variation strain plane (see e.g. Handbook Cross Section Program FAGUS). Thereby the nonlinear stress−strain diagrams for con crete (see e.g. EC2 Figure 4.1) and reinforcing steel (see e.g. EC2 Figure 4.5) as well as the safety factors for the materials are taken form the analysis parameter set ’AP3: Nonlinear analysis’ (see main menu: Settings>Analysis parameter). The characteristic material values (i.e. fck,fyk for conrete and reinf. steel) are taken form the material zone

and the reinforcement field (see D 2.2.1).

Hardening

The hardening behaviour of reinforced concrete sections prevents, among other things, the localization of the deformations after the start of plastification. This allows the plasti fied zones in the slab to spread (i.e. the hinge region will get wider), which reduces correspondingly the required curvatures. As a welcome side effect material models with hardening increase the stability of numerical calculations. The positive hardening para meter m is the only calculation parameter, which has to be provided by the user. It is defined in the bilinear material properties diagram (see D 2.1.1 c) as the ratio of the stif fness after plastification of the cross section and the initial stiffness EI_II and is constant over the slab. Hardening behaviour is implemented as kinematic hardening (see D 2.1.3), whereby the behaviour in the plastic region can be accurately modelled for loading and unloading.

Load History

In CEDRUS5 the user has to specify the load history as a series of different load steps, which are successively applied to the slab and correspond each time to an arbitrary com bination of load cases. As soon as the given load level of a load step has been reached, the next one is applied, whereby the load factor for the last step is increased as far as possible. For ultimate load calculations on actual structures one is often not interested in a global ultimate load factor for all applied loads, but wants to calculate a safety factor, e.g. for a certain loading arrangement acting in addition to the permanent loads. This problem can be solved using different load steps.

Partial Safety Factors

The safety concept of modern building codes is based on probabilistic assumpitions, resulting in partial safety factors for actions and material properties. The safety factors for loads are always to be considered in the load history. In order to account for the safety factors of the materials the user has the choice of two modes:

S Reduction of the material properties

If the material safety factors gc and gs for concrete and reinf. steel are > 1.0 in the

D 4 Ultimate Load Module Part D Reinforcement and

resistance of the cross sections is reduced. As a result the load−deformation curve is pretty far from reality (i.e. yielding of the steel layers occurs far too early). In this case the user does not have to account for the material safety factors in the load his troy, in order to achieve the safety level required in the code.

S Scaling of the applied loads

If the material safety factors gc and gs for concrete and reinf. steel in the used analysis

parameter set are all equal to 1.0, then a more realistic load−deformation curve is found. In order to achieve the safety level required in the code, however the user must account for all safety factors (i.e. for loads and materials) in the load history. Unfortunately this is only possible, if gc and gs do not differ (like in the older codes

SIA 1xx and DIN). Otherwise one has to use the method described above.

The ultimate load module does not automatically account for these factors (as the base and reinforcement moduls of CEDRUS5 do). But the safety factors for actions and ma terials, prescribed by the building codes, are easily introduced in the load history specifi cation (see figure below).

. Example for SIA 1xx (’Scaling of applied loads’): if you want to investigate the structural safety of an existing slab, loaded by self-weight, an applied surface load and a imposed load, you can combine the first two actions into a single load step using the partial safety factor for dead loads (e.g. 1.3 accordingto SIA) times the partial safety factor of the materials (e.g. 1.2 accordingto SIA). The imposed load is introduced into a second load step using the partial safety factor for imposed loads (e.g. 1.5 accordingto SIA) times the factor for the materials. In this way, the dead load is applied first up to the specified level. Then the imposed load is added and increased until the ultimate load is reached. If the resulting load level (=load factor of the second load step) is bigger than 1.0, according to the building codes the structure is safe against collapse.