# Time Dependent Analysis (TDA) (esas.20)

esas.20

Module Time Dependent Analysis (TDA) allows for the time dependent analysis of prestressed concrete, but also composite 2D frame structures, while taking into account the defined stages of construction, creep, shrinkage, and ageing of concrete. The method used for the time-dependent analysis is based on a step-by-step procedure in which the time domain is subdivided by time nodes. The finite element analysis is performed in each time node. Linear ageing viscoelastic theory is applied for the creep analysis.

Highlights
Time-dependent analysis of the structure taking into account the age of concrete.
Calculation of losses due to creep, shrinkage, and relaxation of prestressed tendons.
Calculation of concrete strain using superposition.
Load is changed only in discrete time node.
Numerical method described in annex KK.3 in EN 1992-2.

Due to symmetry of the long-term loads both the structure and the load can be adequately modelled in a vertical plane. On that account the plane frame structural model is used. The finite elements on eccentricity represent e.g. the concrete box girder (or separately concrete webs and layers of deck), prestressed tendons, diaphragms, piers, temporary anchoring ties, non-prestressed reinforcement, etc. All operations in the construction are respected in the structural analysis according to the real production schedule. The elements are installed or removed according to the way of construction. Various operations used in the construction such as addition or removal of segments and prestressed tendons, changes of boundary conditions, loads and prescribed displacements may be modelled.

The prestressed tendons are assumed also as eccentric finite elements. When they are initially stressed, only load terms of the tendons are included in the global equilibrium equations. After anchoring also the stiffness of the tendon is considered. Both, the bonded and unbonded tendons may be modelled. The long-term losses are automatically included in the analysis. If any element is removed or boundary condition is changed, the internal forces of the element and the appropriate reaction are automatically added to the load vector increment.

The total strain of concrete at the time t is subdivided into three parts:

• is the stress-produced strain,
• the shrinkage and
• is the thermal expansion.

Neither shrinkage nor thermal strains are stress-dependent. The shrinkage of structural members is predicted through the mean properties of a given cross-section taking into account the average relative humidity and member size. The stress-produced strain consists of elastic instantaneous strain and creep strain . The development of modulus of elasticity over time due to ageing is respected. The creep prediction model is based on the assumption of linearity between stresses and strains to assure the applicability of linear superposition. The numerical solution is based on the replacement of Stieltjes hereditary integral by a finite sum. The general creep problem is thus converted to a series of elasticity problems. The creep calculation is also based on the mean properties of a given cross-section. The creep, shrinkage and ageing effects may be taken into account according to design recommendations

• EUROCODE 2,
• CSN 73 1201 and CSN 73 6207,
• ÖNORM B4700,
• DIN 1045-1,
• NEN.

The method respects stress history, does not require any iteration in single step and does not restrict the type of creep function.

### Implementation of construction stages and TDA

TDA is closely linked to the Analysis of Construction Stages (ACS) inSCIA Engineer. The difference is that the rheological effects are not considered in ACS. On the other hand "the load-case" and "the combination of load-cases" are the basic "building units" for both TDA and ACS. ACS in fact runs independently on time. It is only a matter of form that each stage is linked to some time node.

The increments of dead load in each building stage (construction or service) and the results (the increments of internal forces and deformations caused by this load) are stored in separate load-cases. This load is assumed to be present (applied on the structure) until infinite time. The unloading must be modelled as a new load with opposite sign. For example - the total internal forces in existing structural members caused by dead loads after third building stage are obtained as the results of the combination of three appropriate load-cases. A load-case representing the life load can be added to this combination.

If any prestressing is applied in the building stage, additional permanent load-case must be applied. Then two permanent load-cases are defined in one building stage – one for the dead load and one for prestressing. The user is not allowed to add loads to prestressing load-case.

One additional (empty) load-case is generated automatically in each building stage in TDA analysis. These load-cases are used for storing of the increments of internal forces and deformations caused by creep and shrinkage calculated during passed time interval. They are marked as creep-load cases in SCIA Engineer.

### A few practical applications of the module:

• Wisconsin Avenue Viaduct in Milwaukee, Wisconsin, USA, by CH2M Hill, Milwaukee, Wisconsin with collaboration of Charles Redfield and Prof. Jiri Strasky.
• Precast segmental structure with replaceable cast-in-place deck slab of a viaduct in Plzen. By Strasky, Husty and Partners, Brno, Czech Republic.
• Gradually prestressed transverse beams in the frame of Sazka Arena in Prague (Ice Hockey World Championship 2004), by PPP Pardubice, Czech republic.

27/01/2015