11th BETONSK DNY (Concrete Days) 2004 Section Posters

EVALUATION OF MODELS, METHODS AND PROGRAMS FOR CALCULATION OF DEVELOPMENT OF STRESS AND DEFORMATIONS IN PRESTRESSED BOX GIRDER BRIDGES OVER TIME

Luk Vrblk, Vladimr Kstek, Roman Lenner

1 Introduction The structure of larger bridges usually passes through several construction stages and a time analysis of development of stresses and strain usually largely influenced by creep or shrinkage of concrete is an important part of static analysis. At present the analysis of effects of concrete shrinkage and creep is usually performed only on the level of technical calculation of integral internal forces and deformations. It is of great importance to realise that this level of calculations cannot find out the actual development of strain and stress in each particular point of a concrete structure, but it can only find integral sectional force and moment components, and possibly also deflection of the structure, and their development over time. Design practice mainly uses three levels (models) of the analysis of effects caused by pure bending of bridge beams (usually accompanied by the shear forces), i.e. without torque.

1) The simplest is the commonly used beam-statics approach, where a bridge beam is modelled by a single beam that is characterised in addition to modulus of elasticity E only by the moment of inertia I and sectional area A. Shear deformation and demonstration of shear weakening are overlooked. This approach is hereinafter called model 1. In design practice, a planar frame (containing various systems, e.g. in bridges it is often a continuous beam) is a typical calculation model for this approach (e.g. (i) due to advantageousness of operations with integral internal forces (i.e. bending moments, shear forces, axial forces) during sizing and also (ii) owing to the need to perform the calculation of the influence of changes in structural scheme in step-by-step construction using the present-day technology and with respect to concrete creep and shrinkage effects). Numerous calculation programs are available and they are frequently used in everyday practice (see below).

2) In an advanced approach, the bridge beam is once again modelled as a single beam, however, account is taken of shear deformation in walls, but shear weakening is not considered (model 2). Shear deformation in walls plays a significant role in box girder bridges, as the sectional area of the walls is small in comparison with the overall sectional area. This level of modelling in addition to earlier stated input characteristics requires also the input of Poisson coefficient (and possibly shear modulus of elasticity G) and sectional area resisting to shear-effects As (in box girder beam the As is taken approximately as the sectional area of the walls (if they are vertical)). Programs used in design practice for the analysis of a complete construction process, including the effects of concrete creep and shrinkage and taking account of shear deformations in walls, are much less frequent but they exist (see below).

3) Nowadays a number of calculation methods based on a shell-like model of individual parts of the box girder beam (model 3) are also available. They take into account the

spatial action of the structure and, among other things, automatically take into consideration not only the effects of shear deformations in walls, but also shear weakening that may have a great impact on deflections and on the character and distribution of axial stresses in cross-sections of box girder bridges. However, these procedures (based on the finite element method or finite strip method, alternatively on folded plate theory) for the analysis of the whole construction process including the effects of concrete creep and shrinkage are not generally available for design practice.

At present structural engineers in Czech conditions can use several calculation programs that provide for prediction of development of stress and deformations in pre-stressed bridges over time. The programs vary in complexity and extent and their calculation procedures are based on different theoretical approaches. The aim of this article is to brief you on the most common programs used in these days for design and scientific purposes. It is not the intention to prefer any of the programs to the others. The goal is just to show their strong and weak points.

Comparison of individual programs

2.1 TM18 A set of programs intended for the design of bridge structures made of pre-stressed concrete (i) that are erected step-by-step, (ii) that change their structural scheme during the construction and (iii) individual concrete parts of which are usually of different age. The program is capable of solving two main problems of static calculation: (i) analysis of statically-indeterminate structure subject to various external loads and pre-stressed by means of post-tensioned tendons and (ii) a basic checking of a cross-section of the structure. The checking covers also the determination of stress-losses that are introduced into the subsequent calculation in the form of pre-stressing with the opposite sign. The solution in program TM18 is based on the force method. The application of program TM18 is limited to structures that can be idealised as a system of beams. The structure is divided into beams with straight axis (which does not have to be identical with the centre of gravity) and it is possible to define variable sectional characteristics (sectional area, moment of inertia, eccentricity of the centre of gravity from the calculation axis). In addition, the program is limited to the calculation of planar frame structures symmetrical in their axis. The solution in program TM18 divides the calculation into a sequence of successive time stages. Change of structural scheme, change of load or change of pre-stress in the structure can be considered in each stage. The calculation the effect of concrete creep on the redistribution of stress state in the structure is solved by the relaxation method [1], [2], which makes it possible to avoid the solution of a system of differential equations. Creep is controlled by the aging theory and creep coefficient, that is the difference between function values of the same function, depends on the age of concrete at the time when the pre-stress has been introduced into the structure.

Program can automatically evaluate all stress losses that are applied in appropriate time stages. The effect of pre-stressing on statically indeterminate quantities is evaluated. Pre-stress is defined through the shape of tendon and its material characteristics. Program automatically calculates (i) short-time losses due to friction and slip in anchors, (ii) short-time losses due to gradual application of stress into the tendon and (iii) long-time losses due to steel relaxation and concrete shrinkage and creep. It is not possible to input directly free tendons and pre-tensioned reinforcement. On the other hand, it is possible to modify the calculation in a suitable way and take into account these structural members. Program TM18 is widespread and popular calculation tool used in civil engineering offices. It is rather simple in use in terms of operation and input of data. The presentation of results is not as clear as in other state-of-the-art calculation programs, but the use of suitable extensions can lead to rather satisfactory results. The transfer of obtained data for the purposes of other design operations is excellent. The program ignores the effect of shear deformations in walls of beams therefore it belongs into the category of models 1 in terms of the sorting presented in the introduction of this article. What is a rather weak point is the calculation of creep using the aging theory (Dischinger) that is completely unsuitable for structures that are subjected to load in a considerable age and that neglects the reversibility of a part of strain, which obviously does not correspond to the reality. This approach reflects the time of the creation of the program and corresponds to the then knowledge level concerning the rheology of concrete old-fashioned creep models however still survive in normative regulations and also in the minds of some structural engineers.

2.2 DOMO This calculation program makes it possible to follow the effect of concrete shrinkage and creep on the structure; the program is limited to systems that can be modelled as planar frames. Version DOMO 98 includes also the effect of shear weakening of a cross-section on the development of deflections of the structure and on the redistribution of internal forces through the reduction of sectional area of cross-section walls in accordance with [3]. Input data for the calculation are divided into two groups. The first group consists of basic data about the structure, materials and time. These values must be input just once at the beginning of the calculation. The second group of input data describes the change of load and structural scheme in individual time steps, the values are input for each part of time discretisation separately. The geometry of the structure is described by means of axes connecting centres of individual beam elements (defined by means of their sectional area and moment of inertia) that are connected in nodes. The position of a node is unambiguously defined by a pair of coordinates. The input of concreting time and number of time steps representing the installation of the element into the structural system provides for the modelling of a step-by-step construction. Material properties are described by means of a creep function. Program DOMO allows for a choice from a great variety of models for the prediction of creep of concrete (B3 Model 1995, ACI Model 1992, GL Model 2001, theory of aging, Eurocode 2, BP-KX Model 1994).

The creep analysis itself is based on two approaches. The first one is time integration that is during the calculation substituted by summation of individual small increments. Strain in time t is determined from the formula:

( ) ( ) ( ) ( )=+=

n

1i, 0,0 0, ittJttJttt

where: is the initial strain (t0) is the initial stress is the stress increment in time step i J(t,ti) is the creep function Such a formulation is applicable to small structures and a low number of time discretisation steps, because the calculation must in every subsequent step go through the whole load history, which significantly increases the time demand on the calculation. An alternative is the application of an incremental method that is suitable for large structural systems and a great number of time steps. It defines the relation between the stress increment i and strain i during one step as:

hp

ii E

+= where: h is the strain increment corresponding to the load from previous time step Ep is the pseudoelastic modulus of elasticity of concrete These parameters are calculated in each analysis step. Only a rather small number of constants independent on the details of time discretisation are stored in the memory of the computer and therefore, the calculation is more effective than in the previous method. The structure can be composed of members made of different concrete, the behaviour of which can be described by means of different creep functions and, if required, different age of concrete of individual parts can be taken into account. As stated earlier, the program makes it possible to model changes in structural scheme during construction and individual members can be added into or removed from the system. Supporting conditions of the structure can be changed the same way. Also the influence of the pre-stress in tendons, which are input in polygonal shape, can be taken into account. The program determines long-time losses of stress caused (i) due to the shrinkage and creep of concrete, (ii) due to deformation of the structure and (iii) due to other pre-stressing of the system. It is possible to add or remove a pre-stressing unit at any time. Results are recorded into an output file usually at the beginning and end of the discretisation step it is not possible to follow simultaneously short-time and long-time deformations. Internal forces (bending moments, shear and axial forces) and forces in pre-stressing units are written into the output file, an extension module can be used to display the distribution of internal forces and

deformations over the whole structure or on its selected parts. Program DOMO is a calculation program taking into account (i) the effects of concrete creep on the structure, (ii) changes of the structural scheme, (iii) effects of temperature and shear deformations of walls therefore, it belongs into the category of models 2 in terms of the sorting presented in the introduction of this article. The effects of shear weakening can be considered approximately by means of reduction of the sectional area of walls in accordance with [3]. A certain disadvantage is the complexity of the transfer of input data and the manipulation with them, which makes the program applicable to scientific purposes rather than to common calculations during the design and checking of structures.

2.3 TDA A calculation program that allows for the analysis of civil engineering structures made as hybrid systems consisting of steel, prefabricated and monolithic concrete. It is possible to follow the development of internal forces and strain in structures composed of concrete parts of different age and quality. The program can be used to examine the effects of creep and shrinkage in 2D frames. The applied method is based on a step-by-step procedure, where the time zone is divided into time nodes. Creep and shrinkage can be analysed according to several standards EC2, CSN 73 1201, CSN 73 6207, NEN1. The program TDA allows for modelling of composite cross-sections (steel-concrete, timber-concrete, concrete-concrete). Function phased cross-sections can be used to model cross-sections composed of different materials e.g. concrete parts of different age. This feature makes it possible to consider the stress redistribution between two different stages (phases) of the cross-section due to creep and shrinkage of concrete. Short-time, long-time and operation (influence of variable load) stress losses are calculated automatically. The total strain of concrete in time t is divided into the strain due to stress (instant elastic strain and creep), shrinkage and strain from temperature. The increase of modulus of elasticity over time due to aging of concrete is taken into account. The model for the calculation of creep is based on the assumption of a linear relation between stress and strain, which means that superposition principle can be applied. At present, the program TDA is integrated into the calculation system NEXIS 32 3.50 i.e. the system of calculation programs for design of structures consisting of beams and walls and plates. This combination ensures an excellent interpretation of results and a link to subsequent stages of structural analysis. 1 SCIA comment: codes available in 2008 version of SCIAESA PT: EN1992-1-1, ENV 1992-1-1, EN 1992-2, ONORM 4750/4700, DIN 1045-1, NEN 6720, SN 73 6207, SN 73 1201.

Program TDA takes into account the effects of shear deformations of walls. The influence of shear weakening can be approximately considered through the reduction of the sectional area of walls in accordance with [3]

2.4 RM2000 The program RM2000, product of Austrian company TDV, is a calculation tool for static and dynamic analysis of spatial frames. It is a rather wide spread program and is recognised as a tool for the design and checking of bridge structures. A structure in RM2000 is defined as a beam-model whose individual parts elements are connected in nodes. The main controlling parameter during the input of the geometry of the system is the line connecting the centres of the nodes. If these lines do not coincide with centroidal axes, eccentricities of individual parts are specified (the distance of the real centroid from the theoretical axis). The structure is supported in nodes with 6 degrees of freedom 3 translations and 3 rotations along and about the coordinate axes. The load defined on the input part of the structure is transferred into nodes as well. Nodal position is uniquely defined by means of three of coordinates. Both internal and external pre-stressing of the structure can be taken into account (the effect of free tendons and tendons sticking out of the cross-section can be thus analysed). An assignment of earlier defined tendons into the cross-section introduces the pre-stress into the structure. The tendons are defined by means of their material characteristics, area and geometry. The same types can be advantageously combined in groups. The program automatically determines stress losses both short-time and long-time including the calculation of relaxation of the pre-stressing reinforcement. Similarly to previous calculation programs, it is possible to model the step-by-step construction, interaction of individual parts made of concrete of different age, changes in the structural scheme during the lifetime, and changes in loading. During the analysis of the effects of creep and shrinkage, one can choose from numerous implemented models used for the calculation of these phenomena. The effects of shrinkage and creep are determined completely for the whole cross-section, not differently for e.g. lower and upper deck of box girder bridges. The program is capable of automatic reduction of the effective width of the deck for a rib with one beam. In case of several beams it is up to the user to take this phenomenon into account through the input of proper values which may be a source of misunderstanding. The results of calculation performed by the program RM2000 are clear and well-interpretable. The results may be only partial, the purpose of which is to make ones opinion about the behaviour of the element, or they may be the final results of the overall analysis. The program RM2000, made by TDV Company, is a calculation tool for the analysis of bridge structures of all types. It makes it possible (i) to model a wide range of structural systems used in bridge industry (composed cross-sections, cable-stayed bridges and suspension bridges), (ii) to

perform their complex analysis, (iii) to observe the effects of creep and shrinkage on the distribution of internal forces and (iv) to monitor the development of deformations over time. It is recognised all over the world and its results served as a basis for the construction of many prestigious bridge structures.

2.5 ATENA Program ATENA is a product of the Czech company ervenka Consulting and it is intended for a non-linear analysis of structures based on the finite analysis method. The program contains tools specially designed for a computer simulation of the behaviour of concrete and reinforced concrete. It is suitable for more complex problems that cannot be covered by usual engineering methods, especially for the determination of various types of damage and loading effects. The program ATENA consists of calculation core-module and user interface. The calculation core-module can perform 2D and 3D analysis of continual structures. It contains libraries of finite elements, material models and calculation methods. The graphical user interface ATENA is a program that allows for the access to the calculation core-module of ATENA. Two versions of user interface are available: 2D environment for modelling of plain-stress projects, plain-strain projects and rotational-symmetry projects, and 3D environment for spatial projects. In addition, the calculation core-module can be used in combination with other programs for geometric modelling and creation of meshes. Two interfaces are available: for program Femap and GID. The main feature that distinguishes ATENA from general programs for the finite element method is the possibility to effectively model concrete reinforced structures, especially to take into account (i) reinforcement bars including various types of pre-stressing and cohesion, (ii) specific concrete properties (cracks and effect of stress on the strength), (iii) influence of time (creep, shrinkage), and environment (temperature, humidity, fire, etc.). Material models are derived from theories of failure, plasticity and elasticity. Non-linear solution can employ several incremental iterative methods (Newton-Raphson, arc-length, line search) and both iterative and direct methods are available for the solution of equation system. The finite element library is based mainly on isoparametric types of elements and contains elements of both lower and higher order, for walls, beams, shells, 3D volumes. The program ATENA is intended for a wide range of structures made of pure, reinforced and pre-stressed concrete, and for structures combining various materials (concrete-steel, concrete-carbon fibres, soil environment, etc.) and for their mutual interaction. It can be used for the analysis of walls, beams, arrangement of reinforcement, tunnels, bridges etc. An extension has been developed that can be used for the calculation of reliability and safety of structures and it integrates the program ATENA and a stochastic program FREET. It is intended for calculations of a global level of safety and for the determination of reliability index of structures.

3 Overall evaluation Program Author Creep

model Changes in structural

system

Temperature effects

Differential shrinkage

Differential creep

Taking account of

shear deformations

of walls

Automatic input of

pre-stressing

effects

Link between

results and subsequent

design operations

TM I. Sita,V. Kstek aging theory

yes no no no no yes excellent

DOMO J. L. Vtek general - arbitrary

yes yes no no yes no requires transfer of results

TDA J. Navrtil general arbitrary

yes yes yes yes yes no2 excellent

RM2000 TDV general arbitrary

yes yes no no partially yes excellent

ERV. V. ervenka general - arbitrary

yes yes yes yes yes yes excellent

4 Conclusion Most often used calculation programs and solution methods for the analysis of bridge structures made of pre-stressed concrete available in the Czech market have been described. It is not our intention to assess and compare them one against the other. More thorough analysis would require an exact solution of precisely defined task performed by all the mentioned programs. Only then and on the basis of obtained results it would be possible to judge the precision of individual methods. The selection of the program depends on the designer, some solutions may be expensive in terms of money and their utilisation may require a great deal of expertise and knowledge. The results have been obtained during the solution of grant projects 103/02/1005 and 103/02/0020 supported by the Grant agency of the Czech Republic.

Literature [1] Kstek, V.: Method of calculation of the influence of creep of concrete on statically indeterminate structures, Acta Technica, Czechoslovak Academy of Sciences, No.1, 1973 [2] merda, Z., Kstek, V.: Creep and shrinkage of concrete members and structures, SNTL Praha, 1978 [3] Kstek, V., Petk, V.: Recommendations for calculation of increase of deflection of box girder bridges, Proceedings of VIIth conference Concrete Days1999, CBZ Pardubice, December 1999, pp. 151 - 154 [4] Navrtil J.: Time-dependent Analysis of Concrete Frame Structures, Stavebnick asopis, 7 (40), 1992, pp. 429-451

2 SCIA comment: Automatic input of pre-stressing effect is possible via Esa Prima Win or SCIA.ESA PT program, into which TDA module has been integrated.

Ing. Luk Vrblk Prof. Ing. Vladimr Kstek, DrSc. Czech Technical University of Prague

Faculty of Civil Engineering Dept. of concrete structures and bridges Thkurova 7 166 29 Praha 6 Czech Republic

Czech Technical University of Prague Faculty of Civil Engineering Dept. of concrete structures and bridges Thkurova 7 166 29 Praha 6 Czech Republic

274 770 428 274 770 428 233 335 797 233 335 797 vrablik@promo.cz kristek@fsv.cvut.cz Ing. Roman Lenner VALBEK, spol. s r. o.

Vaurova 505/17 460 01 Liberec Czech Republic

485 106 447 485 106 447 lenner@valbek.cz Original text written in Czech. Translation to English: Ing. Pavel Roun, CSc., SCIA CZ, s.r.o.