Optimal Slope of Dramix Type Fibers in Reinforced Concrete

6 th World Congresses of Structural and Multidisciplinary Optimization Rio de Janeiro, 3 May - 3 June 25, Brazil Optimal Slope of Dramix Type Fibers in Reinforced Concrete P. Prochazka 1, N. Starikov 2 1 Czech Association of Concrete Engineers, Prague, Czech Republic, petrp@fsv.cvut.cz 2 CTU Prague, Civil Engineering, Prague, Czech Republic, star - 13@mail.ru 1. Abstract Several experimental studies proved that the standard reinforcement in concrete (rebars) could be improved by short fibers to avoid local warping and cracking. In order to restrain the reinforcing rebars from corrosion, the fibers are basically at help during the curing process of concrete. Particularly in geomechanics, or foundation engineering, where one-sided moistening is expected, the reason for usage of the fibers is advantageous due to obviously irregular dehydration. The volume ratio of fibers in these cases is very low, say, about.5 percent. Another one interesting application is in construction of defense walls (bunkers), where the fiber ratio is principally higher. In all cases curvilinear fibers are preferred from then viewpoint of bearing capacity of the system. On the other hand, the financial expenses are also higher. This is why optimal shape design is required for maximum effectiveness of fiber reinforced concrete. Such a typical shape of current most produced fibers of this type is prepared by Dramix. This is why we concentrate our attention on straight fibers and also to the curvilinear fibers. The optimization of slope in Dramix type fibers is solved from pullout problem studied on unit cells of different arrangements. The steel fibers are considered and either linear or elasticplastic Mises material of concrete is taken into account. Mixed variational principle has been applied to enable one to formulate contact problem together with optimization of slopes. Transformation field analysis is employed to make easier the influence of nonlinear behavior of concrete. The critical pullout force is solved from finite element method and is identified from destruction of the aggregate. Several typical examples are solved to show the properties of the composite for various arrangements of fibers in cells. 2. Keywords: Shape optimization, fiber reinforced concrete, Dramix type fibers 3. Introduction The fibers play very important role particularly during curing process of concrete, as they suppress local cracking and warping in the composite structure and avoid a possibility of corrosion of reinforcing steel rods. Very important fact follows from plenty of experimental studies: since the steel fibers are stiffer then concrete itself, pullout problem has to be solved when requiring knowledge about the bearing capacity of the structure. In this study 3D isoparametric finite elements are used for solving the problem on unit cells. The unit impulses are applied to nodes of the 24-degree of freedom elements to create influence matrices. This approach is effective from the point of view of iterative method, which has to be applied to the solution of contact problem. Because the optimization of slope of Dramix type fibers is required, discrete problems are successively solved for different unit cells. Four typical unit cells are studied and discrete fiber slopes are solved, namely admissible pullout forces are determined from the set, 1, 3, 45, 6, and 9 degrees of angle between the axial coordinate and the slope. Interpolation of the influence of slopes on the admissible forces is available, as the results provided from these computations admit approximately continuous curve describing the influence of the slope. Natural question occurs when studying the problem of behavior of curvilinear fibers in concrete matrix. The slope of the fibers in the curvilinear part can change and the response of this change will obviously have an impact to the bearing capacity of the composite aggregate. If we leave the vertical distance between horizontal parts of the fibers constant, the slope of the middle part can change and the design parameter of the optimization is the angle of the slope. From experimental tests it has been proved that the extraction of the fibers out of the concrete are caused by penetrating of the fibers into the concrete due to attaining some admissible stress (compressive stress) or slipping of the fibers along the interfacial zones with the concrete. The optimization criteria follow this observation of experimental results. The first criterion will require that the admissible force cannot exceed the criterion of extraction of the fibers due to violation of Mohr-Coulomb hypothesis. This case prevails particularly in the case of straight or almost straight fibers. The second criterion states that near the fibers no stress attains the compressive strength. 4. Pullout problem of a steel FRC beam The pullout problem has frequently been solved in a problem of cracking of composite structures of several sorts. Previously, several numerical studies were carried out using the FEM, [1], and the BEM, [2], and the results were compared with experimental results from available literature with a good agreement. In the sections from the available literature the topics have mainly been focused on experiments and the theory has been based on various approximate hypotheses, i.e., the results from the theoretical considerations are very approximate. A remarkable feature in [1] and [2] of the mathematical solution occurred: a cracking was initiated not only at the face of the fiber-matrix system, but also inside the trial body. As the previous problems were concentrated only on straight fibers and the nature of the material used was based on classical composites with epoxy matrices and a high bearing capacity and durability, a natural question arises: what happens when the matrix is created from concrete and the aggregate behave realistic, i.e., the shape of the fibers is no more straight and the concrete matrix is taken into account. The answer to this question is the objective of this section. Some numerical experiments show on how the theoretical considerations can be applied in reality. The examples mutually differ by positioning the fibers and by volume fractions. Dramix-type fibers are used in studies of curvilinear fibers. In this study of a lag model of the system concrete matrix - steel fibers of a special shape, the geometry of the problem is considered as three-dimensional and high accurate finite isoparametric elements with 24 degrees of freedom are used together with Uzawa's algorithm, [3], for solving a contact problem fiber - matrix. A finer mesh of the finite elements is necessary for the stability of an iterative process particularly in the neighborhood of expected nonlinear behavior on the contact (interface between fibers and

matrix), which covers the Uzawa's algorithm. A procedure, which is close to TFA, [2], [4], is used. It consists in creation of influence matrices due to unit impulses of tractions on the interface, which are applied at these points of the finite element mesh, which belong on the contact to either boundary of the fibers or the boundary of the concrete. The debonding or slipping" (jumps of responding points of the matrix and fibers boundaries in the tangential direction to the interface) can be caused due to several physical models. First, no tension tractions on the interface may occur. Also, some friction law (such as Coulomb law, Mohr-Coulomb law, shear band strength are often introduced and debond parameters are computed. Since no fiber gripping force from matrix is expected, the general Mohr-Coulomb friction law is restricted to the exclusion of shear stresses exceeding the shear strength in the tangential direction and the tension exceeding the tension strength is also excluded slang the interface between the fiber and the matrix. In our study the grip in normal direction to the fibers is neglected. The problem is obviously three dimensional, so that the most accurate results may only be obtained from very precise numerical models. The behavior of the system starts partially with observations described in [2], where one sided moistening is ensured during the curing process. A periodic system of fibers is assumed in our model and different volume tractions and positioning of the fibers are considered. In order to obtain results comparable with experiments and on the other side results from real composite structures a wide range of examples were analyzed in [1] by applying the FEM and the results from the study put forward consulted them. Our current aim is to show the results from curvilinear shape of fibers and numerical results obtained in [1] and [2] which were in a good agreement with experiments. As the solution of this problem leads to a sequence of iteration steps, an approach similar to the transformation field analysis (TFA), [4], is used for speeding substantially up the iteration. As in recent papers, [1], [2], Uzawa's algorithm is used to the pullout problem of a lag model without proof of solvability and uniqueness. The main goal of this section is to show the behavior of the pullout problem of the lag model in connection with the numerical procedural proposed. Also, Uzawa's algorithm is, in comparison to the previous papers, [1], [2], carried out in the sense of the TFA, i.e., the influence matrices needed are prepared a priori. Since more than 1 iteration steps are expected, the procedure is principally speeded up using the ideas of the TFA. Zone 1 Zone 2 Zone 3 Zone 4 Figure 1. Geometry of the model 4.1 Formulation of the problem Consider a problem of two bodies, the geometries of which are depicted in Fig. 1. The first body (fiber) occupies in undeformed state the domain Ω ' with the interfacial boundary Γ ' (rectangular parallelepiped) and the second body (concrete matrix) occupies the domain Ω' ' with the boundary Γ '', (parallelepiped). There is no external load considered but the pullout force at the face of the rectangular parallelepiped. The bodies are situated in Cartesian coordinate system Oxyz; the axis x is introduced in the radial direction of the lag. Generally, it will be assumed that the tension exceeding the tensile strength in the normal direction to the interface fiber-matrix Γ is not admitted, so that both bodies may disconnect (they mutually debond) in certain region of the common boundary Γ, which is a part of Γ ' and Γ ''. In order to hold the mathematical purity let us consider tensile strength at the contact to be zero. In practical examples, this fact will be overcome by simple introducing the tension strength but theoretical (not practical) difficulties may occur. The Mohr-Coulomb contact conditions will be taken into account. More complex study concerning the contact conditions was carried out in [1]. There a reader can find that four zones can be distinguished on the straight fiber to be pulled out of a surrounding medium. Along the first zone, additional debond can occur, if at the left face of the model described in Fig. 1 no movements in horizontal direction is admissible, for simplicity of description. The second zone is a firm bond, while in the third zone slipping in the tangential direction is possible. The zone belonging to the face (the fourth zone) is a typical opening crack. Similar situation is observed in curvilinear reinforcement. Any of the four zones can disappear but the third, which is the zone in the undeformed state. The load is considered in the following way. The pullout traction p is applied at the face of the fiber in the axial direction, prescribed by a constant value. There is symmetry about the vertical plane containing the horizontal axis x regarding both geometry and loading, so that only one half of the unit cell may be solved. Displacements are described by the vector function u = { u, v, w} of the variable x { x, y, z} =. Denote Ω Ω U Ω. The restriction,,,,,,,, of any function to Ω or Ω is denoted, respectively, by one prime or by two primes, e.g., u / Ω = u and u / Ω = u,. On the boundary displacements and tractions are prescribed in such a manner that a periodicity is assumed at all sides of a "unit cell", The loading is given in both domains due to the pullout traction p, see Fig.1, where only straight fiber is displayed. In practical examples, curvilinear fibers are assumed. Denote the set of admissible displacements u on Ω satisfying the essential boundary conditions by V. The values and the first derivatives are quadratically integrable.,,,

Consider Hooke s law in the form:,,,,,,,,, ε ij ( u' ) = L ijkl σ kl ε ij ( u' ') = L ijkl σ kl (1),,, where L and L are the material stiffness matrices of the fiber and the concrete, respectively. When assuming "small deformation" theory, it may be satisfactory to formulate the essential boundary condition on the contact boundary as follows (Signorini's conditions): Denote,,, [ ] u u < u n = n n a.e. on Γ (2) ;[ u] H { u... in... V n < a.e on Γ } (3) The set H is a cone of admissible displacements with respect to the essential boundary and contact conditions, V are displacements from the space of continuous functions. Suppose that we disconnect both bodies under consideration, but keep the stress and deformation state in them "frozen". Then the p = p, p, p must be introduced and their equilibrium (action and reaction law) says that: vector of contact tractions { } x y z p = p + p,,, = (4) Let us write the total energy J of both bodies assuming them separately: J(u, p ) = Π (u ) I( u, p ), (5) where 1,,, ( u ) = a( u,u) p, u, I ( p,u) = ( p' u' + p'' u'' )dγ, ( u,u) = ( ' ε ' + Π dγ 2 Γ where σ and ε are the stresses and the strains in both bodies. The Mohr-Coulomb contact conditions read as: [] < u n, n <, p [ ] = Γ a σ σ '' ε '')dω (6) Ω u n p n, p t < k ( p n ) c, if pt > k( pt ) c then p t = c (7) and k is Heaviside's function, c is the cohesion (shear strength): Formulation of the contact problem of pull out can be declared as: Find minimum u from H and maximum p of J obeying Eqs. (4) and (7). Such a posted formulation leads to the well-known Uzawa's algorithm; see [3], for example. In this way all examples were solved. Although the Coulomb influence is absent in the law (7), a slip part can occur, Heaviside's function is applied in the formulas given by Eq. (7) because of assuring a natural requirement: once the tensile strength is violated, no shear forces occur either at the trial points. 5 Straight fibers Four examples were tested to show the behavior of different shapes of fibers, which are pulled out of the concrete. In each example the stainless wire fibers are considered, for which (concrete) E' = 179 GPa, v' =.3, (steel) E'' = 41 GPa, v'' =.16 for pseudo-linear computation. The results from the examples are displayed in compact form and described as Experiment-1 to Experiment-4. First, the vertical cut crossing the fibers in the ''unit cell illustrates the geometry of the aggregate in Fig. 2. The vertical size is always identical with the horizontal depth. From the geometry one can calculate the fiber volume fraction. The fiber volume fractions range from.8 to 1.5 percent. First an approximate case is considered for straight fibers imbedded in concrete matrix. We suppose the linear behavior of both fibers and concrete matrices. Moreover, ideal bond along the interfacial zone between fibers and concrete is also assumed. Such an assumption is too rough and this is why additional assumption is introduced to improve the model, which then behaves more versatile: The flaw of various lengths appears at the face of the aggregate. In order to hold the linear behavior in the formulation of pullout problem, the flaws are considered to open successively at nodal points on the interface and the responses of unit pullout force are observed. The unit force is selected as.15 kn. The results are illustrated in the pictures, which follow in this text. In Fig. 1 vertical cut of a typical straight fiber is depicted together with the numbering of nodes, the flaws in which are introduced. The vertical cut of unit cells together with finite element meshing shows the fiber position in the cell in Fig. 2. From this picture it is seen that the position of fibers is symmetric in each cell and one symmetric part (one quarter) is solved in numerical interpretation with appropriate boundary conditions expressing the symmetry. The unit cells are selected in such a manner that typical periodic structures are acquired. Periodicity is introduced along the entire boundary but the face. It means that along both horizontal

boundaries and the left vertical boundary the periodicity conditions are considered in the classical sense, not in the sense of homogenized composites. The right boundary (face) is free but the fiber where the pullout force is applied. Zero in parentheses means the slope of curvilinear fibers, which are generally considered in this paper. 1 2 3 4 5 6 7 8 9 11 13 15 17 19 21 1234 5 6 7 8 9 1 12 14 16 18 2 Figure 1. Shape and numbering of nodes of trial fibers Experiment-1() Experiment-2() 6,5 6 12,5 2 1 12 2 1 12 Experiment-3(),5 16,5 6,5 12,5 8 6 4 4,5 8,5 Experiment-4() 8 1 2 1 12 Figure 2 Vertical cut of four cases of fiber positions in unit cells with FEM meshing In the following four pictures the responses in transversal tractions on unit.15 kn pullout force are shown. Fig. 3 depicts the distribution of transversal tractions for Experiment 1, Fig. 4 for Experiment 2, Fig. 5 for Experiment 3, and Fig. 6 for Experiment 4. On the vertical axis values of tractions are depicted, on the horizontal axis the interface is described and the orientation is enforced by numbers of nodes, see Fig. 1. The curves describe the distribution of transversal tractions on the interface from unit pullout force and various flaws. Series 1 belongs to the flaw at node 2, i.e., the fiber is debonded from points 2 21. Similarly, Series i = 2,,19 belong to the flaws at nodes i + 1. From the pictures one can derive the bearing capacity of the aggregate in the sense, which is explained in what follows: If the tensile strength is, say,.3 MPa, Experiments 1 3 fail, as the tensile strength is exceeded at all nodal points. As for the Experiment 4, the strength is overcome from nodes 9 21, where debond can also be expected. On the other hand, at nodes 1 to 8 the strength is not exceeded and fiber presence increases the bearing capacity of concrete but the crack starting from nodal point 9. If, for example, the tensile strength is.4 MPa, Experiment 1 is undeceived as for the admissible pullout force, in Experiment 2 the real flaw is between nodes 17 and 18, in Experiment 3 the force is pulled out of the aggregate, and the force in Experiment 4 stiffens the aggregate, of course with respect to the previous case (tensile strength is.3). In this manner we can estimate the bearing capacity for other particular cases considered in this study. Similarly, assuming the unit force introduced as.15 kn, and, say, in Experiment 1 we require the fiber is somewhere bonded and interacts with concrete for tensile strength to be equal to.2 MPa, the force must be less then.75 kn (one half of the unit force). In this way we can discuss various cases under study, as the problem is linear, although debond (flaws) is considered.

MPa,7,6,5,4,3,2,1 -,1 -,2 Experiment-1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 Number of nodes Series1 Series2 Series3 Series4 Series5 Series6 Series7 Series8 Series9 Series1 Series11 Series12 Series13 Series14 Series15 Series16 Series17 Series18 Series19 Figure 3. Distribution of transversal tractions - various flaws in Experiment 1 MPa,6,5,4,3,2,1 -,1 Experiment-2 2 3 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 Number of nodes Series1 Series2 Series3 Series4 Series5 Series6 Series7 Series8 Series9 Series1 Series11 Series12 Series13 Series14 Series15 Series16 Series17 Series18 Series19 Figure 4. Distribution of transversal tractions - various flaws in Experiment 2 MPa,8,7,6,5,4,3,2,1 -,1 -,2 Experiment-3 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 Number of nodes Series1 Series2 Series3 Series4 Series5 Series6 Series7 Series8 Series9 Series1 Series11 Series12 Series13 Series14 Series15 Series16 Series17 Series18 Series19 Figure 5. Distribution of transversal tractions - various flaws in Experiment 3

MPa,7,6,5,4,3,2,1 -,1 -,2 Experiment-4 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 Number of nodes Series1 Series2 Series3 Series4 Series5 Series6 Series7 Series8 Series9 Series1 Series11 Series12 Series13 Series14 Series15 Series16 Series17 Series18 Series19 Figure 6. Distribution of transversal tractions for various flaws in Experiment 4 6. Curvilinear fibers - plastic matrix Now we concentrate our attention on nonlinear behavior of concrete according to [3]. Also, the interfacial zones behave nonlinearly according to the law described in chapter 4. Fig. 7 shows the possible geometry and numbering used for denotation of values on the interface. 1 2 3 4 1 2 3 4 1 11 13 15 17 19 21 9 5 6 7 8 1 12 14 16 18 2 9 5 6 7 8 Figure 7. Numbering of nodes of fiber for the slope of 45 Since the optimization problem strongly depends on the meshing, some types of meshing in the neighborhood of fibers are shown in Fig. 8. The meshings for slopes of successively, 1, 3, 45, 6, and 9 degrees are illustrated in these pictures in the vicinity of fibers. () Degree (1) Degree (3) Degree (45) Degree (6) Degree (9) Degree Figure 8. Meshing in the neighborhood of fibers Although the mesh changes sometimes irregularly, the results seem to be relatively reliable and reasonable. In each example the stainless wire fibers are considered. The material properties are described as follows: (concrete) E' = 179 GPa, E' p = 154 GPa, E' r =

1 GPa, v' =.33 v' p =.45, and (steel) E'' = 33 GPa, E'' p = 22 GPa, E'' r = 5 GPa, v'' =.16, v'' =.38. The primed quantities belong to steel and two primed belong to concrete. The subscript p denotes peak and r residual values. Along the interfacial boundary the brittle behavior is considered. It means that ideally elasto-plastic law is obeyed on this boundary. Two cases are studied in details: 1. In the first case, the shear bond strength is c = 4.35 MPa. The tensile strength is.42 Mpa 2. The second case is characterized by interfacial properties given by the shear bond strength c = 2.3 Mpa and the tensile strength is.2 MPa. In both cases the same properties of fibers and concrete matrix are considered, but the interfacial material properties differ basically: In the first case relatively high parameters are implied and in the second case the interface is not as stiff as in the first case. The optimization procedure leads to the admissible pullout forces acting at the face of the samples, the distribution of which is seen in Figs. 9 to 12. In Fig. 9 results from Experiment 1 are depicted, in Fig. 1 the admissible forces from Experiment 2 are described, in Fig. 11 from Experiment 3 and in Fig. 12 forces from Experiment 4 are described, all in dependence of slope angles, which are considered as, 1, 3, 45, 6, and 9 degrees. Assuming the dependence of pullout force slope of the curvilinear fibers to be continuous, in all experiments the straight fibers have the optimal shapes. Similarly, for the second case the distributions of admissible forces are displayed in Figs. 13 to 16. Experiment-1,6,5,4,3,2,1 1 3 45 6 9 Figure 9. Distribution of admissible forces in Experiment 1 the first case Experiment-2,7,6,5,4,3,2,1 1 3 45 6 9 Figure 1. Distribution of admissible forces in Experiment 2 the first case Experiment 1 considers the volume fraction ratio about.7 percent and larger distances between fibers in vertical direction. Long debond zone, but relatively short sliding zones are recorded in this example. Due to the bending of the fiber the axial contact forces are lower in the upper part and vice versa for lower part. This observation is valid for all experiments. Sometimes an additional debond is seen at the points in from of the slope for slopes starting at 4 degrees. Sudden drop of transversal tractions indicates occurrence of a crack.

Experiment-3,5,48,46,44,42,4,38 1 3 45 6 9 Figure 11. Distribution of admissible forces in Experiment 3 the first case Experiment-4,14,12,1,8,6,4,2 1 3 45 6 9 Figure 12. Distribution of admissible forces in Experiment 4 the first case Experiment-1,25,2,15,1,5 1 3 45 6 9 Figure 13. Distribution of admissible forces in Experiment 1 the second case In Experiment 2 we consider relatively higher volume fraction ratio, about 1.5. On the other hand dense distribution of fibers in vertical direction is introduced. The unit cell is the smallest from all four cases. Basically higher stresses are observed at the decisive points. The debonding region is relatively short.

,25,2,15,1,5 Experiment-2 1 3 45 6 9 Figure 14. Distribution of admissible forces in Experiment 2 the second case Experiment-3,1,8,6,4,2, 1 3 45 6 9 Figure 15. Distribution of admissible forces in Experiment 1 the second case Experiment-4,14,12,1,8,6,4,2, 1 3 45 6 9 Figure 16. Distribution of admissible forces in Experiment 4 the second case Experiment 3 has extremely low volume fraction ratio, namely.4, and very long distances between fibers in vertical direction. It leads to decreasing of stresses concentrated at the point of observation. The admissible force appears to be much higher then in the previous case.

Experiment 4 has similar geometry as the first one, and the behavior of the admissible force is also similar. The fibers are almost continuous in horizontal direction. In the latter two cases it is necessary to count with long debonding region in the front part of the fibers. It was shown that in the case of high volume fiber ratio the admissible pullout force is basically lower then in the case of course distribution of fibers. On the other hand, it is necessary to note that the pullout force in this case bears distributed loading, i.e., the tensile force being applied to a cross section of the beam is distributed in such a way that smaller part of the overall loading is applied to this force. Very interesting is a comparison of the previous pseudo-linear case (PL) and the non-linear case (NL) from the last study. The results are summed up in Table 1. Table 1. Comparison of admissible pullout forces for two cases studied in this paper Experiment 1 Experiment 2 Experiment 3 Experiment 4 Pseudo-linear.75.79.68.3 Non-linear.52.6.49.12 Concerning the data from this section, very similar results are obtained for Experiment 2, namely the admissible force in the PL is.79 kn and in the NL.6 kn. The differences in other cases are also closed each other. The higher critical forces in the PL are, of course, due to too stiff concrete matrix consideration. 5. Observations While in the most papers dealing with pullout problem the fibers are straight, in this paper also different shapes of fibers are taken into account. The cohesion of fibers and the concrete matrix is considered and the optimal pullout force is studied. Large test examples were treated and several interfacial conditions were applied. From these examples four experiments are selected to show, first, the material behavior of the interface in the case of the shape envisaged, and second, a possibility of the application of the numerical method introduced at the beginning of this paper. In comparison with the papers [1] and [2] and the papers cited in [1] and [2] it uniquely follows that for the cement-based matrix with steel fibers the shape plays very important role for the interfacial mechanical behavior and, consequently, for the bearing capacity of the aggregate. The optimal slope of Dramix type fibers was the principal goal of chapter 4. It has been shown that the shape of the fibers has very important role in the bearing capacity of fiber-concrete composite. From typical examples of unit cells numerical studies showed important impacts on the behavior of the aggregate. Two cases of interfacial conditions have been studied. In the first case (relatively stiff interface) the best solution is found to be for straight fibers for all arrangements of fibers in unit cells. In the second case (the interfacial zone is chosen in such a way that a relatively lower stiffness is considered), the slope ranging from 45 to 6 degrees seems to be optimal under condition that has been taken into consideration, i.e., the pullout force damages the composite either because of failure of contact conditions or due to attaining the admissible stress (compressive strength). Moreover, the maximum value of admissible force is the higher the course the density of distribution of fibers is. But it is worth noting that the force on the overall cross section is approximately regularly distributed on fibers. The mechanical behavior of fibers is considered as linear elastic, but the material of concrete obeys von Mises Huber Hencky elastic-plastic criterion. This will desire more sophisticated treatment then in the case of straight fibers with linear concrete matrix. It will start with generalized Transformation field analysis, which involves an influence of eigenparameters, see, e.g., [4]. 6. Conclusions For four typical unit cells pullout problem of generally curvilinear fibers out of concrete matrix is formulated and solved in this paper. Starting with modified Uzawa s algorithm and tricks taken from the transformation field analysis extreme pullout force is described for discrete set of slopes of fibers and from this optimal slope is attained. Simplified version is considered for straight fibers as a special case of the general approach. It is seen that the deviations of values from the more accurate computations are not decisive and can be estimated by some constant. Acknowledgment: This paper has been financially supported by the Grand Agency of the Czech Republic, projects No. 13/3/183 and 13/4/1178. 7. References 1. Prochazka P and Sejnoha, M. Development of Debond Region of Lag Model. Int. J. Computers & Structures 1995, 55(2): 249-26 2. Procházka P and Sejnoha M. Pull out problem of lag model. Proc. of BETECH 96, 1996, Honolulu, Hawaii, 132-141 3. Duvaut G and Lions J L. Les inéquations en méchanique et en physique. Paris: Dunod, 1972 4. Dvorak G J Prochazka P. Thick-walled Composite Cylinders with Optimal Fiber Prestress, Composites, Part B, 27B, 1996: 643-649