Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 41, 3, pp. 623-642, Warsaw 2003 ON SURFACE-RELATED SHELL THEORIES FOR THE NUMERICAL SIMULATION OF CONTACT PROBLEMS Rainer Schlebusch Jan Matheas Bernd W. Zastrau Technische Universität Dresden, Dresden, Germany Rainer.Schlebusch@mailbox.tu-dresden.de This paper deals with a surface-related shell theory and its conversion into the finite element method for the investigation of composites and contact problems. In particular, composites made of a textile reinforced concrete are examined for the strengthening of existing shell structures. The interaction between the existing structure and the strengthening layer is considered as a contact problem involving adhesion. Key words: shell theory, enhanced assumed strain method, contact problems 1. Introduction For the design of textile reinforcement for the structural strengthening and restoration in static anddynamic applications suitablemechanicalmodels need to be established. Basic features of the textile reinforced concrete can be found in Curbach (1998). As a possible application, the strengthening of a cylindrical tank, cp. Fig.1, is considered. Reasons for the modification of the existing structural member could be the storing of a substance with a higher density or the increase of the filling height. If the load carrying capacity of the existing structure is not sufficient for the higher loading, an additional layer added to the outer surface might be the best way to improve the tank. Since, generally, the strengthening layer is thin, its mechanical description is favorably based on a shell theory. The present contribution to the develop- ment of a new composite material, textile reinforced concrete, is focused on the derivation of a suitablematerial description and shell theory.With respect 624 R.Schlebusch et al. to the framework of application, the advantage is taken of the free, by princi- ple, choice of the reference surface position by attaching the reference surface to one of the outer surfaces. The shell theory using an outer surface as the reference surface, which is therefore called the surface-related shell theory, is a natural approach for several reasons. Hereby the interface behavior is mo- delled as a contact problem which can take adhesion into account. The idea of surface-related shell theories was already pursued by one of the authors many years ago, cp. Rothert and Zastrau (1981), Zastrau (1983). Unlike in the standard shell theories, which refer to the middle surface of the shell, the models introduced here offer among others the following advantages: • Contact problems can be consideredwithout the usual difficulty ofmap- ping the middle surface data onto the outer shell surface. • The discretization of the contact surface and therewith the complete discretizationmay remain unchanged, in contrast to the classical appro- ach. In an optimization of e.g. the utilization of a material, only the elements of those matrices which belong to elements coordinate to the surface, have to be changed iteratively. • Discontinuities of the stress field, caused e.g. by concentrated loads oc- curring on the surface, which is peculiar to contact problems, can be directly determined where they occur. • Easier detection of compound failure anddelaminationbecomespossible. Fig. 1. Picture of an existing structural member and the model of a cylindrical tank with a strengthening structure made of the textile reinforced concrete In this context the paper by Zastrau et al. (2000) is also referred to, where a very general series expansion in the thickness-direction is proposed. If an On surface-related shell theories... 625 insufficient or a wrong set of terms for the series expansion is used, this so called Naghdi-model does not permit the usage of a three-dimensional, in general, constitutive relation without manipulating the constitutive relation itself. The application of the degeneration concept alone does not permit the usage of a three-dimensional constitutive relation, if for the semidiscretization of the displacements a linear series expansion in the thickness-direction is used.But in combinationwith the enhanced assumed strain (EAS)method its application becomes possible. This combination could beunderstood as a shell theorywhich comprises theminimal number of kinematic equations necessary to operate with a three-dimensional constitutive relation, cp. Bischoff (1999). 2. Continuum mechanical considerations The starting-point for themechanical description of a strengthening struc- ture is the Boltzmann continuum in Lagrangian representation, cp. Eringen (1989).Within the context of this paper, the description of thematerial beha- vior is restricted to isothermal and reversible processes. From this, the number of unknown fields is reduced to three: the displacement field U, the Green- Lagrange strain tensor E and the second Piola-Kirchhoff stress tensor S, where these fields are not independent from each other. They are related to each other by: — kinematic field equation E− 1 2 (F⊤ ·F−G)=E−Eu =0 in B (2.1) — constitutive relation S= ρ0 ∂f ∂E =C :E with C=λG⊗G+2µI in B (2.2) — equilibrium equation Div(F ·S)+ρ0f = ρ0Ü =0 in B (2.3) Herein B is the body in the reference configuration, F = Gradx – the deformation gradient, G – the metric tensor of the reference configuration, E u = (F⊤ ·F−G)/2 – the displacement compatible Green-Lagrange strain tensor, f – the Helmholtz free energy, ρ0 – the density in the reference con- figuration, C – the forth order elasticity tensor of the St. Venant-Kirchhoff 626 R.Schlebusch et al. material, I = G ⊠ G – the forth order identity tensor, and f – the body force per unit mass. In the definition of the identity tensor ⊠ symbolizes the squared tensor product, cp. Halmos (1948), del Piero (1979), defined by A ⊠ B :C := A ·C ·B⊤. Additionally: — dynamic boundary condition F ·S ·N = t0 on ∂σB (2.4) — geometric boundary condition U−U =0 on ∂uB (2.5) are stated to characterize the complete boundary-value problem. In (2.4) ∂σB is the part of the surface of the body in the reference configuration where the surface force per unit area of the boundary t0 is prescribed, and in (2.5) ∂uB is the part of the boundary of the body in the reference configuration where the displacements U are prescribed. The superposed bar indicates prescribed quantities. In addition, N is the outer normal vector to the surface of the body in the reference configuration. Theweak formof the equilibrium equation leads to the principle of virtual displacements,which is the foundation for puredisplacement finite elements. If the kinematic field equation (2.1) and the geometric boundary condition (2.5) are not eliminated but introduced into the principle of theminimumof poten- tial energy by Lagrangian multipliers S and t0, the augmented Hu-Washizu functional is obtained. The demand for a minimum of the total energy in the principle of theminimumof thepotential energy changes to a stationarity con- dition in the Hu-Washizu functional. It describes the following saddle point problem Π(E,U,S,t0)= ∫ B ρ0f dV + ∫ B S : (Eu−E) dV − ∫ B ρ0f ·U dV (2.6) − ∫ ∂σB t0 ·U dA− ∫ ∂uB t0 · (U −U) dA=stat To simplify the four field functional (2.6) and to avoid an explicit interpo- lation for the stress field S the second term on the right-hand side is forced to vanish. This procedure characterizes the above mentioned EAS method which was first presented in the context of a variational formulation of conti- nuum problems by Simo and Rifai (1990), and demands that the additional On surface-related shell theories... 627 strain tensor is orthogonal to the stress tensor. This results in the fact that the orthogonality condition ∫ B S : (E−Eu) dV = ∫ B S : ẼdV =0 with Ẽ=E−Eu (2.7) in the discretized form must be fulfilled. The additional strain tensor Ẽ enri- ches additively the displacement compatible strain tensor Eu to avoid locking phenomena in the shell theory and in the finite element formulation, respecti- vely. This aspectwill bediscussed later on in conjunctionwith the formulation of the shell theory. The discretized additional strain tensor Ẽ has to be linear independent of the displacement compatible strain tensor Eu to avoid singular stiffness matrices, see for details cp. Simo and Rifai (1990). In contrast to the hybrid stress concept, the EAS concept preserves the basic features of pure displacement finite elements, because the resulting additional strain parame- ters neednotbecompatible across the element boundary.They canbe elimina- ted on the element level by condensation. The fulfillment of the orthogonality condition reduces four field Hu-Washizu functional (2.6) to a modified three field functional, which is the underlying principle for the surface-related shell theory and the finite element formulation Π(Ẽ,U,t0)= ∫ B ρ0f dV− ∫ B ρ0f ·U dV− ∫ ∂σB t0·U dA− ∫ ∂uB t0·(U−U) dA=stat (2.8) 3. Surface-related shell theory Since every shell theory is an approximation of the three dimensional con- tinuum theory, basic assumptions have to bemade, cp. Naghdi (1963), Başar andKrätzig (1985). Themost serious assumption is related to the kinematics. Before the chosen kinematics is discussed in detail, the differential geometry of the shell continuum has to be described, cp. Fig.2. To reach this, a general curvilinear, convected coordinate system Θ1, Θ2 and Θ3 is introduced in the shell continuum.With the aid of this parametriza- tion, every point in the shell continuum is uniquely identified by the position vector X. With the position vector X one obtains the base vectors Gi and 628 R.Schlebusch et al. Fig. 2. Differential geometry of the shell Gi as well as the Riemannmetric tensor G of the shell space in the reference configuration as Gi =X,i (3.1) G=Gi ·GjGi⊗Gj =GijGi⊗Gj = δjiGj ⊗G i =Gi⊗Gi where δ j i is the Kronecker symbol and the comma is the partial derivation with respect to the convective coordinate Θi. With Gi ·Gj = δji G=det(Gij) dV = √ GdΘ1dΘ2dΘ3 (3.2) the contravariant base vectors and the volume element of the undeformed shell space are also introduced. Additionally, a space tangent to the shell reference surface is required. Analogous to (3.1) and (3.2) this leads to Aα =X,α (3.3) A=Aα ·AβAα⊗Aβ =AαβAα⊗Aβ = δβαAβ ⊗Aα =Aα⊗Aα On surface-related shell theories... 629 and Aα ·Aβ = δβα A=det(Aαβ) dA= √ AdΘ1dΘ2 (3.4) where X =X(Θα,0) is the position vector of the reference surface, see Fig.2. In order tomake the decomposition of tensorswith components perpendicular to the shell reference surface possible, a third vector is introduced as A3 =X,3 ∣∣∣ Θ3=0 = A1×A2 |A1×A2| (3.5) and is called the unit normal vector. Sincenormal coordinates are used for Θ3, it has the following properties A3 ·Aα =0 A3 ·A3 =1 A3 =A3 (3.6) If thepartial derivative of X =X+Θ3A3 is formed, the following relationship between thebase vectors Gi in the shell space and thebase vectors Ai located in the reference surface can be established Gi =Z ·Ai (3.7) introducing the shell shifter tensor in the following way Z=Gi⊗Ai =µjiAj ⊗A i (3.8) Therewith holds also µ=detZ= √ G A =1−2Θ3H+(Θ3)2K (3.9) and it follows dV =µ √ AdΘ1dΘ2dΘ3 =µdΘ3dA (3.10) where H = (trB)/2 and K = tr + B are the mean curvature H and the Gaussian curvature K of the surface respectively calculated fromthecurvature tensor B=−Aα ·A3,βAα⊗Aβ and its adjoint tensor. The shell shifter tensor is a two point tensor, cp. Ericksen (1960), allowing to replace the base vectors Gi and G i with the base vectors Ai and A i. The shifter tensor is used to define so-called surface-related tensors which are labelled with hats. Firstly, it is used to shift the Green-Lagrange strain tensor E E=Z−⊤ ⊠ Z−⊤ : Ê=EijG i⊗Gj Ê= Z⊤ ⊠ Z⊤ :E=EijA i⊗Aj 630 R.Schlebusch et al. and, secondly, to shift the second Piola-Kirchhoff stress tensor S S= Z⊠ Z : Ŝ=SijGi⊗Gj Ŝ=Z−1 ⊠ Z−1 :S=SijAi⊗Aj The orthogonality condition can be transformed into the following notation ∫ B S : Ẽ dV = ∫ B (Z ⊠ Z : Ŝ) : (Z−⊤ ⊠ Z−⊤ : ̂̃ E) dV = (3.11) = ∫ B Ŝ : ̂̃ E dV = ∫ B SijẼij dV =0 which shows that the shifted quantities can also be used to evaluate the or- thogonality condition. Fig. 3. Undeformed and deformed part of the shell continuum and shell kinematics Due to the displacement field U, the shell continuum is deformed from the reference configuration into the actual configuration, see Fig.3. This general deformation is restricted by a kinematical assumption. Within the scope of the presented shell formulation, it is assumed that the displacement field is a sum of two parts U =V +Θ3W (3.12) first, the displacement field V of the reference surface and, second, a part in the direction of W being linear in the normal coordinate Θ3. Here W is a On surface-related shell theories... 631 vector field describing the displacement of points along the normal vector. In general, the resulting vector a3 is no longer perpendicular to the deformed reference surface a. Introducing the restricted displacement field (3.12) in the definition of the displacement compatible strain tensor, it can be determined as follows Ê u = [αuij +Θ 3βuij +(Θ 3)2γuij]A i⊗Aj =αu +Θ3βu +(Θ3)2γu (3.13) with the sub-strain tensors α u =αuijA i⊗Aj βu =βuijAi⊗Aj (3.14) γu = γuijA i⊗Aj and the sub-strain components αuαβ(V ,W)= 1 2 (V ,α ·Aβ +V ,β ·Aα+V ,α ·V ,β ) βuαβ(V ,W)= 1 2 [V ,α ·(A3,β+W ,β )+V ,β ·(A3,α+W ,α )+W ,α ·Aβ +W ,β ·Aα] γuαβ(V ,W)= 1 2 (W ,α ·A3,β+W ,β ·A3,α+W ,α ·W ,β ) αuα3(V ,W)= 1 2 [V ,α ·(A3+W)+W ·Aα] (3.15) βuα3(V ,W)= 1 2 [W ·A3,α+W ,α ·(A3+W)] γuα3(V ,W)= 0 α u 33(V ,W)= 1 2 W · (2A3+W) βu33(V ,W)= 0 γ u 33(V ,W)= 0 Thedisadvantage of theused 6-parameter shell kinematics is that it suffers fromtheso-calledPoisson thickness locking,whichwillbe shortlydiscussed.To explain the Poisson thickness locking, a beamwith a rectangular cross-section as shown in Fig.4 is examined. Thematerial of the beamhas aPoisson’s ratio being unequal to zero. Due to pure bending a linear normal stress distribution over thebeamthickness is obtained, for instance tensionabove theneutral axis and compressionbelow.This results in transverse contraction in theupperhalf and transverse extension in the bottom half of the cross-section. According to the stress distribution the transverse normal strain is linearly distributed as well, while the overall thickness of the beam or of the shell, respectively, will 632 R.Schlebusch et al. Fig. 4. Description of the origin of the Poisson thickness locking remain unchanged. The material center line, that is the line of material po- ints which are in the undeformed state located on the geometrical center line, moves up and is no longer located on the geometrical center line. Due to the chosen kinematics (3.12) sucha linear distributionof thenormal strain cannot be represented. The corresponding transverse strain component βu33, cp. last Eq. (3.15), is equal to zero. This causes a constraint which prevents themate- rial center line frommotion and produces artificial transverse normal stresses. These so-called parasitical stresses contribute to the internal energy and fi- nally to an increase in the stiffness of the pure displacement finite element. This means that, in bending dominated cases, a relative error of the order of ν2 occurs, even in linear analysis, cp. Büchter (1992), Bischoff (1999). It is obviously not adequate to approximate the deformation of a shell with un- changing thickness by the kinematics (3.12), because this leads to β33 =0.To avoid this locking effect an enhancement of the Green-Lagrange strain tensor is made using the EASmethod. The enhancement is the following Ẽ33 = (H 2 −Θ3 ) β̃33 (3.16) Accordingly, the enrichment of the strain field has only one component. It is important to note that this single component is sufficient to avoid the Poisson thickness locking, because it introduces the missing, in Θ3 linearly varying, expression into the transverse normal strain. Due to the special position of the reference surface, a shift about the half thickness H/2 of the shell is necessary to ensure that the additional strain is zero, if it is calculated in the location of On surface-related shell theories... 633 themiddle surface. Before the additional strain Ẽ33 can be determined, some additional notations and definitions have to be introduced. The integration of the shifted second Piola-Kirchhoff stress tensor Ŝ over the thickness H of the shell defines the stress resultant tensors of the shell theory n=nijAi⊗Aj = H∫ 0 Ŝ(Θ3)0detZ dΘ3 = H∫ 0 Sij(Θ3)0µdΘ3Ai⊗Aj m=mijAi⊗Aj = H∫ 0 Ŝ(Θ3)1detZ dΘ3 = H∫ 0 Sij(Θ3)1µdΘ3Ai⊗Aj(3.17) s= sijAi⊗Aj = H∫ 0 Ŝ(Θ3)2detZ dΘ3 = H∫ 0 Sij(Θ3)2µdΘ3Ai⊗Aj It shouldbementioned that it is characteristic to the presented surface-related shell theory to integrate from zero to H corresponding to the particular po- sition of the reference surface. The stress resultants are called the membrane (n), themoment (m) and the bi-moment (s) stress resultant tensor, respecti- vely. If the constitutive relation (2.2) is also shifted to the reference surface and introduced into the formerly stated definitions of stress resultants (3.17)1 to (3.17)3, the pre-integration over the shell thickness is accomplished and the decomposition into sub-strain tensors (3.13) is used, the constitutive relation can be expressed by the components of the stress resultants and of the sub- strain tensors   nij mij sij  =   D ijkl 0 D ijkl 1 D ijkl 2 D ijkl 1 D ijkl 2 D ijkl 3 D ijkl 2 D ijkl 3 D ijkl 4     αkl βkl γkl   (3.18) Herein, the matrix elements D ijkl 0 to D ijkl 4 are defined by the following inte- grals D ijkl K = H∫ 0 (Θ3)KCijklµdΘ3 with K ∈{0,1,2,3,4} (3.19) For an efficient determination of the integral the determinant of the shell shifter tensor µ is often set equal to one, and the last row and column are also often neglected in the constitutive matrix (3.18). 634 R.Schlebusch et al. With the aid of these definitions, it is now possible to determine the value of the additional strain parameter β̃33 by exploiting thr orthogonality condi- tion (3.11) ∫ V S33Ẽ33 dV =0 (3.20) Furthermore this yields to m33−n33 H 2 =0 (3.21) Theherein enclosed coupling of themembrane (n33) andmoment (m33) stress resultant tensor is characteristic to surface-related shell theories. If the referen- ce surface is identical with themiddle surface, the evaluation of orthogonality condition (3.20) leads to m33 = 0, which is likewise equivalent to the state- ment that in the case of pure bending no transverse normal stresses occur. Consequently, the Poisson thickness locking is avoided with the aid of the EASmethod.As desired no parasitical transverse normal stresses occur in the case of pure bending.With equation (3.21) the value of the additional strain parameter β̃33 can finally be determined from β̃33 = [D33kl1 ,D 33kl 2 ,D 33kl 3 ]− H2 [D 33kl 0 ,D 33kl 1 ,D 33kl 2 ] −D33330 H 2 4 +D33331 H−D33332   αukl βukl γukl   (3.22) Thevalue of β̃33 depends on the elements of the constitutivematrix (3.18) and the displacement compatible sub strains (3.15). Interpreting the additional strain parameter as a kinematical variable one can speak of a 7-parameter shell theory. This is valid, because the strain parameter can be determined in any point of the reference surface. However, it should be mentioned that this seventh parameter can be eliminated on the element level. This results again in a 6-parameter shell theory. Hence, an expansion of the element stiffness matrix and the system stiffness matrix is prevented by the aforementioned elimination. 4. Contact mechanics For the investigation of the behavior of the textile reinforced concrete strengthening of an existing load-carrying structure, the contact mechanics, On surface-related shell theories... 635 which is described extensively inLaursen andSimo (1993),Wriggers (1995), is utilized. The aim is to build the tangential stiffnessmatrix for the normal and tangential contact with the help of contact segments which are placed on the outer surface of the shell element. These segments can be easily used with a body situated in contact that is discretized by volume elements. The difficulty in the middle surface related shell elements is the estimation of the stiffness for tangential and normal displacement degrees of freedom determined by the segments, because they directly effect the stiffness affiliated to the rotational degrees of freedom of the shell element. Using shell theories with higher order kinematics, thismapping from surface data on themiddle surface is connected with considerable effort. Furthermore, it is necessary to know the geometry of the shell element connected with the contact segment. A combination of the contact segment and the shell element becomes unavoidable. Using surface- related shell theories, the assignment of the segments can be performed in the samemanner like using volume elements, because only the stiffness correspon- ding to the three displacements has to be calculated. The coordinate systems of the contact segment as well as of the shell element are both situated in the contact area. The principle of the virtual work can be utilized to derive the tangential stiffness matrix of the shell element because of the normal and tangential contact conditions in the case of sticking contact partners. If the tangential contact stress exceeds a static limit, which can either be dependent on the contact normal pressure p in the form τmax =µpwith µ as the coefficient of friction or be independent with τmax = τstick, then the shear stress τ(gN,gT)=− gT |gT | [µp(gN)+k(gT)] (4.1) is used in the case of slipping. It depends on the gap function gN and the tangential relative displacements represented by the slip vector gT . The first part in (4.1) states Coulomb’s friction law. The resulting stiffness k(gT) can be regarded as the tangential compound stiffness, e.g. adhesion within the contact area. This offers the possibility to describe compoundproblemswithin the scope of contact mechanics. 5. Locking phenomena Among others, the kinematical restriction (3.12) and the interpolation of the degrees of freedom, which is realized by Lagrangian polynomials, are 636 R.Schlebusch et al. the source of several locking phenomena, cp. Bischoff (1999), Bischoff and Ramm(1997). In particular, the following phenomena should benamed in this context: • Poisson thickness locking • membrane locking • volume locking • curvature thickness locking • shear locking Hence, arrangements have to bemade to reduce or avoid these locking effects, because all the artificial stiffening effects cause a significant deviation from the continuummechanical solution. In this context the appropriate concepts shall only be stated. For more details the reader is referred to the cited literature. As described before, in connectionwith the formulation of the shell theory the EAS method is used to prevent the Poisson thickness locking. The described specialization is in close relation to Büchter andRamm (1992), Büchter et al. (1994). Furthermore, the EASmethod is used to prevent or reduce the mem- brane and volume locking. An effective concept against the curvature thick- ness locking is the ANS method having a variational justification, cp. Simo and Hughes (1986), whereas the discrete shear gap (DSG) method, cp. Blet- zinger et al. (1998), shows high efficiency to avoid the shear locking. With a combination of the aforementioned concepts a very efficient finite volume shell element has been developed. The applicability of the presented finite element is demonstrated in the following numerical examples. 6. Numerical examples The first example is a square plate with build-in edges and a central point load. The geometry and material data are given in Fig.5. The geometry, the material data and the value of the central point load are selected in such away that the central deflection of theplate is 0.01m, if theKirchhoffplate theory is used to calculate the deflection.Making use of the symmetry, only one quarter of the plate is dicretized with 8-node shell elements. The number of elements per edge is varied, thus the fineness of thediscretization varies aswell. Because this example represents a bending dominated problem, the Poisson thickness locking and shear locking are expected even in the considered linear analysis. The results of this example are shown in Fig.6. The diagram shows that the On surface-related shell theories... 637 puredisplacement element doesnot reach the exact solution, even ifmore than ten elements per edge are used. That is, a too coarse discretization results in entirely useless results. Fig. 5. Square plate with build-in edges and a central point load Fig. 6. Central deflection of the plate as a function of the number of elements per edge However, the results can obviously be improved, if the EAS method and the DSG method come into operation. A discretization with 2× 2 elements is fine enough to reach very good results in comparison with the analytical solution. Even if only one element is used and the relative error of approxima- tely 10% can be accepted, the element leads to good results in the framework of application. Additionally performed numerical tests show in this example that the developed shell element is also insensitive to distorted discretizations. That is, the element gives reliable results in bendingdominatedplate problems even in coarse as well as in distorted discretizations. For a further numerical test a circular plate with a circular hole as shown in Fig.7 is analyzed, whichwas also examined byBaşar andDing (1990). The 638 R.Schlebusch et al. Fig. 7. Perspective view of a circular plate with a circular hole at the center according to Başar and Ding (1990) material data, the geometry and the load are given inFig.7. The circular plate is coarsely discretized byone element in the radial direction and16 elements in the circumferential direction. The vertical deflection of the points A and B is examined. The calculated results are nearly equal to those given in literature, Büchter (1992), Klinkel (2000). Fig. 8. Diagram of the vertical deflection of the points A and B Themaximum relative deviation is less than 2%. The vertical deflections of the points A and B are depicted inFig.8 as a function of the load factor λ. The reference solution is taken fromBüchter (1992), and is calculated with a discretization of 30×6 4-node shell elements. Figure 9 shows an unscaled drawing of the deformed configuration of the circular plate under the maximum load. The gray shading illustrates the ab- solute value of the displacement. On surface-related shell theories... 639 Fig. 9. Deformed circular plate under maximum load The last example should show the application of the contact mechanics. We choose a square plate, 7.20×7.20×0.20m3, with simply supported edges and a central point load. The plate is made of St. Venant-Kirchhoff material with Young’s modulus E =2.5 ·107kN/m2 and Poisson’s ratio ν =0.2. The plate is strengthenedbyasymmetrically applied textile reinforcedconcreteply, 6.30×6.30×0.02m3, that is pressed against the plate by a contact pressure of 0.1kN/m2 to establish the contact. Fig. 10. Contact area and contact pressure (a) and contact shear stress distribution (b) F =3.75kN For the maximum shear stress the friction coefficient µ0 =µ=0.15 is as- sumed, cp. (4.1). Young’smodulus of the strengthening is E=2.0·107kN/m2 640 R.Schlebusch et al. Fig. 11. Contact area and contact pressure (a) and contact shear stress distribution (b) F =10.0kN and Poisson’s ratio is ν =0.2. Making use of the symmetry, only one quarter of the plate is dicretized with 4-node shell elements. In the first step, the cen- tral point load of F = 3.75kN was applied, see the results in Fig.10, and in the second step to central point load F was 10kN, see the results in Fig.11. The white areas in the contact domain represent delaminated regions of the strengthening layer.The size of thedelaminated regions grows extensivelywith an increase in the load F. Only a quadratic region in the plate center and a strip near the edge stay in contact. 7. Conclusions The formulation of surface-related shell theories allows an efficient simu- lation of the compound behavior of textile reinforced concrete layers for the strengthening of shell structures. Due to the particular position of the refe- rence surface, the well established concepts against the locking phenomena are extended and implemented, which leads to a reliable surface-related finite volume shell element.With the aid of several different nonlinear examples, the efficiency of the developed shell and contact formulation is demonstrated. Acknowledgement The authors gratefully acknowledge the financial support of this research from Deutsche ForschungsgemeinschaftDFGwithin the Sonderforschungsbereich SFB 528 ”Textile Reinforcement for Structural Strengthening andRetrofitting” at Technische Universität Dresden. On surface-related shell theories... 641 References 1. Başar Y., Ding Y., 1990, Finite-rotation shell elements for analysis of finite- rotation shell problems, Second World Congress on Computational Mechanics, 785-788 2. Başar Y., Krätzig W.B., 1985, Mechanik der Flächentragwerke, Friedrich Vieweg & Sohn VerlagmbH, Braunschweig 3. 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ZastrauB., 1983,EinBeitrag zurModifikation der klassischenSchalentheorie für die unmittelbare Berechnung vonKontaktproblemen, Zeitschrift für Ange- wandte Mathematik und Mechanik, 63, 224-225 22. Zastrau B., Schlebusch R., Matheas J., 2000, Surface-related shell the- ories for the treatment of composites and contact problems,Proceedings of the Fourth Colloquium on Computation of Shells and Spatial Structures, 76-84 Powierzchniowe teorie powłok w numerycznej symulacji problemu kontaktowego Streszczenie Wpracy omówiono problem powierzchniowej teorii powłok i jej konwersji dome- tody elementów skończonych w kontekście badań kompozytów i zagadnienia kon- taktowego.W szczególności zajęto się kompozytami osnową cementową wzmacnianą materiałem tekstylnym jako komponentemnośnych konstrukcji powłokowych.Uwagę skoncentrowanona oddziaływaniu, jakie zachodzi pomiędzy strukturą nośną iwzmac- niającą w takich powłokach. Przedstawiono analizę tego oddziaływania jako zagad- nienia kontaktowego z włączeniem zjawiska adhezji. Manuscript received November 28, 2002; accepted for print April 3, 2003