FAILURE ANALYSIS OF CURVED LAYERED TIMBER CONSTRUCTIONS

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1 FAILURE ANALYSIS OF CURVED LAYERED TIMBER CONSTRUCTIONS Stevan Maksimovic 1) and Milorad Komnenovic ) 1) VTI Aeronautical Institute, Ratka Resanovica 1, Belgrade, Serbia and Montenegro s.maksimovic@net.yu ) Faculty of Civil Engineering, Bulevar Revolucije, Belgrade, Serbia and Montenegro Summary This paper is focused on developing of computation method in failure analysis of timber constructions. Timber constructions are considered as layered composite structures. Improved finite elements are included to analyze stresses of layered composite structures. Improved finite elements are based on using in-plane rotations. To determine failure loads of timber structures several initial failure criterions for anisotropic composite structures are included. Combining finite elements in stress analysis with several initial failure criterions of layered timber constructions very useful and efficient method in failure analysis of complex layered timber constructions is obtained. For this purpose several curved layered beams are tested under static loads. Presented computation method and procedures are illustrated and results are compared with experiments. Good agreements between computation and experimental results are obtained. Combining reliable FEM models in stress analyzes, in connection with appropriate failure criterions which are valid for orthotropic materials, reliable and efficient method for complex glulam structures designing is established. Presented computation method can be used for failure analysis of the layered timber structures with general shapes. Key words: Curved laminate timbers, Finite elements, Failure criterions, Computations and experiments 1. Introduction Nowdays, modern wooden structures are providing extraordinary posibilities for designers and architects in shaping up space using most different design forms, achieving both, esthetical and functional ambient requirements, regarding economical and fast building aspects. This is primarily related to glued-laminated timber structures. However, to bring up full advantages of laminated timber structures, it is necessary to have best knowledge and expertize in using of wood properties, as well as 1

2 usage of modern computation methods for stress and stability analyzes, considering orthotropic wood properties together with adequate failure criterions. For stress analyzes in glulam members, analytical and numerical methods are used basically. By analytical methods, designs of simple geometrical shape and border conditions can be adequately solved. However, in complex glulam structures strength analyzes, which besides complex geometrical shapes assume: describing complex border conditions, including orthotropic characteristics of individual lams and often is necessaryto include local mechanical joint strength analyzes, those analyzes are still quite limited. In these situations, it is necessary to use modern numerical methods, like finite element methods (FEM). Main advantage of FEM is that without difficulties it can be used for complex geometrical shapes modeling, and for describing of all kinds of border conditions and different load cases. Subject of this paper are stress analyzes in glulam members, from the aspect of numerical analyzes on one, and by including appropriate criterion for evaluating load levels at wich initial failure is happening on the other side. Because of that, this paper gives significance to application of appropriate FEM model for stress analyzes, especially in curved glulam members, but also in other shapes. Besides numerical modelimg of structures based on FEM, in this elaborate experimental researches of curved glulam members are performed. Comparing of FEM based calculations results with experimental results are basically done for estabilishing appropriate and reliable procedure for failure analyzes and dimensioning of curved glulam members.. Finite Element Analysis In this investigation is used FEM for stress analyzes and displacement of complex glulam structures. Definition of FEM model is based on assumed displacement. Constructional model includes orthotropic characteristics of the material. For problems considered in this paper, improved three and four-node shell finite elements 1 are included, based on Discrete Kirchoff Theory (DKT). These finite elements posses all six degrees of freedom in every node: three translations and three rotations. Those same elements can be used as for working stress determining, so for failure (stability loss) analyzes. Improved behavior of finite elements makes introduced correct degree of rotation (θ z ) in the plane of finite element. Finite element defined in this way possesses exceptionally high precision in describing behavior within its plane, which is primary goal in analyzes of curved glulam members The 4-node shell finite elements [1,] are used in this investgation.these elements present eight degress of freedom at each node: three translaions, three rotations about the nodal x, y and z axes and two higher order terms. Description of layered shell finite element Based on the higher order shear deformation plate theory in the present analysis, a four-noded quadrilateral element with 8 degrees of freedom per node [1] is used. The formulation of a 4-noded shell finite element that can be good enough also if applied to the thin multilayered plates/shells is by no means an easy matter. The authors experience has shown that a good approach to the formulation of a 4-noded shell finite element can be based on the application of the Discrete Kirchhoff s Theory (DKT) [3] for bending behavior. DKT ensures C 1 continuity at discrete points on inter-element boundaries. The improved 4-noded layered shell finite element is derived combining HOST and DKT, Fig.1.

3 y z z 4 s y v 3 w u x x y 1 r Z 1 x X Y Fig. 1: Description of 4-noded shell finite element More details about that element can be found in [1] and [7]. In the C o finite element theory the continuum displacement vector within the element is defined by M a = N ( r, s) a (1) i= 1 i i where N i (r, s) is the interpolation function associated with the node i and expressed through the normalized coordinates (r, s); M is the number of nodes in the element and a i is the generalized displacement vector in the mid-surface. In the case of the negligible mid-surface normal stress σ z the stress-displacement relationships, stress resultants and the constitutive equations associated with the higher-order shear deformation theory are given in [1] and [7]. The total stiffness matrix of the element is obtained by the linear superposition of the following three independent parts: i. Membrane stiffness matrix K M ii. Bending stiffness matrix K B, and iii. Rotational stiffness matrix K Θz Four-node quadrilateral layered shell element, for geometrical nonlinear analysis, is derived combining a higher order shear deformation theory and membrane elements with drilling/rotational degrees of freedom. This finite element has 8 degrees of freedom (DOF) per node a T =(u 0, v 0, w 0, θ x, θ y, θ z, ψ x, ψ y ) () where u 0, v 0, w 0, θ x, θ y represent conventional degrees of freedom, θ z are in-plane vertex rotations. The terms ψ x and ψ y are the corresponding higher order terms in the Taylor s series expansion used in the theory and are also defined at the reference plane. This element is obtained by superposition of the refined membrane element with rotational degrees of freedom and discrete Kirchhoff model for bending. In order to avoid singularity in the assembled matrix using flat elements in a global coordinate system here is used membrane element, which includes in-plane nodal rotations, θ z, as degree of freedom. In this work Allman s approach is used. Allman s approach [4] begins by selecting a quadratic form for 3

4 the normal component of displacement, U n, and a linear form for the tangential component of displacement, U t, along each element edge i j, Fig.. U U n t ξ ξ ξ ξ = 1 U n1 + U n + 1 ( θ z θ z1 ) l ij l ij l ij l ij (3) ξ 1 U ξ = t1 + U t l ij l ij (4) where ξ is the running distance from one end and (U n1, U t1, θ z1 ), (U n, U t, θ z ) are the translational and rotational components displacements at each end of the edge. v n1 z1 z z v t1 v t 1 v n v n4 v t Fig. : Membrane part of shell element As noted Harder and Mac Neal [5] eqns. (3) and (4) can be used to eliminate the transitional displacements at the midpoint of the edge in favor of the degrees of freedom at the adjacent corner points so that, in this way, any eight-noded membrane quadrilateral can be converted into an element with corner translations and rotations as DOF s. 4. FAILURE CRITERION Failure theories are functions of the stresses and strengths of the material are assumed to represent failure under all loading conditions without regard to failure mechanism r failur mode. For isotropic materials there are three well-known strength theories maximum principal stress, maximum shear stress (Tresca), and distortional energy (Mises-Hencky). In each case a function of the stresses is equated with a single parameter, the tensile yeald strength, or the fatigue strength of the material. Recognition of the fact that anisotropic materials have more than one strength parameter has led to numerous proposals for failure theories. More than 40 such theories have been proposed for wood, reinforced plastics, etc. Many of the theories are quadratic functions of the stresses and strengths. Failure theories for plane stress involve σ 1, σ and σ 6. Thorough ivestigation of a failure theory involves comparison of 3 4

experimental data for complex stress loading with the chosen failure surface. This requires an experimental facility capable of changing the relative of σ 1, σ and σ 6 over a wide range.

5 experimental data for complex stress loading with the chosen failure surface. This requires an experimental facility capable of changing the relative of σ 1, σ and σ 6 over a wide range. Along with some practical glulam structures failure estimations, which are mostly based on allowed stresses in fiber direction and perpendicular to fiber direction, there are other failure criterions, based on interactive expresions of work stress and correspondent strength of materials with orthotropic characteristics. For this purpose Tsai-Wu initial failure criterion [6] is used. In tensor notation, initial failure is predicted to occur when failure index (F.I): F.I = F 1 σ 1 + F σ + F 1 σ 1 σ + F 11 σ 1 + F σ + F 66 σ 6 1 (5) where i,j=1,, 6, and F ij are nd and 4 th rank strength tensors, respectively. For the case of plane stress equation (5) can be expressed in the form. The coefficients are described in terms of the strength of timbers. 1 1 F = 1, X 1 1 F t X =, F 11 c XX =, 1 1 F t c YY =, t c Y 1 1 1/ F =, F t Y 66 1 = ( XtXcYY t c ) c T where: X t and X c are the tensile and compressive strength parallel to the direction of the fibre, Y t and Y c are the tensile and compressive strength transverse to the direction of the fiber and T is shear strength. 5. Numerical and experimental results Subject of numerical and experimental study are strength analyzes of curved members under static loads. In practical designing of wooden structures, radial stresses represent critical components for dimensioning. Purpose of these experiments was to perform measurements of stresses analyzis and shifting, which would serve for verification of methods and software for curved members strenth analysis, and beside that, to clarify influence of member curvature to the radial stresses. For this purpose, for testing were defined five groups of specimens. All specimens were of same span L=600 cm and height H=36 cm, and constant cross section 36x8 cm, with variety in curvature, Fig. 3 and Table 1. Fig. 3 Curved timber specimens 5

6 Specimens were made of constant thickness lams (4 lams of 1.5 cm thickness). Lams were made of first class conifer. During testing, measured dampness of wood was %. Figure 4 shows good agreement finite element results with experiments of radial stresses at the curved laminated structures. To determine initial failure loads of curved wood members here Tsai-Wu initial failure criterion is used. Computation failure load level of curved timber member, defined using Tsai Wu criterion, is in good agreement with experiment too (See Table 1 and Figure 5). Table 1: The effect of curvature of specimens (R) on failure load (F f ) Type No. of j f F f [kn] ( j l ) av specimens specimen D D D-1 D ( R=460) D D D D D- D (R=340) D D D D D-3 D (R=4810) D D D D D-4 D (R=9570) D D Straight D member D D-5 D (R= ) D D The corresponding strengths that are necessary for initial failure analyses are experimental determined. In this investigation Tsai-Wu initial failure criterion is used to study initial failure load. X t = 8.7 kn / cm X c = 5.63kN / cm Y t = 0.10 kn / cm Y c = 0.7 kn / cm T = 1.00 kn / cm Combining high-quality shell finite elements and adequate initial failure criterions that is possible to study failure of complex timber structures. This shell finite element can be used in stability analysis of timber structures too. 6

7 H [mm] σ y [kn/cm ] H [mm] D1; j= FEM 1L exp σ y [kn/cm ] R=460 R=340 R=4810 R=9570 Ravan Nosac a b Fig. 4 The effect of radius of curvature (R) on maximal radial stresses using FEM, F=11. kn LAMELIRANI DRVENI NOSAC R=460mm; j=1.7 Koeficijent loma: Tsai-Wu kriterijum Y Z X Failure Indexes (at Failure load level of specimens) F.I (D-1) = 0.80 F.I (D-) = F.I (D-3) = F.I (D-4 ) = F.I (D-5) = 0.85 Fig. 5 Failure index distributions at specimen D-1 using Tsai-Wu criterion for failure load (j=1.7) 6. CONCLUSIONS Inside this investigation is developed numerical procedure for curved members made of glued laminated timber strength calculation. For this purpose is used finite elements method (FEM) for stress analyzes in connection with appropriate criterion for load levels at which initial damage occurs in curved member. Particular attention in this process was aimed on curvature influence determination in those particular specimens, to values of normal stresses perpendicular to fiber direction. Comparations between working stresses obtained by FEM analyzes and experimental results were done. In all those 7

8 analyses was obtained exceptional accordance of numerical and experimental results. That indicates that proposed numerical procedure within the work can be reliably used for working stresses determinations, as well as failure analyzes for the curved glulam members. Combining reliable FEM models in stress analyzes, in connection with appropriate failure criterions, which are valid for orthotropic materials, reliable and efficient method for complex glulam structures designing is established. Presented computation method can be used for failure analysis of the layered timber structures with general shapes. References 1. Maksimovic, S., A., simple and effective finite element for nonlinear analysis of layered composite shells using refined higher-order theory, Proc. 6th World Congress on FEM, Banff, Canada, Ed. J. Robinson, Robinson and Associates, Maksimovic, S. and Komnenovic, M., (1994) Improved Nonlinear Finite Element Analysis of Layered Composite Structures Using Third-order Theory, The Second Int. Conf. on Computational Structures Technology, Athens, Greece, Eds.Topping, B.H.V and Papadrakakis, M., Civil Comp Press. 3. Dhatt, G., Marcotte and Matte, Y., A New Triangular Discrete Kirchhoff Plate/Shell Element, Int.JNME, Vol. 3, , Allman, D.J., A compatible triangular element including vertex rotations for plane elasticity analysis, Comput. Struct., 19, 1-8, Mac Neal, R. H. and Harder, R. L., A refined four-noded membrane element with rotational degrees of freedom, Comput. Struct., 8, 75-84, Tsai, S. W and Wu, E. M., A general theory of strength for anisotropic materials, J. Comp. Mater, Vol. 5, 58-80, Maksimović, S., Rudić, Z., Analysis of buckling of fibrous composite shells by finite element method, World conference on composite structures, Nica, France, Ed. A-Niku Lari, Pergamon Press, London,

EXACT BUCKLING SOLUTION OF COMPOSITE WEB/FLANGE ASSEMBLY

EXACT BUCKLING SOLUTION OF COMPOSITE WEB/FLANGE ASSEMBLY J. Sauvé 1*, M. Dubé 1, F. Dervault 2, G. Corriveau 2 1 Ecole de technologie superieure, Montreal, Canada 2 Airframe stress, Advanced Structures,