3-D Bernoulli Beams within Akantu

Transcription

1 3-D Bernoulli Beams within Akantu Semester Project Fall 2011 Fabian Barras Professor Jean-François Molinari Supervisors Seyedeh Mohadeseh Taheri Mousavi Guillaume Anciaux Nicolas Richart Computational Solid Mechanics Laboratory - LSMS Ecole Polytechnique Fédérale de Lausanne - EPFL

2 Contents 1 Introduction Purposes Some structural mechanics The 3-D Bernoulli Beam Element Definition of the Element From local Archetype Element to the global space Integration Finite Element Model Stiffness Matrix Force Vector Assembly Stresses Post-Processing Patch Test Presentation Analytical Solution Numerical Solution Comments Wood Tower, the study of a complex structure Presentation Mesh loading on Akantu Model Reader Sets assignment Verification Wind Loads Analysis Conclusion 39 1

3 Abstract This report details the implementation of the Finite Element Method dealing with Bernoulli structural mechanics model for 3-D Beams. The 3-D Beam Element is developed within Akantu which is an open source Finite Element library implemented at the Laboratory of Computational Solid Mechanics (LSMS) of EPFL. The aim of Akantu s design is to consider different kinds of models (solid mechanics, structural mechanics, heat transfer, etc...) while staying as generic as possible in the application of the Finite Element Method (FEM). Thus each object or class presented in this report is implicitly related to Akantu s library. 1 Introduction 1.1 Purposes This semester project consists in the extension of Bernoulli model for 3-D beams. According to the general philosophy of Akantu, this implementation should stay as generic as possible. Therefore the 3-D Element should be developed within the structure already defined for the 2-D beams and more generally for all structural mechanics Elements. Before starting the implementation, mechanics and their assumptions should be studied in details, in order to construct an Element consistent with different behaviors. Then the development of the Element according to mechanical assumptions and Finite Element Theory is presented in details in this report. For validation of the model, a patch test was performed in comparison with an analytical case mixing the different behaviors in three dimensions. Finally 3-D Bernoulli Beam Elements are used in the study of a complex tridimensional structure to validate the developed model also on a large number of elements. 1.2 Some structural mechanics In this section, the theoretical model and its assumptions used in the development of the 3-D Beam Element is described. A beam is a tridimensional structure modeled with a curvilinear element suited for structural mechanics. Indeed geometrical and material properties are assigned to its axis. The direct system of coordinates x y z is used in this report. Figure 1 presents 2

4 the convention used for axes related to beams. Figure 1: System of coordinates considered in this section Three different behaviors are considered for the development of the element. They are briefly explained in the following sections, based on the theories presented in [3]. Note: In structural mechanics, stresses can often be considered as integrated on the beam axis. They are thus expressed in unit of load ([N]). Solid mechanics usually considers that stresses are applied on an infinitesimal surface and they are then expressed in terms of pressure ([Pa]). In this report, structural mechanics convention is used. Stresses at the axis are related to solid mechanics stresses by equivalence principles. More details about equivalence principle are given in the end of the present report. Axial Behavior The axial behavior statement is shown in figure 2. The cross section has a constant area A. The displacement field u(x) and the Normal stress N are demanded under a given distributed axial load state q x. 3

5 Figure 2: Axial behavior taken from [4] Kinematic relation is directly derived from Bernoulli s law. It corresponds to the relation expressed as : Planar cross sections which are perpendicular to the axis will be conserved in deformed configuration. This assumption is the core of modeling Beam Element that was also named 3-D Bernoulli Beam Element. It leads to the following kinematic relation: du dx = ε x (1) with ε x the axial strain. Constitutive law which linked the normal stress N to the axial strain is described by the linear elastic Hooke s law : N EA = ε x (2) with E the elastic modulus. Finally, the equilibrium of axial stresses gives the following relation : dn dx = q x (3) Equations 1, 2 and 3 give the following differential equation for the axial behavior : EA d2 u dx 2 + q x = 0 (4) 4

6 Planar Flexural Behavior Flexural behavior into the plane x y can be seen on figure 3. The same beam as before is assumed. Moreover, the beam is subjected to a vertical distributed load q y. The moment of inertia of its cross section is defined by I. Displacements field is described by the vertical displacement v(x) of a point in the axis and the rotation θ(x). Figure 3: Flexural behavior taken from [4] Kinematic conditions are a bit more complex than in the axial behavior. Bernoulli s law is then expressed as : After deformation, cross sections remain planar and perpendicular to the curved axis but also to all other fibers of the beam. Small displacements are also assumed for the model in a way that the neutral axis of the beam conserves the same length after deformation. 5

7 Figure 4: Beam under bending from R.Hooke, taken from [3] Those kinematic conditions are illustrated in figure 4 and bring the following relations for the curvature r : 1 r = dθ dx = d2 v dx 2 (5) Constitutive relation for flexion in the model is also expressed by linear elastic Hooke s law as : 1 r = ψ = M (6) EI Finally, Bernoulli Beams theory belongs to the class of Thin Structures for which the normal planes stay straight and perpendicular to the axis after flexural deformations. In other words, cross sections do not buckle under flexion. Deformations under shear are thus neglected and the shear stress V is directly derived from the moment M. Indeed this three following equilibrium equations are stated : dv dx = q y dm dx = V d2 M dx 2 = q y (7) Differential equation of flexural behavior which is the result of equations 5, 6 and 7 is expressed as : EI d4 v dx 4 = q y (8) 6

8 Torsional Behavior For 3-D Bernoulli Beam Element, only uniform torsion is considered. Assumptions for this behavior are : Cross sections are free to buckle. Torsional resistance is only ensured by shear stresses 1 τ xy,τ xz active in the section s plane. The assumed distortion of the cross section seems to be a priori incompatible with hypothesis of flexural behavior. This inconsistency is discussed and recalled in the following section with the principle of superposition. Figure 5: Torsional kinematic, taken from [3] Under this assumptions, Saint-Venant torsion theory gives the next kinematic relations for χ the torsion angle by unit length, with θ x the torsion angle defined on figure 5. Uniform torsional constitutive law can be expressed as, 1 Solid mechanics stresses expressed in unit of pressure dθ dx = χ (9) T = GJχ (10) 7

9 where G is the shear modulus and J is a geometrical characteristic of the section called constant of torsion. Finally, figure 6 presents the torsional equilibrium relation that is written as, with m x the torsion moment by unit length. dt dx = m x (11) Figure 6: Torsional equilibrium Equations 9, 10 and 11 lead to the Torsional differential equation as following: Principle of superposition GJ d2 θ x dx 2 + m x = 0 (12) This principle is essential in the definition of 3-D Bernoulli Beam Element in Akantu. As expressed in [3], this principle corresponds to : The effect produces by different causes acting together is equal to the sum of the effects produced by each of the causes assumed acting separately To satisfy this principle, two conditions shall be satisfied. Geometric linearization Material linearization 8

10 The first condition means that deformations and rotations requires to be small. This condition is included in the three different behaviors explained. The second one requires a material which follows a linear elastic constitutive law which is also verified since the three behaviors follow Hooke s law. This principle has two major effects in the modeling of the element. Flexural and torsional effects are considered separately. Indeed, this principle allows first to evaluate a cross section that does not buckle under flexural deformations and then a cross section free to buckle under torsion. The final stage is to sum the effects of these two behaviors taken separately whose assumptions were a priori incompatible. By this principle the effects of an oblique flexion can be correctly evaluated by projecting the oblique moment vector M into two moments M y and M z around the principal axes of the beam y and z. Then each of the two moments produces planar flexural behavior that could be studied separately and summed in the end. Since there are two directions of planar flexural behavior, the following convention is adopted in this report : Flexion in x y plane : v(x) is the y directed displacements field and the differential equation becomes EI z d 4 v dx 4 = q y. Flexion in x z plane : w(x) is the z directed displacements field and the differential equation becomes EI y d 4 w dx 4 = q z. Rotation angles, shear stresses and moments take also the index y or z depending on which axis they are expressed. 9

11 2 The 3-D Bernoulli Beam Element This section details the implementation of the presented Beam model within Akantu s library. 2.1 Definition of the Element The Archetype Element deals with four displacements fields separatly defined by the following differential equations. Axial displacements : EA d2 u dx 2 + q x = 0 Transversal displacements along y direction : EI z d 4 v dx 4 q y = 0 Transversal displacements along z direction : EI y d 4 w dx 4 q z = 0 Rotations around the axis : GJ d2 θ x dx 2 + m x = 0 The highest degree of derivatives is commonly designed by 2m. Thus axial fields are of degree two (m = 1) and transversal fields are of degree four (m = 2). Construction of a conform Element Type requires the respect of several rules expressed below. The first convergence criterion imposes that the unknown fields should be of class C m in the element and of class C m 1 at boundaries 2. Thus for the axial fields, continuity C 0 is required at boundaries which means that only the unknown fields needs to be continuous. For the transversal fields, continuity C 1 is needed at boundaries and the unknown fields and their derivatives needs to be continuous. In summary, required continuity through boundaries concerns : Axial displacements : u Transversal displacements along y direction : v and dv dx θ z 2 A function of continuity or class C r is continue and its derivatives are also continue up to degree r. 10

12 Transversal displacements along z direction :w and dw dx θ y Rotations around the axis : θ x Since boundaries of Beam Element are at nodes, those requirements involve the definition of a two nodes Element with six degrees of freedom (dof) u, v, w, θ x, θ y and θ z at each nodes. The vector of the kinematic unknowns takes the following form : d T = {u 1,v 1,w 1,θ x1,θ y1,θ z1,u 2,v 2,w 2,θ x2,θ y2,θ z2 } (13) The second convergence criterion deals with the interpolation. It is satisfied if chosen polynomials are complete up to degree m. The same shape functions defined for the 2-D Element are reused to define the interpolation in the 3-D Element. For recall, they are defined by imposing a unitary displacement at the associated dof and keeping others equal to zero. The next figure summarizes the six interpolation functions obtained. Figure 7: Shape functions definitions, taken from [6] 11

13 Using the same notation as in figure 7, the interpolated fields in the Element are defined as, u(x) v(x) u = w(x) θ x (x) θ y (x) θ z (x) = Nd (14) with N = N N M L 1 0 M L M 1 0 L M 2 0 L N N M 1 0 L M 2 0 L M L 1 0 M L 2 (15) Note the signs of the interpolation functions in (15) which are linked to the rotations around y-axis (5 th and 11 th dof) are inverse compares to rotations around z-axis because of direct coordinates system convention. Since the Finite Element Method only solves the integral forms of the equations, m boundary conditions are needed and they concern the unknown fields and its derivatives up to the degree 2m 1. With the four unknown fields of the Element, required boundary condition are: Axial displacements : One condition on u or du dx N Transversal displacements along y direction : Two conditions on v or dx dv θ z or d2 v M dx 2 z or d3 v V dx 3 y Transversal displacements along z direction : Two conditions on w or dw dx θ y or d2 w M dx 2 y or d3 w dx 3 V z Rotations around the axis : One condition on θ x or dθ x dx T Considering those conditions at the boundaries of the Element represented by two nodes, the following force vector is defined : f T = {N 1,V y1,v z1,t 1,M y1,m z1,n 2,V y2,v z2,t 2,M y2,m z2 } (16) 12

14 2.2 From local Archetype Element to the global space After building correctly the 3-D Bernoulli Beam Element type, the developed object needs to be characterized in a tridimensional space. First, Beam Element is characterized by the position of its two nodes. It gives information about the element s length and the direction of its axis. But the orientation of its cross section is missing. Thus for 3-D Beam Elements a new geometric parameter is needed to describe completely their configuration in space. For this project, a new object has been added to Akantu s class Mesh and it corresponds to an unitary vectors n that gives the direction of the z axis of the cross section. Then the Archetype Element should be clearly defined. It corresponds to a local evaluation of the element where all shape functions are computed before being rotated and assembled in the global space. Similarly to the 2-D Beam Element, the choice of a dimensional Archetype Element is kept for the 3-D case. More details about this choice are explained in Chapter 3 of [2]. As convention, Archetype Element is carried by the x axis between position a to a, with a corresponding to half of the beam length. x direction is given from node 1 to 2 by its axis. z direction is deduced from n and y direction is computed as the cross product of x and z direction. Then the rotation of Archetype Element local axes to the global axes of the structure needs to be defined. Similarly as it was explained for the 2-D Beam Element, rotation must be performed via a rotation matrix. In the bidimensionnal case, rotations are easy to be expressed because only one angle is used to described change of referential. But in a 3-D space, rotations are much more complex to be described. The most common method to rotate an object in 3-D is by using Euler angles. But this method implies very strict and heavy processes to avoid inconsistency such as the gimbals lock. For this reason a simpler process is defined for structural objects in Akantu. It consists in defining two 3x3 matrices P g and P e which characterize respectively coordinates system of the beam in global space and its related Archetype Element. Then the rotation matrix T saved as an object of Akantu s class StructuralMechanicsModel, is constructed with equation 17. P e P 1 g = T (17) 13

15 Left matrices in equation 17 must be made of three row vectors which are linearly independent and are expressed in both coordinates system. It s important that the three vectors conserve their length in both axis in order to compute a consistent rotation matrix. The following convention is chosen: First vector corresponds to the direction of beam axis with size 2a. Second vector corresponds to unitary vector n. Third vector corresponds to the cross-product of the two previous vectors to be sure that we have three linearly independent vectors Constructed with the chosen convention, coordinate system matrices can be expressed in equations 18 and 19: 2a 0 0 P e = 0 0 2a (18) x 2 x 1 n x (y 2 y 1 ) n z (z 2 z 1 ) n y P g = y 2 y 1 n y (z 2 z 1 ) n x (x 2 x 1 ) n z (19) z 2 z 1 n z (x 2 x 1 ) n y (y 2 y 1 ) n x,with (x i,y i,z i ) the coordinates of the i th node and (n x,n y,n z ) the components of n expressed in global coordinates. 3-D Beam Elements bring some changes in the rotation process for structural mechanics Elements. Rotation is no more accomplished in the Element Class class that only deals with operations on Archetype Element. T is now computed in StructuralMechanicsModel class and saved as an object of this class in order to be called by the functions constructing the Finite Element Model presented later in section

16 Figure 8: 3-D Bernoulli Beam Element with its Archetype (in green) Note that rotations are applied on two kinds of vector. Vectors of twelve components related to the dof like d in equation 13 and vectors of six components related to the displacements fields like u in equation 14. The following rotation matrices are also defined as convention for the rest of the report. T T do f = 0 T T 0 (20) T [ ] T 0 T f ields = (21) 0 T 2.3 Integration 3-D Bernoulli Beam Element uses Akantu s GaussIntegrator to perform the different integrations required by the FEM. Three Gauss points are set on the Element. Similarly as for 2-D Beams, the position of the points and their weights are expressed in Archetype Element coordinates system as follow. 15

17 Gauss Point Position Weight 1 a 3 5 5a a 9 3 a Table 1: Positions and weights of Gauss points defined on 3-D Beam Elements 3 5 5a 9 According to Gauss theory, three points integrate exactly polynomials up to degree five. This efficiency is discussed later in the report but the number of Gauss points can be quickly adapted to the requirements of the different analyses. 2.4 Finite Element Model Force-displacement relation Kd = f (22) expresses the Finite Element Equilibrium. d is the vector of unknowns already defined in equation 13. The stiffness matrix K and the force vector f are presented in detail in the next section Stiffness Matrix To construct the Stiffness Matrix, strains vector ε and stresses vector σ need to be defined. According to the hypotheses of the model, strain due to shear stress is neglected. Thus, strains are only due to normal stress, moment and torsion with the following vector, ε x ψ ε = z (23) ψ y χ Kinematic relations from section 1.2 are used to construct shape derivatives matrix B as following: 16

18 ε = du dx d2 v dx 2 d2 w dx 2 dθ x dx = Bd (24) with, N N B = 0 M L 1 0 M L M 1 0 L M 2 0 L N N (25) Then, stresses are related to strains by constitutive relations which are given in 1.2, N M σ = z = Dε (26) M y T with constitutive matrix D constructed according to equation 2, 6 and 10 as, EA D = 0 EI z EI y 0 (27) GJ Finally the Stiffness Matrix of an element is defined in the Finite Element Theory by the following integration according to section 2.3, K = a a B T DBdx (28) Since the highest degree of polynomial in this integral is two, the Gauss Integrator computes exactly the equation Force Vector The force vector on one element is defined by the sum of two integrals in the Finite Element Theory, 17

19 f = Ω N T bdω + Γ N T tdγ (29),with Ω referring to the domain of the element and Γ its boundaries. For the 3-D Beam Element, this equation becomes : f = a a N T bdx +t (30) The right term t contains values of load applied at the nodes and has the same composition as f detailed in equation 16. The term b in the integral represents distributed loads applied on the element. It is also composed similarly as f, however loads are expressed by unit length in function of the position on beam axis, as following: q x (x) q y (x) q b = z (x) (31) m x (x) m y (x) m z (x) As the Stiffness Matrix, the integration for loads vector is performed by the Gauss method presented in section 2.3. Since the highest polynomial in N is three, equation 30 is exactly computed with Gauss Integrator up to quadratic distributed load functions. Load functions need to be evaluated at quadrature point positions which are found by interpolating position fields. This process is similar to the one defined for the 2-D Bernoulli Beam Element (more details are available in section of [2]). A novelty of the 3-D Element is to propose two ways to define a distributed load. Indeed a load case can be defined in the global axes of the structures. An example is the dead weight that is always oriented in the vertical direction. But in other cases, the load case might be defined in the local axes of each element. For example, hydrostatic pressure for water or wind is used to be expressed perpendicular to a given facet. In this case load vectors in local axes b e are expressed in global axes via the rotation matrix, b = T T f ields b e (32) 18

20 2.4.3 Assembly The Assembly is an essential step in the FEM. It consists in assembling the contribution of all elements that are connected to the same nodes. Generic functions of Akantu are defined to assemble matrices and vectors. They are used to assemble K and f before applying Finite Element Equilibrium relation 22. The equilibrium is expressed in the global system of coordinates. Therefore each element must be correctly translated into global axes before the Assembly. This section presents the different procedures defined for Bernoulli Beam Elements within Akantu. For convention matrices and vectors are indexed with e and g, depending if they are expressed in element local axes or into global system of coordinates. First of all, shape functions and their derivatives are pre-computed on quadrature points for every elements. Then shape functions and derivatives matrices N e and B e are constructed into local axes according to the equations 15 and 25. Relations must be defined to construct K g and f g from local matrices. Let s start with the expression of local Finite Element Equilibrium, Relation 17 gives, and Thus With equation 28, or, K e d e = f e K e T do f d g = T do f f g T T do f K et do f d g = f g T T do f K et do f = K g a K g = T T do f [ K g = a a a B T e D e B e dx]t do f (B e T do f ) T D e (B e T do f )dx (33) Let s find the same development for the global force vector. Assuming no load applied directly on nodes, equation 30 becomes in local axes, f e = a a N T e b e dx 19

21 According to the size of each vector, 17 leads to, and, T do f f g = f g = a a a a N T e T f ields b g dx [T T do f NT e T f ields ]b g dx (34) Relations 33 and 34 are used to correctly construct the Stiffness Matrix and the Force Vector of each element in order to be correctly assemble in the global structure by the Assembly functions of Akantu Stresses Post-Processing Finite Element Method computes the unknown displacements in global axes thanks to equation 22. Stresses can be post-processed with the relation 26. Since stresses have only physical sense in local axes of each element, the relation becomes : N M σ e = z = D M e B e T do f d g (35) y T Since B e is evaluated on Gauss points in order to perform Gauss Integration, stresses are also evaluated on each quadrature point which gives three stresses evaluation on every 3-D Bernoulli Beam Element. Note that the sign of transversal moments is depending on the defined axes and therefore depending on the numbering of the nodes and the direction of n. If a positive moment is assumed to apply tension on the upper fiber of the cross section, the convention to define the upper fibers should be set according to 9. 20

22 Figure 9: Fibers subjected to tension under positive transversal moments. 3 Patch Test 3.1 Presentation Since the development of the Element in section 2.1 satisfies the convergence criteria, the patch test is mainly useful to validate the implementation of the model. Different verifications should deal with the torsion behavior, rotations, combination of beams with different orientations, loads integration and stresses postprocessing. The chosen test combines all those difficulties with the advantage of knowing the exact analytical solutions. The example is the exercise taken from [3]. Here the problem is setting: 21

23 Figure 10: Sketch of the patch test. Source [3] The cantilever beam of figure 10 is submitted to distributed loads acting vertically on segment AB. Cross sections are circles of diameter 1.3[cm] made of steel. Dead weight is neglected. 1. Compute vertical displacement w a of point A. 2. Compute rotation θ yb of point B around y axis. 3.2 Analytical Solution The resolution by Displacements Method gives the following analytical solution: w a = qa 4 (69/(24EI) + 5/(GJ)) θ yb = 2qa 3 (1/(EI) + 1/(GJ)) Cross sections properties are evaluated as, E = 2.05e 11 [N/m 2 ] A = 1.33e 4 [m 2 ] I = 2.8e 9 [m 4 ] 22

24 GJ = [m 2 ] thus w a = [m] and θ yb = [rad]. Then stresses can also be evaluated at any points on the beam. Table 2 summarizes the values for the moment and the torsion at Gauss point positions according to same local axes defined later in figure 11. For convention, they are numerated starting from the embedding to the free edge. # of Gauss Point Moment Torsion Table 2: Stresses evaluated analytically, expressed in [Nm] 3.3 Numerical Solution Numerical modeling of the problem is constructed according to figure 11. Node one has all its dof set to zero because of the embedding. Distributed loads are then applied on the z global axis on the third element. 23

25 Figure 11: Problem modeling with Beam Elements w a corresponds to the third dof of the fourth node and is computed as [m]. θ yb corresponds to the fifth dof of the third node and is evaluated as [rad]. Stresses are also post-processed on quadrature points and they are compared to the analytical results in table 3. # of Gauss Point Moment Moment Torsion Torsion Table 3: Stresses evaluated numerically versus analytical results (in yellow), expressed in [Nm] 3.4 Comments 24

26 The model gives back exactly the analytical results for unknown displacements fields. Stresses are also correctly evaluated except flexural moment on the third element. The error is only caused by the approximation of the distributed loads into consistent forces acting on nodes (integral term in (30)). Indeed, distributed loads are integrated exactly by Gauss Integrator into the following consistent nodal forces : V z4 = 15 [N], M x4 = 1.25 [Nm] and V z3 = 15 [N], M x3 = 1.25 [Nm]. Figure 12 illustrates the difference between exact moment under distributed loads case and numerical moment evaluated under consistent forces. Figure 12: loads Numerical approximation of moment repartition under distributed Since on element s length the integral of the consistent flexural moment is equal to integral of the analytical moment, those differences have no effects on the computations of the unknown displacement fields that are exact. To increase the accuracy of stresses evaluation, the mesh should be refined where distributed loads are applied. Finally, the symmetry of the considered cross sections allows to check the stability of the model by changing local y z axes convention according to figure

27 Figure 13: Equivalent problem modeling Because computations give exact results, the 3-D Bernoulli Beam Element implementation can be considered as validated. 4 Wood Tower, the study of a complex structure 4.1 Presentation The last part of this project is to evaluate the 3-D Bernoulli Beam Element by studying a complex 3-D structure. The analyzed structure was designed at I-Bois laboratory of EPFL by Steve Cherpillod in the context of Ateliers Weinand. First of all, I want to thanks I-Bois laboratory and its PhD student Seyed Sina Nabaei for giving me the mesh of the structure and for presenting me their numerical analyses. 26

28 Figure 14: Wood Tower. Source [7] The structure is a tower of 36 meters. It corresponds to a timber structures made of glulam modulus superposed. As presented on figure 15, each modulus is made of three curved vertical elements and one step. 27

29 Figure 15: Details of the timber modulus. Source [7] In the elevation (figure 17), the structure is constructed by the superposition of two helicals containing the two stairs, one for getting up and the other for coming down. The structure is also reinforced by two helicals made of steel. At the top of the tower, a platform is designed to appreciate surroundings. Figure 16: Horizontal section of the tower. Source [7] 28

30 Figure 17: Elevation of the Wood Tower. (Source [7]) In the horizontal section (figure 16), the tower is constituted of the circular juxtaposition of the vertical modulus around a given center. More details on the design of the Wood Tower are given in [7] and [1]. Our interest of this structure stands on the very high number of Beam Elements (almost 20000) and the advantage of owning numerical results performed by I- Bois laboratory on another commercial FEM software, RFEM. RFEM is a 3-D Finite Element Analysis program of the suit Dlubal Software for structural analysis. More details are given on [9]. 4.2 Mesh loading on Akantu Wood Tower Mesh consists of nodes for D Beam Elements made of 4 different cross sections. The mesh of Wood Tower consists in a.xlsx file generated by the software RFEM. This file consists in different spreadsheets. 29

31 Figure 18: MatLab plot of every nodes defined. The first spreadsheet gives node coordinates in global space. Figure 18 gives a MatLab (cf.[10]) plot of every nodes given by the file. Unlike Akantu, some nodes are not connected to the structure. They correspond to help nodes that are used later to orientate beams cross sections. The second spreadsheet gives the lines linking two nodes, which corresponds to the connectivities in Akantu. Two other spreadsheets give the materials and the cross sections used in the structure. They should be entered as StructuralMaterial objects in Akantu. The fifth spreadsheet assigns materials and cross sections to the different elements and gives the orientation of their cross sections by the help nodes. In Akantu, this spreadsheet should be used to assign StructuralMaterial to the elements and to compute their vector n. Finally a spreadsheet gives the number of boundary nodes whose translations are locked. Working with 3-D Bernoulli Beam Elements, those nodes shall have their three first dof set to zero. In summary, three main differences appear between this software and Akantu s mesh construction. The existence of non-connected nodes and the way to define 30

32 cross sections orientation. But also the interaction between mesh and model definition. Thus for this project a new class of readers was developed in Akantu, named ModelIO. Unlike MeshIO class, ModelIO allows to deal with interconnected mesh and model generations. A subclass of ModelIO, called ModelIOIBarras with the name of its creator, is presented in the next sections Model Reader Figure 19 presents an example of the files used by the reader function % Number of nodes % x y z coordinates of node % Number of elements 1 2 % #node1 #node % Number of boudaries 205 % #node where translations are locked % Number of cross section types e e e % E G J Iz Iy A 2.1e+11 8e % #section type #help node % Note: Element are set in order of numbering Figure 19: Format of files needed by the reader Reader s deal with help node is detailed in this section. First, coordinates of all nodes are read and saved in an array named temp nodes. This array contains all the nodes including help nodes. They should be filtered to correctly construct node vectors of Akantu. Inspired by the connectivity vectors, an array named connect to akantu is created. It contents one integer related 31

33 to a given component on temp nodes. If the integer is zero, the given node is a help node and is not saved in Akantu s nodes. Otherwise the node is part of the structure and needs to be saved in vector nodes. Figure 20 illustrates the construction of the different vectors. Figure 20: Filter process to separate help nodes. Then the construction of n needs to be detailed. If the cross section is a circle, the scalar #help node of figure 18 is set to zero and n is defined as some vector perpendicular to beam axis. Otherwise n is constructed according to figure

34 Figure 21: From help node to cross section vector Finally the function is called on a member of StructuralMechanicsModel class. Reader function creates the associated member of class Mesh that can be called in the main routine of an anylsis by the accessor.getfem().getmesh() applied on the StructuralMechanicsModel member Sets assignment For the Wood Tower analysis, a new member of the StructuralMechanicsModel class is defined. set ID corresponds to a scalar used to associate different elements. Sets are useful to assign distributed loads only to particular elements. Sets are also used as filters during results post-processing. Figure 22 presents a structure of files used by the sets assigner function. 33

35 12 % Number of sets % Including elements between 1489 and % Including elements % End of set Figure 22: Format of files needed by sets assigner Note that this function starts by setting every IDs to zero in order to define successively different sets during one analysis Verification To verify that the reader does correctly his job, the writer function of the class MeshIOMSHStruct is used to construct a.msh file from a member of the class StructuralMechanicsModel. Figure 23 and 24 presents the written file open in the software GMSH [8]. 34

36 Figure 23: Mesh of Wood Tower displayed by GMSH Figure 24: Details of Wood Tower displayed by GMSH 35

37 It shows that the mesh of the Wood Tower is correctly read into Akantu and that the help nodes which are visible in figure 18 are correctly filtered. For post-processing visualization, the software Paraview [11] is used for this analysis. At this step, it allows to check the consistency of the normals computations. They should be unitary and orientated radially to the outward direction. Figure 25: Consistency validation of normals computations in a vertical view of the Tower The mesh of the Wood Tower is correctly loaded in Akantu and the analyses should then be defined. 4.3 Wind Loads Analysis Two analyses are performed for the wind load cases defined in [1]. The first load case corresponds only to wind loads. Report [1] considers that wind is acting parallel to z-axis of the outdoor timber beams. Wind loads are also gradually defined in function of considered sets of beams presented on figure

38 Figure 26: Intensity of the wind in function of timber beam sets. Source [1] A second load condition which is similar to the common load combinations used in structural analysis is presented in Annex. The next table summarizes the different loads acting on the structure and their factors of amplitude applied during this second case. Loads Factor Dead Weight 1 Wind 0.5 Exploitation Loads 0.6 Table 4: Loading case defined in [1] for loads combination of the Tower Loads are applied to different sets of beams and defined both in global and local axes. Since the model and the mesh are completely defined by the reader, only the essential boundary conditions, which are meant forces, need to be precised in the main script of the analysis. As example, the following sequences shall be defined to apply correctly the combined loading case. 1. Set ID of every elements are equal to their StructuralMaterial numbers 2. Compute forces vector from Dead Weight function along global z-axis 3. Assign sets from the file containing Exploitation Loads sets 37

39 4. Increment forces vector from Exploitation Loads function along global z-axis 5. Assign sets from the file defining Wind Loads sets 6. Increment forces vector from Wind Loads function along local z-axis 7. Run the solving process After FE analysis is performed, results are transmitted to Paraview through Dumper class. The analysis of results on such a structure could be a project by itself. In the present report the aim is just to verify the consistency of the results. Figure 36 presents the computed displacements for the Tower evaluated by the developed model for the pure wind load case. Figure 27: Deformation of Wood Tower computed with Akantu. Displacements in [m]. Deformations scale 150x The magnitude of displacements is very close to the results obtained with the software RFEM that evaluated the maximum magnitude of displacements at 136[mm] versus 130[mm] with Akantu. The deformed states are also extremely close when comparing Akantu s analysis and RFEM. Figure 28 compares the two deformations. 38

40 Figure 28: Deformation of Wood Tower computed on Akantu (left) and on RFEM (right). Deformations scale 150x. Maximum magnitude of displacements corresponds to 130[mm] with Akantu and 136[mm] with RFEM Those results validate the stability and the robustness of the developed 3-D Bernoulli Beam Element even when used on a real structure containing a great number of elements. Other results of the combined loads analysis are set in Annex of the present report. 5 Conclusion The subject of the present project was the extension of the Bernoulli model for beams in 3-D. Even if the general skeleton needed for Structural Elements was already defined in Akantu and Bernoulli Element were already implemented during previous semester project, many traps were hidden behind those advantages. Paradoxically, the main challenge was perhaps forgetting the developments and the processes defined for the 2-D Elements. Indeed the 3-D extension of the model caused some new problems due to generalization. Rotation in a tridimensional space, axes conventions, combinations of three kinds of nodal rotations, are some of the difficulties which were encountered during this project. After its development, the 3-D Beam Element was evaluated with a patch-test that allowed to show remaining problems and to validate the implementation of Bernoulli Elements in the 3-D space. Finally, the icing on the cake was the possibility to evaluate the robustness of the Element in the study of a complex structure and the comparison with a com- 39

41 mercial FEM software. Results show that the model is able to deal with a great number of elements. Finally, the Wood Tower analysis also shows the great potential of using Akantu library in structural analysis post-processing, compared to the common FE softwares. First, the generation of Paraview files gives good flexibility to process the results independently of the FE computations and on different platforms (Unix, Mac, Windows). Furthermore, it is often faster than post-processing using the commercial softwares. While it takes about three minutes to apply a deformation or to represent stress repartitions in RFEM, Paraview does it almost instantaneously. Figure 29: Principle of equivalence (7 th row) set in [3]. Table 29 brings the last words of this report. It shows the principles of equivalence (cf. Note of page 3) defined for 3-D Beams behaviors. Dumping correctly to Paraview all the parameters defined in the principle of equivalence formulations gives the users the ability to evaluate stresses at any points of beams cross sections. In other words, the users can define analysis tools according to their needs and with more control on them. Besides, it contributes to reduce even more the reluctant problem of black box encountered with Finite Element Analysis. 40

42 References [1] Fares Hobeiche, Stefan Sander, Construction en bois II, Rapport, Tour tressée. I-Bois, EPFL, Semestre de printemps [2] Fabian Barras, 2-D Bernoulli Beams in Akantu. Bachelor s Project, LSMS - EPFL, [3] François Frey, Mécanique des structures. Analyse des structures et milieux continus. Traité de Génie Civil, volume 2. Presses Poytechniques et Universitaires Romandes, [4] François Frey, Mécanique des solides. Analyse des structures et milieux continus. Traité de Génie Civil, volume 3. Presses Poytechniques et Universitaires Romandes, [5] François Frey et Jaroslav Jirousek, Méthode des éléments finis. Analyse des structures et milieux continus. Traité de Génie Civil, volume 6. Presses Poytechniques et Universitaires Romandes, [6] Pierrino Lestuzzi, Polycopiés du cours Mécanique des Structures II. Plaques, Parois, Torsion non uniforme. EPFL-ENAC-SGC, Automne [7] Steve Cherpillod Wooden Waves, a tower for paléo festival. Presentation at the Atelier Weinand. I-Bois, EPFL, [8] [9] [10] [11] 41

43 Annex (Results of combined loads analysis) Figure 30: Axial stresses on the member set printed in magenta on the global structure. Maximum tensile stresses of timber modulus happen at the rear of the Tower, as expected. Units [N] 42

44 Figure 31: Axial stresses on the member set printed in magenta on the global structure. Maximum compressive stresses of timber modulus stand in the front of the Tower, as expected. Units [N] 43

45 Figure 32: Axial stresses on the steps of Tower stairs. Because of the composition of modulus (figure 15), stairs are compressed when vertical stanchion beams are tensed and inversely. 44

46 Figure 33: Flexural Moment on the whole tower. As expected, stiffer elements (steel tubes) take almost all the stresses of the structure. Units [Nm]. Moment corresponds to My 2 + Mz 2. 45

47 Figure 34: Moment filtered on steel tube. High stresses happens in region of curvature changes. Units [Nm]. Moment corresponds to My 2 + Mz 2. 46

48 Figure 35: Concentrations of Torsion at the irregularities in the helicals. Units [Nm]. Torsion corresponds to T. 47

49 Figure 36: Similar Torsion concentrations observed on the Analysis with RFEM 48