3-D Bernoulli Beams within Akantu
Semester ProjectFall 2011
Fabian Barras
Professor Jean-Francois Molinari
SupervisorsSeyedeh Mohadeseh Taheri Mousavi
Guillaume AnciauxNicolas Richart
Computational Solid Mechanics Laboratory - LSMSEcole Polytechnique Federale de Lausanne - EPFL
Contents1 Introduction 2
1.1 Purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Some structural mechanics . . . . . . . . . . . . . . . . . . . . . 2
2 The 3-D Bernoulli Beam Element 102.1 Definition of the Element . . . . . . . . . . . . . . . . . . . . . . 102.2 From local Archetype Element to the global space . . . . . . . . . 132.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . 162.4.2 Force Vector . . . . . . . . . . . . . . . . . . . . . . . . 172.4.3 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.4 Stresses Post-Processing . . . . . . . . . . . . . . . . . . 20
3 Patch Test 213.1 Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Wood Tower, the study of a complex structure 264.1 Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Mesh loading on Akantu . . . . . . . . . . . . . . . . . . . . . . 29
4.2.1 Model Reader . . . . . . . . . . . . . . . . . . . . . . . . 314.2.2 Sets assignment . . . . . . . . . . . . . . . . . . . . . . . 334.2.3 Verification . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Wind Loads Analysis . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Conclusion 39
1
Abstract
This report details the implementation of the Finite Element Methoddealing with Bernoulli structural mechanics model for 3-D Beams. The 3-DBeam Element is developed within Akantu which is an open source FiniteElement library implemented at the Laboratory of Computational Solid Me-chanics (LSMS) of EPFL. The aim of Akantus design is to consider dif-ferent kinds of models (solid mechanics, structural mechanics, heat transfer,etc...) while staying as generic as possible in the application of the FiniteElement Method (FEM). Thus each object or class presented in this reportis implicitly related to Akantus library.
1 Introduction
1.1 PurposesThis semester project consists in the extension of Bernoulli model for 3-D
beams. According to the general philosophy of Akantu, this implementationshould stay as generic as possible. Therefore the 3-D Element should be devel-oped within the structure already defined for the 2-D beams and more generallyfor all structural mechanics Elements.Before starting the implementation, mechanics and their assumptions should bestudied in details, in order to construct an Element consistent with different be-haviors.Then the development of the Element according to mechanical assumptions andFinite Element Theory is presented in details in this report.For validation of the model, a patch test was performed in comparison with ananalytical case mixing the different behaviors in three dimensions.Finally 3-D Bernoulli Beam Elements are used in the study of a complex tridi-mensional structure to validate the developed model also on a large number ofelements.
1.2 Some structural mechanicsIn this section, the theoretical model and its assumptions used in the develop-
ment of the 3-D Beam Element is described. A beam is a tridimensional structuremodeled with a curvilinear element suited for structural mechanics. Indeed geo-metrical 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
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 pre-sented in [3].
Note: In structural mechanics, stresses can often be considered as integratedon the beam axis. They are thus expressed in unit of load ([N]).Solid mechanics usually considers that stresses are applied on an infinitesimalsurface and they are then expressed in terms of pressure ([Pa]).In this report, structural mechanics convention is used. Stresses at the axis arerelated to solid mechanics stresses by equivalence principles. More details aboutequivalence 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 constantarea A. The displacement field u(x) and the Normal stress N are demanded undera given distributed axial load state qx.
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Figure 2: Axial behavior taken from [4]
Kinematic relation is directly derived from Bernoullis law. It corresponds tothe relation expressed as :
Planar cross sections which are perpendicular to the axis will be conserved indeformed configuration.
This assumption is the core of modeling Beam Element that was also named 3-DBernoulli Beam Element. It leads to the following kinematic relation:
dudx
= x (1)
with x the axial strain.Constitutive law which linked the normal stress N to the axial strain is describedby the linear elastic Hookes law :
NEA
= x (2)
with E the elastic modulus.Finally, the equilibrium of axial stresses gives the following relation :
dNdx
=qx (3)
Equations 1, 2 and 3 give the following differential equation for the axial behav-ior :
EAd2udx2
+qx = 0 (4)
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Planar Flexural Behavior
Flexural behavior into the plane x y can be seen on figure 3. The same beamas before is assumed. Moreover, the beam is subjected to a vertical distributedload qy. The moment of inertia of its cross section is defined by I. Displacementsfield is described by the vertical displacement v(x) of a point in the axis and therotation (x).
Figure 3: Flexural behavior taken from [4]
Kinematic conditions are a bit more complex than in the axial behavior. Bernoullislaw is then expressed as :
After deformation, cross sections remain planar and perpendicular to the curvedaxis but also to all other fibers of the beam.
Small displacements are also assumed for the model in a way that the neutralaxis of the beam conserves the same length after deformation.
5
Figure 4: Beam under bending from R.Hooke, taken from [3]
Those kinematic conditions are illustrated in figure 4 and bring the followingrelations for the curvature r :
1r=
ddx
=d2vdx2
(5)
Constitutive relation for flexion in the model is also expressed by linear elasticHookes law as :
1r= =M
EI(6)
Finally, Bernoulli Beams theory belongs to the class of Thin Structures for whichthe normal planes stay straight and perpendicular to the axis after flexural defor-mations. In other words, cross sections do not buckle under flexion. Deformationsunder shear are thus neglected and the shear stress V is directly derived from themoment M. Indeed this three following equilibrium equations are stated :
dVdx =qy dMdx =V d
2Mdx2 =qy (7)
Differential equation of flexural behavior which is the result of equations 5, 6 and7 is expressed as :
EId4vdx4
= qy (8)
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Torsional Behavior
For 3-D Bernoulli Beam Element, only uniform torsion is considered. Assump-tions for this behavior are :
Cross sections are free to buckle.
Torsional resistance is only ensured by shear stresses1 xy,xz active in thesections plane.
The assumed distortion of the cross section seems to be a priori incompatible withhypothesis of flexural behavior. This inconsistency is discussed and recalled inthe following section with the principle of superposition.
Figure 5: Torsional kinematic, taken from [3]
Under this assumptions, Saint-Venant torsion theory gives the next kinematicrelations for the torsion angle by unit length,
ddx
= (9)
with x the torsion angle defined on figure 5.Uniform torsional constitutive law can be expressed as,
T = GJ (10)1Solid mechanics stresses expressed in unit of pressure
7
where G is the shear modulus and J is a geometrical characteristic of the sectioncalled constant of torsion.Finally, figure 6 presents the torsional equilibrium relation that is written as,
dTdx
=mx (11)
with mx the torsion moment by unit length.
Figure 6: Torsional equilibrium
Equations 9, 10 and 11 lead to the Torsional differential equation as following:
GJd2xdx2
+mx = 0 (12)
Principle of superposition
This principle is essential in the definition of 3-D Bernoulli Beam Element inAkantu. As expressed in [3], this principle corresponds to :
The effect produces by different causes acting together is equal to the sum of theeffects produced by each of the causes assumed acting separately
To satisfy this principle, two conditions shall be satisfied.
Geometric linearization
Material linearization
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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 lawwhich is also verified since the three behaviors follow Hookes law.
This principle has two major effects in the modeling of the element.Flexural and torsional effects are considered separately. Indeed, this principleallows first to evaluate a cross section that does not buckle under flexural defor-mations and then a cross section free to buckle under torsion. The final stage is tosum the effects of these two behaviors taken separately whose assumptions werea priori incompatible.
By this principle the effects of an oblique flexion can be correctly evaluated byprojecting the oblique moment vector
M into two moments My and Mz around the
principal axes of the beam y and z.Then each of the two moments produces planar flexural behavior that could bestudied separately and summed in the end.
Since there are two directions of planar flexural behavior, the following conven-tion is adopted in this report :
Flexion in x y plane : v(x) is the y directed displacements field and thedifferential equation becomes EIz d
4vdx4 = qy.
Flexion in x z plane : w(x) is the z directed displacements field and thedifferential equation becomes EIy d
4wdx4 = qz.
Rotation angles, shear stresses and moments take also the index y or z de-pending on which axis they are expressed.
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2 The 3-D Bernoulli Beam ElementThis section details the implementation of the presented Beam model within
Akantus library.
2.1 Definition of the ElementThe Archetype Element deals with four displacements fields separatly defined
by the following differential equations.
Axial displacements : EAd2u
dx2 +qx = 0
Transversal displacements along y direction : EIz d4v
dx4 qy = 0
Transversal displacements along z direction : EIy d4w
dx4 qz = 0
Rotations around the axis : GJ d2x
dx2 +mx = 0
The highest degree of derivatives is commonly designed by 2m. Thus axial fieldsare 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 ex-pressed below.
The first convergence criterion imposes that the unknown fields should beof class Cm in the element and of class Cm1 at boundaries2.
Thus for the axial fields, continuity C0 is required at boundaries which meansthat only the unknown fields needs to be continuous.For the transversal fields, continuity C1 is needed at boundaries and the unknownfields and their derivatives needs to be continuous.In summary, required continuity through boundaries concerns :
Axial displacements : u
Transversal displacements along y direction : v and dvdx z2A function of continuity or class Cr is continue and its derivatives are also continue up to
degree r.
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Transversal displacements along z direction :w and dwdx y Rotations around the axis : x
Since boundaries of Beam Element are at nodes, those requirements involve thedefinition of a two nodes Element with six degrees of freedom (dof) u, v, w, x, yand z at each nodes. The vector of the kinematic unknowns takes the followingform :
dT = {u1,v1,w1,x1 ,y1,z1 ,u2,v2,w2,x2,y2 ,z2} (13)
The second convergence criterion deals with the interpolation. It is satisfiedif chosen polynomials are complete up to degree m.
The same shape functions defined for the 2-D Element are reused to define theinterpolation in the 3-D Element. For recall, they are defined by imposing a uni-tary displacement at the associated dof and keeping others equal to zero. The nextfigure summarizes the six interpolation functions obtained.
Figure 7: Shape functions definitions, taken from [6]
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Using the same notation as in figure 7, the interpolated fields in the Elementare defined as,
u =
u(x)v(x)w(x)x(x)y(x)z(x)
= Nd (14)
with
N =
N1 0 0 0 0 0 N2 0 0 0 0 00 M1 0 0 0 L1 0 M2 0 0 0 L20 0 M1 0 L1 0 0 0 M2 0 L2 00 0 0 N1 0 0 0 0 0 N2 0 00 0 M1 0 L1 0 0 0 M2 0 L2 00 M1 0 0 0 L
1 0 M
2 0 0 0 L
2
(15)
Note the signs of the interpolation functions in (15) which are linked to the ro-tations around y-axis (5th and 11th dof) are inverse compares to rotations aroundz-axis because of direct coordinates system convention.
Since the Finite Element Method only solves the integral forms of the equa-tions, m boundary conditions are needed and they concern the unknown fieldsand its derivatives up to the degree 2m1.
With the four unknown fields of the Element, required boundary condition are:
Axial displacements : One condition on u or dudx N Transversal displacements along y direction :
Two conditions on v or dvdx z or d2v
dx2 Mz or d3v
dx3 Vy Transversal displacements along z direction :
Two conditions on w or dwdx y or d2w
dx2 My or d3w
dx3 Vz Rotations around the axis : One condition on x or dxdx T
Considering those conditions at the boundaries of the Element represented by twonodes, the following force vector is defined :
f T = {N1,Vy1,Vz1,T1,My1,Mz1,N2,Vy2,Vz2,T2,My2,Mz2} (16)
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2.2 From local Archetype Element to the global spaceAfter building correctly the 3-D Bernoulli Beam Element type, the developed
object needs to be characterized in a tridimensional space. First, Beam Element ischaracterized by the position of its two nodes. It gives information about the ele-ments length and the direction of its axis. But the orientation of its cross sectionis missing. Thus for 3-D Beam Elements a new geometric parameter is neededto describe completely their configuration in space. For this project, a new objecthas been added to Akantus class Mesh and it corresponds to an unitary vectorsnthat gives the direction of the z axis of the cross section.
Then the Archetype Element should be clearly defined. It corresponds to a localevaluation of the element where all shape functions are computed before beingrotated 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. Moredetails about this choice are explained in Chapter 3 of [2].As convention, Archetype Element is carried by the x axis between positiona toa, with a corresponding to half of the beam length. x direction is given from node1 to 2 by its axis. z direction is deduced from n and y direction is computed asthe cross product of x and z direction.
Then the rotation of Archetype Element local axes to the global axes of the struc-ture 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 describedchange of referential. But in a 3-D space, rotations are much more complex tobe described. The most common method to rotate an object in 3-D is by usingEuler angles. But this method implies very strict and heavy processes to avoidinconsistency such as the gimbals lock.For this reason a simpler process is defined for structural objects in Akantu. Itconsists in defining two 3x3 matrices Pg and Pe which characterize respectivelycoordinates system of the beam in global space and its related Archetype Element.Then the rotation matrix T saved as an object of Akantus class StructuralMechan-icsModel , is constructed with equation 17.
PeP1g = T (17)
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Left matrices in equation 17 must be made of three row vectors which are linearlyindependent and are expressed in both coordinates system. Its important that thethree vectors conserve their length in both axis in order to compute a consistentrotation 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 ex-pressed in equations 18 and 19:
Pe =
2a 0 00 0 2a0 1 0
(18)Pg =
x2 x1 nx (y2 y1)nz (z2 z1)nyy2 y1 ny (z2 z1)nx (x2 x1)nzz2 z1 nz (x2 x1)ny (y2 y1)nx
(19),with (xi,yi,zi) the coordinates of the ith node and (nx,ny,nz) the components ofn expressed in global coordinates.
3-D Beam Elements bring some changes in the rotation process for structuralmechanics Elements. Rotation is no more accomplished in the Element Classclass that only deals with operations on Archetype Element. T is now computedin StructuralMechanicsModel class and saved as an object of this class in order tobe called by the functions constructing the Finite Element Model presented laterin section 2.4.
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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 com-ponents related to the dof like d in equation 13 and vectors of six componentsrelated to the displacements fields like u in equation 14. The following rotationmatrices are also defined as convention for the rest of the report.
Tdo f =
T 0 0 00 T 0 00 0 T 00 0 0 T
(20)T f ields =
[T 00 T
](21)
2.3 Integration3-D Bernoulli Beam Element uses Akantus GaussIntegrator to perform the
different integrations required by the FEM. Three Gauss points are set on theElement. Similarly as for 2-D Beams, the position of the points and their weightsare expressed in Archetype Element coordinates system as follow.
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Gauss Point Position Weight
1 a
35
5a9
2 0.0 8a93 a
35
5a9
Table 1: Positions and weights of Gauss points defined on 3-D Beam Elements
According to Gauss theory, three points integrate exactly polynomials up todegree five. This efficiency is discussed later in the report but the number ofGauss points can be quickly adapted to the requirements of the different analyses.
2.4 Finite Element ModelForce-displacement relation
Kd = f (22)
expresses the Finite Element Equilibrium. d is the vector of unknowns already de-fined in equation 13. The stiffness matrix K and the force vector f are presentedin detail in the next section.
2.4.1 Stiffness Matrix
To construct the Stiffness Matrix, strains vector and stresses vector need to bedefined.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 followingvector,
=
xzy
(23)Kinematic relations from section 1.2 are used to construct shape derivatives matrixB as following:
16
=
dudxd2vdx2d2wdx2
dxdx
= Bd (24)with,
B =
N1 0 0 0 0 0 N
2 0 0 0 0 0
0 M1 0 0 0 L1 0 M2 0 0 0 L20 0 M1 0 L1 0 0 0 M2 0 L2 00 0 0 N1 0 0 0 0 0 N
2 0 0
(25)
Then, stresses are related to strains by constitutive relations which are given in1.2,
=
NMzMyT
= D (26)with constitutive matrix D constructed according to equation 2, 6 and 10 as,
D =
EA 0 0 00 EIz 0 00 0 EIy 00 0 0 GJ
(27)Finally the Stiffness Matrix of an element is defined in the Finite Element Theoryby the following integration according to section 2.3,
K =aa
BT DBdx (28)
Since the highest degree of polynomial in this integral is two, the Gauss Integratorcomputes exactly the equation 28.
2.4.2 Force Vector
The force vector on one element is defined by the sum of two integrals in theFinite Element Theory,
17
f =
NT bd+
NT td (29)
,with referring to the domain of the element and its boundaries. For the 3-DBeam Element, this equation becomes :
f =aa
NT bdx+ t (30)
The right term t contains values of load applied at the nodes and has the samecomposition as f detailed in equation 16.The term b in the integral represents distributed loads applied on the element. Itis also composed similarly as f , however loads are expressed by unit length infunction of the position on beam axis, as following:
b =
qx(x)qy(x)qz(x)mx(x)my(x)mz(x)
(31)
As the Stiffness Matrix, the integration for loads vector is performed by the Gaussmethod presented in section 2.3. Since the highest polynomial in N is three, equa-tion 30 is exactly computed with Gauss Integrator up to quadratic distributed loadfunctions.Load functions need to be evaluated at quadrature point positions which are foundby interpolating position fields. This process is similar to the one defined for the2-D Bernoulli Beam Element (more details are available in section 3.2.2 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 exampleis the dead weight that is always oriented in the vertical direction. But in othercases, the load case might be defined in the local axes of each element. For exam-ple, hydrostatic pressure for water or wind is used to be expressed perpendicularto a given facet. In this case load vectors in local axes be are expressed in globalaxes via the rotation matrix,
b = TTf ieldsbe (32)
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2.4.3 Assembly
The Assembly is an essential step in the FEM. It consists in assembling the contri-bution of all elements that are connected to the same nodes. Generic functions ofAkantu are defined to assemble matrices and vectors. They are used to assembleK and f before applying Finite Element Equilibrium relation 22.The equilibrium is expressed in the global system of coordinates. Therefore eachelement must be correctly translated into global axes before the Assembly. Thissection presents the different procedures defined for Bernoulli Beam Elementswithin 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 co-ordinates.
First of all, shape functions and their derivatives are pre-computed on quadra-ture points for every elements. Then shape functions and derivatives matrices Neand Be are constructed into local axes according to the equations 15 and 25. Re-lations must be defined to construct Kg and f g from local matrices.Lets start with the expression of local Finite Element Equilibrium,
Kede = feRelation 17 gives,
KeTdo f dg = Tdo f fgand
TTdo f KeTdo f dg = fgThus
TTdo f KeTdo f = KgWith equation 28,
Kg = TTdo f [aa
BTe DeBedx]Tdo f
or,
Kg =aa
(BeTdo f )T De(BeTdo f )dx (33)
Lets find the same development for the global force vector. Assuming no loadapplied directly on nodes, equation 30 becomes in local axes,
fe =aa
NTe bedx
19
According to the size of each vector, 17 leads to,
Tdo f fg =aa
NTe T f ieldsbgdx
and,
fg =aa
[TTdo f NTe T f ields]bgdx (34)
Relations 33 and 34 are used to correctly construct the Stiffness Matrix andthe Force Vector of each element in order to be correctly assemble in the globalstructure by the Assembly functions of Akantu.
2.4.4 Stresses Post-Processing
Finite Element Method computes the unknown displacements in global axes thanksto equation 22. Stresses can be post-processed with the relation 26. Since stresseshave only physical sense in local axes of each element, the relation becomes :
e =
NMzMyT
= DeBeTdo f dg (35)Since Be is evaluated on Gauss points in order to perform Gauss Integration,stresses are also evaluated on each quadrature point which gives three stressesevaluation on every 3-D Bernoulli Beam Element.
Note that the sign of transversal moments is depending on the defined axesand 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 crosssection, the convention to define the upper fibers should be set according to 9.
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Figure 9: Fibers subjected to tension under positive transversal moments.
3 Patch Test
3.1 PresentationSince 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, combi-nation of beams with different orientations, loads integration and stresses post-processing.The chosen test combines all those difficulties with the advantage of knowing theexact analytical solutions.The example is the exercise 12.10.13 taken from [3]. Here the problem is setting:
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Figure 10: Sketch of the patch test. Source [3]
The cantilever beam of figure 10 is submitted to distributed loads acting verti-cally on segment AB. Cross sections are circles of diameter 1.3[cm] made of steel.Dead weight is neglected.
1. Compute vertical displacement wa of point A.
2. Compute rotation yB of point B around y axis.
3.2 Analytical SolutionThe resolution by Displacements Method gives the following analytical solu-
tion:wa =qa4(69/(24EI)+5/(GJ))yB = 2qa3(1/(EI)+1/(GJ))
Cross sections properties are evaluated as,
E = 2.05e11 [N/m2]
A = 1.33e4 [m2]
I = 2.8e9 [m4]
22
GJ = 220.77 [m2]
thus wa =0.1037 [m] and yB = 0.0941 [rad].Then stresses can also be evaluated at any points on the beam. Table 2 summa-rizes the values for the moment and the torsion at Gauss point positions accordingto same local axes defined later in figure 11. For convention, they are numeratedstarting from the embedding to the free edge.
# of Gauss Point Moment Torsion1 5.8097 30.00002 0.0000 30.00003 5.8097 30.00004 26.6190 7.50005 15.0000 7.50006 3.3811 7.50007 5.9040 0.00008 1.8750 0.00009 0.0952 0.0000
Table 2: Stresses evaluated analytically, expressed in [Nm]
3.3 Numerical SolutionNumerical modeling of the problem is constructed according to figure 11.
Node one has all its dof set to zero because of the embedding. Distributed loadsare then applied on the z global axis on the third element.
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Figure 11: Problem modeling with Beam Elements
wa corresponds to the third dof of the fourth node and is computed as0.1037[m].yB corresponds to the fifth dof of the third node and is evaluated as 0.0941 [rad].Stresses are also post-processed on quadrature points and they are compared tothe analytical results in table 3.
# of Gauss Point Moment Moment Torsion Torsion1 5.8097 5.8097 30.0000 30.00002 0.0000 0.0000 30.0000 30.00003 5.8097 5.8097 30.0000 30.00004 26.6190 26.6190 7.5000 7.50005 15.0000 15.0000 7.5000 7.50006 3.3811 3.3811 7.5000 7.50007 5.9040 5.4047 0.0000 0.00008 1.8750 2.5000 0.0000 0.00009 0.0952 0.4047 0.0000 0.0000
Table 3: Stresses evaluated numerically versus analytical results (in yellow), ex-pressed in [Nm]
3.4 Comments
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The model gives back exactly the analytical results for unknown displace-ments 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 con-sistent forces acting on nodes (integral term in (30)). Indeed, distributed loads areintegrated exactly by Gauss Integrator into the following consistent nodal forces :Vz4 =15 [N], Mx4 =1.25 [Nm] and Vz3 =15 [N], Mx3 = 1.25 [Nm].Figure 12 illustrates the difference between exact moment under distributed loadscase and numerical moment evaluated under consistent forces.
Figure 12: Numerical approximation of moment repartition under distributedloads
Since on elements length the integral of the consistent flexural moment isequal to integral of the analytical moment, those differences have no effects onthe computations of the unknown displacement fields that are exact.To increase the accuracy of stresses evaluation, the mesh should be refined wheredistributed loads are applied.Finally, the symmetry of the considered cross sections allows to check the stabilityof the model by changing local y z axes convention according to figure 13.
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Figure 13: Equivalent problem modeling
Because computations give exact results, the 3-D Bernoulli Beam Elementimplementation can be considered as validated.
4 Wood Tower, the study of a complex structure
4.1 PresentationThe 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 Cher-pillod in the context of Ateliers Weinand.First of all, I want to thanks I-Bois laboratory and its PhD student Seyed SinaNabaei for giving me the mesh of the structure and for presenting me their numer-ical analyses.
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Figure 14: Wood Tower. Source [7]
The structure is a tower of 36 meters. It corresponds to a timber structures madeof glulam modulus superposed. As presented on figure 15, each modulus is madeof three curved vertical elements and one step.
27
Figure 15: Details of the timber modulus. Source [7]
In the elevation (figure 17), the structure is constructed by the superposition oftwo helicals containing the two stairs, one for getting up and the other for comingdown. The structure is also reinforced by two helicals made of steel. At the top ofthe tower, a platform is designed to appreciate surroundings.
Figure 16: Horizontal section of the tower. Source [7]
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Figure 17: Elevation of the Wood Tower. (Source [7])
In the horizontal section (figure 16), the tower is constituted of the circular juxta-position 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-DFinite Element Analysis program of the suit Dlubal Software for structural analy-sis. More details are given on [9].
4.2 Mesh loading on AkantuWood Tower Mesh consists of 18660 nodes for 19926 3-D Beam Elements
made of 4 different cross sections. The mesh of Wood Tower consists in a .xlsxfile generated by the software RFEM. This file consists in different spreadsheets.
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Figure 18: MatLab plot of every nodes defined.
The first spreadsheet gives node coordinates in global space. Figure 18 givesa MatLab (cf.[10]) plot of every nodes given by the file. Unlike Akantu, somenodes are not connected to the structure. They correspond to help nodes that areused later to orientate beams cross sections.The second spreadsheet gives the lines linking two nodes, which corresponds tothe connectivities in Akantu.Two other spreadsheets give the materials and the cross sections used in the struc-ture. They should be entered as StructuralMaterial objects in Akantu.The fifth spreadsheet assigns materials and cross sections to the different elementsand gives the orientation of their cross sections by the help nodes. In Akantu,this spreadsheet should be used to assign StructuralMaterial to the elements andto compute their vector n .Finally a spreadsheet gives the number of boundary nodes whose translations arelocked. Working with 3-D Bernoulli Beam Elements, those nodes shall have theirthree first dof set to zero.
In summary, three main differences appear between this software and Akantusmesh construction. The existence of non-connected nodes and the way to define
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cross sections orientation. But also the interaction between mesh and model defi-nition.Thus for this project a new class of readers was developed in Akantu, namedModelIO . Unlike MeshIO class, ModelIO allows to deal with interconnectedmesh and model generations. A subclass of ModelIO , called ModelIOIBarraswith the name of its creator, is presented in the next sections.
4.2.1 Model Reader
Figure 19 presents an example of the files used by the reader function.
18660 % Number of nodes
4.358 17.959 35.805 % x y z coordinates of node
4.051 18.171 35.999
...
19926 % Number of elements
1 2 % #node1 #node2
3 1
...
134 % Number of boudaries
205 % #node where translations are locked
410
...
8 % Number of cross section types
8000000000 3e+09 6.65e-05 1.88e-05 0.000568 0.0352 % E G J Iz Iy A
2.1e+11 8e+10 0.004376 0.002188 0.002188 0.053721
...
2 0 % #section type #help node
5 17808 % Note: Element are set in order of numbering
Figure 19: Format of files needed by the reader
Readers 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 filteredto correctly construct node vectors of Akantu. Inspired by the connectivity vec-tors, an array named connect to akantu is created. It contents one integer related
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to a given component on temp nodes . If the integer is zero, the given node isa help node and is not saved in Akantus nodes . Otherwise the node is partof the structure and needs to be saved in vector nodes . Figure 20 illustrates theconstruction 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 vectorperpendicular to beam axis. Otherwise n is constructed according to figure 21.
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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 calledin the main routine of an anylsis by the accessor .getFEM().getMesh() applied onthe StructuralMechanicsModel member.
4.2.2 Sets assignment
For the Wood Tower analysis, a new member of the StructuralMechanicsModelclass is defined. set ID corresponds to a scalar used to associate different ele-ments. Sets are useful to assign distributed loads only to particular elements. Setsare also used as filters during results post-processing. Figure 22 presents a struc-ture of files used by the sets assigner function.
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12 % Number of sets
1489 1572 % Including elements between 1489 and 1572
7626 8044
10052 10135
18035 % Including elements 18035
18036
8038
...
0 % End of set 1
16608 17133
18142 18153
18155
8156
...
Figure 22: Format of files needed by sets assigner
Note that this function starts by setting every IDs to zero in order to define suc-cessively different sets during one analysis.
4.2.3 Verification
To verify that the reader does correctly his job, the writer function of the classMeshIOMSHStruct is used to construct a .msh file from a member of the classStructuralMechanicsModel . Figure 23 and 24 presents the written file open in thesoftware GMSH [8].
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Figure 23: Mesh of Wood Tower displayed by GMSH
Figure 24: Details of Wood Tower displayed by GMSH
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It shows that the mesh of the Wood Tower is correctly read into Akantu andthat the help nodes which are visible in figure 18 are correctly filtered.For post-processing visualization, the software Paraview [11] is used for this anal-ysis. 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 ofthe Tower
The mesh of the Wood Tower is correctly loaded in Akantu and the analysesshould then be defined.
4.3 Wind Loads AnalysisTwo 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 actingparallel to z-axis of the outdoor timber beams. Wind loads are also graduallydefined in function of considered sets of beams presented on figure 26.
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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 combinationsused in structural analysis is presented in Annex. The next table summarizes thedifferent loads acting on the structure and their factors of amplitude applied duringthis second case.
Loads FactorDead Weight 1
Wind 0.5Exploitation 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 lo-cal axes. Since the model and the mesh are completely defined by the reader, onlythe essential boundary conditions, which are meant forces, need to be precisedin the main script of the analysis. As example, the following sequences shall bedefined 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
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4. Increment forces vector from Exploitation Loads function along globalz-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 Dumperclass. The analysis of results on such a structure could be a project by itself. Inthe present report the aim is just to verify the consistency of the results.Figure 36 presents the computed displacements for the Tower evaluated by thedeveloped 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 soft-ware RFEM that evaluated the maximum magnitude of displacements at 136[mm]versus 130[mm] with Akantu. The deformed states are also extremely close whencomparing Akantus analysis and RFEM. Figure 28 compares the two deforma-tions.
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Figure 28: Deformation of Wood Tower computed on Akantu (left) and on RFEM(right). Deformations scale 150x. Maximum magnitude of displacements corre-sponds to 130[mm] with Akantu and 136[mm] with RFEM
Those results validate the stability and the robustness of the developed3-D Bernoulli Beam Element even when used on a real structure containing agreat number of elements. Other results of the combined loads analysis are set inAnnex of the present report.
5 ConclusionThe 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 Elementswas already defined in Akantu and Bernoulli Element were already implementedduring previous semester project, many traps were hidden behind those advan-tages. Paradoxically, the main challenge was perhaps forgetting the developmentsand the processes defined for the 2-D Elements. Indeed the 3-D extension of themodel caused some new problems due to generalization. Rotation in a tridimen-sional space, axes conventions, combinations of three kinds of nodal rotations, aresome of the difficulties which were encountered during this project.After its development, the 3-D Beam Element was evaluated with a patch-testthat allowed to show remaining problems and to validate the implementation ofBernoulli Elements in the 3-D space.Finally, the icing on the cake was the possibility to evaluate the robustness ofthe Element in the study of a complex structure and the comparison with a com-
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mercial FEM software. Results show that the model is able to deal with a greatnumber of elements.Finally , the Wood Tower analysis also shows the great potential of using Akantulibrary in structural analysis post-processing, compared to the common FE soft-wares. First, the generation of Paraview files gives good flexibility to process theresults independently of the FE computations and on different platforms (Unix,Mac, Windows). Furthermore, it is often faster than post-processing using thecommercial softwares. While it takes about three minutes to apply a deforma-tion or to represent stress repartitions in RFEM, Paraview does it almost instanta-neously.
Figure 29: Principle of equivalence (7th row) set in [3].
Table 29 brings the last words of this report. It shows the principles of equiva-lence (cf. Note of page 3) defined for 3-D Beams behaviors. Dumping correctly toParaview all the parameters defined in the principle of equivalence formulationsgives the users the ability to evaluate stresses at any points of beams cross sec-tions. In other words, the users can define analysis tools according to their needsand with more control on them. Besides, it contributes to reduce even more thereluctant problem of black box encountered with Finite Element Analysis.
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References[1] Fares Hobeiche, Stefan Sander, Construction en bois II, Rapport, Tour
tressee. I-Bois, EPFL, Semestre de printemps 2011.
[2] Fabian Barras, 2-D Bernoulli Beams in Akantu. Bachelors Project, LSMS- EPFL, 2011.
[3] Francois Frey, Mecanique des structures. Analyse des structures et milieuxcontinus. Traite de Genie Civil, volume 2. Presses Poytechniques et Univer-sitaires Romandes, 2006.
[4] Francois Frey, Mecanique des solides. Analyse des structures et milieuxcontinus. Traite de Genie Civil, volume 3. Presses Poytechniques et Univer-sitaires Romandes, 1998.
[5] Francois Frey et Jaroslav Jirousek, Methode des elements finis. Analysedes structures et milieux continus. Traite de Genie Civil, volume 6. PressesPoytechniques et Universitaires Romandes, 2009.
[6] Pierrino Lestuzzi, Polycopies du cours Mecanique des Structures II.Plaques, Parois, Torsion non uniforme. EPFL-ENAC-SGC, Automne 2010.
[7] Steve Cherpillod Wooden Waves, a tower for paleo festival. Presentation atthe Atelier Weinand. I-Bois, EPFL, 2009.
[8] http://geuz.org/gmsh/
[9] http://www.dlubal.com/RFEM-4xx.aspx
[10] http://www.mathworks.ch/products/matlab/
[11] http://www.paraview.org
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Annex (Results of combined loads analysis)
Figure 30: Axial stresses on the member set printed in magenta on the globalstructure. Maximum tensile stresses of timber modulus happen at the rear of theTower, as expected. Units [N]
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Figure 31: Axial stresses on the member set printed in magenta on the globalstructure. Maximum compressive stresses of timber modulus stand in the front ofthe Tower, as expected. Units [N]
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Figure 32: Axial stresses on the steps of Tower stairs. Because of the compositionof modulus (figure 15), stairs are compressed when vertical stanchion beams aretensed and inversely.
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Figure 33: Flexural Moment on the whole tower. As expected, stiffer elements(steel tubes) take almost all the stresses of the structure. Units [Nm]. Momentcorresponds to
M2y +M2z .
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Figure 34: Moment filtered on steel tube. High stresses happens in region ofcurvature changes. Units [Nm]. Moment corresponds to
M2y +M2z .
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Figure 35: Concentrations of Torsion at the irregularities in the helicals. Units[Nm]. Torsion corresponds to ||T ||.
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Figure 36: Similar Torsion concentrations observed on the Analysis with RFEM
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