Paseo de San Francisco, espacios, actores y prácticas

3.1 - INTRODUCTION

The finite element method has been used extensively in simulating mechanically fastened joints in composite laminate materials so that the stress and strain sustained by the material under load may be evaluated. It is an approximate numerical solution to stress analysis problems, the accuracy of which depends upon the assumptions adopted in construction of the model. This chapter gives a brief overview of the finite element procedure and describes its application to contact mechanics. As I-DEAS (1993/1999) software has been used throughout the work reported here, the method of contact used by this software is described.

3.2 - OVERVIEW OF THE FINITE ELEMENT METHOD

The finite element method is now the most popular numerical technique used for the stress analysis of composite structures (Wood, 1994). Many commercial systems are now available incorporating this numerical technique and are becoming more accurate and more widely used with the development of better hardware and software. Each of the packages available have their own methods of data input and solution method, however, the general process by which an engineer carries out an analysis remains the same and follows the simple flow chart given in Figure 3.1.

In a broad sense the finite element method involves three processes. The first is the pre­ processing stage, whereby the data is input, the mesh is constructed and the boundary conditions applied. The next stage is the solution, which the software carries out without any further input from the user. The final stage is post-processing, whereby the results of

Finite element formulations for contact mechanics_______________________________ 74

the analysis can be either extracted to external packages for analysis or can be displayed within the package used, depending upon the capabilities of the chosen software.

I-DEAS (1993/1999) software has been chosen for the simulations throughout the work reported in this thesis as it provides accurate results, as demonstrated in previous work such as Chutima (1996), and quite importantly, it also provides an excellent user interface with good graphics in both the pre- and post-processors. The stages of a simulation are as detailed in the following sections.

3.2.1 Pre-Processing

The first step in finite element modelling is to create the replica geometry of the structure being simulated. The geometry is then discretized into many elements to create a group of elements known collectively as a mesh. Each element in the mesh is connected to the next at nodal points, the number of which depends upon the required mesh refinement, which is usually determined from the shape of the structure, the type of elements used, and the desired accuracy.

To model laminated composites a three-dimensional analysis must be carried out by using a Representative Volume Element (RYE) approach or a replica approach. With the RYE approach global properties are applied to all the brick elements of the structure and are derived from individual fibre/matrix/ply properties using classical laminate theory. With the replica approach orthotropic brick elements are used for each lamina in the structure, whereby the orientation of the material properties is altered for each element of each lamina depending upon the stacking sequence used. This replica modelling method is useful for determining interlaminar and through thickness stresses (Wood, 1994), and will therefore be used throughout the analysis.

Once an appropriate mesh has been constructed boundary conditions such as restraints and loading must be applied, so that the model simulates the problem as accurately as possible. Restraints can be applied to limit nodes from movement in any one or more of their assigned degrees of freedom (the number of d.o.f. assigned to a node, depends upon the type of element attached to that node), to simulate for example symmetry or clamped

Finite element formulations for contact mechanics_______________________________ 75

conditions. Loads are applied to nodes in order to simulate the tension or compression applied to the joint. In the latest version of the I-DEAS software, contact parameters must also be set at this stage, whereby the maximum distance that I-DEAS must search for possible contact between elements must be set, and also the coefficient of friction between contacting surfaces. This is required, as the contact elements are not created physically, as gap elements were in earlier versions of the software but are created using these boundary conditions within the solve routine.

3.2.2 Solution

As detailed in Cook(1995), the finite element method is described as a piecewise polynomial interpolation, which can be carried out on an element, so that a certain quantity for that element can be interpolated from values o f that quantity at each node. As each element is connected in the mesh the quantity can then be interpolated over the entire structure in a piecewise fashion using all the polynomial expressions. The whole process generates a set of simultaneous algebraic equations for values o f the field quantity at nodes. The matrix construction, which represents the equations is as follows:

KD = R (3.1)

where:

K = the stiffness matrix

D = the nodal displacement matrix R = matrix of load on the mesh

The method for solving the simultaneous equations in I-DEAS is Gaussian elimination with a Cholesky decomposition. With Cholesky decomposition the matrix of equations is factorised into a lower and upper triangular matrix, which are a transpose of each other. After factorisation the equations are solved for the unknowns by performing a forward and backward substitution upon the load vector.

3.2.2.1 - Finite element application to contact mechanics

The solution to Equation 3.1 is non-linear in the analysis of a bolted joint connection, as the contact between the bolt and hole changes throughout the application of loading. The

Finite element formulations for contact mechanics_______________________________ 76

non-linear aspect of this analysis is dealt with by the contact solution within the I-DEAS software, whereby an iterative procedure is adopted until the contact conditions have converged.

There are two methods of contact available in the I-DEAS software, the first, which was the only method available in earlier releases of the software, is the ‘gap element’ method and the method more recently available is the ‘contact element’ approach. The method by which the equations are solved is the same in each contact simulation, however, they do differ slightly, as the constraint equations can only be imposed on nodes in the gap element approach, but in the contact element approach, the constraint equations are imposed at the integration points of each element in contact.

In determining the contact conditions during simulations, the I-DEAS software carries out four main stages in the following sequence:

(i) Kinematic equations describing the relative motion between each contacting surface are determined.

(ii) Equilibrium equations containing boundary conditions and contact constraints are then developed.

(iii) The two sets of equations are then transformed into equivalent finite element matrix equations.

(iv) The resulting matrix equations are then solved

Each process is described below:

(i) With reference to Figure 3.2, the penetration of the hitting point into the target surface occurs at the target point and is given by:

p. =P=o+(“h- “t) ' “ (3 2)

where:

Finite element formulations for contact mechanics_______________________________ 77

u „ = the motion of the hitting point Uy = the motion of the target point

n = the normal vector, perpendicular to the hitting surface

When friction forces are also present, kinematic equations must also be determined to compute the relative tangential displacement increment, given by:

Au, = (Au„ - Aut. ) - [ii-(AUh- A Ut.)]u (3.3)

where:

u, = the tangential displacement

(ii) The contact constraints are imposed at the integration points on the finite element faces of the hitting region. The normal contact constraints are given by:

p , < 0 (14)

which states that the penetration must be less than or equal to zero, therefore, surfaces cannot interpenetrate,

t^ = - n - t > 0 (3.5)

which states that the contact pressure ‘tn’, which is defined as the negative of the normal surface traction component, cannot be tensile,

t .p , = 0 (3.6)

which states that if the contact pressure is greater than zero, then the surfaces are in contact and if the penetration is less than zero, then the contact pressure is zero.

Finite element formulations for contact mechanics_______________________________ 78

Pe = the penetration

t

= tangential traction tn = the contact pressure

When friction forces are also present constraint equations must also be developed for Coulomb friction conditions. The additional constraint conditions between the hitting and target surfaces are given as:

^ 0 (3.7)

which imposes the constraint that the magnitude of the in-plane friction traction cannot exceed the coefficient of friction times the contact pressure. When the magnitude of friction force reaches its maximum allowable value, then the function (|) will be equal to zero. Also,

Auj = AÇ (3.8)

which relates the relative tangential displacement increment between the hitting surface and the target surface, to the magnitude of the relative slip increment, which must be non-negative. When,

(|)A4 = 0 (3.10)

the equations imply that if the magnitude of the relative slip increment is greater than zero, then there is slipping and the function (|) must equal zero, and if the relative slip increment is zero, then the surfaces are sticking and the function (|) must be less than zero.

where:

Finite element formulations for contact mechanics_______________________________ 79

tt = the in-plane friction traction

Aut = the relative tangential displacement increment

In summary, as demonstrated in Figure 3.3:

- The maximum tangential traction equals the coefficient of friction times the normal traction force.

- Contacting surfaces stick if the tangential traction is less than the coefficient of friction times the normal traction force.

- Contacting surfaces will slide in the direction of tangential traction if the tangential traction equals the coefficient of friction times the normal traction force.

(iii) The kinematic and equilibrium equations described above for contact must then be transformed into finite element matrix equations using interpolation equations for each element face, so that the equations can be solved with the rest of the model.

(iv) The set of equations are then solved globally using an augmented Lagrangian procedure. This method offers advantages over previously used pure penalty methods and Lagrangian multiplier methods. A penalty stiffness is added but unlike pure penalty methods, the contact constraint solutions can be achieved with almost any required degree of accuracy through a series of contact traction updates. The penalty number can, therefore, be lower than in the penalty method, resulting in better conditioned equations.

An iterative procedure is adopted consisting of an inner loop and an outer loop. The inner loop iteratively updates the contact pressure (using the penalty number) for each active contact element to enforce the zero penetration constraint. The outer loop determines the contact element status, whereby inactive contact

Finite element formulations for contact mechanics_______________________________ ^

elements are activated if penetration between surfaces occurs, and active contact elements are set as sticking or sliding depending upon contact tractions, or are disabled if tensile forces exist.

The contact solution finishes once the contact tractions have converged and the contact status remains unchanged during an iteration.

The steps involved in the overall contact solution are as shown in Figure 3.4.

3.2.3 Post-Processing

The post processing stage involves extracting sample results such as stress or strain, which can be compared to previous work or simple calculations in order to validate the model. If the results are satisfactory, the rest of the results can be extracted or displayed for analysis. This can be carried out within the software used, or the results can be exported to another software package for analysis. However, as the I-DEAS software has an excellent post-processor the results were extracted and displayed within the software throughout this investigation.

The actual value of stress or strain is computed at the integration points of an element within the solution routine, however, during post-processing the values can be extracted at nodes, whereby the values from the integration points of a particular element have been extrapolated, or they can be taken from the element, which is the average between the values at each integration point of that element.

Finite element formulations for contact mechanics 81 NO Are results as expected? YES Gather all information required

for the simulation

A pply boundary conditions and loading

Create replica geometry

S olve the model Select appropriate

elements and discretize the model

Extract results to check validity

D ecide what needs changing and m odify

appropriately

Extract results for assessment

Finite element formulations for contact mechanics 82 H ittin g su rfa c e T a rg e t su rfa c e un -PeO O = ta rg e t p o in t © = h ittin g p o in t Ci/e2= u n it v e c to rs n = n o rm a l v e c to r Peo = in itia l p e n e tra tio n

Un = m o tio n o f h ittin g p o in t U | = m o tio n o f ta rg e t p o in t F ig u re 3.2 - D ia g ra m sh o w in g n o ta tio n u se d fo r th e c o n ta c t a lg o rith m S tic k in g c o n ta c t o c c u rs i f tt < p,t„ / y S lid in g c o n ta c t o c c u rs i f t t = |utn % Fastener F ig u re 3.3 - S u m m a ry o f c o n ta c t b e tw e e n e le m e n ts o f th e fa s te n e r a n d p la te

Finite element formulations for contact mechanics 83

NO Have contact tractions ''"'\converged?^^^^

YES

YES _{Has contact status} NO

changed since las^ ^ ^ te r a tio p ^ ^ Begin contact inner

loop iteration Compute loads for the current load case

Update contact tractions and compute

contact forces Form contact elements by defining

surfaces o f elements in contact

Assem ble contact stiffness for active contact elements in global stiffness

matrix Compute stiffness for

all elements and penalty stiffiiess for

contact elements

Begin contact outer loop iteration - determine contact status depending

on penetration & tractions

Go to next load case

84

In document Espacio público, actores urbanos y sus apropiaciones sociales: un acercamiento antropológico a Paseo De San Francisco (página 91-103)