Structural Analysis by the Finite Element Method

Figure 2.8:NX Motion showing the different menus

• Simcenter NX Motion

This module allows the mechanism motion analysis to obtain displacements, loads, positions, interference or motion ranges, among many others. In order to create a successful kinematic simulation, first the components are defined as motion bodies, joints and the movement is created by adding drivers. Finally, the control can be achieved in several ways.Figure 2.8shows the crane in a motion analysis.

Figure 2.9:NX Motion showing an animation

• NX Animation

The animation is the output for the NX Motion, and it allows the generation of plots and data spreadsheets. An example of the animation can be seen inFigure 2.9In addition, it can generate motion envelopes and measure positions at any step moment.

2.3 Structural Analysis by the Finite Element Method

Structural analysis is a branch of engineering that applies a set of mechanical theories and physical laws with the intention of studying and predicting the behaviour of structures.[25].

A structure can be a bridge, building or a crane, and is defined as a group of links con-nected by joints in a particular configuration and withstanding loads.

A joint restricts the degrees of freedom of a part, depending on the type. Every rigid body has 6 degrees of freedom, as displayed inFigure 2.10and are defined as following:

1. Translation along X-axis 2. Translation along Y-axis 3. Translation along Z-axis 4. Rotation around X-axis 5. Rotation around Y-axis 6. Rotation around Z-axis

Figure 2.10:Degrees of freedom of a rigid body in space

There are several types of joints that restrict the movement of the links and define the load transfer. The two joints that will be used in this project are displayed inFigure 2.11and described as:

• Revolute

Allows the rotation only in the X axis. The rest of the degrees of freedom are fixed.

This constraint or joint is also known as ”pin” or ”hinge”. This joint is used in the crane to move the three angular displacements.

• Slider

Allows the translation only along the X axis. The rest of the degrees of freedom are fixed. This joint will be applied in the crane for all the telescopic parts.

Figure 2.11:Revolute and slider joints

2.3 Structural Analysis by the Finite Element Method To perform an accurate analysis, its necessary to determine the geometry of the part to test, the structural loads, boundary conditions and materials used. The results will contain support reactions, displacements and stresses.

Structural Analysis is often divided in three stages[26]:

• Establishing the boundary conditions and design loads

• Defining acceptance criteria

• Running the analysis

There are many applications of the finite element method such as acoustic simulations, fluid dynamics, thermal analysis and it is particularly applied to the structural analysis due to its capacities for calculating displacements and strains under a set of loads.[27]

2.3.1 Finite Element Method

The Finite Element Method (FEM) is defined as a numerical technique for solving partial differential or integral equations to obtain the evolution in time of the variables that rep-resent the behaviour of a physical system and it is applied by dividing a rigid body into smaller, finite elements. Since the geometry of the studied structure has to be simplified and divided into smaller parts, FEM offers an approximation to the real exact solution. [27]

There are three main error sources that contribute to the complete FEM approximation model, and those are [27]:

• Discretization error:Represents the error created by the mesh finite element and size. It can be reduced by using a finer mesh, an element with more nodes or refine-ment zones. A mesh convergence study will directly impact and reduce this error.

• Modeling error:Represents the error caused by an incorrect model simplification and it can be fixed by increasing the accuracy of the model and assumptions. The evaluation of the original crane parts and an idealized model will provide an idea of the accuracy of the models.

• Numerical error: This error is caused by the use of computers to solve the equa-tions and is usually very small.

Finite Elements

A finite element is a fraction or portion of the complete element or body. [27] There are many different types of finite elements, varying from 1D, 2D and 3D elements. A node is a joint point, and the most simple 1D element consists of two connected nodes. As the number of nodes increase, the computational effort and accuracy of the results increase.

The finite element types are described as a shape followed by the number of nodes, and the list is displayed inTable 2.1. A mesh is defined as the complete set of elements that discretize a 3D model. It can consist of a single element type, or a combination.

Element Type Element Name Number of Nodes Example

1D 1D Mesh ≥2

2D

TRI3 3

TRI6 6

QUAD4 4

QUAD8 8

3D

TET4 4

TET10 10

HEX8 8

HEX20 20

Table 2.1:Finite Element Types

The squared or rectangular elements such asQUADandHEXare most used for defined simple shapes, while the triangular elements such asTRIandTETare used for more irreg-ular shapes. Therefore, for the FEA study case, the focus will be in the triangirreg-ular elements and an approximation using 1D Beam elements will be developed. [28]