Element vs Nodal Stress

Nodal stress output for chasis

One of the primary outputs of FEA is the stress at various points in the body. When a body is subjected to an applied load, a stress state is created within the body. The actual stress values are calculated at specific points within the elements called Gaussian points. The results are then often presented in one of two ways:

a) Element Stress - The values of the stresses at the Gaussian points within the element are averaged and a single stress value is calculated for the entire element. As a result, a discontinuous stress distribution is displayed by the model.

b) Nodal Stress - These values are calculated at the nodes of the element by extrapolation from the values at the Gauss points. Since elements typically share nodes, the value at a shared node is determined by averaging the values from the contributing elements. Nodal stress outputs produce continuous stress distributions in the model.

Errors in FEA

It is important to understand the three main sources of error in FEA and how to control them.

1) Modeling errors - These are errors associated with the model that is used for the analysis. Learning how to build robust and accurate models for complicated problems can only be learned through experience. Correct modeling techniques are required.

2) Discretization errors - The FEA method takes a continuous body and breaks it up into a discrete number of finite elements creating a mesh. The mesh must be of sufficient density to capture the response of interest.

3) Numerical solver errors - The FEA method requires the computer to solve a large system of equations for the unknowns of interest. The errors in the answers obtained are typically very small but are referred to as round off errors.

Common Failure Modes

FEA model of crack growth

Here is a short list of common failure modes encountered in engineering structures.

1) Fracture - Fracture refers to the formation of cracks or the growth of existing cracks in a structure.

2) Yielding - If a body's stresses are higher than the materials yield strength, the body will fail due to yielding if the integrity or function of the part is compromised.

3) Buckling - The sudden loss of stability or stiffness under an applied load.

4) Fatigue - Structures which experience variable loading may fail after a number of loading cycles.

5) Creep - Bodies under a sustained load will deform over time. Empirical creep data is required to compensate for the effect.

6) Lack of Stiffness - Bodies must be stiff enough to support loads and maintain required tolerances.

Material Properties in FEA

Model of a composite material

Material properties can easily be found in a variety of different sources like handbooks or on websites like MatWeb with varying degrees of accuracy. Most materials respond differently depending on conditions such as temperature, processing conditions and strain rates to name just a few.  For this reason, the analyst must be careful to choose properties that are appropriate for the FEA model under study.

Types of Materials

There are three common types of materials that the analyst needs to be aware of:

1) Isotropic - Properties are the same in any direction throughout the body. Common examples of isotropic materials are glass and metals.

2) Anisotropic - Properties differ in two or more directions.  A very common example of an anisotropic material is wood.

3)  Orthotropic - Properties are different in three mutually perpendicular directions. Orthotropic materials are a type of anisotropy. An example of a composite material is fiber reinforced polymers.

Steel Column Buckling and Slenderness Ratio

Models of short, intermediate and long columns

There are three broad classes of steel columns that are defined by the slenderness ratio of the column. The slenderness ratio is the ratio of the length of the column (L) to its radius of gyration (r).  We will assume loading at the centroid of the column and no material imperfections. For steel columns:

a) Short column when L/r <40. The failure mode is by yielding (like a compression specimen).

b) Intermediate column when 40 L/r <120. The failure mode is a combination of yielding and inelastic buckling. The pre-buckled deflections are small but some stresses are beyond the linear range. A nonlinear buckling analysis is required for intermediate columns.

c) Slender (Euler) column when L/r > 120. The failure mode is by elastic buckling. The pre-buckled deflections are small and critical load is reached before the material yields. A linear buckling analysis is required for this Euler column. The long slender member subjected to an axial force will respond by deforming laterally. The actual value of the force required to make the structure buckle can vary by up to a factor of four depending on how the two ends of the column are restrained.

Modal or Natural Frequency Analysis

Cantilevered Beam Model with a natural frequency of vibration of 114 Hz

Modal analysis uses the overall stiffness, mass of a structure and constraints to determine the various frequencies at which it will vibrate.  The mode shapes corresponding to those frequencies are also provided. Once we have these frequencies, we do our best to avoid exposing the structure to these frequencies in order to avoid damaging the structure.  Here are two examples:

a) When designing a bridge in an earthquake zone,  we will design the bridge to have natural frequencies of vibration that are different from the vibrational frequencies of the earthquakes.

b) Unbalanced forces from a motorcycle engine can damage parts mounted to the bike if the natural frequency of vibration of those components are the same as the frequency of vibration of the engine.

The equation used to model natural frequencies is [K - (ω^2 )M]X = 0 where:

K is the stiffness matrix

ω are the natural frequencies of the system (or eigenvalues)

M is the mass matrix

X is the systems eigenmode matrix

Linear Elastic Buckling Analysis

Linear Elastic Buckling Analysis of L bracket

Under certain loading, a structure continues to deform without an increase in the magnitude of loading. The structure has become unstable, it has buckled. Buckling is a failure mode characterized by the sudden loss of stability or stiffness and results are catastrophic. It is most commonly associated with compressive forces but it can also occur in elastic structures due to dead tensile loading.

The types of structures that can buckle are:

1) slender (Euler) columns under axial compressive load.

2) externally loaded thin walled pressure vessels

3) thin plates under edge pressure

4) deep, thin cantilevered beams under transverse loading on the top surface

For linear elastic buckling:

1) there is no yielding of the structure

2) direction of the applied force does not change

The equation used to model linear buckling is ([Ka] + λ [Kd]){U} = λ[Fa] where:

[Ka] is the system stiffness matrix

[Kd] is the differential stiffness matrix (This matrix is necessary to account for the change in      potential energy associated with rotation of continuum elements under load)

{U} is the displacement vector

λ is the eigenvalue

[Fa] is the applied load vector

It is worth noting, the differential stiffness matrix represents a linear approximation of softening (reducing) the stiffness matrix for a compressive axial load and stiffening (increasing) the stiffness matrix for a tensile load.

The output of interest from a linear buckling FEA analysis is the buckling load factor (value of λ for first buckling mode) and first buckling mode shape. The critical applied load which is the load the structure is predicted to buckle at is obtained by multiplying the actual applied load by the buckling load factor. The mode shape shows the instability corresponding to the buckling load factor.

It is very important to realize that the results of a linear buckling analysis is often non conservative. That is, it often over estimates the critical load. Buckling of the real structure may occur at a much lower value.

Linear Static Analysis

Von Mises Stress Contour from Linear Static Analysis of Pipe Support

This most basic type of FEA analysis is also known as small displacement analysis. The matrix equation for this type of analysis can be represented as {F} = [K]{U} where:

{F} is the applied force vector

[K] is the stiffness matrix

{U} is the displacement vector

The following three assumptions must be true for the system to be linear.

1) The material used in the analysis must follow Hooke's Law where the stress is proportional to strain. The constant of proportionality is called Young's modulus.

2) The stiffness matrix does not change throughout the problem.

3) The boundary conditions do not change from the initial load application to the final deformed shape. The loading must be constant in magnitude, orientation and distribution.

Static refers to the assumption that all loads are applied slowly up to their full magnitude.

What is CFD?

Particle Flow Through an Overbend

Introduction

CFD stands for Computational Fluid Dynamics. It is a branch of fluid mechanics that uses numerical methods to analyze and solve fluid flow problems. Computers are used to simulate and solve the interaction between the fluid and engineering surfaces like airplane wings.

What are Fluids?

Fluids are substances which continually deform (flow) under an applied shear stress. Fluids are commonly divided into liquids and gases.

What are the basic equations of CFD?

For any single phase fluid flow, the governing equations are called the Navier-Stokes equations. These equations are derived by applying Newton's second law of motion (F = ma) to viscous fluids.

Why is CFD important when studying industrial flows?

Most industrial flows are turbulent by nature. That is, these flows have chaotic changes in pressure and flow velocity. Solutions to these flow problems can only be approximated using CFD since analytic solutions do not exist. Indeed, finding analytic solutions to turbulent flows remains one of the last great problems in classical physics.

What is FEA?

FEA Mesh of a Heat Sink

Introduction

Finite Element Analysis (FEA) is a computer based numerical method of simulating/analyzing the behavior of engineering structures and components under a variety of loading conditions. It is an advanced engineering tool that is used to design and to augment experimental testing.

FEA Method

1. Pre-processing, in which the analyst develops a finite element mesh of the geometry and applies material properties, boundary conditions and loads.

2. Solution, during which the program derives the governing matrix equations from the model and solves for the displacements, strains and stresses. This is the case in implicit code applications. Alternatively explicit codes can be used, mostly for high strain rate engineering problems.

3. Post- processing, in which the analyst obtains results usually in the form of deformed shapes and contour plots which help to check the validity of the solution.

A variety of reporting tools can be used to illustrate the behavior of the analysis model including color contour and vector plots, section cuts, isosurfaces, animations, graphs and text output.

The Benefits and Applications of FEA

FEA is particularly suitable for:

1. Structural Analysis

FEA is commonly used in structural and solid mechanics for calculating stresses and displacements. These are often critical to the performance of equipment and can be used to predict failure.

2. Thermal Analysis/Heat Transfer

FEA can be used for thermal analysis to evaluate the temperature distribution, and stresses resulting from uneven heating or rapid temperature changes. Thermal analysis may include conduction, convection and thermal radiation.  Analysis types include both steady state and transient analysis.

3. Frequency Analysis

Mechanical vibration characteristics are often important in design. Modal analysis is used to find the natural frequencies of vibration and their associated mode shapes.

4. Fluid Flow

FEA provides insight into complex transient and turbulent flow fields. It allows analysis and optimization of component geometry for efficient fluid flow, as well as allowing users to view velocity, pressure and thermal conditions inherent in the modeled flow fields.

5. Electromagnetics

Electromagnetic compatibility and electromagnetic interference can be important when designing electrical equipment and can be analyzed using FEA.