Chapter 2

ENGINEERING ANALYSIS

Introduction Analysis is the breaking down of an object into its basic

elements to get to its essence.

Studying the nature and identifying its essential features and their relationships.

Tools of analysis are based upon logic and the application of logical systems (e.g. mathematics, physics & mechanics).

The role of analysis in design is a critical one and can be considered the internal guidance system of a project.

A project without analysis is like a sports team without a coaching staff.

The Role of Analysis

The traits of engineers and their relationship with analysis.

What comes first: analysis or experience?

At what point in the evolution of a design should the guidance from theory be given.

priority? Guidance from experiment?

The seamless interplay between hands-on and theoretical components.

Will the application of the tools of engineering remain static?

The Role of Analysis (Cont….)

What happens when theory and experiment do not agree? How is it known that an analysis is flawed? How is a design analyzed? What are the types and levels of analysis? Where does analysis begin? Does a project ever begin based solely upon analysis?

Engineer Traits & Analysis

Universities are developing educational programs to encourage ambidextrous thinking, or both the left & right hemispheres of the brain.

Linear logical process and verbal abilities derive primarily from left side of brain.

Visual spatial properties, intuition and qualitative assessment skills derive primarily from the right side.

An eventual goal as an engineers skill evolves is to move towards “ whole brain” thinking.

Engineer Traits & Analysis (Cont..)

1st year students have significant analytical or creative skills, but lacks broad set of abilities required in engineering.

University education is focused to nurture existing as well as broaden range of talent.

The Role of Analysis in Design Process (Complementary Roles)

Design requires different abilities & perspectives at different stages.

Initially either or both perspectives can derive the choice of project.

Creative brainstorming suggests path investigated.

Next stage involves critical assessment of the possibilities & first level of analysis : resulted in a prioritized list of choices and rational.

After first analysis, followed with iterative & review to include additional avenues.

Creative review the ways of

constructing, testing & implementing the design usually leads to prototypes.

Elements of final design, recommendations for production, use of marketing are key product of the review.

Always an opportunity for another set of iteration and fine tuning.

Iterative Process

Radcliffe [Ref] presents an alternative view of the design process.

Emphasized the iterative nature of design.

New information can be introduced at any design stage, which also necessitates to return to the previous point in the process.

The Design Spiral : Submarine

Designing a submarine is a challenging task.

Design constraints in designing a submarine: Size and weight

Environmental challenges (depth & pressure)

Critical life support needs

Hull design

Mission requirements

Propulsion and energy requirements

Design of each of these parameters effect the other parameters as well.

Different design parameters are methodically integrated with the help of design spiral.

The Design Spiral: Submarine

The Design Spiral (Cont…)

The strong interactions between subcomponents are accounted in all stages.

On moving from outside to inside, each of the systems is revisited in an interactive way, moving toward the final design.

Design of aircrafts, space vehicles, defense vehicles also offer similar challenges.

Analysis can be viewed as an umbrella that protects the whole system-ensuring a minor change in one sector won’t produce disaster changes in another sector.

Interplay Between Theory & Experiments

Design should be a seamless transition between theory and experiments.

An effective engineer respects & understands the relative roles of analysis and practice.

Engineer should be comfortable in handling theory experiments as well as transition between both.

With experience, the choices and application of engineering skills should become a reflex.

Theoretical & Experimental Developments

Evaluation Areas for Theory & Experiments

Several important questions are raised when theory and experiments are not in agreement.

Keys area to evaluate in theory and experiments are tabulated as:

Critical Role of Analysis in Engineering Projects

It is essential for engineers to learn from success & failure.

Success and failure helps to derive valuable lessons by analysis of a great range of designs.

Radio Detection and Ranging

Required an existing base of electronic capabilities to be feasible.

Supporting technology grew to a

threshold to made radar possible. This development eventually led

to parallel remote sensing devices using light and sound.

A successful design exploiting one

area of technology may have fruitful derivatives applications in other areas as well.

Stay on Tabs: Al Cans An example of the value of

being sensitive to a societal need.

Replaced many throw-away

taps and curved pollution. Recovered and recycled

cans along with tabs helps in saving tons of aluminum.

Simple concept with

significant impact.

Boing 777

40% conventional materials are replaced with advanced materials.

Advanced computers and

software's. Networking permitted engineers

world wide to work effectively on the same design.

An example of paperless design

and concurrent engineering. Considered as the most advanced

passenger planes.

Global Positioning System An example of an existing

base of infrastructure and technical capabilities making a concept practical.

Satellite platforms and

electronics enabled the execution of the concept, permitting accurate location world wide.

Example of an area that is

under dynamic expansion with different set of applications.

Tacoma Narrows Bridge Bridge collapsed under

modest wind in 1940. Modest wind exciting a

resonance. An example of extrapolated

engineering, effects of winds were not properly considered.

Engineering design failure

should encourage caution when extending past, seemingly successful, design.

Walkways Regency Hotel

The walkways failed in 1981, resulting in many deaths.

Seemingly non-critical

design change to save time and cost resulted in a weak design of suspension.

Unimportant design

element: NO SUCH THING

Challenger Space Shuttle Exploded on January 28,

1986. Design of O-ring seals failed

at low temperature launch. Seals were critical elements

for separating different stages of rocket.

Decision to use multi-stage

rocket was politically motivated.

This decision made the

design more complicated than necessary, which eventually led to failure.

Three Mile Island Nuclear Plant

Simple component can cause major problems.

Indicates the importance of

working out foolproof displays of system status.

Valve failure led to to

overheating problem. Visual display did not

indicate the actual overheated status of the valve.

Reliability The reliability method of design is one in which we obtain

the distribution of stresses and the distribution of strengths and then relate these two in order to achieve an acceptable success rate.

The reliability R can be expressed by a number having the

range 0 ≤ R ≤ 1 In the reliability method of design, the designer’s task is

to make a judicious selection of materials, processes, and geometry (size) so as to achieve a specific reliability goal.

It is important to note that good statistical data and

estimates are essential to perform an acceptable reliability analysis.

Safety & Liability The strict liability concept of product liability generally prevails

in USA.

It ensures the liability of manufacturer for any damage or harm that results from any defect.

Liability of manufacturer will not be eased if unknown to defects or defective design.

Best approach to the prevention of product liability are: Good engineering in analysis and design.

Quality control.

Comprehensive testing procedure.

Warranties and sales literature should be reviewed carefully.

Statistical Considerations

Introduction

Statistics in mechanical design provides a method of dealing with characteristics whose values are variable.

Products manufactured in large quantities have a life that is variable. One automobile may have so many defects that it must be repaired

repeatedly during the first few months of operation while another may operate satisfactorily for years, requiring only minor maintenance.

The variability inherent in limits and fits, in stress and strength, in

bearing clearances, and in a multitude of other characteristics must be described numerically for proper control.

Evidence gathered from nature by measurement is a mixture of

systematic and random effects. It is the role of statistics to separate these, and, through the sensitive use of data, illuminate the obscure.

Random Variables

Outcome when two dices were tossed:

A Probability Distribution:

A Cumulative Probability Distribution:

Random Variables The strength determined by random experiment

is called a random, or a stochastic, variable.

A probability distribution shows all possible values of a random variable and with the corresponding probabilities.

The probability function p = f (x ) , a function of x, is often called the frequency function or, sometimes, the probability density function (PDF).

A cumulative probability distribution describes the probability that x is less than or equal to a certain value xi.

Arithmetic Mean, Variance , and Standard Deviation

The total number of elements, called the population, may in some cases be quite large.

A small part of the group, called a sample is generally selected for measurement.

Sample mean :

Sample variance :

Sample standard deviation :

Gaussian ( Normal ) Distribution

The Gaussian, or normal, distribution is expressed in terms of its mean μx and its standard deviation σx as

The normally distributed variate x can be expressed as

where N represents the normal distribution function.

To avoid the need for many tables for different values of μ and σ , the deviation from the mean is expressed in units of standard deviation by the transform

Gaussian ( Normal ) Distribution

Probability Distribution Function Cumulative Distribution Function

Lognormal Distribution

The lognormal distribution is one in which the logarithms of the variate have a normal distribution.

Use the transformation

y has a normal distribution

The lognormal distribution has the following two characteristics:

The distribution is asymmetrical about the mean.

The variables have only positive values.

Lognormal Distribution

Probability Distribution Function Cumulative Distribution Function

Uniform Distribution

The uniform distribution is a closed-interval distribution that arises when the chance of an observation is the same as the chance for any other observation.

The probability density function (PDF) for the uniform distribution is

where a is the lower bound and b is the upper bound.

The cumulative density function (CDF) is linear in the range a ≤ x ≤ b given by

The mean and standard deviation are given by

Uniform Distribution

Probability Distribution Function Cumulative Distribution Function

Weibull Distribution

Most reliability information comes from laboratory and field service data, and because of its flexibility, the Weibull distribution is widely used.

The probability density function, for Weibull, is

where x0 = minimum, guaranteed, value of x

θ = a characteristic or scale value (θ ≥ x0)

b = a shape parameter (b > 0)

Weibull Distribution

Probability Distribution Function Cumulative Distribution Function

Propagation of Error

Suppose we wish to add the two variates x and y to form a third variate z.

The mean is given as

The standard deviation

Linear Regression

Statisticians use a process of analysis called regression to obtain a curve that best fits a set of data points.

The process is called linear regression when the best-fitting straight line is to be found.

The standard equation of a straight line is

A correlation coefficient r calculates how well x and y correlate with each other.

Computer Aided Design &

Computer Aided Manufacturing

Introduction to CAD/CAM

• CAD/CAM is commonly used in engineering : Drafting

Design

Simulation and analysis

Manufacturing

• CAD/CAM includes four major areas: Geometric modeling

Computer graphics

Design applications

Manufacturing applications

Product Life Cycle (PLC) (Design Process)

Design Need

Design definitions, specifications

Collecting design information, feasibility study

Design conceptualization

Design modeling & Simulation

Design analysis

Design optimization

Design evaluation

Design communication & documentation

To manufacturing process

Synthesis

Analysis

CAD Process

Product Life Cycle (PLC) (Manufacturing Process)

Process Planning

Production Quality Control

Packaging Design & Procurement of new tools

Order Material

NC,CNC,DNC programming

Shipping Marketing

Production Planning

From design process

CAM Process

Design need (Design Process)

Major Components of PLC Design

Synthesis: Philosophy, functionality and uniqueness is determined. Design takes the form of sketches and layout drawings.

Analysis: Conceptual design in the engineering sciences to evaluate

the performance of the expected product. To finalize the best drawing.

Manufacturing

Process planning: determines the most efficient sequence to manufacture the product. Outcome is production plan, tools procurement, material order and machine programming.

CAD Disciplines

Material properties

Finite element analysis

Dimensioning & tolerances

Assembly modeling

Documentation and drafting

CAD

Geometric modeling

Computer Graphics

Design

CAM Disciplines

Computer aided process planning (CAPP)

NC programming (Numerical modeling) Design of injection molds Coordinate measuring machines

verification (CMM) Inspection Assembly with robots packaging

CAM

CAD Automation

Manufacturing

CAD/CAM Modules

Geometric module

Core of system

Developing, editing and manipulation of geometry

Drafting and documentation

Applications module

Geometry is mean to achieve the goal.

Mass property calculations

Assembly and tolerances analysis

Finite element modeling and analysis

Mechanism analysis

Simulation and analysis of plastic injection molding

CAD/CAM Modules

Programming module

Allows to customize the system

Adapt system with certain design and manufacturing tasks

Communication module

To achieve integration between CAD & CAM, other computer systems and manufacturing facilities

Collaborative module

To established real time connection between design teams working in different geographical locations

Numerical Control

Numerical control (NC) is a form of flexible (programmable) automation in which the process is controlled by numbers, letters, and symbols.

The electronic industries association (EIA) defined NC as

“A system in which actions are controlled by the direct insertion of numerical data at some point. The system must automatically interpret at least some portion of this data.”

53

Basic Components

An NC system consists of the machine tools, the part-program, and the machine control unit (MCU).

54

Machine Tools

The machine tools perform the useful work.

A machine tool consists of.

– A worktable,

– One or more spindles, motors and controls,

– Cutting tools,

– Work fixtures, and.

– Other auxiliary equipment needed in the machining operation.

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The Part-Program

The part-program is a collection of all data required to produce the part. It is arranged in the form of blocks of information.

Each block contains the numerical data required for processing a segment of the work piece.

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The Machine Control Unit

The machine control unit consists of the data processing unit (DPU) and the control loop unit (CLU).

The DPU decodes the information contained in the part-program, process it, and provides instructions to the CLU.

The CLU operates the drives attached to the machine lead screws and feedback signals on the actual position and velocity of each one of the axes. The drive units are actuated by voltage pulses.

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Computer Numerical Control (CNC)

The EIA definition of computer numerical control (CNC). – “A numerical control system wherein a dedicated, stored

program computer is used to perform some or all of the basic numerical control functions in accordance with control programs stored in the read-write memory of the computer.”

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The CNC uses a dedicated microprocessor to perform the MCU functions.

CNC supports programming features not available in conventional NC systems:

– Subroutine macros which can be stored in memory and called by the part-program to execute frequently-used cutting sequence.

– Inch-metric conversions, sophisticated interpolation functions (such as cubic interpolation) can be easily accomplished in CNC.

– Absolute or incremental positioning (the coordinate systems used in locating the tool relative to the work piece).

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– The part-program can be edited (correction or optimization of tool path, speeds, and feeds) at the machine site during tape tryout.

– Tool and fixture offsets can be computed and stored.

– Tool path can be verified using graphic display.

– Diagnostics are available to assist maintenance and repair.

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Direct Numerical Control (DNC)

– “A system connecting a set of numerically controlled machines to a

common memory for part program or machine program storage with provision for on-demand distribution of data to machines.”

– In DNC, several NC machines are directly controlled by a computer, eliminating substantial hardware from the individual controller of each machine tool. The part-program is downloaded to the machines directly (thus omitting the tape reader) from the computer memory.

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Terminology of Solid Models

Coordinate Systems

Model(Master) Coordinate System (MCS)

MCS is the reference space of model with respect to which all the model geometries data is stored. In a CAD system MCS is generally shown by displaying X,Y,Z axis.

Working Coordinate System (WCS)

Portable coordinate system often employed when desired plane of sketching is not easily defined as one of the MCS planes.

Sketching Planes

Are the orthogonal planes

created by the axis of MCS or

WCS. Creating or selecting a

sketch plane is the very first

step toward creating a CAD

model.

Three Modeling Approaches

Primitive Approach

Views a solid model as a combination of simple generic, and standard shapes that can be combined. Primitives include, block (box), cylinder, sphere, cone & tores. These primitives are combined with Boolean operations.

Steps :

i. Create the block using block primitive.

ii. Create a cylinder in the desired location/orientation.

iii. Subtract the cylinder from the block.

Feature Approach

Similar to primitive approach, it replaces primitives with features and embeds Boolean operation in the features definition.

Steps:

i. Create the block using block feature.

ii. Create the hole in the block by creating a hole feature.

Block

Hole

Sketching Approach Sketching

Similar to features approach, with one change . Instead of using predefined shapes only, it allows designers to create much more elaborate & more general features starting from a sketch.

2-D Sketch Solid Model

Modeling 3D Operations

Extrusion

Revolving

Sweep

Loft

Types of geometric Model

21/2 D

Have uniform cross-section and thickness in the direction perpendicular to the plane cross sections. Axisymmetric model also falls in this category. Models made up of many 21/2 D features are called composite 21/2D model.

21/2 D 21/2 D Composite Axisymmetric

3 D Model

Are the ones that do not have a uniform cross section and/or not have constant thickness. Require more than one sketch in different sketch planes.

Visualization

Once the model is created, CAD system allows to view those models in many different ways. Viewing operation in a CAD system can be classified into three groups.

i. View orientation

ii. View modes

iii. View manipulation

View Orientation

Includes standard views such as front, top, right and isometric.

View Modes

Allow us to change the display of the model to different types such as wireframe, hidden & shaded

Wire Frame Hidden Dotted Frame Shaded

View Manipulation Allow us to dynamically rotate, pan and zoom the model to gain better control over its viewing.

Software’s

CATIA

Pro-Engineer

Solid works

ANSYS

Abaqus

I-DEAS

LS-DYNA

Introduction to

Finite Element Method

Introduction to FEM Without numerical techniques, it would be almost

impossible to solve practical engineering problems. Finite Element Method (FEM) is a numerical method

for solving engineering problems. The finite element method has been employed in:

(i) structural analysis (ii) fluid flow (iii) heat transfer

Application of FEM

APPLICATIONS

AEROSPACE

AUTOMOTIVE

BIOMECHANICS

MULTIPHYSICS

FEM in Design : Discretization

FEM in Piping

FEM in Safety

FEM in Crashworthiness

Discretization of Continuum Numerical techniques in continuum mechanics are

based on the principle that a continuum can be divided into an equivalent system of smaller bodies.

These bodies are connected at points (nodes) common

to the sub-regions (smaller bodies called elements). As the size of these small bodies gets smaller, the

numerical solution becomes more accurate. The cost of computation time may become prohibitive.

Elements & Nodes

Advantages of FEM

Accurate representation of complex geometry

Inclusion of dissimilar material properties

Easy representation of the total solution

Capture of local effects.

Element Geometries

The Finite-Element Method

Since the finite-element method is a numerical technique that discretizes the domain of a continuous structure, errors are inevitable.

– Computational errors : due to round-off errors from the computer floating-point calculations

and the formulations of the numerical integration schemes that are employed.

– Discretization errors : The geometry and the displacement distribution of a true structure continuously vary. Using a finite number of elements to model the structure introduces errors in matching geometry and the displacement distribution due to the inherent mathematical limitations of the elements.

Mesh Generation

There are three basic ways to generate an element mesh. – Manual mesh generation : This is how the element mesh was created in

the early days of the finite-element method.

– Semiautomatic mesh generation : this method enable the modeler to automatically mesh regions of the structure that he or she has divided up, using well-defined boundaries.

– Fully automated mesh generation. Many software vendors have concentrated their efforts on developing fully automatic mesh generation,

and in some instances, automatic self-adaptive mesh refinement.

The network of elements and nodes that discretize a region is referred to as a mesh.

Results generally improve when the mesh density is increased in areas of high stress gradients and/or when geometric transition zones are meshed smoothly.

Element Mesh

Mesh in Biomechanics

Meshing Curves

Mesh

MGWS

Overview of the Finite Element Method

Strong

form

Weak

form

Galerkin

approx.

Matrix

form

Axial deformation of a bar subjected to a uniform load

(1-D Poisson equation)

Sample Problem

0p=xp

0

00

2

=dx

duEA

=u

p=dx

udEA

Lx

02

L

L=x 0,

u = axial displacement

E=Young’s modulus = 1

A=Cross-sectional area = 1

Strong Form

The set of governing PDE’s, with boundary conditions, is

called the “strong form” of the problem.

Hence, our strong form is (Poisson equation in 1-D):

0

00

2

=dx

du

=u

p=dx

ud

Lx

02

We now reformulate the problem into the weak form.

The weak form is a variational statement of the problem in

which we integrate against a test function. The choice of test

function is up to us.

This has the effect of relaxing the problem; instead of finding

an exact solution everywhere, we are finding a solution that

satisfies the strong form on average over the domain.

Weak Form

Weak Form

0

0

0

0

2

0

2

2

=vdxpdx

ud

=pdx

ud

p=dx

ud

L

2

2

02

Strong Form

Residual R=0

Weak Form

v is our test function

We will choose the test function later.

Why is it “weak”?

It is a weaker statement of the problem.

A solution of the strong form will also satisfy the weak form,

but not vice versa.

Weak Form

Weak Form

Choosing the test function:

We can choose any v we want, so let's choose v such that it

satisfies homogeneous boundary conditions .

Steps Involved in FEM

• Divide continuum into a collection of pre-selected elements

of simple geometries (triangles, rectangles and quadrilateral

elements)

• Derive element equation for all types of elements involved in

the mesh such that

– equilibrium and compatibility are enforced

– assumed displacement within each element is dependent

upon nodal values

– equivalent nodal loads are established using principle of virtual work

[K]e[u]e = [F]e

• Assemble element equations to obtain the equilibrium equation of the whole problem [k]g{u}g = {F}g

• Impose boundary conditions • Solve the equilibrium equations for the nodal displacement • Calculate stresses and strains and post-process results

Fundamentals of FEM

(i) Idealization of structure

simplify the geometrical features of the structure

(ii) Discretization of structures

subdivide the structure into a system of finite elements. The size and number of elements are dictated by the geometrical features of the structure, applied load and restrains, accuracy and size of computer.

(iii) Choice of interpolation function

assume a trial function for the displacement (e.g. polynomial)

Fundamentals (Contd…) (iv) Derivation of the element stiffness matrix Derive the element stiffness matrix using the principle of minimum of potential energy (equilibrium equation). The derived stiffness relates the nodal displacements to the applied nodal forces. The stiffness matrix is a function of the material and geometric properties of an element. (v) Assembly of global stiffness matrix Assemble the global stiffness matrix from the element stiffness matrices

Fundamentals (Contd…)

(vi) Solution for the unknown nodal displacement

apply boundary conditions

solve the global equilibrium equations that can be

described as [K]{u}={F}

(vii) Computation of element & nodal strains and stresses

calculate the element strains and stresses using the appropriate solid mechanics relations

Features of FEM

have no limitations with regards to geometry, physical composition of domain and nature of loading.

involves a systematic procedure that can be automated for use with digital computers, and

yields approximate analysis by assuming a displacement field (or a stress field)

Pre Processing

General features, nodes, elements,

topology, Co-ordinate axes etc.

Material properties, yield strength,

density, coeff. of thermal expansion

Boundary conditions imposed, mechanical & thermal restraints

Applied loads

Preprocessor Appropriate Input Data File

General purpose FE software

Post Processing

Displacement

General purpose FE software

Strain

Stress

Temperature

Velocity

Post Processing

Result Files

Basic FE Algorithm • Mesh geometry • Element type • Boundary condition • Applied load • Symmetry

Input Data

E.S.M.G.

Assembler

Reducer

Solver

Output data

Apply B.C.

Solve for [ug]

• u • σij

• εij

The Finite-Element Solution Process A truss element is a bar loaded in tension or compression and is of

constant cross-sectional area A, length l, and elastic modulus E.

A truss element can be modeled as a simple linear spring with a spring rate

Assuming all forces f and displacements u directed toward the right as positive, the forces at each node can be written as

Consider a two-spring system, the total force at each node is the external force.

FEM in Design: Free Body Diagram

Example

• We combine the two stiffness matrices into the global matrix.

Now that the displacement at u2 has been obtained, the end forces and stress values can be obtained by reverting back to the individual element stiffness matrices

For the stress, you only need to look at the individual node of the stiffness equation

Reactions

Element Forces

Element Stresses

Software’s

ANSYS

Abaqus

I-DEAS

LS-DYNA