# MECHANICAL ENGINEERING LIBRARY

## MECHANICAL ENGINEERING

• SUBJECT : ENGINEERING MATHEMATICS - THEORY WITH WORKED OUT EXAMPLES SELF AND CLASSROOM PRACTICE QUESTIONS
• PUBLISHERS : ACE ENGINEERING PUBLISHERS
• IDEAL FOR : GATE AND PSU's - MECHANICAL ENGINEERING STREAM

CONTENTS :
1. PROBABILITY AND STATISTICS
2. DIFFERENTIAL EQUATIONS
3. LAPLACE TRANSFORM
4. COMPLEX VARIABLES
5. NUMERICAL METHODS

BRIEF INTRODUCTION :

PROBABILITY BASIC TERMINOLOGY :

Exhaustive events : A set of events is said to be exhaustive,  if it includes all the possible events.  For in tossing a coin there are two exhaustive cases either head or tail and there is no third possibility.

Mutually exclusive events : If the occurrence of one of the events procludes the occurrence of all other then such a set of events is said to be mutually exclusive.  Just as tossing a coin,  either head comes up or the tail and both can't happen at the same time,  i.e.,  these are two mutually exclusive cases.

Equally likely events : If one of the events cannot be expected to happen in preference to another then such events are said to be equally likely.  For instance,  in tossing a coin,  the coming of the head or the tail is equally likely.

PROBABILITY AND SET NOTATIONS

Random experiment : Experiments which are performed essentially under the same conditions and whose results cannot be predicted are known as random experiments.  e g.,  Tossing a coin or rolling a die are random experiments.

Sample space : The set of all possible outcomes of a random experiments called sample space for that experiment and is denoted by S.  The elements of the sample space S are called the sample points e.g.,  On tossing coin,  the possible outcomes are the head(H) and the tail(T)

Thus S={H, T}  a of a sample space S

Event : The outcome of a random experiment is called an event.  Thus every subset is an event.  The null set is also an event and is called an possible event.  Probability of an impossible event is zero

Differential Equations :

Mathematicians have tried in vain to this day to discover some order in the sequence of prime A numbers and we have reason to believe that it is a mystery into which the human mind will never penetrate.

Introduction:  Differential equations arise from many problems in mechanical bending of beams,  conduction of heat,  velocity of t. The chemical reactions etc. Differential equations are used to model the physical systems.  The systems which change con time are called dynamic Systems.  For example, models which describe heat transfer chemical engineering which depend on time,  electronic circuits with time dependent current and voltages are dynamic systems.  The phase A differential equation is an equation which involves differential coefficients or differentials.
A partial differential equation is that in which there are two or more independent variables and partial differential coefficients with respect to any of them.

The degree of a differential equation is the degree of the highest derivative occuring in it, after the equation has been expressed in a form free from radicals and fractions as far as the derivatives are concerned.

Practical approach to differential equations :
• The study of a differential equation consists of three phases.
• Formulation of differential equation from the given physical situation called modelling
• Solution of this differential equation and evaluating the arbitrary constants from the given conditions
• Physical interpretation of the solutions.

Formation of Differential Equation : A differential equation is formed in an attempt to eliminate certain arbitrary constants from a relation in the variables and constants

Solution of a differential equation : A solution if a differential equation is a relation between the variables which satisfies the given differential equation.

The general solution of a differential equation is that in which the number of arbitrary constants is equal to the order of the differential equation.

A particular solution is that which can be obtained from the general solution by giving particular values to the arbitrary constants.

A singular solution is that which cannot be obtained from the general solution by giving particular values to the arbitrary constants.

Laplace Transform

Introduction : Laplace transform is used for solving differential and integral equations. In physics and engineering, it is used for analysis of linear time-invariant systems such as electrical​ circuits, harmonic oscillators,  optical devices and mechanical​ systems.

The Fourier transform named after Joseph finite Fourier,  is a mathematical transformation employed to transform signals between time(or spatial)  domain and frequency domain,  which has many applications in physics and engineering. It is reversible,  being able to transform from either domain to the other.  The term itself refers to both the transform operation and to the function it produces.

In mathematics and signal processing,  the Z transform converts a time domain signal,  which is a sequence of real or complex numbers,  into a complex frequency domain representation. It can be considered as a discrete-time equivalent of the Laplace transform.  This similarity is explored in the theory of time scale calculus.

## MECHANICAL ENGINEERING

• SUBJECT : ENGINEERING MATHEMATICS - THEORY WITH WORKED OUT EXAMPLES SELF AND CLASSROOM PRACTICE QUESTIONS
• PUBLISHERS : ACE ENGINEERING PUBLISHERS
• IDEAL FOR : GATE AND PSU's - MECHANICAL ENGINEERING

CONTENTS :
1. LINEAR ALGEBRA
2. CALCULUS
3. VECTOR CALCULUS

BRIEF INTRODUCTION TO LINEAR ALGEBRA :

INTRODUCTION : Linear algebra comprises of the theory and applications of linear system of equation,  linear transformations and eigen value problems.  In linear algebra,  we make a systematic use of matrices and to a lesser extent determinants and their properties.  Determinants were first introduced for solving linear systems and have important engineering applications in systems of differential equations,  electrical networks,  eigen value problems and so on.  Many complicated expressions occurring in electrical and mechanical systems can be elegantly simplified by expressing them in the form of determinants Cayley matrices in the year 1860.  But it was not the twentieth century was well-  discovered advanced that engineers heard of them.  These days,  however,  matrices have been found to be of great utility in many branches of applied mathematics such as algebraic​ differential equations,  mechanics theory of electrical circuits,  nuclear physics,  aerodynamics and astronomy.  With the advent of computers,  the usage of matrix methods has been facilitated.

RANK OF A MATRIX : If we select any rows and r columns from any matrix A,deleting all the other rows and columns,  then the determinant formed by these r x r elements is called the minor of A of order r.  Clearly,  there will be a number of different minors of the same order,  got by deleting different rows and columns from the same matrix.

Definition : A matrix is said to be of rank r when

(i) it has at least one non-zero minor of order r,  and
(ii) every minor of order higher than r vanishes.  Briefly,  the rank of a matrix is the largest order of any non-vanishing minor of the matrix.
If a matrix has a non-zero minor of order r,  its rank is greater than or equal to r.
If all minors of a matrix of order r + 1 are zero,  its rank is less than or equal to r
The rank of a matrix A shall be denoted by p(A).

Elementary transformation of a matrix :

The following operations,  three of which refer to rows and three to columns are known transformations L The interchange of any two rows(columns).  IT.  The multiplication of any row(column by a non-zero number.  the corresponding elements of III.  The addition of a constant multiple of the elements of any row(column to any other row(column)  Notation.  The elementary row transformations will be denoted by the following symbols:
(i) Rij for the interchange of the ith and jth rows.
(ii)  kRi for multiplication of the ith row by k.
(iii) Ri + pRj for addition to the i'th row,  p times the j'th row.  writing C in place of R

The corresponding column transformation will be denoted by matrix.  While the value of the minors may get. Elementary transformations do not change he order or rank of a changed by the transformation I and II,  their zero or non-zero character remains if one can be obtained from the(3)  Equivalent matrix.  Two matrices A and B are said to be equivalent the same other by a sequence of elementary transformations.  Two equivalent matrices have the same order and rank.  The symbol is used for equivalence

## PRESSURE VESSELS DESIGN AND PRACTICE

CONTENTS
1. OVERVIEW OF PRESSURE VESSELS
2. PRESSURE VESSEL DESIGN PHILOSOPHY
3. STRUCTURAL DESIGN CRITERIA
4. STRESS CATEGORIES AND STRESS LIMITS
5. DESIGN OF CYLINDRICAL SHELLS
6. DESIGN OF HEADS AND COVERS
7. DESIGN OF NOZZLES AND OPENINGS
8. FATIGUE ASSESSMENT OF PRESSURE VESSELS
9. BOLTED FLANGE CONNECTIONS
10. DESIGN OF VESSEL SUPPORTS
11. SIMPLIFIED INELASTIC METHODS IN PRESSURE VESSEL DESIGN
12. CASE STUDIES

Overview of Pressure Vessels

Introduction

Vessels, tanks, and pipelines that carry, store, or receive fluids are called pressure vessels. A pressure vessel is defined as a container with a pressure differential between inside and outside. The inside pressure is usually higher than the outside, except for some isolated situations. The fluid inside the vessel may undergo a change in state as in the case of steam boilers, or may combine with other reagents as in the case of a chemical reactor. Pressure vessels often have a combination of high pressures together with high temperatures, and in some cases flammable fluids or highly radioactive materials. Because of such hazards it is imperative that the design be such that no leakage can occur. In addition these vessels have to be designed carefully to cope with the operating temperature and pressure. It should be borne in mind that the rupture of a pressure vessel has a potential to cause extensive physical injury and property damage. Plant safety and integrity are of fundamental concern in pressure vessel design and these of course depend on the adequacy of design codes. When discussing pressure vessels we must also consider tanks. Pressure vessels and tanks are significantly different in both design and construction: tanks, unlike pressure vessels, are limited to atmospheric pressure; and pressure vessels often have internals while most tanks do not (and those that do are limited to heating coils or mixers).

Pressure vessels are used in a number of industries; for example, the power generation industry for fossil and nuclear power, the petrochemical industry for storing and processing crude petroleum oil in tank farms as well as storing gasoline in service stations, and the chemical industry (in chemical reactors) to name but a few. Their use has expanded throughout the world. Pressure vessels and tanks are, in fact, essential to the chemical, petroleum, petrochemical and nuclear industries. It is in this class of equipment that the reactions, separations, and storage of raw materials occur. Generally speaking, pressurized equipment is required for a wide  range of industrial plant for storage and manufacturing purposes. The size and geometric form of pressure vessels vary greatly from the large cylindrical vessels used for high-pressure gas storage to the small size used as hydraulic units for aircraft. Some are buried in the ground or deep in the ocean, but most are positioned on ground or supported in platforms. Pressure vessels are usually spherical or cylindrical, with domed ends. The cylindrical vessels are generally preferred, since they present simpler manufacturing problems and make better use of the available space. Boiler drums, heat exchangers, chemical reactors, and so on, are generally cylindrical. Spherical vessels have the advantage of requiring thinner walls for a given pressure and diameter than the equivalent cylinder. Therefore they are used for large gas or liquid containers, gas-cooled nuclear reactors, containment buildings for nuclear plant, and so on. Containment vessels for liquids at very low pressures are sometimes in the form of lobed spheroids or in the shape of a drop. This has the advantage of providing the best possible stress distribution when the tank is full.

Pressure Vessels Design Philosophy

General overview

Engineering design is an activity to ensure fitness for service. Within the context of pressure vessel design, this primarily involves strength considerations. The ‘‘total design’’ is a topic with far-reaching ramifications. It might include aspects of fuel system design, reactor design, or thermal hydraulic design. In our subsequent discussions, the underlying philosophy, decisions and calculations related solely to the strength design are referred to the ‘‘pressure vessel design.’’ For certain pressure vessels and related equipment, preliminary design may still be governed by heat transfer and fluid flow requirements. Although the aspect of thermal hydraulic design is intricately related to the structural design, especially for thermal transient loadings, we will not be discussing them in any detail. It will be assumed that the temperature distribution associated with a particular thermal transient has already been evaluated in a typical design application. However, in these cases the designer still has to consider how the desired configurations of the vessel are to be designed from a structural standpoint and how these designs will perform their intended service. The role of engineering mechanics in the pressure vessel design process is to provide descriptions of the pressure vessel parts and materials in terms of mathematical models, which can be analyzed in closed form in a limited number of situations and mostly have to be solved numerically. Even the so-called simple models that can be solved in closed form might involve fairly complex mathematics. In a few isolated instances, intelligent applications of well-known principles have led to simplifying concepts. These concepts have generally eased the designer’s task. However, in a majority of cases, especially when advanced materials and alloys are at a premium, there is a need to make the optimum use of the materials necessitating application of advanced structural analysis. As the complexity of the analysis increases, the aspect of interpretation of the results of the analysis becomes increasingly extensive. Furthermore, a large number of these models approximate the material behavior along with the extent of yielding. As we understand material behavior more and more, the uncertainties and omitted factors in design become more apparent. The improvement will continue as knowledge and cognizance of influencing design and material parameters increase and are put to engineering and economic use.

The safety demands within the nuclear industry have accelerated studies on pressure vessel material behavior and advanced the state of the art of stress analysis. For instance, the nuclear reactor, with its extremely large heavy section cover flanges and nozzle reinforcement operating under severe thermal transients in a neutron irradiation environment, has focused considerable attention on research in this area which has been directly responsible for improved materials, knowledge of their behavior in specific environments, and new stress analysis methods. High-strength materials created by alloying elements, manufacturing processes, or heat treatments, are developed to satisfy economic or engineering demands such as reduced vessel thickness. They are continually being tested to establish design limits consistent with their higher strength and adapted to vessel design as experimental and fabrication knowledge justifies their use. There is no one perfect material for pressure

In assessing the structural integrity of the pressure vessel and associated equipment, an elastic analysis, an inelastic analysis (elastic–plastic or plastic) or a limit analysis may be invoked. The design philosophy then is to determine the stresses for the purpose of identifying the stress concentration, the proximity to the yield strength, or to determine the shakedown limit load. The stress concentration effects are then employed for detailed fatigue evaluation to assess structural integrity under cyclic loading. In some situations a crack growth analysis may be warranted, while in other situations, stability or buckling issues may be critical. For demonstrating adequacy for cyclic operation, the specific cycles and the associated loadings must be known a priority. In this context, it is important for a pressure vessel designer to understand the nature of loading and the structural response to the loading. This generally decides what type of analysis needs to be
performed, as well as what would be the magnitude of the allowable stresses or strains. Generally the loads acting on a structure can be classified as sustained, deformation controlled, or thermal. These three load types may be applied in a steady or a cyclic manner. The structure under the action of these loads may respond in a number of ways

• When the response is elastic, the structure is safe from collapse when the applied loading is steady. When the load is applied cyclically a failure due to fatigue is likely; this is termed failure due to high cycle fatigue.
• When the response is elastic in some regions of the structure and plastic in others, there is the potential to have an unacceptably large deformation produced by both sustained and deformation-controlled loads. Cyclic loads or cyclic temperature distributions can produce plastic deformations that alternate in tension and compression
• and cause fatigue failure, termed low cycle fatigue. Such distribution of loads could be of such a magnitude that it produces plastic deformations in some regions when initially applied, but upon removal these deformations become elastic, and subsequent loading results in predominantly elastic action. This is termed shakedown. Under cyclic loading fatigue failure is likely and because of elastic action, this would be termed as low cycle fatigue.
• When the sustained loading (due to bending or tension) is such that the entire cross-section becomes plastic, gross collapse of the structure takes place.
• Ratcheting is produced by a combination of a sustained extensional load and either a strain-controlled cyclic load or a cyclic temperature distribution that is alternately applied and removed. This produces cycling straining of the material which in turn produces incremental growth (cyclic) leading to what is called an incremental collapse. This can also lead to low cycle fatigue.
• Sustained loads in brittle materials or in ductile materials at low temperatures could result in brittle fracture, which is a form of structural collapse.

Structural Design Criteria

Modes of failure

Two basic modes of failure are assumed for the design of pressure vessels. These are: (a) elastic failure, governed by the theory of elasticity; and (b) plastic failure, governed by the theory of plasticity. Except for thick-walled pressure vessels, elastic failure is assumed. When the material is stretched beyond the elastic limit, excessive plastic deformation or rupture is expected. The relevant material properties are the yield strength and ultimate strength. In real vessels we have a multiaxial stress situation, where the failure is not governed by the individual components of stress but by some combination of all stress components.

Theories of failure

The most commonly used theories of failure are:

Maximum principal stress theory
Maximum shear stress theory

Fatigue assessment of pressure vessels