DESIGN OF TURBINE BLADE USING CATIA.

Abstract

Cooling of gas turbine blades is a major consideration because they are subjected to high

temperature working conditions. Several methods have been suggested for the cooling of

blades and one such technique is to have radial holes to pass high velocity cooling air

along the blade span. The forced convection heat transfer from the blade to the cooling

air will reduce the temperature of the blade to allowable limits.

One of the major challenges in this new century is the efficient use of energy resources as well as

the production of energy from renewable sources. Undoubtedly, researchers from around the

world have shown that global warming has been caused in part by the greenhouse effect which is

largely due to the use of fossil fuels for transportation and electricity. There are several

alternative forms of energy that have already been explored and developed such as geothermal,

solar, wind and hydroelectric power. Moreover, the advancement in renewable energy

technologies has been possible thanks to the vast amount of research performed by scientists and

engineers in order to make them more efficient and most importantly, more affordable. The

affordability and performance of renewable energy technologies is the key to ensure the

availability to the mass market

TURBINE

A turbine, from the Greek τύρβη, tyrbē, ("turbulance"), is a rotary mechanical device that

extracts energy from a fluid flow and converts it into useful work. A turbine is

a turbomachine with at least one moving part called a rotor assembly, which is a shaft or drum

with blades attached. Moving fluid acts on the blades so that they move and impart rotational

energy to the rotor. Early turbine examples are windmills and waterwheels.

Gas, steam, and water turbines usually have a casing around the blades that contains and controls

the working fluid. Credit for invention of the steam turbine is given both to the British

engineer Sir Charles Parsons (1854–1931), for invention of the reaction turbine and to Swedish

engineer Gustaf de Laval (1845–1913), for invention of the impulse turbine. Modern steam

turbines frequently employ both reaction and impulse in the same unit, typically varying

the degree of reaction and impulse from the blade root to its periphery.

The word "turbine" was coined in 1822 by the French mining engineer Claude Burdin from

the Latin turbo, or vortex, in a memoir, "Des turbines hydrauliques ou machines rotatoires à

grande vitesse", which he submitted to the Académie royale des sciences in Paris.[3]Benoit

Fourneyron, a former student of Claude Burdin, built the first practical water turbine.

OPERATION THEORY :

A working fluid contains potential energy (pressure head) and kinetic energy (velocity head).

The fluid may be compressible orincompressible. Several physical principles are employed by

turbines to collect this energy:

Impulse turbines change the direction of flow of a high velocity fluid or gas jet. The resulting

impulse spins the turbine and leaves the fluid flow with diminished kinetic energy. There is no

pressure change of the fluid or gas in the turbine blades (the moving blades), as in the case of a

steam or gas turbine, all the pressure drop takes place in the stationary blades (the nozzles).

Before reaching the turbine, the fluid's pressure head is changed to velocity head by accelerating

the fluid with a nozzle. Pelton wheelsand de Laval turbines use this process exclusively. Impulse

turbines do not require a pressure casement around the rotor since the fluid jet is created by the

nozzle prior to reaching the blading on the rotor. Newton's second law describes the transfer of

energy for impulse turbines.

Reaction turbines develop torque by reacting to the gas or fluid's pressure or mass. The pressure

of the gas or fluid changes as it passes through the turbine rotor blades. A pressure casement is

needed to contain the working fluid as it acts on the turbine stage(s) or the turbine must be fully

immersed in the fluid flow (such as with wind turbines). The casing contains and directs the

working fluid and, for water turbines, maintains the suction imparted by the draft tube. Francis

turbines and most steam turbinesuse this concept. For compressible working fluids, multiple

turbine stages are usually used to harness the expanding gas efficiently. Newton's third

law describes the transfer of energy for reaction turbines.

FIGURE : IMPULSE AND REACTION TURBINE

TYPES OF TURBINES :

Steam turbines are used for the generation of electricity in thermal power plants, such as

plants using coal, fuel oil or nuclear power. They were once used to directly drive

mechanical devices such as ships' propellers (for example the Turbinia, the first turbine-

powered steam launch) but most such applications now use reduction gears or an

intermediate electrical step, where the turbine is used to generate electricity, which then

powers an electric motor connected to the mechanical load. Turbo electric ship machinery

was particularly popular in the period immediately before and during World War II,

primarily due to a lack of sufficient gear-cutting facilities in US and UK shipyards.

Gas turbines are sometimes referred to as turbine engines. Such engines usually feature an

inlet, fan, compressor, combustor and nozzle (possibly other assemblies) in addition to one

or more turbines.

Water turbines

Pelton turbine, a type of impulse water turbine.

Francis turbine, a type of widely used water turbine.

Kaplan turbine, a variation of the Francis Turbine.

Turgo turbine, a modified form of the Pelton wheel.

Cross-flow turbine, also known as Banki-Michell turbine, or Ossberger turbine.

Wind turbine. These normally operate as a single stage without nozzle and interstage guide

vanes. An exception is the Éolienne Bollée, which has a stator and a rotor.

INLETCASING

COMPRCASING

CDC

TURBINECASE EXHAUST

FRAME

INTRODUCTION TO GAS TURBINE

Gas turbines in simple cycle mode

The gas turbine is the most versatile item of turbomachinery today.It can be used in several

different modes in critical industries such as power generation, oil and gas ,process plants,

aviation,as well domestic and smaller related industries.A gas turbine essentially brings together

air that it compresses in its compressor module ,and fuel,that are then ignited. Resulting gases are

expanded through a turbine. That turbine’s shaft continues to rotate and drive the compressor

which is on the same shaft, and operation continues. A separator starter unit is used to provide

the first rotor motion, until the turbine’s rotation is up to design speed and can keep the entire

unit running. The compressor module, combustor module and turbine module connected by one

or more shafts are collectively called the gas generator. The figure below illustrate a typical gas

turbine sectionalview.

A. General

A single-shaft gas turbine , is mounted on a platform or base which supports the basic gas turbine unit. The various assemblies , systems and components that comprise the compressor, combustion and turbine sections of the gas turbine are described in the text which follows.

B. Detail Orientation

By definition ,the air inlet of the gas turbine is the forward end ,while the exhaust is the aft end. The forward and aft ends of each component are determined in like manner wuth respect to its orientation within the complete unit.The right and left sides of the turbine or of a particular component are determined by standing forward and looking aft.

Fig. the basic gas turbine cycle [ Brayton cycle]

1. COMPRESSOR SECTION

A. General

The axial-flow compressor section consists of the compressor rotor and the compressor casing .Within the compressor casing are the variable inlet guide vanes, the various stages of rotor and stator blading , and the exit guide vanes.

In the compressor , air is confined to the space between the rotor and stator where it is compressed in stages by a series of alternate rotating(rotor) and stationary(stator) airfoil-shaped blades.

The rotor blades supply the force needed to compress the air in each stage and the stator blades guide the air so that it enters the following rotor stage at the proper angle. The compressed air exits through the compressor dischargs casing to the combustion chambers.

B. Rotor

The compressor portion of the gas turbine rotor is an assembly ofwheels, a speed ring, ties bolts, the compressor rotor blades ,and a forward stub shaft.Each wheel hasslots broached around its periphery.

Fig. compressor rotor

The rotor blades and spacers are inserted into these slots and held in axial position by staking at each end of the slot.The wheels are assembled to each other with mating rabbets for

concentricity control and are held together with tie bolts.Selective positioning of the wheels is made during assembly to reduce balance correction.

After assembly, the rotor is dynamically balanced. The forward stub shaft is machined to provide the thrust collar which carries the forward and aft thrust loads. The stub shaft also provides the journal for the No.1 bearing, the sealing surface for the No.1bearing oil seals and the compressor low-pressure air seal. The stage 17 wheel carries the rotor blades and also providesthe sealing surface for the high-pressure air seal and the compressor-to-turbine marriage flange.

C. Stator

1. General

The casing area of the compressor section is composed of three major sections . These are :

a. Inlet casingb. Compressor casingc. Compressor discharge casing

a. Inlet casing

The inlet casing is located at the forward end of the gas turbine .Its prime function is to uniformly direct air into the compressor. The inlet casing also supports the #1 bearing assembly. The #1 bearing lower half housing is integrally cast with the inner bell mouth. The upper half bearing housing is a separate casting, flanged and bolted to the lower half. The inner bell mouth is positioned to the outer bell mouth by nine airfoil-shaped radial struts.

b. Compressor casing

The forward compressor casing contains the stage 0 through stage 4 compressor stator stages . The compressor casing lower half is equipped with two large integrally cast trunnions which are used to lift the gas turbine when it is separated from its base. The aft compressor casing contains stage 5 through stage 12 compressor stator stages. Extraction ports in aft casing permit removal of 13th stage compressor air. This air is used for cooling functions and is also used for pulsation control during startup and shutdown.

c. Compressor discharge casing

The compressor discharge casing is the final portion of the compressor section. It is the longest single casing, situated at mid point—between the forward and aft supports—and is, in effect, the keystone of the gas turbine structure.

2. Multi Nozzle Quiet Combustor Combustion systemA. General

The combustion system is of the reverse flow type with the 6 combustion chambers arranged around the periphery of the compressor discharge casing as shown on figure1.combustion chambers are numbered counterclockwise when viewed looking downstream and starting from the top of machine.This system also includes the fuel nozzles ,a spark plug ignition system,flame detectors and cross fire tubes.Hot gases, generated from burning fuel in the combustion chambers, flow through the impingement cooled transition pieces to the turbine.

Figure : multi nozzle combustion system arrangement

B. Outer Combustion Chambers and Flow Sleeves

The outer combustion casings act as the pressure shells for the combustors. They also provide flanges for the fuel nozzle-end cover assemblies ,crossfire tube flanges and where called for, spark plugs, flame detectors and false start drains. The flow sleeves form an annular space around the cap and linear assemblies that directs the combustion and cooling air flows into the reaction region.

C. Crossfire Tubes

All combustion chambers are interconnected by means of crossfire tubes. The outer chambers are connected with an outer crossfire tube and the combustion linear primary zones are

connected by the inner crossfire tubes.

Figure 2: crossfire tube

D. Fuel Nozzle End Covers

The multi nozzle combustor utilizes six fuel nozzles in each combustion end cover in conjunction with provisions for water injection. Typical fuel nozzle and linear arrangements for the end cover equipped with water injection. On the multi nozzle combustor, the fuel nozzle is functionally integrated with the combustor end cover as shown.

E. Cap and Liner Asssemblies

The combustion liners use conventional cooling slots. The cap has six floating collars to engage each of the six fuel nozzle tips. It is cooled by a combination of film cooling and impingement cooling and has thermal barrier coating on the inner surfaces.

Figure 3: cap and liner assemblies

F. Spark Plugs

Combustion is intiated by means of the discharge from two retractable spark plugs which are bolted to flanges on the combustion casings and centered within the liner and flow sleeve in adjacent combustion chambers. A typical spark plug arrangement is shown in figure 4. These plugs receive their energy from high energy-capacitor discharge power supplies. At the time of firing, a spark at one or both of these plugs ignites the gases in a chambe ; the remaining chambers are ignited by crossfire through the tubes that interconnect the reaction zone of the remaining chambers.

Figure 4: Multi combustor gas turbine

G. Ultraviolet Flame Detectors

During the starting sequence , it is the essential that an indication of the presence or absence of flame be transmitted to the control system. For this reason, a flame monitoring system is used consisting of multiple flame monitoring system is used consisting of multiple flame detectors located as shown on figure 4. The gas within this detector is sensitive to the presence of ultraviolet radiation which is emitted by a flame.

A DC voltage, supplied by the amplifier, is impressed across the detector terminals. If flame is present, the ionization of the gas in the detector allows conduction in the circuit which activities the electronics to give an output indicating flame. Conversely, the absence of flame will generate an output indicating flame. The signals from the four flame detectors are sent to the control system which uses an internal logic system to determine whether a flame or loss of flame condition exists.

3. TURBINE SECTION

A. General

The three-stage turbine section is the area in which energy in the form of high temperature pressurized gas, produced by the compressor and combustion sections, is converted to mechanical energy.

Gas turbine hardware includes the turbine rotor, turbine casing, exhaust frame, exhaust diffuser, nozzles, and shrouds.

B. Turbine Rotor

1. Structure

The turbine rotor assembly consists of the forward and aft turbine wheel shafts and the first, second and third stage turbine wheel assemblies with spacers and turbine buckets. Concentricity control is achieved with mating rabbets on the turbine wheels, wheel shafts and spacers. Selective positioning of rotor members is performed to minimize balance corrections.

Figure: Turbine section rotor

2. Wheel shafts

The turbine rotor distance piece extends from the first-stage turbine wheel to the aft flange of the compressor rotor assembly. The turbine rotor aft shaft includes the #2 bearing journal.

3. Wheel Assemblies

Spacers between the first and second between the second and third-stage turbine wheels determine the axial position of the individual wheels. These spacers carry the diaphragm sealing lands. The 1-2 spacer forward and aft faces include radial slots for cooling air passages. Turbine buckets are assembled in the wheels with fir-tree-shaped dovetails that fit into matching cut outs in the turbine wheel rims. All three turbine stages have precision investment-cast, long shank buckets.

4. Cooling

The turbine rotor is cooled to maintain reasonable operating temperatures and, therefore, assure a longer turbine service life. Cooling is accomplished by means of a positive flow of cool air extracted from the compressor and discharged radially outward through a space between the turbine wheel and stator, into the main gas stream.

5. First-stage Wheel Spaces

The first-stage forward wheel space is cooled by compressor discharge air. A honeycomb seal is installed at the aft end of the compressor rotor between the rotor and inner barrel of the compressor discharge casing. The leakage through this seal furnishes the air flow through the first-stage forward wheel space. This cooling air flow discharges into the main gas stream aft of the first-stage nozzle. The first-stage aft wheel space is cooled by 9th stage extraction air ported through the 2nd stage nozzle. This air returns to the gas path forward of the 2nd stage nozzle.

6. Second-Stage Wheel spaces

The second-stage forward wheel space is cooled by leakage from the first-stage aft wheel space through the inter stage labyrinth. This air returns to the gas path at the entrance of the second-stage buckets. The second-stage aft wheel space is cooled by 13th stage extraction air ported through the 3rd stage nozzle. Air from this wheel returns to the gas path at the third-stage entrance.

7. Third-Stage Wheel Spaces

The third-stage forward wheel space is cooled by leakage from the second-stage aft wheel space through the inter stage labyrinth. This air renters the gas path at the third-stage bucket entrance. The third-stage aft wheel space obtains its cooling air from the discharge the third-stage aft wheel space, and into the gas path at the entrance to the exhaust diffuser.

8. Buckets

Air is introduced into each first-stage bucket through a plenum at the base of the bucket dovetail. It flows through serpentine cooling holes extending the length of the bucket and exits at the trailing edge and the bucket tip. The holes are spaced and sized to obtain optimum cooling of the airfoil with minimum compressor extraction air.

C. Turbine Stator

1. Structure

The turbine casing and the exhaust frame constitute the major portion of the gas turbine stator structure. The turbine nozzles, shrouds and turbine exhaust diffuser are internally supported from these components.

2. Turbine Casing

The turbine casing controls the axial and radial positions of the shrouds and nozzles. It determines turbine clearances and the relative positions of the nozzles to the turbine buckets. This positioning is critical to gas turbine performance.

3. Nozzles

In the turbine section there are three stages of stationary nozzles which direct the high-velocity flow of the expanded hot combustion gas against the turbine buckets causing the turbine rotor to rotate. Because of the high pressure drop across these nozzles, there are seals at both the inside and the outside diameters to prevent loss of system energy by leakage.

4. First-stage Nozzle

The first-stage nozzle receives the hot combustion gases from the combustion system via the transition pieces. The transition pieces are sealed to both outer and inner sidewalls on entrance side of the nozzle; this minimizes leakage of compressor discharge air into nozzles.

5. Second-Stage Nozzle

Combustion air exiting from first stage buckets is again expanded and redirected against the second-stage turbine buckets by the second-stage nozzle. This nozzle is made of cast segments, each with two partitions or airfoils. The nozzle segments are held in a circumferential position by radial pins from the shell into axial slots in the nozzle outer side wall.

6. Third-Stage Nozzle

The third-stage nozzle receives the hot gas as it leaves the second-stage buckets, increases its velocity by pressure drop, and directs this flow against the third-stage buckets. The nozzle consists of cast segments, each with three partitions or airfoils. It is held at the outer sidewall forward and aft sides in grooves in the turbine shrouds in a manner similar to that used on the second-stage nozzle.

7. Diaphragm

Bolted to the inside diameters of both the second and third-stage nozzle segments are the nozzle diaphragms. These diaphragms prevent air leakage past the inner sidewall of the nozzles and the turbine rotor. A honeycomb labyrinth seal is brazed into the inside diameter of the diaphragm. They mate with opposing sealing teeth on the turbine rotor.

8. Shrouds

Unlike the compressor blading, the turbine bucket tips do not run directly against an integral machined surface of the casing against annular curved segments called turbine shrouds. The shrouds primary function is to provide a cylindrical surface for minimizing bucket tip clearance leakage and its secondary function is to provide a high thermal resistance between the hot gases and the comparatively cool turbine casing.

9. Exhaust Frame

The exhaust frame is bolted to the aft flange of the turbine casing. Structurally, the frame consists of an outer cylinder and an inner cylinder interconnected by the radial struts. The #2 bearing is supported from the inner cylinder. The exhaust diffuser located at the aft end of the turbine is bolted to the exhaust frame. Gases exhausted from the third turbine stage enter the diffuser where velocity is reduced by diffusion and pressure is recovered. At the exit of the diffuser, the gases are directed into the exhaust plenum.

APPLICATIONS OF GAS TURBINES

Direct drive and mechanical drive

With land-based industries, gas turbines can be used in either direct drive or mechanical drive application. With power generation, the gas turbine shaft is coupled to the generator shaft, either directly or via a gearbox “ direct drive ” application. A gearbox is necessary in applications where the manufacturer offers the package for both 60 and 50 cycle applications. The gear box will use roughly 2 percent of the power developed by the turbine in their cases.

Fig : 100MW simple cycle gas turbine plant

Power generation applications extend to offshore platform use. Minimizing weight is a major consideration for this service and the gas turbines used are generally “ aeroderivatives ”. For mechanical drive applications, the turbine module arrangement is different. In these cases, the combination of compressor module, combustor module and turbine module is termed the gas generator. Beyond the turbine end of the gas generator is a freely rotating turbine. It may be one or more stages. It is not mechanically connected to the gas generator, but instead is mechanically coupled, sometimes via a gearbox, to the equipment it is driving.

Fig : Gas turbine in offshore service Fig : Marine gas turbine for Indian navy

SOLIDWORKS

Solid Works is mechanical design automation software that takes advantage of the familiar

Microsoft Windows graphical user interface.

It is an easy-to-learn tool which makes it possible for mechanical designers to quickly sketch

ideas, experiment with features and dimensions, and produce models and detailed drawings.

A Solid Works model consists of parts, assemblies, and drawings.

Typically, we begin with a sketch, create a base feature, and then add more features to

the model. (One can also begin with an imported surface or solid geometry).

We are free to refine our design by adding, changing, or reordering features.

Associativity between parts, assemblies, and drawings assures that changes made to one

view are automatically made to all other views.

We can generate drawings or assemblies at any time in the design process.

The SolidWorks software lets us customize functionality to suit our needs.

INTRODUCTION TO SOLIDWORKS :

Solidworks mechanical design automation software is a feature-based,parametric solid modeling

design tool which advantage of the easy to learn windowsTM graphical user interface. We can

create fully associate 3-D solid models with or without while utilizing automatic or user defined

relations to capture design intent.

Parameters refer to constraints whose values determine the shape or geometry of the model or

assembly. Parameters can be either numeric parameters, such as line lengths or circle diameters,

or geometric parameters, such as tangent, parallel, concentric, horizontal or vertical, etc.

Numeric parameters can be associated with each other through the use of relations, which allow

them to capture design intent.

Design intent is how the creator of the part wants it to respond to changes and updates. For

example, you would want the hole at the top of a beverage can to stay at the top surface,

regardless of the height or size of the can. Solid Works allows you to specify that the hole is a

feature on the top surface, and will then honour your design intent no matter what the height you

later gave to the can.several factors contribute to how we capture design intent are Automatic

relations,Equations,added relations and dimensioning.

Features refer to the building blocks of the part. They are the shapes and operations that

construct the part. Shape-based features typically begin with a 2D or 3D sketch of shapes such as

bosses, holes, slots, etc. This shape is then extruded or cut to add or remove material from the

part. Operation-based features are not sketch-based, and include features such as fillets,

chamfers, shells, applying draft to the faces of a part, etc.

Building a model in Solid Works usually starts with a 2D sketch (although 3D sketches are

available for power users). The sketch consists of geometry such as points, lines, arcs, conics

(except the hyperbola), and splines. Dimensions are added to the sketch to define the size and

location of the geometry. Relations are used to define attributes such as tangency, parallelism,

perpendicularity, and concentricity. The parametric nature of Solid Works means that the

dimensions and relations drive the geometry, not the other way around. The dimensions in the

sketch can be controlled independently, or by relationships to other parameters inside or outside

of the sketch.

Several ways a part can be builded like

1. Layer-cake approach :

The layer-cake approach builds the part one piece at a time ,adding each layer,orfeature,onto the

previous one.

2. Potter’s wheel approach :

The potter’s wheel approach builds the part as a single revolved feature.As a single sketch

representing the cross section includes all the information and dimensions necessary to make the

part as one feature.

3. Manufacturing approach :

The manufacturing approach to modeling mimics the way the part would be

manufactured.Forexample,if the stepped shaft was turned a lathe ,we would start with a piece of

bar stock and remove material using a series of cuts.

In an assembly, the analogue to sketch relations is mates. Just as sketch relations define

conditions such as tangency, parallelism, and concentricity with respect to sketch geometry,

assembly mates define equivalent relations with respect to the individual parts or components,

allowing the easy construction of assemblies. Solid Works also includes additional advanced

mating features such as gear and cam follower mates, which allow modelled gear assemblies to

accurately reproduce the rotational movement of an actual gear train.

Finally, drawings can be created either from parts or assemblies. Views are automatically

generated from the solid model, and notes, dimensions and tolerances can then be easily added to

the drawing as needed. The drawing module includes most paper sizes and standards.

A Solid Works model consists of parts, assemblies, and drawings.

(1) Part: Individual components are drawn in the form of part drawings.

(2) Assembly: The individual parts are assembled in this region.

(3) Drawings: This contains detailed information of the assembly.

HISTORY OF SOLIDWORKS :

SolidWorks Corporation was founded in December 1993 by Massachusetts Institute of

Technology graduate Jon Hirschtick; Hirschtick used $1 million he had made while a member of

the MIT Blackjack Team to set up the company. Initially based in Waltham, Massachusetts,

USA, Hirschtick recruited a team of engineers with the goal of building 3D CAD software that

was easy-to-use, affordable, and available on the Windows desktop. Operating later from

Concord, Massachusetts, SolidWorks released its first product SolidWorks 95, in 1995. In 1997

Dassault, best known for its CATIA CAD software, acquired SolidWorks for $310 million in

stock.

SolidWorks currently markets several versions of the SolidWorks CAD software in addition to

eDrawings, a collaboration tool, and DraftSight, a 2D CAD product.

SolidWorks was headed by John McEleney from 2001 to July 2007 and Jeff Ray from 2007 to

January 2011. The current CEO is Bertrand Sicot.

Solidworksversions :

Name/Version Version History Value Release DateSolidWorks 95 44 1995SolidWorks 96 243 1996SolidWorks 97 483 1996SolidWorks 97Plus 629 1997SolidWorks 98 817 1997SolidWorks 98Plus 1008 1998SolidWorks 99 1137 1998SolidWorks 2000 1500 1999SolidWorks 2001 1750 2000SolidWorks 2001Plus 1950 2001SolidWorks 2003 2200 2002SolidWorks 2004 2500 2003SolidWorks 2005 2800 2004SolidWorks 2006 3100 2005SolidWorks 2007 3400 2006SolidWorks 2008 3800 July 1, 2007SolidWorks 2009 4100 January 28, 2008SolidWorks 2010 4400 December 9, 2009SolidWorks 2011 4700 June 17, 2010SolidWorks 2012 5000 September, 2011SolidWorks 2013 6000 September, 2012SolidWorks 2014 7000 October, 7, 2013

SOLIDWORKS 3-D MECHANICAL DESIGN APPLICATIONS :

SolidWorks Standard

SolidWorks Professional

SolidWorksPremium :provides a suite of product development tools mechanical design, design

verification, data management, and communication tools. SolidWorks Premium includes all of

the capabilities of SolidWorks Professional as well as routing and analysis tools, including

SolidWorks Routing, SolidWorks Simulation, and SolidWorks Motion.

SolidWorks Education Edition :provides the same design functionality but is configured and

packaged for engineering and industrial design students.

DESIGN VALIDATION TOOLS :

SolidWorks Simulation is a design validation tool that shows engineers how their designs will

behave as physical objects.

SolidWorks Motion is a virtual prototyping tool that provides motion simulation capabilities to

ensure designs function properly.

SolidWorks Flow Simulation is a tool that tests internal and external fluid-flow simulation and

thermal analysis so designers can conduct tests on virtual prototypes.

SolidWorks Simulation Premium is a Finite Element Analysis (FEA) design validation tool

that can handle some multiphysics simulations as well as nonlinear materials.

PRODUCT DATA MANAGEMENT TOOLS :

SolidWorks Workgroup PDM is a PDM tool that allows SolidWorks users operating in teams

of 10 members or less to work on designs concurrently. With SolidWorks PDM Workgroup,

designers can search, revise, and vault CAD data while maintaining an accurate design history.

SolidWorks Enterprise PDM is a PDM tool that allows SolidWorks users operating in teams at

various separate facilities to work on designs concurrently. With SolidWorks Enterprise PDM,

designers can search, revise, and vault CAD data while maintaining an accurate design history.

Enterprise PDM maintains an audit trail, is compatible with a variety of CAE packages

(AutoDesk, Siemens, PTC, Catia, etc.) to maintain interfile relations, and will manage the

revisions of any document saved in the vault. Enterprise PDM also uses a workflow diagram to

automatically notify team members when a project moves from one stage to the next, as well as

tracking comments. Enterprise PDM is capable of interfacing with various MRP/ERP systems

and can be used online to interface with customers and the supply chain.

DESIGN COMMUNICATION AND COLLABORATION TOOLS :

eDrawings Professional :An e-mail-enabled communication tool for reviewing 2D and 3D

product design data across the extended product development team. eDrawings generates

accurate representations of DWGgateway is a free data translation tool that enables any

AutoCAD software user to open and edit any DWG file, regardless of the version of AutoCAD it

was made in.

Mobile eDrawings

SolidWorksViewer : is a free plug-in for viewing SolidWorks parts, assemblies, and drawings.

'3DVIA Composer', now known as 'SolidWorks Composer', is a technical communications

software that allows 3D views of models to be integrated into documents such as work

instructions, internal or external manuals, marketing materials, or web applications. The 3D

views can be updated automatically when the design updates, reducing the workload of the

employee creating the technical document, as editing for changes is not as severe.

CAD PRODUCTIVITY TOOLS :

SolidWorks Toolbox is a library of parts that uses "Smart Part" Technology to automatically

select fasteners and assemble them in the desired sequence.

SolidWorks Utilities is software that lets designers find differences between two versions of the

same part, or locate, modify, and suppress features within a model.

FeatureWorks is feature recognition software that lets designers make changes to static

geometric data, increasing the value of translated files. With FeatureWorks, designers can

preserve or introduce new design intent when bringing 3D models created in other software into

the SolidWorks environment.

SOLIDWORKS INTERFACE :

Figure 15 :solidworks interface

Feature bar :

Figure 16 : feature bar

Some of the features are:

1. Extrude feature :

once the sketch is completed ,it can be extruded to create the first feature . This feature can be a base,boss or cut feature

2. Revolve feature :

The revolve options enablesus to create a feature from an axisymmetricsketch and an axis.This feature can be a base,boss or cut feature..The axis can be centerline ,line,lineredge,axis or temporary axis.

3. Swept feature :

sweep creates feature fro m two sketches : a sweep section and sweep path.The section is moved along the path,creating the feature.

4. Loft feature :

Loft creates a feature by making transitions between profiles. A loft can be a base, boss, cut, or surface. This feature can be a base,boss or cut feature..

5. Boundary feature :

Boundary tools produces very high quality, accurate features useful for creating complex shapes for markets focused on consumer product design, medical, aerospace, and molds.

6. Fillet feature :

Fillet/Round creates a rounded internal or external face on the part. You can fillet all edges of a face, selected sets of faces, selected edges, or edge loops.

7. Pattern feature :

Patterns are the best method for creating multiple instances of one feature or more features.some of types of patterns are Linear,circular,mirror,tabledriven,sketchdriven ,curvedrivenand fill driven.

FEATURE MANAGER DESIGN TREE :

Figure 17 : feature manager design tree

The FeatureManager design tree on the left side of the SolidWorks window provides an outline

view of the active part, assembly, or drawing. This makes it easy to see how the model or

assembly was constructed or to examine the various sheets and views in a drawing.

PROPERTY MANAGER IN FEATURE MANAGER DESIGN TREE :

The Property Manager is a means to set properties and other options for many

SolidWorkscommands.ThePropertyManager appears on the PropertyManager tab in the

panel to the left of the graphics area. It opens when you select entities or commands defined in

the PropertyManager.

MODELLING OF GAS TURBINE BLADE

The orthographic views of gas turine blade is as follows:

Figure : orthographic views of gas turbine blade

Figure : 3D model of gas turbine blade

Figure : four different views of gas turbine blade

Now the gas turbine blade with holes 7 ,8 ,9and 10 has been modelled as shown below.

Figure : gas turbine blade of 7 holes

Figure: gas turbine blade of 8 holes

Figure : blade of 9 holes

Figure : gas turbine blade of 10 holes

Now the thermal analysis is performed on gas turbine blade with different holes and then

structural analysis is carried out.

INTRODUCTION TO SOLIDWORKS SIMULATION :

SolidWorks® Simulation is a design analysis system fully integrated with SolidWorks.

SolidWorks Simulation provides simulation solutions for linear and nonlinear static, frequency,

buckling, thermal, fatigue, pressure vessel, drop test, linear and nonlinear dynamic, and

optimization analyses.

Powered by fast and accurate solvers, SolidWorks Simulation enables you to solve large

problems intuitively while you design. SolidWorks Simulation comes in two bundles:

SolidWorks Simulation Professional and SolidWorks Simulation Premium to satisfy your

analysis needs. SolidWorks Simulation shortens time to market by saving time and effort in

searching for the optimum design.

Figure 46 : simulation example

Benefits of Simulation:

After building your model, you need to make sure that it performs efficiently in the field. In the

absence of analysis tools, this task can only be answered by performing expensive and time-

consuming product development cycles. A product development cycle typically includes the

following steps:

1. Building your model.

2. Building a prototype of the design.

3. Testing the prototype in the field.

4. Evaluating the results of the field tests.

5. Modifying the design based on the field test results.

This process continues until a satisfactory solution is reached. Analysis can help you accomplish

the following tasks:

Reduce cost by simulating the testing of your model on the computer instead of

expensive field tests.

Reduce time to market by reducing the number of product development cycles.

Improve products by quickly testing many concepts and scenarios before making a final

decision, giving you more time to think of new designs.

Basic Concepts of Analysis :

The software uses the Finite Element Method (FEM). FEM is a numerical technique for

analyzing engineering designs. FEM is accepted as the standard analysis method due to its

generality and suitability for computer implementation. FEM divides the model into many small

pieces of simple shapes called elements effectively replacing a complex problem by many simple

problems that need to be solved simultaneously.

CAD model of a part Model subdivided into small pieces (elements)

Elements share common points called nodes. The process of dividing the model into small pieces

is called meshing.

The behavior of each element is well-known under all possible support and load scenarios. The

finite element method uses elements with different shapes.

The response at any point in an element is interpolated from the response at the element nodes.

Each node is fully described by a number of parameters depending on the analysis type and the

element used. For example, the temperature of a node fully describes its response in thermal

analysis. For structural analyses, the response of a node is described, in general, by three

translations and three rotations. These are called degrees of freedom (DOFs). Analysis using

FEM is called Finite Element Analysis (FEA).

A tetrahedral element. Red dots represent nodes. Edges of an element can be curved or straight.

The software formulates the equations governing the behavior of each element taking into

consideration its connectivity to other elements. These equations relate the response to known

material properties, restraints, and loads.

Next, the program organizes the equations into a large set of simultaneous algebraic equations

and solves for the unknowns.

In stress analysis, for example, the solver finds the displacements at each node and then the

program calculates strains and finally stresses.

The software offers the following types of studies:

Study type Study icon    

Static Modal Time History

Frequency Harmonic

Buckling Random Vibration

Thermal Response Spectrum

Design Study Drop Test

Nonlinear Static Fatigue

Nonlinear Dynamic Pressure Vessel Design

Analysis Steps :

The steps needed to perform an analysis depend on the study type. You complete a study by

performing the following steps:

Create a study defining its analysis type and options.

If needed, define parameters of your study. A parameter can be a model dimension,

material property, force value, or any other input.

Define material properties.

Specify restraints and loads.

The program automatically creates a mixed mesh when different geometries (solid, shell,

structural members etc.) exist in the model.

Define component contact and contact sets.

Mesh the model to divide the model into many small pieces called elements. Fatigue and

optimization studies use the meshes in referenced studies.

Run the study.

View results.

Specific capabilities of SolidworksSimulation :

1. Static Analysis :When loads are applied to a body, the body deforms and the effect of loads is

transmitted throughout the body. The external loads induce internal forces and reactions to render the body into a state of equilibrium. Linear Static analysis calculates displacements, strains, stresses, and reaction forces under the effect of applied loads.

2. Thermal Stress Analysis :

Changes in temperature can induce substantial deformations, strains, and stresses. Thermal stress analysis refers to static analysis that includes the effect of temperature.

Perform thermal stress analysis using one of the following options:

Using a uniform rise or drop in temperature for the whole model. Using a temperature profile resulting from a steady state or transient thermal

analysis. Using a temperature profile from Flow Simulation.

3. Frequency analysis :If the design is subjected to dynamic environments, static studies cannot be used to evaluate the response. Frequency studies can help you avoid resonance and design vibration isolation systems. They also form the basis for evaluating the response of linear dynamic systems where the response of a system to a dynamic environment is assumed to be equal to the summation of the contributions of the modes considered in the analysis.

4. Dynamic analysis :

Dynamic analysis include:

Design structural and mechanical systems to perform without failure in dynamic environments.

Modify system's characteristics (i.e., geometry, damping mechanisms, material properties, etc.) to reduce vibration effects.

5. Buckling analysis :Used to calculate the buckling loads and determine the buckling mode shape. Both linear (Eigen value) buckling and nonlinear buckling analyses are possible.

6. Non-linear static analysis :All real structures behave nonlinearly in one way or another at some level of loading. In some cases, linear analysis may be adequate. In many other cases, the linear solution can produce erroneous results because the assumptions upon which it is based are violated. Nonlinearity can be caused by the material behavior, large displacements, and contact conditions. We can use a nonlinear study to solve a linear problem. The results can be slightly different due to different procedures.In the nonlinear static analysis, dynamic effects like inertial and damping forces are not considered.

7. Drop test studies :Drop test studies evaluate the effect of the impact of a part or an assembly with a rigid or flexible planar surface. Dropping an object on the floor is a typical application and hence the name. The program calculates impact and gravity loads automatically. No other loads or restraints are allowed.

8. Fatigue Analysis :Fatigue is the prime cause of the failure of many objects, especially those made of metals. Examples of failure due to fatigue include, rotating machinery, bolts, airplane wings, consumer products, offshore platforms, ships, vehicle axles, bridges, and bones. Linear and nonlinear structural studies do not predict failure due to fatigue. They calculate the response of a design subjected to a specified environment of restraints and loads. If the analysis assumptions are observed and the calculated stresses are within the allowable limits, they conclude that the design is safe in this environment regardless of how many times the load is applied.

Results of static, nonlinear, or time history linear dynamic studies can be used as the basis for defining a fatigue study. The number of cycles required for fatigue failure to occur at a location depends on the material and the stress fluctuations. This information, for a certain material, is provided by a curve called the SN curve.

9. Pressure vessel Design study :In a Pressure Vessel Design study, you combine the results of static studies with the desired factors. Each static study has a different set of loads that produce corresponding results. These loads can be dead loads, live loads (approximated by static loads), thermal

loads, seismic loads, and so on. The Pressure Vessel Design study combines the results of the static studies algebraically using a linear combination or the square root of the sum of the squares (SRSS).

THERMAL ANALYSIS OF GAS TURBINE BLADE

Firstly thermal analysis is carried out on gas turbine blade of 7 holes.

Study PropertiesStudy name Thermal 1Analysis type Thermal(Steady state)Mesh type Solid MeshSolver type FFEPlusSolution type Steady stateContact resistance defined? NoResult folder SolidWorks document

UnitsUnit system: SI (MKS)Length/Displacement mmTemperature KelvinAngular velocity Rad/secPressure/Stress N/m^2

Material PropertiesModel Reference Properties Components

Name: Commercially Pure CP-Ti UNS R50400 (SS)

Model type: Linear Elastic Isotropic

Default failure criterion:

Max von Mises Stress

Thermal conductivity: 16.4 W/(m.K)Mass density: 4510 kg/m^3

SolidBody

Thermal Loads

Load name Load Image Load Details

Temperature-1

Entities: 1 face(s)Temperature: 800 Kelvin

Temperature-2

Entities: 2 face(s)Temperature: 300 Kelvin

800k temperature is applied at the leading edge of turbine blade and 300k is applied at the

cooling inlet passages face as shown above.

Mesh information

Mesh type Solid MeshMesher Used: Standard meshAutomatic Transition: OffInclude Mesh Auto Loops: OffJacobian points 4 PointsElement Size 6 mmTolerance 0.42672 mmMesh Quality High

Figure: meshed model of gas turbine blade with 7 holes

Study Results

Name Type Min MaxThermal TEMP: Temperature 300 Kelvin

Node: 1800 KelvinNode: 792

Figure : temperature ditribution in gas turbine blade 0f 7 holes

Figure : temperature values at the trailing edge

The avaerge value of temperature at the trailing edge is 516.11K.

THERMAL ANALYSIS OF GAS TURBINE BLADE 0F 8 HOLES

The same procedure is carried out for gas turbine blade with 8 holes.the temperature distribution

result is as follows.

Study Results

Name Type Min MaxThermal TEMP: Temperature 300 Kelvin

Node: 1800 KelvinNode: 1225

Figure : temperature distribution of gas turbine blade with 8 holes

Figure : temperature values at the trailing edge

The avaerge value of temperature at the trailing edge is 515K.

THERMAL ANALYSIS OF GAS TURBINE BLADE 0F 9 HOLES

The same procedure is carried out for gas turbine blade with 9 holes.the temperature distribution

result is as follows.

Study Results

Name Type Min MaxThermal TEMP: Temperature 300 Kelvin

Node: 1800 KelvinNode: 1378

Figure : temperature distribution of gas turbine blade with 9 holes

Figure : temperature values at the trailing edge

The avaerge value of temperature at the trailing edge is 514K.

THERMAL ANALYSIS OF GAS TURBINE BLADE 0F 10 HOLES

The same procedure is carried out for gas turbine blade with 10 holes. The temperature

distribution result is as follows.

Study Results

Name Type Min MaxThermal TEMP: Temperature 300 Kelvin

Node: 1800 KelvinNode: 1531

Figure : temperature distribution of gas turbine blade with 10 holes

Figure : temperature values at the trailing edge

The avaerge value of temperature at the trailing edge is 513K.

STRUCTURAL ANALYSIS OF GAS TURBINE BLADE OF 7 HOLES

Study PropertiesStudy name Static 1Analysis type StaticMesh type Solid MeshThermal Effect: OnThermal option Include temperature loadsZero strain temperature 298 KelvinInclude fluid pressure effects from SolidWorks Flow Simulation

Off

Solver type FFEPlusInplane Effect: OffSoft Spring: OffInertial Relief: OffIncompatible bonding options AutomaticLarge displacement OffCompute free body forces OnFriction OffUse Adaptive Method: OffResult folder SolidWorks document

Material PropertiesModel Reference Properties Components

Name: Commercially Pure CP-Ti UNS R50400 (SS)

Model type: Linear Elastic Isotropic

Default failure criterion:

Max von Mises Stress

Yield strength: 3.7e+008 N/m^2Tensile strength: 3.44e+008 N/m^2Elastic modulus: 1.05e+011 N/m^2Poisson's ratio: 0.37

Mass density: 4510 kg/m^3Shear modulus: 4.5e+010 N/m^2

Thermal expansion coefficient:

9e-006 /Kelvin

SolidBody

Loads and FixturesFixture name Fixture Image Fixture Details

Fixed-1

Entities: 2 face(s)Type: Fixed Geometry

Load name Load Image Load Details

Force-1

Entities: 1 face(s)Type: Apply normal force

Value: 1000 NPhase Angle: 0

Units: deg

Mesh InformationMesh type Solid MeshMesher Used: Standard meshAutomatic Transition: OffInclude Mesh Auto Loops: OffJacobian points 4 PointsElement Size 6 mmTolerance 0.3 mmMesh Quality High

Figure : meshed model 0f 7 holes

Study Results

Name Type Min MaxStress1 VON: von Mises Stress 184.996 N/m^2

Node: 14821.56189e+007 N/m^2Node: 38512

Figure : von misses stress developed in blade of 7 holes The maximum stresses developed is 15Mpa which is less than the materials yield strength of 370Mpa.

Name Type Min MaxDisplacement URES: Resultant Displacement 0 mm

Node: 17020.0321405 mmNode: 2383

Name Type Min MaxStrain1 ESTRN: Equivalent Strain 2.14166e-009

Element: 178260.00013509 Element: 20078

Name Type Min Max

Name Type Min Max

A factor of safety less than 1 at a location indicates that the material at that location has failed.A factor of safety of 1 at a location indicates that the material at that location has just started to fail. A factor of safety greater than 1 at a location indicates that the material at that location is safe.

Hence our blade has no areas less than 1 of factor of safety. So our design is safe.

Name Type Min MaxFactor of Safety Automatic 23.6893

Node: 385122.00004e+006 Node: 1482

STRUCTURAL ANALYSIS OF GAS TURBINE BLADE OF 8 HOLES

Now the same procedure of structural analysis is carried out on gas turbine blade of 8 holes .

Study Results

Name Type Min MaxStress VON: von Mises Stress 185.872 N/m^2

Node: 396221.96286e+007 N/m^2Node: 50890

Figure : von misses stress developed in blade of 8 holes The maximum stresses developed is 19.6Mpa which is less than the materials yield strength of 370Mpa.

Name Type Min MaxDisplacement URES: Resultant Displacement 0 mm

Node: 4600.0322284 mmNode: 2541

Name Type Min MaxStrain ESTRN: Equivalent Strain 1.83521e-009

Element: 176570.000137517 Element: 18314

Name Type Min Max

Name Type Min MaxFactor of Safety Automatic 18.85

Node: 508901.99061e+006 Node: 39622

Name Type Min Max

A factor of safety less than 1 at a location indicates that the material at that location has failed.A factor of safety of 1 at a location indicates that the material at that location has just started to fail. A factor of safety greater than 1 at a location indicates that the material at that location is safe.

Hence our blade has no areas less than 1 of factor of safety. So our design is safe.

STRUCTURAL ANALYSIS OF GAS TURBINE BLADE OF 9 HOLES

Now the same procedure of structural analysis is carried out on gas turbine blade of 9 holes

Study Results

Name Type Min MaxStress1 VON: von Mises Stress 147.585 N/m^2

Node: 390871.77925e+007 N/m^2Node: 38560

Figure : von misses stress developed in blade of 9 holes The maximum stresses developed is 17.7Mpa which is less than the materials yield strength of 370Mpa.

Name Type Min MaxDisplacement URES: Resultant Displacement 0 mm

Node: 9190.0322555 mmNode: 2702

Name Type Min MaxStrain1 ESTRN: Equivalent Strain 1.83181e-009

Element: 189280.000145974 Element: 19623

Name Type Min MaxFactor of Safety Automatic 20.7953

Node: 385602.50703e+006 Node: 39087

A factor of safety less than 1 at a location indicates that the material at that location has failed.A factor of safety of 1 at a location indicates that the material at that location has just started to fail. A factor of safety greater than 1 at a location indicates that the material at that location is safe.

Hence our blade has no areas less than 1 of factor of safety. So our design is safe.

STRUCTURAL ANALYSIS OF GAS TURBINE BLADE OF 10 HOLES

Now the same procedure of structural analysis is carried out on gas turbine blade of 10 holes

Study Results

Name Type Min MaxStress VON: von Mises Stress 0.000175636 N/mm^2

(MPa)Node: 2134

28.6292 N/mm^2 (MPa)Node: 51675

Name Type Min MaxDisplacement1 URES: Resultant Displacement 0 mm

Node: 6130.032347 mmNode: 2866

Name Type Min MaxStrain1 ESTRN: Equivalent Strain 1.882e-009

Element: 201490.000140078 Element: 3938

RESULTS AND DISCUSSIONS

In thermal analysis of gas turbine blades with holes 7,8,9and 10 the average temperature

distribution at the trailing edge are as follows :

Blade with holes Average temperature at the trailing edge

7 holes blade 516.1k

8 holes blade 515.12k

9 holes blade 514.1k

10 holes blade 513.2k

Table : comparision of average temperatures at the trailing edge of blades with different holes

From the above results it is observed that the temperature distribution increases as the number of

holes increases.The blade with 10 holes has more temperature distribution than holes with 7,8

and 9 .

Similarly in structural analysis of blades is tabulated as below.

Blade with holes Von misses stresses [Mpa]

7 holes 15.6

8 holes 19.6

9 holes 17.7

10 holes 28.6

Table : comparision of von misses stresses developed in blades

From the above results it is observed that the stresses developed in blade is max. in blade with 10

holes.i.e,28.6Mpa.Therfore the blade with 9 holes indicating that it has optimum performance for

prescribed loading conditions.

CONCLUSIONS

In this project using finite element analysis as a tool ,the thermal and structural analysis is carried

out sequentially. The blade with different no. of holes 7, 8, 9and 10 were used for analysis.

The gas turbine blade is modeled in a 3D cad tool called Solidworks 2014 by using

extrude feature.

Then gas turbine blade with different holes such as 7,8,9 and 10 has been modeled on the

blade span.

The blade with different no. of holes 7, 8, 9and 10 were used for thermal analysis in

solidworks simulation tool. It is observed that as the no. of holes increases the

temperature distribution increase.

The structural analysis is carried out after the thermal analysis in SOLID WORKS

SIMULATION TOOL. It is observed that blade with 10 holes has showing more

stresses than the remaining blades.

Finally the blade with 9 holes has giving optimum performance for prescribed loading

conditions with average temperature of 514.1K at the trailing edge and von misses

stresses as 17.7 Mpa.

REFERENCES

[1] Gowreesh, S., Sreenivasalu Reddy, N. and Yogananda Murthy, NV. 2009. Convective Heat

Transfer Analysis of a Aero Gas Turbine Blade Using Ansys, International Journal of Mechanics

and Solids. 4: 39-46.

[2] Facchini, B. and Stecco. S.S. 1999. Cooled expansion in gas turbines: a comparison of

analysis methods, Energy Conversion and Management. 40: 1207-1224.

[3] Mohammad, H., Albeirutty., Abdullah, S., Alghamdi., Yousef, S. Najjar. 2004. Heat transfer

analysis for a multistage gas turbine using different blade-cooling schemes, Applied Thermal

Engineering. 24: 563-577.

[4] Mahfoud, K. and George, B. 1997. Computational study of turbine blade cooling by slot-

injection of a gas, Applied Thermal Engineering. 17: 1141-1149.

[5] Moyroud, F., Fransson, T. and Jacquet-Richardet, G. 2002. A comparison of two finite

element reduction techniques for mistuned bladed-disks, Journal of Engineering for Gas

Turbines and Power. 124: 942-953.

[6] Giovanni, C., Ambra, G., Lorenzo, B. and Roberto, F. 2007 Advances in effusive cooling

techniques of gas turbines, Applied Thermal Engineering. 27: 692-698.

[7] Cun-liang, L., Hui-ren, Z., Jiang-tao, B. and Du-chun, X. 2010. Film cooling performance of

converging slot-hole rows on a gas turbine blade, International Journal of Heat and Mass

Transfer. 53: 5232-5241.

[8] Zhang, JJ., Esat, II. and Shi, YH. 1999. Load Analysis with Varying Mesh Stiffness,

Computers and Structures. 70: 273-280.

[9] Hildebrabd, FB. 1997. Introduction to Numerical Analysis, McGraw-Hill, New York.

[10] Moussavi Torshizi, SE., Yadavar Nikravesh, SM. and Jahangiri, A. 2009. Failure analysis of

gas turbine generator cooling fan blades, Engineering Failure Analysis. 16: 1686-1695.

[11] Cleeton, JPE., Kavanagh, RM. and Parks, GT. 2009. Blade cooling optimisation in humid-

air and steam-injected gas turbines, Applied Thermal Engineering. 29: 3274-3283.

[12] Krishnamoorthy, C. 1994. Finite Element Analysis Theory and Programming,Tata

McGraw-Hill, New Delhi.

[13] Martin, HC. and Carey, GF. 2006. Introduction to the Finite Element Analysis,McGraw Hill

Publishing Co Ltd, New Delhi.

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