Analysis of stress and strain in three dimensions




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MCE/SE/101 THEORY OF ELASTICITY


Internal examination : 2 hours University examination : 3 hours


  1. Analysis of stress and strain in three dimensions

Stress at a point – components of stress; Principal stresses; Stress ellipsoid and stress director surface; Determination of principal stresses; Stress invariants; Determination of maximum shear stresses; Octahedral shear stress; strain at a point – Components of strain; Differential equations of equilibrium ; Conditions of compatibility; Generalised Hooke’s law


  1. Strain energy methods

Total strain energy; Principle of virtual work; Griffith’s theory of rupture; Castigliano’s theorem; Principle of least work (Stationary potential energy)


  1. Two-dimensional problems in rectangular coordinates

Plane stress ; Plane strain; Differential equations of equilibrium; Boundary conditions; Compatibility equations; Stress function; Governing differential equation; Solution by polynomials; End effects – Saint-Venant’s Principle; Determination of displacements; Bending of a cantilever loaded at the end; Bending of a beam by uniform load



  1. Two-dimensional problems in polar coordinates

General equations in polar coordinates; Stress distribution symmetrical about an axis; Effect of circular holes on stress distribution in plates; Concentrated force at a point of a straight boundary; Concentrated force acting on a beam; Stresses in a circular disc


  1. Torsion

Torsion of straight bars – Saint Venant’s theory; Elliptic cross section; Membrane analogy; Torsion of a bar of narrow rectangular cross-section; Torsion of rolled profile sections; Torsion of thin tubes


Text Books

  1. Theory of elasticity by S.P.Timoshenko & J.N.Goodier, McGraw-Hill,1970.

  2. Mechanics of deformable bodies by I.H.Shames , Prentice-Hall of India, 1965.

  3. Introduction to Solid Mechanics by I.H.Shames & Pitarresi , Prentice-Hall of India, 2003.

  4. Engineering mechanics of solids by E.P.Popov, Prentice-Hall of India, 2005.


MCE/SE/102 DYNAMICS OF STRCUTURES


Internal examination : 2 hours University examination : 3 hours


1. Introduction


Fundamental objective of structural dynamics; Types of prescribed loadings ; Essential characteristics of a dynamic problem; Methods of descritisation – Lumped, Generalised displacements, Finite element concept; Formulation of equation of motion ; Dynamic equilibrium equation using D’Alembert’s Principle


2. Single-Degree-of-Freedom Systems

Force-displacement relation – Linear elastic systems; Damping force; Equation of motion-external force; Mass-spring-damper system; Undamped free vibration; Viscously damped free vibration; Harmonic vibration of undamped and viscously damped systems; Response to periodic excitation; Response to unit impulse; Response to arbitrary force; Response to step force; Response to rectangular pulse force; Numerical evaluation of dynamic response – Newmark’s method


Earthquake response of linear systems – Earthquake excitation , Equation of motion, Response quantities, Time history analysis using Newmark’s method for a particular ground motion (EI CENTRO), Response spectrum concept , Pseudo acceleration response spectrum, Peak structural response from the response spectrum, Elastic design spectrum


3. Multi-Degree-Of-Freedom Systems


Undamped free vibrations – Analysis of vibration frequencies, analysis of vibration mode shapes, orthogonality conditions


Analysis of dynamic response – Normal coordinates, Uncoupled equations of motion (undamped and viscously damped), Mode (displacement) superposition analysis – Viscously damped


Numerical evaluation of dynamic response - Newmark’s method


4. Systems with Distributed Mass And Elasticity


Undamped and viscously damped free vibration of beams ; Analysis of dynamic response – Normal coordinates, uncoupled flexural equations of motion (undamped and viscously damped)


Analysis dynamic response – Normal coordinates; Uncoupled flexural equations of motion (undamped and viscously damped)


Text Books :

  1. Dynamics of Structures by R.W. Clough and P.E. Penzien , McGraw-Hill, 1993.

  2. Dynamics of Structures by A.K.Chopra , Prentice-Hall of India, 2001.

  3. Structural Dynamics by Mario Paz, CBS Publishers, 1987.


MCE/SE/103 MATRIX METHODS OF STRUCTURAL ANALYSIS


Internal Examination : 2 hours University Examination : 3 hours



  1. Basic Concepts of Structural Analysis :

Introduction; Types of Framed Structures; Deformations in Framed Structures; Actions and Displacements; Equilibrium; Compatibility; Static and Kinematic Indeterminacy; Structural Mobilities; Principle of Superposition; Action and Displacement Equations; Flexibility and Stiffness Matrices; Equivalent Joint Loads; Energy Concepts; Virtual Work.


  1. Fundamentals of the Flexibility Method:

Introduction; Flexibility Method; Temperature changes; Prestrains and Support Displacements; Joint Displacements; Member End Actions and support reactions; Flexibilities of prismatic members; Formalization of the Flexibility method.


  1. Fundamentals of the Stiffness Method :

Introduction; Stiffness Method; Temperature changes; Prestrains and Support Displacements; Stiffness of Prismatic Members; Formalization of the Stiffness Method.


  1. Computer Oriented Direct Stiffness Method :

Introduction; Direct Stiffness Method; Complete Member Stiffness Matrices; Formation of Joint Stiffness Matrix; Formation of Load Vector; Rearrangement of Stiffness and Load Arrays; Calculation of Results; Analysis of Continuous Beams; Plane Truss Member Stiffness; Analysis of Plane Trusses; Rotation of Axes in Two Dimensions; Application to Plane Truss Members; Rotation of Axes in Three Dimensions; Plane Frame Member Stiffness; Analysis of Plane Frames.


  1. Computer Programs for Framed Structures:

Flow Chart for the analysis of the following structures:

  1. Continuous Beam

  2. Plane Truss

  3. Plane Frame




  1. Miscellaneous:

Analysis of large structures; Sub-structuring ; Static condensation procedure; Non – prismatic and curved members.


Text Books :

  1. Matrix Analysis of Framed Structures by W. Weaver & J.M.Gere, CBS Publishers, 1986.

  2. Computational structural mechanics by S.Rajasekharan and G. Sankarasubramanian, Prentice Hall of India , 2001.

  3. Matrix and finite element analyses of structures by M.Mukhopadhay and A.H.Sheikh, Ane Books, 2004.



MCE/SE/104 ADVANCED THEORY AND DESIGN OF RCC STRUCTURES


Internal Examination : 2 hours University Examination : 3 hours


1. Behaviour of RCC members in Shear and Torsion

Kani’s theory for shear; Skew bending theory for torsion; Different modes of failure;

Design of beams in combined shear, bending and torsion


2. Detailing of RCC structures

Basic principles of detailing – Truss analogy, Directional changes, General layout of

reinforcement; Beam-column joints – Strut- and-Tie model, Detailing ; Beam-to-

girder joints; Corners and T-Joints; Brackets and corbels


3. Reinforced concrete deep beams

Introduction; Minimum thickness; Steps of designing of deep beams; Design by

IS456;Checking for local failures; Detailing of deep beams


4. Design of shear walls

Introduction; Classification of shear walls; Classification according to behaviour;

Loads on shear walls; Design of rectangular and flanged shear walls


5. Flat slabs

Shear in flat slabs and flat plates – One-way shear, Two-way (punching) shear, Shear

due to unbalanced moment, Shear reinforcement design; Equivalent frame analysis of

flat slabs – Historical development and definition of equivalent frame, Moment of

inertia of slab-beams, Theoretical column stiffnesses, Use of published data for flat \

slabs, equivalent column method, arrangement of live load, Reduction in negative

moments, Design procedure


6. Yield line analysis of slabs

Introduction; Upper and lower bound theorems; Rules for yield lines; Analysis by

segment equilibrium; Analysis by virtual work; Orthotropic reinforcement and skewed

yield lines; special conditions at edges and corners; Fan patterns at concentrated loads;

Limitations of yield line theory


7. Design of statically indeterminate RC structures

Development of moment curvature diagrams; Moment redistribution in RC structures;

Baker’s method of design; Ductility of RC members; Confined concrete; Cambridge

method of design ; Generation of load-deflection diagrams


Text books :



  1. Advanced reinforced concrete design by P.C.Varghese, Prentice-Hall of India, 2005.

  2. Reinforced concrete structural elements by P.Purushothaman, Tata McGraw-Hill, 1984.

  3. Reinforced concrete design by S.U. Pillai and D.Menon, Tata McGraw-Hill, 2003.

  4. Design of concrete structures by A.H.Nilson, McGraw-Hill, 1997.

  5. Reinforced concrete structures by R.Park and T.Paulay, John Wiley & Sons, 1975.

  6. Reinforce and Pre-stressed concrete structures by Kong and Evans, ELBS,1995.


MCE/SE/105A CONCRETE TECHNOLOGY


Internal Examination : 2 hours University Examination : 3 hours


  1. Proportioning of Concrete Mixes

Variables in concrete proportioning; Effects of variables; Methods of proportioning; Fineness modulus method; ACI method; British method; Design of high strength concrete mix; Design of high performance concrete mix; Design of self compacting concrete mix



  1. Properties of hardened concrete

Behaviour under compressive stress; Factors affecting strength; Statistical analysis of test results; Material properties – Young’s modulus and Poisson’s ratio; Other properties of concrete


3. Fatigue and Creep

General principles; Applications to concrete; Endurance limit; Factors affecting fatigue and creep in concrete; Mechanism of creep; Allowance in design


4. Durability of concrete

Strength and durability relationship; Definition of durability and its significance; Various factors affecting durability



  1. Special types of concrete

Light weight concrete; Silicafume concrete; Fly ash concrete; Polymer concrete; Shotcrete and Gunite; Fibre reinforced concrete


7. Non-destructive testing methods

Schmidt’s rebound hammer test; Penetration techniques; Pullout test; Dynamic or vibration method; Pulse velocity method; Applications


8. Introduction to fracture mechanics of concrete

Structural failure based on material performance; Concepts of linear elastic fracture mechanics; Fracture mechanics of concrete


9. Principles of linear elastic fracture mechanics

Airy stress functions for problems in elasticity; Complex stress function; Elastic stress

and displacement fields at crack tip; Stress intensity factors and crack opening

displacements for useful geometries; Superposition of stress intensity factors; Plastic

zone at crack tip; Griffith’s fracture theory; Strain energy release rate for crack

propagation; Relationship between stress intensity factor and strain energy release

rate; Design based on linear elastic fracture mechanics


10. Principles of non-linear fracture mechanics

Energy principles for crack propagation in non-linear materials; J-integral for non-

linear elastic materials; Fracture resistance (R curve); Crack tip opening displacement;

11. Structure and fracture process of concrete

Constituents and microstructure of concrete; Fracture behaviour and strain

localization of concrete; Fracture process zone and toughening mechanisms;

Experimental determination of fracture zone; Influence of fracture process zone on

fracture behaviour of concrete


Text books :

  1. Concrete Technology by M.S.Shetty, S.Chand & Company Ltd., 2005.

  2. Concrete Technology by A.M.Neville and J.J.Brooks, Pearson Education, 2005.

  3. Fracture Mechanics of Concrete: Applications of Fracture Mechanics to Concrete,

Rock and Other Quasi-Brittle Materials by Surendra P. Shah, Stuart E. Swartz,

Chengsheng Ouyang, Publisher : Wiley , 1995.

4. Analysis of Concrete Structures by Fracture Mechanics by by L. Elfgren,

Publisher: Routledge, 1990.

5. Fracture mechanics – Applications to concrete , Edited by Victor C.Li and

Z.P.Bazant, ACI SP118.

6. Elements of fracture mechanics by Prashant Kumar, Wheeler Publishing, 1999.


MCE/SE/105B FIBRE REINFORCED PLASTIC COMPOSITES


Internal Examination : 2 hours University Examination : 3 hours



  1. Introduction

Definition; History of fibre reinforced composites; Constituent materials – Fibres, Polymeric matrix, Prepregs; Lamina and Laminate; General characteristics of FRP; Micromechanics and macromechanics; Properties of typical composite materials; Applications of FRPs in Civil engineering


  1. Processing of FRP Composites

Contact moulding; Compression moulding methods ; Filament winding


3. Macromechanical behaviour of a lamina

Introduction; Stress-strain relations of a lamina with respect to its principal axes;

Stress-strain relations of an arbitrarily oriented lamina; Typical elastic properties of a

unidirectional lamina


4. Macromechanical behaviour of a laminate

Introduction; Classical lamination theory – Lamina stress strain behaviour, Strain and

stress variation in a laminate, Resultant laminate forces and moments; Special cases of

laminate stiffnesses


5. Design of FRP structures

Introduction; Composite structural design; The design spiral; Design criteria; Design

allowables; Material selection; Selection of configuration and manufacturing process;

Laminate design – selection of laminate, laminate design problem, laminate design

Procedure; Mathematical analysis of the laminate – estimation of shear force,

estimation of deflection, mathematical algorithm; Design examples – design of

tension member, laminate design for strength, laminate design for stiffness


6. Composite Joints

Introduction; Classes of laminate joints; Bonded joints- stress distribution, modes of

failure, Merits and demerits of adhesive bonded joints; Mechanical joints – failure

modes, advantages and disadvantages


Text books :


  1. Mechanics of composite materials and structures by Madhujit Mukhopadhay,

Universities Press, 2004.

2. Mechanics of composite materials by R.M.Jones, Publisher : Taylor & Francis, 1998.


MCE/SE/105C EXPERIMENTAL STRESS ANALYSIS AND

MOTION MEASUREMENT


Internal Examination : 2 hours University Examination : 3 hours


  1. Introduction to Strain Measurements

Experimental determination of strain; Properties of strain gage systems; Types of strain gages

2. Strain Measurement using Electrical Resistance Strain Gages

Introduction; Strain sensitivity in metallic alloys; Gage construction; Strain gage

adhesives and moulding methods; Gage sensitivities and gauge factor; The

Wheatstone bridge ; Wheatstone bridge sensitivity; Temperature compensation ;

Static recording and data logging – Manual strain indicators, Automatic data

acquisition systems, PC based data acquisition systems; Strain analysis methods –

Three element rectangular rosette


3. Stress analysis using Photoelasticity

Wave theory of light; Refraction of light; The Polariscope – Plane polarisers, wave

plates; Plane polariscope; Circular polariscope; Diffused light polariscope; The stress

optic law for two-dimensional plane-stress bodies; Two-dimensional photoelastic

stress analysis – Isochromatic fringe patterns, Isoclinic fringe patterns, Calibration

methods, Principal stress separation methods, Scaling model-to-prototype stresses;

Materials for two dimensional photoelasticity; Three-dimensional photoelasticity –

Stress freezing


4. Model analysis of Structures

Introduction – Objectives of structural model studies, Some basic definitions, Types

of similitude , Classification of model studies, Model materials, Size effects;

Principles of similitude – Dimensional analysis, Buckingham π Theorem, Variables in

structural behaviour; Requirements of similitude; Direct approach


5. Motion Measurement

Introduction; Vibrometers and Accelerometers; The seismic instrument; General

theory of the seismic instrument; The seismic accelerometer; Practical

accelerometers


Text Books :


1. Experimental Stress Analysis by Dally and Riley, McGraw-Hill, 1991.

2. Mechanical measurements by Bechwith, Merangoni & Lienhard, Pearson Education,

2003.

3. Model analysis of Structures by T.P.Ganesan, Universities Press, 2000.


MCE/SE/106A CONSTRUCTION ENGINEERING AND MANAGEMENT


Internal Examination : 2 hours University Examination : 3 hours


1. Introduction

Classification of construction works; Various stages in the construction of a project


2. Construction equipment

Introduction; Classification of construction equipment; Earthmoving equipment; Hauling

equipment; Hoisting equipment; Conveying equipment; Aggregate and concrete

production equipment; Pile driving equipment; Tunneling and drilling equipment;

Pumping and dewatering equipment


3. Management of Construction

Introduction; Management requirement; Need for mechanization; Financial aspects of

construction plants and equipment; Factors affecting selection of construction equipment;

Planning of construction equipment; Factors affecting the cost of owning and operating

construction equipment; Planning of infrastructure for mechanisation; Role of operations

research; Equipment management


4. Materials Management

Importance; Objectives; Costs; Functions of materials management department; Uses of

materials management; Stores management; Materials procurement Materials handling


5. Construction Safety Management

Introduction; Importance of safety ; Causes of accidents; Responsibility for safety; Safety

measures; Role of various parties in safety management; Measures to improve safety in

construction ; Prevention of fires at construction sites


6. Quality control in construction

Importance of quality; Elements of quality- quality characteristics, design quality, quality

of conformance; Organisation for quality control; Quality assuarance techniques-

Inspection, Testing, Sampling; Documentation; Quality control circles


7. Human Factors in Construction

Qualities of efficient construction managers; Personality; Ethics and integrity; Personal

drive; Multidisplinary capability; Human relations


8. Value Engineering

Definition ; Value engineering job plan; Life cycle costing; Value engineering

Applications


9. Management Information Systems

Introduction; Definition of organization; Definition of management; Definition of

management information system; Computer as information system; Use of computer in

construction industry; Requirements of management information system; A data base

approach; Salient features of some software packages used in construction industry


10. Information Technology in Construction Industry

Introduction; Information flow and communication; Knowledge data base; Learning

organization attributes; Use of information technology in construction industry; Role of

artificial intelligence and expert systems


Text Books :

  1. Construction Engineering and Management by S.Seetharaman, Umesh Publications, 2003.

  2. Construction planning, Equipment and methods by R.L.Peurifoy, C.J.Schexnayder and Aviad Shaptra, McGraw-Hill, 2005.

  3. Construction management by D.W.Halpin, Publisher : Wiley, 2005.

  4. Project management in construction by S.M.Levy, McGraw-Hill Professional, 2006.

  5. Total construction project management by G.J.Ritz, McGraw-Hill Professional, 1993.

  6. Construction management fundamentals by C.J.Schexnayder and R.Mayo, McGraw-Hill Science/Engg./Math, 2003.



MCE/SE/106B DESIGN OF TALL BUILDINGS


Internal Examination : 2 hours University Examination : 3 hours


1. General Considerations

Introduction; Definition of a tall building ; Lateral load design philosophy; Concept of

premium for height; Factors responsible for slimming down the weight of structural

frame; Development of high-rise architecture; structural concepts


2. Wind effects

Design considerations; Nature of wind; Extreme wind conditions; Characteristics of

wind; Provisions of IS875(Part3); Wind tunnel engineering – Introduction, Description,

of wind tunnels; Objectives of wind tunnel tests, Rigid model studies, Aeroelastic study


3. Seismic Design

Introduction; Tall building behaviour during earthquakes ; Philosophy of earthquake

design; Provisions of IS1893(Part1).


4. Lateral Systems for Steel Buildings

Introduction; Semi-rigid frames; Rigid frames; Braced frames; Interacting system of

braced and rigid frames


5. Lateral Systems for Concrete Buildings

Introduction; Frame action of column and slab systems; Flat slab and shear walls; Flat

slab, shear walls and columns; Coupled shear walls; Rigid frame; Widely spaced

perimeter tube; Core-supported structures; Shear-wall frame interaction


6. Lateral Systems for Composite Construction

Introduction; Composite elements; Composite systems


7. Gravity Systems

Concrete floor systems; Prestressed concrete systems; Composite metal decks


8. Structural Analysis

Introduction; Partial computer models; General computer analysis techniques; Special

techniques for planar shear walls; Finite element analysis


Text Books :

  1. Structural Analysis and design of tall buildings by B.S.Taranath, McGraw-Hill, 1988.

  2. Steel,Concrete and Composite design of tall buildings by B.S.Taranath, McGraw-Hill, 1997.

  3. Tall building structures by B.S.Smith and A.Coull, John Wiley & Sons, 1991.



MCE/SE/106C ADVANCED GEOTECHNICAL ENGINEERING


Internal Examination : 2 hours University Examination : 3 hours



  1. Bearing Capacity of shallow foundations subjected to special loading and ground conditions


Effect of eccentric loading, inclined load, inclination of base of foundation, sloping ground; Bearing Capacity of stratified soils; Meyerhof analysis, Vesic’s analysis and Hansen’s analysis.


  1. Settlement analysis


Contact pressure, sources of settlement, uniform settlement, differential settlement, construction practices to avoid differential settlement, allowable bearing pressure of sands from SPT, immediate settlement in sands and clays- Terzaghi and Janbu’s methods for clays , Schmertmann and Hartman method for cohesionless soils; consolidation settlement.



  1. Machine Foundation


Fundamentals of Vibration; Free and Forced Vibration with and without damping; Natural frequency of foundation; Soil system; Dynamic soil properties; Vibration Isolation; Types of machines and machine foundation; I.S. Code of practice for design and construction of block foundation for reciprocating and impact type machines and framed foundations for high speed rotary machines.



  1. Geotechnical Earthquake Engineering

Effect of type soil on the response spectrum; Liquefaction – Definition and types,

Effect of liquefaction on built environment, Evaluation of liquefaction

susceptibility, liquefaction hazard mitigation; Seismic slope stability –

Introduction, Pseudo-static analysis, sliding block method.

.


  1. Foundations in Expansive Soils

Problems with expansive soils, Identification of expansive soils, Field observation to identify expansive soils, classification of expansive soils, swell pressure – Free swell method, constant volume method and method of different surcharges; Under- reamed pile foundations, CNS layer method, lime stabilization of expansive soils, lime- slurry pressure injection method.


Text books :


  1. Hand book of machine foundations by Sreenivasulu, P & Vaidyanathan,C.V.

  2. Foundation Analysis & Design by Bowles, J.E., McGraw- Hill Book Company.

  3. Principles of Foundation Engineering, B.M. Das., PWS Publishing Company, ITP An International Thomson Publishing Company.

  4. Geotechnical Earthquake Engineering by Steven L. Kramer, Pearson Education.

  5. Basic and Applied Soil Mechanics by Gopal Ranjan and ASR Rao, Wiley Eastern Limited, New Delhi.

  6. Geotechnical Engineering by SK Gulati & Manoj Datta, Tata McGraw- Hill Publishing Company Limited.

  7. Foundations on Expansive Soils by F.H. Chen.

  8. Soil dynamics and Machine Foundations by Swami Saran, Galgotia Publications Pvt. Ltd., New Delhi.



MCE/SE/151 STRUCTURAL ENGINEERING LABORATORY


University Examination : 3 hours


Any 10 of the following experiments are to be carried out :



  1. Study of the effect of water/cement ratio on workability and strength of

Concrete.


  1. Study of the effect of aggregate /cement ratio on strength of concrete




  1. Mix design methods using a) I.S. Code method b) ACI Code method




  1. A study of correlation between cube strength, cylinder strength, split tensile strength and modulus of rupture




  1. A study of behaviour of under-reinforced and over-reinforced beams




  1. A study on the effect of span to depth ratio on the failure pattern of RC beams




  1. A study of the effect of pre-stressing on the flexural behaviour of beams




  1. Measurement of static strain by electrical resistance strain gauge




  1. Determination of the material fringe value of a given photo elastic material.




  1. Determination of principal stress difference in a circular disc subjected to diametrical compression.




  1. Determination of principal stresses in a bar subjected to axial tension.




  1. Determination of stress concentration factor




  1. Dynamics of a three storey building frame subjected to harmonic base motion




  1. Dynamics of three storey building frame subjected to non-harmonic (periodic) base motion




  1. Dynamics of a one-storey building frame with planar asymmetry subjected to harmonic base motion



MCE/SE/201 FINITE ELEMENT ANALYSIS OF STRUCTURES


Internal Examination : 2 hours University Examination : 3 hours


1. Basic Principles

Equilibrium equations; Strain-displacement relations; Linear constitutive relations;

Principle virtual work; Principle of stationary potential energy


2. Element Properties

Different types of elements; Displacement models; Relation between nodal degrees of

freedom and generalized coordinates; Convergence requirements; Compatibility

requirement; Geometric invariance; Natural coordinate systems; Shape functions; Element

strains and stresses; Element stiffness matrix; Element nodal load vector


Isoparametric elements – Definition, Two-dimensional isoparametric elements – Jacobian

transformation, Numerical integration


3. Direct Stiffness method and Solution Technique

Assemblage of elements–Obtaining Global stiffness matrix and Global load vector;

Governing equilibrium equation for static problems; Storage of Global stiffness matrix in

banded and skyline form; Incorporation of boundary conditions; Solution to resulting

simultaneous equations by Gauss elimination method


4. Plane-stress and Plane-strain analysis

Solving plane stress and plane-strain problems using constant strain triangle and four

nodded isoparametric element


5. Analysis of plate bending

Basic theory of plate bending; Shear deformation plates; Plate bending analysis using four

noded isoparametric element


6. Analysis of shells

Degenerated shell elements – Evaluation of element stiffness matrix and load vector for

eight noded isopametric shell element


Text books :

  1. Finite element analysis by C.S.Krishnamurthy, Tata-McGraw-Hill, 1994.

  2. Matrix and finite element analyses of structures by M.Mukhopadhay and A.H.Sheikh, Ane Books, 2004.

  3. Concepts and applications of finite element analysis by R.D.Cook et.al., John Wiley and Sons, 1989.



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