e-proceeding

INTERNATIONAL CONFERENCE ON STRUCTURAL ENGINEERING AND MECHANICS (ICSEM 2013) December 20-22, 2013

Co-Editors: Pradip Sarkar, Robin Davis P. and Srinivas Sriramula

ISBN 978-93-80813-26-4

DEPARTMENT OF CIVIL ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA ODISHA - 769 008, INDIA

TABLE OF CONTENTS Paper Authors No.

Title

Page No.

002

R. P. Khandelwal; Anupam Chakrabarti and Pradeep Bhargava

Calculation of inter-laminar shear stresses in laminated composite shallow shell using least square of error (LSE) method

003

Ajay Kumar; Anupam Chakrabarti and Pradeep Bhargava

Finite element analysis of free vibration of laminated composite spherical shells with cut-outs

12

004

S. N. Patel

Nonlinear response of laminated composite stiffened plates

19

006

Manav Mittal and Pijush Samui

Revived technique for compressive strength of CFRP confined concrete cylinders using Gaussian process regression

36

008

M.N.A. Gulshan Taj and Anupam Chakrabarti

Bending analysis of functionally graded sandwich plates

41

009

V. Chitra and R.S. Priyadarsini

Effect of lay-up sequences and imperfections on the dynamic buckling of CFRP cylindrical shells

48

010

Suprabhat. Jakati

Evaluation of shear strength of RC members using different methods

58

012

Shivaji T Bidgar and Partha Bhattacharya

The study of tension stiffening and crack band width in reinforced concrete beam under flexure

65

015

Soumi Bhattacharyya and Aparna (Dey) Ghosh

Effect of mass ratio on the performance of a TMD with non-optimal damping

71

016

Chhabirani Tudu and Asha Patel

Study of torsional behaviour of rectangular RC beam wrapped with GFRP

79

017

Jyotirmoya Dutta Majumdar and Aparna (Dey) Ghosh

Control of wind induced vibration in transmission line towers using tuned liquid column damper

88

018

Swetapadma Panda; Joygopal Jena and Bidyadhar Basa

Stress analysis around spiral casing of Francis Turbine of a Hydel power house by finite element method

96

i

1

Paper Authors No.

Title

Page No.

019

Benazir F. Ahmed and Kaustubh Dasgupta

Influence of location of staircase on seismic behaviour of RC flat slab building

105

020

Arijit Acharjya and Kaustubh Dasgupta

Influence of staircase and elevator core on twisting behaviour of RC wall-frame buildings

113

021

P. Bhattacharya and S. Karar

Random vibration analysis of RCC structure using pseudoexcitation method

121

022

A. Roy and Aparna (Dey) Ghosh

Tuned liquid damper system for seismic vibration control of elevated water tanks

129

023

Manish K. Singh; Sailesh Adhikari; Sumit Mazumder and Amiya K. Samanta

Study on stability of natural draft hyperbolic RC cooling tower shell under wind induced flow field

137

025

R. Velmurugan and Jafar Sadique P

An experimental and numerical study of natural-fiber composites subjected to different environmental conditions

145

026

N Murli Krishna and Md Masihuddin Siddiqui

Evaluation of performance point using energy based pushover analysis

154

028

Arghya Sengupta and Rana Roy

Seismic behaviour of R/C frames with biaxial interaction

164

029

Pawan Thakur and Rana Roy

Seismic behaviour of plan-asymmetric structures under spectrally matched records

174

031

Syed H. Basha and Hemant B. Kaushik

Influence of masonry properties on lateral load response of reinforced concrete frames

185

032

Somdatta Goswami; Subrata Chakraborty and Shymal Ghosh

Adaptive response surface method in structural response approximation under uncertainty

194

035

Avaya K. Baliarsingh; P. K. Ray; B. B. Verma and Deepak K. Agarwalla

Prediction of fatigue crack propagation life in single edge notched beam using exponential model

203

036

Deepak K. Agarwalla; Dayal R. Parhi and Amiya K. Dash

Fuzzy logic based fault diagnosis of structures using vibration responses

210

ii

Paper Authors No.

Title

Page No.

037

Komathi M. and Amlan K.Sengupta

Strengthening of columns for shear in reinforced concrete buildings

222

038

Shakti P. Jena and Dayal R. Parhi

Dynamic analysis of cantilever beam with moving mass

230

039

Saswati De and K. Bhattacharya

Behaviour of two-span continuous steel box RC deck composite bridges

238

042

Shantaram Parab; Jatin Alreja and Pijush Samui

Prediction of hysteretic energy demand in steel moment resisting frames using multivariate adaptive regression splines

250

043

H.D. Chalak; Anupam Chakrabarti; Mohd. Ashraf Iqbal and Abdul Hamid Sheikh

Buckling analysis of laminated soft core sandwich plate

258

044

A. Subbulakshmi and P. Jayabalan

Nonlinear analysis of square concrete filled steel tubes under constant axial load and monotonic lateral loading

266

045

Bijan Kumar Roy and Subrata Chakraborty

Reliability based design of TMD system considering system parameter uncertainty in seismic vibration control

275

046

Bulu Pradhan

Performance evaluation of concrete against rebar corrosion in composite chloride-sulphate exposure conditions

284

048

Fouzia Shaheen and Bulu Pradhan

Potentiodynamic polarization study on bare steel in concrete powder solution extracts contaminated with chloride and sulphate ions

291

049

Arun Mukherjee; Sreyashi Das (nee Pal) and A. Guha Niyogi

Non-linear vibration and dynamic response of epoxy-based laminated composite plate with square cut-out at elevated temperature

300

053

M. V. N. Sivakumar and B. Vineel Kumar

Seismic vulnerability assessment of Indian standard code designed midrise and high-rise RC framed structures using fragility analysis

308

054

Snehal Kaushik and Kaustubh Dasgupta

Seismic behaviour of slab-structural wall junction in rc building

316

055

Sanjay Goswami; Partha Bhattacharya and Subham Rath

Investigating effectiveness of dynamic strain response for damage detection in composite beams

324

iii

Paper Authors No.

Title

Page No.

056

A. Vanuvamalai and K.P. Jaya

Finite element analysis of road tunnel

332

059

Tushar Kanti Dey , Anupam Chakrabarti and Umesh Kumar Sharma

Optimization of a fiber reinforced polymer web core skew bridge

339

060

G. V. Rama Rao; M. Pavan Kumar and Sk. Rasool

Cost effective structural systems for high rise buildings subjected to seismic load

347

061

Soumya Bhattacharjya; Urmi Saha; Asish Modak and Baidyanath Sarker

Probabilistic assessment of safety, economy and robustness of steel structure including parameter uncertainty in is: 8002007 format

356

062

A. K. Nayak; R. A. Shenoi and J. I. R. Blake

A computer aided fem based numerical solution for transient response of laminated composite plates with cutouts

363

063

C. Mohanlal and N. Murali Krishna

Optimal design of a retractable roof structure using real coded genetic algorithm

372

064

Nityananda Nandi

The change in stress distribution pattern within the earthen dam due to the effect of seepage

381

065

Bhaskar Ghosh and Subhashish Roy Chowdhury

Fundamental period of vertically irregular frames – a modification over the codal stipulation

394

066

Maganti Janardhana; Jogi Naidu P. and Hymavathy K.

Structural behaviour of an RC building on sloping ground under earthquake load

403

067

V. R. Kar and S. K. Panda

Thermal stability analysis of functionally graded panels

416

068

M. Mishra and K. C. Panda

Properties of rubberised fly ash concrete

422

073

G Muthukumar and Manoj Kumar

Influence of opening on the dynamic structural response of rectangular slender shear wall for different damping ratios

430

074

S S S Sastry; Anasuyeswar K; Ayush Mathur and Shailaja B

Studies on the channel fitting in aircraft using classical and finite element methods

447

076

P.V. Katariya and S.K. Panda

Modal analysis of laminated composite spherical shell panels using finite element method

454

iv

Paper Authors No.

Title

Page No.

077

Hipparagi A. K.; Injaganeri S.S. and Mahadevgouda H.

Enhancement of shear ductility for HSC beams with minimum shear reinforcement

463

083

J S Ali and S Gupta

Spectral strain energy based approach for system identification in structures

471

085

A. Syed Mohamed and C. O. Arun

Towards an eco-friendly concrete with waste glass and rice husk ash

478

086

K. V. S. N. Murthy and B. Dean Kumar

A study on analysis of turbo generator foundation resting on pile foundations for various earthquake zones in india

486

087

Rajiv Verma and Puneet Mathur

Transient analysis of an elliptical journal bearing

493

088

A.S. Balu and B.N. Rao

Response surface based analysis of structures with fuzzy variables

503

090

Shiv Shanker Ravichandran and Richard E. Klingner

Seismic design factors for steel moment frames with masonry in-fills: Part 1

510

091

Shiv Shanker Ravichandran and Richard E. Klingner

Seismic design factors for steel moment frames with masonry in-fills: Part 2

521

092

K. Sharma; V. Bhasin; I. V. Singh; B. K. Mishra and R. K. Singh

Simulation of bi-metallic interfacial crack using EFGM and XFEM

532

093

Moon Banerjee; N. K. Jain and S. Sanyal

Evaluation of stress concentration for a simply supported laminated composite plate with a centre circular hole under uniform transverse loading

540

096

Revathi P.; Ramesh, R. A. and Lavanya, K.

Effect of treatment methods on the strength characteristics of recycled aggregate concrete

548

097

Shashank Pandey and S. Pradyumna

Finite element analysis of sandwich plates with functionally graded material core using a layerwise theory

556

099

Irshad A Khan and Dayal R Parhi

Diagnosis of multiple crack of cantilever composite beam by vibration analysis and hybrid AI technique

564

109

Rehan A. Khan and T. Naqvi

Performance based design of rcc building under earthquake forces

571

v

Paper Authors No.

Title

Page No.

111

A.K.L. Srivastava

Effect of stiffened cut-out on vibration and parametric excitation

580

112

H. S. Panda; S. K. Sahu and P. K. Parhi

Modal analysis of delaminated woven fibre composite plates in moist environment

585

117

Abhishek Kumar; V. S. Phanikanth and K. Srinivas

Seismic analysis of reinforced concrete ventilation stack using simplified modal analysis technique

593

118

G Srikar; B Gopi and S Suriya Prakash

Effect of temperature on compressive behaviour of concrete reinforced with structural polypropylene fibres

604

119

C Sreenivasulu; R Mehar Babu; S Suriya Prakash and K.V.L. Subramaniam

Behaviour of masonry assemblages made of soft brick under compression

614

120

K.V.V. Sumanth and S. Suriya Prakash

Effect of tension stiffening on behaviour of concrete columns under pure torsion

624

121

M. R. Das and S. Samal

Optimal earthquake resistant design of a fixed beam using a simple optimization tool

633

122

Souvik Chakraborty and Rajib Chowdhury

Uncertainty propagation using hybrid HDMR for stochastic field problems

642

123

P. Dinakar and Manu S. Nadesan

Design and development of high strength self compacting concrete using metakaolin

657

124

Monalisa Priyadarshini; Robin Davis; Pradip Sarkar and Haran Pragalath D C

Seismic reliability assessment of R/C stepped frames

663

126

P.R. Ravi Teja and Sasmita Sahoo

Major indian earthquakes and earthquake risk assessment – a case study

671

128

P. R. Maiti; Bhawesh Madhukar and Satyam Mandloi

Analysis of plate resting on Pasternak and Winkler foundation due to moving load

685

129

Pardeep Kumar and R. K. Sharma

Determination of dynamic modulus of elasticity of concrete

695

vi

Paper Authors No.

Title

Page No.

130

S. C. Choudhury; A. Simadri Dora; M. K. Tripathy and P. M. Tripathy

Construction of high level bridge over river Mahanadi at 6th km of Sambalpur-Sonepur road to Chadheipank-Binca road - a case study

704

131

D. Jena and K.C. Biswal

Performance evaluation of tuned liquid damper on response of high rise structure under harmonic load

707

132

Ganesh R.; Haran Pragalath D.C.; Robin Davis; Pradip Sarkar and S. P. Singh

Seismic fragility analysis of axially loaded single pile

713

133

Priyadarshi Das and ManasRanjan Das

Optimal design of a tuned mass damper using a simple optimization tool

720

134

Sourabh S. Deshpande and R. L. Wankhade

Analysis of thick beams using first order shear deformation theory

728

135

Jitendra Kumar Meher and Manoranjan Barik

Free vibration of multiple-stepped Bernoulli-Euler beam by the spectral element method

737

136

Istiyak Khan; M. K. Agrawal and S. B. Chafle

Attenuation of seismic response of structures using passive devices

745

137

N. Trivedi and R. K. Singh

Experimental and analytical estimation of concrete creep for large prototype structure

753

138

N. Trivedi and R. K. Singh

Numerical simulation for VTT impact test under iris program

760

Synthetic ground motion generation using a semi analytical model

766

Experimental and analytical studies of ratcheting in pressurized piping system under seismic load

774

139

140

A. Ravi Kiran; M. K. Agrawal; G. R. Reddy; R. K. Singh and K. K. Vaze A. Ravi Kiran; P. N. Dubey; M. K. Agrawal; G. R. Reddy; R. K. Singh and K. K. Vaze

141

M. Bandyopadhyay and A. K. Banik

Numerical analysis of semi-rigid jointed steel frame using rotational springs

782

142

Kotla Shiva Kumar and Diptesh Das

Passive control of structures subjected to earthquake excitation using particle swarm optimization

791

vii

Paper Authors No.

Title

Page No.

143

Jagdish Malav; Swapnil Takle; Kali P. Sethy; Dinakar. P and U. C. Sahoo

Properties of high strength concrete containing ultrafine slag

799

144

A. K. Banik and M. Bandyopadhyay

Progressive collapse study of semi-rigid jointed frame structures: a state-of-the art review

807

145

M. K. Agrawal; A. Ravi Kiran; G. R. Reddy; R.K. Singh; K. K. Vaze; D. K. Sakhrodia; B. Biswas and A. Bhowmick

Re- evaluation of industrial equipments for earthquake and wind loads

827

146

R. K. Sharma and Pardeep Kumar

Utilization of dredged material from reservoir of hydro project as fine aggregate in concrete

834

147

D. V. Prasada Rao and G. V. Sai Sireesha

Effect of silica fume on strength of Partially used recycled coarse aggregate concrete

843

148

Pratika Preeti

Seismic response of vertically irregular structures: issues to be addressed

850

149

Tarapada Mandal and Sanjay Sengupta

Slope stability analysis by static and dynamic method

858

150

Karthik Subhash A.; U. K. Mishra and Ganesh R.

Damage detection in simply supported beam - an ANN approach

865

151

Kirtikanta Sahoo; Pradip Sarkar and Robin Davis P.

Behaviour of recycled aggregate concrete

871

viii

International Conference on Structural Engineering and Mechanics December 20-22, 2013, Rourkela, India Paper No.: 134

ANALYSIS OF THICK BEAMS USING FIRST ORDER SHEAR DEFORMATION THEORY 1

2

Sourabh S. Deshpande and R. L. Wankhade 1

2

PG Student, Dept. of Applied Mechanics, Government College of Engineering Karad, Maharashtra, India Asst. Professor, Dept. of Applied Mechanics, Government College of Engineering Karad, Maharashtra, India Email: [email protected], [email protected]

ABSTRACT : A first order shear deformation beam theory is employed here for the static analysis of thick beams. The limitations of classical theory of beam bending developed by Euler and Bernoulli forced for the refinement over these classical theories. After refinement discrepancies of the classical beam theory are eliminated and hence first order shear deformation theory is developed. The assumption of Euler-Bernoulli beam theory stating that the plane section remains perpendicular to the neutral axis of the section after the deformation is modified by Timoshenko. Timoshenko beam theory assumes that the section which was normal to the neutral axis before deformation does not remain normal to the neutral axis after the deformation. He considers the rotation of the normal to be the combined effect of the rotation due to bending and shear effects. But, Timoshenko theory does not remove another discrepancy of the classical theory i.e. normal section remains plane after bending. Timoshenko also assumes that plane section before bending remains plane after bending also. Thus, He assumes the constant shear strain across the section and thus rectangular distribution of the shear stress across the section, which is not in accordance with the actual distribution of the shear stress which is parabolic across the section. Thus, shear correction factor is required in this theory to modify the result to get the fairly accurate result. KEYWORDS:

Shear deformation theory, Thick beams, Shear stress

1. INTRODUCTION The elementary theory of beam bending introduced by Euler and Bernoulli is the simplest theory of beam bending which neglects the effects of the shear deformation. The theory assumes that the plane section which is normal to the neutral axis before bending remains plane and normal to the neutral axis even after deformation of the beam. That means shear stresses are not considered in the hypothesis. Effect of rotary inertia and shear deformation are included in the theory of beam by Rayleigh (1877) and Timoshenko (1921) as the pioneer investigators. To remove the discrepancies in the classical and first order shear deformation or Timoshenko theories, higher order shear deformation theories were developed by various researchers. Levinson (1981), Bickford (1982), Rehfield and Murty (1982) Krishna Murty (1984), Bhimaraddi and Chandrashekhara (1993) are some of the investigators who presented parabolic shear deformation theories assuming a higher variation of the axial displacement in terms of the thickness coordinate for the analysis of thick beams. Finite element method is used by Kant and Gupta (1988), Heyliger and Reddy (1988) based on higher order shear deformation of uniform rectangular beams. Further refined shear deformation theories for thick beams including sinusoidal functions in terms of the thickness coordinate in displacement field are developed by Vlasov and Leont’ev (1996), Stein (1989). The drawback of these theories was that shear stress free boundary conditions were not satisfied at top and bottom

surfaces of the beam. Thus, a refined trigonometric shear deformation theory has been developed by Ghugal and Shimpi (2001) to accurately predict shear stress distribution across the section of the beam and which also satisfies the free boundary conditions at top and bottom surfaces of the beam. In this paper, first order shear deformation theory is developed and various non dimensional parameters are introduced. These parameters are applied to numerical examples to give values of these parameters for the specific cases and then these values are plotted on the graph to show the variation of these parameters across the section of the beam.

2. EQUATIONS AND DEVELOPEMENT OF THEORYThe beam under consideration is shown in figure. It occupies in 0 − x − y − z Cartesian coordinate system in the region;

0 ≤ x ≤ L;

0 ≤ y ≤ b;

−h ≤ z≤h 2 2

Where x, y, z are Cartesian coordinates, L is the length of the beam in x direction, b is the width of the beam in y direction and h is the thickness of the beam in z direction. The beam is made up of homogeneous, linearly elastic isotropic material.

h L

2.1 The displacement field The displacement field of the present beam theory is given by;

u ( x , z ) = − zφ

(1)

w( x, z ) = w(x) Where u is the axial displacement in x direction and w is the transverse displacement in z direction of the beam. The function φ represents total rotation of the beam at neutral axis, which is an unknown function to be determined. The normal and shear strains obtained within the framework of linear theory of elasticity using displacement field given by above equation is as followsNormal strain: ε x =

Shear strain:

γ zx =

∂u dφ = −z ∂x dx

(2)

∂u ∂w ∂w + = −φ + ∂z ∂x ∂x

(3)

729

The stress-strain relationship used is as follows-

σx = εx ⋅E

τ zx = γ zx ⋅ G

(4)

The above relationships are used to obtain the governing differential equations of the present beam theory by substituting these equations in the principle of virtual work. The procedure is explained in the next section. 2.2 Governing equations and boundary conditions Using the expressions for strains and stresses (2) through (4) and using the principle of virtual work, variationally consistent governing differential equations and boundary conditions for the beam under consideration can be obtained. The principle of virtual work, when applied to the beam leads to; h x = L y =b z = 2

x=L

x =0 y =0 z =−

x =0

∫ ∫ ∫h (σ xδε x + τ zxδγ zx )dxdydz − ∫ q(x )δwdx = 0 2

(5)

Where, symbol δ denotes the variational operator. Substituting the values of stresses and strains in the above equation, we get the coupled Euler-Lagrange equations which are the governing differential equations and associated boundary conditions of the beam. The governing differential equations are obtained as underh x = L y =b z = 2

x=L

x =0 y =0 z = −

x=0

∫ ∫ ∫h (Eε xδε x + Gγ zxδγ zx )dxdydz = ∫ qδwdx 2

Putting the values of stresses and strains from the above equations and then integrating and equating the coefficients of δw and δφ , we get governing differential equations of this theory as follows-

dφ d 2 w GAK s − 2 = q( x ) dx dx

(A)

d 2φ dw GAK s − φ + EI 2 = 0 dx dx

(B)

2.3 The general solution of governing equilibrium equations of the beamUsing above obtained differential equations, we can obtain the general equation of this beam theory in terms of rotation of the beam. From equation (B),

EI

d 2φ dw − GAK s φ − =0 2 dx dx

Differentiate w. r. t. x, we get,

dφ d 2 w d 3φ EI 3 − GAK s − 2 = 0 dx dx dx

∴ From equation (A),

730

∴ EI

d 3φ = q(x ) dx 3

This is the general equation of this beam theory in terms of the rotation of the beam at the neutral axis. Using this equation, we can obtain the expression for transverse displacement of the beam by substituting the value of rotation of the beam in either of the equations (A) or (B). 3. ILLUSTRATIVE EXAMPLES AND DIAGRAMSIn order to apply above theory to various boundary conditions, we have considered three examples here. With the help of these examples, we can predict the distribution of various non dimensional parameters across the section of the beam. Material properties of the beam are as follows-

E = 210GPa

µ = 0.25

ρ = 7800 kg

Where, E =modulus of elasticity of the material, the beam material

m3

µ =Poisson’s ratio for the beam material and ρ =density of

Example 1-simply supported beam with uniformly distributed load over the entire span-

b

L

d 3φ EI 3 = q ( x ) dx By taking integration of above equation thrice and substituting appropriate boundary conditions in the equations; we get the expression for the rotation of the beam at the neutral axis. It is given as-

∴ EIφ =

qx 3 qLx 2 qL3 − + 6 4 24

This expression gives the rotation of the simply supported beam carrying uniformly distributed load over the entire span at any section of the beam along its length. Now, this expression is used to find out the expression for the transverse displacement of the beam at any section along the length of the beam. From equation (A),

d 2 w dφ q = − 2 dx GAK s dx

731

Putting the value of φ in the above equation and then integrating twice with respect to x by putting appropriate boundary conditions in the equations, we get the expression for the transverse displacement of the beam at neutral axis. It is given by-

q w= 2 EI

x 4 Lx 3 qx 2 qL3 x qLx − − + + 6 2GAK s 24 EI 2GAK s 12

After modification, the final expression for the transverse displacement is given by,

w=

qL4 24 EI

x4 x3 x qL2 x x 4 − 2 3 + + 1 − L 2GAK s L L L L

For the maximum transverse displacement, put x = L

wmax =

Put

2

in above expression, we get,

5 qL4 384 EI 1 + 384 EI 40 L2 GAK s

E = 2(1 + µ ) and simplify the equation, we get the final expression for maximum transverse G

displacement as,

wmax

2 5 qL4 h = 1 + 1.6(1 + µ ) 384 EI L

Non dimensional transverse displacement is given by,

w=

w h 4 Eb h L4 q

h ∴ w = 1 + 1.6(1 + µ ) L

2

From the above values of transverse displacement and rotation of the beam at the neutral axis, we can find out the remaining values of axial stress in x-x direction, shear stress in x-z plane, stress in z-z direction and axial displacement u in the form of non dimensional parameters. These can be found out as followsExpression for axial displacement u-

zq L u=− 2 Eb h

3

x3 x2 4 3 − 6 2 + 1 L L

732

Expression for axial stress sigma xx-

q z L σ x = −6 b h h

2

x2 x 2 − L L

Expression for transverse shear stress using constitutive relationship τ zxCR It may be noted that it is possible to obtain transverse shear stress τ zx , by using either constitutive relationship or the equilibrium equation of theory of elasticity. Notion τ zxCR denotes τ zx obtained by using constitutive relationship.

τ zx =

qL x 1 − bh L 2

Expression for transverse shear stress τ zxEE , and transverse normal stress, σ z , obtained from equation-

equilibrium

The following equilibrium equations of two dimensional elasticity ignoring body forces are used to obtain transverse shear and transverse normal stresses.

∂σ x ∂τ zx + =0 ∂x ∂z ∂τ xz ∂σ z + =0 ∂x ∂z

To get τ zx , substitute the expression obtained for axial stress σ x in equilibrium Equation above and integrate with respect to thickness coordinate z and impose the following boundary condition at bottom of beam

[τ zx ]z =± h / 2 = 0

To get constant of integration. The expression obtained for transverse shear stress using this procedure in its final form is as follows:

τ zx = −

3 q L x 1 z2 − 1 − 4 2 b h L 2 h 2

To get σ z , substitute the expression obtained for shear stress, in the equilibrium Equation above and integrate it with respect to thickness coordinate z and impose the following boundary condition at bottom of the beam

[σ z ]z =h / 2 = 0

To get the constant of integration. The expression obtained for transverse normal stress σ z is as follows:

σ zz

q1 3 z z3 = − + 2 3 b2 2 h h

733

Example 2-simply supported beam carrying uniformly varying load over entire span-

h

L

In this beam, the loading is expressed in terms of maximum intensity of loading q 0 and distance of section along the length of beam i.e. x . The expression is given by,

EI

d 3φ x = q 0 1 − 3 dx L

After integrating w. r. t. x thrice by applying appropriate boundary conditions to find constants of integrations, we get the expression for rotation of this beam at the neutral axis. This expression is given by,

q 0 L3 φ= 6 EI

x4 x3 x2 3 − 4 + 3 − 2 + L L 17 4L

This expression is used to find the expression for transverse displacement. From equation (A), we can write,

q0 d 2 w dφ x = − 1 − 2 dx GAK s L dx

Putting the value of φ in above equation and then integrating twice w. r. t. x by applying appropriate boundary conditions to find out the constants of integration, we get the expression for transverse displacement of beam at any section along the length of the beam. It is given by,

q0 L4 w= 6 EI

x5 x4 x3 3 x q0 L2 x 2 x3 x − + − + − − − 0.42 5 4 3 2 3 L 20L 4 L 3L 17 L GAK s 2 L 6 L

Non dimensional parametersExample 1-

σx =

σ xb q

τ zx =

τ zx b q

σ zz =

σ zz b q

734

Example 2-

σx =

σ x bL

τ zx =

P

Example 3-

σx =

σ xb

τ zx bL P

τ zx =

q0

τ zx b q0

σ zz =

σ zz bL

σ zz =

P

σ zz b q0

Fig 1-variation of axial stress ( σ x ) through the

Fig 2-variation of shear stress

thickness of simply supported beam

thickness of the simply supported beam

(At x = L

2

,

(x = 0, z ) , when subjected to uniformly distributed

z ) when subjected to uniformly

load for aspect ratio 4 and 10, by using equilibrium equation from 2-D elasticity problem

distributed load for aspect ratio 4 and 10.

Fig 3-variation of shear stress thickness

(x = 0, z )

of

the

simply

τ zx

through the

supported

τ zx through the

Fig 4-variation of transverse stress σ zz through the thickness of simply supported beam when subjected to uniformly distributed load

beam

, when subjected to uniformly distributed load for aspect ratio 4, by using 1-D constitutive law of elasticity

735

CONCLUSION1. Analysis of thick beams is carried out using first order shear deformation theory. Simply supported beam carrying uniformly distributed load, carrying point load at the centre and a beam carrying uniformly varying load are solved using the present theory. 2. The results of axial stress for the beam under given loading, obtained by the present theory are more accurate than the Euler and Bernoulli theory. 3. The transverse shear stress obtained from constitutive relation and equilibrium using present theory gives less error than the Euler and Bernoulli theory. 4. The governing differential equations and the associated boundary conditions presented are variationally consistent. REFERENCES -

[1] Lord Rayleigh, J.W.S. (1877): The Theory of Sound , Macmillan Publishers, London. [2] Levinson. M. (1981): A new rectangular beam theory. J. Sound and Vibration. 74(1), 81-87 [3] Bickford, W.B. (1982): A consistent higher order beam theory, In: Proceeding of Dev. In Theoretical And applied Mechanics ,SETAM,11 137-150

[4] Rehfield, L.W. and Murthy, P.L.N (1982): Toward a new engineering Theory of Bending: fundamentals. AIAA Journal. 20(5), 693-699

[5] Krishna Murthy, A.V. (1984): Towards a consistent beam theory, AIAA Journal. 22 (6),811-816 [6] Bhimaraddi, A., Chandrashekhara K. (1993): Observations on higher order beam Theory. ASCEJ. Aerospace Engineering. 6(4), 408-413

[7] Heyliger, P.R, Reddy, J.N. (1988): A higher order beam finite element for bending and vibration problems. J. Sound and Vibration. 126(2),309-326

[8] Vlasov, V. Z., Leont’ev, U.N. (1966) : Beams, Plates and Shells on Elastic Foundations Moskva, Chapter 1, 1-8. Translated form the Russian by A. Barouch , and T. Plez , Iseral Program for Scientific Translation Ltd., Jerusalem

[9] Stein, M. (1989): Vibration of beams and plate strips with three dimensional flexibility. ASME J. App. Mech. 56(1), 228-231

[10] Ghugal, Y.M., Shmipi, R. P. (2001): A review of refined shear deformation theories for isotropic and anisotropic laminated beams. Journal of Reinforced Plastics And Composites,20(3),255-272

[11] Timoshenko, S. P. (1921): On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars. Philosophical Magazine, Series 6, Vol. 41, 742-746

[12] Timoshenko, S. P., Goodier, J. N. (1970): Theory of Elasticity, McGraw-Hill, 3rd Int. ed., Singapore.

736

INTERNATIONAL CONFERENCE ON STRUCTURAL ENGINEERING AND MECHANICS (ICSEM 2013) December 20-22, 2013

Co-Editors: Pradip Sarkar, Robin Davis P. and Srinivas Sriramula

ISBN 978-93-80813-26-4

DEPARTMENT OF CIVIL ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA ODISHA - 769 008, INDIA

TABLE OF CONTENTS Paper Authors No.

Title

Page No.

002

R. P. Khandelwal; Anupam Chakrabarti and Pradeep Bhargava

Calculation of inter-laminar shear stresses in laminated composite shallow shell using least square of error (LSE) method

003

Ajay Kumar; Anupam Chakrabarti and Pradeep Bhargava

Finite element analysis of free vibration of laminated composite spherical shells with cut-outs

12

004

S. N. Patel

Nonlinear response of laminated composite stiffened plates

19

006

Manav Mittal and Pijush Samui

Revived technique for compressive strength of CFRP confined concrete cylinders using Gaussian process regression

36

008

M.N.A. Gulshan Taj and Anupam Chakrabarti

Bending analysis of functionally graded sandwich plates

41

009

V. Chitra and R.S. Priyadarsini

Effect of lay-up sequences and imperfections on the dynamic buckling of CFRP cylindrical shells

48

010

Suprabhat. Jakati

Evaluation of shear strength of RC members using different methods

58

012

Shivaji T Bidgar and Partha Bhattacharya

The study of tension stiffening and crack band width in reinforced concrete beam under flexure

65

015

Soumi Bhattacharyya and Aparna (Dey) Ghosh

Effect of mass ratio on the performance of a TMD with non-optimal damping

71

016

Chhabirani Tudu and Asha Patel

Study of torsional behaviour of rectangular RC beam wrapped with GFRP

79

017

Jyotirmoya Dutta Majumdar and Aparna (Dey) Ghosh

Control of wind induced vibration in transmission line towers using tuned liquid column damper

88

018

Swetapadma Panda; Joygopal Jena and Bidyadhar Basa

Stress analysis around spiral casing of Francis Turbine of a Hydel power house by finite element method

96

i

1

Paper Authors No.

Title

Page No.

019

Benazir F. Ahmed and Kaustubh Dasgupta

Influence of location of staircase on seismic behaviour of RC flat slab building

105

020

Arijit Acharjya and Kaustubh Dasgupta

Influence of staircase and elevator core on twisting behaviour of RC wall-frame buildings

113

021

P. Bhattacharya and S. Karar

Random vibration analysis of RCC structure using pseudoexcitation method

121

022

A. Roy and Aparna (Dey) Ghosh

Tuned liquid damper system for seismic vibration control of elevated water tanks

129

023

Manish K. Singh; Sailesh Adhikari; Sumit Mazumder and Amiya K. Samanta

Study on stability of natural draft hyperbolic RC cooling tower shell under wind induced flow field

137

025

R. Velmurugan and Jafar Sadique P

An experimental and numerical study of natural-fiber composites subjected to different environmental conditions

145

026

N Murli Krishna and Md Masihuddin Siddiqui

Evaluation of performance point using energy based pushover analysis

154

028

Arghya Sengupta and Rana Roy

Seismic behaviour of R/C frames with biaxial interaction

164

029

Pawan Thakur and Rana Roy

Seismic behaviour of plan-asymmetric structures under spectrally matched records

174

031

Syed H. Basha and Hemant B. Kaushik

Influence of masonry properties on lateral load response of reinforced concrete frames

185

032

Somdatta Goswami; Subrata Chakraborty and Shymal Ghosh

Adaptive response surface method in structural response approximation under uncertainty

194

035

Avaya K. Baliarsingh; P. K. Ray; B. B. Verma and Deepak K. Agarwalla

Prediction of fatigue crack propagation life in single edge notched beam using exponential model

203

036

Deepak K. Agarwalla; Dayal R. Parhi and Amiya K. Dash

Fuzzy logic based fault diagnosis of structures using vibration responses

210

ii

Paper Authors No.

Title

Page No.

037

Komathi M. and Amlan K.Sengupta

Strengthening of columns for shear in reinforced concrete buildings

222

038

Shakti P. Jena and Dayal R. Parhi

Dynamic analysis of cantilever beam with moving mass

230

039

Saswati De and K. Bhattacharya

Behaviour of two-span continuous steel box RC deck composite bridges

238

042

Shantaram Parab; Jatin Alreja and Pijush Samui

Prediction of hysteretic energy demand in steel moment resisting frames using multivariate adaptive regression splines

250

043

H.D. Chalak; Anupam Chakrabarti; Mohd. Ashraf Iqbal and Abdul Hamid Sheikh

Buckling analysis of laminated soft core sandwich plate

258

044

A. Subbulakshmi and P. Jayabalan

Nonlinear analysis of square concrete filled steel tubes under constant axial load and monotonic lateral loading

266

045

Bijan Kumar Roy and Subrata Chakraborty

Reliability based design of TMD system considering system parameter uncertainty in seismic vibration control

275

046

Bulu Pradhan

Performance evaluation of concrete against rebar corrosion in composite chloride-sulphate exposure conditions

284

048

Fouzia Shaheen and Bulu Pradhan

Potentiodynamic polarization study on bare steel in concrete powder solution extracts contaminated with chloride and sulphate ions

291

049

Arun Mukherjee; Sreyashi Das (nee Pal) and A. Guha Niyogi

Non-linear vibration and dynamic response of epoxy-based laminated composite plate with square cut-out at elevated temperature

300

053

M. V. N. Sivakumar and B. Vineel Kumar

Seismic vulnerability assessment of Indian standard code designed midrise and high-rise RC framed structures using fragility analysis

308

054

Snehal Kaushik and Kaustubh Dasgupta

Seismic behaviour of slab-structural wall junction in rc building

316

055

Sanjay Goswami; Partha Bhattacharya and Subham Rath

Investigating effectiveness of dynamic strain response for damage detection in composite beams

324

iii

Paper Authors No.

Title

Page No.

056

A. Vanuvamalai and K.P. Jaya

Finite element analysis of road tunnel

332

059

Tushar Kanti Dey , Anupam Chakrabarti and Umesh Kumar Sharma

Optimization of a fiber reinforced polymer web core skew bridge

339

060

G. V. Rama Rao; M. Pavan Kumar and Sk. Rasool

Cost effective structural systems for high rise buildings subjected to seismic load

347

061

Soumya Bhattacharjya; Urmi Saha; Asish Modak and Baidyanath Sarker

Probabilistic assessment of safety, economy and robustness of steel structure including parameter uncertainty in is: 8002007 format

356

062

A. K. Nayak; R. A. Shenoi and J. I. R. Blake

A computer aided fem based numerical solution for transient response of laminated composite plates with cutouts

363

063

C. Mohanlal and N. Murali Krishna

Optimal design of a retractable roof structure using real coded genetic algorithm

372

064

Nityananda Nandi

The change in stress distribution pattern within the earthen dam due to the effect of seepage

381

065

Bhaskar Ghosh and Subhashish Roy Chowdhury

Fundamental period of vertically irregular frames – a modification over the codal stipulation

394

066

Maganti Janardhana; Jogi Naidu P. and Hymavathy K.

Structural behaviour of an RC building on sloping ground under earthquake load

403

067

V. R. Kar and S. K. Panda

Thermal stability analysis of functionally graded panels

416

068

M. Mishra and K. C. Panda

Properties of rubberised fly ash concrete

422

073

G Muthukumar and Manoj Kumar

Influence of opening on the dynamic structural response of rectangular slender shear wall for different damping ratios

430

074

S S S Sastry; Anasuyeswar K; Ayush Mathur and Shailaja B

Studies on the channel fitting in aircraft using classical and finite element methods

447

076

P.V. Katariya and S.K. Panda

Modal analysis of laminated composite spherical shell panels using finite element method

454

iv

Paper Authors No.

Title

Page No.

077

Hipparagi A. K.; Injaganeri S.S. and Mahadevgouda H.

Enhancement of shear ductility for HSC beams with minimum shear reinforcement

463

083

J S Ali and S Gupta

Spectral strain energy based approach for system identification in structures

471

085

A. Syed Mohamed and C. O. Arun

Towards an eco-friendly concrete with waste glass and rice husk ash

478

086

K. V. S. N. Murthy and B. Dean Kumar

A study on analysis of turbo generator foundation resting on pile foundations for various earthquake zones in india

486

087

Rajiv Verma and Puneet Mathur

Transient analysis of an elliptical journal bearing

493

088

A.S. Balu and B.N. Rao

Response surface based analysis of structures with fuzzy variables

503

090

Shiv Shanker Ravichandran and Richard E. Klingner

Seismic design factors for steel moment frames with masonry in-fills: Part 1

510

091

Shiv Shanker Ravichandran and Richard E. Klingner

Seismic design factors for steel moment frames with masonry in-fills: Part 2

521

092

K. Sharma; V. Bhasin; I. V. Singh; B. K. Mishra and R. K. Singh

Simulation of bi-metallic interfacial crack using EFGM and XFEM

532

093

Moon Banerjee; N. K. Jain and S. Sanyal

Evaluation of stress concentration for a simply supported laminated composite plate with a centre circular hole under uniform transverse loading

540

096

Revathi P.; Ramesh, R. A. and Lavanya, K.

Effect of treatment methods on the strength characteristics of recycled aggregate concrete

548

097

Shashank Pandey and S. Pradyumna

Finite element analysis of sandwich plates with functionally graded material core using a layerwise theory

556

099

Irshad A Khan and Dayal R Parhi

Diagnosis of multiple crack of cantilever composite beam by vibration analysis and hybrid AI technique

564

109

Rehan A. Khan and T. Naqvi

Performance based design of rcc building under earthquake forces

571

v

Paper Authors No.

Title

Page No.

111

A.K.L. Srivastava

Effect of stiffened cut-out on vibration and parametric excitation

580

112

H. S. Panda; S. K. Sahu and P. K. Parhi

Modal analysis of delaminated woven fibre composite plates in moist environment

585

117

Abhishek Kumar; V. S. Phanikanth and K. Srinivas

Seismic analysis of reinforced concrete ventilation stack using simplified modal analysis technique

593

118

G Srikar; B Gopi and S Suriya Prakash

Effect of temperature on compressive behaviour of concrete reinforced with structural polypropylene fibres

604

119

C Sreenivasulu; R Mehar Babu; S Suriya Prakash and K.V.L. Subramaniam

Behaviour of masonry assemblages made of soft brick under compression

614

120

K.V.V. Sumanth and S. Suriya Prakash

Effect of tension stiffening on behaviour of concrete columns under pure torsion

624

121

M. R. Das and S. Samal

Optimal earthquake resistant design of a fixed beam using a simple optimization tool

633

122

Souvik Chakraborty and Rajib Chowdhury

Uncertainty propagation using hybrid HDMR for stochastic field problems

642

123

P. Dinakar and Manu S. Nadesan

Design and development of high strength self compacting concrete using metakaolin

657

124

Monalisa Priyadarshini; Robin Davis; Pradip Sarkar and Haran Pragalath D C

Seismic reliability assessment of R/C stepped frames

663

126

P.R. Ravi Teja and Sasmita Sahoo

Major indian earthquakes and earthquake risk assessment – a case study

671

128

P. R. Maiti; Bhawesh Madhukar and Satyam Mandloi

Analysis of plate resting on Pasternak and Winkler foundation due to moving load

685

129

Pardeep Kumar and R. K. Sharma

Determination of dynamic modulus of elasticity of concrete

695

vi

Paper Authors No.

Title

Page No.

130

S. C. Choudhury; A. Simadri Dora; M. K. Tripathy and P. M. Tripathy

Construction of high level bridge over river Mahanadi at 6th km of Sambalpur-Sonepur road to Chadheipank-Binca road - a case study

704

131

D. Jena and K.C. Biswal

Performance evaluation of tuned liquid damper on response of high rise structure under harmonic load

707

132

Ganesh R.; Haran Pragalath D.C.; Robin Davis; Pradip Sarkar and S. P. Singh

Seismic fragility analysis of axially loaded single pile

713

133

Priyadarshi Das and ManasRanjan Das

Optimal design of a tuned mass damper using a simple optimization tool

720

134

Sourabh S. Deshpande and R. L. Wankhade

Analysis of thick beams using first order shear deformation theory

728

135

Jitendra Kumar Meher and Manoranjan Barik

Free vibration of multiple-stepped Bernoulli-Euler beam by the spectral element method

737

136

Istiyak Khan; M. K. Agrawal and S. B. Chafle

Attenuation of seismic response of structures using passive devices

745

137

N. Trivedi and R. K. Singh

Experimental and analytical estimation of concrete creep for large prototype structure

753

138

N. Trivedi and R. K. Singh

Numerical simulation for VTT impact test under iris program

760

Synthetic ground motion generation using a semi analytical model

766

Experimental and analytical studies of ratcheting in pressurized piping system under seismic load

774

139

140

A. Ravi Kiran; M. K. Agrawal; G. R. Reddy; R. K. Singh and K. K. Vaze A. Ravi Kiran; P. N. Dubey; M. K. Agrawal; G. R. Reddy; R. K. Singh and K. K. Vaze

141

M. Bandyopadhyay and A. K. Banik

Numerical analysis of semi-rigid jointed steel frame using rotational springs

782

142

Kotla Shiva Kumar and Diptesh Das

Passive control of structures subjected to earthquake excitation using particle swarm optimization

791

vii

Paper Authors No.

Title

Page No.

143

Jagdish Malav; Swapnil Takle; Kali P. Sethy; Dinakar. P and U. C. Sahoo

Properties of high strength concrete containing ultrafine slag

799

144

A. K. Banik and M. Bandyopadhyay

Progressive collapse study of semi-rigid jointed frame structures: a state-of-the art review

807

145

M. K. Agrawal; A. Ravi Kiran; G. R. Reddy; R.K. Singh; K. K. Vaze; D. K. Sakhrodia; B. Biswas and A. Bhowmick

Re- evaluation of industrial equipments for earthquake and wind loads

827

146

R. K. Sharma and Pardeep Kumar

Utilization of dredged material from reservoir of hydro project as fine aggregate in concrete

834

147

D. V. Prasada Rao and G. V. Sai Sireesha

Effect of silica fume on strength of Partially used recycled coarse aggregate concrete

843

148

Pratika Preeti

Seismic response of vertically irregular structures: issues to be addressed

850

149

Tarapada Mandal and Sanjay Sengupta

Slope stability analysis by static and dynamic method

858

150

Karthik Subhash A.; U. K. Mishra and Ganesh R.

Damage detection in simply supported beam - an ANN approach

865

151

Kirtikanta Sahoo; Pradip Sarkar and Robin Davis P.

Behaviour of recycled aggregate concrete

871

viii

International Conference on Structural Engineering and Mechanics December 20-22, 2013, Rourkela, India Paper No.: 134

ANALYSIS OF THICK BEAMS USING FIRST ORDER SHEAR DEFORMATION THEORY 1

2

Sourabh S. Deshpande and R. L. Wankhade 1

2

PG Student, Dept. of Applied Mechanics, Government College of Engineering Karad, Maharashtra, India Asst. Professor, Dept. of Applied Mechanics, Government College of Engineering Karad, Maharashtra, India Email: [email protected], [email protected]

ABSTRACT : A first order shear deformation beam theory is employed here for the static analysis of thick beams. The limitations of classical theory of beam bending developed by Euler and Bernoulli forced for the refinement over these classical theories. After refinement discrepancies of the classical beam theory are eliminated and hence first order shear deformation theory is developed. The assumption of Euler-Bernoulli beam theory stating that the plane section remains perpendicular to the neutral axis of the section after the deformation is modified by Timoshenko. Timoshenko beam theory assumes that the section which was normal to the neutral axis before deformation does not remain normal to the neutral axis after the deformation. He considers the rotation of the normal to be the combined effect of the rotation due to bending and shear effects. But, Timoshenko theory does not remove another discrepancy of the classical theory i.e. normal section remains plane after bending. Timoshenko also assumes that plane section before bending remains plane after bending also. Thus, He assumes the constant shear strain across the section and thus rectangular distribution of the shear stress across the section, which is not in accordance with the actual distribution of the shear stress which is parabolic across the section. Thus, shear correction factor is required in this theory to modify the result to get the fairly accurate result. KEYWORDS:

Shear deformation theory, Thick beams, Shear stress

1. INTRODUCTION The elementary theory of beam bending introduced by Euler and Bernoulli is the simplest theory of beam bending which neglects the effects of the shear deformation. The theory assumes that the plane section which is normal to the neutral axis before bending remains plane and normal to the neutral axis even after deformation of the beam. That means shear stresses are not considered in the hypothesis. Effect of rotary inertia and shear deformation are included in the theory of beam by Rayleigh (1877) and Timoshenko (1921) as the pioneer investigators. To remove the discrepancies in the classical and first order shear deformation or Timoshenko theories, higher order shear deformation theories were developed by various researchers. Levinson (1981), Bickford (1982), Rehfield and Murty (1982) Krishna Murty (1984), Bhimaraddi and Chandrashekhara (1993) are some of the investigators who presented parabolic shear deformation theories assuming a higher variation of the axial displacement in terms of the thickness coordinate for the analysis of thick beams. Finite element method is used by Kant and Gupta (1988), Heyliger and Reddy (1988) based on higher order shear deformation of uniform rectangular beams. Further refined shear deformation theories for thick beams including sinusoidal functions in terms of the thickness coordinate in displacement field are developed by Vlasov and Leont’ev (1996), Stein (1989). The drawback of these theories was that shear stress free boundary conditions were not satisfied at top and bottom

surfaces of the beam. Thus, a refined trigonometric shear deformation theory has been developed by Ghugal and Shimpi (2001) to accurately predict shear stress distribution across the section of the beam and which also satisfies the free boundary conditions at top and bottom surfaces of the beam. In this paper, first order shear deformation theory is developed and various non dimensional parameters are introduced. These parameters are applied to numerical examples to give values of these parameters for the specific cases and then these values are plotted on the graph to show the variation of these parameters across the section of the beam.

2. EQUATIONS AND DEVELOPEMENT OF THEORYThe beam under consideration is shown in figure. It occupies in 0 − x − y − z Cartesian coordinate system in the region;

0 ≤ x ≤ L;

0 ≤ y ≤ b;

−h ≤ z≤h 2 2

Where x, y, z are Cartesian coordinates, L is the length of the beam in x direction, b is the width of the beam in y direction and h is the thickness of the beam in z direction. The beam is made up of homogeneous, linearly elastic isotropic material.

h L

2.1 The displacement field The displacement field of the present beam theory is given by;

u ( x , z ) = − zφ

(1)

w( x, z ) = w(x) Where u is the axial displacement in x direction and w is the transverse displacement in z direction of the beam. The function φ represents total rotation of the beam at neutral axis, which is an unknown function to be determined. The normal and shear strains obtained within the framework of linear theory of elasticity using displacement field given by above equation is as followsNormal strain: ε x =

Shear strain:

γ zx =

∂u dφ = −z ∂x dx

(2)

∂u ∂w ∂w + = −φ + ∂z ∂x ∂x

(3)

729

The stress-strain relationship used is as follows-

σx = εx ⋅E

τ zx = γ zx ⋅ G

(4)

The above relationships are used to obtain the governing differential equations of the present beam theory by substituting these equations in the principle of virtual work. The procedure is explained in the next section. 2.2 Governing equations and boundary conditions Using the expressions for strains and stresses (2) through (4) and using the principle of virtual work, variationally consistent governing differential equations and boundary conditions for the beam under consideration can be obtained. The principle of virtual work, when applied to the beam leads to; h x = L y =b z = 2

x=L

x =0 y =0 z =−

x =0

∫ ∫ ∫h (σ xδε x + τ zxδγ zx )dxdydz − ∫ q(x )δwdx = 0 2

(5)

Where, symbol δ denotes the variational operator. Substituting the values of stresses and strains in the above equation, we get the coupled Euler-Lagrange equations which are the governing differential equations and associated boundary conditions of the beam. The governing differential equations are obtained as underh x = L y =b z = 2

x=L

x =0 y =0 z = −

x=0

∫ ∫ ∫h (Eε xδε x + Gγ zxδγ zx )dxdydz = ∫ qδwdx 2

Putting the values of stresses and strains from the above equations and then integrating and equating the coefficients of δw and δφ , we get governing differential equations of this theory as follows-

dφ d 2 w GAK s − 2 = q( x ) dx dx

(A)

d 2φ dw GAK s − φ + EI 2 = 0 dx dx

(B)

2.3 The general solution of governing equilibrium equations of the beamUsing above obtained differential equations, we can obtain the general equation of this beam theory in terms of rotation of the beam. From equation (B),

EI

d 2φ dw − GAK s φ − =0 2 dx dx

Differentiate w. r. t. x, we get,

dφ d 2 w d 3φ EI 3 − GAK s − 2 = 0 dx dx dx

∴ From equation (A),

730

∴ EI

d 3φ = q(x ) dx 3

This is the general equation of this beam theory in terms of the rotation of the beam at the neutral axis. Using this equation, we can obtain the expression for transverse displacement of the beam by substituting the value of rotation of the beam in either of the equations (A) or (B). 3. ILLUSTRATIVE EXAMPLES AND DIAGRAMSIn order to apply above theory to various boundary conditions, we have considered three examples here. With the help of these examples, we can predict the distribution of various non dimensional parameters across the section of the beam. Material properties of the beam are as follows-

E = 210GPa

µ = 0.25

ρ = 7800 kg

Where, E =modulus of elasticity of the material, the beam material

m3

µ =Poisson’s ratio for the beam material and ρ =density of

Example 1-simply supported beam with uniformly distributed load over the entire span-

b

L

d 3φ EI 3 = q ( x ) dx By taking integration of above equation thrice and substituting appropriate boundary conditions in the equations; we get the expression for the rotation of the beam at the neutral axis. It is given as-

∴ EIφ =

qx 3 qLx 2 qL3 − + 6 4 24

This expression gives the rotation of the simply supported beam carrying uniformly distributed load over the entire span at any section of the beam along its length. Now, this expression is used to find out the expression for the transverse displacement of the beam at any section along the length of the beam. From equation (A),

d 2 w dφ q = − 2 dx GAK s dx

731

Putting the value of φ in the above equation and then integrating twice with respect to x by putting appropriate boundary conditions in the equations, we get the expression for the transverse displacement of the beam at neutral axis. It is given by-

q w= 2 EI

x 4 Lx 3 qx 2 qL3 x qLx − − + + 6 2GAK s 24 EI 2GAK s 12

After modification, the final expression for the transverse displacement is given by,

w=

qL4 24 EI

x4 x3 x qL2 x x 4 − 2 3 + + 1 − L 2GAK s L L L L

For the maximum transverse displacement, put x = L

wmax =

Put

2

in above expression, we get,

5 qL4 384 EI 1 + 384 EI 40 L2 GAK s

E = 2(1 + µ ) and simplify the equation, we get the final expression for maximum transverse G

displacement as,

wmax

2 5 qL4 h = 1 + 1.6(1 + µ ) 384 EI L

Non dimensional transverse displacement is given by,

w=

w h 4 Eb h L4 q

h ∴ w = 1 + 1.6(1 + µ ) L

2

From the above values of transverse displacement and rotation of the beam at the neutral axis, we can find out the remaining values of axial stress in x-x direction, shear stress in x-z plane, stress in z-z direction and axial displacement u in the form of non dimensional parameters. These can be found out as followsExpression for axial displacement u-

zq L u=− 2 Eb h

3

x3 x2 4 3 − 6 2 + 1 L L

732

Expression for axial stress sigma xx-

q z L σ x = −6 b h h

2

x2 x 2 − L L

Expression for transverse shear stress using constitutive relationship τ zxCR It may be noted that it is possible to obtain transverse shear stress τ zx , by using either constitutive relationship or the equilibrium equation of theory of elasticity. Notion τ zxCR denotes τ zx obtained by using constitutive relationship.

τ zx =

qL x 1 − bh L 2

Expression for transverse shear stress τ zxEE , and transverse normal stress, σ z , obtained from equation-

equilibrium

The following equilibrium equations of two dimensional elasticity ignoring body forces are used to obtain transverse shear and transverse normal stresses.

∂σ x ∂τ zx + =0 ∂x ∂z ∂τ xz ∂σ z + =0 ∂x ∂z

To get τ zx , substitute the expression obtained for axial stress σ x in equilibrium Equation above and integrate with respect to thickness coordinate z and impose the following boundary condition at bottom of beam

[τ zx ]z =± h / 2 = 0

To get constant of integration. The expression obtained for transverse shear stress using this procedure in its final form is as follows:

τ zx = −

3 q L x 1 z2 − 1 − 4 2 b h L 2 h 2

To get σ z , substitute the expression obtained for shear stress, in the equilibrium Equation above and integrate it with respect to thickness coordinate z and impose the following boundary condition at bottom of the beam

[σ z ]z =h / 2 = 0

To get the constant of integration. The expression obtained for transverse normal stress σ z is as follows:

σ zz

q1 3 z z3 = − + 2 3 b2 2 h h

733

Example 2-simply supported beam carrying uniformly varying load over entire span-

h

L

In this beam, the loading is expressed in terms of maximum intensity of loading q 0 and distance of section along the length of beam i.e. x . The expression is given by,

EI

d 3φ x = q 0 1 − 3 dx L

After integrating w. r. t. x thrice by applying appropriate boundary conditions to find constants of integrations, we get the expression for rotation of this beam at the neutral axis. This expression is given by,

q 0 L3 φ= 6 EI

x4 x3 x2 3 − 4 + 3 − 2 + L L 17 4L

This expression is used to find the expression for transverse displacement. From equation (A), we can write,

q0 d 2 w dφ x = − 1 − 2 dx GAK s L dx

Putting the value of φ in above equation and then integrating twice w. r. t. x by applying appropriate boundary conditions to find out the constants of integration, we get the expression for transverse displacement of beam at any section along the length of the beam. It is given by,

q0 L4 w= 6 EI

x5 x4 x3 3 x q0 L2 x 2 x3 x − + − + − − − 0.42 5 4 3 2 3 L 20L 4 L 3L 17 L GAK s 2 L 6 L

Non dimensional parametersExample 1-

σx =

σ xb q

τ zx =

τ zx b q

σ zz =

σ zz b q

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Example 2-

σx =

σ x bL

τ zx =

P

Example 3-

σx =

σ xb

τ zx bL P

τ zx =

q0

τ zx b q0

σ zz =

σ zz bL

σ zz =

P

σ zz b q0

Fig 1-variation of axial stress ( σ x ) through the

Fig 2-variation of shear stress

thickness of simply supported beam

thickness of the simply supported beam

(At x = L

2

,

(x = 0, z ) , when subjected to uniformly distributed

z ) when subjected to uniformly

load for aspect ratio 4 and 10, by using equilibrium equation from 2-D elasticity problem

distributed load for aspect ratio 4 and 10.

Fig 3-variation of shear stress thickness

(x = 0, z )

of

the

simply

τ zx

through the

supported

τ zx through the

Fig 4-variation of transverse stress σ zz through the thickness of simply supported beam when subjected to uniformly distributed load

beam

, when subjected to uniformly distributed load for aspect ratio 4, by using 1-D constitutive law of elasticity

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CONCLUSION1. Analysis of thick beams is carried out using first order shear deformation theory. Simply supported beam carrying uniformly distributed load, carrying point load at the centre and a beam carrying uniformly varying load are solved using the present theory. 2. The results of axial stress for the beam under given loading, obtained by the present theory are more accurate than the Euler and Bernoulli theory. 3. The transverse shear stress obtained from constitutive relation and equilibrium using present theory gives less error than the Euler and Bernoulli theory. 4. The governing differential equations and the associated boundary conditions presented are variationally consistent. REFERENCES -

[1] Lord Rayleigh, J.W.S. (1877): The Theory of Sound , Macmillan Publishers, London. [2] Levinson. M. (1981): A new rectangular beam theory. J. Sound and Vibration. 74(1), 81-87 [3] Bickford, W.B. (1982): A consistent higher order beam theory, In: Proceeding of Dev. In Theoretical And applied Mechanics ,SETAM,11 137-150

[4] Rehfield, L.W. and Murthy, P.L.N (1982): Toward a new engineering Theory of Bending: fundamentals. AIAA Journal. 20(5), 693-699

[5] Krishna Murthy, A.V. (1984): Towards a consistent beam theory, AIAA Journal. 22 (6),811-816 [6] Bhimaraddi, A., Chandrashekhara K. (1993): Observations on higher order beam Theory. ASCEJ. Aerospace Engineering. 6(4), 408-413

[7] Heyliger, P.R, Reddy, J.N. (1988): A higher order beam finite element for bending and vibration problems. J. Sound and Vibration. 126(2),309-326

[8] Vlasov, V. Z., Leont’ev, U.N. (1966) : Beams, Plates and Shells on Elastic Foundations Moskva, Chapter 1, 1-8. Translated form the Russian by A. Barouch , and T. Plez , Iseral Program for Scientific Translation Ltd., Jerusalem

[9] Stein, M. (1989): Vibration of beams and plate strips with three dimensional flexibility. ASME J. App. Mech. 56(1), 228-231

[10] Ghugal, Y.M., Shmipi, R. P. (2001): A review of refined shear deformation theories for isotropic and anisotropic laminated beams. Journal of Reinforced Plastics And Composites,20(3),255-272

[11] Timoshenko, S. P. (1921): On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars. Philosophical Magazine, Series 6, Vol. 41, 742-746

[12] Timoshenko, S. P., Goodier, J. N. (1970): Theory of Elasticity, McGraw-Hill, 3rd Int. ed., Singapore.

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