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
