STATIC AND DYNAMIC BEHAVIOR OF TAPERED BEAMS ON TWO-PARAMETER FOUNDATION · 2018-09-14 · In the present work, the stability and vibration behavior of axially-loaded tapered beams

IJRRAS 14 (1) ● January 2013 www.arpapress.com/Volumes/Vol14Issue1/IJRRAS_14_1_ 20.pdf

176

STATIC AND DYNAMIC BEHAVIOR OF TAPERED BEAMS ON

TWO-PARAMETER FOUNDATION

Mohamed Taha Hassan & Mohamed Nassar

Dept. of Eng. Math and Physics, Faculty of Eng., Cairo University, Giza.

ABSTRACT

The static and dynamic behaviors of tapered beams resting on two-parameter foundations are studied using the

differential quadrature method (DQM). The governing differential equations are derived and discretized; then the

appropriate boundary conditions are discretized and substituted into the governing differential equations yielding a

system of homogeneous algebraic equations. The equivalent two-parameter eigenvalue problem is obtained and

solved for critical loads in the static case (=0) and natural frequencies in the dynamic case with a prescribed value

of the axial load (PoPcr). The obtained solutions are found compatible with those obtained from other techniques. A

parametric study is performed to investigate the significance of d ifferent parameters.

Keywords: Tapered beams, two-parameter foundation, differential quadrature, critical load and natural

frequencies.

1. INTRODUCTION

Nonprismat ic elements are commonly used in many practical applicat ions to optimize weight or materials. The static

and dynamic behavior of such elements need design criteria to identify the optimal configurations. The analytical

treatments of such elements are intractable due to the complicated governing equations whereas the numerical

techniques offer tractable alternatives. Different configurations are studied by many researchers to obtain stability

and/or vibration behaviors of such structural elements. Closed forms and analytical solutions for simple cases of

prismat ic and non-pris matic elements are found in literature. Taha and Abohadima [1-2] studied the free vibrat ion of

non-uniform beam resting on elastic foundation using Bessel functions. Taha [3] investigated the nonlinear v ibration

of initially stressed beam resting on elastic foundation by employing the elliptic integrals. Ruta [4] used the

Chebychev series to obtain solutions for non-prismatic beam vibration. Asymptotic perturbation has been used by

Maccari [5] to analyze the nonlinear dynamics of continuous systems. Sato [6] reported the transverse vibration of

linearly tapered beams using Ritz method. He studied the effect of end restraints and axial fo rce on the vibration

frequencies. Numerical methods such as the FEM [7-9], the differential transform methods [10, 11] and the

differential quadrature method [12-14] are used to study certain configurations of such elements.

The free vib ration of tapered beams with nonlinear elastic restraints was studied by Naidu [9] using the FEM, and

the effect of tapering rat io and end restraints were analyzed.

The behavior of non-prismat ic beams resting on elastic foundations had received a little attention in literature due to

the complexity in its mathemat ical treatment and most researches in that area were carried out to investigate special

cases.

In the present work, the stability and vibration behavior of axially -loaded tapered beams resting on a two-parameter

foundation will be investigated using the DQM. The present work differs than Naidu work [9] in implementing the

two-parameter foundation and axial compression load.

The governing equations are formulated in dimensionless form, d iscretized over the studied domain; and the

boundary conditions are discretized and substituted into governing equations yielding a system of homogeneous

algebraic equations. Using eigenvalue analysis yields the critical loads in static case (=0) and natural frequencies

for a prescribed axial load value (Po Pcr). The obtained solutions will be verified and the effects of different

parameters related to the studied model on the stability and frequency parameters will be illustrated.

2. FORMULATION OF THE PROBLEM

2.1 Vibration equation

The free vibration equation of a non-prismatic beam axially-loaded by Po and resting on a two-parameter foundation

shown in Fig.(1) is given as:

2 2 2 2

2 12 2 2 2( ) ( ) ( ) 0o

Y Y YEI X P k A X k Y X

X X X t

(1)

where I(X) is the moment of inertia of the beam cross section at X; is the mass density per unit volume; E is

modulus of elasticity; A(X) is the area of cross section at X; Y(X, t) is the lateral displacement; Po is the axial load

IJRRAS 14 (1) ● January 2013 Hassan & Nassar ● Static and Dynamic Behavior of Tapered Beams

177

acting on the beam; k1 and k2 are the foundation stiffness per unit length of the beam; X is the distance along the

beam; and t is the time.

Figure (1): The axia lly -loaded tapered beam on a two-parameter foundation

Using dimensionless parameters x=X/L and y=Y/L, eqn. (1) can be expressed in dimensionless form as: 2 2 2 2

212 3 2 2 2

( )( )( ) ( ) 0oP kEI x y y y

LA x k Ly xLx L x x t

(2)

The solution of eqn. (2) depends on the boundary conditions at the beam ends

2.2 Boundary conditions

The boundary conditions at the beam ends depend on the type of support at the ends. For clamped support (C) at

location x (x=0, 1) which prevents translation and rotation, the boundary conditions can be expressed as:

( , ) 0 and ( , ) 0y

y x t x tx

(3.a)

For pinned support (P) at location x (x=0, 1) which prevents translation and allows rotation, the boundary conditions

are expressed as: 2

2

( , )( , ) 0 and 0

y x ty x t

x

(3.b )

For free support (F) at location x (x=0, 1) which allows translation and rotation the boundary conditions are

expressed as: 2 3

2 3

( , ) ( , )0 and 0

y x t y x t

x x

(3.c)

2.3 Mode functions

Equation (2) is a fourth-order linear d ifferential equation with variab le coefficients; hence, the separation of

variables technique can be addressed. Let the solution of eqn. (2) be assumed as:

( , ) ( ) ( )oy x t y x t (4)

Where (x) is the mode function, (t) is a function representing the variation of lateral displacement with time; and

yo is the dimensionless vibration amplitude (obtained from the initial conditions). Substituting eqn. (4) into eqn. (2),

then eqn. (2) can be separated into: 2 4 4 22 2 2

2 1

2 2 2

( ) ( )( ) ( ) 0oP k L k L L A xd d d

I x xd x d x E d x E

(5) 2

2

2( ) 0

dt

dt

(6)

where is the separation constant.

The solution of eqn. (6) is:

( ) sin( ) cos( )t A t B t (7)

where A and B are constants obtained from the initial conditions and is the natural frequency of the beam

vibration.


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The general solution of eqn. (5) depends on the distribution of the section geometry along the beam. Figure (1)

shows the case of a symmetric tapered beam, where the depth of the beam increases lineally from do at x=0 to d1 at

x=0.5, then decreases linearly form d1 at x=0.5 to do at x=1, while the width of the beam b is assumed constant, then:

( ) ( )od x d x (8.a)

where:

1-2x(1- ) for 0.0 x 0.5

(x) = 2 +2x(1- )-1 for 0.5 x 1.0

(8.b )

and =d1 / do is the tapering ratio.

Using the distribution of section geometry expressed in eqn. (8), the distribution of the area and moment of inertia of

the beam cross section with x are g iven as: 3( ) ( )

( ) ( )

o

o

I x I x

A x A x

(9)

where Ao and Io are the area and the second moment of area of the beam cross section respectively.

Substitution of eqn. (8) and eqn. (9) into eqn. (5) y ields:

2 4 2 44 / 3 / / / 2 2

2 1

4 3 2 3 2 3

( )6 3 6( ) 0o o

o o

P k L k L A Ld d dx

d x d x EI d x EI

(10)

where prime stands for differentiation w.r.to x.

Equation (10) is a fourth-order differential equation with variab le coefficients, which is difficult to be solved

analytically. However, solving eqn. (10) considering =0 (static case) yields the critical (buckling) loads Pcr, while

solving the equation with a prescribed value of Po (less than critical load) yields the natural frequencies of the free

vibration of axially-loaded tapered beams. The dimensionless boundary conditions at x=0, 1 can be rewritten as:

For clamped support:

( , ) 0 and ( , ) 0d

x t x tdx

(11.a)

For pinned support: 2

2

( , )( , ) 0 and 0

d x tx t

d x

(11.b)

and for free support: 2 3

2 3

( , ) ( , )0 and 0

d x t d x t

d x d x

(11.c)

3. SOLUTION OF THE PROBLEM

3.1 Differential Quadrature Method (DQM)

The solution of eqn. (10) is obtained using the differential quadrature method (DQM), where the solution domain is

discretized into N sampling points and the derivatives at any point are approximated by a weighted linear summation

of all the functional values at the other points as [13]:

( )

,

1

( )( ), ( 1, ), ( 1, )

i

m Nm

i j jmjX

d f xC f x i N m M

dx

(12)

where M is the order of the highest derivative in the governing equation, f(xj) is the functional value at point x=xj

and

( )m

ijCis the weighting coefficient relating the functional value at x=xj to the m-derivative of the function f(x)

at x=xi. To obtain the weighting coefficients, many polynomials with different base functions are commonly used to

approximate the functional values. Using the Lagrange interpolat ion formula, the functional value at a point x can be

approximated by all the functional values f(xk ) , (k=1, N) as:


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1 1

N N

j=1 1 k k=1

( )( ) ( )

( ) ( )

where: L(x)= ( ), L (x )= ( ) , ( , 1, )

N

k

k k k

j i k

L xf x f x

x x L x

x x x x i k N

(13)

Substitution of eqn. (13) into eqn. (12) y ields the weighting coefficients of the first derivative as [13]:

1 )(1)

,

1

(for( ) and ( , 1, )

( ) ( )

i

i j

i j j

L xC i j i j N

x x L x

(14a)

(1) 1

, ,

1,

for( ) and ( , 1, )N

i j i j

j j i

C C i j i j N

(14b)

Applying the chain rule onto eqn. (12), the weighting coefficients of the m-order derivative are related to the

weighting coefficients of (m-1) o rder derivative as:

( ) (1) ( 1)

, , ,

1

, ( , 1, ), ( 1, )N

m m

i k i k i k

k

C C C i k N m M

(15)

The DQM is a numerical method, hence the accuracy of the obtained results are affected by both the number and the

distribution of descretization points. Moreover, in boundary value problems, it is known that the irregular

distribution of the discretized points with smaller mesh spaces near the boundaries to cope the rapid variation near

the boundaries is more adequate. One of the frequently used distributions for mesh points generation is the

normalized Gauss-Chebychev – Lobatto distribution given as:

1 11 cos , ( 1, ).

2 1i

ix i N

N

(16)

3.2 Implementation of the Boundary Conditions

The boundary conditions due to support at x=0 can be discretized as:


(1)

1 1,

1

0 and ( ) 0N

j j

j

C x

(17a)

For pinned support:

(2)

1 1,

1

0 and ( ) 0N

j j

j

C x

(17b)

For free support:

(2) (3)

1, 1,

1 1

( ) 0 and ( ) 0N N

j j j j

j j

C x C x

(17c)

Also, the boundary conditions due to support at x=1are d iscretized as:


(1)

,

1

0 and ( ) 0N

N N j j

j

C x

(18a)

For pinned support:

(2)

,

1

0 and ( ) 0N

N N j j

j

C x

(18b)

For free support:

(2) (3)

, ,

1 1

( ) 0 and ( ) 0N N

N j j N j j

j j

C x C x

(18c)


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Rearranging the terms in eqn. (17) and eqn. (18), the unknowns 1, 2, N-1 and N can be obtained in terms of the

other unknowns i, i=3, N-2 as: 2

1 1,

3

N

i i

i

2

2 2,

3

N

i i

i

2

1 1,

3

N

N N i i

i

2

,

3

N

N N i i

i

(19)

where are known numerical coefficients which depend on the type of end supports and i are the required

unknowns.

3.3Descretization of Governing Equation

The mode shape differential equation (eqn.10) may be rewritten as:

1

4 3 2

2 34 3 2( ) ( ) ( ) ( ) 0

d d dx x x x

d x d x d x

. (20)

where;

2 4 4 2/ / / / 2

2 11 2 32 2 3 3

( )6 3 6( ) , ( ) and ( )o o

o o

P k L k L A Lx x x

EI EI

Using the DQM, eqn. (20) can be discretized at sampling point xi as:

2 2 2 2(4) (3) (2)

, 1, , 2, , 3,

3 3 3 3

, ( 3, 2)N N N N

i j j i i j j i i j j ij j j

j j j j

C C C i N

(21)

where ij is the Kronecr delta. Substituting eqn. (19) into the governing differential eqn. (21), one obtains: 2

1, 1, 2, 2, 1, 1, , , ,

3

0 , ( 3, 2)N

i i i i N i N i N i N i i j i

j

i N

(22)

Equation (22) represents a system of N-4 homogeneous algebraic equations in N-4 unknowns in addition to Po and

[14]. The eigenvalue analysis can be addressed to calculate the critical loads in the static case (=0) and to obtain

the natural frequencies n for a given value of the axial load Po<Pcr. Furthermore, knowing the natural frequencies of

the beam, the functional values i, i=1, N can be obtained and mode shapes can be illustrated.

3.4 Verification of the present solution

The calculated values of both the fundamental stability parameter b and the fundamental frequency parameter for

prismat ic beams using the present work and those obtained from closed -form solutions are presented in Table (1).

The fundamental stability parameter b and the fundamental frequency parameter (fundamental means first or

lowest value and simply called the stability or frequency parameter) are defined as: 2 2 2

2 4 1andcr ob

o o

P L A L

E E

(23)

where Pcr is a critical value of the axial load after which the beam losses its stability theoretically (also ca lled

buckling or Euler’s load). It is clear that the two approaches produce close results, which validates the present

solution.

Table (1): Values of and b for pris matic beam

Supports P-P P-C C-C Analysis

b 4.488 2 Closed Form

3.1413 4.4938 6.2643 Present

3.9266 4.73 Closed Form

3.1413 3.918 4.726 Present

Moreover, values of the frequency parameter for prismat ic beam resting on a two-parameter foundation obtained

from the present solution are presented in Table (2) compared to those obtained from the FEM [7]. The results are

calculated for different values of the foundation stiffness parameters (k1 andk2 ) and loading ratio . It is clear that

the results drawn from the two approaches are in close agreement. The foundation parameters and loading ratio are

defined as:


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4 2

1 21 2 2

, and O

cr

Pk L k Lk k

EI EI P

(24)

where k1 and k2 are the foundation stiffnesses, Po is the applied axial.

Table (2): Values of for a prismat ic beam on a two-parameter foundation.

Supports k1 k2=0 k2=1 k2=2.5

FEM Present FEM Present FEM Present

P-P

0

0.0 3.1415 3.1402 3.7306 3.7342 4.2970 4.2949

0.4 2.7705 2.7622 3.2947 3.2859 3.7893 3.7819

0.8 2.1257 2.1020 2.5270 2.4990 2.9050 2.8787

102

0.0 3.7483 3.7475 4.1437 4.1414 4.5824 4.5819

0.4 3.3055 3.3010 3.6541 3.6479 4.0408 4.0318

0.8 2.5350 2.5129 2.8014 2.7765 3.0964 3.0683

C-C

0

0.0 4.7300 4.7186 5.3183 5.3060 5.3183 5.3060

0.4 4.1611 4.1630 4.6829 4.6917 4.6829 4.6917

0.8 3.1205 3.1560 3.5178 3.5706 3.5178 3.5706

102

0.0 4.9504 4.9403 5.4773 5.4660 5.4773 5.4660

0.4 4.3591 4.3502 4.8272 4.8277 4.8272 4.8277

0.8 3.2764 3.3100 3.6344 3.6805 3.6344 3.6805

Furthermore, values of the frequency parameter for tapered beams obtained from the present solution are

presented in Table (3) against those obtained from the FEM [10] for d ifferent values of the tapering ratio and

found in close agreement.

Table (3): Values of the frequency parameter for the tapered P-P beam

= d1/do 1.0 1.1 1.2 1.3 1.4 1.5 Analysis

3.141 3.248 3.349 3.449 3.534 3.620 FEM

3.141 3.283 3.392 3.496 3.540 3.588 Present

4. NUMERICAL RES ULTS

A simple MATLAP code is designed and used to calculate the numerical results. The number of sampling points

that achieve the required accuracy (0.5%) was found to be 15 points [14].

The influences of the foundation parameters (k1 andk2) on the stability parameter b for different supporting

conditions and different values of tapering ratio =d1/do are shown in Figures (2) to (4). The figures indicate that the

stability parameter increases as the overall stiffness of the beam-foundation system increases. The overall stiffness

of the beam-foundation system is an integrated resultant of the support stiffness, the foundation stiffness and the

flexural rigid ity of the beam. It is known that the flexural rig idity of the beam increases as increases. The variation

of the stability parameter b with the tapering ratio is shown in Fig. (5). It is observed from figures (2) to (5) that the

influence of tapering ratio on the stability parameter is more noticeable in the case of clamped support than the case

of pinned support. In addition, the effects of foundation parameters on the stability parameter are more noticeable in

the pinned support case than the clamped support case.

The variations of the frequency parameter with different characteristics of beam and foundation parameters are

shown in Figures (6) to (11). It is obvious that the frequency parameter increases as the overall stiffness of the beam-

foundation system increases. The Figures indicate that the frequency parameter of the system decreases as the axial

compression load increases. Moreover, as the axial compression load approaches a certain value (crit ical load), the

system is transformed into aperiodic one and no free vibrat ion occur. The effect of tapering ratio on the frequency

parameter is negligible for the case of P-C beam with no axial load.

The effect of tapering ratio on the frequency parameter increases as the axial load increases. The effect of fou ndation

parameters on the frequency parameter is more significant for the small values of tapering rat io .


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Figure (2): Influence of the foundation parameters (k1,k2)

on the stability parameter b for (P-P) beams.


on the stability parameter b for (P-C) beams.


183


on the stability parameter b for (C-C) beams.

Figure (5): Influence of the tapering ratio on

the stability parameter b .


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Figure (6): Influence of the load parametersPo on the frequency

parameter for (P-P) beams( k1 =0 and k2 =0.5).

Figure (7): Influence of the load parameters Po on the frequency

parameter for (P-C) beams (k1 =0 and k2 =0.5).


185

Figure (8): Influence of the load parameters Po on the frequency

parameter for (C-C) beams; k1 =0 and k2 =0.5

Figure (9): Influence of the load parametersPo

on the frequency parameter ( =1 andk1 =0).


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on the frequency parameter ( =1.2 andk1 =0).


on the frequency parameter ( =1.5 andk1 =0).

5. CONCLUS ION

The stability and vibrational behavior of axially-loaded tapered beams resting on two-parameter foundations are

investigated using the DQM. The governing differential equations with variable coefficient are derived and

discretized at sampling points; and the boundary conditions are discretized and substituted into the discretized

governing equations. Then, the governing differential equation is transformed into a system of N-4 homogeneous

algebraic equations in N-4 functional values of mode functions in addition to the two parameters Po and . Using the


187

eigenvalue analysis, a MATLAP code is designed to calculate either the critical loads (Pcr) for the static case (=0)

or the natural frequencies n for a prescribed value of the axial load Po<Pcr. It is found that the natural frequencies

and the critical loads for the tapered beams increase as the stiffness of the beam-foundation system increases. In

addition, it is found that the natural frequencies of the beam-foundation system decrease as the axial compression

load increases.

6. REFERENCES

[1] M.H. Taha, and S. Abohadima, Mathemat ical model for v ibrations of nonuniform flexural beams, Eng. Mech.;

15 (1): 3-11, (2008).

[2] S. Abohadima and M.H. Taha, 2009. Dynamic analysis of non-uniform beams on elastic foundations. TOAJM,

3, 40-44, (2009).

[3] M.H Taha, Nonlinear vib ration model for init ially stressed beam-foundation system, TOAJM, 6, 23-31 (2012).

[4] Ruta, Application of Chebychev series to solution of non-prismatic beam vibration problems, Journal of Sound

and Vibration 227(2) 449-467 (1999).

[5] A. Maccari, The asymptotic perturbation method from nonlinear continuous systems, Nonlinear dyn. ; 19: 1-18

(1999).

[6] K. Sato, Transverse vibrations of linearly tapered beams with ends restrained elastically against rotation

subjected to axial force, International journal of Mechanical Science.; 22, 109-115 (1999).

[7] N.R. Naidu, and G.V. Rao, Vibrat ions of initially stressed uniform beams on A two-parameter elastic foundation,

Computers & Structures, 57(5), 941-943 (1995).

[8] J.R. Banerjee, H. Su, and D.R. Jackson, Free vibration of rotating tapered beams using the dynamic stiffness

method, Journal of Sound and Vibrat ion, 298: 1034-1054 (2006).

[9] N.R. Naidu, G.V. Rao, and K.K. Raju, Free vibrat ions of tapered beams with nonlinear elastic restraints, Journal

of Sound and Vibration; 240(1): 195-202 (2001).

[10] S.H. Ho, and C.K. Chen, Analysis of general elastically end restrained non -uniform beams using differential

transform. Applied Mathemat ical Modeling, 22: 219-234 (1998).

[11] Seval, Çatal, Solution of free vibration equations of beam on elastic soil by using differential transform

method. Applied Mathemat ical Modeling, 32: 1744-1757 (2008).

[12] C. Bert, X. Wang, and A. Striz, Static and free vibrat ional analysis of beams and plates by differential

quadrature method. ACTA Mechanics; Vol.102(1-4), p.p. 11-24 (1994).

[13] C. Shu,” Differential quadrature and its application in engineering”. Springer, Berlin (2000).

[14] M.A. Essam,” Analys is of stability and free vibration behavior of tapered beams on two parameter foundation

using differential quadrature method”. Master degree thesis Submitted to Dept. of Eng. Math. and Physics Faculty of

Eng., Cairo University (2012).

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STATIC AND DYNAMIC BEHAVIOR OF TAPERED BEAMS ON TWO-PARAMETER FOUNDATION · 2018-09-14 · In the present work, the stability and vibration behavior of axially-loaded tapered beams