STATIC AND DYNAMIC BEHAVIOR OF TAPERED BEAMS ON TWO-PARAMETER FOUNDATION

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1 0.pdf STATIC AD DYAMIC BEHAVIOR OF TAPERED BEAMS O TWO-PARAMETER FOUDATIO Mohamed Taha Hassan & Mohamed assar 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 aial load (P o P cr ). The obtained solutions are found compatible with those obtained from other techniques. A parametric study is performed to investigate the significance of different parameters. Keywords: Tapered beams, two-parameter foundation, differential quadrature, critical load and natural frequencies. 1. ITRODUCTIO onprismatic elements are commonly used in many practical applications 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 prismatic and non-pris matic elements are found in literature. Taha and Abohadima [1-] studied the free vibration of non-uniform beam resting on elastic foundation using Bessel functions. Taha [3] investigated the nonlinear vibration 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 aial force on the vibration frequencies. umerical methods such as the FEM [7-9], the differential transform methods [10, 11] and the differential quadrature method [1-14] are used to study certain configurations of such elements. The free vibration of tapered beams with nonlinear elastic restraints was studied by aidu [9] using the FEM, and the effect of tapering ratio and end restraints were analyzed. The behavior of non-prismatic beams resting on elastic foundations had received a little attention in literature due to the compleity in its mathematical treatment and most researches in that area were carried out to investigate special cases. In the present work, the stability and vibration behavior of aially -loaded tapered beams resting on a two-parameter foundation will be investigated using the DQM. The present work differs than aidu work [9] in implementing the two-parameter foundation and aial compression load. The governing equations are formulated in dimensionless form, discretized 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 aial load value (P o P cr ). 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.. FORMULATIO OF THE PROBLEM.1 Vibration equation The free vibration equation of a non-prismatic beam aially-loaded by P o and resting on a two-parameter foundation shown in Fig.(1) is given as: Y Y Y EI ( X ) Po k A( X ) k 1Y ( X ) 0 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; P o is the aial load 176

2 Hassan & assar Static and Dynamic Behavior of Tapered Beams acting on the beam; k 1 and k 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 aially -loaded tapered beam on a two-parameter foundation Using dimensionless parameters =X/L and y=y/l, eqn. (1) can be epressed in dimensionless form as: EI ( ) y ( Po k) y y ( ) LA k 3 1Ly ( ) 0 L L t () The solution of eqn. () depends on the boundary conditions at the beam ends. Boundary conditions The boundary conditions at the beam ends depend on the type of support at the ends. For clamped support (C) at location (=0, 1) which prevents translation and rotation, the boundary conditions can be epressed as: y y (, t ) 0 and (, t ) 0 (3.a) For pinned support (P) at location (=0, 1) which prevents translation and allows rotation, the boundary conditions are epressed as: y (, t ) y (, t ) 0 and 0 (3.b) For free support (F) at location (=0, 1) which allows translation and rotation the boundary conditions are epressed as: 3 y (, t ) y (, t ) 0 and Mode functions Equation () is a fourth-order linear differential equation with variable coefficients; hence, the separation of variables technique can be addressed. Let the solution of eqn. () be assumed as: y (, t ) y ( ) ( t ) o (4) Where () is the mode function, (t) is a function representing the variation of lateral displacement with time; and y o is the dimensionless vibration amplitude (obtained from the initial conditions). Substituting eqn. (4) into eqn. (), then eqn. () can be separated into: 4 4 d d ( Po k ) L d k 1L L A( ) I ( ) ( ) 0 d d E d E (5) d dt ( t ) 0 where is the separation constant. The solution of eqn. (6) is: ( t ) A sin( t ) B cos( t ) (7) where A and B are constants obtained from the initial conditions and is the natural frequency of the beam vibration. (3.c) (6) 177

3 Hassan & assar Static and Dynamic Behavior of Tapered Beams 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 d o at =0 to d 1 at =0.5, then decreases linearly form d 1 at =0.5 to d o at =1, while the width of the beam b is assumed constant, then: d ( ) d ( ) where: o 1-(1- ) for () = +(1- )-1 for (8.b) and =d 1 / d o is the tapering ratio. Using the distribution of section geometry epressed in eqn. (8), the distribution of the area and moment of inertia of the beam cross section with are given as: I I 3 ( ) o ( ) A ( ) A ( ) o (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) yields: 4 / 3 / / / 4 4 d 6 d 3 6 ( Po k ) L d k 1L Ao L ( ) d d EI o d EI o (10) where prime stands for differentiation w.r.to. Equation (10) is a fourth-order differential equation with variable coefficients, which is difficult to be solved analytically. However, solving eqn. (10) considering =0 (static case) yields the critical (buckling) loads P cr, while solving the equation with a prescribed value of P o (less than critical load) yields the natural frequencies of the free vibration of aially-loaded tapered beams. The dimensionless boundary conditions at =0, 1 can be rewritten as: For clamped support: d (, t ) 0 and (, t ) 0 d For pinned support: d (, t ) ( t, ) 0 and 0 d and for free support: 3 d t d t (, ) (, ) 0 and 0 3 d d (8.a) (11.a) (11.b) (11.c) 3. SOLUTIO 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 sampling points and the derivatives at any point are approimated by a weighted linear summation of all the functional values at the other points as [13]: m d f ( ) m d X i C f ( ), ( i 1, ),( m 1, M ) 1 ( m ) i, where M is the order of the highest derivative in the governing equation, f( ) is the functional value at point = ( m ) C i and is the weighting coefficient relating the functional value at = to the m-derivative of the function f() at = i. To obtain the weighting coefficients, many polynomials with different base functions are commonly used to approimate the functional values. Using the Lagrange interpolation formula, the functional value at a point can be approimated by all the functional values f( k ), (k=1, ) as: (1) 178

4 Hassan & assar Static and Dynamic Behavior of Tapered Beams L ( ) f ( ) f ( ) k k 1 ( k) L1( k) =1 1 k k=1 i k i k (13) where: L()= ( ), L ( )= ( ),(, 1, ) Substitution of eqn. (13) into eqn. (1) yields the weighting coefficients of the first derivative as [13]: L (1) 1 ( i ) C i, for( i ) and ( i, 1, ) ( ) L ( ) i 1 (1) 1 i, i, 1, i C C for( i ) and ( i, 1, ) (14b) Applying the chain rule onto eqn. (1), the weighting coefficients of the m-order derivative are related to the weighting coefficients of (m-1) order derivative as: ( m) (1) ( m 1) i, k i, k i, k k 1 C C C, ( i, k 1, ), ( m 1, 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 i 1 i 1 cos, ( i 1, ). 1 (16) (14a) 3. Implementation of the Boundary Conditions The boundary conditions due to support at =0 can be discretized as: For clamped support: (1) 1 C1, 1 0 and ( ) 0 For pinned support: () 1 C1, 1 0 and ( ) 0 For free support: () (3) C1, C1, 1 1 ( ) 0 and ( ) 0 Also, the boundary conditions due to support at =1are discretized as: For clamped support: (1) C, 1 0 and ( ) 0 For pinned support: () C, 1 0 and ( ) 0 For free support: () (3) C, C, 1 1 ( ) 0 and ( ) 0 (17a) (17b) (17c) (18a) (18b) (18c) 179

5 Hassan & assar Static and Dynamic Behavior of Tapered Beams Rearranging the terms in eqn. (17) and eqn. (18), the unknowns 1,, -1 and can be obtained in terms of the other unknowns i, i=3, - as: i 1 1, i i 3 i, i i 3 1 1, i i i 3, i i i 3 (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: 4 3 d d d ( ) ( ) 3( ) ( ) 0 d d d where;. (0) / / / / ( Po k ) L k 1L AoL 1( ), ( ) and 3 3( ) 3 EI o EI o Using the DQM, eqn. (0) can be discretized at sampling point i as: (4) (3) () i, 1, i i,, i i, i 3, C C C, ( i 3, ) (1) where i is the Kronecr delta. Substituting eqn. (19) into the governing differential eqn. (1), one obtains: 1, i1, i, i, i 1, i 1, i, i, i i, i 0, ( i 3, ) 3 () Equation () represents a system of -4 homogeneous algebraic equations in -4 unknowns in addition to P o 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 aial load P o <P cr. Furthermore, knowing the natural frequencies of the beam, the functional values i, i=1, 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 prismatic 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: P L A L cr 4 o 1 b and E o Eo (3) where P cr is a critical value of the aial load after which the beam losses its stability theoretically (also called 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 Closed Form b Present Closed Form Present Moreover, values of the frequency parameter for prismatic beam resting on a two-parameter foundation obtained from the present solution are presented in Table () compared to those obtained from the FEM [7]. The results are calculated for different values of the foundation stiffness parameters (k 1 andk ) 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: 180

6 Hassan & assar Static and Dynamic Behavior of Tapered Beams k k L k L PO, and EI EI P k where k 1 and k are the foundation stiffnesses, P o is the applied aial. cr (4) Table (): Values of for a prismatic beam on a two-parameter foundation. Supports k 1 k =0 k =1 k =.5 FEM Present FEM Present FEM Present P-P C-C 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 different values of the tapering ratio and found in close agreement. Table (3): Values of the frequency parameter for the tapered P-P beam = d 1 /d o Analysis FEM Present 4. UMERICAL 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 (k 1 andk ) on the stability parameter b for different supporting conditions and different values of tapering ratio =d 1 /d o are shown in Figures () 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 fleural rigidity of the beam. It is known that the fleural rigidity 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 () 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 beamfoundation system increases. The Figures indicate that the frequency parameter of the system decreases as the aial compression load increases. Moreover, as the aial compression load approaches a certain value (critical load), the system is transformed into aperiodic one and no free vibration occur. The effect of tapering ratio on the frequency parameter is negligible for the case of P-C beam with no aial load. The effect of tapering ratio on the frequency parameter increases as the aial load increases. The effect of fou ndation parameters on the frequency parameter is more significant for the small values of tapering ratio. 181

7 Hassan & assar Static and Dynamic Behavior of Tapered Beams Figure (): Influence of the foundation parameters (k 1,k ) on the stability parameter b for (P-P) beams. Figure (3): Influence of the foundation parameters (k 1,k ) on the stability parameter b for (P-C) beams. 18

8 Hassan & assar Static and Dynamic Behavior of Tapered Beams Figure (4): Influence of the foundation parameters (k 1,k ) on the stability parameter b for (C-C) beams. Figure (5): Influence of the tapering ratio on the stability parameter b. 183

9 Hassan & assar Static and Dynamic Behavior of Tapered Beams Figure (6): Influence of the load parametersp o on the frequency parameter for (P-P) beams( k 1 =0 and k =0.5). Figure (7): Influence of the load parameters P o on the frequency parameter for (P-C) beams (k 1 =0 and k =0.5). 184

10 Hassan & assar Static and Dynamic Behavior of Tapered Beams Figure (8): Influence of the load parameters P o on the frequency parameter for (C-C) beams; k 1 =0 and k =0.5 Figure (9): Influence of the load parametersp o on the frequency parameter ( =1 andk 1 =0). 185

11 Hassan & assar Static and Dynamic Behavior of Tapered Beams Figure (10): Influence of the load parametersp o on the frequency parameter ( =1. andk 1 =0). Figure (11): Influence of the load parametersp o on the frequency parameter ( =1.5 andk 1 =0). 5. COCLUS IO The stability and vibrational behavior of aially-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 -4 homogeneous algebraic equations in -4 functional values of mode functions in addition to the two parameters P o and. Using the 186

12 Hassan & assar Static and Dynamic Behavior of Tapered Beams eigenvalue analysis, a MATLAP code is designed to calculate either the critical loads (P cr ) for the static case (=0) or the natural frequencies n for a prescribed value of the aial load P o <P cr. 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 aial compression load increases. 6. REFERECES [1] M.H. Taha, and S. Abohadima, Mathematical model for vibrations of nonuniform fleural beams, Eng. Mech.; 15 (1): 3-11, (008). [] S. Abohadima and M.H. Taha, 009. Dynamic analysis of non-uniform beams on elastic foundations. TOAJM, 3, 40-44, (009). [3] M.H Taha, onlinear vibration model for initially stressed beam-foundation system, TOAJM, 6, 3-31 (01). [4] Ruta, Application of Chebychev series to solution of non-prismatic beam vibration problems, Journal of Sound and Vibration 7() (1999). [5] A. Maccari, The asymptotic perturbation method from nonlinear continuous systems, onlinear dyn. ; 19: 1-18 (1999). [6] K. Sato, Transverse vibrations of linearly tapered beams with ends restrained elastically against rotation subected to aial force, International ournal of Mechanical Science.;, (1999). [7].R. aidu, and G.V. Rao, Vibrations of initially stressed uniform beams on A two-parameter elastic foundation, Computers & Structures, 57(5), (1995). [8] J.R. Baneree, H. Su, and D.R. Jackson, Free vibration of rotating tapered beams using the dynamic stiffness method, Journal of Sound and Vibration, 98: (006). [9].R. aidu, G.V. Rao, and K.K. Rau, Free vibrations of tapered beams with nonlinear elastic restraints, Journal of Sound and Vibration; 40(1): (001). [10] S.H. Ho, and C.K. Chen, Analysis of general elastically end restrained non -uniform beams using differential transform. Applied Mathematical Modeling, : (1998). [11] Seval, Çatal, Solution of free vibration equations of beam on elastic soil by using differential transform method. Applied Mathematical Modeling, 3: (008). [1] C. Bert, X. Wang, and A. Striz, Static and free vibrational analysis of beams and plates by differential quadrature method. ACTA Mechanics; Vol.10(1-4), p.p (1994). [13] C. Shu, Differential quadrature and its application in engineering. Springer, Berlin (000). [14] M.A. Essam, Analysis 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 (01). 187