Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions
We study a dynamic fourth-order Euler-Bernoulli partial differential equation having a constant elastic modulus and area moment of inertia, a variable lineal mass density , and the applied load denoted by , a function of transverse displacement . The complete Lie group classification is obtained for different forms of the variable lineal mass density and applied load . The equivalence transformations are constructed to simplify the determining equations for the symmetries. The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arbitrary applied load, general power-law, exponential, and log type of applied loads for different forms of . For the linear applied load case, we obtain an infinite-dimensional Lie algebra. We recover the Lie symmetry classification results discussed in the literature when is constant with variable applied load . For the general power-law and exponential case the group invariant solutions are derived. The similarity transformations reduce the fourth-order partial differential equation to a fourth-order ordinary differential equation. For the power-law applied load case a compatible initial-boundary value problem for the clamped and free end beam cases is formulated. We deduce the fourth-order ordinary differential equation with appropriate initial and boundary conditions.
Daniel Bernoulli and Leonard Euler developed the theory of the Euler-Bernoulli beam problem. Let be the transverse displacement at time and position from one end of the beam taken as the origin, the flexural rigidity, and the lineal mass. The transverse motion of an unloaded thin beam is represented by the following fourth-order partial differential equation (PDE):
Euler-Bernoulli beam equation (1) has been frequently studied in the literature. Gottlieb  studied the isospectral properties of this equation and its nonhomogeneous variants with and . Soh  considered the equivalence problem for an Euler-Bernoulli beam utilizing the Lie symmetry approach. Later on Morozov and Soh  attempted the problem with the aid of Cartan’s equivalence method. Recently, Ndogmo  obtained the complete equivalence transformations of the Euler-Bernoulli equation which were initially considered in the work  in terms of some undetermined set of functions. Özkaya and Pakdemirli , using the symmetry method, investigated the transverse vibrations of a beam moving with time-dependent axial velocity and obtained approximate solutions for an exponentially decaying and harmonically varying problem.
Now let be the elastic modulus, let be the area of inertia, let be the mass per unit length, let be the transverse displacement at time and position , and let be the applied load. The transverse motion of a loaded thin elastic beam is governed by the following dynamic beam fourth-order PDE :where the applied load is a function of . Bokhari et al.  studied the following dynamic Euler-beam equation from the symmetry viewpoint with , , as constants and dependent on :A complete group classification was obtained for (3). The symmetry reductions were derived to reduce the fourth-order PDE to fourth-order ordinary differential equations (ODEs). For the power-law load function, compatible initial-boundary value problems corresponding to clamped end and free end beams were formulated and the reduced fourth-order ODEs were determined. The static beam problem was discussed by Bokhari et al. .
The dynamic fourth-order Euler-Bernoulli PDE having a constant elastic modulus and area moment of inertia, a variable lineal mass density , and the applied load denoted by , a function of transverse displacement , is given by
In this paper we study dynamic Euler-Bernoulli beam equation (4) from the symmetry point of view.
We give a complete classification of the Lie symmetries for dynamic Euler-Bernoulli beam equation (4). The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arbitrary applied load, general power-law, exponential, and log type applied loads for different forms of (see Table 1). For the linear applied load case, we obtain an infinite-dimensional Lie algebra. We recover the Lie symmetry classification results discussed by Bokhari et al.  when is a constant with variable applied load . We derive the group invariant solutions for the general power-law and exponential cases. The fourth-order PDE reduces to a fourth-order ODE with the help of similarity transformations. For the power-law applied load case compatible initial-boundary value problems for the clamped and free end beam cases are formulated. We deduce the corresponding fourth-order ODE with appropriate initial and boundary conditions. We show that the solution fails to satisfy the initial or boundary conditions for the exponential and logarithmic cases.
The paper is organized as follows. In Section 2, the complete Lie point symmetry classification up to equivalence transformations is presented. The nontrivial symmetry reductions and initial-boundary value problems which correspond to clamped and free end beams are discussed in Section 3. The conclusions are summarized in the last section.
2. Complete Lie Symmetry Classification
We derive the equivalence transformations which are important for the simplification of the determining equations and for obtaining disjoint classes . Equivalence transformations of the PDE (4) are point transformations in the space of independent and dependent variables of the equation and these point transformations leave invariant family (4). That is, the equivalence transformations transform any equation (4) with arbitrary functions and into the same family (4) with, in general, different functions and . Equivalence transformations of the PDE (4) are easy to obtain although the computations are tedious. These arewhere , , and are constants and .
The Lie point symmetry generatorof dynamic Euler-Bernoulli equation (4), is derived by solvingwhere is the fourth prolongation of the operator . The fourth prolongation of the generator is defined aswhere can be determined fromin whichis the total derivative operator. Equation (7) is separated according to the derivatives of and an overdetermined system of partial differential equations for the unknown coefficients , , and is obtained. The determining equations finally yieldand the following classification equations for and :where are constants. For the case, arbitrary in and arbitrary in , the only symmetry iswhich constitutes the one-dimensional principal algebra of (4). Now we investigate all the possibilities of and for which an extension of the principal algebra is possible.
Differentiating classification equation (12) with respect to , we haveFurther differentiating (15) twice with respect to yieldsEquation (17) after using from (16) becomesNow differentiating (15) with respect to , we haveWe at once look at the possible cases of from (18) and (19).
Case 2 ( and ). If and then (18) yields the following possible forms of :(i) for , ;(ii), ;(iii), ,where , , , , are constants and from (19), . It is worth mentioning here that the principal algebra extends for these forms of .
After equivalence transformations, the simplified forms for are(i) for ,(ii),(iii).
Case 2.1 (). Equation (12) for with results in the following operators:with from (13) satisfyingThe solution of (22) is . When then in (21) and (22).
Case 2.2 (). Equation (12) for with results in the following operators:with satisfying (22). When then in (22) and (23).
Case 2.3 (). In this case , we havewith from (13) satisfyingThe solution of (25) is . We deduce the same results for both cases and .
Case 3 (). For the case , the equivalence transformations yield , , and , where .
Case 3.1 (). Equations (12) and (13) yieldwhere and satisfy
Case 3.2 (). From (12) and (13), we havewith , satisfying
Case 3.3 (). For this case the symmetry generator is the same as that given in (28) with satisfying (29) and :
Remark 1. The Lie algebras for all the nonlinear cases in Table 1 are easily seen by inspection. For the linear cases, they are infinite-dimensional and well known (see also ). Also the solution of the linear equation in the final linear case of Table 1 is easy to obtain and is not given as we do not use it here.
Now we work out the equivalence transformations for different forms of arising from Cases 2 and 3. After equivalence transformations becomes as follows:(i), , becomes for , ; if , then ;(ii), , , becomes , ; if , then with ;(iii)if solves then after equivalence transformations it reduces to with solution (iv)if satisfies then after equivalence transformations it becomes where is arbitrary.
The Lie symmetries for the simplified forms of and are presented in Table 1 and all cases discussed in  are recovered for .
3. Symmetry Reductions and Boundary Value Problems
Now we find the symmetry reductions. The initial conditions areThe four types of boundary conditions (see ) are as follows: hinged end: clamped end: free end: sliding end: consider the power-law case:If we take a linear combination of the symmetries , the initial condition and boundary condition are left invariant only by the scaling symmetry . The group invariant solution corresponding to iswhere is the similarity variable. The substitution of (42) into (41) yields the following ODE:
For the following power-law case, the PDE isand reduction via yields the invariant solutionHere the second boundary condition for the clamped end is not satisfied.
An asymptotic solution was found by Bokhari et al.  corresponding to the clamped or free end case for the following power-law case
For the remaining cases the solution does not satisfy the initial or boundary conditions. We take one example below.
Consider the following exponential case:The group invariant solution of (49) is of the formThe substitution of (50) into (49) yields the following fourth-order ODE: and this ODE fails to satisfy the boundary conditions at .
Similarly for the rest of the cases, the initial or boundary conditions are not satisfied when the load is of exponential or logarithmic form. We have ignored the linear cases as much attention has been already focused on these cases.
4. Concluding Remarks
We have performed the complete Lie symmetry classification of the dynamic fourth-order Euler-Bernoulli PDE having a constant elastic modulus and area moment of inertia, a variable lineal mass density , and the applied load denoted by , a function of transverse displacement . The equivalence transformations are constructed to simplify the determining equations for the symmetries. The simplified forms of lineal mass density and applied load are constructed via equivalence transformations. The principal algebra is one-dimensional for arbitrary and . The principal algebra is extended to a two- and three-dimensional algebra for arbitrary applied load, general power-law, exponential, and log type of applied loads for different forms of whereas an infinite-dimensional algebra is obtained for the linear applied load case (see Table 1). We recover the Lie symmetry classification results discussed in the literature when is constant with variable applied load . The similarity transformations reduce the fourth-order PDE to a fourth-order ODE. Only for one case with the applied load power-law, compatible initial-boundary value problems for the clamped and free end beam cases are formulated. We deduce the fourth-order ODE with appropriate initial and boundary conditions.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
F. M. Mahomed thanks the National Research Foundation (NRF) of South Africa for a research grant that has facilitated this research through the Unique Grant no. 92857, 2014. R. Naz thanks Lahore School of Economics for providing funds to complete this work.
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