New Analytical Approach to Nonlinear Behavior Study of Asym-metrically LCBs on Nonlinear Elastic Foundation under Steady Axial and Thermal Loading

In this paper, nonlinear behavior analysis of an asymmetrically laminated composite beam (LCB) on nonlinear foundation under axial and in-plane thermal loading is considered. To solve the obtained governing equation, a novel method based on Laplace transform is used. The resulted approximate analytical solution allows us the parametric study of diﬀerent parameters which inﬂuence the nonlinear behavior of the system. The numerical results illustrate that proposed technique yields a very rapid convergence of the solution as well as low computational eﬀort. The accuracy of the proposed method is veriﬁed by those available in literatures.

In this paper, geometrically nonlinear vibration and post-buckling analysis of asymmet-35 rically LCB on nonlinear foundation under axial and in-plane thermal loading is considered. 36 First, Galerkin method is used and the governing nonlinear partial differential equation is re-37 duced to a single nonlinear ordinary differential equation. Afterwards, a novel method based Andz measures the distance of beam's material element from midline, α th is coefficient of 57 thermal expansion. Moreover, ∆T 0 is temperature variation at midline of the beam and ∆T 1 58 stands for temperature difference between top and bottom sides and they can be presented as: The force and moment resultants per unit length based on the classical laminate beam 60 theory can be written as [10,18]: where its stiffness coefficients are given as follows [10,18]: Each layer k is referred to by thez coordinates of its lower face (h k−1 ) and upper face (h k ) k L andk N L are linear and nonlinear elastic foundation coefficients,k Sh is the shear stiffness 70 of the elastic foundation.

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By defining non-dimensional variables it can be written in a simple form as where r is the radius of gyration of the beam's cross-section, and To achieve the aims of the paper, the solution of Eq. (9) is assumed to be where φ(x) is the first normal mode of the beam [17] that is defined for simply supported and 76 fixed-fixed boundary conditions in Table 1 and η(t) is an unknown time dependent function. 77 Table 1 The first normal modes for beam with various boundary conditions Applying the Galerkin method [17], Eq. (9) yields Now, it can be assumed that the beam is subjected to an initial displacement according 80 to its first modal shape and zero initial velocity. So, the initial conditions of Eq. (12) can be 81 presented as where according to the Fig. 2, A denotes the non-dimensional maximum amplitude of oscilla-83 tion at the beam's center.

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Based on the Eq. (12) the nonlinear post-buckling load of the considered LCB can be 85 written as Neglecting the A in Eq. (15), the linear buckling load will be derived as The next step is to find the natural frequency of the system. Since the governing equation 88 Eq. (12) is nonlinear, the free vibration of the system has a nonlinear natural frequency 89 which is introduced by ω N L . Indeed, the nonlinear free vibration response of the system η(t) 90 and its nonlinear natural frequency ω N L depend on the system parameters, the boundary 91 condition and the initial conditions. Eq. (12) is strongly nonlinear and nobody can find an 92 exact analytical closed form solution for η(t) and ω N L . Although numerical methods can be 93 implemented to get over this problem but, they cannot offer any suitable way for parametric to present an accurate solution for nonlinear differential equations. To clarify the basic ideas 100 of proposed method consider the following second order differential equation, with artificial zero initial conditions and N is the nonlinear operator. Adding and subtracting 102 the term ω 2 u(t), the Eq. (17) can be written in the form where L is the linear operator and Taking Laplace transform of both sides of the Eq. (18) in the usual way and using the 105 homogenous initial conditions gives where s and I are the Laplace variable and operator, correspondingly. Therefore it is obvious where 112 Substituting Eq. (19) and (24) into (23) gives Now, the actual initial conditions must be imposed. Finally the following iteration formu-114 lation can be used [5] Knowing the initial approximation u 0 , the next approximations u n , n > 0 can be determined Applying the proposed method, the following iterative formula is assembled Eq. (28) will be homogeneous, if f (η(t)) is considered to be zero. So, its homogeneous is considered as the zero approximation for using in iterative Eq.(30).

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Expanding f (η 0 (τ )), we have: Foundation under Steady Axial and Thermal Loading Considering the relation: To avoid secular terms in the next iterations, the coefficient of the cos(ωt) in f (η 0 (τ )) 130 should be vanished. So the first approximation of the frequency is obtained as: Substituting Eq. (31) into (30) and neglecting the secular terms that are the coefficient of where I i are given in Appendix.

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To illustrate the robustness of the proposed LIM method and to compare with other methods, 142 some cases are studied. First, an isotopic beam in two cases of simply supported and fixed-143 fixed boundary conditions is taken. In these cases, the effects of thermal loading and elastic 144 foundation are ignored. The amounts of the nonlinear to the linear frequency ratio ω N L /ω L are 145 derived for four non-dimensional amplitudes A. Table 2   In the second step, it is assumed that the composite beam is made by AS4/3501 Graphite-  Now, the axial loading is applied. Figure 8 shows the variation of the nonlinear to the 169 linear frequency ratio ω N L /ω L due to change in the axial loading P . It shows that axial 170 loading amplifies the nonlinear frequency ratio of the LCB.

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Finally, the thermal loading is considered. As it is seen in Figure 9, thermal loading 172 increases the nonlinear to the linear frequency of the considered LCB. The results show that the 173 linear and nonlinear natural frequencies decrease by increasing the thermal loading however, 174 the decreasing rate of nonlinear frequency is less than linear natural frequency.

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In the previous steps, the effect of each factor was studied, independently. So in the last walk, effects of all factors are implemented simultaneously. Table 3 and 4 show the results for 177 cases with simply supported and fixed-fixed boundary conditions, respectively.
178 Table 3 Comparison of nonlinear frequency (ω N L ) and nonlinear to linear frequency ratio (ω N L /ω L ) due to change of different factors for S-S LCB, [0/90/0/90] lay-up configuration, A=2 and L/h=50 In this paper, the effects of different parameters such as vibration amplitude, nonlinear elastic