Dynamic Stability of Beam on Elastic Foundation Including Higher Transition Foundation

Abstract

A precise analytical solution has been developed using Energy Method to predict the dynamic instability bounds of simply supported beams on elastic foundation, with a focus on the impact of the higher transition foundation on dynamic stability boundaries. To determine the dynamic instability zones, a trigonometric function with a single term that meets the geometric boundary criteria to represent lateral deflection is used. For the analysis, Euler-Bernoulli beam theory is employed. Numerical results are presented in non-dimensional form in both digital and/or analogue forms, with varying foundation parameters below and above the transition foundation values of an elastic foundation. When compared to those produced using the finite element approach, the current findings exhibit a fair degree of consistency. The impact of the elastic foundation’s first and higher transition foundation values on dynamic stability behavior is amply demonstrated in the current study. According to the studies, when the elastic foundation parameter value increases, the width of the dynamically unstable zones decreases, making the beam less susceptible to the dynamic stability phenomena under periodic loads. By using the precise non-dimensional parameters for the imposed periodic load and its radian frequency, the presence of the master dynamic instability curves is demonstrated in the current work.

1. Introduction

An essential input for the structural design engineers is the prediction of the dynamic stability bounds of structural components exposed to periodic axial or in-plane loads. In his seminal work, Bolotin [1] covers in detail the dynamic instability of structural components/elements exposed to axial or in-plane periodic loads. In Ref. [2], a finite element technique is used to explore the dynamic instability of thin bars that are periodically exposed to intense axial loads at the free ends. In this paper, it is demonstrated intuitively that, provided the stability and vibration mode forms are same, the dynamic stability bounds derived with adequate non-dimensional parameters are the equal regardless of the boundary conditions. In [3], it is suggested that these non-dimensional parameters may be rigorously determined, and for the majority of structural members, it is shown that there are master dynamic instability curves that are valid for all structural members, regardless of boundary condition or complicating effects. If the requirement of the exactness of the mode shapes, but not the similarity as mentioned in [2], is broken while trying to get the master dynamic stability curves, the error involved in the analysis is dependent on the deviation of these mode shapes and may be evaluated by calculating the corresponding L₂ norms [4].

In several engineering areas, structural components like homogenous plates and beams on elastic foundations are frequently used. In Ref. [5], the impact of an elastic base on the dynamic stability of columns is examined. According to an instability discovery in Ref. [5], the zones of dynamic stability move away from the vertical axis and become narrower as the elastic foundation parameter rises, making the beam less susceptible to periodic loads. It has been demonstrated in Ref. [2] that the value of the foundation stiffness parameter affects the mode forms of columns under on elastic foundation. There is a value for transition foundation stiffness, and once this value is exceeded, the mode form of the buckling changes. For instance, when the foundation parameter is less than the initial/first transitional foundational parameter in the case of a simply supported beam, the stability and vibration mode forms are the equal (half of a sine wave along the length). When the foundation parameter is bigger than the initial transition value, these mode forms are different (full sinusoidal waves for the stability and half sinusoidal wave for the free vibration problems, respectively). The foundational parameters for this transition are discovered to be 4, 36, 144…for (first, second, third…) for a simply supported beam/column [6]. Furthermore, it is demonstrated that the vibration mode forms of uniform beams supported by an elastic foundation do not correspond to such a transition foundation parameter. In Ref. [7], similar tests on rectangular plates with simple supports and biaxial periodic compressive stress on a homogenous elastic basis were described.

A beam that is axially loaded and rests on an elastic base with dampening is examined for dynamic stability in Ref. [8]. It has been demonstrated that raising the damping or stiffness of the foundation raises the critical dynamic load and moves the unstable zones to a higher/greater applied frequency. The dynamic instability of a tapered cantilever beam on an elastic basis was studied by Lee [9]. The dynamic instability of beams on elastic basis was explored by Subba Ratnam et al. [10]. By taking into account the reference buckling and frequency characteristics, Subba Ratnam et al. recently explored the dynamic instability of structural elements with secondary effects [11, 12, 13, 14, 15].

Wachirawit et al. [16] applied the Ritz and Newmark techniques for the Timoshenko beams to study the free vibrations and dynamic behavior of the functionally graded sandwich lying on an elastic base/foundation. Based on the Floquet theory, Fourier series, and matrix eigenvalue analysis, Ying et al. [17] studied multi-mode coupled periodically supported beams vibrating under generic harmonic excitations. Subbaratnam et al. [18] used the variational method to analyze the impact of dynamic instability boundaries of SS beams resting on elastic foundation under periodic loads caused by higher transition foundation. Using the Runge-Kutta method and the Floquet theorem, Chao Xu et al. [19] investigated dynamic instability zones of a simply supported beam under multi-harmonic parametric excitation Using the single matrix approach, Jian Deng et al. [20] investigated the dynamic stability and reactions of beams on elastic foundations under pulsing axial parametric load. The study and consequences of numerous factors, such as foundation basis models, damping, and static and dynamic loads, are taken into account using the Winkler, Pasternak, and Hetenyi models. Youqin Huang et al. [21] based on Reddy’s beam theory, the dynamic instability of nanobeams was studied.

This study’s major contribution is to examine how higher transition foundations affect the dynamic instability areas of SS beams lying on elastic foundations while being subjected to axial periodic loads. This study highlights the ease of employing a particular type of non-dimensional parameters employed, in order to show the results, and discusses the influence of altering mode forms for stability and vibration on the dynamic stability zones, with rising values of foundation parameter. This study demonstrates that the breadth of the zones of dynamic stability diminishes as the foundation increases, making a beam less responsive to the dynamic instability phenomena under periodic loads.

2. Formulation

Here, we briefly describe the mathematical formulation for SS beam on elastic foundation based on the variational principle and the assessment of the dynamic instability bounds.

When a regular beam of length L rests on elastic basis that is also uniformly distributed, it is subject to a end-axial periodic load P(t)., as shown in the Figure 1, the potential energy $\left(\prod \right)$ is given, by

Figure 1.

Uniform SS beam lying on foundation subjected to periodic load.

$$\prod =U+U_F-W-T$$E1

where T is kinetic energy, W is work produced by the external axial periodic load, U_F is the energy stored in the elastic foundation, and U is the strain energy. The formulas for U, U_F, T, and W are provided by

$$U=\frac{\mathit{EI}}{2}\underset{0}{\overset{L}{\int }}{\left(\frac{d^{2}w}{{\mathit{dx}}^2}\right)}^2\mathit{dx}$$E2

$$U_F=\frac{k}{2}\underset{0}{\overset{L}{\int }}{w}^2\mathit{dx}$$E3

$$T=\frac{m{\omega }^2}{2}\underset{0}{\overset{L}{\int }}{w}^2\mathit{dx}$$E4

and

$$W=\frac{P\left(t\right)}{2}\underset{0}{\overset{L}{\int }}{\left(\frac{\mathit{dw}}{\mathit{dx}}\right)}^2\mathit{dx}$$E5

where E stands for Modulus of Elasticity, I for moment of inertia, m is the mass per unit length, w for lateral displacement, P(t) for axial compressive periodic load, and ω is the natural radian frequency.

P(t) represents

$$P\left(t\right)=P_S+P_{t}cos\theta t$$E6

where P_S is the periodic component of P(t), Pt is its periodic component, and θ is the compressive load’s radian frequency. In terms of the buckling/critical load parameter Pcr, the numbers Ps and Pt are stated.

The differential equation controlling the dynamic stability problem of beams on elastic basis is expressed in terms of energy by substituting Eqs. (2)–(5) in Eq. (1).

$$\mathrm{\Pi }=\frac{\mathit{EI}}{2}\underset{0}{\overset{L}{\int }}{\left(\frac{d^{2}w}{{\mathit{dx}}^2}\right)}^2\mathit{dx}+\frac{k}{2}\underset{0}{\overset{L}{\int }}{w}^2\mathit{dx}-\frac{P\left(t\right)}{2}\underset{0}{\overset{L}{\int }}{\left(\frac{\mathit{dw}}{\mathit{dx}}\right)}^2\mathit{dx}-\frac{m{\omega }^2}{2}\underset{0}{\overset{L}{\int }}{w}^2\mathit{dx}$$E7

As ω = θ/2, Eq. (7) becomes

$$\mathrm{\Pi }=\frac{\mathit{EI}}{2}\underset{0}{\overset{L}{\int }}{\left(\frac{d^{2}w}{{\mathit{dx}}^2}\right)}^2\mathit{dx}+\frac{k}{2}\underset{0}{\overset{L}{\int }}{w}^2\mathit{dx}-\frac{P\left(t\right)}{2}\underset{0}{\overset{L}{\int }}{\left(\frac{\mathit{dw}}{\mathit{dx}}\right)}^2\mathit{dx}-\frac{m}{2}\frac{{\theta }^2}{4}\underset{0}{\overset{L}{\int }}{w}^2\mathit{dx}$$E8

When we replace the expression for P(t) from Eq. (6) in Eq. (8), we obtain

$$\mathrm{\Pi }=\frac{\mathit{EI}}{2}\underset{0}{\overset{L}{\int }}{\left(\frac{d^{2}w}{{\mathit{dx}}^2}\right)}^2\mathit{dx}+\frac{k}{2}\underset{0}{\overset{L}{\int }}{w}^2\mathit{dx}-\left(P_S+P_{t}cos\theta t\right)\frac{1}{2}\underset{0}{\overset{L}{\int }}{\left(\frac{\mathit{dw}}{\mathit{dx}}\right)}^2\mathit{dx}-\frac{m}{2}\frac{{\theta }^2}{4}\underset{0}{\overset{L}{\int }}{w}^2\mathit{dx}$$E9

The boundary between Eq. (9)‘s stable and unstable solutions are periodic solutions with periods of 2π/θ and 4π/θ according to the theory of linear equations with periodic coefficients [1]. The unstable solutions that are constrained by the solutions with period’s 4π/θ are those that have the most practical significance. The need for these boundary solutions, as a first approximation, is

$$\mathrm{\Pi }=\frac{\mathit{EI}}{2}\underset{0}{\overset{L}{\int }}{\left(\frac{d^{2}w}{{\mathit{dx}}^2}\right)}^2\mathit{dx}+\frac{k}{2}\underset{0}{\overset{L}{\int }}{w}^2\mathit{dx}-\left(\alpha \pm \frac{\beta }{2}\right)\frac{P_{\mathit{cr}}}{2}\underset{0}{\overset{L}{\int }}{\left(\frac{\mathit{dw}}{\mathit{dx}}\right)}^2\mathit{dx}-\frac{m}{2}\frac{{\theta }^2}{4}\underset{0}{\overset{L}{\int }}{w}^2\mathit{dx}$$E10

The boundary between the stable and unstable solutions of Eq. (8) are periodic solutions with periods of 2π/θ and 4π/θ according to the theory of linear equations with periodic coefficients [1].

where α is the fraction of buckling load ($=\frac{P_s}{P_{\mathit{cr}}}$) and β is the fraction of periodic buckling load

$$\left(=,\frac{P_t}{P_{\mathit{cr}}},.\right)$$E11

It should be noted that Eq. (10), which yields the two borders of the zones of instability, combines two requirements in the plus or minus sign. The beam’s static buckling load factor is used as the standard in this research.

2.1 Buckling (critical) load parameter and frequency parameter of SS beam on elastic foundation

The fundamental equation for dynamic stability provides the corresponding solutions for the Euler buckling loads and frequency parameters as functions of the wave number n, taking into account the influence of the elastic foundation. This is done by non-dimensionalizing all length quantities by the length of the beam L and neglecting the kinetic energy term (T) and work done by the periodic load (W), respectively.

Using the single term standard exact trigonometric admissible function, for a SS beam, for lateral deflection, as

$$w=asin\frac{\mathit{n\pi x}}{L}$$E12

where a is the indeterminate coefficient, n is the mode shape number, (denoted by n_b for the buckling issue and n_f for the free vibration problem), w is the lateral deflection, and x is the axial coordinate, respectively.

In terms of the mode numbers n_b and n_f, this admissible function provides the buckling loads and the frequency parameters as

$${\lambda }_b=\frac{P_{\mathit{cr}}{L}^2}{\mathit{EI}}={\pi }^2\left(n^2+\frac{\gamma }{n^2}\right)$$E13

and

$${\lambda }_f=\frac{m{\omega }^2{L}^4}{\mathit{EI}}={\pi }^4\left(n^2+\frac{\gamma }{n^2}\right)$$E14

where $\gamma$ is the foundation parameter ($\gamma =\frac{{\mathit{kL}}^4}{{\pi }^4\mathit{EI}}$), ${\lambda }_b$ is buckling/critical load parameter ($=\frac{P_{\mathit{cr}}{L}^2}{\mathit{EI}}$) and ${\lambda }_f$ is frequency parameter ($=\frac{m{\omega }^2{L}^4}{\mathit{EI}}$).where P_cr is critical load and ω is natural frequency.

2.2 Foundation parameter transition values

The transitional foundation parameter value γ_Ti (i varies from 1, 2, 3…. and with reference to buckling mode number n_b), where buckling load parameter λ_b changes from mode n_b to mode (n + 1)_b. The transition value γ_Ti is evaluated from Eq. (13), using the condition that the buckling load parameters are the same for the two consecutive modes of buckling n_b and (n + 1)_b, as

$${\lambda }_{\mathit{bn}}={\lambda }_{b\left(n+1\right)}$$E15

The values of the γ_Ti are

$${\gamma }_{\mathit{Ti}}={n_b}^{\mathit{2}}{{\left(n+\mathit{1}\right)}_{\mathrm{b}}}^2,\left(i,n_b=1,2,3,4,5\dots .\right)$$E16

Two noteworthy findings from the assessment of the γ_Ti are:

1. the contribution of γ_Ti for buckling problem,

2. a similar equation as Eq. (16) for the free vibration issue, where the mode forms vary, the phenomena of transition does not occur.

For buckling issue, the values of γ_Ti are 4, 36, 144… for (i = 1, 2, 3,4…).

The impact of the initial and second transition foundation parameters γ_T1 = 4 and γ_T2 = 36, respectively and its impact on dynamic instability behavior of SS beam are both thoroughly explored in the current work.

2.3 Variation of λ_b and λ_f for different values of n_b and n_f

Eqs. (13) and (14), provide expressions for ${\lambda }_b$ and ${\lambda }_f$ the various values of n_b, n_f and $\gamma$. In light of these values of $\gamma$, n_b and n_f, succinct evaluations of the formulations for λ_b and λ_f are given here;

2.3.1 For $\gamma$ < ${\gamma }_{T1}$ (n_b = n_f = 1)

The λ_b and λ_f terms are

$${\lambda }_b=\frac{P_{\mathit{cr}}{L}^2}{\mathit{EI}}={\pi }^2\left[1+\gamma \right]$$E17

and

$${\lambda }_f=\frac{m{\omega }^2{L}^4}{\mathit{EI}}={\pi }^4\left(1+\gamma \right)$$E18

2.3.2 For $\gamma$ > ${\gamma }_{T1}$ and $\gamma$ < ${\gamma }_{T2}$ (n_b = 2, n_f = 1)

The λ_b and λ_f terms are

$${\lambda }_b=\frac{P_{\mathit{cr}}{L}^2}{\mathit{EI}}={\pi }^2\left[4+\frac{\gamma }{4}\right]$$E19

and

$${\lambda }_f=\frac{m{\omega }^2{L}^4}{\mathit{EI}}={\pi }^4\left(1+\gamma \right)$$E20

2.3.3 For $\gamma$ > ${\gamma }_{T1}$ and $\gamma$ < ${\gamma }_{T2}$ (n_b = 2, n_f = 2)

The λ_b and λ_f terms are

$${\lambda }_b=\frac{P_{\mathit{cr}}{L}^2}{\mathit{EI}}={\pi }^2\left[4+\frac{\gamma }{4}\right]$$E21

and

$${\lambda }_f=\frac{m{\omega }^2{L}^4}{\mathit{EI}}={\pi }^4\left(16+\gamma \right)$$E22

2.3.4 For $\gamma$ > ${\gamma }_{T2}$ and $\gamma$ < ${\gamma }_{T3}$ (n_b = 3, n_f = 1)

The λ_b and λ_f terms are

$${\lambda }_b=\frac{P_{\mathit{cr}}{L}^2}{\mathit{EI}}={\pi }^2\left[9+\frac{\gamma }{9}\right]$$E23

and

$${\lambda }_f=\frac{m{\omega }^2{L}^4}{\mathit{EI}}={\pi }^4\left(1+\gamma \right)$$E24

2.3.5 For $\gamma$ > ${\gamma }_{T2}$ and $\gamma$ < ${\gamma }_{T3}$ (n_b = 3, n_f = 3)

The λ_b and λ_f terms are

$${\lambda }_b=\frac{P_{\mathit{cr}}{L}^2}{\mathit{EI}}={\pi }^2\left[9+\frac{\gamma }{9}\right]$$E25

and

$${\lambda }_f=\frac{m{\omega }^2{L}^4}{\mathit{IE}}={\pi }^4\left(81+\gamma \right)$$E26

For the sake of completeness, the expressions for λ_b and λ_f for the three cases that are being considered here are given, but any combination of n_b, n_f and $\gamma$, the buckling and frequency parameters can be immediately obtained for any given value of these four parameters by taking into account transition value of foundation parameter for $\gamma$.

3. Formulas for dynamic stability

The following formulae are short and to the point for forecasting the dynamic instability areas of the simply supported beam on the foundation:

The dynamic stability formula is determined by replacing Eqs. (12), (17), and (18) in Eq. (10) and utilizing the first mode of buckling and free vibration (nb = nf = 1) as the reference values, where $P_{\mathit{cr}}=\frac{{\pi }^2\mathit{EI}}{L^2}$ and $m=\frac{{\pi }^4\mathit{EI}}{{\omega }^2{L}^4}$ without taking into account the impact of foundation.

$$1+\gamma -\left[\alpha \pm \frac{\beta }{2}\right]-\frac{{\theta }^2}{4{\omega }^2}=0$$E27

$$\alpha =\frac{P_s}{P_{\mathit{cr}}}\text{and}\beta =\frac{P_t}{P_{\mathit{cr}}}.$$E28

or

$$\frac{\theta }{\omega }=\mathrm{\Omega }=2\sqrt{\left(1-\alpha \right)\left(1\pm \mu \right)+\gamma }$$E29

where μ is the defined non-dimensional parameter, $\mu =\frac{\beta }{2\left(1-\alpha \right)}$ used in some earlier studies on the topic of dynamic stability [2].

The dynamic stability formula is formulated as follows for n_b = 2 and n_f = 1 when Eqs. (12), (19), and (20) are substituted in Eq. (10)

$$1-\frac{\left(\alpha \pm \frac{\beta }{2}\right)P_{\mathit{cr}}.4\frac{{\pi }^2}{L^2}}{\frac{{\pi }^4\mathit{EI}}{L^4}\left[16+\frac{{\mathit{kL}}^4}{{\pi }^4\mathit{EI}}\right]}-\frac{\frac{m{\theta }^2}{4}}{\frac{{\pi }^4\mathit{EI}}{L^4}\left[1+\frac{{\mathit{kL}}^4}{{\pi }^4\mathit{EI}}\right]}=0$$E30

Substituting the corresponding reference values of $P_{\mathit{cr}}=\frac{4{\pi }^2\mathit{EI}}{L^2}$ and $m=\frac{{\pi }^4\mathit{EI}}{{\omega }^2{L}^4}$ without considering the effect of the elastic foundation in Eq. (30) becomes

$$\mathrm{\Omega }=\frac{\theta }{\omega }=2\sqrt{\left[\left(\left(1-\alpha \right)\left(1\pm \mu \right)+\frac{\gamma }{16}\right)\left(\frac{\left(1+\gamma \right)}{\left(1+\frac{\gamma }{16}\right)}\right)\right]}$$E31

In the same way, the dynamic stability formula is as follows when Eqs. (12), (23), and (24), respectively, are substituted for n_b = 3 and n_f = 1 in Eq. [10], as

$$1-\frac{\left(\alpha \pm \frac{\beta }{2}\right)P_{\mathit{cr}}.9\frac{{\pi }^2}{L^2}}{\frac{{\pi }^4\mathit{EI}}{L^4}\left[81+\frac{{\mathit{kL}}^4}{{\pi }^4\mathit{EI}}\right]}-\frac{\frac{m{\theta }^2}{4}}{\frac{{\pi }^4\mathit{EI}}{L^4}\left[1+\frac{{\mathit{kL}}^4}{{\pi }^4\mathit{EI}}\right]}=0$$E32

Eq. (32) is expressed as follows when the reference values $P_{\mathit{cr}}=\frac{9{\pi }^2\mathit{EI}}{L^2}$ and $m=\frac{{\pi }^4\mathit{EI}}{{\omega }^2{L}^4}$ are substituted and the foundation is ignored.

$$\mathrm{\Omega }=\frac{\theta }{\omega }=2\sqrt{\left[\left[\left(1-\alpha \right)\left(1\pm \mu \right)+\frac{\gamma }{81}\right]\left[\frac{\left(1+\gamma \right)}{\left(1+\left(\gamma /81\right)\right)}\right]\right]}$$E33

These three dynamic instability formulae are constructed using the modes n_b and n_f for initial and second transition foundation parameter values γ smaller or greater than ${\gamma }_{T1}$ and ${\gamma }_{T2}$.

4. Master dynamic stability formula

Interestingly, the master dynamic stability formula can be obtained by substituting Eq. (12) and the reference buckling load Pcr and frequency ω into Eq. (10), along with half sinusoidal wave for both buckling and vibration problems (nb = nf = 1), a full sinusoidal wave for both buckling and vibration problems (nb = nf = 2), and three half sinusoidal waves for both the stability and vibration problems (nb = nf = 3), as

$$\frac{\theta }{\omega }=\mathrm{\Omega }=2\sqrt{\left(1-\alpha \right)\left(1\pm \mu \right)}$$E34

Use of these specific reference values of λ_b and λ_f the master dynamic stability curves turn out to be simple to use, exactly the same and are independent of the foundation parameter for the same values of n_b and n_f (either 1, 2 or 3). It should be noticed that, unlike preceding formulations where the foundation parameters appeared clearly, the master dynamic stability formula shown in Eq. (34) does not explicitly include the foundation parameter (γ).

5. Results and discussion

A homogeneous simply supported beam that is sitting on elastic basis and being subjected to axial concentrated periodic load is shown in Figure 1. The equation created in the current work to examine the dynamic instability limits of a SS beam sitting on an elastic basis and subjected to end axial periodic concentrated load is highly universal. The fundamental frequency and the static buckling load parameters, which are characteristic values of the beam, do not clearly exist in the non-dimensional form. However, when non-dimensional parameters $\mu$ and Ω. are defined, their characteristic values are referenced implicitly.

The values of the stability boundaries Ω₁and Ω_2, between which it is dynamically unstable with variable for α = 0.0, 0.5 & 0.8 with mode form of a half sinusoidal wave for both stability and vibration problems for γ = 1, are shown in Table 1. Table 2 gives the values of the stability boundaries Ω₁and Ω₂ with varying β, instead of μ, for α = 0.0, 0.5 & 0.8 for mode form of one half sinusoidal wave for both stability & vibration problems for γ = 2. The dynamic instability results of a slender beam on foundation [5] are also included in this Table. Since the non-dimensional parameter given in Ref. [5] is β instead of μ proposed here, and as it is difficult to convert β to μ due to lack of information, the values of μ is converted in to β for the results presented in the Table. The results are in good accord with results obtained from present formula and those given in Ref. [5] obtained using finite approach for α = 0.5. Table 3 gives the values of the stability boundaries Ω₁and Ω_{2, b}etween which it exhibits dynamic instability with variable μ for α = 0.0, 0.5 and 0.8, and a half sinusoidal wave mode form for both stability and vibration issues γ = 3. The instability boundaries Ω₁and Ω₂ given in Tables 1–3 are for α = 0.0, 0.5 and 0.8 with γ = 1, 2 and 3 respectively. For a better understanding of the zones of dynamic stability, Figures 2–4 illustrate the boundaries of the instability that are described in Tables 1–3. Additionally, it can be shown that the zones of dynamic instability shifts away from the vertical axis and reduce as the elastic foundation parameter increases, making the dynamic stability phenomena less sensitive to periodic loads.

$\mu$	$\alpha =0.0$		$\alpha =0.5$		$\alpha =0.8$
	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$
0.0	2.8284	2.8284	2.4494	2.4494	2.1908	2.1908
0.1	2.7568	2.8982	2.4083	2.4899	2.1725	2.2090
0.2	2.6832	2.9664	2.3664	2.5298	2.1540	2.2271
0.3	2.6076	3.0331	2.3237	2.5690	2.1354	2.2449
0.4	2.5298	3.0983	2.2803	2.6076	2.1166	2.2627
0.5	2.4494	3.1622	2.2360	2.6457	2.0976	2.2803

Table 1.

Variation of Ω₁ and Ω₂ for half sinusoidal waves in both stability and vibration for $\gamma =\mathit{1.0}$ (γ < γ_T1)*.

^*

λ_b and λ_f values for $\gamma =0$.

$\mu$	$\alpha =0.0$		$\alpha =0.5$		$\alpha =0.8$
	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$
0.0	3.4641	3.4641	3.1622 (3.1439)$	3.1622 (3.1439)	2.9664	2.9664
0.2	3.4058	3.5213	3.0983 (3.0384)	3.2249 (3.1818)	2.8982	3.0331
0.4	3.3466	3.5777	3.0331 (3.0000)	3.2863 (3.2575)	2.8284	3.0983
0.6	3.2863	3.6331	2.9664 (2.9090)	3.3466 (3.3333)	2.7568	3.1622
0.8	3.2249	3.6878	2.8982 (2.8787)	3.4058 (3.4090)	2.6832	3.2249
1.0	3.1622	3.7416	2.8284 (2.8181)	3.4615 (3.4545)	2.6076	3.2863

Table 2.

Variation of Ω₁and Ω₂ for half sinusoidal waves in both stability and vibration for $\gamma =\mathit{2.0}$ (γ < γ_T1)*.

^$

Values in the parenthesis are read from Ref. [5].

^*

λ_b and λ_f values for $\gamma =0$.

$\mu$	$\alpha =0.0$		$\alpha =0.5$		$\alpha =0.8$
	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$
0.0	4.0000	4.0000	3.7416	3.7416	3.5777	3.5777
0.1	3.9496	4.0496	3.7148	3.7682	3.5665	3.5888
0.2	3.8987	4.0987	3.6878	3.7947	3.5552	3.6001
0.3	3.8471	4.1472	3.6606	3.8209	3.5441	3.6111
0.4	3.7947	4.1952	3.6331	3.8471	3.5327	3.6221
0.5	3.7416	4.2426	3.6055	3.8729	3.5213	3.6331

Table 3.

Variation of Ω₁and Ω₂ for half sinusoidal waves in both stability and vibration for $\gamma =3.0$ (γ < γ_T1)*.

^*

λ_b and λ_f values for $\gamma =0$.

Figure 2.

Dynamic stability bounds for half sinusoidal wave in both stability and vibration for $\gamma =1$($\gamma <4$).

Figure 3.

Dynamic stability bounds for half sinusoidal wave in both stability and vibration for $\gamma =\mathit{2}$ ($\gamma <\mathit{4}$).

Figure 4.

Dynamic stability bounds for half sinusoidal wave in both stability and vibration for on for $\gamma =\mathit{3}$ ($\gamma <\mathit{4}$).

Tables 4–7 gives the values of the stability boundaries Ω₁and Ω_2, within which it is dynamically unstable with change μ for α = 0.6, 0.7 & 0.8 with the mode form of full sinusoidal wave for the stability problem and half sinusoidal wave for the vibration problem for γ > 4 (γ > γ_T1). The instability boundaries Ω₁and Ω₂ given in Tables 4–7 are for α = 0.6, 0.7 and 0.8 with γ = 5 to γ = 35. For an improved understanding of the zones of dynamic stability, Figures 5–8 illustrate the instability boundaries Ω₁and Ω₂ provided in Tables 4–7. Additionally, it can be shown that when the foundation parameter increases, the zones of dynamic instability move away from the vertical axis. The areas of dynamic instability for γ = 5 to γ = 35.0 decrease from above the initial transition foundation to below second transition foundation parameter value, which is an important finding in this study.

$\mu$	$\alpha =0.6$		$\alpha =0.7$		$\alpha =0.8$
	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$
0.0	3.6095	3.6095	3.3466	3.3466	3.0612	3.0612
0.1	3.5067	3.7094	3.2636	3.4276	3.0009	3.1204
0.2	3.4008	3.8067	3.1784	3.5067	2.9393	3.1784
0.3	3.2915	3.9016	3.0910	3.5840	2.8765	3.2355
0.4	3.1784	3.9942	3.0009	3.6598	2.8122	3.2915
0.5	3.0612	4.0848	2.9081	3.7340	2.7464	3.3466

Table 4.

Variation of Ω₁and Ω₂ for full sinusoidal wave for stability and half sinusoidal wave for vibration for $\gamma =\mathit{5.0}$ (γ > γ_T1)*.

^*

λ_b and λ_f values for $\gamma =0$.

$\mu$	$\alpha =0.6$		$\alpha =0.7$		$\alpha =0.8$
	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$
0.0	6.6468	6.6468	6.3935	6.3935	6.1297	6.1297
0.1	6.5467	6.7455	6.3155	6.4705	6.0756	6.1834
0.2	6.4450	6.8427	6.2366	6.5467	6.0210	6.2366
0.3	6.3416	6.9386	6.1566	6.6219	5.9659	6.2893
0.4	6.2366	7.0331	6.0756	6.6963	5.9102	6.3416
0.5	6.1297	7.1265	5.9935	6.7699	5.8541	6.3935

Table 5.

Variation of Ω₁and Ω₂ for full sinusoidal wave for stability and half sinusoidal wave for vibration for $\gamma =\mathit{15.0}$ (γ > γ_T1)*.

^*

λ_b and λ_f values for $\gamma =0$.

$\mu$	$\alpha =0.6$		$\alpha =0.7$		$\alpha =0.8$
	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$
0.0	8.9246	8.9246	8.6942	8.6942	8.4576	8.4576
0.1	8.8331	9.0151	8.6239	8.7640	8.4095	8.5054
0.2	8.7408	9.1047	8.5530	8.8331	8.3611	8.5530
0.3	8.6474	9.1934	8.4816	8.9018	8.3124	8.6003
0.4	8.5530	9.2812	8.4095	8.9699	8.2634	8.6474
0.5	8.4576	9.3683	8.3368	9.0375	8.2142	8.6942

Table 6.

Variation of Ω₁and Ω₂ for full sinusoidal wave for stability and half sinusoidal wave for vibration for $\gamma =\mathit{25.0}$ (γ > γ_T1)*.

^*

λ_b and λ_f values for $\gamma =0$.

$\mu$	$\alpha =0.6$		$\alpha =0.7$		$\alpha =0.8$
	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$
0.0	10.8117	10.8117	10.6007	10.6007	10.3855	10.3855
0.1	10.7278	10.8950	10.5366	10.6645	10.3419	10.4289
0.2	10.6433	10.9776	10.4721	10.7278	10.2981	10.4721
0.3	10.5580	11.0596	10.4072	10.7908	10.2541	10.5151
0.4	10.4721	11.1410	10.3419	10.8534	10.2100	10.5580
0.5	10.3855	11.2218	10.2761	10.9157	10.1656	10.6007

Table 7.

Variation of Ω₁and Ω₂ for full sinusoidal wave for stability and half sinusoidal wave for vibration for $\gamma =\mathit{35.0}$ (γ > γ_T1)*.

^*

λ_b and λ_f values for $\gamma =0$.

Figure 5.

Dynamic stability bounds for full sinusoidal wave for stability and half sinusoidal wave for vibration for $\gamma =\mathit{5.0}$ (γ > γ_T1).

Figure 6.

Dynamic stability curves for full sinusoidal wave for stability and half sinusoidal wave for vibration for $\gamma =\mathit{15.0}$ (γ > γ_T1).

Figure 7.

Dynamic stability curves for full sinusoidal wave for stability and half sinusoidal wave for vibration for $\gamma =\mathit{25.0}$ (γ > γ_T1).

Figure 8.

Dynamic stability curves for full sinusoidal wave for stability and half sinusoidal wave for vibration for $\gamma =\mathit{35.0}$ (γ > γ_T1).

Tables 8–10 provide the values of the stability boundaries Ω₁and Ω₂that, for three half-sinusoidal waves for stability problems and half-sinusoidal wave for vibration problems, respectively, are dynamically unstable above the value of the second transition foundation parameter (γ > 36) for γ = 45.0 to γ = 60.0. It is noted that the regions of dynamic instability are pushed away from the vertical axis and the instability zones are initially bigger above the second transition foundation value. Another noteworthy finding is that the dynamic instability areas for values above the second transition foundation and below the third transition foundation value decrease with increasing foundation value. For a better understanding of the limits/bounds of dynamic instability, Figures 9–11 demonstrate the instability boundaries Ω₁and Ω₂ provided in Tables 8 and 10. Figure 12 displays the dynamic instability curves for full sinusoidal waves for stability and vibration problems, with reference buckling load and reference frequency parameters taking into account elastic foundation for the foundation parameter below the second transition value (γ < 36), and three half-sinusoidal waves for stability and vibration problems, with reference buckling load and reference frequency parameters taking into account foundation for the foundation parameter above the second transition value (γ > 36) obtained from Eq. (34).

$\mu$	$\alpha =0.6$		$\alpha =0.7$		$\alpha =0.8$
	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$
0.0	10.6314	10.6314	10.0598	10.0598	9.4536	9.4536
0.1	10.4065	10.8517	9.8818	10.2346	9.3276	9.5779
0.2	10.1767	11.0675	9.7006	10.4065	9.2000	9.7006
0.3	9.9415	11.2793	9.5160	10.5757	9.0705	9.8218
0.4	9.7006	11.4871	9.3276	10.7421	8.9391	9.9415
0.5	9.4536	11.6912	9.1354	10.9060	8.8058	10.0598

Table 8.

Variation of Ω₁and Ω₂ for three half sinusoidal waves for stability and half sinusoidal wave for vibration for $\gamma =\mathit{45.0}$ (γ > γ_T2)*.

^*

λ_b and λ_f values for $\gamma =0$.

$\mu$	$\alpha =0.6$		$\alpha =0.7$		$\alpha =0.8$
	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$
0.0	11.9980	11.9980	11.4285	11.4285	10.8291	10.8291
0.1	11.7735	12.2184	11.2520	11.6023	10.7052	10.9516
0.2	11.5446	12.4348	11.0728	11.7735	10.5798	11.0728
0.3	11.3112	12.6476	10.8905	11.9423	10.4530	11.1926
0.4	11.0728	12.8568	10.7052	12.1087	10.3246	11.3112
0.5	10.8291	13.0627	10.5166	12.2728	10.1945757	11.4285

Table 9.

Variation of Ω₁and Ω₂ for three half sinusoidal waves for stability and half sinusoidal wave for vibration for $\gamma =\mathit{55.0}$ (γ > γ_T2)*.

^*

λ_b and λ_f values for $\gamma =0$.

$\mu$	$\alpha =0.6$		$\alpha =0.7$		$\alpha =0.8$
	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$
0.0	12.6450	12.6450	12.0781	12.0781	11.4831	11.4831
0.1	12.4213	12.8648	11.9027	12.2509	11.3604	11.6046
0.2	12.1936	13.0809	11.7247	12.4213	11.2364	11.7247
0.3	11.9614	13.2935	11.5440	12.5895	11.1109	11.8437
0.4	11.7247	13.5027	11.3604	12.7554	10.9840	11.9614
0.5	11.4831	13.7088	11.1738	12.9192	10.8557	12.0781

Table 10.

Variation of Ω₁and Ω₂ for three half sinusoidal waves for stability and half sinusoidal wave for vibration for $\gamma =\mathit{60.0}$ (γ > γ_T2)*.

^*

λ_b and λ_f values for $\gamma =0$.

Figure 9.

Dynamic stability curves for three half sinusoidal waves for stability and half sinusoidal wave for vibration for $\gamma =\mathit{45.0}$ (γ > γ_T2).

Figure 10.

Dynamic stability curves for three half sinusoidal waves for stability and half sinusoidal wave for vibration for $\gamma =\mathit{55.0}$ (γ > γ_T2).

Figure 11.

Dynamic stability curves for three half sinusoidal waves for stability and half sinusoidal wave for vibration for $\gamma =\mathit{60.0}$ (γ > γ_T2).

Figure 12.

Master dynamic stability curves.

6. Conclusions

To accurately forecast the dynamic stability behavior of SS beam resting on elastic basis under periodic axial load, closed form solutions are derived. The geometric boundary criteria are satisfied by a one-term trigonometric admissible function. Suitable non-dimensional parameters come out of the present formulation following the Variational method that is the basis for the standard Rayleigh-Ritz technique. The existence of transition, where the mode form changes for the stability/buckling problem, is discussed. The numerical findings from the current formulation and those from the finite element approach are in good agreement. It is also proven how higher transition foundations affect the dynamic stability zones and borders. This study presents the findings obtained with different first and second foundation parameters, both below and above this transition threshold. The breadth/width of the zones of dynamic stability diminishes as elastic basis does, making the beam less susceptible to the dynamic stability phenomena under periodic loads. When the reference values of the buckling load and radian frequency are evaluated taking into consideration the effect of the foundation, which is not recognized by the earlier researchers, the existence of the master dynamic stability curves for the same mode numbers for the buckling and free vibration problems is established. The versatility of our proposed method is highlighted by its potential extension to the analysis of dynamic instability boundaries for different boundary conditions, such as Pasternak foundation, Timoshenko beams, and nanobeams, all subjected to axial periodic loads.

Acknowledgments

The author expresses gratitude to the management of Malla Reddy Engineering College and Management Sciences for their continuous support throughout this work.