Preface

*Welcome to Challenges in Foundation Engineering – Case Studies and Best Practices*. Within these pages, you will find a curated collection of chapters that delve deep into the multifaceted realm of foundation engineering. Each chapter offers invaluable insights, drawing from real-world case studies and best practices to tackle the contemporary challenges facing the field.

This book covers a diverse array of topics, including shallow and deep foundations, soil–structure interaction, and reinforced soil. It serves not merely as a compilation of case studies but also as a comprehensive synthesis of the current state of research in foundation engineering. The interdisciplinary nature of the topics reflects the collaborative spirit necessary to navigate the complexities inherent in the field.

In the first chapter following the introduction of our book "Dynamic Stability of Beam on Elastic Foundation Including Higher Transition Foundation", we explore the dynamic stability of beams on elastic foundation. Employing the energy method, a precise analytical solution has been crafted to anticipate the dynamic instability bounds of simply supported beams on elastic foundations. This analysis zeroes in on the influence of higher transition foundations on dynamic stability boundaries. Findings indicate that as the elastic foundation parameter value increases, the width of dynamically unstable zones decreases, thereby reducing the susceptibility of the beam to dynamic stability phenomena under periodic loads. Additionally, the study demonstrates the existence of master dynamic instability curves by utilizing precise non-dimensional parameters for the imposed periodic load and its radian frequency.

In Chapter 3, "Integrity Assessment for Drill Shafts Foundation in Public and Private Works with Available Technologies in the Twenty-First Century", we focus on integrity control tests to detect anomalies and construction defects promptly. These tests provide precise information on pile geometry using methods like pile impedance, ultrasonic logging, and thermal profiling. They facilitate the identification of anomalies in 2D and 3D, allowing evaluation of concrete placement integrity during hardening. Despite their benefits, these tests also have limitations, which we discuss based on field experience and research findings from global experts and the Deep Foundations Institute.

In Chapter 4, "Soil-Structure Interaction: Understanding and Mitigating Challenges", we delve into the intricate dynamics of soil–structure interaction (SSI). The chapter wraps up by delving into innovative mitigation strategies for challenges associated with SSI, including the utilization of novel materials and advanced computational models. Emphasizing sustainability and resilience, the discussion centers on practices designed to endure the impacts of time and climate change. This comprehensive overview of SSI aims to equip readers with the insights and tools necessary to understand and address the complexities of this crucial aspect of geotechnical engineering. Serving as

a valuable reference, the chapter caters to a diverse audience of researchers, engineers, students, and practitioners, facilitating the creation of safer, more resilient structures while minimizing environmental repercussions.

Finally, Chapter 5, "Geosynthetic Reinforcement Applications", serves to introduce readers to various applications of reinforcement. The chapter provides a brief overview of geosynthetics before delving into specific examples such as geosynthetics-reinforced soil retaining walls, which offer benefits under earthquake loading conditions. Additionally, the chapter discusses using geosynthetic reinforcement below foundations to enhance bearing capacity and explores the use of geosynthetics as encasements around stone columns, showcasing improvements in seismic performance compared to traditional stone columns. Through these examples, readers gain insight into the versatility and effectiveness of geosynthetic reinforcement across different engineering contexts.

As you embark on this intellectual journey, I encourage you to adopt a discerning perspective, embracing the dynamic nature of foundation engineering. This book represents more than just a collection of chapters; it stands as a testament to the collective pursuit of knowledge and excellence in geotechnical engineering.

May the wealth of knowledge and insights gleaned from these pages catalyze the development of groundbreaking solutions in foundation engineering. As we navigate the complexities of this field, let us remain committed to fostering sustainable practices and embracing innovation at every turn. By continually pushing the boundaries of our understanding and leveraging the latest advancements in technology, we can forge a path toward a future where foundation engineering not only meets the demands of today but also anticipates and addresses the challenges of tomorrow. Together, let us strive to shape a more resilient and sustainable built environment for generations to come.

> **Dr. Mohamed Ayeldeen** Technical Director, Arcadis Consulting, London, United Kingdom

## **Chapter 1**

## Introductory Chapter: Challenges in Foundation Engineering – Case Studies and Best Practices

*Mohamed Ayeldeen*

## **1. Introduction**

In the ever-evolving landscape of civil engineering, foundation engineers find themselves grappling with an array of challenges that have assumed new dimensions in the twenty-first century. While the global economy traverses through periods of uncertainty, the domain of foundation engineering is not insulated from its far-reaching impacts. Economic challenges have given rise to an intensified focus on efficiency, cost-effectiveness, and the imperative to develop infrastructure that stands resilient in the face of adversities. Consequently, the demand for smart and sustainable solutions has reached unprecedented levels, challenging engineers to innovate and adapt.

Foundations come in a variety of forms, each tailored to suit the specific requirements of the structure and the characteristics of the underlying soil or rock strata. Broadly categorized, foundations can be classified into shallow foundations and deep foundations. Shallow foundations, as the name suggests, are those that typically penetrate only a few meters into the ground and spread their loads over a larger area. These include footings, raft foundations, and mat foundations, among others. Deep foundations, on the other hand, are designed to transfer structural loads to deeper, more competent soil or rock layers. These may include piles, drilled shafts, and caissons, depending on the site conditions and engineering considerations [1, 2].

Regardless of their type, the fundamental function of foundations remains consistent: to distribute the loads from the structure safely into the ground, ensuring stability and preventing settlement or failure. Whether supporting towering skyscrapers in urban landscapes or resilient infrastructure in remote regions, foundations play a pivotal role in the built environment, often operating silently beneath our feet but serving as the critical backbone of our civilization.

## **2. Geotechnical engineering and its applications**

Geotechnical engineering is the cornerstone of foundation design and construction, focusing on the behavior of earth materials—soil and rock—and their interaction with civil engineering structures. Understanding the properties of soil and rock formations is paramount in ensuring the stability, safety, and longevity of foundations. Soil types vary widely, ranging from cohesive clays to granular sands and everything in between. Each type possesses distinct characteristics that influence its behavior

under load, moisture content, and other environmental factors. Cohesive soils, such as clay, tend to exhibit high plasticity and low permeability, making them susceptible to swelling, shrinkage, and consolidation. Granular soils, like sand and gravel, offer better drainage but may experience settlement and instability if not properly compacted or reinforced [3].

During foundation construction and design, various soil-related challenges can arise, presenting engineers with complex problems to overcome. One common issue is soil settlement, which occurs when the soil beneath a foundation compress under the weight of the structure, leading to uneven settling and potential structural damage. This phenomenon is particularly prevalent in areas with expansive clay soils or poorly compacted fill materials. Another challenge is soil liquefaction, a phenomenon in which saturated granular soils lose their strength and stiffness during seismic events, behaving more like a liquid than a solid. Liquefaction can cause catastrophic failure of foundations, especially in regions prone to earthquakes, posing significant risks to structures and occupants alike. Additionally, soil erosion and instability can pose challenges during foundation construction, especially in areas with steep slopes, high groundwater tables, or heavy rainfall. Erosion control measures and slope stabilization techniques are essential to mitigate these risks and ensure the long-term stability of foundations [4].

Geotechnical engineers employ a variety of techniques and methodologies to address these challenges, including soil testing and analysis, ground improvement methods, and innovative foundation design approaches. By understanding the unique properties and behavior of soil and rock formations, engineers can develop effective solutions to safeguard structures against the unpredictable forces of nature and ensure the resilience and sustainability of our built environment [5–7].

### **3. Geotechnical engineering and climate change**

As we confront the challenges of climate change and global warming, the role of geotechnical engineering in sustainable development has never been more critical. The construction and infrastructure sectors are significant contributors to greenhouse gas emissions, primarily through energy-intensive processes and the extraction of raw materials. Geotechnical engineering, with its focus on the built environment's interaction with the natural landscape, plays a pivotal role in mitigating these impacts and promoting environmentally responsible practices. One of the key considerations in addressing global warming within the geotechnical industry is the reduction of carbon emissions associated with construction activities. This can be achieved through the adoption of green engineering principles, which prioritize energy efficiency, resource conservation, and environmental stewardship throughout the project lifecycle. By optimizing construction methods, materials selection, and site management practices, engineers can minimize the carbon footprint of foundation projects while maintaining structural integrity and safety [8, 9].

The choice of foundation systems and construction techniques can significantly impact a project's sustainability. Sustainable foundation solutions aim to minimize environmental disruption, conserve natural resources, and enhance the resilience of built structures to climate-related hazards. Innovative approaches such as recycled materials for backfilling, prefabricated foundation components, and bio-based soil stabilization techniques offer sustainable alternatives to traditional construction methods, reducing both environmental impact and project costs [9, 10].

*Introductory Chapter: Challenges in Foundation Engineering – Case Studies and Best Practices DOI: http://dx.doi.org/10.5772/intechopen.114827*

### **4. Research and development importance**

Research and development (R&D) serve as the cornerstone of innovation and progress in all fields, including geotechnical engineering. In the realm of foundation engineering, the pursuit of new knowledge and the application of cutting-edge technologies are essential for addressing the complex challenges facing the industry. However, for R&D efforts to have a meaningful impact, it is crucial to bridge the gap between academic research and practical applications, ensuring that innovative solutions are not only technically sound but also economically viable, time-saving, and environmentally sustainable [11].

However, the translation of academic research into practical solutions faces several challenges, including the complexity of real-world engineering problems, the limitations of existing technologies, and the constraints of time and budget. Bridging this gap requires collaboration and knowledge exchange between academia and industry, fostering a symbiotic relationship that leverages the strengths of both sectors. Practical solutions derived from academic research offer several advantages over conventional approaches. First and foremost, they are often more efficient and cost-effective, leveraging innovative technologies and methodologies to streamline construction processes and reduce project timelines. By optimizing material usage, construction techniques, and equipment utilization, these solutions can yield substantial savings in both time and resources, enhancing the overall competitiveness and profitability of construction projects.

Practical solutions derived from academic research have the potential to minimize environmental impact and promote sustainability. By incorporating green engineering principles, such as the use of recycled materials, energy-efficient construction methods, and sustainable land use practices, these solutions can reduce carbon emissions, conserve natural resources, and protect ecosystems. In doing so, they contribute to the long-term health and resilience of the built environment while mitigating the adverse effects of climate change.

To realize the full potential of academic research in driving practical solutions, it is essential to foster collaboration and knowledge exchange between researchers, practitioners, and industry stakeholders. This can be achieved through initiatives such as joint research projects, technology transfer programs, and professional development opportunities that facilitate the exchange of ideas, expertise, and best practices. By reducing the barriers between academia and industry, we can accelerate the pace of innovation, drive economic growth, and create a more sustainable future for generations to come.

In recent years, the field of geotechnical engineering has witnessed the emergence of promising new topics that hold significant potential for revolutionizing traditional practices and addressing contemporary challenges. Among these, ground improvement techniques and geothermal piles stand out as particularly promising areas of exploration.

Ground improvement techniques encompass a variety of methods aimed at enhancing the engineering properties of soils to meet the requirements of construction projects. These techniques offer alternative solutions to mitigate soil-related challenges such as low bearing capacity, excessive settlement, or poor drainage. Methods such as soil compaction, vibro-compaction, deep soil mixing, and grouting have been increasingly utilized to improve soil strength, stability, and drainage characteristics. Ground improvement techniques not only optimize the performance of foundation systems but also offer sustainable alternatives to traditional soil

stabilization methods, reducing the need for excessive excavation and the use of non-renewable materials [5].

Geothermal piles represent another innovative approach gaining traction in geotechnical engineering. These specialized foundation elements integrate geothermal heat exchange systems into traditional pile foundations, allowing for the extraction or injection of thermal energy from or into the ground. By harnessing the stable thermal properties of the subsurface, geothermal piles offer opportunities for sustainable heating and cooling solutions in buildings and infrastructure projects. Beyond providing structural support, geothermal piles contribute to energy efficiency, reducing reliance on fossil fuels and lowering carbon emissions associated with heating, ventilation, and air conditioning systems [12, 13].

As these new topics continue to evolve, there is growing interest in their application across a wide range of construction projects, from high-rise buildings and transportation infrastructure to renewable energy installations and sustainable development initiatives. However, their successful implementation requires interdisciplinary collaboration, technical expertise, and rigorous performance evaluation to ensure reliability, efficiency, and environmental sustainability [14].

### **5. Artificial intelligence revolution**

In the realm of geotechnical and foundation engineering, the fusion of scientific inquiry with practical experience is essential for tackling the evolving challenges faced by the construction industry while safeguarding the environment. This entails directing research and development efforts toward addressing practical challenges encountered in the field, such as soil stabilization, foundation design optimization, and construction material efficiency. Collaborative partnerships between academia, industry, and government agencies are instrumental in channeling resources toward research initiatives that yield tangible benefits for the construction sector and society at large. The emerging artificial intelligence, *AI* revolution, holds immense promise for revolutionizing the geotechnical industry. Advancements in machine learning, data analytics, and predictive modeling offer unprecedented opportunities to enhance accuracy, efficiency, and innovation across the entire project lifecycle. By leveraging AI algorithms to analyze vast datasets, simulate complex scenarios, and optimize design parameters, engineers can make more informed decisions, mitigate risks, and optimize resource utilization, ultimately leading to more sustainable and resilient infrastructure solutions [15].

Nonetheless, it is crucial to ensure that AI technologies are deployed ethically and responsibly, with due consideration given to issues of data privacy, algorithmic bias, and societal impact. Collaboration between AI experts, domain specialists, and stakeholders is key to developing AI-driven solutions that not only optimize performance but also uphold ethical principles and promote equitable outcomes.

Through this holistic approach that integrates scientific research, practical experience, and cutting-edge technologies, the geotechnical and foundation engineering community can drive innovation, sustainability, and resilience in the face of evolving challenges, paving the way toward a more sustainable and prosperous future for generations to come.

*Introductory Chapter: Challenges in Foundation Engineering – Case Studies and Best Practices DOI: http://dx.doi.org/10.5772/intechopen.114827*

## **Author details**

Mohamed Ayeldeen1,2

1 Arcadis Consulting, London, United Kingdom

2 Higher Institute of Engineering and Technology (THIET), Tanta, Egypt

\*Address all correspondence to: ayeldeen.moh@gmail.com

© 2024 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Bowles JE, Guo Y. Foundation Analysis and Design. Vol. 5. New York: McGraw-Hill; 1996. p. 127

[2] Poulos HG, Davis EH. Pile Foundation Analysis and Design. Vol. 397. New York: Wiley; 1980

[3] William Powrie. Soil Mechanics: Concepts and Applications. 2018

[4] Ayeldeen M, Negm A, El Sawwaf M, Gädda T. Laboratory study of using biopolymer to reduce wind erosion. International Journal of Geotechnical Engineering. 4 May 2018;**12**(3):228-240

[5] Ayeldeen M et al. Enhancing mechanical behaviors of collapsible soil using two biopolymers. Journal of Rock Mechanics and Geotechnical Engineering. 2017;**9**(2):329-339

[6] Ayeldeen M, Kitazume M. Using fiber and liquid polymer to improve the behaviour of cement-stabilized soft clay. Geotextiles and Geomembranes. 2017;**45**(6):592-602

[7] Azzam W, Ayeldeen M, El Siragy M. Improving the structural stability during earthquakes using in-filled trench with EPS geofoam—Numerical study. Arabian Journal of Geosciences. Jul 2018;**11**(14):1

[8] Basu D, Misra A, Puppala AJ. Sustainability and geotechnical engineering: Perspectives and review. Canadian Geotechnical Journal. 2014;**52**(1):96-113

[9] Jefferis SA. Moving Towards Sustainability in Geotechnical Engineering. In: GeoCongress 2008: Geosustainability and Geohazard Mitigation; 2008. pp. 844-851

[10] Lee M, Basu D. An integrated approach for resilience and sustainability in geotechnical engineering. Indian Geotechnical Journal. 2018;**48**(2): 207-234

[11] Morgenstern NR. Geotechnical engineering and frontier resource development. Geotechnique. 1981;**31**(3):305-365

[12] Cunha RP, Bourne-Webb PJ. A critical review on the current knowledge of geothermal energy piles to sustainably climatize buildings. Renewable and Sustainable Energy Reviews. 2022;**158**:112072

[13] Ghasemi-Fare O, Basu P. A practical heat transfer model for geothermal piles. Energy and Buildings. 2013;**66**:470-479

[14] De Moel M et al. Technological advances and applications of geothermal energy pile foundations and their feasibility in Australia. Renewable and Sustainable Energy Reviews. 2010;**14**(9):2683-2696

[15] Baghbani A et al. Application of artificial intelligence in geotechnical engineering: A state-of-the-art review. Earth-Science Reviews. 2022;**228**:103991

## **Chapter 2**

## Dynamic Stability of Beam on Elastic Foundation Including Higher Transition Foundation

*Bhavanasi Subbaratnam*

## **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.

**Keywords:** dynamic instability, uniform beams, energy method, periodic loads, elastic foundation, transition foundation

## **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 inplane 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 [11–15].

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

#### *Dynamic Stability of Beam on Elastic Foundation Including Higher Transition Foundation DOI: http://dx.doi.org/10.5772/intechopen.113009*

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 Q ð Þ is given, by

$$
\prod = U + U\_F - W - T \tag{1}
$$

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

$$U = \frac{EI}{2} \int\_0^L \left(\frac{d^2 w}{dx^2}\right)^2 dx \tag{2}$$

$$U\_F = \frac{k}{2} \int\_0^L w^2 dx \tag{3}$$

**Figure 1.** *Uniform SS beam lying on foundation subjected to periodic load.*

*Challenges in Foundation Engineering – Case Studies and Best Practices*

$$T = \frac{m\nu^2}{2} \int\_0^L w^2 d\mathbf{x} \tag{4}$$

and

$$\mathcal{W} = \frac{P(t)}{2} \int\_0^L \left(\frac{dw}{d\mathbf{x}}\right)^2 d\mathbf{x} \tag{5}$$

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

$$P(t) = P\_S + P\_I \cos\theta t \tag{6}$$

where PS 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).

$$\Pi = \frac{EI}{2} \int\_0^L \left(\frac{d^2 w}{d\mathbf{x}^2}\right)^2 d\mathbf{x} + \frac{k}{2} \int\_0^L w^2 d\mathbf{x} - \frac{P(t)}{2} \int\_0^L \left(\frac{dw}{d\mathbf{x}}\right)^2 d\mathbf{x} - \frac{m\sigma^2}{2} \int\_0^L w^2 d\mathbf{x} \tag{7}$$

As ω = θ/2, Eq. (7) becomes

$$\Pi = \frac{EI}{2} \int\_0^L \left(\frac{d^2 w}{dx^2}\right)^2 dx + \frac{k}{2} \int\_0^L w^2 dx - \frac{P(t)}{2} \int\_0^L \left(\frac{dw}{dx}\right)^2 dx - \frac{m}{2} \frac{\theta^2}{4} \int\_0^L w^2 dx \tag{8}$$

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

$$\Pi = \frac{EI}{2} \int\_0^L \left(\frac{d^2 w}{dx^2}\right)^2 dx + \frac{k}{2} \int\_0^L w^2 dx - \left(P\_S + P\_t \cos\theta t\right) \frac{1}{2} \left[\left(\frac{dw}{dx}\right)^2 dx - \frac{m}{2} \frac{\theta^2}{4} \int\_0^L w^2 dx\right] \tag{9}$$

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

$$\Pi = \frac{EI}{2} \int\_0^L \left(\frac{d^2 w}{dx^2}\right)^2 dx + \frac{k}{2} \int\_0^L w^2 dx - \left(a \pm \frac{\beta}{2}\right) \frac{P\_{cr}}{2} \int\_0^L \left(\frac{dw}{dx}\right)^2 dx - \frac{m}{2} \frac{\theta^2}{4} \int\_0^L w^2 dx \tag{10}$$

*Dynamic Stability of Beam on Elastic Foundation Including Higher Transition Foundation DOI: http://dx.doi.org/10.5772/intechopen.113009*

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 <sup>α</sup> is the fraction of buckling load (<sup>¼</sup> *Ps Pcr*) and β is the fraction of periodic buckling load

$$\left(= \frac{P\_t}{P\_{cr}}.\right) \tag{11}$$

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

$$w = a \sin \frac{n \pi x}{L} \tag{12}$$

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

In terms of the mode numbers *nb* and *nf,* this admissible function provides the buckling loads and the frequency parameters as

$$
\lambda\_b = \frac{P\_{cr}L^2}{EI} = \pi^2 \left( n^2 + \frac{\mathcal{Y}}{n^2} \right) \tag{13}
$$

and

$$
\lambda\_f = \frac{m\alpha^2 L^4}{EI} = \pi^4 \left( n^2 + \frac{\chi}{n^2} \right) \tag{14}
$$

where *<sup>γ</sup>* is the foundation parameter (*<sup>γ</sup>* <sup>¼</sup> *kL*<sup>4</sup> *<sup>π</sup>*4*EI*), *λ<sup>b</sup>* is buckling/critical load parameter (<sup>¼</sup> *PcrL*<sup>2</sup> *EI* ) and *<sup>λ</sup><sup>f</sup>* is frequency parameter (<sup>¼</sup> *<sup>m</sup>ω*2*L*<sup>4</sup> *EI* ).where *Pcr* is critical load and ω is natural frequency.

#### **2.2 Foundation parameter transition values**

The transitional foundation parameter value *γTi* (*i* varies from 1, 2, 3 … . and with reference to buckling mode number *nb*), where buckling load parameter *λ<sup>b</sup>* changes from mode *nb* to mode (*n* + 1)*b*. The transition value *γTi* is evaluated from Eq. (13),

using the condition that the buckling load parameters are the same for the two consecutive modes of buckling *nb* and (*n* + 1)*b*, as

$$
\lambda\_{bn} = \lambda\_{b(n+1)} \tag{15}
$$

The values of the *γTi* are

$$\left(\boldsymbol{\gamma}\_{T\bar{\imath}} = \boldsymbol{n}\_b\right)^2 \left(\boldsymbol{n} + \mathbf{1}\right)\_\mathbf{b}^2, \left(\boldsymbol{i}, \boldsymbol{n}\_b = \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5} \dots \right) \tag{16}$$

Two noteworthy findings from the assessment of the *γTi* are:


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

The impact of the initial and second transition foundation parameters *γT*<sup>1</sup> = 4 and *γT*<sup>2</sup> = 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<sup>b</sup>* **and** *λ<sup>f</sup>* **for different values of** *nb* **and** *nf*

Eqs. (13) and (14), provide expressions for *λ<sup>b</sup>* and *λ<sup>f</sup>* the various values of *nb, nf* and *γ*. In light of these values of *γ*, *nb* and *nf*, succinct evaluations of the formulations for *λ<sup>b</sup>* and *λ<sup>f</sup>* are given here;

*2.3.1 For γ < γ<sup>T</sup>*<sup>1</sup> *(*nb = nf *= 1)*

The *λ<sup>b</sup>* and *λ<sup>f</sup>* terms are

$$
\lambda\_b = \frac{P\_{cr}L^2}{EI} = \pi^2[1+\chi] \tag{17}
$$

and

$$
\lambda\_f = \frac{m\alpha^2 L^4}{EI} = \pi^4 (1+\chi) \tag{18}
$$

*2.3.2 For γ > γ<sup>T</sup>*<sup>1</sup> *and γ < γ<sup>T</sup>*<sup>2</sup> *(*nb = *2,* nf *= 1)*

The *λ<sup>b</sup>* and *λ<sup>f</sup>* terms are

$$
\lambda\_b = \frac{P\_{cr}L^2}{EI} = \pi^2 \left[4 + \frac{\mathcal{Y}}{4}\right] \tag{19}
$$

and

$$
\lambda\_f = \frac{m\alpha^2 L^4}{EI} = \pi^4 (1+\chi) \tag{20}
$$

*Dynamic Stability of Beam on Elastic Foundation Including Higher Transition Foundation DOI: http://dx.doi.org/10.5772/intechopen.113009*

## *2.3.3 For γ > γT*<sup>1</sup> *and γ < γT*<sup>2</sup> *(*nb = *2,* nf *= 2)*

The *λ<sup>b</sup>* and *λ<sup>f</sup>* terms are

$$
\lambda\_b = \frac{P\_{cr}L^2}{EI} = \pi^2 \left[ 4 + \frac{\mathcal{Y}}{4} \right] \tag{21}
$$

and

$$
\lambda\_f = \frac{m\alpha^2 L^4}{EI} = \pi^4 (16+\chi) \tag{22}
$$

*2.3.4 For γ > γ<sup>T</sup>*<sup>2</sup> *and γ < γ<sup>T</sup>*<sup>3</sup> *(*nb = *3,* nf *= 1)*

The *λ<sup>b</sup>* and *λ<sup>f</sup>* terms are

$$
\lambda\_b = \frac{P\_{cr}L^2}{EI} = \pi^2 \left[ \Re + \frac{\mathcal{Y}}{\mathfrak{Y}} \right] \tag{23}
$$

and

$$
\lambda\_f = \frac{m\alpha^2 L^4}{EI} = \pi^4 (1+\chi) \tag{24}
$$

*2.3.5 For γ > γ<sup>T</sup>*<sup>2</sup> *and γ < γ<sup>T</sup>*<sup>3</sup> *(*nb = *3,* nf *= 3)*

The *λ<sup>b</sup>* and *λ<sup>f</sup>* terms are

$$
\lambda\_b = \frac{P\_{cr}L^2}{EI} = \pi^2 \left[\Theta + \frac{\chi}{9}\right] \tag{25}
$$

and

$$
\lambda\_f = \frac{m\alpha^2 L^4}{I\mathcal{E}} = \pi^4 (8\mathbf{1} + \boldsymbol{\gamma})\tag{26}
$$

For the sake of completeness, the expressions for *λ<sup>b</sup>* and *λ<sup>f</sup>* for the three cases that are being considered here are given, but any combination of *nb*, *nf* and *γ*, 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 *γ*.

### **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 *Pcr* <sup>¼</sup> *<sup>π</sup>*2*EI <sup>L</sup>*<sup>2</sup> and *<sup>m</sup>* <sup>¼</sup> *<sup>π</sup>*4*EI <sup>ω</sup>*2*L*<sup>4</sup> without taking into account the impact of foundation.

*Challenges in Foundation Engineering – Case Studies and Best Practices*

$$\mathbf{1} + \mathbf{y} - \left[a \pm \frac{\beta}{2}\right] - \frac{\theta^2}{4a^2} = \mathbf{0} \tag{27}$$

$$a = \frac{P\_s}{P\_{cr}} \text{ and } \beta = \frac{P\_t}{P\_{cr}}.\tag{28}$$

or

$$\frac{\theta}{a\rho} = \mathfrak{Q} = 2\sqrt{(1-a)(1\pm\mu)+\gamma} \tag{29}$$

where <sup>μ</sup> is the defined non-dimensional parameter, *<sup>μ</sup>* <sup>¼</sup> *<sup>β</sup>* 2 1ð Þ �*<sup>α</sup>* used in some earlier studies on the topic of dynamic stability [2].

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

$$\mathbf{1} - \frac{\left(a \pm \frac{\theta}{2}\right) P\_{cr} \cdot \mathbf{4} \frac{\pi^2}{L^2}}{\frac{\pi^4 EI}{L^4} \left[\mathbf{1} \mathbf{6} + \frac{kL^4}{\pi^4 EI}\right]} - \frac{\frac{m\theta^2}{4}}{\frac{\pi^4 EI}{L^4} \left[\mathbf{1} + \frac{kL^4}{\pi^4 EI}\right]} = \mathbf{0} \tag{30}$$

Substituting the corresponding reference values of *Pcr* <sup>¼</sup> <sup>4</sup>*π*2*EI <sup>L</sup>*<sup>2</sup> and *<sup>m</sup>* <sup>¼</sup> *<sup>π</sup>*4*EI <sup>ω</sup>*2*L*<sup>4</sup> without considering the effect of the elastic foundation in Eq. (30) becomes

$$\Omega = \frac{\theta}{a\nu} = 2\sqrt{\left[ \left( (1-a)(1\pm\mu) + \frac{\chi}{16} \right) \left( \frac{(1+\chi)}{\left(1+\frac{\chi}{16}\right)} \right) \right]} \tag{31}$$

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

$$\mathbf{1} - \frac{(\alpha \pm \frac{\beta}{2})P\_{cr} \cdot \mathbf{9} \frac{\pi^2}{L^2}}{\frac{\pi^4 EI}{L^4} \left[ \mathbf{81} + \frac{kL^4}{\pi^4 EI} \right]} - \frac{\frac{m\theta^2}{4}}{\frac{\pi^4 EI}{L^4} \left[ \mathbf{1} + \frac{kL^4}{\pi^4 EI} \right]} = \mathbf{0} \tag{32}$$

Eq. (32) is expressed as follows when the reference values *Pcr* <sup>¼</sup> <sup>9</sup>*π*2*EI <sup>L</sup>*<sup>2</sup> and *<sup>m</sup>* <sup>¼</sup> *<sup>π</sup>*4*EI ω*2*L*<sup>4</sup> are substituted and the foundation is ignored.

$$\Omega = \frac{\theta}{a\nu} = 2\sqrt{\left[\left[ (\mathbf{1} - a)(\mathbf{1} \pm \mu) + \frac{\chi}{8\mathbf{1}}\right] \overline{\left[ \frac{(\mathbf{1} + \chi)}{(\mathbf{1} + (\chi/8\mathbf{1}))} \right]}\right]}\tag{33}$$

These three dynamic instability formulae are constructed using the modes *nb* and *nf* for initial and second transition foundation parameter values γ smaller or greater than *γ<sup>T</sup>*<sup>1</sup> and *γ<sup>T</sup>*2*.*

### **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 *Dynamic Stability of Beam on Elastic Foundation Including Higher Transition Foundation DOI: http://dx.doi.org/10.5772/intechopen.113009*

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

$$\frac{\theta}{\omega} = \Omega = 2\sqrt{(1-a)(1\pm\mu)}\tag{34}$$

Use of these specific reference values of *λ<sup>b</sup>* and *λ<sup>f</sup>* 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 *nb* and *nf* (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 nondimensional form. However, when non-dimensional parameters *μ* and Ω. are defined, their characteristic values are referenced implicitly.

The values of the stability boundaries Ω1and Ω2, 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 Ω<sup>2</sup> 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


#### **Table 1.**

*Variation of <sup>Ω</sup><sup>1</sup> and <sup>Ω</sup><sup>2</sup> for half sinusoidal waves in both stability and vibration for <sup>γ</sup>* <sup>¼</sup> *<sup>1</sup>:0 (<sup>γ</sup> <sup>&</sup>lt; <sup>γ</sup>T1)\* .*


#### **Table 2.**

*Variation of <sup>Ω</sup>1and <sup>Ω</sup><sup>2</sup> for half sinusoidal waves in both stability and vibration for <sup>γ</sup>* <sup>¼</sup> *<sup>2</sup>:0 (<sup>γ</sup> <sup>&</sup>lt; <sup>γ</sup>T1)\* .*


#### **Table 3.**

*Variation of <sup>Ω</sup>1and <sup>Ω</sup><sup>2</sup> for half sinusoidal waves in both stability and vibration for <sup>γ</sup>* <sup>¼</sup> <sup>3</sup>*:*<sup>0</sup> *(<sup>γ</sup> <sup>&</sup>lt; <sup>γ</sup>T1)\* .*

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 Ω2, between 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 Ω<sup>2</sup> 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.

**Tables 4**–**7** gives the values of the stability boundaries Ω1and Ω2, 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 > γT1). The instability boundaries Ω1and Ω<sup>2</sup> 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 Ω<sup>2</sup>

*Dynamic Stability of Beam on Elastic Foundation Including Higher Transition Foundation DOI: http://dx.doi.org/10.5772/intechopen.113009*

**Figure 2.** *Dynamic stability bounds for half sinusoidal wave in both stability and vibration for γ* ¼ 1*(γ* <4*).*

**Figure 3.** *Dynamic stability bounds for half sinusoidal wave in both stability and vibration for γ* ¼ *2 (γ* <*4).*

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.

**Figure 4.** *Dynamic stability bounds for half sinusoidal wave in both stability and vibration for on for γ* ¼ *3 (γ* <*4).*


#### **Table 4.**

*Variation of Ω1and Ω<sup>2</sup> for full sinusoidal wave for stability and half sinusoidal wave for vibration for γ* ¼ *5:0 (γ > γT1)\* .*


#### **Table 5.**

*Variation of Ω1and Ω<sup>2</sup> for full sinusoidal wave for stability and half sinusoidal wave for vibration for <sup>γ</sup>* <sup>¼</sup> *<sup>15</sup>:0 (<sup>γ</sup> <sup>&</sup>gt; <sup>γ</sup>T1)\* .*

*Dynamic Stability of Beam on Elastic Foundation Including Higher Transition Foundation DOI: http://dx.doi.org/10.5772/intechopen.113009*


#### **Table 6.**

*Variation of Ω1and Ω<sup>2</sup> for full sinusoidal wave for stability and half sinusoidal wave for vibration for <sup>γ</sup>* <sup>¼</sup> *<sup>25</sup>:0 (<sup>γ</sup> <sup>&</sup>gt; <sup>γ</sup>T1)\* .*


#### **Table 7.**

*Variation of Ω1and Ω<sup>2</sup> for full sinusoidal wave for stability and half sinusoidal wave for vibration for <sup>γ</sup>* <sup>¼</sup> *<sup>35</sup>:0 (<sup>γ</sup> <sup>&</sup>gt; <sup>γ</sup>T1)\* .*

#### **Figure 5.**

*Dynamic stability bounds for full sinusoidal wave for stability and half sinusoidal wave for vibration for γ* ¼ *5:0 (γ > γT1).*

*Dynamic stability curves for full sinusoidal wave for stability and half sinusoidal wave for vibration for γ* ¼ *15:0 (γ > γT1).*

*Dynamic stability curves for full sinusoidal wave for stability and half sinusoidal wave for vibration for γ* ¼ *25:0 (γ > γT1).*

**Figure 8.**

*Dynamic stability curves for full sinusoidal wave for stability and half sinusoidal wave for vibration for γ* ¼ *35:0 (γ > γT1).*

*Dynamic Stability of Beam on Elastic Foundation Including Higher Transition Foundation DOI: http://dx.doi.org/10.5772/intechopen.113009*


#### **Table 8.**

*Variation of Ω1and Ω<sup>2</sup> for three half sinusoidal waves for stability and half sinusoidal wave for vibration for <sup>γ</sup>* <sup>¼</sup> *<sup>45</sup>:0 (<sup>γ</sup> <sup>&</sup>gt; <sup>γ</sup>T2)\* .*


#### **Table 9.**

*Variation of Ω1and Ω<sup>2</sup> for three half sinusoidal waves for stability and half sinusoidal wave for vibration for <sup>γ</sup>* <sup>¼</sup> *<sup>55</sup>:0 (<sup>γ</sup> <sup>&</sup>gt; <sup>γ</sup>T2)\* .*


#### **Table 10.**

*Variation of Ω1and Ω<sup>2</sup> for three half sinusoidal waves for stability and half sinusoidal wave for vibration for <sup>γ</sup>* <sup>¼</sup> *<sup>60</sup>:0 (<sup>γ</sup> <sup>&</sup>gt; <sup>γ</sup>T2)\* .*

**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 Ω<sup>2</sup> provided in **Tables 8** and **10**. **Figure 12** displays the dynamic instability curves for full

#### **Figure 9.**

*Dynamic stability curves for three half sinusoidal waves for stability and half sinusoidal wave for vibration for γ* ¼ *45:0 (γ > γT2).*

#### **Figure 10.**

*Dynamic stability curves for three half sinusoidal waves for stability and half sinusoidal wave for vibration for γ* ¼ *55:0 (γ > γT2).*

*Dynamic Stability of Beam on Elastic Foundation Including Higher Transition Foundation DOI: http://dx.doi.org/10.5772/intechopen.113009*

#### **Figure 11.**

*Dynamic stability curves for three half sinusoidal waves for stability and half sinusoidal wave for vibration for γ* ¼ *60:0 (γ > γT2).*

**Figure 12.** *Master dynamic stability curves.*

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 halfsinusoidal 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).

### **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.

## **Acknowledgements**

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

## **Author details**

Bhavanasi Subbaratnam Department of Mechanical Engineering, Malla Reddy Engineering College and Management Sciences, Kistapur, Medchal, Hyderabad, India

\*Address all correspondence to: b\_subbaratnam@yahoo.com

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Dynamic Stability of Beam on Elastic Foundation Including Higher Transition Foundation DOI: http://dx.doi.org/10.5772/intechopen.113009*

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## **Chapter 3**
