Abstract
This paper investigates the plane stress behavior of a cantilever beam made of functionally graded material (FGM) subjected to combined shear and bending loading. A new formulation of the elastic modulus of FGM materials is proposed, incorporating two graduation parameters: the first representing the material gradation and the second the ratio of the longitudinal elastic modulus. Three models of the mechanical property distribution through the beam thickness are considered: the classical variation, the new model-1, and the new model-2. The mathematical formulation, based on the static equilibrium equations, is developed to predict the distribution of stresses and displacements in the beam subjected to a tangential force applied at its free end. The influence of the graduation parameters k1, k2 and the length-to-thickness ratio L/h on the stress and displacement response is also
analyzed.
Key Words
cantilever beam; material gradation; shear and bending loading; static analysis
Address
Benferhat Rabia: Department of Civil Engineering, Ibn Khaldoun University of Tiaret, Algeria; Laboratory of Geomatics and Sustainable Development LGéo2D, University of Tiaret, Algeria
Tahar Hassaine Daouadji: Department of Civil Engineering, Ibn Khaldoun University of Tiaret, Algeria; Laboratory of Geomatics and Sustainable Development LGéo2D, University of Tiaret, Algeria
Abdelaziz Hadj Henni: Department of Civil Engineering, Ibn Khaldoun University of Tiaret, Algeria
Abstract
This study presents a novel analytical framework for modeling unsteady gas flow in long-distance pipelines laid across nonuniform terrain. Unlike existing approaches that typically neglect gravitational and geometric complexities, the proposed model incorporates slope-dependent gravitational effects through exponential modulation coefficients and accounts for transient leakage using a time-dependent source term located at an arbitrary point along the pipeline. Using Charny's linearization method, the nonlinear governing equations are reduced to a solvable diffusion-type partial differential equation. A closed-form solution for pressure distribution P(x,t) is derived by applying a strategic substitution that separates terrain and friction effects, enabling Laplace-based inversion to yield physically interpretable results. The derived pressure expression reveals how leakage intensity (Gut), terrain slope (a1), and leak location (l) interact to produce non-intuitive pressure patterns-such as pressure inversion, where downstream pressure loss becomes smaller than that at the inlet. The model is validated through extensive simulations, and key pressure dynamics are analyzed across multiple operating scenarios. Results indicate that increasing slope angle magnifies the gravitational effect, leading to early pressure crossover phenomena and stronger downstream stability, even for high leakage rates. This finding has substantial implications for sensor deployment, leak localization, and the development of SCADA-integrated decision-making tools. This work constitutes the first known analytical solution to the unsteady gas dynamics problem in sloped pipeline systems with leakage and provides a critical foundation for next-generation monitoring and control strategies in energy transport infrastructure.
Key Words
analytical model; Charny linearization; pressure inversion; relief-influenced pipelines; SCADA/IoT integration; unsteady gas flow
Address
Ilgar G. Aliyev: Head of Operation and Reconstruction of Buildings and Facilities Department, Azerbaijan University
Architecture and Construction, Baku, Azerbaijan
Abstract
In this article, the mechanical and thermal buckling analysis of simply-supported functionally graded plates resting on an elastic foundation is conducted using an innovative higher shear deformation theory (HSDT) in conjunction with the Airy stress function method. The key novelty of this work lies in the exact resolution of the
equilibrium equations through the Airy stress function, eliminating the need for shear correction factors, and in the introduction of a new transverse shear function that ensures a parabolic variation of transverse shear stresses across the thickness while naturally satisfying the stress-free boundary conditions at the surfaces. Three types of thermal loads are considered: uniform, linear, and nonlinear distributions through the thickness. The material properties of the plate vary according to a power-law distribution based on the volume fraction of its constituents. Numerical results are presented to assess the effects of the power-law index the foundation stiffness and geometric ratios on the critical buckling load and the critical buckling temperature, highlighting the accuracy and efficiency of the proposed methodology.
Key Words
airy stress function; analytical modeling; buckling; computational modeling; functionally graded plate; refined plate theory
Address
Ahmed Bakoura: Material and Hydrology Laboratory, Civil Engineering Department, Faculty of Technology, University of Sidi Bel Abbes, Algeria; Département de Génie Civil, Faculté d'Architecture et de Génie Civil, Université des Sciences et de la Technologie d'Oran, BP 1505 El M'naouer, USTO, Oran, Algeria
Aicha Remil: Laboratoire d'Etude des Structures et de Mécanique des Matériaux, Département de Génie Civil, Faculté des Sciences et de la Technologie, Université Mustapha Stambouli, B.P. 305, R.P. 29000 Mascara, Algeria
Aicha Bessaim: Département de Génie Civil, Faculté d'Architecture et de Génie Civil, Université des Sciences et de la Technologie d'Oran, BP 1505 El M'naouer, USTO, Oran, Algeria; Laboratoire d'Etude des Structures et de Mécanique des Matériaux, Département de Génie Civil, Faculté des
Sciences et de la Technologie, Université Mustapha Stambouli, B.P. 305, R.P. 29000 Mascara, Algeria
Mohammed Sid Ahmed Houari: Laboratoire Signaux et Images (LSI), University of Science and Technology of Oran, Mohammed Boudiaf, Bir El Djir 31000, Algeria
Sahla Meriem: Laboratoire d'Etude des Structures et de Mécanique des Matériaux, Département de Génie Civil, Faculté des Sciences et de la Technologie, Université Mustapha Stambouli, B.P. 305, R.P. 29000 Mascara, Algeria
Ibka Mohamed Soufiane: Laboratoire Signaux et Images (LSI), University of Science and Technology of Oran, Mohammed Boudiaf, Bir El Djir 31000, Algeria
Abdelouahed Tounsi: Material and Hydrology Laboratory, Civil Engineering Department, Faculty of Technology, University of Sidi Bel Abbes, Algeria
Abstract
This paper explores the plane stress problem for a cantilever beam under shear loading that causes bending, specifically focusing on functionally graded materials (FGMs). These materials exhibit a gradual change in mechanical properties, promoting enhanced performance over traditional homogeneous materials. The authors introduce a new formulation for the elastic modulus of FGM, incorporating two graduation parameters: one for material composition variation along the beam and another for the longitudinal elastic modulus ratio, which characterizes stiffness variation. The study formulates the static equilibrium equations that relate external forces, stresses, and internal deformations. Analytical expressions derived from these equations predict stress and displacement distributions within the beam, particularly under a tangential load at the free end, which induces shear and bending deformations. The analysis reveals how the gradation parameters influence the beam's structural response, demonstrating the effect of elastic modulus variation on stress distribution and deflection behavior. The study culminates in an analytical framework for assessing FGM cantilever beams under shear loading, enhancing understanding of material gradation impacts on structural behavior and aiding in the design and optimization of advanced engineering structures utilizing FGMs.
Address
Hassaine Daouadji Tahar, Abdelaziz Hadj Henni: Civil Engineering Department, University of Tiaret, Algeria; Laboratory of Geomatics and Sustainable Development, University of Tiaret, Algeria
Abstract
A complete laboratory-based seismic testing is the ideal standard qualification method to determine the amplified response accelerations of the electric power utility equipment. As an alternative to the complete experimental procedures, an analytical methodology with partial experimentation is proposed to assess the structural performance prior to the seismic qualification experiments in the present analogy. The proposed approach is based on the free vibration tests conducted at manufacturing unit, an analytical study of acceleration transmissibility and phase shifts of individual modes of the structure. This method also focuses on understanding the state of the system at post resonance. Different types of substation equipment are analyzed for the dynamic response and a case study of 36 kV circuit breaker is presented in this work. Harmonic vibrations of 0.2 g ground accelerations applied to index the modal parameter i.e., damping ratios at corresponding natural frequencies. Force transmission and phase lag at individual modes of the system opposing random ground accelerations are computed. The plots of frequency response curves of majority of the tested substation equipment showed the response acceleration is low at higher frequency ratios. Also, the response acceleration is not influenced by damping of the structure in the region of high frequency ratios. The presented methodology reduces the experimental complexities in the laboratories and the final qualification can be done in single attempt.
Key Words
electric substation equipment; frequency response; harmonic vibrations; phase shift; seismic qualification
Address
N. Srujana: Structural Health Monitoring Division, KDM Engineers India Private Limited, Hyderabad, India
Shakti P Jena: Department of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Chennai, India
open access