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.
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.
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.
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.
This research presents Structural Health Monitoring (SHM) techniques that employ static, modal, harmonic, and transient analyses of functionally graded material (FGM) cracked plates, modeled using First-Order Shear Deformation Theory (FSDT). Crack effects are represented using an equivalent stiffness-reduction method, enabling efficient damage modeling without introducing geometric discontinuities. The study analytically investigates static deflection under concentrated loading, free vibration, harmonic response at resonance, and transient response to impulsive excitation. The primary objective is to predict plate behavior and assess damage history using SHM methodologies, validated by monitoring changes in natural frequencies and dynamic responses of damaged thin plates. Finite element models are developed for cracked steel plates with varying crack lengths and orientations. Results indicate that stress increases with crack length but decreases as the crack orientation aligns more closely with the plate axis (y-axis). Both crack length and orientation significantly influence static compliance, natural frequencies, resonance amplitudes, and transient decay, highlighting the sensitivity of dynamic response parameters to damage severity. The combined use of static and dynamic indicators provides a comprehensive framework for SHM of functionally graded material plates, supporting effective damage detection and integrity assessment. Analytical results exhibit strong agreement with ANSYS simulations, with discrepancies remaining below 1% at a/c=0.01 for all crack angles considered. These findings establish a quantitative relationship between crack parameters and frequency reduction, confirming the model's applicability for vibration-based structural health monitoring of porous FGM plates.
