Session: 09-05-01: Applied Mechanics, Dynamic Systems, Experimental and Computational Methods, Modeling and Virtual Simulations of Dynamic Structures, Advanced Materials and Testing

Paper Number: 143842

143842 - Thermally Induced Stress and Deflection in a Cantilever Beam 

Temperature gradients, in the absence of mechanical loads, could cause significant stresses in a solid body which, in turn, could lead to failure of mechanical and structural parts. In a typical undergraduate mechanical engineering curriculum, students are introduced to the concept of mechanical stress in solid bodies during sophomore or early junior year of their studies. They have no exposure to heat transfer and temperature field in a solid body until later in the curriculum. As such, students’ study of thermally induced stresses is limited just to statically indeterminant axial loads due to uniform temperature rise, and in the absence of temperature gradient in the body. As such, they are not aware of thermal stresses induced solely by temperature gradients in a solid body and the harmful effects they can cause even in the absence of mechanical loads. The work we present in this paper would make for a great term-long project in either heat transfer class or machine design course to introduce our students to this important aspect of design of structural parts and mechanical components. It is believed that students, in either of the above-mentioned senior level courses, possess the technical maturity both in stress and heat conduction analyses to handle such a project.

In this project, students consider analysis and design of a simply supported beam subject to a transient temperature gradient field in the transverse direction of the beam -- with or without transverse mechanical loads. The transient transversal temperature distribution in the beam, due to its thermal interaction with its surroundings, is obtained by application of the heat diffusion equation to the beam and consideration of its initial and boundary conditions. The temperature distribution in the beam will cause beam’s fibers to deform longitudinally, hence bending the beam and causing a constant curvature to the longitudinal axis of the beam- even in the absence of any mechanical bending load. The derivation of the governing equations for thermal and mechanical stresses and lateral deflection equations of the beam is then obtained, using Euler-Bernoulli assumption along with linear and small deformation theory of elasticity. Although the temperature distribution could be considered as a two-dimensional field, for simplicity, only one-dimensional temperature field is considered in the project. Thus, the curvature of the beam axis is also considered in a single transverse plane of the beam. It is also assumed that the longitudinal centroidal axis of the beam is one of the principal directions of the cross section thereby simplifying further the governing equations of stress and deflections. In conclusion, in design and analysis of mechanical and structural components, stresses due to the existence of temperature gradient in the part are significant and cannot be overlooked.

Presenting Author: Ali Mohammadzadeh Grand valley State University

Presenting Author Biography: Ali R. Mohammadzadeh is associate professor of mechanical engineering at Grand Valley State University's Padnos school of engineering and computing. He earned his PhD in mechanical engineering from the University of Michigan in Ann Arbor. His research interest is in the field of fluid-solid interactions.

Authors:

Ali Mohammadzadeh Grand valley State University
Salim Haidar Grand Valley State University
Mahdi Norouzi Grand Valley State University