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A New Approach for Teaching In-Plane Principal Stresses, Principal Directions and Maximum Shear Stress for Plane Stress

Abstract

The topic of in-plane principal stresses, principal directions and maximum shear stress for a state of plane stress is typically taught in introductory mechanics of materials and solid mechanics courses using the following approach. First, the equations for the normal and shear stresses on an inclined plane are obtained applying the principle of static equilibrium to an infinitesimal wedge element. Then, the derivatives of the equations for the normal and shear stresses on the inclined plane with respect to the angle used to define that plane are found and set equal to zero to find the angles corresponding to the maximum and minimum algebraic values for the normal and shear stresses. Based on the expressions obtained for those angles, equations to find the in-plane principal stresses and the maximum in-plane shear stress are presented and the relative orientation of the planes corresponding to those stresses is discussed.

In this paper, a different approach is used to obtain the equations for the in-plane principal stresses, the maximum in-plane shear stress and the angles corresponding to the planes in which those stresses occur. Using the idea of a phase angle that is commonly employed in the analysis of signals or physical quantities that involve a linear combination of sine and cosine functions of the same angle, the equations for the normal and shear stress on an inclined plane are expressed in terms of trigonometric functions of twice the difference between the angle that defines the inclined plane and a phase angle. Plots of the resulting expressions are presented and used to quickly obtain formulas for the values and orientations of the planes for the stresses in question. The approach, although relatively simple, is mathematically rigorous and allows students to visualize in a convenient way the relationship between the normal and shear stress on an inclined plane. Furthermore, it can be used to show in a straightforward fashion that the two equations can be combined to represent the equation of a circle and thus serve as convenient starting point to introduce the concept of Mohr’s circle.

Introduction

Exploring different ways of presenting topics covered in introductory mechanics of materials and solid mechanics courses is of particular importance. The concepts taught in those courses serve as the foundation over which students build additional knowledge and constitute an integral part of their professional expertise. Efforts to improve student learning typically focus on using new pedagogical strategies and methods to present the concepts as well as their practical application. However, in most cases the activities selected for the learning process are based on the same mathematical derivations that have been employed in the past. Sometimes the approach used to obtain certain equations is difficult for the students to understand and obscures the underlying concept. As a “solution” to this situation, the emphasis is shifted to the use the final formulas taking them as definitions. It is a good exercise for instructors to take a given derivation and explore if there are new alternatives that can be used to obtain the same formulas but that are easier for the students to comprehend.

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Muci-Küchler, K., & Kjerengtroen, L. (2007, June), A New Approach For Teaching In Plane Principal Stresses, Principal Directions And Maximum Shear Stress For Plane Stress Paper presented at 2007 Annual Conference & Exposition, Honolulu, Hawaii. 10.18260/1-2--2244