Illustrating Rotating Principal Stresses in a Materials Science Course

This work constitutes a laboratory component of a junior level materials science course and illustrates the importance of rotating principal stresses in the design of components such as the automotive crankshaft. The activity is centered on Mohr’s circle for biaxial stress situations involving time varying normal and shear stresses. A number of dynamic situations have been considered, namely. (a) sinusoidally varying normal and shear stresses that are in phase, (b) sinusoidally varying normal and shear stresses that are 90° out of phase, (c) constant normal stress and sinusoidally varying shear stress, and (d) sinusoidally varying normal stress and constant shear stress. Employing a graphical approach, the diameter of Mohr’s circle (the absolute difference between the two principal stresses) as well as the principal stress directions is monitored. The students see that for certain stress situations the principal stress directions remain unchanged while for others the principal stress directions change with time (rotating principal stresses). In general, the size of Mohr’s circle changes with time. The plotting option of the Matlab code has been employed to construct three dimensional plots for the indicated stress situations with the normal and stresses respectively in the x and y directions and time in the z-direction. The plots show how the principal directions change with time, along with the size of Mohr’s circle. The students are made aware of the fact that rotating principal stresses play a very important role in designing components that are subjected to biaxial or multiaxial fatigue, such as the crankshaft. Also the diameter of Mohr’s circle can be directly related to Tresca or von Mises theories of failure. INTRODUCTION This study constitutes a laboratory component of the Mechanics of Materials courses taught to engineering students at the sophomore or junior levels. It is important that the students learn how the external loads combine to produce stresses in a critical location of a structure or a component. This is fundamental to the understanding of the response of a structural component to a combined system of loads that result in normal and shear stresses. Mohr’s circle is an invaluable tool for this purpose, especially in its ability to determine principal stresses and principal directions for combined load situations. Mohr’s circle can be used to study a number of situations involving multiaxial stress states. The principal stresses can be used to evaluate material failure using appropriate failure criteria, and the nature of loading plays an important role in this process. It is desirable that the students learn the concepts of proportionality and nonproportionality of various loadings, since these are important in the design of automotive components, such as connecting rods and crankshafts. A sample problem involving combined bending and torsion of a shaft under steady and harmonic loadings is employed for this purpose. Proportional loading is defined as any state of time varying stress where the orientation of the principal stress axes does not change with respect to the axis of the shaft. Non-proportional loading is defined as any state of time varying stress where the orientation of the principal axes changes with respect to the shaft axis. The students study the “proportionality” of loadings using Mohr’s circle for four specific cases for a shaft under combined bending and torsion, which are: 1. Time harmonic bending moment and time harmonic torsion that are in phase. 2. Time harmonic bending moment and time harmonic torsion that are 90 out of phase. 3. Time harmonic torsion and steady bending moment. 4. Steady torsion and time harmonic bending moment. Although the fatigue failure is typically not addressed in Mechanics of Materials course, the students will be made aware of the fact that in many situations involving complex loadings, the locations of the critically stressed areas are not known in advance. For such cases appropriate and efficient methods are needed for fatigue analysis. The complications arise due to complex geometries and complex non-proportional loads acting on such structures. MOHR’S CIRCLE Transformation of stress among coordinate systems is important in structural analysis. More than 140 years ago, Mohr came up with a graphical construction (Mohr’s circle) to assist with this process [Mohr, 1882]. In this paper the transformation of stresses is not specifically addressed, but the principal stresses and the associated principal directions are obtained for the four biaxial stress situations identified above. Mohr’s circle is one of the most difficult topics in Mechanics of Materials course. A number of issues appear in the area of student learning on Mohr’s circle, namely, (a) Identification of the relationship of the load on a member and the state of stress at a point. (b) Confusion between the stress axes and the spatial coordinate axes (c) Inability to perceive rotation of the principal axes. (d) Relevance of Mohr’s circle without reference to yield and fracture criteria. (e) The presentation of identical information in different coordinate systems. The procedure for construction of Mohr’s circle is outlined in various texts, such as [Mott, 2008]. For the four biaxial cases considered in this paper, we essentially have a bending stress, x due to bending moment and a shear stress, xy due to torsion. The principal stresses and principal directions are given by: