Introduction To Mechanical Engineering P2 2E – Michael Clifford

However, there is also a resultant moment about the y-­axis, which is as follows: σ = − = − = − ∫ ∫ . . . . y xy xdA c y xdA c I where xy xydA = ∫ is the Product Moment of Area [note the −ve sign arising because a positive stress results in a –ve moment about the y-­axis] Thus, in general, a moment has to be applied about the y-­axis as well as the x-­axis to produce bending about the x-­axis only.

A +ve moment is required about the y-­axis to counterbalance the –ve moment set up by the stresses arising from Mx. This is not the case if Ixy is zero, that is, for sections which are symmetric about the y-­axis. To ensure bending about the x-­axis only, a resultant moment y = + must be applied at angle, θ, given by tan tan y xy θ − − −     = =         The resultant moment is only applied to the x-­axis when Ixy = 0.

Figure 3.97 illustrates the effect for the z-­section. If a bending moment is applied about the x-­axis only, then the stresses in the flanges will create a resulting moment about the y-­axis. Consequently, bending will take place about both the x-­ and y-­axes. This is a consequence of Ixy not being zero for this asymmetric section. To avoid this moment coupling effect, it is usually convenient to solve bending problems by considering bending about the principal axes of a section for which the product moment of area is zero.

Mx y Tensile Stresses Compressive Stresses FIGURE 3.97 Z-section. 274 Introduction to Mechanical Engineering 3.9.2.1 Solving Asymmetrical Bending Problems Consider the arbitrary asymmetric section shown in Figure 3.98(a). O is the centroid and O-­p and O-­q are the principal axes of the section. The principal axes are inclined at angle θ to the x-­y axes.

The components of an applied moment M, that is Mx and My, act about the O-­x and O-­y axes, respectively.

Introduction to Mechanical Engineering: Part 2 is the essential text for all second-­year undergrad- uate students as well as those studying foundation degrees and Higher National Diplomas. Written by an experienced team of lecturers at the internationally renowned University of Nottingham, the text provides thorough coverage of the following core engineering topics, fully updated for the Second Edition: • Fluid dynamics • Thermodynamics • Solid mechanics • Electromechanical drive systems • Feedback and control theory • Structural vibration As well as mechanical engineers, the text will be highly relevant to automotive, aeronautical/aero- space and general engineering students.

All units include questions, with Units 4 and 5 including enhanced, detailed solutions online as a bonus feature. Introduction to Mechanical Engineering Introduction to Mechanical Engineering Part 2 Second Edition Edited by Michael Clifford Designed cover image: www.shutterstock.com Second edition published 2025 by CRC Press 2385 NW Executive Center Drive, Suite 320, Boca Raton FL 33431 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2025 selection and editorial matter, Michael Clifford; individual chapters, the contributors First edition published by CRC Press 2010 Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use.

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