Machine Elements in Mechanical Design FIFTH EDITION CHAPTER

Machine Elements in Mechanical Design FIFTH EDITION CHAPTER 4 Combined Stresses and Mohr's Circle Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

General Case of Combined Stress • Normal Stresses § Due to direct tensile force or due to bending • Shear Stress § Due to direct shear, torsional shear, vertical shear stress Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

General Case of Combined Stress • Positive Shear Stress § Tends to rotate stress element clockwise • Necessary to determine magnitudes and signs of each of these stresses to show them properly on stress element Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

General Case of Combined Stress • Maximum Normal Stresses: Principal Stresses § Maximum principal stress: combination of applied normal and shear stresses that produces maximum normal stress § Minimum principal stress: combination of applied stresses that produces minimum normal stress § Principal planes: angle of inclination of planes on which principal stresses act Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

General Case of Combined Stress • Procedure for Analyzing Combined Stresses § Decide for which point you want to compute the stresses. § Clearly specify coordinate system for the object, free-body diagram, magnitude/direction of forces. § Compute the stresses on selected point due to the applied forces. Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

General Case of Combined Stress • Procedure for Analyzing Combined Stresses § Show stresses acting on a stress element at desired point with careful attention to directions. § Compute principal stresses on the point and directions in which they act. § Draw stress element on which principal stresses act; show its orientation relative to original x-axis. Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

General Case of Combined Stress • Procedure for Analyzing Combined Stresses § Compute maximum shear stress on the element and orientation of the plane on which it acts. § Draw stress element on which maximum shear stress acts; show its orientation to original x-axis. Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

General Case of Combined Stress • Procedure for Analyzing Combined Stresses § Recommended maximum shear stress element be drawn beside maximum principal stress. Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

Figure 4 -6 Relationships among original stress element, principal stress element, and maximum shear stress element for a given loading Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

Analysis of Complex Loading Conditions and von Mises Stress • Methods of Analysis § Experimental stress analysis § Finite-element analysis § von Mises stress • von Mises stress for biaxial stresses • Alternate form for von Mises stress • Three-dimensional stresses and von Mises stress Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

Mohr's Circle • Mohr's Circle § Combination of graphical aids/simple calculations § Finding maximum and minimum principal stresses and directions in which they act § Finding maximum shear stresses and orientation of planes on which they act Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

Mohr's Circle • Mohr's Circle § Finding the value of normal stresses that act on planes where maximum shear stresses act § Finding the values of normal and shear stresses that act on an element with any orientation Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

Mohr's Circle • Mohr's Circle § Plot of the combinations of normal and shearing stresses that exist on a stress element for all possible angles of orientation of the element § Valuable in experimental stress analysis work Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

Case When Both Principal Stresses Have the Same Sign • True maximum shear stress on the element will not be found if the two principal stresses are of the same sign. § Examples: various forms of pressure vessels. § Tensile stresses in two directions: tangent to its circumference; axially, parallel to axis of cylinder. Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

Case When Both Principal Stresses Have the Same Sign • True maximum shear stress on the element will not be found if the two principal stresses are of the same sign. § Tangential stress: hoop stress. § Axially directed stress: longitudinal stress. Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

Case When Both Principal Stresses Have the Same Sign • Three-Dimensional Stress Element § There will be one orientation of the element in which there are no shear stresses acting. § Normal stresses on three perpendicular faces are three principal stresses. Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved

Mohr's Circle for Special Stress Conditions • • Pure uniaxial tension Pure uniaxial compression Pure torsional shear Uniaxial tension combined with torsional shear Machine Elements in Mechanical Design , Fifth Edition Robert L. Mott Copyright © 2014 by Pearson Education, Inc. All Rights Reserved