Budynas-Nisbett:Shigley's I.Basics 3.Load and Stress Analysis T©The McGraw-Hill 83 Mechanical Engineering Companies,2008 Design,Eighth Edition 78 I Mechanical Engineering Desigr In a similar manner the two extreme-value shear stresses are found to be -Ov T,T2= 2 十品 (3-14) Your particular attention is called to the fact that an extreme value of the shear stress may not be the same as the actual maximum value.See Sec.3-7. It is important to note that the equations given to this point are quite sufficient for performing any plane stress transformation.However,extreme care must be exercised when applying them.For example,say you are attempting to determine the principal state of stress for a problem where ox=14 MPa,oy=-10 MPa,and txy =-16 MPa. Equation (3-10)yieldsp=-26.57 and 63.43 to locate the principal stress surfaces, whereas,Eq.(3-13)gives o1 =22 MPa and o2 =-18 MPa for the principal stresses. If all we wanted was the principal stresses,we would be finished.However,what if we wanted to draw the element containing the principal stresses properly oriented rel- ative to the x,y axes?Well,we have two values ofp and two values for the princi- pal stresses.How do we know which value of op corresponds to which value of the principal stress?To clear this up we would need to substitute one of the values ofp into Eq.(3-8)to determine the normal stress corresponding to that angle. A graphical method for expressing the relations developed in this section,called Mohr's circle diagram,is a very effective means of visualizing the stress state at a point and keeping track of the directions of the various components associated with plane stress.Equations (3-8)and(3-9)can be shown to be a set of parametric equations for o and t,where the parameter is 20.The relationship between o and t is that of a cir- cle plotted in the o,plane,where the center of the circle is located at C=(o.r)= [(ox +o)/2.0]and has a radius of R =/[(ox-o)/212+.A problem arises in the sign of the shear stress.The transformation equations are based on a positive o being counterclockwise,as shown in Fig.3-9.If a positive t were plotted above the o axis,points would rotate clockwise on the circle 2 in the opposite direction of rotation on the element.It would be convenient if the rotations were in the same direction.One could solve the problem easily by plotting positive t below the axis. However,the classical approach to Mohr's circle uses a different convention for the shear stress. Mohr's Circle Shear Convention This convention is followed in drawing Mohr's circle: Shear stresses tending to rotate the element clockwise (cw)are plotted above the o axis. Shear stresses tending to rotate the element counterclockwise(ccw)are plotted below the o axis. For example,consider the right face of the element in Fig.3-8b.By Mohr's circle con- vention the shear stress shown is plotted below the o axis because it tends to rotate the element counterclockwise.The shear stress on the top face of the element is plotted above the o axis because it tends to rotate the element clockwise. In Fig.3-10 we create a coordinate system with normal stresses plotted along the abscissa and shear stresses plotted as the ordinates.On the abscissa,tensile (positive) normal stresses are plotted to the right of the origin O and compressive(negative)nor- mal stresses to the left.On the ordinate,clockwise (cw)shear stresses are plotted up; counterclockwise (ccw)shear stresses are plotted down.Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition I. Basics 3. Load and Stress Analysis © The McGraw−Hill 83 Companies, 2008 78 Mechanical Engineering Design In a similar manner the two extreme-value shear stresses are found to be τ1, τ2 = ±σx − σy 2 2 + τ 2 xy (3–14) Your particular attention is called to the fact that an extreme value of the shear stress may not be the same as the actual maximum value. See Sec. 3–7. It is important to note that the equations given to this point are quite sufficient for performing any plane stress transformation. However, extreme care must be exercised when applying them. For example, say you are attempting to determine the principal state of stress for a problem where σx = 14 MPa, σy = −10 MPa, and τxy = −16 MPa. Equation (3–10) yields φp = −26.57◦ and 63.43° to locate the principal stress surfaces, whereas, Eq. (3–13) gives σ1 = 22 MPa and σ2 = −18 MPa for the principal stresses. If all we wanted was the principal stresses, we would be finished. However, what if we wanted to draw the element containing the principal stresses properly oriented rel￾ative to the x, y axes? Well, we have two values of φp and two values for the princi￾pal stresses. How do we know which value of φp corresponds to which value of the principal stress? To clear this up we would need to substitute one of the values of φp into Eq. (3–8) to determine the normal stress corresponding to that angle. A graphical method for expressing the relations developed in this section, called Mohr’s circle diagram, is a very effective means of visualizing the stress state at a point and keeping track of the directions of the various components associated with plane stress. Equations (3–8) and (3–9) can be shown to be a set of parametric equations for σ and τ , where the parameter is 2φ. The relationship between σ and τ is that of a cir￾cle plotted in the σ, τ plane, where the center of the circle is located at C = (σ, τ ) = [(σx + σy )/2, 0] and has a radius of R = [(σx − σy )/2]2 + τ 2 xy . A problem arises in the sign of the shear stress. The transformation equations are based on a positive φ being counterclockwise, as shown in Fig. 3–9. If a positive τ were plotted above the σ axis, points would rotate clockwise on the circle 2φ in the opposite direction of rotation on the element. It would be convenient if the rotations were in the same direction. One could solve the problem easily by plotting positive τ below the axis. However, the classical approach to Mohr’s circle uses a different convention for the shear stress. Mohr’s Circle Shear Convention This convention is followed in drawing Mohr’s circle: • Shear stresses tending to rotate the element clockwise (cw) are plotted above the σ axis. • Shear stresses tending to rotate the element counterclockwise (ccw) are plotted below the σ axis. For example, consider the right face of the element in Fig. 3–8b. By Mohr’s circle con￾vention the shear stress shown is plotted below the σ axis because it tends to rotate the element counterclockwise. The shear stress on the top face of the element is plotted above the σ axis because it tends to rotate the element clockwise. In Fig. 3–10 we create a coordinate system with normal stresses plotted along the abscissa and shear stresses plotted as the ordinates. On the abscissa, tensile (positive) normal stresses are plotted to the right of the origin O and compressive (negative) nor￾mal stresses to the left. On the ordinate, clockwise (cw) shear stresses are plotted up; counterclockwise (ccw) shear stresses are plotted down