Budynas-Nisbett:Shigley's I.Basics 3.Load and Stress Analysis T©The McGraw-Hill Mechanical Engineering Companies,2008 Design,Eighth Edition Load and Stress Analysis 79 Figure 3-10 Mohr's circle diagram. (g-0) 5-" 2 y H g E 2中 D 0 (g,TW x+dy Using the stress state of Fig.3-8b,we plot Mohr's circle,Fig.3-10,by first look- ing at the right surface of the element containing o to establish the sign of or and the cw or ccw direction of the shear stress.The right face is called the x face where =0.If ox is positive and the shear stress txy is ccw as shown in Fig.3-8b,we can establish point A with coordinates (ox.ew)in Fig.3-10.Next,we look at the top y face,where =90,which contains oy,and repeat the process to obtain point B with coordinates (oy.)as shown in Fig.3-10.The two states of stress for the element are△p=90°from each other on the element so they will be2△中=l80°from each other on Mohr's circle.Points A and B are the same vertical distance from the o axis. Thus,AB must be on the diameter of the circle,and the center of the circle C is where AB intersects the o axis.With points A and B on the circle,and center C,the complete circle can then be drawn.Note that the extended ends of line AB are labeled x and y as references to the normals to the surfaces for which points A and B represent the stresses. The entire Mohr's circle represents the state of stress at a single point in a struc- ture.Each point on the circle represents the stress state for a specific surface intersect- ing the point in the structure.Each pair of points on the circle 180 apart represent the state of stress on an element whose surfaces are 90 apart.Once the circle is drawn,the states of stress can be visualized for various surfaces intersecting the point being ana- lyzed.For example,the principal stresses o and o2 are points D and E,respectively. and their values obviously agree with Eq.(3-13).We also see that the shear stresses are zero on the surfaces containing o and o2.The two extreme-value shear stresses,one clockwise and one counterclockwise,occur at F and G with magnitudes equal to the radius of the circle.The surfaces at F and G each also contain normal stresses of (ox+o)/2 as noted earlier in Eq.(3-12).Finally,the state of stress on an arbitrary surface located at an angle o counterclockwise from the x face is point H.Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition I. Basics 3. Load and Stress Analysis 84 © The McGraw−Hill Companies, 2008 Using the stress state of Fig. 3–8b, we plot Mohr’s circle, Fig. 3–10, by first look￾ing at the right surface of the element containing σx to establish the sign of σx and the cw or ccw direction of the shear stress. The right face is called the x face where φ = 0◦. If σx is positive and the shear stress τxy is ccw as shown in Fig. 3–8b, we can establish point A with coordinates (σx , τ ccw xy ) in Fig. 3–10. Next, we look at the top y face, where φ = 90◦, which contains σy , and repeat the process to obtain point B with coordinates (σy , τ cw xy ) as shown in Fig. 3–10. The two states of stress for the element are φ = 90◦ from each other on the element so they will be 2φ = 180◦ from each other on Mohr’s circle. Points A and B are the same vertical distance from the σ axis. Thus, AB must be on the diameter of the circle, and the center of the circle C is where AB intersects the σ axis. With points A and B on the circle, and center C, the complete circle can then be drawn. Note that the extended ends of line AB are labeled x and y as references to the normals to the surfaces for which points A and B represent the stresses. The entire Mohr’s circle represents the state of stress at a single point in a struc￾ture. Each point on the circle represents the stress state for a specific surface intersect￾ing the point in the structure. Each pair of points on the circle 180° apart represent the state of stress on an element whose surfaces are 90° apart. Once the circle is drawn, the states of stress can be visualized for various surfaces intersecting the point being ana￾lyzed. For example, the principal stresses σ1 and σ2 are points D and E, respectively, and their values obviously agree with Eq. (3–13). We also see that the shear stresses are zero on the surfaces containing σ1 and σ2. The two extreme-value shear stresses, one clockwise and one counterclockwise, occur at F and G with magnitudes equal to the radius of the circle. The surfaces at F and G each also contain normal stresses of (σx + σy )/2 as noted earlier in Eq. (3–12). Finally, the state of stress on an arbitrary surface located at an angle φ counterclockwise from the x face is point H. Load and Stress Analysis 79 x y (x – y) O x + y 2 x – y 2 F (y , xy cw) (x , ccw) y B C G D H E xy 2 y  x 1 2 A 2p xy x cw ccw x – y 2 + 2 xy  2 xy Figure 3–10 Mohr’s circle diagram.