The following list gives a summary of the techniques pre sented in this section

The following list gives a summary of the techniques

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The following list gives a summary of the techniques pre- sented in this section; it also outlines the general procedure for applying the techniques to a given stress analysis problem. Angle for Principal Stress Element uni03D5 σ = 1 2 arctan [2 τ xy /( σ x - σ y )] (4–3) The angle uni03D5 σ is measured from the positive x -axis of the original stress element to the maximum principal stress, σ 1 . Then the minimum principal stress, σ 2 , is on the plane 90° from σ 1 . When the stress element is oriented as discussed so that the principal stresses are acting on it, the shear stress is zero. The resulting stress element is shown in Figure 4–4. Maximum Shear Stress On a different orientation of the stress element, the maxi- mum shear stress will occur. Its magnitude can be computed from Maximum shear stress τ max = C a σ x - σ y 2 b 2 + τ xy 2 (4–4) The angle of inclination of the element on which the maxi- mum shear stress occurs is computed as follows: Angle for Maximum Shear Stress Element uni03D5 τ = 1 2 arctan [ - ( σ x - σ y )/2 τ xy ] (4–5) The angle between the principal stress element and the max- imum shear stress element is always 45°. On the maximum shear stress element, there will be normal stresses of equal magnitude acting perpendicular to the planes on which the maximum shear stresses are acting. These normal stresses have the value Average Normal Stress σ avg = ( σ x + σ y )/2 (4–6) Note that this is the average of the two applied normal stresses. The resulting maximum shear stress element is shown in Figure 4–5. Note, as stated above, that the angle between the principal stress element and the maximum shear stress element is always 45°. FIGURE 4–4 Principal stress element y x H9268 avg H9268 avg H9268 avg H9268 avg H9270 max H9270 max H9270 max H9270 max H9278 H9270 FIGURE 4–5 Maximum shear stress element GENERAL PROCEDURE FOR COMPUTING PRINCIPAL STRESSES AND MAXIMUM SHEAR STRESSES 1. Decide for which point you want to compute the stresses. 2. Clearly specify the coordinate system for the object, the free-body diagram, and the magnitude and direction of forces. 3. Compute the stresses on the selected point due to the applied forces, and show the stresses acting on a stress element at the desired point with careful attention to directions. Figure 4–3 is a model for how to show these stresses. 4. Compute the principal stresses on the point and the di- rections in which they act. Use Equations (4–1), (4–2), and (4–3). 5. Draw the stress element on which the principal stresses act, and show its orientation relative to the original x -axis. It is recommended that the principal stress element be drawn beside the original stress element to illustrate the relationship between them. 6. Compute the maximum shear stress on the element and the orientation of the plane on which it acts. Also, com- pute the normal stress that acts on the maximum shear stress element. Use Equations (4–4), (4–5), and (4–6).
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