## 311 Principal Stresses And Maximum Shear Stress

When stress components relative to a defined set of axes are given at any point in a condition of plane stress or plane strain (see Art. 3.10), this state of stress may be expressed with respect to a different set of axes that lie in the same plane. For example, the state of stress at point O in Fig. 3.15a may be expressed in terms of either the x and y axes with stress components, fx, fy, and v^ or the x' and y' axes with stress components fx,, fy,, and vxy (Fig. 3.15b). If stress components fx, fy, and vy are given and the two orthogonal coordinate systems differ by an angle a with respect to the original x axis, the stress components fx', f y,, and vxy can be determined by statics. The transformation equations for stress are fx' = 1/2(fx + fy) + X/2(fx - fy) COS 2a + V„ Sin 2a

fy' = 1/2(fx + fy) - 1/2(fx - fy) COS 2a - V^ Sin 2a

From these equations, an angle ap can be chosen to make the shear stress vxy equal zero. From Eq. (3.43c), with vxy = 0,

This equation indicates that two perpendicular directions, «p and «p + (W2), may be found for which the shear stress is zero. These are called principal directions. On the plane for which the shear stress is zero, one of the normal stresses is the maximum stress fV and the other is the minimum stress /2 for all possible states of stress at that point. Hence the normal stresses on the planes in these directions are called the principal stresses. The magnitude of the principal stresses may be determined from fx + fy (/ x fy f = ^ ± VI 2 j + < ^

where the algebraically larger principal stress is given by fj and the minimum principal stress is given by f2.

Suppose that the x and y directions are taken as the principal directions, that is, vxy = 0. Then Eqs. (3.43) may be simplified to fx, = j + f2) + 11 (f - f2) cos 2« (3.46a)

By Eq. (3.46c), the maximum shear stress occurs when sin 2« = tt/2, i.e., when « = 45°. Hence the maximum shear stress occurs on each of two planes that bisect the angles between the planes on which the principal stresses act. The magnitude of the maximum shear stress equals one-half the algebraic difference of the principal stresses:

If on any two perpendicular planes through a point only shear stresses act, the state of stress at this point is called pure shear. In this case, the principal directions bisect the angles between the planes on which these shear stresses occur. The principal stresses are equal in magnitude to the unit shear stress in each plane on which only shears act.

3.12 MOHR'S CIRCLE

Equations (3.46) for stresses at a point O can be represented conveniently by Mohr's circle (Fig. 3.16). Normal stress / is taken as the abscissa, and shear stress v is taken as the ordinate. The center of the circle is located on the / axis at (/1 + /2)/2, where /1 and /2 are the maximum and minimum principal stresses at the point, respectively. The circle has a radius of (/1 _ /2)/2. For each plane passing through the point O there are two diametrically opposite points on Mohr's circle that correspond to the normal and shear stresses on the plane. Thus Mohr's circle can be used conveniently to find the normal and shear stresses on a plane when the magnitude and direction of the principal stresses at a point are known.

Use of Mohr's circle requires the principal stresses /1 and /2 to be marked off on the abscissa (points A and B in Fig. 3.16, respectively). Tensile stresses are plotted to the right of the v axis and compressive stresses to the left. (In Fig. 3.16, the principal stresses are indicated as tensile stresses.) A circle is then constructed that has radius (/1 + /2)/2 and passes through A and B. The normal and shear stresses /x, /y, and vxy on a plane at an angle a with the principal directions are the coordinates of points C and D on the intersection of

the circle and the diameter making an angle 2a with the abscissa. A counterclockwise angle change a in the stress plane represents a counterclockwise angle change of 2a on Mohr's circle. The stresses fx, vxy, and fy, vyx on two perpendicular planes are represented on Mohr's circle by points (fx, - vxy) and (fy, vyx), respectively. Note that a shear stress is defined as positive when it tends to produce counter-clockwise rotation of the element.

Mohr's circle also can be used to obtain the principal stresses when the normal stresses on two perpendicular planes and the shearing stresses are known. Figure 3.17 shows construction of Mohr's circle from these conditions. Points C (fx, vxy) and D (fy, - vxy) are plotted and a circle is constructed with CD as a diameter. Based on this geometry, the abscissas of points A and B that correspond to the principal stresses can be determined.

(I. S. Sokolnikoff, Mathematical Theory of Elasticity; S. P. Timoshenko and J. N. Goodier, Theory of Elasticity; and Chi-Teh Wang, Applied Elasticity; and F. P. Beer and E. R. Johnston, Mechanics of Materials, McGraw-Hill, Inc., New York; A. C. Ugural and S. K. Fenster, Advanced Strength and Applied Elasticity, Elsevier Science Publishing, New York.)

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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