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type of failure because mechanical properties of gear materials are well known, and the design equations are sufficiently accurate. The analysis of bending stresses is as follows: 
       In transmitting power, the driving, force acts along the line-of-action, and the tooth senses a moving force acting from the tip to the base, as shown in Figure 1.49. The load can be resolved into a tangential force, W1, causing bending, and a normal force, WN, causing compression. These are shown in Figure 1.50 along the corresponding net stresses.
       Based upon the above static analysis, Wilfred Lewis, in 1892, presented his expression for tooth beam strength which is now reknowned as the classic Lewis equatien:
                   W_t   =  SFY                                
                               P_d

As a static beam resisting a fixed load in position and magnitude, this equation is usually adequate. However, it does not take into account the dynamics of meshing teeth. In that regard, later investigators have modified and improved the original Lewis equation.

Beam Strength (Figure 1.51)
Improved results can be obtained by use of Barth’s modified Lewis formula, which takes velocity into consideration but not wear. Impact and fatigue stresses become more pronounced as pitch-line velocity increases. The formula includes a velocity factor and is satisfactory for commercial gears at pitch-line velocities up to 1,500 fpm:

For non-metallic gears, the velocity factor is changed from (   600  ) to  (    150     +  0.25 )
                                                                                     600+V           200+V           
       The Lewis factor is dimensionless and independent of tooth size, and a function only of shape. Lewis factors for standard teeth are given in Table 1.11.
       A safe stress level depends upon the material and the number of stress cycles to which the teeth are subjected. This can be evaluated from an S-N curve, modified Goodman diagram, Soderberg line, or equivalent data. Reference 6 contains helpful information on fatigue stress analysis.

Table 1.12 gives safe stresses for a number of engineering materials. An estimate for the maximum allowable bending stress, S in equation 52, can then be obtained by multiplying the stress given in Table 1.12 by two factors: a service factor given in Table 1.13 and a lubrication factor given in Table 1.14.
       Use of a proper limiting stress value, S_e in equation 52, results in a calculated tooth load, W0, based on beam strength. For acceptable designs, W_b>= W_t
       The tangentially transmitted load is calculated from the transmitted horsepower as follows:

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