Machinery's Handbook, 31st Edition
SIMPLE STRESSES
217
Table 2. Table of Simple Stresses Table 2. (Continued) Table of Simple Stresses
Type of Loading
Stress Distribution
Stress Equations
Case
Illustration
F 2
F 1
–
M
I My
Z M
(11)
3
Bending
! ! σ= =
x
+
Bending moment diagram Neutral plane
For beams of rectangular cross section: (12) For beams of solid circular cross section: (13) For wide flange and I beams (approximately): (14) A V 2 3 τ = A V 3 4 τ = a V τ =
F 1
F 2
R 1
R 2
R 1 V
F 1
4
Shear
R 2 x
F 2
Shearing force diagram Neutral plane
F
A F τ =
Direct shear
(15)
5
Uniform
F
T
Zp T J Tc τ = =
Torsional Shear
(16)
6
For direct tension and direct compression loading, Cases 1 and 2 in the table on page 216 , the force F must act along a line through the center of gravity of the section at which the stress is calculated. The equation for direct compression loading applies only to members for which the ratio of length to least radius of gyration is relatively small, ap- proximately 20; otherwise the member must be treated as a column. The table Stresses and Deflections in Beams starting on page 257 give equations for calculating stresses due to bending for common types of beams and conditions of loading. Where these tables are not applicable, stress may be calculated using Equation (11) in the table on page 216 . In using this equation it is necessary to determine the value of the bend ing moment at the point where the stress is to be calculated. For beams of constant cross- section, stress is ordinarily calculated at the point coinciding with the maximum value of bending moment. Bending loading results in the characteristic stress distribution shown in the table for Case 3. It will be noted that the maximum stress values are at the surfaces farthest from the neutral plane. One of the surfaces is stressed in tension and the other in compression. It is for this reason that the ± sign is used in Equation (11). Numerous tables for evaluating section moduli are given in the section starting on page 239 . Shear stresses caused by bending have maximum values at neutral planes and zero val- ues at the surfaces farthest from the neutral axis, as indicated by the stress distribution dia- gram shown for Case 4 in the Table of Simple Stresses . Values for V in Equations (12), (13) and (14) can be determined from shearing force diagrams. The shearing force diagram shown in Case 4 corresponds to the bending moment diagram for Case 3. As shown in this diagram, the value taken for V is represented by the greatest vertical distance from the
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