ABSTRACT
Contents 4.1 Introduction 90 4.2 Properties of Beam Cross Sections 90 4.3 Normal Stress and Strain 94 4.4 Torsion 98 4.5 Bending Stress and Strain 99 4.6 Transverse Shear Stress and
Strain 104 4.7 Summary 109
Examples 4.1 Determining the Centroid of an
Area 91 4.2 Area and Polar Moment of Inertia 91 4.3 Moment of Inertia and Centroid
from Parallel Axis Theorem 92 4.4 Parallel Axis Theorem 93 4.5 Radius of Gyration 93 4.6 Normal Stress, Deformation, and
Spring Rate 97 4.7 Elongation and Spring Rate in
Tension 97 4.8 Angle of Twist and Spring Rate in
Torsion 98 4.9 Power Transmitted by a Shaft 99 4.10 Stress in Bending 101 4.11 Stress in Curved Member 103 4.12 Stress Due to Transverse Shear 105 4.13 Critical Location in a Beam 106
Case Study Design of a Shaft for a Coil Slitter 107
Symbols A cross sectional area, m2
A′ partial cross sectional area, m2
a width, m b height, m c distance from neutral axis to outer fiber
of solid, m dx, dy distance between two parallel axes, one
of which contains centroid of area, m E modulus of elasticity, Pa e eccentricity, distance separating centroidal
and neutral radii of curved member, m G shear modulus of elasticity, Pa h height of triangular cross-sectional area, m hp power, W I area moment of inertia, m4
Im mass moment of inertia, kg-m2
J polar area moment of inertia, m4
J¯ polar area moment of inertia about centroidal coordinates, m4
k spring rate, N/m ka angular spring rate, N-m l length, m M bending moment, N-m ma mass, kg P force, N Q first moment about neutral axis, m3
r radius, m r¯ centroidal radius, m rg radius of gyration, m s length of a line segment, m T torque, N-m u velocity, m/s V transverse shear force, N wt width, m x, y, z Cartesian coordinate system, m x¯, y¯, z¯ centroidal coordinate system, m x′, y′ coordinates parallel to x-and y-axes Zm section modulus, I/c, m3
γ shear strain δ deformation, m normal strain ρ density, kg/m3
θ angle of twist, rad σ normal stress, Pa τ shear stress, Pa ω angular velocity, rad/s
Subscripts i inner o outer x, y, z Cartesian coordinates x¯, y¯, z¯ centroidal coordinates x′, y′ coordinates parallel to x-and y-axes
Normal, torsional, bending, and transverse shear loadings were described in Section 2.3. This chapter describes the stresses and strains resulting from these types of loading while making use of the general Hooke’s law relation developed in Chapter 3. The theory developed in this chapter is applicable to any machine element. For the purposes of this
A
C
x
x
y
y
dA
x
y
Figure 4.1: Centroid of area. The centroid is at point C with coordinates (x¯, y¯).
