Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.29
Circular Sections. If a circular shaft (hollow or solid) is twisted, a section that is
plane before twisting remains plane after twisting. Within the proportional limit,
the shearing unit stress at any point in a transverse section varies with the distance
from the center of the section. The maximum shear, psi, occurs at the circumference
and is given by
where T � torsional moment, in-lb
r � radius of section, in
J � polar moment of inertia, in 4
Tr
v � (5.43)
J
Polar moment of inertia of a cross section is defined by
2 J � � � dA (5.44)
where � � radius from shear center to any point in the section
dA � differential area at the point
In general, J equals the sum of the moments of inertia above any two perpendicular
axes through the shear center. For a solid circular section, J � �r 4 /2. For a hollow
circular section with diameters D and d, J � �(D 4 � d 4 )/32.
Within the proportional limits, the angular twist between two points L inches
apart along the axis of a circular bar is, in radians (1 rad � 57.3�):
TL
� � (5.45)
GJ
where G is the shearing modulus of elasticity (see Art. 5.2.4).
Noncircular Sections. If a shaft is not circular, a plane transverse section before
twisting does not remain plane after twisting. The resulting warping increases the
shearing stresses in some parts of the section and decreases them in others, compared
wit the sharing stresses that would occur if the section remained plane. Consequently,
shearing stresses in a noncircular section are not proportional to distances
from the share center. In elliptical and rectangular sections, for example, maximum
shear occurs on the circumference at a point nearest the shear center.
For a solid rectangular section, this maximum may be expressed in the following
form:
T
v � (5.46)
2 kb d
where b � short side of rectangle, in
d � long side, in
k � constant depending on ratio of these sides;
d/b � 1.0 1.5 2.0 3 4 5 10 �
k � 0.208 0.231 0.246 0.258 0.267 0.282 0.291 0.312 0.333
(S. Timoshenko and J. N. Goodier, ‘‘Theory of Elasticity,’’ McGraw-Hill Publishing
Company, New York.)