Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.55
I�R
yo � (5.79)
2 I� � AR
Since y o is positive, the neutral axis shifts toward the center of curvature.
5.6.2 Curved Beams with Various Cross Sections
Equation (5.78) for bending stresses in curved beams subjected to end moments in
the plane of curvature can be expressed for the inside and outside beam faces in
the form:
Mc
ƒ � K (5.80)
I
where c � distance from the centroidal axis to the inner or outer surface. Table 5.4
gives values of K calculated from Eq. (5.78) for circular, elliptical, and rectangular
cross sections.
If Eq. (5.78) is applied to 1 or T beams or tubular members, it may indicate
circumferential flange stresses that are much lower than will actually occur. The
error is due to the fact that the outer edges of the flanges deflect radially. The effect
is equivalent to having only part of the flanges active in resisting bending stresses.
Also, accompanying the flange deflections, there are transverse bending stresses in
the flanges. At the junction with the web, these reach a maximum, which may be
greater than the maximum circumferential stress. Furthermore, there are radial
stresses (normal stresses acting in the direction of the radius of curvature) in the
web that also may have maximum values greater than the maximum circumferential
stress.
A good approximation to the stresses in I or T beams is as follows: for circumferential
stresses, Eq. (5.78) may be used with a modified cross section, which is
obtained by using a reduced flange width. The reduction is calculated from b� �
�b, where b is the length of the portion of the flange projecting on either side from
the web, b� is the corrected length, and � is a correction factor determined from
equations developed by H. Bleich, � is a function of b2 /rt, where t is the flange
thickness and r the radius of the center of the flange:
b 2 /rt � 0.5 0.7 1.0 1.5 2 3 4 5
� � 0.9 0.6 0.7 0.6 0.5 0.4 0.37 0.33
When the parameter b 2 /rt is greater than 1.0, the maximum transverse bending
stress is approximately equal to 1.7 times the stress obtained at the center of the
flange from Eq. (5.78) applied to the modified section. When the parameter equals
0.7, that stress should be multiplied by 1.5, and when it equals 0.4, the factor is
1.0 in Eq. (5.78), I� for I beams may be taken for this calculation approximately
equal to
2 c � 2�
I� � I 1 � (5.81)
R