FLEXURAL STRESS IN A CURVED MEMBER
The ring in Fig. 33 has an internal diameter of 12 in (304.8 mm) and a circular cross sec-
tion of 4-in (101.6-mm) diameter. Determine the normal stress at A and at B (Fig. 33).
Calculation Procedure:
- Determine the geometrical properties of the cross section
The area of the cross section is ,4 = 0.7854(4)
2
= 12.56 in
2
(81.037 cm
2
); the section mod-
ulus is S = 0.7854(2)
3
= 6.28 in
3
(102.92 cm
3
). With c = 2 in (50.8 mm), the radius of cur-
vature to the centroidal axis of this section is R = 6 + 2 = 8 in (203.2 mm).
- Compute the R/c ratio and determine the correction factors
Refer to a table of correction factors for curved flexural members, such as Roark—For-
mulas for Stress and Strain, and extract the correction factors at the inner and outer sur-
face associated with the RIc ratio. Thus RIc = 8/2 = 4; kt =l.23;k 0 = 0.84.
- Determine the normal stress
Find the normal stress at A and B caused by an equivalent axial load and moment. Thus fA
= PIA + kt(M/S) = 9000/12.56 + 1.23(9000 x 8)76.28 = 14,820-lb/in^2 (102,183.9-kPa)
compression;/* = 9000/12.56 - 0.84(900Ox 8)76.28 = 8930-lb/in^2 (61,572.3-kPa) tension.
SOfL PRESSURE UNDER DAM
A concrete gravity dam has the profile
shown in Fig. 34. Determine the soil
pressure at the toe and heel of the dam
when the water surface is level with the
top.
Calculation Procedure:
- Resolve the dam into
suitable elements
The soil prism underlying the dam may
be regarded as a structural member sub-
jected to simultaneous axial load and
bending, the cross section of the member
being identical with the bearing surface
of the dam. Select a 1-ft (0.3-m) length
of dam as representing the entire struc-
ture. The weight of the concrete is 150
lb/ft
3
(23.56kN/m
3
).
Resolve the dam into the elements
AED and EBCD. Compute the weight of
each element, and locate the resultant of
the weight with respect to the toe. Thus
W 1 =
1
/ 2 (12)(20)(150) = 18,000 Ib (80.06
(b) Soil pressure under dam
FIGURE 34
(a) Loads on dam
Water surface
