Page 69 - Handbook of Civil Engineering Calculations, Second Edition

1.52 STRUCTURAL STEEL ENGINEERING AND DESIGN

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:

1. Determine the geometrical properties of the cross section
2
2
The area of the cross section is A 0.7854(4) 12.56 sq.in. (81.037 cm ); the section
3
3
3
modulus is S 0.7854(2) 6.28 in (102.92 cm ). With c 2 in. (50.8 mm), the radius
of curvature to the centroidal axis of this section is R 6 2 8 in. (203.2 mm).
2. 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 R/c ratio. Thus R/c 8/2 4; k i 1.23; k o 0.84.
3. Determine the normal stress
Find the normal stress at A and B caused by an equivalent axial load and moment. Thus f A
P/A k i (M/S) 9000/12.56 1.23(9000 8)/6.28 14,820-lb/sq.in. (102,183.9-
kPa) compression; f B 9000/12.56 0.84(9000 8)/6.28 8930-lb/sq.in. (61,572.3-
kPa) tension.

SOIL PRESSURE UNDER DAM
A concrete gravity dam has the profile
shown in Fig. 34. Determine the soil pres-
sure at the toe and heel of the dam when
the water surface is level with the top.

Calculation Procedure:

1. 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 structure.
The weight of the concrete is 150 lb/ft 3
3
(23.56 kN/m ).
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 lb
FIGURE 34 (80.06 kN); W 2 3(20)(150) 9000 lb