Page 132 - Civil Engineering Formulas

74 CHAPTER TWO

2
2
where A cross-sectional area, in (mm )
c distance from neutral axis to outermost fiber, in (mm)
4
4
I moment of inertia of cross section about neutral axis, in (mm )
r radius of gyration 2 I/A , in (mm)
Figure 2.1 gives values of the radius of gyration for several cross sections.
If there is to be no tension on the cross section under a compressive load, e
2
should not exceed r /c. For a rectangular section with width b, and depth d, the
eccentricity, therefore, should be less than b/6 and d/6 (i.e., the load should not
be applied outside the middle third). For a circular cross section with diameter D,
the eccentricity should not exceed D/8.
When the eccentric longitudinal load produces a deflection too large to be
neglected in computing the bending stress, account must be taken of the addi-
tional bending moment Pd, where d is the deflection, in (mm). This deflection
may be closely approximated by
4eP/P c
d (2.36)

(1 P/P c )
2
2
P c is the critical buckling load
EI/L , lb (N).
If the load P, does not lie in a plane containing an axis of symmetry, it pro-
duces bending about the two principal axes through the centroid of the section.
2
The stresses, lb/in (MPa), are given by
P Pe x c x Pe y c y
f (2.37)
A I y I x
2
2
where A cross-sectional area, in (mm )
e x eccentricity with respect to principal axis YY, in (mm)
e y eccentricity with respect to principal axis XX, in (mm)
c x distance from YY to outermost fiber, in (mm)
c y distance from XX to outermost fiber, in (mm)
4
4
I x moment of inertia about XX, in (mm )
4
4
I y moment of inertia about YY, in (mm )
The principal axes are the two perpendicular axes through the centroid for
which the moments of inertia are a maximum or a minimum and for which the
products of inertia are zero.
NATURAL CIRCULAR FREQUENCIES AND NATURAL
PERIODS OF VIBRATION OF PRISMATIC BEAMS

Figure 2.26 shows the characteristic shape and gives constants for determi-
nation of natural circular frequency and natural period T, for the first
four modes of cantilever, simply supported, fixed-end, and fixed-hinged
beams. To obtain , select the appropriate constant from Fig. 2.26 and