CIVIL ENGINEERING FORMULAS

74 CHAPTER TWO

whereAcross-sectional area, in^2 (mm^2 )

cdistance from neutral axis to outermost fiber, in (mm)

Imoment of inertia of cross section about neutralaxis, in^4 (mm^4 )

rradius of gyration , 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

should not exceed r^2 /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 dis the deflection, in (mm). This deflection

may be closely approximated by

(2.36)

Pcis the critical buckling load 

2 EI/L^2 , 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.

The stresses, lb/in^2 (MPa), are given by

(2.37)

where Across-sectional area, in^2 (mm^2 )

execcentricity with respect to principal axis YY, in (mm)

eyeccentricity with respect to principal axis XX, in (mm)

cxdistance from YYto outermost fiber, in (mm)

cydistance from XXto outermost fiber, in (mm)

Ixmoment of inertia about XX, in^4 (mm^4 )

Iymoment of inertia about YY, in^4 (mm^4 )

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

f

P

A



Pexcx

Iy



Peycy

Ix

d

4 eP/Pc

(1P/Pc)

2 I/A