Page 131 - Civil Engineering Formulas

BEAM FORMULAS 73

minimum value of M Pd is zero. The deflection d for axial compression and
bending can be closely approximated by

d 0
d (2.32)
1 (P/P c )
where d 0 deflection for the transverse loading alone, in (mm); and P c criti-
2
2
cal buckling load
EI / L , lb (N).
UNSYMMETRICAL BENDING

When a beam is subjected to loads that do not lie in a plane containing a princi-
pal axis of each cross section, unsymmetrical bending occurs. Assuming that
the bending axis of the beam lies in the plane of the loads, to preclude torsion,
and that the loads are perpendicular to the bending axis, to preclude axial com-
2
ponents, the stress, lb/in (MPa), at any point in a cross section is

M x y M y x
f (2.33)
I x I y
where M x bending moment about principal axis XX,
in lb (Nm)
M y bending moment about principal axis YY,
in lb (Nm)
x distance from point where stress is to be computed to YY axis,
in (mm)
y distance from point to XX axis, in (mm)
4
I x moment of inertia of cross section about XX, in (mm )
4
I y moment of inertia about YY, in (mm )
If the plane of the loads makes an angle with a principal plane, the neutral
surface forms an angle with the other principal plane such that
tan I x tan (2.34)
I y

ECCENTRIC LOADING

If an eccentric longitudinal load is applied to a bar in the plane of symmetry, it
produces a bending moment Pe, where e is the distance, in (mm), of the load P
from the centroidal axis. The total unit stress is the sum of this moment and the
stress due to P applied as an axial load:
P Pec P ec
f 1 2 (2.35)
A I A r