CIVIL ENGINEERING FORMULAS

BEAM FORMULAS 73

minimum value of MPdis zero. The deflection dfor axial compression and
bending can be closely approximated by

(2.32)

whered 0 deflection for the transverse loading alone, in (mm); and Pccriti-
cal buckling load
2 EI/L^2 , 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-
ponents, the stress, lb/in^2 (MPa), at any point in a cross section is

(2.33)

whereMxbending moment about principal axis XX,
in lb (Nm)
Mybending moment about principal axis YY,
in lb (Nm)
xdistance from point where stress is to be computed to YYaxis,
in (mm)
ydistance from point to XXaxis, in (mm)
Ixmoment of inertia of cross section about XX, in (mm^4 )
Iymoment of inertia about YY, in (mm^4 )

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

(2.34)

ECCENTRIC LOADING

If an eccentric longitudinal load is applied to a bar in the plane of symmetry, it
produces a bending moment Pe, where eis 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 Papplied as an axial load:

f (2.35)

P

A



Pec

I



P

A

1 

ec

r^2 

tan

Ix

Iy

tan

f

Mxy

Ix



Myx

Iy

d

d 0

1 (P/Pc)