Strength of Timbers - Carpenter's and Builder's Math, Plans, and Specifications - page 99

DOWNWARD THRUST
OF FIXED SUPPORT

L

DEFLECTION FIXED ENDS

TANGENT

L' COMPOUND

CURVE

F'

STIFFENING
COUPLE

F

UPWARD THRUST
OF FIXED SUPPORT

DEFLECTION
SUPPORTED ENDS

Figure 4-10 One end of a beam illustrating the stiffening ef-
fect of fixed ends as compared with supported ends. When the
ends are fixed, the deflection of the beam will be resisted by
an upward thrust, indicated by F, and a downward thrust, L.
These thrusts form a resisting or stiffening couple that holds
the portion of the beam embedded in the bearing in a horizon-
tal position, thereby causing the beam to deflect in a compound
curve, which increases its stiffness. The dotted lines show the
excess deflection for the same load if the beam were simply
supported at the ends.

If depths and widths are the same, strengths vary inversely as
the lengths of the spans.
Their stiffness, or resistance to deflection, will vary inversely
as the cubes of their spans, other factors being equal.
Their stiffness will vary directly as the cubes of their depths,
other factors being equal.
Other factors being equal, stiffness will vary directly as their
widths.
If a beam is split horizontally, and the two halves are laid side
by side, they will carry only one-half as much loading as the
original beam.

These relations are not strictly true for I-beams because of their
irregular shapes, but they are approximately true for all types of
beams. It is usually most economical with materials to use as deep a
beam as can conveniently be employed. Note that double 2-inch

×
4-inch trimmers over window or door heads, if set edge up, are
8 times as strong and 32 times as stiff as when placed flatwise.