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254 STRENGTH OF MATERIALS.
calculated load on B from the total load the load on A is ob-
tained. By subtracting the calculated load at A from the total
load, the load on B is obtained.
To each load as calculated above for each support also add
half the weight of the beam.
To Figure Sizes of Beams When Placed in an Inclined
Position.
FIG. 23 Figure all calculations concerning
the transverse strength from the dis-
tance S, and leave the length L out of
consideration. If the distance S cannot
be obtained by measurement it may
be found by multiplying L by cosine of
angle a.
DEFLECTION IN BEAMS WHEN LOADED
TRANSVERSELY.
Experiments and theory both prove that if the span is
increased and the width of the beam increased in the same pro-
portion the transverse strength of the beam is unchanged ; but
such is not the case with its stiffness. If a beam is to have the
same stiffness its depth must be increased in the same ratio as
the span, providing the width is unchanged. Within the
elastic limit of the beam the deflection is directly proportional
to the load; that is, half the load produces half the deflection,
but doubling the load will double the deflection.
Deflection is proportional to the cube of the span ; that is,
with twice the length of span the same load will, when the other
dimensions of the beam are unchanged, produce eight times as
much deflection.
Deflection is inversely as the cube of the depth (thickness)
of the beam. For instance, if the depth of a beam is doubled
but the length of span and the width of beam is unchanged, the
same load will produce only one-eighth as much deflection.
Deflection is inversely as the width of the beam ; for instance,
when a beam is twice as wide as another beam of the same
material but all the other dimensions are unchanged, the same
load will produce only half as much deflection.
The deflection in a beam caused by various modes of load-
ing is calculated by the following formulas :
—
For beams laid in a horizontal position and loaded trans-
versely, fastened at one end and loaded at the other : (See Fig. 6).
s _ PxL?
SXE X /
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