(1) If a beam supported at both ends and loaded uniformly will safely carry an amount of load =u; then will the same beam:

(2) if both ends are built in solidly and load uniformly distributed, carry 1 1/2. u,

(3) if one end is supported and other built in solidly and load uniformly distributed, carry 1. u,

(4) if both ends are built in solidly and load applied in centre, carry 1. u,

(5) if one end is supported and other built in solidly and load applied in centre, carry 2/3. u,

(6) if both ends are supported and load applied in centre, carry 1/2. u,

(7) if one end is built in solidly and other end free (cantilever) and load uniformly distributed, carry 1/4. u,

(8) if one end is built in solidly and other end free (cantilever) and load applied at free end, carry 1/8. u.

That is, in cases (1), (3) and (4) the effect would be the same with the same amount of load; in case (2) the beam could safely carry 11/2 times as much load as in case (1); in case (5) the beam could safely carry only § as much as in case (1), etc., provided that the length of span is the same in each case.2

If the amount of deflection in case (1) were δ, then would the amount of deflection in the other cases be as follows:

Case

(2)

δ11

=

1/5

δ1

Case

(3)

δ111

=

2/5

δ1

Case

(4)

δ1v

=

2/3.δ1

Case

(5)

δv

=

3/4.δ1

Case

(6)

δv1

=

13/5.δ1

Case

(7)

δv11

=

9 3/5.δ1

Case

(8)

δv111

=

25 3/5.δ

2To count on the end of a beam being built in solidly would be very bad practice in most cases of building construction; as, for instance, a wooden beam with end built in solidly could not fall out in case of fire, and would be apt to throw the wall. Even where practicable, it would require very careful supervision to get the beam built in properly; then, too, it causes upward strains which must be overcome, complicating the calculations unnecessarily. In most cases where it is necessary to " build in" beam ends, the additional strength and diminished deflection thereby secured had better be credited as an additional margin of safety. The above rules for deflection do not hold good if the beam is not of uniform cross-section throughout; the deflection being greater as the variation in cross-section is creator.