is Z,and x, be the strength of a column whose length is L1 ,then we have approximately x: x1 = L12:L2, or x1 = x.L2/L12 (34)

Where x1, = approximately the strength of a column, L, feet long. Where x = the strength (previously ascertained or known), of a column of same cross-section, and L feet long.

Where L and L1= the respective lengths of columns in feet. The nearer L and L1 are to each other the closer will be the result. The comparative strength per square inch of cross-section of columns of same length, but of different cross-sections, is, approximately, as their least outside diameter, or side, or x: x1 = b:b1 or xl = x.b1/b (35)

Where x1 = approximately the strength of a column, per square inch, whose least side or diameter (outside) is = b,.

Where x = the strength per square inch (previously ascertained or known) of a column of same length, but whose least side or diameter (outside) is = b.

The more similar and the nearer in size the respective cross-sections are, the closer will be the result. That is, the comparison between two circular columns, each 1" thick, will be very much nearer correct than between two circular columns, one 3/4" thick and the other 2" thick, or between a square and a circular column. The thicker the shell of a column the less it will carry per square inch. The formulae (34 and 35) are hardly exact enough for safe practice, but will do for ascertaining approximately the necessary size of column, before making the detailed calculation required by formula (3).

The approximate thickness required for the flanges of plate girders is as follows:

Approximate thickness of flange of plate girders.

x = r/d-a1/b (36)

Where x= the approximate thickness, in inches, of either flange of a riveted girder.

Where b = the breadth of flange, less rivet holes, all in inches.

Where d=the total depth of girder in inches.

Where r = the moment of resistance in inches.

Where a, = the area (less rivet holes) of cross-section of both angles at flange, in square inches.

Fig. 11

Deflection

The derivation of the following formulae would be too lengthy to go into here, it will suffice for all practical purposes to give them. They are all based on the moduli of elasticity of the different materials, and are as follows:

For A Cantilever, Uniformly Loaded

(37)

For A Cantilever, Loaded At Free End

Fig. 12.

δ=1/3.w.l3/e.i (38)

For A Ream, Uniformly Loaded

Fig. 13.

For A Beam, Loaded At Centre

Fig. 14.

δ = 1/4/. w.l3/e.i (40)

For A Beam, Loaded At Any Point

Greatest deflection is near the centre, not at the point where load is applied.1

(41)

Where u = uniform load, in pounds.

Where w = concentrated load, in pounds.

Where 1 = length of span, in inches.

Where e=the modulus of elasticity, in pounds-inch, of the material, see Tables IV and V.

Where i = the moment of inertia, of cross-section, in inches.

Where m and n = the respective distances to supports, in inches.

Where δ = the greatest deflection, in inches (see Formulae 28 and 29).

For A Cantilever, Loaded At Any Point

Greatest deflection is at free end; if y = distance from support to load, in inches, then:2

δ = 1/3. w.y.l2/e.i (42)