When a Girder ok Beam is Fixed at one End, and Loaded at the other.

1. When The Depth Is Uniform Throuyhout The Entire Length

The section at every point must be in proportion to the product of the length, breadth and square of the depth, and as the square of the depth is in every point the same, the breadth must vary directly as the length; consequently, each aide of the beam must be a vertical plane, tapering gradually to the end.

2. When The Breadth Is Uniform Throughout The Entire Length

The depth must vary as the square root of the length; hence the upper or lower sides, or both, must be determined by a parabolic curve.

3. When the section at every point is similar - that is, a Circle, an Ellipse, a Square, or a Rectangle, the sides of which bear a fixed proPortion To Each Other.

The section at every point being a regular figure, for a circle, the diameter at every point must be as the cube root of the length; and for an ellipse, or a rectangle, the breadth and depth must vary as the cube root of the length.

When a Girder or Beam is Fixed at one End and Loaded.

Uniformly Throughout Its Length

1. When The Depth Is Uniform Throughout Its Entire Length

The breadth must increase as the square of the length.

2. When The Breadth Is Uniform Throughout Its Entire Length

The depth will vary directly as the length.

3. When The Section At Every Point Is Similar, As A Circle, Ellipse, Square, And Rectangle

The section at every point being a regular figure, the cube of the depth must be in the ratio of the square of the length.

When A Girder Or Beam Is Supported At Both Ends

1. When Loaded In The Middle

The constant of the beam, or the product of the breadth and the square of the depth, must be in proportion to the distance from the nearest support; consequently, whether the lines forming the beam are straight or curved, they meet in the centre, and of course the two halves are alike: the beam, therefore, may be considered as one half the length, the supported end corresponding with the free end in the case of beams, one end being fixed, and the middle of the beams similarly corresponding with the fixed end.

2. When The Depth Is Uniform Throughout

The breadth must be in the ratio of the length.

3. When The Breadth Is Uniform Throughout

The depth will vary as the square root of the length.

4. When the section at every point is similar, as a Circle, Ellipse.

Square, And Rectangle

The section at every point being a regular figure, the cube of the depth will be as the square of the distance from the supported end.

When a Girder or Beam is Supported at roth Ends, and Loaded uniformly throughout its Length.

L. When The Depth Is Uniform

The breadth will be as the product of the length of the beam and the length of it on one side of the given point, less the square of the length on one side of the given point.

2. When The Breadth Is Uniform

The depth will be as the square root of the product of the length of the beam and the lenght of it on one side of the given point, less the square of the length on one side of the given point.

3. When The Section At Every Point Is Similar, As A Circle, Ellipse, Square, And Rectanqle

The section at every point being a regular figure, the cube of the depth will be as the product of the length of the beam and the length of it on one side of the given point, less the square of the length on one side of the given point.

General Deductions from the Experiments of Stephen-box, FAIRBAIRN, CUBITT, HUGHES, etc. Fairbairn shows in his experiments that with a stress of about 12,320 lbs. per square inch on cast iron, and 28,000 lbs. on wrought iron, the sets and elongations are nearly equal to each other.

A cast-iron beam will be bent to one-third of Its breaking weight if the load is laid on gradually; and one-sixth of it, if laid on at once, will produce the same effect, if the weight of the beam is small compared with the weight laid on. Hence beams of east iron should be made capable of bearing more than 6 times the greatest weight which will be laid upon them.

In wrought-iron beams, if fixed at both ends, the upper flange should be larger than the lower, in the ratio of 1.35 to 1.

The breaking weights in similar beams are to each other as the squares of their like linear dimensions; that is, the breaking weights of beams are computed by multiplying together the area of their section, their depth, and a constant, determined from experiments on beams of the particular form under investigation, and dividing the product by the distance between the supports.

Cast and wrought iron beams, having similar resistances, have weights nearly as 2.44 to 1.

The range of the comparative strength of girders of the same depth, having a top and bottom flange, and those having bottom flange alone, is from having but a little area of bottom flange to a large proportion of it, from ½ to ¼ greater strength.

A box beam or girder, constructed of plates-of wrought iron, compared to a single rib and flanged beam I, of equal weights, has stance as 100 to 93.

The resistance of beams or girders, where the depth is greater than their breadth, when supported at top, is much increased. In some cases the difference is fully one third.

When a beam is of equal thickness throughout its depth, the curve should be an ellipse to enable it to support a uniform load with equal resistance in every part; and if the beam is an open one, the curve of equilibrium, for a uniform load, should be that of a para-bola. Hence, when the middle portion is not wholly removed, the curve should be a compound of an ellipse and a parabola, approaching nearer to the latter as the middle part is decreased.

Girders of cast iron, up to a span of 40 feet, involve a less cost than of wrought iron.

Cast iron beams and girders should not be loaded to exceed.....1 one-fifth of their breaking weight; and when the Strain is attended with concussion and vibration, this proportion must be increased.

Simple cast iron girders may be made 50 feet in length, and the best form is that of Hodgkinson: when subjected to a fixed load, the flange should be as 1 to 6, and when to a concussion, etc., as 1 to 4.

The forms of girders for spaces exceeding the limit of those of simple cast iron are various; the principal ones adopted are those of the straight or arched cast iron girders in separate pieces, and bolted together - the Trussed, the Bow-string, and the wrought iron Box and Tubular,

A Straight or Arched Girder is formed of separate castings, and is entirely dependent upon the bolts of connection for its strength.

A Trussed or Bow-string Girder is made of one or more castings to a single piece, and its strength depends, other than upon the depth or area of it, upon the proper adjustment of the tension, or the initial strain, upon the wrought iron tarn.

A Box or Tubular Girder is made of wrought iron, and is best constructed with cast iron tops, in order to resist compression: this form of girder is best adapted to afford lateral stiffness.