A Beam fixed at both Ends - that is, so fixed that the ends cannot tilt up when the beam is loaded - is shown in Fig. 437.

Such a beam is in the condition of two cantilevers, Af and Bi, carrying a beam fi between them, which is supported at its ends f and i by hanging from the ends f and i of the cantilevers.

From the figure it will be seen that the upper portion of the beam is in tension from A to f and from B to i; the remainder from i to f is in compression.

Fig. 437.

The lower portion of the beam is under compression from A to f and B to i, the central portion if being in tension.

It will be noticed that at the points i and / the nature of the stress in each case changes; i and f are called the points of contra-flexure, and their distances from A and B depend upon the form of section of the beam, and the distribution of the load, etc. Roughly speaking, the points of contraflexure are generally distant about ¼ of the span from the abutments.

A Beam fixed at one End and supported at the other (Fig. 438) is like a combination of a cantilever Af and a supported beam fB; and the portions in tension and compression respectively are shown by the letters ttt and ccc.

A continuous Beam is one that extends without break in itself over two or more spans.

Fig. 439.

If the ends are fixed the compressions and tensions will be as shown by ccc and ttt in Fig. 439, resembling those of two fixed beams.

Fig. 440.

If the ends are supported the stresses will be as shown in Fig. 440, the arms in each span being like those of a beam fixed at one end and supported at the other. (Fig. 438.)

C. " Difference in Strength of a Girder carrying a given Load at its Centre or Uniformly Distributed."

## On Beams

A beam that can bear a given load concentrated at its centre can bear twice that load uniformly distributed over its length.

Thus if the beams in Figs. 428, 429 are similar, and the one in Fig. 428 could bear a concentrated load of 400 lbs., that in Fig. 429 could bear a distributed load of 800 lbs.

## On Cantilevers

Similarly a cantilever that can just bear a given load suspended from its outer end can bear twice that load if it is distributed over its length.

Difference in Strength between Beams of uniform Section supported at both Ends and those fixed at both Ends, or fixed at one End and supported at the other.

A beam fixed at both ends, with a concentrated load at the centre is twice as strong as the same beam supported at both ends and similarly loaded.

A beam fixed at both ends, with a uniform load throughout its length is l½ times as strong as the same beam supported at both ends and similarly loaded.

A beam fixed at one end, and supported at the other, with a concentrated load in the centre is 1 1/3 times as strong as the same beam supported at both ends and similarly loaded.

A beam fixed at one end and supported at the other, with a uniform load throughout its length is of the same strength as the same beam supported at both ends and similarly loaded.

D. "Best Forms for Struts, Ties, and Beams, such as floor joists exposed to transverse Stress."

Best Form for Struts.

Timber Struts should be rectangular in section, and of the same section throughout.

Cast-iron Struts may be of these cross sections, and tapering in their length, widening from one end to the other as in a column, or from both ends.

Wrought-Iron Struts are often of these cross sections.

Fig. 441.

Fig. 442.

Fig. 443.

Fig. 444.

Fig. 445.

Fig. 446.

Fig. 447.

Fig. 448.

Fig. 449.

Of these c, d, and g, are the best. Fig. 450 is an elevation of g, for the other forms the section is uniform throughout the length of the strut; g is very useful for struts of roofs.

Long Struts or Compression Bars are those which are so long in proportion to their width that they fail by bending before crushing.

Short Struts or Compression Bars are those which do not bend under the load, but fail by actual crushing.

Fig. 450.

Long Struts fixed at the Ends are much stronger than those of which the ends are hinged or rounded. If both ends are fixed they are 3 times, if one end only is fixed, 1½ times, as strong.

Best Form for Ties.

Any cross section is suitable for a rod or bar in tension whether it be made of timber or wrought iron. Cast iron should never be used for ties.

Best Form for Beams subject to Transverse Stress.

Timber Beams may be of rectangular cross section uniform throughout their length. The deeper they are the better both for strength and stiffness.

Iron Girders are of a section roughly resembling an I, the upper and lower horizontal portions are called the flanges, and the upright portion the web.

Cast-Iron Beams should have a cross section in which the lower flange to resist tension should have an area from four to six times as great as that of the upper flange which is to resist compression (see Parts I. and IV.)

Fig. 451 shows a section with flanges having areas as 6 to 1, and Fig. 452 with flanges as 4 to 1.

Fig. 451.

Fig. 452.

And Figs. 453 to 456 show plans and elevations of cast-iron girders for uniformly distributed loads.

The flanges are sometimes made to differ in thickness as in Fig. 451, the web tapering from one to the other, or the metal may be of equal thickness throughout as in Fig. 452. ELEVATION.

Figs. 453, 454, are the elevation and plan of a girder of uniform width, the depth being varied according to the stress to be borne. Figs: 455, 456, are the elevation and plan of a girder of uniform depth, the width of the flanges being varied to suit the stress.

Eolled-Iron Joists are of uniform section like Fig. 457 throughout, the flanges being similar and of equal area.

Plate Girders are also of a general I form, built up with a plate and angle irons, riveted together as in Figs. 458-460, or, where additional strength is required, with one or more plates in the flanges (one plate is shown in Fig. 460), and stiffeners to support the web.

Fig. 457. Section.

Fig. 458. Elevation.

Fig. 459. Section.

Fig. 460.

E. "In the ordinary kinds of Wooden or Iron Roof Trusses and Framed Structures of a similar description to distinguish the Members in Compression from those in Tension."

There is no very short and simple method for ascertaining whether the members of a truss are in tension or compression under a given load in any position.

The information can be easily obtained, but the methods employed cannot be explained in these very short notes. They are fully explained in Part IV.

When the loads vary from time to time in position, a member which may with one position of the loads be in tension may with another position of the load have no stress upon it, or even one of an opposite nature.

Thus in an ordinary king-post roof (see Fig. 464) the wind blowing from the right causes a compressive stress upon the strut on that side, but no stress whatever on the other strut, and when the wind is from the left, the stresses on the struts are just reversed.

The student can, however, easily learn and carry in his head the nature of the stresses to which each member of a roof truss is practically subjected.

In Plate IV. and in Figs. 461 to 465, which give a great many forms of roof trusses, each of the members shown in thick lines is in compression, and each of those shown in thin lines is in tension, under all loads that can practically come upon the roof, such as the weight of the roof, of snow lying upon it, and the pressure of the wind upon both sides in turn.