SEMI-BEAMS OF UNIFORM STRENGTH LOADED AT THE END.

By a semi-beam is meant a beam fixed at one end and free at the other; as it represents the half of a beam supported at both ends.

The moments of stress due to the weight on the end, at any section of the beam, increase directly as the distance of the section from the end of the beam. Let cb, Fig. 149, be a rectangular beam, fixed at the base cd,

Fig. 149.-Stress in a Semi-beam
loaded at the end.

and loaded at the end b. Draw the diagonal straight line bd, then the ordinates to the triangle bcd, represent proportionally the moment of stress at all parts of the length; and the moments of stress vary directly as the length. Now, the ultimate moments of resistance at any section are as the square of the depth, when the breadth is uniform; and it follows, conversely, that the depth of the beam, of uniform strength, must vary as the square root of the distance from the end b. Take, for instance, the section d'ďat the half-length of the beam; the moment of stress at cd is to that of the stress at dd', as I to 1⁄2; and the required depth at the origin cd, is to the depth at 'd', as I to 2, or as I to .707. The depth cc", equal to .707, would be the depth of a beam of uniform strength, at that section. The depth for uniform strength at any other section may be calculated in the same way; and the form of the lower side of the beam, of uniform strength, is that of a parabola, bc" d, of which the vertex is at the end b. With respect to transverse resistance, then, the semi-beam would be equally strong if the lower portion bb'd were removed.

The semi-beam, rectangular in section, of uniform strength, fixed at one end, and loaded at the other end, having the breadth constant, may therefore be moulded in depth to any of the parabolic outlines, Figs. 150.

When the depth of the semi-beam, rectangular in section, is constant, the breadth is in simple proportion to the distance from the end of the beam, as in Fig. 151, and the beam is triangular in plan.

When the section of the semi-beam is double-flanged, or is hollow rectangular, and the breadth is constant, the flanges are assumed to be of a constant sectional area.

Leaving out of the calculation the strength of the vertical

web, and calculating only for the flanges, the moment of resistance at any section is as the depth, and the form of the beam is triangular, as in Fig. 152, which shows a semi-beam with double flanges.

If the strength of the vertical web be taken into the calculation, the form of the beam is intermediate between the triangular and the parabolic.

Fig. 151.

Fig. 152.

Fig. 153.

Semi-beams loaded at one end.

When the section of the semi-beam is double-flanged, or is hollow-rectangular, and the depth is constant, calculating only for the flanges, Fig. 153, their sectional area increases uniformly with the distance from the end, and if their thickness be uniform, they are triangular in plan, as shown.

If the web be taken into the calculation, it is calculated as a solid semibeam rectangular in section, and the thickness should increase as the distance from the end. The web would, therefore, be triangular in plan.

When the section of the semi-beam is circular, the moment of resistance varies as the cube of the diameter, and the cube of the diameter is therefore as the distance from the end; or, inversely, the diameter is as the cube root of the distance, and the outline of the semi-beam may be formed by the revolution of a cubic parabola on its axis, Fig. 154.

When the section of the semi-beam is annular; when the thickness is uniform and small in proportion to the diameter, the square of the diameter varies as the distance from the end, or the diameter varies as the square root of the distance, and the semi-beam is formed by the revolution of a parabola on its axis.

If the thickness varies with the diameter, the diameter varies as the cube root of the distance from the end, and the semi-beam is cubic-parabolic, like Fig. 154.

When the section of the semi-beam is elliptical, the sections being similar at all points of the length, the cube of the

depth varies as the distance from the end, or the depth varies as the cube root of the distance, and the elevation of the beam is cubic-parabolic, like Fig. 154.

When the section of the semi-beam is hollowelliptical, the beam being of similar sections throughout. When the thickness is uniform, and is small in proportion to the depth, the square of the depth varies as the distance from the end, or the depth varies as the square root of the distance, and the side elevation of the beam is parabolic.

Fig. 154.-Semi-beam ioaded at one end.

If the thickness varies with the depth, the depth varies as the cube root of the distance from the end, and the beam is cubic-parabolic in side elevation, like Fig. 154.

SEMI-BEAMS OF UNIFORM STRENGTH UNIFORMLY LOADED.

The moment of stress due to the weight when uniformly distributed increases as the square of the distance from the end of the beam, as will be shown in the following case:

When the semi-beam is rectangular in section, and its breadth is constant. Suppose the load equally divided and distributed as a great number of weights, W', W", W", &c., Fig. 155; and suppose the beam to be divided into an equal number of corresponding sections at c', c", c", &c. The loads supported by the successive sections are W', 2 W', 3 W', &c.; the distances of the centres of gravity of these loads, from the respective sections, are as

Figs. 155, 156.-Semi-beams uniformly loaded.

1, 2, 3, &c. Therefore, the moments of stress at the successive intersections, c, d, d", &c., are as 12, 22, 32, &c., or as the square of the distance from the end. But the moments of resistance at the intersections are as the squares of the depths at c', c", c", &c.; and so the square of the depth is as the square of the distance, or the depth is as the distance from the end. The beam is therefore triangular in elevation.

When the semi-beam is rectangular in section, and has the depth constant, Fig. 156. As the depth is constant, the breadth must increase as the square of the distance; and it may be, in outline, of the form of two parabolas bc, be, back to back, touching each other at their vertices at b; the axes being perpendicular to the length.

When the section of the semi-beam is hollow-rectangular, or is double-flanged; and the breadth is constant. Calculating the strength of the upper and lower members, or flanges, only, and supposing the thickness to be uniform, the moment of resistance is as the depth; the depth is, therefore, as the square of the distance from the end, and is of the form of a parabola, Fig. 157, of which the vertex is at b, and the axis is perpendicular to the length.

Calculating the strength of the vertical webs or rib only, the beam would be triangular in side elevation.

Combining the webs and the flanges in the calculation, the form of the beam would be intermediate between the parabolic and the triangular.

2d. When the depth is constant. Calculating for the flanges only, the thickness being uniform; the breadth of the flanges is as the square of the distance from the end, Fig. 158, the same as in Fig. 156.

If the vertical web or rib, of uniform thickness, be included in the calculation, it does not materially modify the form of the flange.

When the section of the semi-beam is circular. The moment of resistance is as the cube of the diameter, and the moment of stress is as the square of the length; therefore the cube of the diameter is as the square of the

length, or the diameter is as the cube root of the square of the length, or as the 2/3 power, or .666 power of the length. The solid is formed by the revolution of a semi-cubic parabola on its axis.

When the section of the semi-beam is annular, the thickness being uniform and small compared to the diameter. The moment of resistance of any section is as the square of the diameter. The square of the diameter is, therefore, as the square of the length, or the diameter is as the length, and the semi-beam is triangular or conical in elevation, Fig. 159.

When the thickness diminishes with the diameter, the moments of resistance of sections are as the cubes of the diameters, and the diameter varies as the 2/3 power, or .666 power of the length, as with a solid circular section, and the form is derived from the revolution of a semi-cubic parabola on its axis.

When the section of the semi-beam is elliptical. The moment of resistance of a section is as the cube of the depth, and the form is the same as that of a circular beam.

Fig. 159.-Annular Semi-beam uniformly loaded.

When the section of the semi-beam is hollowelliptical. The form is the same as that of a beam of annular section.

BEAMS OF UNIFORM STRENGTH SUPPORTED AT BOTH ENDS.

The forms of beams supported at both ends, and loaded at the middle, are simply doubles of the forms of semi-beams, or such as are fixed at one end and unsupported at the other end. In the beam of rectangular section, for example, A B, Fig. 160, the diagonal lines, ca and cb, from the top at the middle to the supports at each end, are simply doubles of the diagonal bd, in the semi-beam, Fig. 149, and represent the graduated moment of bending stress from the middle, where it is a maximum, to the ends, where it vanishes; and the parabolic curves ca and cb, meeting base to base at the middle cd, form the outline of the rectangular beam of uniform strength, when the breadth is constant.

The beam, rectangular in section, of uniform strength, loaded at the middle,

and having breadth constant, may therefore be moulded according to any of the parabolic forms, Figs. 161, 162, 163, having the axes horizontal, and the vertices at the points of support.

B

a

d

Fig. 160.-Stress in rectangular beam.
supported at both ends.

Fig. 161.-Beam loaded at the middle.

When a rectangular beam, with a constant breadth, is loaded uniformly; referring to formula (5), page 508. When the weight is constant, together with the breadth b, and the length 7, the square of the depth, d2, varies as

Figs. 162, 163.-Beams loaded at the middle.

the products, mn, of the segments, m and n, of the length of the span at any point of the length. Or, the depth varies as the square root of the product of the segments, and the form of the beam, Fig. 164, is a semiellipse. It may be a complete ellipse, Fig. 165.

Figs. 164, 165.-Beams uniformly loaded.

For a rectangular beam, with a constant depth, and loaded at the middle, the form of the breadth, Fig. 166, is a double of Fig. 151, page 519; consisting of two triangles, in plan, united at their base.

Fig. 166.-Beam loaded at the middle.

Fig. 167.-Beam uniformly loaded.

When a rectangular beam, with a constant depth, is uniformly loaded; referring to the formula ( 5 ) above noticed, the variables are the breadth b and the product mn, and the breadth varies as the product of the