CHAPTER XVII

SECTIONS OF SOLIDS

207. Sections of Solids.-Many objects of which mechanical drawings have to be made are of such a form that their construction is not completely shown by outside views only. The construction of the interior of a house, for instance, cannot be seen from the outside. In order to exhibit the interior of such an object it is imagined to be cut in pieces by planes and these pieces are then represented separately. But in representing an object of comparatively simple form, the addition of a sectional drawing of it often adds very much to the illustration of it, although such a sectional drawing may not be absolutely necessary for the complete representation of the object.

That surface which is produced when a plane cuts a solid is called a section, and if that part of the solid which is below or behind the cutting plane is also shown on the projection of a section the projection is called a sectional plan or a sectional elevation. But in the application of these terms to architectural and engineering drawings the term section is often used in the same sense as sectional plan or sectional elevation.

The projection of a section is distinguished in various ways, one way, used in this work, is by drawing across it parallel diagonal lines at equal distances apart. These lines are called section lines.

If the true form of a section is required it must be projected on a plane parallel to that of the section.

In this chapter sections of solids having plane faces are considered. Sections of the sphere, cylinder and cone are considered in chapter XVIII.

208. Section of a Prism by a Plane perpendicular to one of the Planes of Projection.-A plane section of a prism will be a rectilineal figure whose angular points are at the points where the plane of section cuts the edges of the prism. Since the plane of section is perpendicular to one of the planes of projection one of the projections of the section will be a straight line coinciding with the trace of the plane of section on the plane of projection to which the plane of section is perpendicular.

The determination of sections of a right square prism is illustrated by Fig. 471. The plan (1) and the elevation (2) represent the prism

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when standing with one end on the horizontal plane. PQ is the horizontal trace of a vertical plane of section. This plane of section cuts two of the vertical faces

of the prism in vertical lines of which the points a and b are the plans, and a'a' and b'b' perpendiculars to XY, the elevations. The same plane intersects the ends of the prism in horizontal straight lines of which ab is the plan and a'b' and a'b' the elevations. The rectangle a'b'b'a' is the complete elevation of the section.

RS is the vertical trace of another plane of section. This second plane of section is perpendicular to the vertical plane

(3)

FIG. 471.

of projection. On the plan (1) this second section appears as the figure cdec. Another plan (3) of this second section is shown on a ground line XY parallel to RS. The plan (3) shows the true form of the section of the prism by the plane RS. The whole solid is also shown in plan (3), the part above the section by the plane RS being represented by thin dotted lines

209. Section of a Pyramid by a Plane perpendicular to one of the Planes of Projection.-The first paragraph of the preceding Article on the section

of a prism applies also to the section of a pyramid, and all that need be done here is to give an example which the student should work out.

In Fig. 472, (1) is the plan and (2) the elevation of a right hexagonal pyramid when standing on the horizontal plane with two sides at right angles to XY. (3) is a plan of the pyramid when one triangular face is horizontal. The plan (3) is projected from the elevation (2) as shown. (4) is an elevation projected from (3) on a ground line X,Y, parallel to XY. The straight line PQ is the horizontal trace of a plane

of section which is perpendicular to the plane of the plan (3). PQ is parallel to XY and passes through e, the plan (3) of one angular point of the base of the pyramid. The section of the pyramid by the

plane PQ is shown projected on to each of the other views of the pyramid, and in each view the part of the solid between the vertex and the section is shown by thin dotted lines. The side of the hexagonal base may be taken 1.25 inches long.

210. Sections of Mouldings.-A moulding is an ornament of uniform cross section which may be formed on a piece of a structure or it may be a separate piece attached to the structure for ornament only, or it may be the main part of the structure as in a picture frame. Mouldings are of frequent occurrence in cabinet making, joinery, and

masonry.

The principal problem in the geometry of mouldings is the determination of the cross section of one moulding which will mitre correctly

with another when faces of the one are in different planes from corresponding faces of the other.

Given the cross section of a moulding the determination of any other section is a simple problem and is worked exactly as for a prism. In Fig. 473, (1) is the true form of the cross section of a straight piece of moulding. (2) is a plan of the moulding, the long edges being horizontal. PQ is the horizontal trace of a vertical plane of section, and (3) is the elevation of the section PQ on a ground line X,Y, parallel to PQ.

The same moulding is shown in Fig. 474. (1) is the cross section, (2) is a side elevation, and (3) is a plan when the long edges are parallel to the vertical plane of projection but inclined to the horizontal plane. RS is the horizontal trace of a vertical plane of section. a'b' is the projection of the section RS on the plane of the elevation (2), and a 'b' is a projection of the same section on a plane parallel to it and which therefore shows the true form of the section.

In Fig. 474 the projections (1), (2), (3), and (4) have been drawn in the order in which they are numbered, but a study of the figure will show that if the section a,b,' by the plane RS be given instead of the cross section (1) the latter may be found by working backwards.

In Fig. 475 the section a' is the cross section and a is the plan of a straight piece of moulding whose long edges are horizontal and which is fixed to the face of a vertical wall whose plan is rs. b is the plan and b' the elevation of another straight piece of moulding whose long edges are inclined to the horizontal plane and which is fixed to the face of a vertical wall whose plan is st. The faces of the two walls are at right angles to one another, and the two mouldings intersect in a vertical plane, whose horizontal trace, or plan, is the straight line ns which bisects the angle between sr and st. A moulding such as bb'

whose long edges are inclined to the horizontal is called a raking moulding.

The problem is now to find the true form of the cross section of the raking moulding. The condition which the raking moulding must satisfy is that its section by the plane ns of the joint with the other moulding must be the same as the section of the other moulding by that plane. From this condition it follows that the corresponding longitudinal edges or lines on the two mouldings must intersect in the plane of the joint. The projections of the longitudinal lines on the raking moulding may therefore now be drawn and the cross section c' determined as shown.

The case illustrated by Fig. 476 differs from that illustrated by Fig. 475 in that the angle rst between the faces of the walls in Fig. 476 is greater than a right angle. a, the cross section of the horizontal

moulding, is shown rabatted into the vertical plane of projection. It will be seen that in order that the elevations of the longitudinal lines on the raking moulding may be drawn the elevation n's' of the joint must first be determined.

Two straight mouldings, of the same cross section, may be mitred correctly together if they are attached to a plane surface, or if they are attached to plane surfaces inclined to one another, provided that the longitudinal lines of the two mouldings are perpendicular to the line of intersection of the plane surfaces to which they are attached.

Two curved mouldings of the same cross section but of different curvatures and attached to a plane surface may be jointed to

FIG. 477.

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gether but the joint is a curved surface. Fig. 477 shows a curved moulding A jointed to a straight moulding B both having the same cross section C. The two mouldings are supposed to be attached to the plane of the paper. The projection of the joint on the plane of the paper is the curved line DEF determined as shown.

211. Geometry of Rafters.-Practical problems on the sections of prisms occur in timber construction, and illustrations will now be given of these problems as applied to a timber roof frame.

Fig. 478 shows plan and elevation line diagrams of part of what is called a hipped roof. The various members of the frame of this roof are mainly of rectangular cross section. vu, v'u' is the ridge piece. vr, v'r' is the hip or hip rafter which passes from one end of the ridge piece to the top corner of two intersecting walls. cd, c'd' is one of the jack rafters which lie between the hip rafter and the wall plate on the top of one of the walls. ef, e'f' is one of the common rafters which lie between the ridge piece and a wall plate.

The dihedral angle between the two roof surfaces intersecting in the line VR is determined by the construction of Art. 191, p. 224, or by that construction slightly modified as shown here. vv is drawn at right angles to vr and is made equal to the height of 'above r'. Joining v'r determines the true length of VR. mn is drawn at right angles to vr to meet the horizontal traces of the roof surfaces at the level of R at m and n. mn is the horizontal trace of a plane at right angles to VR. This plane intersects the roof surfaces whose line of intersection is VR in two straight lines the angle between which is the angle required. 08' is drawn at right angles to vi'r and oS is made equal to os'. Joining S to m and n determines mSn the dihedral angle required.

At h and k are shown two forms of the cross-section of the hip rafter to an enlarged scale. TP and TQ are parallel to Sm and Sn respectively. This determines the angle which the carpenter requires