Elementary graphic statics

PREFACE

In lecturing on the subject of Graphic Statics to first-year students, the Author has always felt the want of a suitable text-book which could be recommended to students in search of a working knowledge of the application of graphical methods to the solution of the simpler problems met with in Engineering and Building Construction practice.

It is not given to everyone to be mathematically brilliant, but this is no reason why the young engineer should be denied the privilege of studying much in engineering which interests and appeals to him. In the solution of engineering problems, the science of Graphics presents a ready means of circumventing the many intricate and cumbrous mathematical equations which can, all too easily, clothe comparatively simple problems with an air of mystery and difficulty.

There are those who scoff at graphical methods, forgetting that there are hundreds of first-class practical engineers who are daily solving some of the most useful problems of everyday life by such methods. The work of such men is alone a sufficient justification of the existence of graphical methods in their application to engineering problems.

It only lies with the student to realize that every problem set out in the following pages must be carefully worked out and thoroughly understood. Mere reading of the subject matter is worse than useless.

The Student who conscientiously works through all the problems and examples which follow, will have acquired much that will stand him in good stead in the performance of useful engineering work.

JOHN T. WIGHT.

Framed Structures. In dealing with the more practical application of Graphics to the determination of the stresses induced in the various members of roof trusses and bridge girders by the application of external loads, we have to consider what are known as framed structures.

A Frame is a structure consisting of several bars jointed together at their ends by pins, which allow of free motion in one plane round their centres. The several bars composing a frame are known as members.

Fig. 53 shows a frame composed of four members, and it is obvious that such a contrivance could be readily adapted to carry a load W across the space S. Such an arrangement is, however, open to this objection, that, should any lateral force, for instance due to wind, be applied to it, the members BD and AC would tend to rotate about their bottom pins D and C, and would consequently displace the load. To render the frame suitable for its work under these conditions we must therefore adopt some means of preventing this motion, we must prevent this deformation by stiffening, the joints in some way and making the frame stable under all loads.

In practice, the frames used in roofs and bridges do not fulfil the conditions theoretically required of a frame, owing to the different arrangement of the parts, and hence frames used in this way are known as trusses.

Types of Frames. Frames may be divided roughly into three distinct classes

(a) Firm Frames.
(6) Deficient Frames.
(c) Redundant Frames.

Firm Frames. An example of a firm frame is shown in Fig. 54 (a). It will be noticed that such a frame possesses just sufficient members to prevent any appreciable deformation under any load in the plane of the frame, provided the members are not stressed beyond their limit of safety. It should also be noticed that any one member may be lengthened or shortened without, in any way, affecting the stresses in the other two members.

Such a frame is stable for all loads within the breaking load.

Deficient Frames. An example of a deficient frame is. shown, in Fig. 54 (b). Such a frame is not possessed of sufficient members to prevent deformation on the application of a load. If we refer to our remarks on the funicular polygon we will readily see that there will be, at least, one system of forces which will keep the members of the frame in the given shape, but should one of these forces be altered, the effect would be such that the frame, as presently outlined by the members, would no longer coincide with the link or funicular polygon, and hence an alteration in shape would occur. Such a frame would therefore not be stable under all loads, but it could be made so by the application of another member, as shown dotted. Such an addition would convert it into a firm frame, possessing stability under any system of loading, in the plane of the frame, and also possessing the further advantage that any one member may yet be lengthened or shortened without in any way affecting the stresses in the others.

Redundant Frames. An example of this type is shown in Fig. 54 (c). A redundant frame may possess one or more members more than is necessary for stability, but it is stable under all loads, although possessed of this disadvantage, that any alteration in the length of one member affects the stresses in all the others. Such a frame would therefore be affected by bad workmanship, and carelessness in marking off the lengths of the various members would result in initial stressing when the frame was fitted together. Such frames are sometimes spoken of as self -strained frames.

If n number of joints in a frame, then a perfect frame should have (2n 3) members.

Although roof and bridge trusses do not always meet the theoretical requirements of a frame, yet it is found advantageous to assume that, in the generality of cases, the theoretical conditions are satisfied. In the truss shown in Fig. 55, for example, the rafters AC and CE are all in one piece, instead of being jointed at B and D respectively, while the tie-rod, instead of being formed of two separate members AF and FE, is all in one length. The joints B, C, D and F would be securely bolted or riveted, but, for our purpose, we assume that pin-joints exist at these points, and that AB, BC and such parts are all separate members. Such assumptions admit of the frame adjusting itself, when loaded, so that all members are stressed axially. Pin-jointed structures are more in use in America than in this country, where the securely riveted or bolted joint finds most favour. Rigid joints give rise to bending stresses when the load comes on the structure, but, generally speaking, unless the structure be a very important one, such stresses are neglected.

Definitions. The primary duty of a roof truss is to provide some means of support for the roofing materials as used to exclude the weather from a building. Trusses may be composed entirely of wood or iron, or they may be combined of both materials. A few terms in common use in connection with roof trusses may be worthy of short explanations.

Roof Truss, or Principal, is the name given to the framed structure complete without any roofing material.

Pitch of the Principals is the horizontal distance P between the centre lines of the trusses, as indicated in the plan in Fig. 56.

Rafter is the name given to the parts marked R1,  R2, etc., extending from the abutment to the apex and providing support for the purlins. In the majority of cases the rafters are in compression.

Purlins are the cross roof -bearers which are attached to the rafters, the attachment being usually made by means of small pieces of angle-iron, known as cleats.

Struts and Ties. Strut is the name given to all members subjected to a compressive stress, while all members in tension are known as ties.

Pitch is the slope of the rafter measured in degrees from the horizontal. It depends on the nature of the material used in the covering. A table of weights of various roofing materials, together with minimum pitches, is given later.