# Elementary graphic statics

ELEMENTARY GRAPHIC STATICS

BY JOHN T. WIGHT,
Lecturer in Machine Design and Prime Movers, Heriot-Watt College, Edinburgh

Honours Medallist and Prizeman in Mechanical Engineering
(City and Guilds of London Instititte}

WHITTAKER & CO.; LONDON; 1913

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.

CONTENTS

CHAPTER I
INTRODUCTION
Definition and Specification of a Force
Graphic Representation of a Force
Coplanar Forces
Concurrent and Non-concurrent Forces
Composition and Resolution of Forces
Resultant
Equilibrant Units

CHAPTER II
COMPOSITION AND RESOLUTION OP FORCES
Forces Acting in the same Straight Line
Resultant of Two Concurrent Forces
Parallelogram of Forces
Triangle of Forces
Polygon of Forces
Examples Worked Out
Bow's Notation

CHAPTER III
SIMPLE PRACTICAL PROBLEMS
Triangular Frame
Shear Legs
Tripod
Simple Crane
Simple Crane with Lifting Chain
Rotating Crane
Warehouse Crane

CHAPTER IV
COMPOSITION OP NON-CONCURRENT FORCES
Funicular Polygon
Resultant of Non-concurrent Forces
Resultant of Like Parallel Forces
Resultant of Unlike Parallel Forces
Moments
Graphic Representation of Moments
Moment of a System of Non-parallel Forces
Moment of a System of Parallel Forces
Couples

CHAPTER V
BENDING MOMENT AND SHEARING FORCE DIAGRAMS
Definitions
Construction of Parabola
Supported Beam (Concentrated Load)
Supported Beam (Distributed Load)
Scales of B.M. and S.F.
Overhung Beam (Concentrated Loads)

CHAPTER VI
BEAMS WITH ROLLING LOADS
Single Concentrated Rolling Load
Uniformly Distributed Rolling Load

CHAPTER VII
Frames
Definitions
Types of Roofs
Plain Rafters without Tie-rod
Plain Rafters with Tie-rod
Simple King-rod Truss
Simple Swiss Truss
Compound Swiss Truss
Compound Swiss Truss (Right-angled Struts)
Simple King-rod Truss (Single Struts)
King-rod Truss (Single Struts and Inclined Ties)
Belgian Truss
English Truss
Simple Queen post Truss
Compound Queen-post Truss
French Truss
Mansard Truss

CHAPTER VIII
Northern Lights Roof
Overhung Roof
Overhung Roof with Pillar
Island Station Roof

CHAPTER IX
ROOFS WIND PRESSURE
Calculation of Wind Pressure
Experimental Results
Maximum Wind Pressure
Determination of Stress due to Wind
Comparison of Methods
Roof with Free End
Saw-tooth or Northern Lights Roof
Island Station Roof

CHAPTER X
BRACED BEAMS AND GIRDERS
Braced Beam (Single Stanchion)
Braced Beam (Double Stanchion)
Trapezoidal Truss
Fink Truss
Bollman Truss
Warren Girder L
inville Girder
Pratt Truss
Lattice Girders
Cantilever Pier

CHAPTER XI
CENTRE OF GRAVITY - NEUTRAL AXIS - RESISTANCE FIGURES - MOMENTS OF
Gravity
Experimental Method of finding the C.G.
Centre of Gravity of a Triangle
Centre of Gravity of a Parallelogram
Centre of Gravity of Irregular Surfaces
Funicular Polygon Method
Cast-iron Beam Section
Neutral Axis
Resistance Figures
Modulus of Section
Moment of Inertia
Moment of Inertia of a Particle
Moment of Inertia of a System of Forces
Mohr's Method
Moment of Inertia of Ferro-concrete Section

CHAPTER XII
RETAINING WALLS
Definitions
Simple Case of Water Pressure
Example
Earth Pressure
Rankine's Theory
Coulomb's Theory
Example
Rebhann's Method

CHAPTER VII

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.