A text book on graphic statics

A TEXT BOOK ON GRAPHIC STATICS

BY CHARLES W. MALCOLM, C. E.,
Assistant Professor of Structural Engineering, University of Illinois; Associate Member American Society of Civil Engineers; Member Society for Promotion of Engineering Education.

NEW YORK AND CHICAGO;THE MYRON C. CLARK PUBLISHING CO.; 1911

A text book on graphic statics

PREFACE.

This text was prepared for the author's students in elementary Graphic Statics and Stresses. It has not been the object of the writer to discover new principles, but rather to present the subject clearly and logically. Many texts on Graphic Statics are open to the criticism that they have a tendency to state principles and give constructions without proofs, and the student is therefore compelled to memorize propositions and constructions without being taught the underlying principles. It has been the aim of the writer to give the proofs, together with full explanations of the constructions. No attempt has been made to give elaborate solutions which have little or no practical applications. Particular attention has been given to the order of presentation; and the text has been divided into chapters, articles, and numbered sections to facilitate easy reference.

Most of the material in Part I and Part II has been used in printed form by the author's students for several years. It is hoped that the material in Part II, Part III, and Part IV will be found of assistance to the practicing engineer.

CONTENTS.

PART I. GENERAL PRINCIPLES.

CHAPTER I. DEFINITIONS.

CHAPTER II, CONCURRENT FORCES.

CHAPTER III. NON-CONCURRENT FORCES.
Art. 1. Composition of Non-concurrent Forces
Art. 2. Resolution of Non-concurrent Forces
Art. 3. Equilibrium of Non-concurrent Forces
Art. 4. SPECLA.L Constructions for Funicular Polygons

CHAPTER IV. MOMENTS.
Art. 1. Moments of Forces and of Couples
Art. 2. Graphic Moments

CHAPTER V. CENTER OF GRAVITY OF AREAS.

CHAPTER VI. MOMENT OF INERTIA.
Art. 1. Moment of Inertia of Parallel Forces
Art. 2. Moment of Inertia of Areas

PART II FRAMED STRUCTURES - ROOF TRUSSES.

CHAPTER VII. DEFINITIONS.

CHAPTER IX. REACTIONS.
Art. 2. Reactions for Wind Loads

CHAPTER X. STRESSES IN ROOF TRUSSES.
Art. 1. Definitions and General Methods for Determining Stresses
Art. 2. Stresses by Algebraic Moments
Art. 3. Stresses by Graphic Moments
Art. 4. Stresses by Algebraic Resolution
Art. 5. Stresses by Graphic Resolution

Art. 1. Both Ends of Truss Fixed - Reactions Parallel
Art. 2. Both Ends of Truss Fixed - Horizontal Components of Reactions Equal
Art. 3. Leeward End of Truss on Rollers
Art. 4. Windward End op Truss on Rollers

CHAPTER XII. STRESSES IN CANTILEVER AND UNSYMMETRICAL TRUSSES – MAXIMUM STRESSES.
Art. 1. Stresses in Cantilever and Unsymmetrical Trusses
Art. 2. Maximum Stresses

CHAPTER XIII. COUNTERBRACING.
Art. 1. Definitions AND Notation
Art. 2. Stresses in Trusses with Counterbracing - Separate Stress Diagrams
Art. 3. Stresses in Trusses with Counterbracing - Combined Stress Diagram

CHAPTER XIV. THREE-HINGED ARCH.

CHAPTER XV. STRESSES IN A TRANSVERSE BENT OF A BUILDING.

CHAPTER XVI. MISCELLANEOUS PROBLEMS.

PART III. BEAMS.

CHAPTER XVII. BENDING MOMENTS, SHEARS, AND DEFLECTIONS IN BEAMS FOR FIXED LOADS.
Art. 1. Bending Moments and Shears in Cantilever, Simple and Overhanging Beams
Art. 2. Graphic Method for Determining Deflections in Beams
Art. 3. Bending Moments, Shears, and Deflections in Restrained Beams

CHAPTER XVIII. MAXIMUM BENDING MOMENTS AND SHEARS IN BEAMS FOR MOVING LOADS.

PART IV. BRIDGES.

CHAPTER XIX. TYPES OF BRIDGE TRUSSES.

CHAPTER XXI STRESSES IN TRUSSES DUE TO UNIFORM LOADS.
Art. 1. Stresses in a Warren Truss by Graphic Resolution
Art. 2. Stresses in a Pratt Truss by Graphic Resolution
Art. 3. Stresses by Graphic Moments and Shears
Art. 4. Stresses in a Bowstring Truss - Triangular Web Bracing.
Art. 5. Stresses in a Parabolic Bowstring Truss
Art. 6. Wind Load Stresses in Lateral Systems
Art. 7. Stresses in Trusses with Parallel Chords by the Method of Coefficients

CHAPTER XXII. INFLUENCE DIAGRAMS, AND POSITIONS OF ENGINE AND TRAIN LOADS FOR MAXIMUM MOMENTS, SHEARS, AND STRESSES.

CHAPTER XXIII. MAXIMUM MOMENTS, SHEARS, AND STRESSES DUE TO ENGINE AND TRAIN LOADS.
Art. 1. Maximum Moments, Shears, and Stresses in Any Particular Girder or Truss
Art. 2. Maximum Moments, Shears, and Stresses in Girders and Trusses op Various Types and Spans

PART IV - BRIDGES

CHAPTER XIX - TYPES OF BRIDGE TRUSSES

Bridge trusses are comparatively recent structures, the ancient bridges being pile trestles or arches. Somewhat later, a combination of arch and truss was used, although the principles governing the design were not understood. It was not until 1847 that the stresses in bridge trusses were fully analyzed, although trusses were constructed according to the judgment of the builder before this date. In 1847, Squire Whipple issued a book upon bridge building, and he was the first to correctly analyze the stresses in a truss. Soon afterward, the solution of stresses became very generally understood, wooden trusses were discarded for iron ones, and still later, steel replaced iron as a bridge-truss material. From this time, the development of bridge building was very rapid, culminating in its present high state of efficiency.

Through and Deck Bridges. Bridges may be grouped into two general classes, viz.: through bridges and deck bridges.

A through bridge is one in which the floor is supported at, or near, the plane of the lower chords of the trusses (see Fig. 114, e). The traffic moves through the space between the two trusses. Except in the case of a pony truss (one in which there is no overhead bracing), a system of overhead lateral bracing is used.

A deck bridge is one in which the floor is supported directly upon the upper chords of the trusses. In this type, the trusses are below the floor (see Fig. 114, d).

161. Types of Bridge Trusses. In Fig. 114 are shown several types of bridge trusses that have been very generally used.

Fig. 1 14, a shows a Warren truss. This truss is still used for short spans, but has the disadvantage that the intermediate web members are subjected to reversals of stress.

Fig. 114, b shows a Howe truss. This truss was in favor when wood was extensively used as a building material for trusses, but is little used at present. It has the disadvantage of having long compression web members.

Fig. 114, c shows a Pratt truss. This form of truss is extensively used, both for highway and railroad bridges, up to about a 2OO-foot span. It is economical and permits of good details.

Fig. 114, d shows a Baltimore deck truss, and Fig. 114, e, a Baltimore through truss. These trusses are used for comparatively long spans, and have short compression members.

Fig. 114, f shows a Whipple truss. This truss was quite extensively used, but is now seldom employed. It is a double intersection truss, and has a redundancy of web members. The stresses are indeterminate by ordinary graphic methods.

Fig. 114, g shows a Camels-Back truss. This truss is used both for short and long spans.

Fig. 114, h and Fig. 114, i show Parabolic Bowstring trusses. The upper chord panel points are on the arc of a parabola. A great disadvantage of these types is that the upper chord changes direction at each panel point and that the web members change both their angle of inclination and length at the panel points.

This type is sometimes modified by placing the panel points on the arc of a circle.

Fig. 114, shows a Petit truss. This truss is quite extensively used for long spans, and is economical.

Members of a Truss. The general arrangement of members is given in the through Pratt railroad truss shown in Fig. 115. The arrangement of members in the various types of trusses is somewhat similar to that shown. In this truss, the tension members are shown by light lines, and the compression members by heavy lines.

Main Trusses. Each truss consists of a top chord, a bottom chord, an end post, and web members. The web members may be further subdivided into hip-verticals, intermediate posts, and diagonals. The diagonals may be divided into main members and counters, the main members being those stressed under a dead load, and the counters those stressed only under a live load.

Lateral Bracing. The bracing in the plane of the upper chord (Fig. 115) is called the top lateral bracing; and that in the plane of the lower chord, the bottom lateral bracing. The members of the lateral systems are stressed by wind loads and by the vibrations due to live loads. The top lateral system is com- posed of top lateral struts and ties. The floor beams act as the struts in the lower lateral system.

Portals. In through bridges, the trusses are held in position and the bridge made rigid by a system of bracing in the planes of the end posts. This system of bracing is called the portal bracing, or portal.

Knee-braces and Sway Bracing. The braces connecting the top lateral struts and intermediate posts (see Fig. 115), in the plane of the intermediate posts, are called knee-braces. When greater rigidity is required, a system of bracing somewhat similar to the portal bracing is used instead of the knee-braces. This bracing is called the sway bracing. The top lateral strut is also the top strut of the sway bracing. Knee-braces and sway bracing are often omitted on small span highway bridges.

Floor System. The floor systems of ordinary highway bridges differ considerably from those of railroad bridges. Both types, however, have cross-beams running from one hip vertical or intermediate post to the opposite one. These beams are called floor beams. The beams at the ends of the bridge are called the end floor beams, and those at the intermediate posts, intermediate floor beams. The end floor beams are usually omitted in highway bridges, and an end strut, or joist raiser, is substituted.

In railroad bridges, there are beams which run parallel to the chords and are connected at their ends to the floor beams. These beams are called stringers.

In highway bridges, there are several lines of beams which run parallel to the chords and which rest upon the floor beams. These beams are called joists.

In railroad bridges, the ties, which support the rails, rest directly upon the stringers ; and in highway bridges, the floor surface is supported directly by the joists.

Pedestals. The supports for the ends of the trusses are called pedestals. For spans over about 70 feet, the pedestals at one end of the bridge are provided with rollers, to allow for expansion and contraction.

Connections. The members of the truss may be either riveted together or connected by pins. In the former case, the truss is said to be a riveted truss, and in the latter, a pin-connected truss. Riveted trusses are often used for short spans and are very rigid. Pin-connected trusses are easy to erect, and are used for both short and long spans.