# Elementary statics of two and three dimensions

ELEMENTARY STATICS OF TWO AND THREE DIMENSIONS

BY R. J. A. BARNARD
PROFESSOR OF MATHEMATICS, ROYAL MILITARY COLLEGE OF AUSTRALIA

MACMILLAN AND CO.; LONDON; 1921

Elementary statics of two and three dimensions

PREFACE

The reception of my Elementary Dynamics has encouraged me to write a Statics of a similar standard of difficulty, and I have followed the same principles as in that book. With reference to the contents I may mention the following:

-    The introduction of examples on friction early, in dealing with forces at a point,
-    The insertion of a chapter on Shearing Stress and Bending Moments.
-    The use of the methods of the elementary differential calculus in the chapter on Virtual Work, and of the integral calculus for finding the centres of mass of a number of geometrical areas and volumes.
-    Three-Dimensional Statics are dealt with more fully than usual,
-     A chapter on Vectors in Space, including vector products, has been included.

It is hoped that the last-named chapter, with the chapter on vectors in my Dynamics, will make a satisfactory introduction to Vector Algebra and its applications.

As in my Dynamics, a number of the more difficult examples come from papers set at the Melbourne University. Nearly all the others have been constructed by myself for the book or for my classes.

R. J. A. BARNARD., 1921.

CONTENTS

CHAPTER
I. FOUNDATIONS OF MECHANICS
II. FORCES AT A POINT
III. PARALLEL FORCES
IV. WORK. SIMPLE MACHINES
V. COUPLES. ANY FORCES IN A PLANE
VI. CONNECTED BODIES
VII. WEIGHTLESS RODS, FRAMES. GRAPHICAL STATICS OF WEIGHTLESS FRAMES
VIII. SHEARING STRESS AND BENDING MOMENT
IX. MISCELLANEOUS QUESTIONS
X. CENTRES OF MASS
XI. VIRTUAL WORK
XII. GRAPHICAL STATICS
XIII. FORCES AT A POINT IN THREE DIMENSIONS
XIV. ANY FORCES IN THREE DIMENSIONS
XV. VECTORS IN SPACK
APPENDIX

CHAPTER VIII - SHEARING STRESS AND BENDING MOMENT

It has been seen that in a stretched string or in a rod, if we consider any section, there is a stress exerted between the two portions on each side of the section, so that the portion, say on the right of the section, exerts a force on the portion on the left, and the portion on the left exerts an equal and opposite force on the portion on the right. In the case of the string and the weightless frameworks this mutual stress consists simply of a tension or thrust in the direction of the length.

But in other cases, when the external forces acting on the rod are not tensions or thrusts in the direction of the length, the stresses brought into play at each section will also be more complicated. The same principle is applied as to the string in Art. 7, namely that if we take any portion of the rod, it will be in equilibrium under the stresses exerted on its ends by the adjoining portions of the rod, and any external forces such as its weight which may act on it.

Thus, if we consider a plane section AB of a rectangular bar perpendicular to its length, we see that at any element P of the section there will be equal and opposite forces acting on the portions of the bar on each side of the section. If we confine ourselves to the forces acting on one side (which we will call the right hand) of the section, the force at P may be resolved into two components, one perpendicular to the section, and the other in it.

The forces perpendicular to the section on the different elements, like P, can be compounded into a single force T at acme point, which again may be replaced by a force T at the centre of the section and a couple M. Of course either T or M may be zero.

T is the tension or thrust, as the case may be.

M is called the Bending Moment.

There will, of course, be the equal and opposite force T and couple M acting on the left hand portion of the bar.

Similarly, if we take component forces in the section, these can also be compounded into a single force S acting at the centre of the section and a couple N.

S is called the Shearing Stress.

N is called the Torsion Couple.

As before, there are an equal and opposite force and couple S and N acting on the left-hand portion.

We will not deal with any questions involving a torsion couple, as they do not occur in ordinary statical questions on frameworks, but a knowledge both of the shearing stress and bending moment, as well as of the tension or thrust, is important in deciding on the size and shape of cross section of a bar in a framework designed for any particular purpose.

Let us now consider one or two very simple cases to illustrate the method of obtaining the shearing stress and bending moment.