The elementary principles of graphic statics - Hardy
THE ELEMENTARY PRINCIPLES OF GRAPHIC STATICS
Specially prepared for students of science and technical schools, and those entering for the examinations of the board of education in building construction, machine construction, drawing, applied mechanics, and for other similar examinations
BY EDWARD HARDY
Teacher of Building Construction;
LONDON; B. T. BATSFORD; 1904
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The elementary principles of graphic statics
PREFACE
The following chapters are placed before students of Building Construction, Applied Mechanics, and Machine Construction and Drawing, in the hope that they may be of service to those who desire aid in the study of the "Statics" branch of these subjects.
It should be stated that, in the chapter on Graphic Arithmetic, only such matter has been introduced as is deemed necessary for the study of the succeeding chapters.
The author desires to express his gratitude to Pro- fessor Henry Adams, M.I.C.E.. M.I.M.E., F.S.I., etc., for his kindness in reading through the MS., and for his valuable help and advice.
EDWARD HARDY.
CONTENTS
CHAPTER I
Graphic Arithmetic
Graphic Representation of Quantities - Advantage of Decimally-divided Scales - Addition - Subtraction - Similar Triangles - Multiplication - Division - Proportion - Examples
CHAPTER II
Force
Definition, how Measured and how Represented - Resultant - Equilibrium - Equilibrant - Parallel Forces - Reaction - Moments and how Measured - Point of Application of the Resultant of Parallel Forces - The Three Orders of Levers - Solution of Levers - Cranked or Bent Levers - Examples
CHAPTER III
Centre of Gravity
Of a Parallelogram - Of a Triangle - Of a Trapezium - Of any Quadrilateral Figure - A Door as a Lever - Bow's Notation - Load - Stress - Strain - Examples
CHAPTER IV
Non-Parallel Forces
Parallelogram of Forces - Triangle of Forces - Resolution of Forces - Inclined Plane - Bent Levers - Reaction of Door Hinges - Lean-to Roofs – Retaining Walls for Water and Earth - Polygon of Forces - Examples
CHAPTER V
Funicular Polygon
Links, Pole, and Polar Lines or Vectors - Solution of Parallel Forces - Reactions of the Supports of Framed Structures - The Load Supported by a Roof Truss, and how it is Conveyed to it - Centre of Pressure of Irregular Masses - Examples
CHAPTER VI
Graphic Solution of Bending Moment
How to Obtain the B.M. Scale - Cantilevers Loaded at Different Points - Beams with a Uniformly Distributed Load and Supported at Both Ends - How to draw a Parabola - Cantilevers with a Uniformly Distributed Load - Cantilevers and Beams Supported at Both Ends with the B.M. Diagrams for Concentrated and Uniformly Distributed Loads Combined - Shearing Force - S.F. Diagrams: for Cantilever Loaded at Different Points - for Cantilevers with Uniformly Distributed Loads - for Cantilevers with Combined Concentrated and Uniformly Distributed Loads - for Beams Supported at Both Ends with Concentrated Loads, with Uniformly Distributed Loads, and with Concentrated and Uniformly Distributed Loads Combined - Examples
CHAPTER VII
Explanation op Reciprocal or Stres3 Diagrams
Rules for Drawing Stress Diagrams - Span Roof - Couple Close - Couple Close with a King-rod Added - King-post Truss - Other Forms of Roof Trusses - Framed Cantilevers - Apportioning Distributed Loads - How to Obtain the Magnitude of the Stresses of the Members of Framed Cantilevers and Girders from the Stress Diagrams - Warren Girder with a Concentrated Load on the Top Flange – Warren Girder with a Uniformly Distributed Load on the Top Flange - Warren Girder with a Uniformly Distributed Load on the Bottom Flange - N Girder with a Uniformly Distributed Load on the Top Flange - N Girder with a Concentrated Load on the Top Flange - N Girder with Concentrated Loads on the Bottom Flange - Lattice Girder Without Verticals - Lattice Girder With Verticals - Examples
Answers to Examples
Graphic Arithmetic
Graphic Representation of Quantities - Advantage of Decimally-divided Scales - Addition - Subtraction - Similar Triangles - Multiplication - Division - Proportion - Examples
CHAPTER II
Force
Definition, how Measured and how Represented - Resultant - Equilibrium - Equilibrant - Parallel Forces - Reaction - Moments and how Measured - Point of Application of the Resultant of Parallel Forces - The Three Orders of Levers - Solution of Levers - Cranked or Bent Levers - Examples
CHAPTER III
Centre of Gravity
Of a Parallelogram - Of a Triangle - Of a Trapezium - Of any Quadrilateral Figure - A Door as a Lever - Bow's Notation - Load - Stress - Strain - Examples
CHAPTER IV
Non-Parallel Forces
Parallelogram of Forces - Triangle of Forces - Resolution of Forces - Inclined Plane - Bent Levers - Reaction of Door Hinges - Lean-to Roofs – Retaining Walls for Water and Earth - Polygon of Forces - Examples
CHAPTER V
Funicular Polygon
Links, Pole, and Polar Lines or Vectors - Solution of Parallel Forces - Reactions of the Supports of Framed Structures - The Load Supported by a Roof Truss, and how it is Conveyed to it - Centre of Pressure of Irregular Masses - Examples
CHAPTER VI
Graphic Solution of Bending Moment
How to Obtain the B.M. Scale - Cantilevers Loaded at Different Points - Beams with a Uniformly Distributed Load and Supported at Both Ends - How to draw a Parabola - Cantilevers with a Uniformly Distributed Load - Cantilevers and Beams Supported at Both Ends with the B.M. Diagrams for Concentrated and Uniformly Distributed Loads Combined - Shearing Force - S.F. Diagrams: for Cantilever Loaded at Different Points - for Cantilevers with Uniformly Distributed Loads - for Cantilevers with Combined Concentrated and Uniformly Distributed Loads - for Beams Supported at Both Ends with Concentrated Loads, with Uniformly Distributed Loads, and with Concentrated and Uniformly Distributed Loads Combined - Examples
CHAPTER VII
Explanation op Reciprocal or Stres3 Diagrams
Rules for Drawing Stress Diagrams - Span Roof - Couple Close - Couple Close with a King-rod Added - King-post Truss - Other Forms of Roof Trusses - Framed Cantilevers - Apportioning Distributed Loads - How to Obtain the Magnitude of the Stresses of the Members of Framed Cantilevers and Girders from the Stress Diagrams - Warren Girder with a Concentrated Load on the Top Flange – Warren Girder with a Uniformly Distributed Load on the Top Flange - Warren Girder with a Uniformly Distributed Load on the Bottom Flange - N Girder with a Uniformly Distributed Load on the Top Flange - N Girder with a Concentrated Load on the Top Flange - N Girder with Concentrated Loads on the Bottom Flange - Lattice Girder Without Verticals - Lattice Girder With Verticals - Examples
Answers to Examples
Chapter II - FORCE
24. Force is (a) that which tends to move a body, or (6) that which tends to stop a body when it is moving, or (c) that which tends to change the direction of a body when it is moving. In this work it is only intended to deal with force as defined in (a). No reference will be made to velocity, and only bodies which are in a state of rest relatively to neighbouring bodies will be treated upon,
25. Force is measured in units of lbs., cwts., or tons.
26. We have already seen that lbs., etc., can be represented by lines drawn to scale. Hence, if the magnitude of a force be known, a line may be drawn whose length will be proportional to the force.
27. Force must be exerted in a certain direction; the line representing it must, therefore, be drawn in that direction.
28. An arrow can be placed on the line indicating the sense of the force, that is, showing in which direction along the line the force is acting.
29. The point of application (i.e. the place at which the force is applied) must be known.
30. When the magnitude, direction, sense, and point of application of a force are known, the force is said to be known.
These four points should be clearly understood, and always kept in mind. In determining a force the student must see that he finds all four.
31. Resultant. - If a number of forces (whether parallel or otherwise) act on a body, and move it in a certain direction, it is evident that another force could be found, which, acting in that direction, would do the same work.
This force is called the resultant of the others.
32. Equilibrium. - If the resultant of a number of forces be zero, then they are said to be in equilibrium. If these forces be applied to a body in a state of rest, then it will still remain at rest.
33. Equilibrant. - When a body is not in equilibrium it moves in a certain direction with a force which has a resultant. Another force equal to the resultant in magnitude, acting in the same line and opposite in sense, would produce equilibrium.
This force is called the equilibrant.
34. The equilibrant and the resultant of a system of forces are always equal in magnitude, act in the same line of direction, and are of opposite sense.
35. If a number of forces be in equilibrium, any one of them is the equilibrant of the others, and if the sense be reversed, it will represent their resultant.
For, if a system of forces be in equilibrium, each force helps to maintain it, and the removal of any one of them would cause the remainder to move along the line on which it was acting, and in the opposite direction.
36. Parallel Forces. - Forces are said to be parallel when the lines along which they act are parallel.
Suppose forces equal to 5, 3, and 8 lbs. to be acting in one direction, and forces equal to 4, 7, and 2 lbs. in the opposite direction. In the one direction a force equal to 16 lbs. would be acting, whilst against that a force equal to 13 lbs. would be exerted. The whole system will have a resultant of 3 lbs. acting in the first direction. The 3 lbs. represents the resultant of the 6 forces, and acts in the direction of the 16 lbs.
If a force equal to 3 lbs. be added to the second set of forces, or taken from the first, then the whole system would be in equilibrium.
37. In dealing with parallel forces those acting in one direction are taken as positive, and those acting in the opposite direction as negative. The algebraical sum of all the forces will represent the resultant, and it will act in the direction of those whose sum is the greater.
39. Reaction. - When a force acts on a body it produces a resisting force from that body. This second force is always equal to and opposite to the first.
The beam with its load (Fig. 20) exerts a force on the walls, and this produces a resistance from each wall equal to the portion of the weight it has to carry. These resistances are called the reactions of the wall.
40. Moments. - It now becomes necessary to ascertain why the point of application of a force should be known. Place a book or similar object at A B C D (Fig. 21) on a table. Apply a force, as shown at P. This will cause the book to rotate clockwise, i.e. in the same direction as the hands of a clock. If the force be applied near A, it will rotate in the opposite direction, or anti-clockwise. By applying the force at different points, the student will find that to move the book forward he must apply a force in a direction which, if produced, would pass through the point G.
Again, let him take a lath, holding it horizontally by one end. Place a 1 lb. weight 1 ft. from the hand. He will find that the weight causes the lath to try to rotate with his hand as the centre of rotation, and he also experiences a difficulty in counteracting this rotation. If he moves the weight 2 ft. from the hand, he will find the tendency to rotate twice as great, and that it is twice as difficult to keep the weight in position.
25. Force is measured in units of lbs., cwts., or tons.
26. We have already seen that lbs., etc., can be represented by lines drawn to scale. Hence, if the magnitude of a force be known, a line may be drawn whose length will be proportional to the force.
27. Force must be exerted in a certain direction; the line representing it must, therefore, be drawn in that direction.
28. An arrow can be placed on the line indicating the sense of the force, that is, showing in which direction along the line the force is acting.
29. The point of application (i.e. the place at which the force is applied) must be known.
30. When the magnitude, direction, sense, and point of application of a force are known, the force is said to be known.
These four points should be clearly understood, and always kept in mind. In determining a force the student must see that he finds all four.
31. Resultant. - If a number of forces (whether parallel or otherwise) act on a body, and move it in a certain direction, it is evident that another force could be found, which, acting in that direction, would do the same work.
This force is called the resultant of the others.
32. Equilibrium. - If the resultant of a number of forces be zero, then they are said to be in equilibrium. If these forces be applied to a body in a state of rest, then it will still remain at rest.
33. Equilibrant. - When a body is not in equilibrium it moves in a certain direction with a force which has a resultant. Another force equal to the resultant in magnitude, acting in the same line and opposite in sense, would produce equilibrium.
This force is called the equilibrant.
34. The equilibrant and the resultant of a system of forces are always equal in magnitude, act in the same line of direction, and are of opposite sense.
35. If a number of forces be in equilibrium, any one of them is the equilibrant of the others, and if the sense be reversed, it will represent their resultant.
For, if a system of forces be in equilibrium, each force helps to maintain it, and the removal of any one of them would cause the remainder to move along the line on which it was acting, and in the opposite direction.
36. Parallel Forces. - Forces are said to be parallel when the lines along which they act are parallel.
Suppose forces equal to 5, 3, and 8 lbs. to be acting in one direction, and forces equal to 4, 7, and 2 lbs. in the opposite direction. In the one direction a force equal to 16 lbs. would be acting, whilst against that a force equal to 13 lbs. would be exerted. The whole system will have a resultant of 3 lbs. acting in the first direction. The 3 lbs. represents the resultant of the 6 forces, and acts in the direction of the 16 lbs.
If a force equal to 3 lbs. be added to the second set of forces, or taken from the first, then the whole system would be in equilibrium.
37. In dealing with parallel forces those acting in one direction are taken as positive, and those acting in the opposite direction as negative. The algebraical sum of all the forces will represent the resultant, and it will act in the direction of those whose sum is the greater.
39. Reaction. - When a force acts on a body it produces a resisting force from that body. This second force is always equal to and opposite to the first.
The beam with its load (Fig. 20) exerts a force on the walls, and this produces a resistance from each wall equal to the portion of the weight it has to carry. These resistances are called the reactions of the wall.
40. Moments. - It now becomes necessary to ascertain why the point of application of a force should be known. Place a book or similar object at A B C D (Fig. 21) on a table. Apply a force, as shown at P. This will cause the book to rotate clockwise, i.e. in the same direction as the hands of a clock. If the force be applied near A, it will rotate in the opposite direction, or anti-clockwise. By applying the force at different points, the student will find that to move the book forward he must apply a force in a direction which, if produced, would pass through the point G.
Again, let him take a lath, holding it horizontally by one end. Place a 1 lb. weight 1 ft. from the hand. He will find that the weight causes the lath to try to rotate with his hand as the centre of rotation, and he also experiences a difficulty in counteracting this rotation. If he moves the weight 2 ft. from the hand, he will find the tendency to rotate twice as great, and that it is twice as difficult to keep the weight in position.
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