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CD.

component of the deflection of D. Find also the angular deflection of [B.E.] 14. A crane has the form shown in Fig. 232. A load W causes the bars to alter in length by the amounts written thereon, the minus sign denoting shortening. The joint A being fixed and the joint B being free to move horizontally, find the deflection of D and measure its vertical and horizontal components. What is the angular deflection of the bar CD, if its actual length is 100 inches? [B.E.]

+0-025"

15. Forces, not shown, acting on the given roof truss (Fig. 233), cause the

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B

FIG. 231.

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bars to alter in length by the amounts written therecn, the minus sign denoting shortening. If the end A is fixed, and B is free to move horizontally, find the displacement of B.

What is the angular displacement of the horizontal bar CD if its actual length is 110 inches? [B.E.] 16. The diagram (Fig. 234) shows a simple roof truss, resting on walls and loaded as shown, the span being 15 feet and the inclination of the rafters 39°. (a) Find and measure the horizontal thrust on the walls.

(b) Draw diagrams showing the thrust, shearing force and bending moment throughout the length of one of the rafters. Measure the maximum values of these quantities.

[B.E.]

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17. A braced arch, hinged at the crown and at each springing, is loaded as shown in Fig. 235. Determine the horizontal thrust on the abutments. Find the forces in all the members of the structure. What is the shearing force on the pin at the crown? [B.E.] 18. An arched rib in the form of a circular arc ACB, is hinged at each end A, B, and at the crown C. The span AB is 150 feet, and the rise 30 feet. Draw the curve of the arch to a scale of inch to 10 feet.

A load of 10 tons passes over the arch. Confine your attention to one position only of this load, that for which its horizontal distance from A is 50 feet. Determine and measure the horizontal thrust of the arch in tons. Draw a diagram of bending moment on the rib, and measure the maximum bending moment in ton-feet. Determine also and measure the greatest thrust and the

greatest shear in the rib, and state the horizontal distances from A of the places where these occur. [B.E.] 19. The form of an arched rib is a circular arc of 100 feet span and 16 feet rise, the supports being at the same level. It is hinged at the ends and loaded with a weight of 12 tons at a horizontal distance of 30 feet from one end. The horizontal thrust of the arch is known to be 12.5 tons.

Draw a diagram of bending moment for the arch. Indicate the places where the shearing force, thrust, and bending moment on the rib have their maximum values, and give these values.

20. A flying buttress has the form and weight shown in Fig. 236.

[B.E.]

It is sub

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ject to a thrust P as indicated. Show the lines of the forces acting across the several joints. State the magnitude of the force on the bed joint BB. Adopt a force scale of inch to 1 ton.

[B.E.] 21. ABDC, Fig. 237, represents the section of a half-arch. The span of the half-arch is 40 feet, rise 20 feet, thickness of key-stone 6 feet, thickness of archring at abutment, AB, 9 feet.

The loads on the half-arch are supposed concentrated as shown.

The dotted arcs bd and ac represent the limits of safety, and are at one-third of the thickness of the arch-ring from the extrados and intrados respectively. The horizontal thrust at the crown is assumed to be along the line dt, and the curve of thrust is further assumed to pass through the point e where the vertical through the abutment FA meets the dotted arc ac. Determine and draw the curve of thrust between the points d and e, by means of a funicular polygon, and give the value of the horizontal thrust at d.

Scale of lengths, 10 feet to an inch.
Scale of loads, 10 tons to an inch.

[B.E.]

CHAPTER IX

PLANE CO-ORDINATE GEOMETRY

110. Co-ordinates of a Point.-The position of a point P in a plane (the plane of the paper) may be fixed by giving its distances x and y from two fixed intersecting straight lines OX and OY (Figs. 238 and 239), the distances x and y being measured parallel to OX and OY respectively as shown.

The fixed lines OX and OY are called the axes. The axes are of unlimited length and each extends both ways from O. The point O where the axes intersect is called the origin. The distance x is called the abscissa and the distance y the ordinate of the point P, and these two distances together are called the co-ordinates of the point P. A point P whose co-ordinates are x and y may be referred to as the point x,y.

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When the axes are at right angles to one another (Fig. 238) x and y are the rectangular co-ordinates of the point P, and when the angle between the axes is not a right angle (Fig. 239) x and y are the oblique co-ordinates of the point P.

The abscissa x is positive (+) or negative (-) according as the point P is to the right or left respectively of the axis OY, and the ordinate y is positive or negative according as the point P is above or below the axis OX.

The position of a point P in a plane (the plane of the paper) may be fixed in the manner illustrated by Fig. 240. O is a fixed point called the pole, and OX is a fixed straight line called the initial line, or line of reference. The point P is joined to O and the line OP whose length is r is called the radius vector. The angle which OP makes with OX, measured from OX in the anti-clockwise direction is called the vectorial angle. r and are called the polar co-ordinates of

the point P, and these co-ordinates fix the position of the point P. As defined above r and are both positive. If is measured in the clockwise direction from OX then it is negative. The angle whether it is positive or negative fixes the position of the positive radius vector OP. If the radius vector is negative then produce PO to P' and make OP' equal to OP, then OP being the positive radius vector, OP' is the negative radius vector. A point P whose polar co-ordinates are r and may be referred to as the point r,0.

Rectangular co-ordinates are the most common in practical problems and when co-ordinates are referred to, without qualification, rectangular co-ordinates will be understood.

111. The Straight Line.-AB (Figs. 241 and 242) is a straight line which intersects the axis OY at C. Let OC = c. Take any point P in AB. Draw the ordinate PM and through C draw CL parallel to

PL
CL

OX to meet PM at L. Then is a measure of the slope of AB to OX, and since the slope of AB is independent of the position of the point P, But PL PMLM = PM OC = y — c,

PL

CL

= constant = m, say.

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=

-

=m, or y = mx + c, and this is the

B B

equation to the straight line AB. The meaning of this equation is that, the constants m and c being known, the co-ordinates of every point in the straight line satisfy the equation. The distance OC = c is called the intercept of AB on the axis of y. For the line shown

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A

A

M

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c = 5, hence the equa

tion to AB is y=x+5.

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It will be found that for the point Q, x 6, and y = 8. Inserting these values in the equation, 8×6+58, and the equation is satisfied. Again, for the point R, 3, and y = 3. Inserting these values in the equation, 3×3 +5=13+5= 3, and the equation is again satisfied, and so it will be found for every point in AB.

The line AB (Fig. 241) is called the graph of the equation y=x+5. Conversely if any pair of values of x and y which satisfy the equation y=x+5 be taken as the co-ordinates of a point, and this point be plotted, all such points will lie in the straight line AB which is the graph of the equation.

An equation containing the variables x and y in the first power only is called an equation of the first degree. The general form of the equation of the first degree, with two variables, is Ax + By + C = 0, where A, B, and C are constants which are finite or zero and may be positive or negative. It may be proved that the graph of an equation

K

Put

of the first degree is a straight line, and since a straight line is fixed by two points in it, the graph of such an equation may be drawn as soon as two values of x and the two corresponding values of y are known. For example, take the equation - 2x + 3y - 90. x = 0, then y = 3. Put x = 10, then y = 93. Plotting the points x = 0, y = 3, and x = 10, y = 9, and joining them determines the graph of the equation -2x+3y-9= 0.

If a straight line is parallel to OX and at a distance b from it, then for every point in the line y b, and this is the equation to the line. The distance b is positive or negative according as the line is

110

=

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above or below OX. If a straight line is parallel to OY and at a distance a from it, then for every point in the line x = a, and this is the equation to the line. The distance a is positive or negative according as the line is to the right or left of OY.

For problems in co-ordinate geometry "squared paper" will be found very convenient. Fig. 243 represents a piece of squared paper on which are drawn eleven straight lines numbered 1 to 11.

The equations to the lines shown in Fig. 243 are given below and the student should carefully study these, comparing them with their graphs. The unit used is the length of side of the small squares.

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