CHAPTER VIII. THE STABILITY OF WALLS. THE general principles which regulate the relations that exist between the dimensions of a wall and the pressure it can sustain on its summit have been already discussed (Arts. 42, 43); in the present chapter we shall extend the application of the same principles to a few other cases. Several questions intimately connected with the subject of the present chapter are not discussed, as being too difficult for a purely elementary work-such are the conditions of the equilibrium of arches, vaults, domes, the more complicated forms of roofs, &c. 93. The Line of Resistance. - Let A BLM represent any structure divided into horizontal courses by the lines CD, EF, GH....and let it be subjected FIG. 121. (R2) of R1, and the weight of CDFE, which will cut EFat a determinate point c, between b' and F; in the same manner, the pressure on the joint GH will act through a determinate point d, and on LM through a pointe. Now if we join the points a, b, c, d... we shall obtain a polygonal line which cuts each joint in the point through which the direction of the resultant pressure on that joint passes; if now we suppose the number of joints to be indefinitely great, the polygonal line will become a curved line, which is then called the line of resistance. It will be remarked that the directions of the resultants do not coincide with the sides of the polygon ab, bc, and therefore the line of resistance determines only the point at which the pressure on each joint acts, not the direction of the pressure at that point. ..... The line of resistance can be determined without much difficulty in a large number of cases: when this has been done, the condition of equilibrium-so far as the tendency of the structure to turn round any of its joints is concerned -is that this line cut each joint at a point within the structure; and, of course, the stability of a structure about any joint will be greater or less according as the intersection of the line of resistance with the joint is at a greater or less distance within the surface to which it is nearest. It is plain that since the resultant of the pressures that act on a wall passes through the point of intersection of the line of resistance with its base, the algebraical sum of the moments of the pressures acting on the wall taken with respect to that point must equal zero. It may also be remarked that, in the case of most walls of ordinary shapes, the line of resistance continually approaches the extrados or outward surface; and hence, if the wall possess a certain degree of stability with reference to its lowest joint, it will possess a greater degree of stability with reference to any higher joint; most of the following questions can, accordingly, be solved without the actual determination of the line of resistance. FIG. 122. A X Ex. 433. A wall of Portland stone 30 ft. high and 2 ft. thick has to sustain on each foot of its length a thrust equal to the weight of 3 cubic feet of stone acting in a direction inclined to the vertical at an angle of 45°. Find the point of a bracket to which this force must be applied that the line of resistance may cut the base 6 in. within the extrados. [Let the annexed figure represent a section of the wall; let the force act along the line XN, and let AX equal x; take BQ equal to 6 inches; then the condition of equilibrium is that the moments of the force and of the weight of the wall round be equal. Draw QN perpendicular to XN; it can be easily shown that N QN=AC COS AXN-QC sin AXN-AX sin AXN B C It may be remarked that the determination of a perpendicular resembling QN occurs in many of the following questions. It may also be added that it is sometimes convenient to resolve the thrust into its horizontal and vertical components at x and obtain the moment of each.] Ex. 434. Determine the point of application of the thrust in the last article if the line of resistance cut the base 3 in. within the extrados. Ans. 7.04 ft. Ex. 435. A roof, whose average weight is 20 lbs. per square foot, is 40 ft. in span and has a pitch of 30°, i. e. the rafters make an angle of 30° with the horizon; the walls of the building are of brickwork, and are 50 ft. high and 2 ft. thick; they are supported by triangular buttresses reaching to the top of the wall; the buttresses are 2 ft. wide, and 20 ft. apart from centre to centre. Determine their thickness at the bottom that the line of resistance may fall 6 in. within their extrados: determine also the answer that results from neglecting the weight of the buttress. Ans. (1) 1·1675 ft. (2) 1.1754 ft. Ex. 436. A roof weighing 20 lbs. per square foot has a pitch of 60°; the distance between the walls that support it is 30 ft.; they are of Portland stone and are 2 ft. thick; the pressure of the roof being received on the inner edge of the summit, what is the extreme height to which the walls can be built? Ans. The wall can be carried to any height whatever. Ex. 437. If the weight of each square foot of a roof is 15 lbs., its pitch 22°, and the length of the rafters 30 ft., determine (1) the thrust along the rafters, supposing them to be 4 ft. apart; (2) the strain upon the tiebeam if one is introduced; (3) the magnitude and direction of the pressure on each foot of the length of the wall-plate,* if there is no tie-beam; (4) the thickness of the wall, which is of brickwork and 20 ft. high, when the line of resistance cuts the base 2 in. within the extrados, the pressure of the roof being received on the inner edge of the summit; (5) the distance from the axis of the wall at which the pressure of the roof must act if the line of resistance cuts the base of the wall 3 in. within the extrados. Ans. (1) 2352 lbs. (2) 2173 lbs. (3) 705 lbs. at an angle of (4) 3 ft. (5) 2.7 ft. Ex. 438. If w is the weight supported by each rafter of an isosceles roof whose pitch is a, show that the thrust on each rafter is W 2 sin a and the 94. The Pressure produced against a Wall by Water. -The following construction can be easily proved from of a prism of water whose base is CBF and height the length of the wall; or, in other words, the pressure on each foot of the length of the wall will be the weight of as many cubic feet of water as the triangle BCF contains square * The wall-plate is the beam on which the feet of the rafters rest: its office is to distribute the pressure along the wall. feet; this pressure will act perpendicularly to the face of the wall through a point p, where BP= Bс. Ex. 439. There is a wall supporting the pressure of water against its vertical face; determine the pressure produced by the water on each foot of its length when 20 ft. of its height are covered. Ans. 12,500 lbs. Ex. 440. In the last case determine the pressure on the lower 10 ft. of the wall. Ans. 9375 lbs. Ex. 441. An embankment of brickwork has a section whose form is a right-angled triangle ABC; the base BC is 6 ft. long; the height AB is 14 ft.; will the embankment be overthrown when the water reaches to the top, if AB is the face which receives the pressure? Ans. Yes; the excess of the moment of pressure of water is 9767. Ex. 442. In the last case will the embankment be overthrown if AC is the face which receives the pressure? Ans. Yes; excess of moment of weight of water 8675. Ex. 443. In Ex. 441 what horizontal pressure applied at a would keep the embankment steady? Ans. 698 lbs. Ex. 444. If the section of a river wall of brickwork have the form shown in the accompanying diagram, in which AB=5 ft., DC=15 ft., and Bc equals 50 ft.; Bc being vertical, and the angles B and c right angles, find the height to which the water must rise against BC to overturn it. Ans. 37.2 ft. Ex. 445. If in the last Example the dimensions were Bc equal to 30 ft., AB equal to 3 ft., and DC equal to 10 ft., would the wall be overthrown if the water rose to the summit? Ans. Yes. FIG. 124. A B C Ex. 446. There is a cofferdam sustaining a pressure of 26 ft. of water, supported by props 20 ft. long, D 20 ft. apart, one end of each is placed below the surface of the water and the other end on the ground; determine the thrust on each prop. Ans. 468,800 lbs. Ex. 447. If the section of an embankment of brickwork were of the form shown in fig. 124, and the dimensions were AB equal to 4 ft., DC equal to 12 ft., and Bc equal to 24 ft., would it support the water when it rises to the top and presses on the face AD? Ans. Yes; excess of moment of weight of wall 5184. Ex. 448. If the coefficient of friction between the courses of brickwork in the last Example be 0.75, will the wall slide on its lowest section? Ans. No; defect of horizontal pressure 2628 lbs. Ex. 449.-In Ex. 446 what vertical pressure must by some means be supplied that equilibrium may be possible? Ans. 203,100 lbs. |