il in =f (7) 3 -cARO MO VT EA 3 1293 10612 2777 Michigan State University MSU RETURNING MATERIALS: Place in book drop to LIBRARIES remove this checkout from A your record. FINES will ee ee eee _—- 6x 3.25 = 19.25 8q- ft. (3 x 6) $2 9.9 . Total area of ITI 58.5 e 19.5 x 1.625 = 31.69 9x 4.25 = 38.25 aot 69.94 $ 26.5 © 2.45 ft. = 9xis SE 19.5 x 3 = 88.5 04.5 225.52 ft. 22 Then C.G. of Sec. II is at 2.46° al where the xx° axis es dg and the yy* axis £,P3 Then next step is to combine the centers of gravity of the two parbs I and II and thus get ©: Te the C.C. of the whole section showm in Fig. I. Area tft <= 173.36 * II = 28.50 BOI.eo 173.2 x 4.96 5 859.0 268.5 x (17 .0=-2.45) = 414.9 1273.9 % 201.86 = 6.53 Fte go Deing xx° axis 86.60 x 7.88* = 1366.0 28.5 2x3.-S1 @ 94.38 1460.5 2 2017 = 7.26 ft. 2 Z where be is the yy® exis. Thus the C.G. of the total section is mt 7.26, 6.556 OVERTURWING MOMENT OF DAM. Assuming the dam is unable to slide (as it mist be), the pressure of the water against the up- stream face tends to rotate the cam about the point &, in a counter-clockwise direction. The weight of the dam, acts downward thru its center of gravity andtends to hold the dam in place, that is, it exerts @ clockwise direction, moment Let ¥ = total water pressure on upstream face. P = wsf where Bg the wt. of water per cue. ft. * 72 = distance to the C.G. of the area. * F = the area which water is azainst. In this case 1* along the dem is teken. Thus: wf Ps = 64.5 x 9.8 zx 19 =~ ll, 642.25 lbs To take the most extreme case, no counter- acting tall water pressure has been used. Let W = the weight of a portion of the cam 1* al its length. Assuming masonry as wei 150¢s / cuefte3 7 = 201.86 x 1x 150 = 350,255/8. W.=- the effective wt. of the dam. If the footings unfer the line gc do not cut off all see page the dam will be beuyed up by the amount of the water displaced, which will extend up to the height of the tell water. ‘This amount must be subtracted 8. from the wt. of the dam, to give the force effective to prevent or counteract overturning. Then W,> 30,255 -(5 x 17 ¢ 3.25 © 5.75)150 - 3 30, 255-5 ,485 B ett2 #8 Then taking moments about g3 Resisting W. x 17-7.26 24,772 x 9.74 240,500 ft. peunds. Overturning mom. g P x 6.333 211,642 x 6.338 @75,700 foot pounds. Thus in the most extreme csse, as taken, the moment to prevent overturning, is more than 3 times the moment which would tend to cause 23.28 (38) Then Factor of Safety @ 240,500: or 5.26 Gdoviously the dam is amply safe against overturning fArr TY AGAINST SLIDING. moment Case XY. At somes horisontal plane between the base and the crest. It can be seen from Fig.l that below elevation 85* the tail water would tend to keep the dam trom shearing off, so cvidently somewhere above here lies the most dangerous sectien. 1 above here at elevation 86" the dam has a much emaller horizontal cross section, while the pressure 4s nearly as great as at 85°. Let S @ the shesring stress, caused by water pressure down to elev. 86°. P e WZF | s §452.2#8 = Se Let R = the foree which will tend to prevent horizontal shear. R = U(F.)d Where Us the coefficient of friction. With masonry on masonry U- .65 With masonry on rock 0 ge -50 Then KR s 065(10.85 9 4.5 x 15)150 x 1 9,816 To prevent failure by shear k gust be greater than S. 9,810 is greater than 5452.2 Hence the dam is safe at this most precarious secticne SAFELY AGAINST SLIDING Case II. At the base. fhe extreme case has been used, that the footings below go heve already been sheer, or at least not in the same integral mass as the rest of the dam. Also no tail pressure has been fizured, which makes it the rare case, where no water has been allowed to escepe by the dem for some tine. (Never has occured). . . . ““e E42 = 28.7 772 x 5 ” 42° "386 #8 R is greater than s, hence dam will not slide. However in this extreme case the dem has a factor of safety of only a little over 1, which is not suf- ficient. However the footings should never become injured in so small a dem. CRUSHING STRENGTH. To get the resistance ti crushing at the downstream toe of the dam. Base of dam gc g 17° a 24,772 #s. Taking moments about the upstream else ec, we have; (11 642 x 6.55) (24, ext 7.26)al6ure. Where F, «a whe erushing force exerted equare Ft. of conerete to the left. 255,1899.5 =z 16.6 F, or Fee 15 545 According to Ketchum, in his “structural Engineer’ s Handbook," the allowable crushing strength of gravel agzregate concrete is 2000 #s/sq." or 288,000 /‘3/sq.° (on a 1:2:4:mix} Thus this dem is way sare as re- gards danger of concrete being crushed. Thig completes the investication of the section noel for the 3 main factors. There are sev- eralminor ones however, as mentioned before, which ere worthy of irvesticaticn. Seepage e The dam is safe as regards seepage under the dam. We have already allowed for scepage in computing for overturning moment hence tie only thing yet to investigate, is danger of seepage become ing excessive underneath the dam and undermining it, This hes beem guarded against in 5 ways3 (1) The dem is brot down to the rock which is but, about 2* below river bed. This will prevent 10. wearing away, by water action, underneath the dam. (2) A gravel f111, or embankment, has been made on the upstream face. It is 7.75 ft. high next to the dam and has a 2 on 1 slope downward, upstream. This alse prevents washing neer the toe, as it eze cludes the water from that vicinity. (3) At the heel a 1* eut-off arm or sill has been built intecral with the dam proper- which is 5° thick at the base. At the toe there is one, having a depth of 2° and 2.25° thick at the base. These two “curtains” prevent any percolation of water toru the roek at the vase. These projections beneath the base of the dem alse make the dam safer against sliding off, its base, The factor of safety against overturning is large enuf to take care of the effect of ice expansion. The dem being only 19° from crest to base, the consideration of the green concrete being crushed, at the bottom, by the addition of the top layers, while if was 3 in process of construction, need not be MAAS This dam is proviced with flocdegates (Fig. B.- Plate 1) and herce in flood time, the flow over the crest is not much greater than ordinary times. The slight vacuum effect that the sucking of the wate er as it leaves the rollways, produces, need not be considered. GRAPHICAL SOLUTIO¥ YOR OVERTURHING MOMENT. This solution is given on Fig. l, Plate 1. To find the Center of Gravity of See. I (abed) lay off db*® to the left of d » ab be® * © right "b= de Connect e* with b* HWext connect the middle of the de with the middle point of ab. Where this line intersects b*e’, there is the ©.G. of section I. Section Ig is worked in the same manner ef? being laid off horisontally to the left of gss 12. ef and ed* » gde These tro centers of gravity are then combined as in the previous solution. It is foune that this center: of gravity occurs 1/3 of the distance from the bottom, that is 19 & 5 or 6.55. Also, the center of pressure, of any submerzed area, is 1/3 the distance from the bottom. Hence P (the water pressure) acts thru the C.G. of the dem as does also the weicghte ‘This is the best possible place for P to act as it thus produces no secondary stress- OS Then using the scale: 5000 ## = 1°P was laid off to the left thru the 6.C.ard Ww vertically downward thru the ©.C. ‘Then the resultant of P and W was drawn. This cuts the base @eCe, 7-9 from ge If this resultant falls within the middle third of the base, no cverturning monent will be pro- @uced by the conditions, relating to the resultant. In other words, no tension has been placed in the cone erete, in the right hand 1/3 of the base Daugherty Hydrau (pp.cf cnd 335) 17 23 g— 5.66 Thus this dem 4s very safe and is realiy not the most economical dam that could have been built, for the Resultant falls 2.2 inside the uwiddle third. Thug have found the folloring things cone cerning the dam; That the ©C.G. has been made on the lime of application of the total water pressure, P}; that the rasultant of P and W, falls well within the middle thiré ( shown by -rephical solution); and thet the factor of safety azainst over-turning is ample, teing 5.25 for section which is tne most peri-g louse ‘The dam is also sufe against sliding in the most extreme casese also the alloweble crughing etrength of concrete is many times, what the outside sections of this com cre called upon to stand. The downstream iace of the dam however should nave been given a greater horizontal pitch as in flood season the water leaves the face of the dam at the crest and encounters it again abour ¢' lower, As this unalysis does not attempt to go inte the economical side of construction, or shape, of this dam, it has not heen concidered necessary to further analyze the other two sections. Their bases are the same and of the same ccmensione Certainly the flood gete section shown in fige ®@ will not over-turn as the 12 e height is much less, and hence P is applied much lower comm, relative to che Cou. The section between the fleod gates is ehown in Fig. 4, plate 1. This section will be more safe than the spillway section, as to overturning, ae we will be much greatere Also will be safer aguinst shear’. | 3. FRESENT POWER CONTITIONS. The condition of tke dam, as it is at pree gent has been ;iven in the forex