
small (250x250 max)
medium (500x500 max)
Large
Extra Large
large ( > 500x500)
Full Resolution


t  t : STRESSES IN !!!!: l MULTIPLE ARCHED DAM. The design of the Multiple Arched Dam was the result of the necessity of getting some form of structure in which the dangers of overturning and slIding could be minimized The problem r:~ was to got at the design that would at once have the rewest objections, and would be free from the upward pressure of the water under the foundations, danger from sliding, danger from overturning, that would be evenly and lightly loaded on 1ta foundations, and that would be safe from derangement from any extraneous causes, and strong enough to resist anyone of them, and at the same time would be the most economical to build. The flat deck or Ambursen type of dam overcame many of these objections, but still was at fault 1n some of them. These faults particularly are, the lack of flex1bility as to the de Sign, the reinforced concrete slab deck soon reaching its limitations of height, both by reason of the restriction of the span, and the required thickness of the slab, also the ability to use a reinforcement to take the load1ng, all of which tended to limit it to a low structure. Another serious objection was the plaCing of the tens11e stress in the water face of the structure, and relying on steel alone to carry the water load. Both of these objections are at once gotten r1d of by the use of the arch 1n the Multiple Arched type. and a structure produced 1n which there is not a 1. sinsle tensile stress that 1s not off set by a greater compressive stress, resulting 1n there being no tension in the structure whatever. The proper loading for concrete is in compresslon~ While curing the last of the objections to the Ambursen type, it has also been found possible to make the Multiple Arched type 9. universal design, this being accomplished by a proper modIfication to fit any character of foundation, and any size or shape of opening. Then in addItion, as a final proof of its efficiency, it can be built for less money than any type of dam for which to close the same opening. The cost of the Multiple Arched Dam, when compared to other types, will be approximately 80% of the cost of the Ambursen deck type, 60% of the cost of loose rock or earth filled and concrete core wall, and 50% of the cost of the Cyclopean, masonry, gravity type, to close the same opening. The ruling feature Of my design is the arch. There can be provided no more perfect loading for a cirOular ring arch, than a water load, for the reason that the pressures are constant for a given depth, and are always acting exaotly normal through the surface, and, therefore, are radial. The loading being radial, the shape of the surface, to make a balanced loading, must be cylindrical. The stresses sot up in the arch are dependant upon & number of conditions that tend Lo influence it, among which are,  the matarial of which it comppsed, the position of the arch with regard to the vertical er hOrizontal, the thickness of the arch 2. ring, and the various loads and other stresses it 1s to carry. All of these affect. in a varying degree. the stresses set up in the arch ring, and their distribution in it, and each of these must be given its true magnitude. 1n order to reach the true resultant loading. The inf" luence of the strosses and their magn1tude a. re determined by the position in space of the arch ring, and also whether or not, the refining counte~ influences are taken into consideration. The simplest position in which to consider the action of a water load on a cylindrical surface is that which it w1l1 take 1n a vortical cylinder, in which but two forces act, namely,  the tangential forces set up 1n the arch ring by the water pressure, and the weIght load of the masonry superimposed on any part of the cylinder. These conditions are baaed on the assumption that the cylinder will be submerged, so that its top will be even with the water Burface, and that its bottom w11l rest on a firm foundation. and that the thickness of the walls is such as to resist the water pressure for any given depth, with any given unit stress or sate load. If the weight of the water is taken at 62.5 pounds per cubic foot, and 1f the weight of concrete is taken at 150 pounds. per cubic foot, the safe load at 16 tons per square foot, and the radius of the extradosof the cylinder at 16 feet, then tho wall will be increased by the ratio of the increase of the water pressure due to the depth, and the tangential forces w111 be constant throughout. If the weight of the water is 62 5 pounds per cubic foot, it 3. , w111 be the equivalent to .03125 tons per square foot. which weight per square foot in tons, let US call p; the depth of water over any givon point hI the rad1us of the extrados r; the tangential stresses set up in the arch r1ng T; the safe load or unit stre •• 5, and the roquired thiCkness t; then, the thickness requirod at any given depth tor a sare load will equal phr = T and T = t. S Take aa an' example: Let h = 100 feet p • .03125 torts per square foot r = 16 feet 8 = 16 tons pe~ square foot Tl', len pbr • 50 tona, the amount ot the tangential thrust T and 50 tona div1ded b1 16, the unit stress allowed per square foot = 3.125 the required thlokness t in feet. The tangential thrust T 1s the resultant water load stroa. only. nnd it tends to deform any ring of the cylinder by reducing the length of' lts diameter. radius and clrcumf'erence. The computed detormation tor auoh a loading by the formula for the computation tor deformation in concrete would be the correct answer. prov1dIng it were the only foroe act1na, and prov1ding further, that the deformatIon dId not alter the dimensions ot any of the tactors entering into the calculations. The reduced radlus, or the new radIus, that the rlne would have after the calculated deformation in the length of the circumference plus the horizontal deformatIon due to th18 same . trell., which would again tend to lengthen the 4. ){ new radius, would actually ba sho~ ter than before the loading. and hence woUld reduce the unit water pressure proportionately to the extent of its influence 1n determ1ning the tangential stress, which in turn, would reduce the tangential stres8, wh1ch in turn, would a180 leasen both the longitudinal and lateral deformatIons, and so on, till equilIbrium were reached. The deformation of any r! ng 1n the oylinder, caused by the weight of the super1mposed mason~ y, wl11 aleo influenoe the length ot the final rad1us, and wlll have a variable influence in this direct10n depending on its distance trom the surface, but will always act to increase the length of the f1nal radius. Now, it w111 be seen that startIng at the water surface, Where all of the elements in the problem are aero, we have a8 variant. the wate~ pressure, the weIght of the masonry and the radlus, and a constant. consl sting of the wll t stre •• , and that from thl s poInt a depar~ ure begIns and a difforent ratio 1n all of thea., excepting 1n the unit stress, it we oonsider the unit trees as not being 1nfluenced by the chanS6s In dimensions due to the deformations. The water pressure will incroase 8S the depth Increases at the rate of .03125 tons per square foot. The weight ot the masonry rIng w1l1 inorease at the rat. ot .075 tons per cubic toot, and will itself create a deformation of any part of the aroh r1ng 1n a varying degree, proportionately as the rate of such increase exceeds the rata of increase 1n thlckne.. or the arch ring .• We thus have the tangential forces eat up by the water pressure actIng to deform the ring by shortening its circumferential length, and tendlng to thloken it in all directiona, and at the same tlme, 5. the masonry weIght load tending to counteract. neutralize or exceed this foroe on the top and bottom of any ring depending on the depth at whiCh the section under consideration 1s taken, which 1n turn would inorease the lateral deformation to the degree dependent on the locatio~ of the section, whether above, at the neutral point or below It. this deformation being met on the side or the ring toward the intradoe by no resistance except a counter balanced atmospheric pressure., a. ndtoward the extrados by the water pressure So muoh tor the simplest torm of water loaded aroh, the submerged cylinder in whIch all of the elements that can be harmonized, have been reduced to their simplest form, in whloh there are apparent many Influences to make a slight variation in the amount of the streBses in the areh ring Now, in the submersed cylInder. when the new radii and the new stresses take equilibrium, due to the counter balancing torces and influences. the oylinder wl11 not remain a true cylinder, but will always be a cl11ndrtcal f1gure, and will be doformed in 1t. cUtferent parts in proportion as the varying forces and res1stanoes influenoe such change ot shape, but in actual practice, this variation will be so small In amount as to be impossible to deteet, and each part of the cylIndrloal flsure wl1l st1ll ret. aln ita circular form at a given depth, and would be practically equally stressed throughout its c1rcumferencet regardless of the deformations. The point where the oy11ndrlcal flg~., with dimension. as given above, would become 8011d, would be 512 teat below the wator surface, a. nd at that point the apex of the inverted cone, representing the shape ot the internal spaoe of the cylinder, 6. f I would be found. Vrhen the cy11ndrlcal f1gure beComes a 80114, the pressure on any halt 18 equivalent to the pr•• s~ e on any other halt, and la, in all cas •• , equal to the pressure due to the depth presslns on a pl_. Burtaoe, the w1dth of whlch 19 equal to the d1ametor of the cylinder, and thla 18 true trom the surface down, whether the cylinder Is hollow or solid, hence whether hollow or on 8011d, the pressure on any halt, andAthe surfaoe of any ring 1n equal to that on the OPPOsite half, and equal allover such surface, therefope, the centor of pra8sure 1n a water loaded aroh r1na 1s always at the point midway between the lntrado8 and extrados or the arch ring. Now, 1f the cylinder wore assumed to be tilted at an angle between the vertioal and horizontal, at eaOh pos1tion, there . ouid take place in it a different set of strela. a, but no one ot theae are of Inter•• t. except those that would occur In the upper half of the cylinder so tilted, and these again would be 80 altered by tho tensile stresses of the weight component and unequal water pressure on the under aid., that it 1s better to omit the discussion of the8e cond1tlen , as not tending to lea4 1* the direction of the practical solution of the probl~, which 1s to determine the stre ••• a 1n an inclined water loaded arch. Let us then continue the discusslon or the V6l'tlcal cylinder considered a8 an arch, assum1ng that any halt of it 18 a fixed abutment. but that the halt cyl.• 1nder acting as the arch were free ~ o move on its abutments without friction at the spring linea, then each of the faroes acting a8 7. before mentioned, would produce the same result, and the half cylinder would assume the normal shape, due to the forces acting upon it, which shape would be the same as if it were a part of the complete cylindero If nowwe go a step further, and fix the half cylinder to immovable abutments at the spring line, and apply the loads as before, we will introduce into our problem, a new factor, for it will now be impossible for uniform deformation to take place throughout the entire length of the arch ring. It is at this stage of the consideration, where we have introduced the condition of the practical arch, and that we find all of the stresses tending to deformation, more or less neutralized, by counter forces set up by the resistance to bending at trle spring line, the magnitude of which depends on the elements of thickness of the arch ring or the relative thickness of the arch ring to the length of its arco At the water surface, where the ~ rch ring would be infinitely thin owing to the load being zero, the resistance to bending and its influence would be the minimum, and this resistance to bending and its influence will increase as the ratio of thickness to length of arc increases. In the case of a very thin arch, the resistance to bending is a minimum, as such an arch will be hinged at the spring line on a knife edge, and the loading will shorten it, causing it to become an arc having a smaller included angle, but at the same time having a longer radius, and it will, therefore, be 8. subjected to a still greater stress after deformation due to such increase of radius, which brings a reversed condition to that where thin the infinitely~ arch is free to move at the sp~ ing lines, or as in , I tf the case of a complete cylinder. Calline the tendency of an arch to assume a new position without changing its shape, arch action, true arch action or complete arch action 1s only possible in a complete cylinder or in an arch freG to move at ita ends, in which arch action takes place for all thicknesses, but as soon as the ends are fixed, full arch action ceases. but will be nearest approached in an infinitely thin arch with fixed ends and gradually diminished to disappearance as the fixed end arch incre, aseS 1n thickness to a solid. The tendency to be$ d1ng at the spring Itncs 1s to be considered not as tension and compression acting about a neutral axis, but merely as a tendoncy towards tens~ on or a diminution of the compressive stresses due to the water load on the outer spring line of the arch ring and an increase 1n the compressive stresses on the inner spring line, tending to change the center of pressure at the abutments. The amount of this tendency toward tension and increase of compression will depend on the extent of the loads applied and upon the thickness ot the arch ring being the maximum with P ~"",:, ythin arch subject to a load1ng to cause an equal deformation at the crovm line that will take plaoe in a thick arch, therefore, the ratio of this change in rate of loading at the abutments would become more marked and will increase as the allowed unit stress increases unt1l it would actually become a tensile stress 1n the fiber~ at the spring line of the extrados of a thin arch. The tendency to deformation of an arch along the line of its arc 1s al~ ays off set and neutralized, more or less, by the weight load 9. , t \ . \ '. I l \ \, counter acting this tendency, and hence if the weight load Is sufficient. it will balance it, which it actually does in practicable uses. The theoretical consideratlomof extreme thim19Ss and extreme thickness are only needed to arrive at the true limits of the influences of the force ~~ t1a! acting in an arch ring, for in the buttresses at all, we at once introduce a change of action in practice, neither extreme is called into service In a design. The above dIscussion is for the vertIcal arch only, for if we inclined the masonry element, and also in the water pz easur e, This change will commence at the vertical posItion, a. nd will continue to change as the slope 18 ibcreased, finally reaching the .:: 8.;~ imum at a horizontal position. This change will consist in a change in the water pressure at the cro~ n line and at the spring lines of the extrados, and also in the direction and magnitude of components of the water load of the masonry_ If an arch is vertical" the water pressure at the crown 11ne and the spring lines 1s equal, and is equal at all points around the cylinder at any given dp~+~, and the weight load of the masonry is all supported on the ne& t succ~ eding part belowt mld the required thickness of the arch ring would be in direct proportion to the depth of the portion dor consideration, but if' thIs same arch were laid horIzontal at the bottom of a body of water, the water pressure at the crown line and at the spring l1nes would be different by the amount or length of the radius for a semi circular arch being the greatest that they can be at the spring line, and diminished u. s the depth was diminished at the crown line. It can be ~ eadily seen that tills 10. load, and that the water load against the vertical arch is an absolutely balanced load. It will also be seen that the wei~ t components of the masonry will gradually change, until with the arch in position will give the greatest eccentricity possjble with the water the horizontal position, all of the weight load will be carried by of varying thicknesses, all of the arch ring or cylinder would have to be of a uniform dimension to take the stresses applied to it.• the abutments, and no portion of it by the succeeding part of the archo It will also be seen that instead of having an arch ring It will alSO be seen that the shape of the arch, in order to carry to the abutments the distorted water pressure load and also the / vertical weight load of the masonry r would need to be a departure from the true cylindrical shape, depending on the rise of the arch and the dept~ of water beneath which it lies for the reason that the difference in pressure at the crown and at the sprins lines are a percentage of the total head of water over the c~ own line or the spring lines, and consequently the r e Lationshlp off th se two dimensions come into the problem in this position. It will also be evident that the change taking place in the excentric loading would be directly pr opor t. Lona t e to the departure from the vertical. The range through which a water loaded arch could be taken, would be, in any event, only 45° from the vertical, and in practice with the multiple arched deSign, the departure is very much less than thiS, having been found for nearly all cases to have reached its best economy with a slope of ~ to 1, or at a departure of 36° 521 from the vertical. The weight load of the arch ring itself; when the arch has a slope at any angle between the ve. rtical and II. , horizontal, is divided into two component parts, one of which acts along a line parallel with the crown line of the arch, and the other normal to the crown line, so that a port1on of the we1ght load of the masonry 1s always taken by the wall itself, and another portion of it is actually carried to the abutments, and acts in conjunction with the tangential'loads set up by the water load. In all of the. e d1scussions, an arc ~ f 1800 has been considered, for it 1s necessary to use this complete semi circular arc 1n order that the forces and the stresses set up by it may be carried to fit complete conclusion. AF etat~ u above, the water load sets up stresses in the arch ring tending to deform it by the reduction of the length of the arc, and that these again set up bending resistant stresses tending to counteract the deformation oaused by the water load stresses, and that the weight load of the masonry also tends to neutralize the deformation set up by the water load stresses, hence it will be seen, that the weight load of the masonry would always act 1n conjUDGt1on with the resistance to bending at the spring lines to neutrali' 7f.? the deformation set up by the water load forces. Now, if an arch consisting of concrete masonry, and fixed at abutments is taken as being horizontal, and consist1ng of a barrel of any length, the shape of the thrust l1ne, set up by this weight load of the arch alone, lselllpticalJ and the tendency of the thrust line 1s to pass out of the arch as the spring lines 12. are approached. Now, if this arcll ring rotated until it were in a vertical position, this tendency to form an eliptlcal thrust line will disappoar in direct proportion as the angle of its slope decreases until at the vertical pos1tion it would be zero and in the vertical position, the line would pass at all parts of the arch ring through the center of gravity of any radial plane, passing through the arch ring. Hence, in the vertical position, the 11ne of pressure will be in the exact center of the arch ring. N" Oll, if this same horizontal arch were considered as being loaded with a superimposed water load, and if the weight load were eliminated, it can be seen that the ring would necessarily be thiner at. the crown line than a. t the spring lines, the shape of the intra( los being a pointed ell1pse and of the extrados a semi circular arc. so that the tendency would be twward the reduotion to the thickness at tho crown line, gradually increasing to the sprlrlg lines when there is any departure from a vertical arch and this difference in thicknes3 that would carry the load at a given unit strose would be the maximum in the horizoltal arch. In the case of an arch having a croWn line slope of 36° 52' J the water load at the crown 11ne and at the sprlne lines varies as the taneent of the angle of the slope. The stresses eet up by the water load In any portion of the arch rine will, therefore, vary in a like proportion. The normal or radial component of the weight load will be the maximum at the crown line and dimin1sh to zero ~ t the spring lines. The component of the weight load parallel to the crown line will be a minimum at the crown line~ and will be a maximum at the spring line. The angle of departure of tho arch from the 13. vertical being less than 45°, at which the components of the weight load will become equalJ the radial component is always less or greater than the crown line component in proportion to the angle of the slope of the arch. With the new consideration of. position of the arch ring, we have a water pressure increasing from the crown line setting up corresponding increasing tangential stresses, and we have the normal or radial component of the weight load acting in conjunction with the water load but as its magnitude is maximum at the crown line where the water pressure has its minimum, and as the thrust line set up by the weiGht load component tends toward the extrados of the arch ring, and as the thrust line set up by the water load tends toward the intrados due to the tendency to bending at the spring lines caused by the deformation of the arch ring at the crown line, and as the crown line component of the weight load becomes more active, being minimum at the crown line and maximum at the spring line, the, tendency is to decrease the deformation and to balance the load on the abutments. In all of the considerations above, the full semi circular arch has been conSidered, but in practice the arch having an arc, known as the " economic arc", is used, which arc will vary with the other influences determining it from about 136° to 144 u • The use of a smaller arc cuts off the legs of the semi circular arch at such a point that the tendenCies to departure of the thrust lines at the spring line are very much smaller than they are with a full semi circular arch, and also the ratio of the thickness of the arch ring to the radius has been increased likewise, consequently, the tendency of the normal weight load thrust line l4D toward the outside of the arch ring at this new position of the spring linea is much reduced, the tendency to deformation has been decreased by the ratio of the arc length to the radius, thus reducing the tendency to unbalanced water loading, the actual defor~ matton, and consequently, bending action action 1s also reduced, so that with the slope adopted, the thrust line will follow so nearly the exact middle or the arch ring for all of the loads applied, that the departure is negligible. In the foregoing discussion, only the weight load of the arch ring of the masonry and the water prABsure have been considered, In addition to then:; l forces ctlng upon an arch r tng , there will be also introduoed initial stresses, and stresses due to, tempera~ ture changes. Considering first the init. ial stresses, whicn. w111 depend on the method of performine the work of laying the material, and this will be very slight in amount in any event. If concrete is laid and is allowect to set under water, the rise in the tempera~ ture due to the settio2 is mimimized by the cooling effect of the water. Concrete setting under water has been determined by the best author! 1.166 to increase s1ightly in bulk during tho pt'ooess of setting, an~ if setting in the air, and if it 1s allowed to dry out during the process or setting, it will diminish in volume sllehtly) so that if the concrete ware laid in such a manner that it was always wot after laying, 1t would swell slieht1y, but if left to dry after setting, it would shrink slightly. Now, if the concrete were kept at a cortain degree of moisture during the process of setting, which 1s the condition that applies in actual practice in this type of construction, thnn t would neither ewell 15. nor ahrink in the process and there would be no initial straIns set up. These initial strains, in any event, baing so near the ne. ut. r a L point. of expansion and contraction, depending on the degree of moisture, applied during the pr oce ss of setting, are ve!,~ r small. In addition to the above, the stresses that can take place, are thus set up by temporature change. in the concrete, and their dogree will depend on the range of the temperature to be considered. The range of temperature, setting up temperature stresses, will be t. he departure froom the nean temperature at which the concr et e took its final set, anc' the amount of the stresses will depend on the magnitude of the range of tem, erattwe. A rise in temperature above th07 normal at which the concrete was placed would hnve a tendency to incroase the bulk of the material in the arch ring, and to oppose the tendency of the water load and of the normal weight component to deform it, and hence it would act with the crown line component of the water load to maitain Q balan ced condition of loading. A fall in temperaturo will cause a decrease 1n the dimensions in the material in the arch ring, and will have a tendency to increase the deformation due to the influance of the water load and of the normal compo~ ent of the we1ght load, and would be assisted or neutralized in its action by the increased bending reaction, and also by the cr0~ n line component of the weight lOF. d. It can readIly be seen that the action of any foroe or set of forces on the arch ring 1s confined to one arch as the resultant of these in each arch is counterbalanced by those in any other adjacent arch. The expansion or contraction due to a temperature Change can only be throughout the arc or one arch and 1s not cumu~ at1ve. 16. The range of temperature of the portion of the wall always under water can only be the amount of the departure from the mean at which the concrete took its set to that of the mean annual temperature of the water itself, for the wall in contact with the water will take the temperature of the water, and will be but slightly influenced by the temperature of the air on the outside, by reason of its inability to cormnunicate its heat to the wall. Under actual conditions, however, there would be a slight increase in temperature in the intrados of the wall over that at the water face, and this unequality will tend toward a greater degree of expansion at the intrados than at the extrados, thus the water loaded portion WOUld, by this unequal · d~ stribution of the temperature stresses, be assisted in the prevention of deformation. The extreme range of the winter and summer temperatures come only against those portions of the wall that are exposed during the entire year above water, and as these portions of the wall have no stresses in them, excepting the weight load and the temperature stress, all of which combined are but a very small portion of the allowable maximum stress due to the water load, they are never stressed to anywhere near the amount they are calculated to bear when loaded with water. The range of temperature tending to set up stresses will be the departure from the lUcan temperature at which the concrete took its final set. Let us take as an example the ranee of temperature and conditions at the site of the Big Meadows Dam, Plumas COlmty, California, being erected for the Great Western Power Company. The mean temperature of the setting of the concrete resulting from 17. the mean of the day and night temperat~ combined with the mean day and night temperatures of the water with which the concre~ is mixed w1ll give the final mean temperature at which the concrete takes its final set. The mean of all of these at the Big Meadows Site wIll be very close to 50° F., and the range of temperature from mean to extreme of the portions of the dam out of the water and exposed to the air on both sides would be about 40° F. above and 60° F. below this mean. The extreme range of the air temperatures 1s from  20° F. to + lOOoF., a maximum for summer conditions, but the change of the temperature 1n the walls would not reach these extremes of air temperature either way because they will always retain some ot the heat of the daytime throughout the night and vice verea, and will never reach the full degree of the air surrounding them. These would be the conditions fGr the portions out of water throughout all of the seasons, which meane, of course, the extreme top where all of the walls are very much over sized for the ptresses. The portions of the wall under water at all times of the year would range from the minimum or + 32° F. to a possible + 50oF., so that tho only change in these portions wOl~ d be a drop from the mean temperature of the setting time + 50 o F. to + 32° F., a difference or 18° F., so that the extreme condItions of shrinkage would be that set up by a temperature change of 18° F. Should the concrete be laid in the Spring or Fall, when the water and air both have a lower mean temperature, and such is now the c&. e of the concrete being laid at the present time, the mean temperature will be nearer 45° F., which in the case of shrinkage would make a differenoe of 13° F. as the range. This range of temperature w111 18. I I ,, cause the arch rings, as designed at the Big Meadows Dam, to shorten in their length .0027 of 1 foot or. 032 of an inch, whereas they are all reinforced to stand a change of temperature of 250 o F. without danger of localizing cracks. The maximum change either way that could take place in any part of the structure would be on those portions of the dam that extended above the water level and were subjected to the full range of temperature of winter and summer having to carry no other load would be .005 of a toot or .06 of an inch 1n the length of any other ring. It would be practically impossible to measure the deflection at the crown llne at this maximum range of temperature. It must also be borne 1n mind that the thickness of the arch ring itself alSO affects the resultant deflection due to the temperature changes and the resistance to bending of the arch ring becomes greater with its thickness. The stresses Bet up by the water load and by the weight or the masonry are affected by the length of arc and the position of the arch in space, but the temperature change stresses, whether plus or minus, w11l act equally for any position of the mater1als and are only affected as they are more or less neutralized by the thickness of the arch ring. The rate of expansion or contraction of conorete is given by the authorities as .0000055 for a unit of length for 1° F., and of steel .000006 for the same conditions, so that it is universally recognized that steel and concrete pract~ cally expand and contract at an equal rate, and that if reinforcement be introduced 1n the concrete having an area equal to 1 of the con 275 crete, no cracking will take place regardless of the change in temperature. 19. Returning to the stresses in the arch ring, the simplest form of a water loaded arch ring that can be considered is that of a vertIcal cylinder assumed to rest on a solid base at its bottom, the top of which is even with the water surface, and whereiq the water pressure and thickness are aero at the top, and wherein the thickness increases to a solid in proportion to the 1ncrease of the water pressure for the depth by using a predetermined radius of extrados and unit stress due to the water pressure in the arch rings. Let us consider the stresses in such a cylinder having for its dimensions a radius of extrados of 16 feet, and a safe load or unit stress of 16 tons per square foot. We will then have a " . thickness that w111 increase at the rate of the water pressure. In the nomenclature, let p = water pressure on one sq. ft. in tons ; .01235 the weight of one cu. ~ t. of water 1n tons. h : the depth of water at any seotion. p = the pressure per sq. ft. 1n tons for any given height h = ph r = the radius of extrados of the cylinder = 16 T = the tangential thrus. t 1n the ar ch ring due to the water pressure P S = the safe load or unit stress in the concrete per sq. ft. in tons :: 16 t ::: the thickness at any section required to resist a thrust T at the safe load or unit stress C ::: the weight of a cu ft. of concrete in tons = .075 W ::: the weight of any section of wall above the section under consideration Then, for any given value of h, ph ::: P ' 1' :: Pr, t =....!!:., w :: th G S 2' 20. . ( i 1 • / I i rate, ph increases Let us consider that the water pressure p~ at a certain and that W, the masonry we1ght load increases at the rate Now for any given case p 1s a constant r " " » 5 ft " " C " " h of 2th C. Hence using the above numerical values: Let P = Klh and W = K2h 2, the values of Kl and K2 being respectively Kl = P = .03125 and K2 = pre or in numerical values of p S and r ( as above) = 25 ,001172 W = .001172 h2 ( 1) P = .03125 h ( 2) In order to find the depth at wh1ch the water pressure e~ uals the weight load of the masonry superimposed, we equate ( 1) and ( 2) and solve for h. h = .03125 .001172 = 26.6 ft., the depth of the neutral point at which the weight load and water pressure are aqual. Let Fig. I ( a) represent a horizontal section of a vertical cylinder on a line an an A A and Fig. I ( b) a vertical section of the same on the line B B. Since the water pressure on a curved surface 1s equivalent to the pressure on the orthographic ~ rojeotlon of the surface on a plane area and 1s uniformly distributed, then if a plane 1s conoeived to cut the cylinder through the center 21. \ , \ \ ,\ vertically, the pressure on the section cut off' by this plane of the cylinder will be equally divided and uniformly distributed from extrados to intradoB, provided the seml circular arch thuB created were free to move on the other half of the cylinder along this plane area due to the deformation caused by the radial water pressure of the load, the remaining half of the cylinder being assumed to be resolved into fixed abutments. The weight load of the masonry of the arch will evidently have no resultant effective pressure against the spring l1nes of the arch because its load Is applied vertically. The water pressure, however, 1s transmitted to the abutments as tangential forces set up by the water pressure on the curved surface. As these tangential torces vary as the tangential angle, and as this' angle varies as the radius of the eztrados, the tangential forces set up for the water pressure due to any depth vary as the radius of the extrados of the arch ring varies. The water pressure acting through the tangential forces tends to compress all parts of the arch ring under the above named • condItions uniformly, tending to shorten its length and to thicken it in all directions and to reduce lts radii. The Weight load alSO acts to deform any given section and acts vertically tend1ng to counteract the upward and downward deformation due to the water pressure ' and to increase the radial deformation. If any arch ring one unit 1n height were conSidered, and its position were above the neutral height, the water pressure would be in excess of the weight load, and at the neutral point the pressure wIll be eq. al_ but for any point below the neutral point, as determined above, the weight pre~ sure will exceed the water pressure so that 22. !! the resultant deformation would depend on the position of the ring considered 1n its relat10n to the depth below the surface of the water. The tangential forces act along the 11ne normal to the center of any radial plane through the arch ring, or the surface of any assumed voussoir, wh1le the weight load acts vertically at the l1ne between the outer and middle third of the ring and under the center of gravity of the triangular surface representing a section of this wall, thUB creating an eccentric loading of the arch ring vert1cally, the degree ot eccentricity affecting the deformation, depending on the relative influence of the water pressure and the weight load. The influence of the weight load deformation should be considered first and that of the water load deformation and its influence to act with or counteract the deformation due to the weight load should be next considered. Let us take as a numerical example a submerged cylinder with the dimensions named above and considering the stresses as taking place in a ring of this cylinder. the ring to be 1 foot high, and the center of the ring to be 100 feet below the surface of the water, the problem being to find the deformation due to the weight load alone. 23. the resultant deformation would depend on the position of the ring considered in its relation to the depth below tile surface of the water. The tangential forces act along the line normal to the center of any radial plane through the arch ring, or the surface of any assumed voussoir, while the weieht load acts vertically at the line between the outer and middle third of the r ing and under the center of gravity of the triangular surface representing a section of this wall, thus creating an eccentric loading of ~ he arch ring vertically, the degree of eccentricity affecting the deformation, depending on the relative influence of the water pressure and the weight load. The influence of the weight load deformation should be considered first and that of the water load deformation and its influence to act with or counteract the deformation due to the weight load should be next considered. Let us take as a numerical example a submerged cylinder with the dimensions named above and considering the stresses as taking place in a ring of this cy1inder, the ring to be 1 foot high, and the center of the ring to be 100 feet below the surface of the water, the problem being to find the deformation due to the weight load alone. 23. the resultant deformation would depend on the position of the rIng considered In 1ts rolntlon to the depth below the surtace at the wat.,. The tangential foroes act alone the line normal to the center or any radial plane throUGh tho arch ring. or the surtao. or any assumed vou. eolr, while the weight load aot. vertically lat the line between the outer and middle third of the ring and under the center of sravlty or the triangular surrace repreeen'toins a section or th1s wall. thus creatln& an ecoentrlc load. lng; ot the arcb rins vert1oally. the 4egree of eccentricity affeotlng the detormatlon, depending on the relative influence of the water pressure anel tho woigbt load. The influence of the w@ lght load cieformatlon ohould. be considered first. and that ot the water load deformation and Ita influence to act wit. h or counteract the deformat1on due to the walght load should be next oonsidered. Let us take as a numerical example a submerged cylinder with the dimensions named above and considering the stresses aa taking place In 0. ring or th18 cylinder, tho ring to be 1 toot hiBb. and the center ot the rins to be 100 feet below the surface of the water, tho problem belnc to find the deformation due to the weight load alon.•• 23. . . • r Let rl ::: Radius of intrados ::: 16 ft. r 12.875 ftD 2 :;: Radius of extrados ;: h ::: Depth of center of section under consideration :;: 100 ft. E ::: Modulus of elasticity::: 180,000 tons per sq. ft. A :;: Area of inside circle before compression B :;: Area of outside circle before compression . . AI :;: Added area intrados after compression II II extrados II Thickness of ring = 3.125 ft. s :;: Safe working stress :;: 16 tons per sq. ft. W :;: Weight of Cylind, er Z ::: Total cross section of ring after compression Dl = Decrease of rl at intrados D2 = Increase of r2 at extradoso Let us consider a slice of the cylinder one foot long whose mean thickness lies at a depth of 100 feet. Then t = (. 03125 x 100 x 16) = 30125 16 The dead load of the superimposed masonry above the section under consideration tends to deform that section by shortening it longitudinally and thickening it radially. From the equation of elongation 24. r water Resources Center ArchiveS EASr:' 08D 3 June 29, 19l7. f Eastwood Multiple Arched Reinforced concrete Dams Safest, strongest and most economical and scient. ific type of structure for impo'J. ndine wat. er t In Service and Bu1lding Height feet Bume LRke Dam, near ? resno, Cal.  ~~ 61 Big Bear Valle~ yanDaEme, rnanredQirneRedGloR. n, dsG, al. Los Verj~ lS Dam, near Oroville, p. utte C,,), , Gal. Kennedy Dam, nekr . Ttlckson, ArnA. dorCo., Cal. Argonaut Dam, dittO Mountain Dell Dam, S~ lt Lak~ City) utah. Malad Dam, Malad, Idaho Murray Dam, near 58. n Di"~ o, Ga 1. Carroll Dam, nSe~ anr DiTie: gsocoGndol. d, o, Gal. San Dieguito Dam, ditto IMITATIONS Reclamation Se~ vice Dam, Oregon 2 Small dams in Sierra Nevada Mountains, ca. lifornia Rock Creek D& m, near ~ uburn, Cal. ' ll; Small dams in :;. uchigal1 677 92 363 60 350 64 50 450 150 630 110 50S 112 ]. 30 650 50 700 r. apacity of Reservoir in millions of Gallonfl 4,600 22,880 846 Debris dam Df'JbriS dam \
Click tabs to swap between content that is broken into logical sections.
Rating  
Title  Stresses in the multiple arched dam. 
Subject  Arch damsDesign and constructionAnalysis.; Structural analysis (Engineering); Dam safetyAnalysis.; Concrete damsDesign and constructionAnalysis.; EASTWOOD 3 WRCA 
Description  Typescript (carbon), in EASTWOOD box 1.; "June 29, 1917." 
Publisher  s.n 
Type  Text 
Language  eng 
Relation  Also available on the Internet.; http://worldcat.org/oclc/606639373/viewonline 
DateIssued  1917 
FormatExtent  25 leaves ; 28 cm. 
RelationRequires  Mode of access: World Wide Web. 
Transcript  t  t : STRESSES IN !!!!: l MULTIPLE ARCHED DAM. The design of the Multiple Arched Dam was the result of the necessity of getting some form of structure in which the dangers of overturning and slIding could be minimized The problem r:~ was to got at the design that would at once have the rewest objections, and would be free from the upward pressure of the water under the foundations, danger from sliding, danger from overturning, that would be evenly and lightly loaded on 1ta foundations, and that would be safe from derangement from any extraneous causes, and strong enough to resist anyone of them, and at the same time would be the most economical to build. The flat deck or Ambursen type of dam overcame many of these objections, but still was at fault 1n some of them. These faults particularly are, the lack of flex1bility as to the de Sign, the reinforced concrete slab deck soon reaching its limitations of height, both by reason of the restriction of the span, and the required thickness of the slab, also the ability to use a reinforcement to take the load1ng, all of which tended to limit it to a low structure. Another serious objection was the plaCing of the tens11e stress in the water face of the structure, and relying on steel alone to carry the water load. Both of these objections are at once gotten r1d of by the use of the arch 1n the Multiple Arched type. and a structure produced 1n which there is not a 1. sinsle tensile stress that 1s not off set by a greater compressive stress, resulting 1n there being no tension in the structure whatever. The proper loading for concrete is in compresslon~ While curing the last of the objections to the Ambursen type, it has also been found possible to make the Multiple Arched type 9. universal design, this being accomplished by a proper modIfication to fit any character of foundation, and any size or shape of opening. Then in addItion, as a final proof of its efficiency, it can be built for less money than any type of dam for which to close the same opening. The cost of the Multiple Arched Dam, when compared to other types, will be approximately 80% of the cost of the Ambursen deck type, 60% of the cost of loose rock or earth filled and concrete core wall, and 50% of the cost of the Cyclopean, masonry, gravity type, to close the same opening. The ruling feature Of my design is the arch. There can be provided no more perfect loading for a cirOular ring arch, than a water load, for the reason that the pressures are constant for a given depth, and are always acting exaotly normal through the surface, and, therefore, are radial. The loading being radial, the shape of the surface, to make a balanced loading, must be cylindrical. The stresses sot up in the arch are dependant upon & number of conditions that tend Lo influence it, among which are,  the matarial of which it comppsed, the position of the arch with regard to the vertical er hOrizontal, the thickness of the arch 2. ring, and the various loads and other stresses it 1s to carry. All of these affect. in a varying degree. the stresses set up in the arch ring, and their distribution in it, and each of these must be given its true magnitude. 1n order to reach the true resultant loading. The inf" luence of the strosses and their magn1tude a. re determined by the position in space of the arch ring, and also whether or not, the refining counte~ influences are taken into consideration. The simplest position in which to consider the action of a water load on a cylindrical surface is that which it w1l1 take 1n a vortical cylinder, in which but two forces act, namely,  the tangential forces set up 1n the arch ring by the water pressure, and the weIght load of the masonry superimposed on any part of the cylinder. These conditions are baaed on the assumption that the cylinder will be submerged, so that its top will be even with the water Burface, and that its bottom w11l rest on a firm foundation. and that the thickness of the walls is such as to resist the water pressure for any given depth, with any given unit stress or sate load. If the weight of the water is taken at 62.5 pounds per cubic foot, and 1f the weight of concrete is taken at 150 pounds. per cubic foot, the safe load at 16 tons per square foot, and the radius of the extradosof the cylinder at 16 feet, then tho wall will be increased by the ratio of the increase of the water pressure due to the depth, and the tangential forces w111 be constant throughout. If the weight of the water is 62 5 pounds per cubic foot, it 3. , w111 be the equivalent to .03125 tons per square foot. which weight per square foot in tons, let US call p; the depth of water over any givon point hI the rad1us of the extrados r; the tangential stresses set up in the arch r1ng T; the safe load or unit stre •• 5, and the roquired thiCkness t; then, the thickness requirod at any given depth tor a sare load will equal phr = T and T = t. S Take aa an' example: Let h = 100 feet p • .03125 torts per square foot r = 16 feet 8 = 16 tons pe~ square foot Tl', len pbr • 50 tona, the amount ot the tangential thrust T and 50 tona div1ded b1 16, the unit stress allowed per square foot = 3.125 the required thlokness t in feet. The tangential thrust T 1s the resultant water load stroa. only. nnd it tends to deform any ring of the cylinder by reducing the length of' lts diameter. radius and clrcumf'erence. The computed detormation tor auoh a loading by the formula for the computation tor deformation in concrete would be the correct answer. prov1dIng it were the only foroe act1na, and prov1ding further, that the deformatIon dId not alter the dimensions ot any of the tactors entering into the calculations. The reduced radlus, or the new radIus, that the rlne would have after the calculated deformation in the length of the circumference plus the horizontal deformatIon due to th18 same . trell., which would again tend to lengthen the 4. ){ new radius, would actually ba sho~ ter than before the loading. and hence woUld reduce the unit water pressure proportionately to the extent of its influence 1n determ1ning the tangential stress, which in turn, would reduce the tangential stres8, wh1ch in turn, would a180 leasen both the longitudinal and lateral deformatIons, and so on, till equilIbrium were reached. The deformation of any r! ng 1n the oylinder, caused by the weight of the super1mposed mason~ y, wl11 aleo influenoe the length ot the final rad1us, and wlll have a variable influence in this direct10n depending on its distance trom the surface, but will always act to increase the length of the f1nal radius. Now, it w111 be seen that startIng at the water surface, Where all of the elements in the problem are aero, we have a8 variant. the wate~ pressure, the weIght of the masonry and the radlus, and a constant. consl sting of the wll t stre •• , and that from thl s poInt a depar~ ure begIns and a difforent ratio 1n all of thea., excepting 1n the unit stress, it we oonsider the unit trees as not being 1nfluenced by the chanS6s In dimensions due to the deformations. The water pressure will incroase 8S the depth Increases at the rate of .03125 tons per square foot. The weight ot the masonry rIng w1l1 inorease at the rat. ot .075 tons per cubic toot, and will itself create a deformation of any part of the aroh r1ng 1n a varying degree, proportionately as the rate of such increase exceeds the rata of increase 1n thlckne.. or the arch ring .• We thus have the tangential forces eat up by the water pressure actIng to deform the ring by shortening its circumferential length, and tendlng to thloken it in all directiona, and at the same tlme, 5. the masonry weIght load tending to counteract. neutralize or exceed this foroe on the top and bottom of any ring depending on the depth at whiCh the section under consideration 1s taken, which 1n turn would inorease the lateral deformation to the degree dependent on the locatio~ of the section, whether above, at the neutral point or below It. this deformation being met on the side or the ring toward the intradoe by no resistance except a counter balanced atmospheric pressure., a. ndtoward the extrados by the water pressure So muoh tor the simplest torm of water loaded aroh, the submerged cylinder in whIch all of the elements that can be harmonized, have been reduced to their simplest form, in whloh there are apparent many Influences to make a slight variation in the amount of the streBses in the areh ring Now, in the submersed cylInder. when the new radii and the new stresses take equilibrium, due to the counter balancing torces and influences. the oylinder wl11 not remain a true cylinder, but will always be a cl11ndrtcal f1gure, and will be doformed in 1t. cUtferent parts in proportion as the varying forces and res1stanoes influenoe such change ot shape, but in actual practice, this variation will be so small In amount as to be impossible to deteet, and each part of the cylIndrloal flsure wl1l st1ll ret. aln ita circular form at a given depth, and would be practically equally stressed throughout its c1rcumferencet regardless of the deformations. The point where the oy11ndrlcal flg~., with dimension. as given above, would become 8011d, would be 512 teat below the wator surface, a. nd at that point the apex of the inverted cone, representing the shape ot the internal spaoe of the cylinder, 6. f I would be found. Vrhen the cy11ndrlcal f1gure beComes a 80114, the pressure on any halt 18 equivalent to the pr•• s~ e on any other halt, and la, in all cas •• , equal to the pressure due to the depth presslns on a pl_. Burtaoe, the w1dth of whlch 19 equal to the d1ametor of the cylinder, and thla 18 true trom the surface down, whether the cylinder Is hollow or solid, hence whether hollow or on 8011d, the pressure on any halt, andAthe surfaoe of any ring 1n equal to that on the OPPOsite half, and equal allover such surface, therefope, the centor of pra8sure 1n a water loaded aroh r1na 1s always at the point midway between the lntrado8 and extrados or the arch ring. Now, 1f the cylinder wore assumed to be tilted at an angle between the vertioal and horizontal, at eaOh pos1tion, there . ouid take place in it a different set of strela. a, but no one ot theae are of Inter•• t. except those that would occur In the upper half of the cylinder so tilted, and these again would be 80 altered by tho tensile stresses of the weight component and unequal water pressure on the under aid., that it 1s better to omit the discussion of the8e cond1tlen , as not tending to lea4 1* the direction of the practical solution of the probl~, which 1s to determine the stre ••• a 1n an inclined water loaded arch. Let us then continue the discusslon or the V6l'tlcal cylinder considered a8 an arch, assum1ng that any halt of it 18 a fixed abutment. but that the halt cyl.• 1nder acting as the arch were free ~ o move on its abutments without friction at the spring linea, then each of the faroes acting a8 7. before mentioned, would produce the same result, and the half cylinder would assume the normal shape, due to the forces acting upon it, which shape would be the same as if it were a part of the complete cylindero If nowwe go a step further, and fix the half cylinder to immovable abutments at the spring line, and apply the loads as before, we will introduce into our problem, a new factor, for it will now be impossible for uniform deformation to take place throughout the entire length of the arch ring. It is at this stage of the consideration, where we have introduced the condition of the practical arch, and that we find all of the stresses tending to deformation, more or less neutralized, by counter forces set up by the resistance to bending at trle spring line, the magnitude of which depends on the elements of thickness of the arch ring or the relative thickness of the arch ring to the length of its arco At the water surface, where the ~ rch ring would be infinitely thin owing to the load being zero, the resistance to bending and its influence would be the minimum, and this resistance to bending and its influence will increase as the ratio of thickness to length of arc increases. In the case of a very thin arch, the resistance to bending is a minimum, as such an arch will be hinged at the spring line on a knife edge, and the loading will shorten it, causing it to become an arc having a smaller included angle, but at the same time having a longer radius, and it will, therefore, be 8. subjected to a still greater stress after deformation due to such increase of radius, which brings a reversed condition to that where thin the infinitely~ arch is free to move at the sp~ ing lines, or as in , I tf the case of a complete cylinder. Calline the tendency of an arch to assume a new position without changing its shape, arch action, true arch action or complete arch action 1s only possible in a complete cylinder or in an arch freG to move at ita ends, in which arch action takes place for all thicknesses, but as soon as the ends are fixed, full arch action ceases. but will be nearest approached in an infinitely thin arch with fixed ends and gradually diminished to disappearance as the fixed end arch incre, aseS 1n thickness to a solid. The tendency to be$ d1ng at the spring Itncs 1s to be considered not as tension and compression acting about a neutral axis, but merely as a tendoncy towards tens~ on or a diminution of the compressive stresses due to the water load on the outer spring line of the arch ring and an increase 1n the compressive stresses on the inner spring line, tending to change the center of pressure at the abutments. The amount of this tendency toward tension and increase of compression will depend on the extent of the loads applied and upon the thickness ot the arch ring being the maximum with P ~"",:, ythin arch subject to a load1ng to cause an equal deformation at the crovm line that will take plaoe in a thick arch, therefore, the ratio of this change in rate of loading at the abutments would become more marked and will increase as the allowed unit stress increases unt1l it would actually become a tensile stress 1n the fiber~ at the spring line of the extrados of a thin arch. The tendency to deformation of an arch along the line of its arc 1s al~ ays off set and neutralized, more or less, by the weight load 9. , t \ . \ '. I l \ \, counter acting this tendency, and hence if the weight load Is sufficient. it will balance it, which it actually does in practicable uses. The theoretical consideratlomof extreme thim19Ss and extreme thickness are only needed to arrive at the true limits of the influences of the force ~~ t1a! acting in an arch ring, for in the buttresses at all, we at once introduce a change of action in practice, neither extreme is called into service In a design. The above dIscussion is for the vertIcal arch only, for if we inclined the masonry element, and also in the water pz easur e, This change will commence at the vertical posItion, a. nd will continue to change as the slope 18 ibcreased, finally reaching the .:: 8.;~ imum at a horizontal position. This change will consist in a change in the water pressure at the cro~ n line and at the spring lines of the extrados, and also in the direction and magnitude of components of the water load of the masonry_ If an arch is vertical" the water pressure at the crown 11ne and the spring lines 1s equal, and is equal at all points around the cylinder at any given dp~+~, and the weight load of the masonry is all supported on the ne& t succ~ eding part belowt mld the required thickness of the arch ring would be in direct proportion to the depth of the portion dor consideration, but if' thIs same arch were laid horIzontal at the bottom of a body of water, the water pressure at the crown line and at the spring l1nes would be different by the amount or length of the radius for a semi circular arch being the greatest that they can be at the spring line, and diminished u. s the depth was diminished at the crown line. It can be ~ eadily seen that tills 10. load, and that the water load against the vertical arch is an absolutely balanced load. It will also be seen that the wei~ t components of the masonry will gradually change, until with the arch in position will give the greatest eccentricity possjble with the water the horizontal position, all of the weight load will be carried by of varying thicknesses, all of the arch ring or cylinder would have to be of a uniform dimension to take the stresses applied to it.• the abutments, and no portion of it by the succeeding part of the archo It will also be seen that instead of having an arch ring It will alSO be seen that the shape of the arch, in order to carry to the abutments the distorted water pressure load and also the / vertical weight load of the masonry r would need to be a departure from the true cylindrical shape, depending on the rise of the arch and the dept~ of water beneath which it lies for the reason that the difference in pressure at the crown and at the sprins lines are a percentage of the total head of water over the c~ own line or the spring lines, and consequently the r e Lationshlp off th se two dimensions come into the problem in this position. It will also be evident that the change taking place in the excentric loading would be directly pr opor t. Lona t e to the departure from the vertical. The range through which a water loaded arch could be taken, would be, in any event, only 45° from the vertical, and in practice with the multiple arched deSign, the departure is very much less than thiS, having been found for nearly all cases to have reached its best economy with a slope of ~ to 1, or at a departure of 36° 521 from the vertical. The weight load of the arch ring itself; when the arch has a slope at any angle between the ve. rtical and II. , horizontal, is divided into two component parts, one of which acts along a line parallel with the crown line of the arch, and the other normal to the crown line, so that a port1on of the we1ght load of the masonry 1s always taken by the wall itself, and another portion of it is actually carried to the abutments, and acts in conjunction with the tangential'loads set up by the water load. In all of the. e d1scussions, an arc ~ f 1800 has been considered, for it 1s necessary to use this complete semi circular arc 1n order that the forces and the stresses set up by it may be carried to fit complete conclusion. AF etat~ u above, the water load sets up stresses in the arch ring tending to deform it by the reduction of the length of the arc, and that these again set up bending resistant stresses tending to counteract the deformation oaused by the water load stresses, and that the weight load of the masonry also tends to neutralize the deformation set up by the water load stresses, hence it will be seen, that the weight load of the masonry would always act 1n conjUDGt1on with the resistance to bending at the spring lines to neutrali' 7f.? the deformation set up by the water load forces. Now, if an arch consisting of concrete masonry, and fixed at abutments is taken as being horizontal, and consist1ng of a barrel of any length, the shape of the thrust l1ne, set up by this weight load of the arch alone, lselllpticalJ and the tendency of the thrust line 1s to pass out of the arch as the spring lines 12. are approached. Now, if this arcll ring rotated until it were in a vertical position, this tendency to form an eliptlcal thrust line will disappoar in direct proportion as the angle of its slope decreases until at the vertical pos1tion it would be zero and in the vertical position, the line would pass at all parts of the arch ring through the center of gravity of any radial plane, passing through the arch ring. Hence, in the vertical position, the 11ne of pressure will be in the exact center of the arch ring. N" Oll, if this same horizontal arch were considered as being loaded with a superimposed water load, and if the weight load were eliminated, it can be seen that the ring would necessarily be thiner at. the crown line than a. t the spring lines, the shape of the intra( los being a pointed ell1pse and of the extrados a semi circular arc. so that the tendency would be twward the reduotion to the thickness at tho crown line, gradually increasing to the sprlrlg lines when there is any departure from a vertical arch and this difference in thicknes3 that would carry the load at a given unit strose would be the maximum in the horizoltal arch. In the case of an arch having a croWn line slope of 36° 52' J the water load at the crown 11ne and at the sprlne lines varies as the taneent of the angle of the slope. The stresses eet up by the water load In any portion of the arch rine will, therefore, vary in a like proportion. The normal or radial component of the weight load will be the maximum at the crown line and dimin1sh to zero ~ t the spring lines. The component of the weight load parallel to the crown line will be a minimum at the crown line~ and will be a maximum at the spring line. The angle of departure of tho arch from the 13. vertical being less than 45°, at which the components of the weight load will become equalJ the radial component is always less or greater than the crown line component in proportion to the angle of the slope of the arch. With the new consideration of. position of the arch ring, we have a water pressure increasing from the crown line setting up corresponding increasing tangential stresses, and we have the normal or radial component of the weight load acting in conjunction with the water load but as its magnitude is maximum at the crown line where the water pressure has its minimum, and as the thrust line set up by the weiGht load component tends toward the extrados of the arch ring, and as the thrust line set up by the water load tends toward the intrados due to the tendency to bending at the spring lines caused by the deformation of the arch ring at the crown line, and as the crown line component of the weight load becomes more active, being minimum at the crown line and maximum at the spring line, the, tendency is to decrease the deformation and to balance the load on the abutments. In all of the considerations above, the full semi circular arch has been conSidered, but in practice the arch having an arc, known as the " economic arc", is used, which arc will vary with the other influences determining it from about 136° to 144 u • The use of a smaller arc cuts off the legs of the semi circular arch at such a point that the tendenCies to departure of the thrust lines at the spring line are very much smaller than they are with a full semi circular arch, and also the ratio of the thickness of the arch ring to the radius has been increased likewise, consequently, the tendency of the normal weight load thrust line l4D toward the outside of the arch ring at this new position of the spring linea is much reduced, the tendency to deformation has been decreased by the ratio of the arc length to the radius, thus reducing the tendency to unbalanced water loading, the actual defor~ matton, and consequently, bending action action 1s also reduced, so that with the slope adopted, the thrust line will follow so nearly the exact middle or the arch ring for all of the loads applied, that the departure is negligible. In the foregoing discussion, only the weight load of the arch ring of the masonry and the water prABsure have been considered, In addition to then:; l forces ctlng upon an arch r tng , there will be also introduoed initial stresses, and stresses due to, tempera~ ture changes. Considering first the init. ial stresses, whicn. w111 depend on the method of performine the work of laying the material, and this will be very slight in amount in any event. If concrete is laid and is allowect to set under water, the rise in the tempera~ ture due to the settio2 is mimimized by the cooling effect of the water. Concrete setting under water has been determined by the best author! 1.166 to increase s1ightly in bulk during tho pt'ooess of setting, an~ if setting in the air, and if it 1s allowed to dry out during the process or setting, it will diminish in volume sllehtly) so that if the concrete ware laid in such a manner that it was always wot after laying, 1t would swell slieht1y, but if left to dry after setting, it would shrink slightly. Now, if the concrete were kept at a cortain degree of moisture during the process of setting, which 1s the condition that applies in actual practice in this type of construction, thnn t would neither ewell 15. nor ahrink in the process and there would be no initial straIns set up. These initial strains, in any event, baing so near the ne. ut. r a L point. of expansion and contraction, depending on the degree of moisture, applied during the pr oce ss of setting, are ve!,~ r small. In addition to the above, the stresses that can take place, are thus set up by temporature change. in the concrete, and their dogree will depend on the range of the temperature to be considered. The range of temperature, setting up temperature stresses, will be t. he departure froom the nean temperature at which the concr et e took its final set, anc' the amount of the stresses will depend on the magnitude of the range of tem, erattwe. A rise in temperature above th07 normal at which the concrete was placed would hnve a tendency to incroase the bulk of the material in the arch ring, and to oppose the tendency of the water load and of the normal weight component to deform it, and hence it would act with the crown line component of the water load to maitain Q balan ced condition of loading. A fall in temperaturo will cause a decrease 1n the dimensions in the material in the arch ring, and will have a tendency to increase the deformation due to the influance of the water load and of the normal compo~ ent of the we1ght load, and would be assisted or neutralized in its action by the increased bending reaction, and also by the cr0~ n line component of the weight lOF. d. It can readIly be seen that the action of any foroe or set of forces on the arch ring 1s confined to one arch as the resultant of these in each arch is counterbalanced by those in any other adjacent arch. The expansion or contraction due to a temperature Change can only be throughout the arc or one arch and 1s not cumu~ at1ve. 16. The range of temperature of the portion of the wall always under water can only be the amount of the departure from the mean at which the concrete took its set to that of the mean annual temperature of the water itself, for the wall in contact with the water will take the temperature of the water, and will be but slightly influenced by the temperature of the air on the outside, by reason of its inability to cormnunicate its heat to the wall. Under actual conditions, however, there would be a slight increase in temperature in the intrados of the wall over that at the water face, and this unequality will tend toward a greater degree of expansion at the intrados than at the extrados, thus the water loaded portion WOUld, by this unequal · d~ stribution of the temperature stresses, be assisted in the prevention of deformation. The extreme range of the winter and summer temperatures come only against those portions of the wall that are exposed during the entire year above water, and as these portions of the wall have no stresses in them, excepting the weight load and the temperature stress, all of which combined are but a very small portion of the allowable maximum stress due to the water load, they are never stressed to anywhere near the amount they are calculated to bear when loaded with water. The range of temperature tending to set up stresses will be the departure from the lUcan temperature at which the concrete took its final set. Let us take as an example the ranee of temperature and conditions at the site of the Big Meadows Dam, Plumas COlmty, California, being erected for the Great Western Power Company. The mean temperature of the setting of the concrete resulting from 17. the mean of the day and night temperat~ combined with the mean day and night temperatures of the water with which the concre~ is mixed w1ll give the final mean temperature at which the concrete takes its final set. The mean of all of these at the Big Meadows Site wIll be very close to 50° F., and the range of temperature from mean to extreme of the portions of the dam out of the water and exposed to the air on both sides would be about 40° F. above and 60° F. below this mean. The extreme range of the air temperatures 1s from  20° F. to + lOOoF., a maximum for summer conditions, but the change of the temperature 1n the walls would not reach these extremes of air temperature either way because they will always retain some ot the heat of the daytime throughout the night and vice verea, and will never reach the full degree of the air surrounding them. These would be the conditions fGr the portions out of water throughout all of the seasons, which meane, of course, the extreme top where all of the walls are very much over sized for the ptresses. The portions of the wall under water at all times of the year would range from the minimum or + 32° F. to a possible + 50oF., so that tho only change in these portions wOl~ d be a drop from the mean temperature of the setting time + 50 o F. to + 32° F., a difference or 18° F., so that the extreme condItions of shrinkage would be that set up by a temperature change of 18° F. Should the concrete be laid in the Spring or Fall, when the water and air both have a lower mean temperature, and such is now the c&. e of the concrete being laid at the present time, the mean temperature will be nearer 45° F., which in the case of shrinkage would make a differenoe of 13° F. as the range. This range of temperature w111 18. I I ,, cause the arch rings, as designed at the Big Meadows Dam, to shorten in their length .0027 of 1 foot or. 032 of an inch, whereas they are all reinforced to stand a change of temperature of 250 o F. without danger of localizing cracks. The maximum change either way that could take place in any part of the structure would be on those portions of the dam that extended above the water level and were subjected to the full range of temperature of winter and summer having to carry no other load would be .005 of a toot or .06 of an inch 1n the length of any other ring. It would be practically impossible to measure the deflection at the crown llne at this maximum range of temperature. It must also be borne 1n mind that the thickness of the arch ring itself alSO affects the resultant deflection due to the temperature changes and the resistance to bending of the arch ring becomes greater with its thickness. The stresses Bet up by the water load and by the weight or the masonry are affected by the length of arc and the position of the arch in space, but the temperature change stresses, whether plus or minus, w11l act equally for any position of the mater1als and are only affected as they are more or less neutralized by the thickness of the arch ring. The rate of expansion or contraction of conorete is given by the authorities as .0000055 for a unit of length for 1° F., and of steel .000006 for the same conditions, so that it is universally recognized that steel and concrete pract~ cally expand and contract at an equal rate, and that if reinforcement be introduced 1n the concrete having an area equal to 1 of the con 275 crete, no cracking will take place regardless of the change in temperature. 19. Returning to the stresses in the arch ring, the simplest form of a water loaded arch ring that can be considered is that of a vertIcal cylinder assumed to rest on a solid base at its bottom, the top of which is even with the water surface, and whereiq the water pressure and thickness are aero at the top, and wherein the thickness increases to a solid in proportion to the 1ncrease of the water pressure for the depth by using a predetermined radius of extrados and unit stress due to the water pressure in the arch rings. Let us consider the stresses in such a cylinder having for its dimensions a radius of extrados of 16 feet, and a safe load or unit stress of 16 tons per square foot. We will then have a " . thickness that w111 increase at the rate of the water pressure. In the nomenclature, let p = water pressure on one sq. ft. in tons ; .01235 the weight of one cu. ~ t. of water 1n tons. h : the depth of water at any seotion. p = the pressure per sq. ft. 1n tons for any given height h = ph r = the radius of extrados of the cylinder = 16 T = the tangential thrus. t 1n the ar ch ring due to the water pressure P S = the safe load or unit stress in the concrete per sq. ft. in tons :: 16 t ::: the thickness at any section required to resist a thrust T at the safe load or unit stress C ::: the weight of a cu ft. of concrete in tons = .075 W ::: the weight of any section of wall above the section under consideration Then, for any given value of h, ph ::: P ' 1' :: Pr, t =....!!:., w :: th G S 2' 20. . ( i 1 • / I i rate, ph increases Let us consider that the water pressure p~ at a certain and that W, the masonry we1ght load increases at the rate Now for any given case p 1s a constant r " " » 5 ft " " C " " h of 2th C. Hence using the above numerical values: Let P = Klh and W = K2h 2, the values of Kl and K2 being respectively Kl = P = .03125 and K2 = pre or in numerical values of p S and r ( as above) = 25 ,001172 W = .001172 h2 ( 1) P = .03125 h ( 2) In order to find the depth at wh1ch the water pressure e~ uals the weight load of the masonry superimposed, we equate ( 1) and ( 2) and solve for h. h = .03125 .001172 = 26.6 ft., the depth of the neutral point at which the weight load and water pressure are aqual. Let Fig. I ( a) represent a horizontal section of a vertical cylinder on a line an an A A and Fig. I ( b) a vertical section of the same on the line B B. Since the water pressure on a curved surface 1s equivalent to the pressure on the orthographic ~ rojeotlon of the surface on a plane area and 1s uniformly distributed, then if a plane 1s conoeived to cut the cylinder through the center 21. \ , \ \ ,\ vertically, the pressure on the section cut off' by this plane of the cylinder will be equally divided and uniformly distributed from extrados to intradoB, provided the seml circular arch thuB created were free to move on the other half of the cylinder along this plane area due to the deformation caused by the radial water pressure of the load, the remaining half of the cylinder being assumed to be resolved into fixed abutments. The weight load of the masonry of the arch will evidently have no resultant effective pressure against the spring l1nes of the arch because its load Is applied vertically. The water pressure, however, 1s transmitted to the abutments as tangential forces set up by the water pressure on the curved surface. As these tangential torces vary as the tangential angle, and as this' angle varies as the radius of the eztrados, the tangential forces set up for the water pressure due to any depth vary as the radius of the extrados of the arch ring varies. The water pressure acting through the tangential forces tends to compress all parts of the arch ring under the above named • condItions uniformly, tending to shorten its length and to thicken it in all directions and to reduce lts radii. The Weight load alSO acts to deform any given section and acts vertically tend1ng to counteract the upward and downward deformation due to the water pressure ' and to increase the radial deformation. If any arch ring one unit 1n height were conSidered, and its position were above the neutral height, the water pressure would be in excess of the weight load, and at the neutral point the pressure wIll be eq. al_ but for any point below the neutral point, as determined above, the weight pre~ sure will exceed the water pressure so that 22. !! the resultant deformation would depend on the position of the ring considered 1n its relat10n to the depth below the surface of the water. The tangential forces act along the 11ne normal to the center of any radial plane through the arch ring, or the surface of any assumed voussoir, wh1le the weight load acts vertically at the l1ne between the outer and middle third of the ring and under the center of gravity of the triangular surface representing a section of this wall, thUB creating an eccentric loading of the arch ring vert1cally, the degree ot eccentricity affecting the deformation, depending on the relative influence of the water pressure and the weight load. The influence of the weight load deformation should be considered first and that of the water load deformation and its influence to act with or counteract the deformation due to the weight load should be next considered. Let us take as a numerical example a submerged cylinder with the dimensions named above and considering the stresses as taking place in a ring of this cylinder. the ring to be 1 foot high, and the center of the ring to be 100 feet below the surface of the water, the problem being to find the deformation due to the weight load alone. 23. the resultant deformation would depend on the position of the ring considered in its relation to the depth below tile surface of the water. The tangential forces act along the line normal to the center of any radial plane through the arch ring, or the surface of any assumed voussoir, while the weieht load acts vertically at the line between the outer and middle third of the r ing and under the center of gravity of the triangular surface representing a section of this wall, thus creating an eccentric loading of ~ he arch ring vertically, the degree of eccentricity affecting the deformation, depending on the relative influence of the water pressure and the weight load. The influence of the weight load deformation should be considered first and that of the water load deformation and its influence to act with or counteract the deformation due to the weight load should be next considered. Let us take as a numerical example a submerged cylinder with the dimensions named above and considering the stresses as taking place in a ring of this cy1inder, the ring to be 1 foot high, and the center of the ring to be 100 feet below the surface of the water, the problem being to find the deformation due to the weight load alone. 23. the resultant deformation would depend on the position of the rIng considered In 1ts rolntlon to the depth below the surtace at the wat.,. The tangential foroes act alone the line normal to the center or any radial plane throUGh tho arch ring. or the surtao. or any assumed vou. eolr, while the weight load aot. vertically lat the line between the outer and middle third of the ring and under the center of sravlty or the triangular surrace repreeen'toins a section or th1s wall. thus creatln& an ecoentrlc load. lng; ot the arcb rins vert1oally. the 4egree of eccentricity affeotlng the detormatlon, depending on the relative influence of the water pressure anel tho woigbt load. The influence of the w@ lght load cieformatlon ohould. be considered first. and that ot the water load deformation and Ita influence to act wit. h or counteract the deformat1on due to the walght load should be next oonsidered. Let us take as a numerical example a submerged cylinder with the dimensions named above and considering the stresses aa taking place In 0. ring or th18 cylinder, tho ring to be 1 toot hiBb. and the center ot the rins to be 100 feet below the surface of the water, tho problem belnc to find the deformation due to the weight load alon.•• 23. . . • r Let rl ::: Radius of intrados ::: 16 ft. r 12.875 ftD 2 :;: Radius of extrados ;: h ::: Depth of center of section under consideration :;: 100 ft. E ::: Modulus of elasticity::: 180,000 tons per sq. ft. A :;: Area of inside circle before compression B :;: Area of outside circle before compression . . AI :;: Added area intrados after compression II II extrados II Thickness of ring = 3.125 ft. s :;: Safe working stress :;: 16 tons per sq. ft. W :;: Weight of Cylind, er Z ::: Total cross section of ring after compression Dl = Decrease of rl at intrados D2 = Increase of r2 at extradoso Let us consider a slice of the cylinder one foot long whose mean thickness lies at a depth of 100 feet. Then t = (. 03125 x 100 x 16) = 30125 16 The dead load of the superimposed masonry above the section under consideration tends to deform that section by shortening it longitudinally and thickening it radially. From the equation of elongation 24. r water Resources Center ArchiveS EASr:' 08D 3 June 29, 19l7. f Eastwood Multiple Arched Reinforced concrete Dams Safest, strongest and most economical and scient. ific type of structure for impo'J. ndine wat. er t In Service and Bu1lding Height feet Bume LRke Dam, near ? resno, Cal.  ~~ 61 Big Bear Valle~ yanDaEme, rnanredQirneRedGloR. n, dsG, al. Los Verj~ lS Dam, near Oroville, p. utte C,,), , Gal. Kennedy Dam, nekr . Ttlckson, ArnA. dorCo., Cal. Argonaut Dam, dittO Mountain Dell Dam, S~ lt Lak~ City) utah. Malad Dam, Malad, Idaho Murray Dam, near 58. n Di"~ o, Ga 1. Carroll Dam, nSe~ anr DiTie: gsocoGndol. d, o, Gal. San Dieguito Dam, ditto IMITATIONS Reclamation Se~ vice Dam, Oregon 2 Small dams in Sierra Nevada Mountains, ca. lifornia Rock Creek D& m, near ~ uburn, Cal. ' ll; Small dams in :;. uchigal1 677 92 363 60 350 64 50 450 150 630 110 50S 112 ]. 30 650 50 700 r. apacity of Reservoir in millions of Gallonfl 4,600 22,880 846 Debris dam Df'JbriS dam \ 
OCLC number  606639373 



W 


