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TEMPERATURE CONTROL OF CONCRETE-Sec. 7-12 115

exposure temperatures as sinusoidal waves with applicable periods of 1 day, 1 week or 2 weeks depending upon the severity of the weather to be used for the design, and 1 year. Solar radiation is then added to obtain the final range of mean concrete temperature.For average (mean) weather conditions, theambient air temperatures are obtained from aplotting of the mean monthly air temperatureson a year scale. For usual and extreme weather conditions, the above ambient air temperatures are adjusted for a 7-day period and a 14-day period, respectively, at the high and low points of the annual curve. The amount of the adjustment for these weather conditions is described in subsection (a) below.The thickness of section for these studies ismeasured along lines normal to the exposedsurfaces, the intersection of the normals beingequidistant from the two faces.(a) Ambient Air Temperatures. -Whencomputing the range of mean concretetemperature, mean daily, mean monthly, andmean annual air temperatures are used. Thetheory applies the daily and annual airt em peratures as sinusoidal variations oftemperature, even though the cycles are nottrue sine waves. The annual and dailyamplitudes are assumed to be the same for allweather conditions.To account for the maximum and minimumrecorded air temperatures, a third andsomewhat arbitrary temperature cycle isassumed. This temperature variation isassociated with the movements of barometricpressures and storms across the country. Plotsthroughout the western part of the UnitedStates show from one to two cycles per month.Arbitrarily, this third temperature variation isassumed as a sine wave with either a 7-day or14-day period for usual weather conditions and extreme weather conditions, respectively. For extreme weather conditions, the amplitudes of the arbitrary cycle are assigned numerical values which, when added to the amplitudes of the daily and annual cycles, will account for the actual maximum and minimum recorded air temperatures at the site. For usual weather conditions, these amplitudes are assigned values which account for temperatures halfway

between the mean monthly maximum (minimum) and the maximum (minimum)recorded. When computing the mean concretetemperature condition, no third cycle is used.(b) Reservoir Water Temperatures. -Thereservoir water temperatures used indetermining the range of mean concretetemperature for a proposed dam are thosetemperatures which will occur after thereservoir is in operation. These reservoir water temperatures vary with depth, and for all practical purposes can be considered to have only an annual cycle. For preliminary studies, the range of mean concrete temperature with full reservoir is the normal condition. For final designs, stage construction should be taken into consideration and the design reservoir operation used. When the reservoir is to befilled or partially filled before concretetemperatures have reached their final stage oftemperature equilibrium, further studies areneeded for the particular condition.( c ) S o l a r R a d i a t i o n E f f e c t . - T h edownstream face of a dam, and the upstreamface when not covered by water, receives anappreciable amount of radiant heat from thesun. This has the effect of warming theconcrete surface above the surrounding airtemperature. The amount of this temperaturerise above the air temperature was recorded onthe faces of several dams in the western portionof the United States. These data were thencorrelated with theoretical studies which tookinto consideration varying slopes, orientationof the exposed faces, and latitudes. The results of these studies are presented in reference [ 11 .l These theoretical temperature rises due to solarradiation should be corrected by a terrain factor obtained from an east-west profile of the site terrain. This is required because thetheoretical computations assumed a horizontalplane at the base of the structure, and theeffect of the surrounding terrain is to block out some hours of sunshine. This terrain factor will vary with elevation and from abutment to abutment.

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‘Numbers in brackets refer to items in bibliography, sec.7-31.

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116 DESIGN OF GRAVITY DAMS (d) Amplitudes of Concrete Temperatures.-T h e r a n g e o r a m p l i t u d e o f c o n c r e t e temperatures is determined by applying the above-described external sinusoidal air andwater temperatures to the edges of atheoretical flat slab, the width of the slab being equal to the thickness of the dam at theelevation under consideration. The problem isidealized by assuming that no heat flows in adirection normal to the slab. The law ofsuperposition is used in that the finalamplitude in the concrete slab is the sum of the amplitudes obtained from the differentsinusoidal variations.To apply the theoretical heat flow in apractical manner, unit values are assumed forthe several variables and a curve is drawn toshow the ratio of the variation of the meantemperature of the slab to the variation of theexternal temperature. Figure 7-1 shows therelationship thus derived for temperaturevariations in flat slabs exposed to sinusoidalvariations for h* = 1.00 square foot per day, aperiod of 1 day, and a thickness of slab of II . A correlation equation is given to take into account the actual thickness of dam, diffusivity constant, a n d p e r i o d of time. The computations are shown in figures 7-2 and 7-3.* For the actual thickness of dam, I,, a value of I, is obtained from the correlationequation for each of the air temperature cycles.For each value of I, , a ratio of the variation ofmean concrete temperature to the variation ofexternal temperature is obtained. The sums ofthe products of these ratios and their respectiveamplitudes are algebraically added to andsubtracted from the mean annual airtemperature to obtain mean concretetemperatures for the condition of air on bothfaces. Mean concrete temperatures are thenobtained in the same manner for a fictitiouscondition of water on both faces, and the twoconditions are simply averaged together toobtain the condition of air on one face andwater on the other. Solar radiation values arethen added to obtain the final range of meanconcrete temperatures.2 These and sev~al other figures and tables in this chapterwere reprinted from Bureau of Reclamation EngineeringMonograph No. 34, listed as reference ] 11 in the bibliography, sec. 7-3 1.

7-13. Temperature Gradients. –Temperature distributions in a mass where boundary conditions vary with time are easily determined by the Schmidt method. (See references [ 11, [23, [3], [4] .) This method is generally used for temperature studies of mass concrete structures when the temperature gradient or distribution across the section is desired. The depth of freezing, and temperature distribution after placement are typical of the solutions which can be obtained by this step-by-step method. Different exposure temperatures on the two faces of the theoretical slab and the autogenous heat of hydration are easily taken into consideration.An early objection to the Schmidt methodof temperature computation was the timerequired to complete the step-by-stepcomputation. This has been overcome by theuse of electronic data processing machineswhich save many man-hours of work. Programs have been developed which will take into consideration any thickness of section, varying exposures on the two faces of the slab, variable initial temperatures, a varying heat of hydration with respect to time, and increasing the thickness of slab at regular intervals as would occur when lifts of concrete are placedon previously placed lifts.A s e c o n d m e t h o d o f t e m p e r a t u r ecomputation in mass concrete which isparticularly adaptable to thick walls andplacement lifts near the rock foundation wasdevised by R. W. Carlson. This method isdescribed in reference [ 51. It, like the Schmidtmethod, is essentially a step-by-step integrationwhich can be simplified by selection of certainvariables. Conditions such as initialtemperature distributions, diffusivity, andadiabatic temperature rise must be known orassumed. Carlson’s method can also bemodified to take into account the flow of heatbetween different materials. This would be thecase where insulated or partially insulatedforms are used, or where concrete lifts areplaced on rock foundations.T h e v a r i a t i o n i n t e m p e r a t u r e i n asemi-infinite solid at any particular point canalso be estimated from figure 7-4. Thisillustration gives the ratio of the temperature