# Design Constants for Interior Cylindrical Concrete Shells

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• IEI I I PORTLAND CEMENT

ASSOCIATION

Design Constants for Interior Cylindrical

Concrete Shells

• Design Constants forInterior Cylindrical Concrete Shells

In the discussion of the ACI article Cylindrical SheJJAnalysis Simplified by Beam Method by James Chinn,design constants based on a linear transverse distributionof longitudinal strains, or in other words based on the

assumption that the shells behave like a beam, were

presented by Messrs. Parme and Comer, These constantsprovided a convenient method of readily evaluating the

internal forces and moments created in long and inintermediate length cylindrical shells by uniform arrddead load, While these constants are perfectlysatisfactory for long shells and were recommended in thisrange, some vagueness regnrding the applicable limit fnrintermediate length shells existed. This uncertainty was

caused primarily because the validity of the assumption

of linear strain depends not only on the ratio of radius to

longitudinal span but is as well a function of the subtended

@Portand Cement AssoclaUon196o

angle and the ratio of thickness to radius. Because of theinterdependence of the effect of these factors, no preciselimits for the beam method could be given.

To remove this uncertainty and at the same time reducethe labor involved in the design of cylindrical shellswhich cannot be adequately treated by the beam method,

a new series of compwable constants are presented inTable 1. These constants have been computed on the

basis of the shell theory expounded in ASCE Manual No.31 Design of Cylindrical Concrete Shell Roofs,Consequently these newer constants in contrast to thosepreviously given are a function of r/L and r/t as well as

the subtended angle, @k. To avoid interpolation as muchas possible, values are given for the three r/t values of 100,

200, and 300 and for six values of r/L with r/L varying

from a low of 0.4 to a high of 2.6, For @k less than 45

• deg., it was found that the modified beam method was

sufficiently accurate for all values of r/L less than 0.6.Thus for the portion of Table 1 dealing with @k less than45 deg., the internal forces ure only given for values of

r/L greater than 0.6. When @kis greater than 45 deg., itwas found necessary to include an r/L as low as 0.4 toprovide a gond transition from values as computed by thebearrr method to those computed by shell theory.

It should be noted that although values are tabulatedfor r/t = 300, which represents a shell beyond practicallimit, they have been included to avoid extrapolation forcases of r/t beyond 200. Likewise the selection of r/L =2.6 represents an arbhry limit. For values of r/L greater

than those listed the internal forces are concentrated near

the edge. For this reason, the arrangement of Table 1 isnot suitable for such shells.

Values have been given only for load vnrying as thedeud weight. This is due to the fact that numerouscomparisons made with different r/L values indicate thatthe effect of a uniform load could be very closely

approximated by an equivalent dead weight by the simpleexpression that

[)

sirrOkPd=Pu ~

The constants have been determined on the basis thattransverse and horizontal displacement of thelongitudinal edges of the shell are prevented, They arethus applicable to interior barrels in which restraint tosuch movement is provided by adjacent barrels. However

they can be applied with tolerable accuracy to the interiorhalf of the exterior bay since the effect of disturbance ofloads on the far edge has only minor influence on the first

interior valley. This is especially true since to preventexcessive deflection of the free edge an edge beam should

always be provided (except for long shells with shortchord width) at the exterior edge.

Detcrrnination of the internal forces in cylindrical

shells subject to uniform longitudinal loading by the shelltheory requires that the actual load he approximated asthe sum of pnrtial loads varying sinusoidally according toa Fourier Series in the longitudinal direction. From a

practical point of view generally only the first or at mosttwo partial loads are used with adjustments made

especially to the value of shear on the basis of staticalrequirements. However since Table 1 was prepared bymeans of an electronic computer, the algebraic sum offour partial loads was used to avoid the need of any

adjustment. Even with this number of loads to achievesufficient accuracy it was found necessary in some cases

to employ Eulers convergence technique, The use ofsuch care should not hc interpreted however as needed orjustified on the basis uf underlying assumptions. Its worthrests solely on the fact that it permitted a more accurate

comparison of values as the parameters r/t and r/L arevaried, and enabled a more precise examination of the

variation of the internal forces in the longitudinaldirection.

In this connection, the constants in Table 1 give only

the transverse distribution of forces at midspan and at the

support as noted by the footnote in Table 1, with no

indication of the longitudinal distribution of forces. Thereason for this is that the exact expression for longitudinaldistribution even for simply supported shells is highly

complex involving four functions. Fortunately within the

range of the tabulated values the longitudinal dktributioncan be approximated by well recognized relationship.

For example, as shown in Fig. 1, the distribution of T.

as might be anticipated follows very closely that given bya parabolic distribution as the case of a uniform load onabeam even for widely different shells represented by r/L

equal to 0.6 and 2.6. Although the curves shown in Fig.1 havehcen computed on thehasis of Ok = 27.5 deg., theyare typical of those for other angles. A sinusoidal

distribution of T, would also be satisfactory.With respect to To, for design purposes this force can

be assumed to be uniform in the longitudinal direction ascan be inferred from Fig. 2, Because the analysis has beenbased on the prescribed boundary condition that the shell

is supported by a rigid member at x = O and x = L, thevalue of T\$ decremes theoretically to zero at the support.The transition from zero to the full value however takesplace over a very short interval. Thus, especially furvalues near the crown, the assumption of a unifomrdistribution of T+, is justified. Distribution of T+ in the

valley can also be considered uniform even though acareful evaluation of the distribution in this area indicatessome departure from uniform distribution near thesupport. The computed variation near the support may

be due however to the sensitivity of the results to the

number of load terrrrs used. This is primarily due to thetlct that the absolute value is generally quite smallcompared to the crown value with the final result equalto the difference of almost like vahres, Because the valuesare small and have almost no effect on the design, theassumption of uniform distribution of T+ in the valley is

justified.As in the case of the distribution of the Tx furces, the

distribution of shear can be assumed to be like that i abeam with the shear varying Iirrcwly from a maximum

value at the support to zero value at midspan. As shownin Fig. 3, the distribution as computed by the shell theorygives slightly higher values, but the variation from the

linear distribution is negligible.

There is one important aspect of shear distributionwhich warrants some comments, As shown in Fig. 4, inwhich a plot of the transverse distribution nf shear at

various sections along the shell are superimposed on eachother, the shear tends to be concentrated towards the

valley as the support is approached, From this plot itshould not be inferred that the magnitude of shear doesnot decrease proportionally to the distance from thesupport. For purpose of clarity in presentation of the

variation in transverse distribution, all values have been

plotted in terms of the value of shear at CT= 0.5 @k. The

values hence are all relative. While this change in

transverse shear distribution is insignificant with respectto its effect on the direct stresses in the shell, it has a

pronounced effect on the longitudinal moment

distribution.

2

• As in the case of To, the boundary condition of

supports rigid in the transverse dkection leads to zero

moment at the support. For long shells as discussed inReference 1, the moment increases at a vuriable rate fromzero at the support to a maximum value near the quarter

point, and there remains essentially uniform to midspan.On the other hand, for shells in the range covered by Table

1, the magnitude of the moment increases almostparabolically from the support to midsparr as shown inFig. 5, especially for the moment at the crown. At the

valley, the moment increases at a slightly faster rate for

smaller r/L values as can be seen by a comparison of thecurves of Fig. 5 and Fig. 6. In determining the amount oftransverse reinforcement for shells with rlL about 1.0, dueaccount should be taken of the greater curvature of thelongitudinal distribution of M+.

ContinuityThe design consta;ts of Table 1 are for simply supportedshells, i.e., the supports are assumed to offer no lateral

restraint. Thus it will be found that taking the summationof the moment of Tx forces at midspan about any axis will

equal to wL2/8, Nevertheless the constants can be appliedwithout any great loss of accuracy to shells continuous in

the longitudhral direction. The effect of continuity as one

might expect from

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