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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 S’heJJAnalysis 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 perfectly
satisfactory 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 Euler’s 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 beam behavior is to mdically changethe magnitude and sense of the T. forces without
affecting greatly the other internal forces such as TO andM+ However, while continuity alters greatly the
longitudinal distribution of T, forces, previousinvestigations have shown that only minor change in the
transverse distribution uccurs.Without becoming involved in complex mathematics,
a qualitative appraisal Of the effect Of continuity On thetransverse distribution can be made by recalling that thetmasverse distribution of Tx is a function solely of the
relative proportions of transverse to longitudinal
displacement. When as in the case of long shell, thevertical deflection of the edge measured with respect to
the crown of a unit strip at midspan is small compared tothe deflection of the same point measured longitudinally,the distribution of TX in the transverse direction is linearand thus is similar to that of the fiber stress in a beam. As
the relative displacement in the transverse direction tothat in the longitudinal direction increases, the transverse
distribution of T, departs from a linear pattern becomingcurvilinear with a decrease in the slope of the stress curvebelow the neutral axis. Since continuity decreases thedeflection of the section at midspan with respect to thesupport, the effect of continuity is to increase the ratio of
transverse to longitudinal deflection.From this it follows that the transverse distribution of
T. forces in a continuous shell has slightly greatercurvature than that of a simply supported shell of the same
span aud radius. An inkling of the relative difference
between the two distributions can be obtained bycomparing the design constants in Table 1 for any two r/L
values with one r/L being 1.4 times the other. A pint of
the two transverse distribution curves will show that
while there might be significant change in the magnitude
of Tx at the edge of the shell, the total urea below theneutral axis will be about the same for both curves. Ingeneral, the difference will not be greater than 3 or 4percent. Because of this, it is sufficiently accurate to use
the transverse distribution of stresses of a simplysupported shell, irrespective of the degree of continuity.
As shown by Dr. Olev Olsen in the article CcmrirumusShells in the Proceedings of the Second Symposium ofConcrete Shell Roof Construction, the transversedistribution of T. for all practical purposes is uniform
tbroughuut the length of the shell.By similar deductive reasoning, the longitudinal
distribution can also be accurately estimated. In longbarrel shells, because the transverse distribution is almost
linear, it is apparent that the magnihrde of T. at anysection will be to the Tx in a simply supported shell as the
ratio of the moment in a continuous beam of equal lengthand support condition is to the simple beam bending
moment. For short barrel shells, because of the effect ofsheur strain, the longitudinal stresses over the support willbe somewhdt greater than that indicated by the analogy
to a continuous beam. This increase, which will be slightfor the range of shells covered in Table 1, is of littleconsequence since an underestimate of the intensity of the
forces at the support will be compensated by anoverestimate of the forces in the region of positivemoment. Consequently proportioning the longitudinal
forces on the basis of the variation of the momentoccurring in a continuous beam can be applied withoutany decrease in the ultimate capacity.
The change in the transverse distribution of the T.
forces caused by continuity will naturally be reflected inthe transverse distribution of the shearing forces.However because very slight change in the location of tbeneutral axis occurs, the position of the peak shear willundoubtedly be quite insensitive to the effect of
continuity, and may therefore be considered to occur at
the same phace as in a simply supported shell. On theother hand the dnwnward drift of the tensile forces willcause the sh~ar curve to have more of a bulge near thevalley. Since the shear stresses in this region are notgenerally the critical ones, inaccuracy in this area is
relatively unimportant.With respect to the longitudinal distribution of shear,
the reasoning presented for TX applies. Refinementsaimed at increasing the accuracy of determining the
intensity of the shear forces are hardly warranted in viewof the cnmmon practice of providing shear resistance.Generally to avoid vuriable spacing, shear reinforcement
is placed uniformly and thus leads always to overdesignbecause of the large number of bdrs crossing a section of
principal stresses. For this reason, modification of tbeshear forces in a shell to correspond to the total shear in
a continuous beam is satisfactory.
ExampleThe ease with which the internal forces can be cumputed
makes the use of Table 1 self-explanatory. In all cases,
3
the internal force is equal to the product of a multiplier
and the design constants. The multiplier shown in the
third row of Table 2 equals the product of the load timesvarious powers of the dimension indicated in the headingof Table 1. However to avoid misinterpretation thecomputation required for a typical interior sheIl will beoutlined. From the dimensions given in Table 2
If the shell is continuous in the longitudinal direction,
the forces detemrined in Table 2 cam be modified as
previously discussed. For example if two 50-ft long shells
we continuous over a central arch, then the forces aremultiplied by the ratio of moments in a beam of similarcontinuity to the moment in a simply supported beam.Since the moment over a central support is -wL2/8,
obviously the ratio is -1.0. The ratio to be applied to the
forces at midspan isr/t=45 X 12/4= 135
WL2/~ 6— = 0.50WL218
r/L = 45/50= 0.90
Inspctimr of the constants in Table 1 show that thereis only slight differences in the constants for vafrres of r/tand r/L in the range with%= 25 deg. and 27.5 deg. Assuch, the design cnnstants will k selected from r/L= 1,0and r/t = 100. But interpolation for the specific @k isrecommended. To simplify this task, advantage will be
taken of the fact that linear interpolation can be aehlevedby adding algebraically a fixed ratio of the two adjacentvalues. For this example, tbe constants for @k = 25 deg.
are multiplied by
Similarly, the shear forces are rdtered by the ratio ofcontinuous beam shear at the interior support to that in asimple beam. The ratio is
%= 1.25
The shear forces at the outer support are given by thefollowing ratio:
3wLJ8 = ,75
wfJ2
As discussed above, continuity does not cause T$ andMO to change significantly.while the constants for 27,5 deg. are multiplied by
1- 0.45= 0.55 Notationh —
.
.
.
total vertical height of shell from edge to crown
vertical height of shell measured from edgelength of shell between supportscenterline radius of shellthickness of shell
longitudinal distance measured from the left
Thus the design constant for Tx at the crown for~k = 26,4
deg., r/L= 1,0 andr/t = ltXl is YL
-(4.482 X 0.45 + 3.509X 0.55) = -3,947 rt
xwhich is recorded in the first row of numbers, secondcnlumn of Table 2. The other coefficients me obtained
in a similar manner.In accordance with the formula given on page 2 and
the intensity of load listed in Table 2, the equivalent dead
load for which the shell is to be designed is
supportangle measured from the right edge of shellangle subtended by the edge of shell measured
.
from the centerline axisintensity of uniform load on unit meaintensity of dead load on unit area
the direct force component in the transversedirection, considered positive when tensile
puPdT(j
.
.pd=50+30~=79pSf
The multiplier for TX therefore is
L2 2~ pd = ~ X 79 = 4390 lb/ft
TO at midspan of the shell
the direct force component in the longitudinaldirection, considered positive when tensile
.
In a similar mamner the other multipliers can be
obtained as readily. The product of these and thetabulated constants gives the internal forces in the shellwhich appear in the columns marked Force.
A graphical representation of the tabulated values forT, and MO is given in l+g. 7 for comparison with valuesas obtained by the beam methed. As to be expected, thevalue of Tx as computed by the shell theory is slightly
larger while the value of the moment M+ is slightly less.For design purposes the difference is negligible,
However, thk gond agreement holds only for the interiorshells. If the outer edge of the exterior shell is not
stiffened by an edge beam, marked increase in theintensity of Tx will occur at the edge,
Tx at midspan of the shell
. the tangential shearing force, considered positive
when it creates tension in the direction ofincreasing values of x and CDS at the transverse supportthe moment in the transverse direction,considered positive when it produces tension in
the inner fibers
MO at midspan of the shell.
load per foot of length of shell
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.062
. 0
00
- .0
0277
*She
-or
forc
es
ore
o+
wpp
orts
, ot
hers
or
e 01
mid
span
.
Tab
le1
Inte
rnal
Fo
rces
ino
Mu
ltip
leC
ylin
dri
cal
Sh
ell
Du
eto
Dee
dL
oad
+,’
55°
TX,
$[1
PdC
ol.(
1)[1
S*=
-L
PdC
Ol (
3)-’d
~
.*A
~
To
,r
[’,
]C
ol.
(2)
M+=
‘[p,
co,.
w]
r/L .4 .6 1.0 1.4
1.8
22
I,O
o$k
.75
$K.5
aqk
.25.
$~
o
Ioa$
k.7
5@
k.w
$~.2
5@k
o
I.oa
6k.7
5Q
k.3
0+k
.25Q
ko
Ioa+
k.7
5.$
k,5
a~.2
5& o
I.~$
k.7
5.$
kW
@k
.25$
ko
I,O
c+k
.75$
k.W
$k.2
5$k
o
*Sh
eer
r/+
z10
0
Tx
T+
s‘$
(1)
(21
(3)
(4)
-.6
40.
.764
-.6
42.5
322.
847
-.0
32.5
52-1
.157 .1
403.
977
.179
.231
-1.3
69 .599
5.74
8
.092
-.0
49-1
.083
-1.3
017.
382
.264
-.0
13-
.752
-1.8
468.
970
.310
.059
-.4
95-2
.201
10.4
67
1.31
81.
143
-.6
22.0
11.1
91
-1.
097
-1.
097
-.8
36-
.170
.147
.951
1.02
81.
001
.384
.103
.%6
.990
-1.
006
.509
.084
.989
-.9
73-
.983
.592
.073
-1.
001
-.9
69-
.959
-.6
42.0
65
.000
.528
1.17
41.
525
.000
.000
.193
.922
1.70
1.0
00
.000
.003
.616
1.80
1.0
00
.000
.081
.435
1.75
1.0
00
.000
.159
.342
1.65
b.0
00
.000
.202
.313
1.55
3.0
00
..0
121E
..0
049(
.009
9:.0
1053
..0
29%
.004
79.0
0298
.003
83.0
0728
.019
88
.000
11.0
0105
.OO
Q48
.003
85.0
1553
.00+
33:
-.0
003:
.00C
81.0
0226
-.0
073:
.00+
316
.000
0?.0
006(
.001
3!-
.005
5$
.000
0:.0
0001
.000
3!.0
008i
-.0
044;
rces
ore
0?su
pp
ort
s,o
ther
sar
eo
frn
idsp
on
T‘$
s‘Q
(1;
(2)
(3)
(4)
-.0
48.
1.10
9-
.585
1.10
6-1
.143
-.8
15.2
21-
.142
3.77
2.1
54
.298
-.9
41.3
13-1
.047
-1.
488
-.9
97.3
04-
.322
5.05
4.1
13
.189
-.9
66.0
55-
.974
-1.
086
-1.0
21-1
.439
-.5
177.
308
.083
.437
-1.0
08.0
57-
.953
.483
-.9
73-
2.37
6-
.645
9.54
0.0
58
.376
-1.0
15.1
19-
.959
-.0
43-
.933
-2.9
64-
.723
11.6
97.0
50
.285
-1.0
12-
.253
-.9
68.1
86-
.9C
8-3
.252
-.7
6813
.734
.053
.000
.215
.939
1.69
7.O
oo
.000
-.023 .6
791.
820
.0+3
0
.000
.097
.359
1.77
5.0
00
.000
.230
.220
1.63
5.0
00
.000
.268
.218
1.46
7.0
C4
.000
.265
.264
1.30
2.0
00
..0
060:
-.0
0344
.004
%.0
079:
-.0
2171
-.0
0037
-.0
+316
5.0
0005
.005
00-
.012
93
.000
51-
.000
31-
.001
08.0
0244
-.0
0737
.Ooo
os.O
oow
-.0
0068
.001
16-.
0050
$S
-.0
0002
.000
0$-.0
0035
.000
54-.0
0397
-.000
02.0
0005
-.000
16.0
0023
–.0
0311
rlt
=3o
a
Tx
T+
s‘Q
(1)
(2)
(3)
(4)
.243
-.9
91.O
oo.
.002
4(-
.449
.1.
076
.IM
5-
.002
5(-1
.411
-.9
32.7
96.0
0194
-.0
13-
.240
1.78
4.0
C63
$4.
395
.130
.000
-.0
1672
.192
.140
-1.4
63 .671
5.75
5
-.4
48.1
31.6
97-2
.098
8.51
9
.407
.108
.006
-3.0
5111
.136
-.2
46-
.328
.368
-3.5
5213
.680
-.2
00-
.380
.426
–3.6
6416
.157
.924
-1.
013
-1.
037
.396
.100
-1.
002
.951
.995
.602
.073
-1.
019
.955
.934
.721
.Cbo
-1.
012
-.9
71-
.897
-.7
85.0
52
-1.
007
-.9
79.8
80.8
19.0
47
.Ow
-.02
6 .547
1.82
6.0
00
.000
.205
.232
1.69
3.0
00
.000
.284
.190
L.4
79.O
oo
.000
.274
.260
1.25
5.O
CQ
.Ooo
.250
.340
1.05
7.0
00-
.001
M6
-.001
05-.0
0098
.004
06-.0
1033
.000
22.0
0001
-.00
+394
.001
66-.0
0592
-.000
04.0
0013
-.000
44.0
0%5
-.0
C40
8
-.000
03.0
0006
-.000
15.0
0020
-.003
11
-.000
01.0
0001
-.000
03.0
0000
‘.002
51
Tabl
e 1
lnte
rnol
Fo
rces
in
o M
ultip
le
Cyl
indr
ical
St
-11
nalp
+nn
anA
ln&
h=60
° 10
1.
““.a
.” 1-
v”
I--
s*=
- L
[p,
WI.
(3)
1 *p
d lrL
J&m
T,
= C
OI. (
211
MQ
= ,'[
pd
cd.
(411
“L .4
.6
1.0
1.4
I.6
22
I .O
OQ
k “=
4h
-=o+
k 2=
Qk
0
-.358
-
1.21
8 .o
oo
- .0
1041
-.5
97
- 1.
113
.392
m
.004
65
-.705
-
.691
1.
022
.008
62
.358
-
.056
1.
474
.010
3e
2.71
5 .1
44
.ooo
-
-029
12
.083
-1
.012
-
.4w
-1
.062
-1
.092
-
.896
-
.wv
- .2
38
3.72
3 .lO
b
l.OO
q,
-75%
=+
k 25
’+k
0
- .O
lO
- .9
52
- .1
3&
- .9
98
1.04
2 -
.986
-
.7a2
-
.424
5.
279
.075
.ooo
' -.W
256
.113
-
.002
49
.793
.0
0195
1.
622
.OO
b64
.OO
O
-.017
69
.OW
.W
OS
O
.076
-.O
W73
-5
35
-.000
95
1.65
1 .W
338
.WO
-.0
0964
lDO
Qk
75+k
.-Q
k .2
5&
0
- .2
55
- .9
85
.WO
.o
w27
-
.038
-
-967
.1
76
-.OW
lL
- .7
Oa
- .9
65
.391
-.O
Om
2 1.
333
- .5
41
1.57
5 .W
181
6.80
4 .0
62
.OO
O
-.OO
b62
I.00 .75
k -
.323
-,1
.003
.w
o .W
OO
7 -
.074
-
-960
=Q
k .2
32
.OO
OO
Z -
.420
-
.937
.2
5+k
.344
-.O
OO
Sl
-1.7
53
- -6
13
.ooo
99
0 1.
470
8.25
2 .0
53
.ooo
-.W
SW
l.wk
- .3
09
-1.0
06
.*+k
.WO
.o
owl
- .1
43
- .%
2 =Q
k .2
51
.OO
OW
-
.234
-
.915
=‘
+k
.347
-.0
0029
-1
.986
-
.653
0
1.36
1 .0
0055
9.
616
.a7
.ooo
-.w
399
+She
or
fa ~r
ces
are
01
supp
orts
, ot
hers
or
e at
m
idq
+ ‘/t
=
I00
TX
T*
S M
* (I
1 (2
) (3
) (4
1
f/+
= ZC
KI
TX
T*
S M
Q
(1)
(2)
(3)
(4)
-099
-
1.01
2 .o
oo
- .W
336
- -4
28
_ 1.
070
-117
-
.w29
1 -1
.107
-
-;&6
.808
.0
0272
.0
64
_ .2
15
1.62
4 .0
0731
3.
546
.llO
.w
o _
.019
28
,120
-
.932
-
.194
-1
.012
-1
.192
-
-998
-
.432
-
.364
I+
.617
.0
83
.ow
.o
oo%
-0
29
-.001
25
,594
-.W
O88
1.
677
.w53
.o
oo
- .0
1170
- .3
60
- .9
95
.ooo
.0
0030
.0
3e
- .9
52
.208
-.O
OO
OS
-
.656
-
-970
.3
25
-.OO
ll?2
-1.4
94
- -5
53
1.58
3 .w
193
b.73
3 .0
61
.oa
-A06
64
- .3
81
-1.0
16
.ooo
-
.103
-
.951
.2
84
- .1
44
- .9
18
-274
- .o
ooo2
.0
0012
-.W
Oso
-
2.21
3 -
.662
1.
418
.wo7
9 8.
739
-050
.o
w
-.Ow
+55
- .2
76
-1.0
12
.ooo
-.W
W3
- -2
61
- .%
3 .2
85
.WO
O7
.136
-
.884
.3
18
- .0
0020
-
2.57
9 -
.722
1.
244
.000
28
10.6
71
.043
30
0 -.w
345
- .2
31
-1.0
08
.wo
-.OO
Wl
- .3
15
- .9
71
-269
.o
ooo2
.2
07
- .8
69
.375
-.O
OO
%
-2.6
75
- .7
53
1.09
0 .W
OO
s 12
.536
.0
38
.OO
O
-.002
77
8.
yt
= 30
0
TX
TQ
S
M*
(1)
(2)
(3)
(4)
.205
-
.941
-
.312
-
1.04
0 -1
.234
-
.%8
- .1
4b
- .2
V4
4.03
2 .o
tw
,000
-
.ooo
, .0
26
- .0
020
,697
.o
oo2
1.67
0 .W
58,
.ooo
-
.014
3
- -0
86
- ,9
49
- -0
47
- .9
79
-1.0
57
-1.0
07
- -8
15
- A
37
5.29
1 .0
73
.ooo
.o
ooa:
.0
85
-.OO
%i
.471
-.w
13:
1.65
7 .w
351
.ow
-
.W94
!
- .4
40
-1.0
18
- -0
16
- .%
A
- .2
90
- .9
36
-2.0
33
_ .6
29
7.80
9 .O
%
- .2
70
-1,0
14
- -2
84
- .9
63
-210
-
.881
-2
.703
-
.725
10
.176
.o
w,
- .2
02
-1.W
6 -
.364
-
-974
.3
22
- .8
58
-2.9
06
- .7
70
12.4
69
.038
- -2
22
-1.0
05
- .3
24
- .9
76
^__
.ooo
.o
ooo2
.2
80
.000
1:
.259
-.0
007:
1.
481
.w12
c .o
w
-.w53
c
.m
-.ooo
o5
-295
.W
olC
.3
01
-.OW
25
1.24
9 .0
0034
.o
oo
-.003
65
.ow
-.O
OO
Ol
.268
,3
89
-:iE
E
1.03
5 .W
OQ
l .o
oo
-.002
77
.wo
.ooo
oa
.248
-.O
OW
l .2
16
- .tJ
x .4
55
.WO
o3
-2.7
87
- .7
87
.862
-.O
Ool
l 14
.699
.0
34
.ooo
-.0
0223
-,--
-,,
..
..
.,,..
...
..
.,.,
lam
ez
-L
olcu
larl
onoT
Tor
tes
Ina
sim
ply
supp
orT
eain
ferio
rcy
linar
lcal
snel
l
Giv
en: t=4
in.
+,=
26.4
°0
L=50
ft.P
d
r=
45ft.
P:
❑30
psf
‘50
psf
Wti
For
ce
Mul
tiplie
r
4
($:”
,
.751
pk
.50+
k
.25#
k
(va?
ley)
Tx
(L2/
r)pd
=43
90
Con
stan
t
-3.9
47
-3.4
55
–1.5
51
2.69
3
10.1
11
For
ce(lb
./ft.)
-173
00
–15
,200
-6/
300
I1,8
00
44,4
00
vi”
T+
sM
+
(r)
pd=
3560
-(L)
Pd
=-3
950
(rz)
pd=
160,
000
Con
stan
t
-1.4
41
–1.2
04
–.5
95
.083
.375
Far
ce(l
b./ft
.)C
onst
ant
::yf;,
Con
stan
t~f
p;:,;
f+.:
1–5
130
o0
-,00
385
-620
–42
901.
415
-559
0–.
0014
6-
230
–212
02.
57I
-10,
160
.003
2151
0
300
2.75
9-1
0,90
000
324
520
I340
00
-.00
970
-I5
50
Lo
0.8
r/L
=2.6
0.6
Par
ob
oti
cd
istr
ibu
tio
n04
-
0.2 0 0
0.1
0.2
0.3
0.4
0.5
x/L
Fig
.I
-Lo
ngitu
dina
ldi
strib
utio
nof
Tx
otvo
lley
‘6
N.*L
m
o al (Q (y- C3 o z o
*s/s
-4’u-lN
oII
I.C
O.E
0.6
c \ %
0.4
0.2 c
,75
.50
*“ ;
r/t
=13
5r/
L=0.
6+,
=27
.5°
0.2
0.4
0.6
0.8
1.0
s‘s
o.50
+k
Fig
.4R
elat
ive
dist
ribut
ion
ofsh
ear
k%3
-J
(0.*L
M+ (k - ft. /ft.)
I .0-i
r
.75
0.8
-0.5 0 0.5 Lo
L@ ‘ < ~
---M~.50
\ \
y~ by beam
method
0.6~“
s\ /
.
o.4~5 h ‘~\ //
TX by beam method
Y \r/t = 135
0.2 Ar/L= 0,9
+, =26.4°
o-6 -4 -2 0 2 4
—, .,.
TX (k /ft.)
Fig. 7 Transverse distribution of TX and M+ for interior shell example
This publication is based on the facts, tests, and authorities stated herein It is intended for the use of professional personnel competent to evaluate the significance and limitations of the reported findings and who will accept resposibility for the application of the material it contains. The Portland Cement Association disclaims any and all responsibility for application of the stated principles or for the accu- racy of any of the sources other than work performed or information developed by the Association.
Printed in the U.S.A. EB020.01 D
