Journal of the American Concrete Institute, Vol. 17, No. 3

J O U R N A L

of the

AMERICAN C O N C R ET E INSTITUTE

(A CI PROCEEDINGS Vol. 42)

V o l. 17 January 1946 No. 3

CONTENTS

Papers and| Reports... ’... ... 165-264

Shrinkage Stresses in Concrete... G ERALD PICKETT 165

Floating Block Theory in Structural Analysis. .S T A N L E Y U. BEN SC O TER 205

Shrinkage and Plastic Flow of Pre-Stressed Concrete

H O W A R D R. ST A LEY and D EA N PE A B O D Y , JR ... 229

Proposed Minimum Standard Requirements for Precast Concrete Floor Units

Report of A C I Committee 711, F. N . M enefee, C h airm an ... 245

Proposed Recommended Practice for the Construction of Concrete Farm Silos

Report of A C I Committee 714, William W . Gurney, Chairman... 261

Current Reviews...265-276 News Letter... 1-12

A C I Convention, Buffalo, N. Y . , Feb. 19-21,1946

See First Four News Letter pages

Who's W ho # New Members # Honor Roll # John J. Earley

to p ro vid e a com radeship in finding the best w ays to do co n crete work o f a ll kinds an d in sp re ad in g th at know led ge

A D D R E S S , 7 4 0 0 S E C O N D B O U L E V A R D , D E T R O I T 2 , M I C H .

S1.50 per copy

Extra co p ies to m em bers S I .0 0

DISCUSSION

Discussion closes M arch 1, 1946 „ „

' Sept, Jfc *45

Concrete Construction in the National Forest*— Clifford A . Betts

Lapped Bar Splices in Concrete Beams— Ralph W . Kluge and Edward C . Tuma Tests of Prestressed Concrete Pipes Containing A Steel Cylinder— Culbertson W . Ross Field Use of Cement Containing Vinsol Resin— Charles E. W uerpel

Nov. JL 45 Maintenance and Repair of Concrete Bridges on the Oregon H ighw ay System

— G . S. Paxson

Should Portland Cement Be Dispersed?— T. C. Powers

A n Investigation of the Strength of W elded Stirrups in Reinforced Concrete Beams—

Oreste Moretto

Discussion closes A p ril 1,1946

Ja n . Jl. '46 Shrinkage Stresses in Concrete— Gerald Pickett

Floating Block Theory in Structural Analysis— Stanley U. Benscofer

Shrinkage and Plastic Flow of Pre-Stressed Concrete— H o w ard R. Staley and Dean Peabody, Jr.

Proposed Minimum Standard Requirements for Precast Concrete Floor Units— A C I Gommittee 711

Proposed Recommended Practice for the Construction of Concrete Farm Silos—■ A C I Committee 714

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V o l. 17 No. 3 7 4 0 0 S E C O N D B O U L E V A R D , D ET R O IT 2, M I C H I G A N January 1946

Shrin kag e Stresses in Concrete*

By GERALD PICKETTf

M e m b e r A m erican C o n cre te Institute

S Y N O P S I S

Theoretical expressions for deformations of concrete beams and slabs th a t occur during the course of drying an d expressions for distribution of th e accompanying shrinkage stresses are derived in P a rt 1. These expressions are derived on the assum ption th a t the laws governing the developm ent of shrinkage stresses in concrete during drying are analogous to those governing the developm ent of therm al stresses in an ideal body during cooling. T hree cases are considered:

(a) slab or beam drying from one face only;

(b) slab or beam drying from two opposite faces; and (c) prism drying from four faces.

The applicability of the equations to concrete is considered in P a rt 2 (to appear ACI Jo u r n a l, F ebruary 1949). I t is shown th a t the course of shortening of prisms is in very good agreem ent w ith the theoretical equations and th a t from a te s t on one prism the shortening versus period of drying of other prisms of the same m aterial differing in size and num ber of sides exposed to drying can be predicted w ith fair accuracy if the differences in size arc not too great. However, it is shown th a t th e theory m ust be modified to take into account inelastic deforma­

tion an d to perm it the supposed constants of th e m aterial to v ary w ith m oisture content and size of specimen if the theory is to be in agree­

m ent w ith all results on all types of specimen of a given concrete.

Various tests are described which, when used in conjunction w ith th e theory, provide a m eans for studying some of th e more fundam ental properties of concrete and for predicting the performance of concrete under some conditions in th e field.

IN T R O D U C T IO N

C oncrete, like m a n y o th e r m ateria ls, gains or loses w a te r w ith changes in a m b ie n t conditions. W ith each change in w a te r c o n te n t th e concrete

* R eceiv ed b y th e I n s titu te , A p ril 30, 1945.

1 P ro fesso r of A p p lie d M ech an ics, K a n s a s S ta te C ollege, M a n h a tta n , K a n s .; fo rm e rly P o r tla n d C e m e n t A sso ciatio n R e s e a rc h L a b o r a to r y , C h icago.

(165)

te n d s to sh rin k or swell. As a resu lt of th ese changes in v o lu m e, stresses are p ro d u ced t h a t m a y affect th e perform ance of th e co n crete s tru c tu re concerned.

In th e design of concrete stru c tu re s some consideration is u su a lly given to th e p o ssib ility of su b se q u e n t shrinkage or swelling, b u t th e c o m p u ta ­ tio n s for stresses u su ally include only th e stresses p ro d u ced b y loads. I he c o m p u ta tio n s are m ad e b y m eans of form ulas ta k e n from th e science of m echanics of m a te ria ls a n d th e com puted stresses so o b ta in e d a re com ­ p a re d w ith allow able stresses, considering th e ty p e of s tr u c tu r e a n d location of th e concrete in th e stru c tu re . T h e basic a ssu m p tio n s for th e fo rm u las are t h a t th e m aterial is hom ogeneous, iso tro p ic, free from self-strain a n d obeys H o o k e’s law.

C oncrete is n o t hom ogeneous; b y n a tu re it is h etero g en eo u s, even including th e b in d in g m edium itself, h a rd e n e d cem en t p a ste . I t is n o t isotropic. F a c to rs such as sed im en tatio n before h a rd e n in g te n d to d e­

stro y w h a t iso tro p y th e re m ig h t h av e been. I t does n o t obey H o o k e ’s law except p erh ap s u n d e r in sta n ta n e o u s strain s. I t is n o t free fro m self­

stra in a t a n y tim e ; th e h ard e n e d p a ste m a y be sh rin k in g w hile th e aggregate m a y be resistin g a change in v o lu m e; th e regions n e a r th e surface m a y be fa irly d ry a n d te n d in g to sh rin k w hereas th o se f a r th e r in w ard m a y be m u ch w e tte r a n d te n d in g to re sist a re d u c tio n in v o lu m e.

In ad d itio n , concrete m a y change w ith tim e, becom ing s tro n g e r a n d m ore rigid if conditions are such as to p ro m o te a d d itio n a l h y d ra tio n .

N o t only is concrete a m u ch d ifferen t m a te ria l fro m t h a t a ssu m ed in th e d eriv atio n s of ele m e n ta ry form ulas b u t th e co n d itio n s of lo ading, th e tendencjr to w ard re d u n d a n c y (sta tic a lly in d e te rm in a te ), a n d th e stru c tu ra l shapes of con crete m em bers a re o ften such as to m a k e th e ele m e n ta ry form ulas only ro u g h a p p ro x im a tio n s co m p ared w ith w h a t th e sam e fo rm u las w ould be fo r th e u su a l co n d itio n s of lo ad in g a n d s tru c tu ra l shapes of steel s tru c tu re s w hich a re u s u a lly less re d u n d a n t.

I t is n o t to be in ferred t h a t co n crete w ould n e cessarily b e a b e tte r m a te ria l were all of its p ro p e rtie s like th o se a ssu m ed b y th e design form ulas. On th e c o n tra ry , its a b ility to relieve stre ss b y creep or p la stic flow, for exam ple, p a r tly co m p en sates fo r its in h e re n tly low te n sile stre n g th an d for u n c e rta in tie s arisin g from re d u n d a n c y . T o b e re m e m ­ bered also is th e fa c t th a t, in sp ite of th e deficiencies of d esig n -fo rm u las, concrete stru c tu re s on th e w hole p erfo rm th e ir in te n d e d fu n c tio n ’.

N evertheless, it should be e v id e n t t h a t co n crete c a n n o t be u se d as in- te lh g e n t’y as it m ig h t be a n d ca n n o t be s tu d ie d effectively w ith o u t a b e tte r know ledge as to th e m ag n itu d e a n d d is trib u tio n of stresses w ith in it.

T h e p urpose of th is p a p e r m a y be s ta te d as follow s:

F irs t, to derive on th e basis of sim plifying a ssu m p tio n s in re g a rd to th e p ro p erties of concrete— expressions fo r: (a) d e fo rm a tio n s of, a n d (b)

166 JO U RN A L OF THE AMERICAN CONCRETE INSTITUTE January 1946

SHRINKAGE STRESSES IN CONCRETE

th e d is trib u tio n of sh rin k ag e stresses in, con crete beam s a n d slabs d u rin g th e course of d ry in g .

Second, to show b y m ean s of d a ta from specim ens u n d e r con tro lled c o n d itio n s th e m a n n e r an d degree to w hich th e e q u a tio n s a p p ly to con­

crete.

T h ird , to suggest m e th o d s for stu d y in g som e of th e m ore fu n d a m e n ta l p ro p e rtie s of d ry in g concrete.

N o a tte m p t will be m ad e here to give a com plete an aly sis of stresses in concrete. In p a rtic u la r, th e effect of ag g reg ate p a rticles on th e stresses w ith in th e h a rd e n e d p a ste will n o t be considered.

B efore expressions for sh rin k ag e stresses in concrete can be d erived, a ssu m p tio n s m u s t be m ad e in re g ard to th e re la tio n b etw een shrin k ag e a n d m o istu re c o n te n t a n d th e law s con tro llin g th e flow of m o istu re in co n crete as well as th e re la tio n betw een stress a n d strain .

T h e a c tu a l relatio n sh ip s are n o t as sim ple as could be desired. If th e flow of w a te r w ere e n tire ly b y v a p o r diffusion, if th e v a p o r pressure of th e w a te r in th e con crete w ere p ro p o rtio n a l to th e m o istu re -c o n te n t, a n d if p e rm e a b ility w ere in d e p e n d e n t of th e m o istu re -c o n te n t, th e n th e differ­

e n tia l e q u a tio n for th e flow of w a te r w ould be a p a rtia l-d iffere n tia l e q u a tio n kno w n in physics a n d m a th e m a tic s e ith e r as th e d iffusion equation or as th e equation o f heat conduction. C a rlso n ,1* in a s tu d y of d is trib u tio n of m o istu re in concrete, assum ed t h a t th is e q u a tio n applies.

If th e flow of w a te r could be expressed b y th e diffusion e q u a tio n a n d if th e sh rin k a g e (or swelling) te n d e n c y f of each elem en tal volum e were lin early re la te d to th e m o istu re -c o n te n t, th e u n re stra in e d sh rin k ag e (or swelling) could also be expressed b y th e diffusion eq u atio n . T h is possi­

b ility w as also considered b y C a rlso n . 1 B u t th e flow of w a te r is different from t h a t in d ic a te d b y th e diffusion eq u atio n , a n d th e relatio n sh ip b e tw een th e change in m o istu re -c o n te n t a n d u n re stra in e d sh rin k ag e is n o t lin e a r as re q u ired b y th ese eq u atio n s. M oreover, sa tisfa c to ry expressions for e ith e r th e flow of w a te r or th e m o istu re-sh rin k ag e re la tio n h av e n o t been found.

I t is believed t h a t m o istu re in concrete flows p a rtly as liq u id in c a p illar­

ies, p a r tly as v ap o r, a n d p a rtly as ad so rb ed liquid on th e surface of th e colloidal p ro d u c ts of h y d ra tio n . W hile d ry in g progresses, th e v a p o r p ressu re of th e w a te r rem ain in g in th e region losing w a te r decreases progressively w ith th e m o istu re co n te n t. T h is change in v a p o r pressure w ith change in m o istu re c o n te n t is n o t lin ear w ith re sp ect to m o istu re

*See refere n ces a t e n d of te x t of P a r t 1.

f B y s h rin k a g e (or sw elling) te n d e n c y is m e a n t th e u n it lin e a r d e fo rm a tio n d u e to a n y cau se o th e r t h a n stre s s t h a t w o u ld occu r in a n in fin ite sim al e le m e n t if th e e le m e n t w ere u n re s tra in e d b y n e ig h b o rin g elem en ts.

I t is n o t to b e co n fu sed w ith th e average u n it d e fo rm a tio n , co m m o n ly called sh rin k ag e , of a so -called u n ­ r e s tra in e d sp e cim en , n o r w ith th e r e s u lta n t lin e a r u n it d efo rm atio n w h ich for th e x -d ire ctio n w ill b e d e­

s ig n a te d ex. H e re in a fte r, th e lin e a r u n it sh rin k a g e te n d e n c y w ill b e referre d to e ith e r as u n re s tra in e d s h rin k ag e , fo r c la r ity , o r m e rely as sh rin k ag e , fo r b re v ity .

168 JO U RN A L OF THE AMERICAN CONCRETE INSTITUTE January 1946 c o n te n t. N e ith e r is th e ra te of flow p ro p o rtio n al to th e g rad ien t in a apoi pressure. T h e shapes a n d re la tiv e p ro p o rtio n s of th e spaces occupied by liq u id a n d b y v a p o r change as d ry in g proceeds. T his fact, as w ell as th e n o n -u n ifo rm ity of th e spaces, is believed to be p a rtly responsible lo r th e w ay in w hich v a p o r pressure depends on m o istu re c o n te n t an d th e \v a y in w hich r a te of flow d epends on g ra d ie n t of v a p o r pressure.

T h e volum e-change-vs.-w eight-loss rela tio n is different fo r d ifferen t concretes d ep en d in g on th e com position of th e concrete a n d th e co n d itio n s of curing. F o r th e sam e concrete it is different d u rin g first sh rin k ag e from w h a t it is d u rin g th e second or su b seq u en t volum e changes. I f a s a tu ra te d prism of concrete is allow ed to d ry , th e ra tio of change of len g th to loss of w a te r increases as d ry in g proceeds. A t first, co m p arativ ely sm all changes of volum e occur p er u n it loss of w eight. T h e hig h er th e w a te r-c em e n t ra tio a n d th e sh o rte r th e period of curing th e sm aller th e change d u rin g th e in itia l stages of drying. L a te r th e ra tio becom es m u ch larg er a n d rem ains alm o st c o n sta n t for som e tim e, a fte r w hich it m ay e ith e r increase or decrease as th e specim en approaches its final w eight. I t is believed t h a t th is ra tio , a t a n y stag e of drying, depends u p o n (a) th e shape, size, an d degree of u n ifo rm ity of th e spaces t h a t hold th e w a te r; (b) th e shape, size, rig id ity , a n d spacing of th e solid p a rtic le s; a n d (c) th e s tre n g th of th e bo n d s b etw een particles.

T h e relatio n s betw een stress an d stra in m u st be considered in a n y s tu d y of volum e changes resu ltin g from m o istu re changes in con crete because a n y te n d e n c y for a change in volum e t h a t progresses from th e su rface in w ard alw ays develops stresses. T h e stresses in tu rn , th ro u g h th e stress- stra in relatio n , m o d ify th e re s u lta n t deform ations. F o r low stresses b o th th e elastic an d inelastic s tra in prod u ced b y stress are ap p ro x im a te ly p ro p o rtio n a l to th e stress, p e rm ittin g H o o k e’s law to be assum ed, b u t u n d e r m o st conditions of d ry in g th e shrinkage stresses, e ith e r alone or in co m b in atio n w ith stresses from o th e r sources, m ay be large en o u g h to cause cracks a n d s tru c tu ra l dam age w ith in th e concrete an d for such stresses th e p ro p o rtio n a lity does n o t hold. M oreover, th e a p p a re n t p la stic ity of an elem ent* is g re a te r d u rin g th e tim e th e elem ent is d ry in g for th e first tim e th a n a t a n y o th e r tim e. T h e relativ e positions of th e colloidal gel p a rticles are no d o u b t changed b y drying an d w hile th e changes are ta k in g place sm all re s u lta n t stresses on an elem ent will p ro ­ duce re lativ ely large in elastic deform ations.

In sp ite of th e a p p a re n t difficulties of o b tain in g a sa tisfacto ry so lu tio n to th e problem of d efo rm atio n s an d stresses in concrete exposed to changes in a m b ie n t conditions, a re la tiv ely sim ple procedure has p ro v ed to be ra th e r successful. T h e procedure is to assum e as C arlson d id t h a t

th * T a6teeferenCe 'S t0 a n e Ie m en t of h y d r a te d Pa s te of Ju st sufficient size to b e a re p re s e n ta tiv e s a m p le of

SHRINKAGE STRESSES IN CONCRETE

th e diffusion e q u a tio n applies to sh rin k ag e even th o u g h th e sim ple re la ­ tio n s t h a t are im p lied b y t h a t a ssu m p tio n a re c o n tra ry to fact. I t is fu r­

th e r assu m ed t h a t co n crete follows H o o k e’s law. T h e d e riv a tio n s given in P a r t 1 are b ased u p o n th ese assu m p tio n s.

Since in P a r t 1 th e d e riv a tio n s for d efo rm atio n s a n d stresses are b ased on th e assu m p tio n s t h a t sh rin k ag e follows th e diffusion e q u a tio n a n d th e m a te ria l follows H o o k e’s law , th e e q u a tio n s are even m ore ap p licab le to th e rm a l stresses in m e ta ls th a n to sh rin k ag e stresses in concrete. I n fa c t, m u c h of th e m a th e m a tic a l w ork given here w as ta k e n from th e lite ra tu re on diffusion of h e a t an d on th e rm a l stresses, as th e references w ill show . H ow ever, c e rta in corresp o n d in g coefficients in th e tw o pro b lem s are of an e n tire ly d ifferen t order of m ag n itu d e. F o r exam ple, th e n u m e rical v alu e of th e th e rm a l d iffu siv ity for steel expressed in sq u a re inches per second is a p p ro x im a te ly th e sam e as th e n u m erical v alu e of th e sh rin k ag e diffu­

s iv ity of con crete expressed in sq u a re inches per day. B ecause of th e re la tiv e ly slow diffusion of sh rin k ag e th e ap p lic a tio n of th e h y p o th e sis to th e sh rin k ag e of concrete n e c e ssita te s th e s tu d y of e arly tra n s ie n t condi­

tio n s (u su ally ignored in th e tr e a tm e n t of h e a t ) .

P A R T 1— S H R IN K IN G (O R S W E L L IN G ), ITS EFFECT U P O N D IS P L A C E M E N T S A N D STRESSES IN S L A B S A N D B E A M S O F H O M O G E N E O U S , IS O T R O P IC , E L A S T IC

M A T E R IA L Notation

S = free, unrestrained u n it linear shrinkage-strain

—S = free, unrestrained unit linear swelling-strain

So, = final shrinkage-strain under fixed am bient conditions, value of S when t = œ Sav — average shrinkage over th e volume of the specimen, the same as average shorten­

ing per u n it length if the m aterial follows Hooke’s law t = tim e in days

k = diffusivity coefficient of shrinkage in sq. in. per day

/ = surface factor, characteristic of the m aterial and the boundary conditions, in in. per day

a, b, c, d, I = distances related to the dimensions of the specimen in inches B = fb /k , a non-dimensional param eter

T = kt/b 2, a non-dimensional param eter

B c an d T c, non-dimensional param eters corresponding to B and T and used when a second characteristic dimension of the specimen m ust be considered

x, y, z = rectangular coordinates p„ = n th root of p tan p = B

pm = same as p„ except used in connection w ith c, whereas pn is used in connection w ith b

A n — Fourier coefficient

2 B _ 2 B c

170 JO U RN A L OF THE AMERICAN CONCRETE INSTITUTE January 1946

\ a » A . 2 ) j

2 r

*(*) = - i = a/ -tt 7! '

<rx, <jv, az = norm al components of stress parallel to i - , y — , and 2—axes positive i tensile, negative if compressive.

ex, ev, ez = elongations in x —, y —, and 2—directions Txy, Txz, tvz = shearing-stress components

y xv, "Yxz, y yx = shearing-strain components E = Young’s modulus in psi.

n — Poisson’s ratio

v = deflection in inches, displacement of the elastic line in the y-direction N = th e norm al to th e surface directed outw ard

2 f x

P (x ) — — e dx, th e probability integral V TT O

2 r & -x2

= e dx = 1 - P{x) V it X '

« -T&l COS fin ~ 06 = 1 — 2 e Fn —

1 cosJ(3„

2

00 -Tc&2m œs 0** ~ 0e = 1 - 2 e f ra -

1 cos 0m

» - r t f

= 1 - 2 e B

1

» - r t f . / / . = 1 - 2 e Hm

1

Equation for diffusion of unrestrained shrinkage

T h e diffusion e q u a tio n is a m a th e m a tic a l s ta te m e n t of th e fa c t t h a t for each infinitesim al volum e of a b o d y th e excess of th e su b sta n c e in q u estio n flowing in over t h a t flowing o u t p er u n it of tim e is e q u al to th e ra te of increase of th e su b stan c e in t h a t volum e. W h en sim ilar assu m p tio n s are m ade in reg ard to shrinkage, shrinkage th u s being tr e a te d as if it w ere a

“ su b sta n c e ” ju s t as h e a t is so tre a te d , th e re su lt is2

~dV5 d*S 3*5 "I d S

_ d x 2 d y 2 dz2 J dt ^

w here k is th e d iffusivity of shrinkage.

T h e e q u a tio n becom es

* 1 - / & - - * > ( 2 )

k

SHRINKAGE STRESSES IN CONCRETE a t exposed b o u n d a rie s a n d

^ = 0 ...(3) d N

a t sealed b o u n d aries w here

N is th e n o rm a l to th e surface, d ire c te d aw ay from th e b o d y / is th e su rface fa c to r

Sco is th e v a lu e t h a t S will e v e n tu a lly reach u n d e r fixed a m b ie n t conditions.

E q u a tio n s 2 a n d 3 correspond, respectively, to N e w to n ’s law of cooling a t exposed b o u n d a rie s an d to no flow of h e a t p a s t p e rfectly in su la te d b o u n d a rie s in th e analogous p ro b lem of flow of h e a t.

If th e b o u n d a rie s of th e b o d y are n o t p arallel planes, a tra n sfo rm a tio n of E q u a tio n 1 fro m a n expression in re c ta n g u la r co o rd in ates to som e o th e r form is u su a lly desirable. F o r exam ple, if th e b o d y is a circu lar cy lin d er, E q u a tio n 1 is b e s t tra n sfo rm e d to

fc f— + I ^ +

Ldr2 r dr r 2 d 9 2 d z 2 J dt

w here r, 9 , a n d z are cy lin d rical coo rd in ates. F re q u e n tly , th e co n d itio n dS— = 0 a t som e b o u n d a rie s or som e o th e r c o n d itio n s w ill m ak e S inde- d N

p e n d e n t of c e rta in co o rd in ates a n d th e re b y sim plify E q u a tio n 1.

Since th e fo rm of th e so lu tio n fo r S d ep en d s u p o n th e fo rm of th e d ifferen tial eq u a tio n , th e fo rm of th e so lu tio n is d e p e n d e n t u p o n th e b o u n d a ry c o n d itio n a n d th e sh ap e of th e b o d y u n d e r in v e stig a tio n . T h e in itia l co n d itio n s (values of S a t t = 0) a n d a n y v a ria tio n in b o u n d a ry co n d itio n s w ith tim e wall also affect th e form of th e solution.

Assumptions as to elastic properties

A fte r a sa tisfa c to ry so lu tio n for S h a s been o b ta in e d , th e n d isplace­

m e n ts a n d stresses w ill be fo u n d b y th e a p p lic a tio n of c e rta in fu n d a ­ m en ta ls of th e th e o ry of e la sticity . T h e solutions for stresses are here re s tric te d to hom ogeneous iso tro p ic solids t h a t follow H o o k e’s law.

Also, as w ill be b ro u g h t o u t below , th e effect of P o isso n ’s ra tio will be n eg lected in som e cases.

Effect of shape of body on relative values of principal stresses

T h e s ta te of stress a t a n y p o in t in a b o d y is defined b y th e d irectio n s a n d m a g n itu d e s of th e th re e p rin c ip a l stresses. T h e th re e p rin c ip a l stresses in w ide slabs a n d in n a rro w beam s w ill be in th e d irectio n s of le n g th , w id th , a n d d e p th , resp ectiv ely , if th e bodies are u n d e r u n ifo rm exposure e ith e r fro m one or from tw o opposite faces a n d are w ith o u t e x te rn a l r e s tra in t. T h e p rin cip al stress in th e d irectio n of d e p th (norm al

172 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 to the exposed face) will be zero. The other two principal stresses will be equal in a wide slab (width large compared to depth) if the slab is either n o t restrained or is restrained equally in the directions of length and w idth ; b u t in a narrow beam (width small compared to depth) th e princi­

p al stress in the direction of w idth will be negligible, and the stress in the direction of length will be (1 — g) tim es th e corresponding stress in a corresponding slab. A corresponding slab differs from the beam ju st described prim arily only in the m atter of width. Beams whose w idths are not small compared to their depths will have longitudinal stresses inter­

m ediate between those of slabs and narrow beams. Corresponding slabs and beams would, of course, have th e same longitudinal stresses regard­

less of w idths if Poisson’s ratio were zero. M athem atical analyses will be m ade for the three cases shown in Fig. 1.*

To simplify th e m athem atical work th e effects of Poisson’s ratio will be neglected in m ost of the derivations for Cases I and II. W hen these effects are neglected, the results will be strictly correct only if Poisson’s ratio is zero or if th e beam is very narrow.

C A SE I— SLAB OR BEA M DRYING FROM O N E FACE O N L Y Shrinkage

Solution by Fourier series. T he exposed face of th e slab or beam will be tak en as th e plane y = b, and the opposite face will be taken as the plane y = 0 as shown in Fig. 1 for Case I. F or this case th e diffusion equation reduces to

... (la) dy- dt

The equation becomes

— = £ (S co - S ) ... (2a)

dy k

a t the exposed boundary y = b and

f = 0 ... (3a) dy

a t th e sealed boundary y — 0.

A general solution for S satisfying E quations la and 3a is kt 2

00 Q

<— b2 n o y

S = S m ^ A n e cos n b ...(4) 1

*A11 figures an d tab les p ertain in g to P a r t I will be found on pages 196 to 204.

173 E quation 2a is also satisfied if j8„ is the n th root of

fb

k ...

t onB=’~ ... (2b) i e -> dn tan pn = F ...(2c)

k

The above statem ents m ay be verified by substituting S from Equation 4 into E quations la , 2a and 3a.

F or tim e t= °°, E quation 4 reduces to S = S<x>, which is in accord with the definition of S a>. An infinite series of term s such as the trigonometric series in E quation 4 is necessary to give an arb itrary distribution of shrinkage a t tim e t = 0.

If the initial conditions are such th a t S = 0 when t = 0, then the Fourier coefficients A n are given by*

2 f~b

S ^co k

COS 3n ( f b V fb

\ k / k I t therefore follows th a t

+

COS —

S ^ - ~ T Pn b

= 1 - > e Fn a (5)

« cos

where

rp kt

~ b - 2

F = ___

1 n

B'- + B + , B = fb

k

E quation 5 (in slightly different form) and similar equations for other shapes and other conditions, applied to analogous phenomena, m ay be found in the literature of m athem atical physics such as the textbooks of Byerly, Carslaw, and Ingersoll and Zobel. Various tables and diagrams have been prepared from which the numerical relationship of the four non-dimensional quantities S /S * ,, y/b, B, and T m ay be found, such as Fig. 4, page 841 of P erry ’s Chemical Engineer’s Handbook (1934).

*The general procedure of o btaining F o u rier coefficients to satisfy initial conditions som ew hat analogous to th e p resen t problem is given in A rticles 66 to 68 of B yerly’s Fourier Series and Spherical Harmonics B oston: G inn & Co., 1893).

174 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 T o use more th a n a few of th e term s in E quation 5 for the evaluation of S / S » is very laborious because of th e difficulty in evaluating /3„ and F n.

The num ber of term s required for a given degree of precision will depend somewhat upon the param eters B and y /b b u t is chiefly controlled by the param eter T. C om putations show th a t very little error is introduced by neglecting all term s in the series except the first if T is more th an about 0.2; b u t m any term s are needed for the usually desired precision if T is less th a n 0.02,— th e smaller th e value of T th e greater the num ber of term s needed. Very precise values of S / S c for small values of T m ay be found w ithout th e tedious com putation indicated in E quation 5 by using another expression which will now be derived.

Solution in terms of the probability integral. As long as the shrinkage a t th e sealed surface rem ains negligible, th e distribution of shrinkage from th e exposed surface inw ard will be nearly independent of th e distance betw een th e tw o surfaces. Suppose th a t instead of considering th e surface a t y = 0 to be sealed, th e body is considered to be extended to infinity in a negative «/-direction. T hen instead of th e boundary condition — = 0 a t ydS

dy

= 0, th e requirem ent will be

S = 0 ...(6) a t y = — co.

T he solution* satisfying E quations la , 2a and 6 and giving S i '= 0 is

0 when

S_

S c

y

-\ r - n

1 _ b

— </>

. 2 VT7"- _2 -

5 ( 1 y

- ) + 5 2T b

■ (7)

where <f> (x ) is - 1=.

f

V X J

/£ 2 . kt

e dx and T is again used in place of —.

o2

The q u an tity 1 — 4> (x), or —

f

V x qJ

dx, is known as the proba-

bility integral. Values of <t> (x ) m ay be readily found by using a table of th e probability integral.

Num erical calculations show th a t E quation 7 gives values th a t differ from those given by E quation 5 by an am ount less th a n th e value of S / S c a t y = 0 ; therefore, E quation 7 m ay be used in place of E quation 5 whenever T is so small th a t S / S c a t y = O is less th an the permissible error.

*This solution is very sim ilar to th a t given for an analogous problem b y C arslaw in A rticle 25 of The Conduction o f H eat, (M acm illan & Co., L td ., 2d ed., 1921).

175 Table 2 and Fig. 7, showing S / S co in term s of y/b, k t/b 2, a n d / 6//c, were prepared from Equations 5 and 7.

Stresses and strains

Continuity, Hooke’s law and equilibrium. As stated previously, the solutions for stresses are here restricted to homogeneous, isotropic solids th a t follow Hooke’s law. Equations for th e stresses th a t would be pro­

duced in such a body by the shrinkages given by Equations 5 or 7 will now be derived.

The shrinkage S has been defined as the linear u n it deformation th a t would occur if each infinitesimal elem ent were unrestrained. However, th e properties of a continuous solid will not perm it an arb itrary distribu­

tion of deform ations; therefore, unless the distribution of shrinkage given b y E quation 5 happens to be compatible w ith the conditions of continuity, stresses will be produced th a t will modify the deformations so as to make them compatible. Although in general six partial differential equations are required for a complete m athem atical statem ent of th e conditions of com patibility,3 these are reduced to

a- % - 0 (8)

d y2

for either long narrow beams (plane stress) or slabs (plane strain) if the stresses are considered to be independent of th e longitudinal coordinate x.

The term ex is defined as the resultant u n it deformation in the x-direc- tion (the direction of length). I t is therefore th e algebraic sum of shrink­

age, S, and th e strain produced by stresses. <r„ is obviously zero; and if Poisson’s ratio is zero or if th e discussion is confined to narrow beams, az is negligible. Therefore,

«, = ~ ~ S ... (9) A

or, solving for stress,

<?x — E f e + S ) ... (10) where E is Young’s modulus.

The restriction imposed by E quation 8 requiring th a t th e expression for longitudinal deform ation contain no term s in y other th an the first power (second derivative equal to zero) is equivalent to th e assum ption usually made in the elem entary theory of beams th a t “plane cross-sections rem ain plane.” If longitudinal restraint is complete, then ex is zero and it follows from E quation 10 th a t <rx = E S . If, however, longitudinal short­

ening is perm itted b u t complete restrain t against bending is provided, then ex is not zero b u t is still independent of y. If the beam has no external restraint, th e non-sym m etrical distribution of shrinkage causes it to warp, m aking ex a linear function of y. For no external restraint the equations of

176 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 equilibrium (sum m ation of forces in th e x-direction equal to zero and sum m ation of m om ents about th e z-axis equal to zero) become

b

ax dy = 0 ...(11) 0

/ * and

I

I t m ay be shown by substituting E quation 10 into both Equations 11 and 12 th a t if shrinkage (S) is either independent of or a linear function of y, an unrestrained beam will be free of stress (ex = — S and <rx = 0). For any other variation of shrinkage a stressed condition m ust result because the restriction on ez (E quation 8) will n o t perm it it to be equal and oppo­

site to S if shrinkage is a non-linear function of y.

The only solution for ex th a t satisfies E quations 8, 10, 11 and 12 is

b b

e*

l ~ 4) i I s

( 6 “ 12l ) i I

0 0

W hen this value of ex is substituted into E quation 10, th e result is

r h h "I

ax = E 'S + ( 6 f - 4 ) y f S d y + ( g - 12 y J S y dy (14)

0 0 J

Finally, S from E quation 5 m ay be substituted into E quation 14, thus giving stress in an unrestrained beam as a function of th e param eters y/b , k t/b 2, fb /k , S <= and of Young’s modulus. This substitution will not be made until later, because it seems advisable a t this tim e to consider another approach.

Solution by superposition. Although the above derivation is short and is in th e simplest form for checking the m athem atical correctness of the equation, a derivation in which elem entary solutions are superposed is also desirable because it will be easier in general to understand and because the final expressions are in more usable forms. In this second derivation the resultant stress ax is considered as consisting of three parts. The first p a rt is th a t stress which would be produced by complete restraint against longitudinal deform ation; the second p a rt is a uniform stress equal to and opposite in sign to the average of the first p a rt; the th ird p a rt is a stress

resulting from a simple m om ent th a t is equal to and opposite in sign to the m om ent produced by the sum of the first two parts. T h at is, the first p a rt alone a j would result from complete restraint, the sum of the first and second parts a " would result from restraint against warping only;

th e sum of all three parts, i.e., <jx, would result if no external restrain t were applied during shrinkage.

Although in this derivation an expression for ax appears to be the ultim ate goal, expressions for a j and for <r" are also desirable. The stress

<rx m ay be representative of the stress in pavem ent slabs or building walls th a t are restrained from shortening and the stress a " is representa­

tive of an unrestrained wall drying equally from two opposite sides (Case I I discussed later).

Since for complete longitudinal restraint ex = 0, it follows from Equa­

tion 10 th a t the first p a rt of the stress is

<rJ = E S ... (15)

1 r Since the average value of <jx is — I

b

a j dy, the second p a rt of 0

ax is E r

I S dy; therefore, the sum of the first and second parts 0

(<tx") is given by

S

-

I f sd*

0

(16)

M

- / y dy = E

The m om ent produced by <rx" is the m om ent necessary to prevent w arp­

ing. This mom ent per u n it w idth of beam is found by m ultiplying E qua­

tion 16 by y dy and integrating. This gives

b r- b b

J

^■J'

*-0 O '

For no external restraint this m om ent m ust be removed by superposing an equal and opposite moment. The stress resulting from a moment — M is given by the elem entary theory of beams as

M (y - 5/2) 1/12

178 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 This becomes

< ⁶ - i ² r) ^{I f}

0 0

when M from E quation 17 is substituted. W hen this stress, the th ird p a rt of <tx is added to th e sum of th e first and second p arts of E quation 16, th e result is E quation 14 previously derived.

Stresses in terms of shortening and warping. The shortening of th e beam can be considered as due to the second p a rt of th e stress since it is the addition of this p a rt th a t removes longitudinal restrain t and thereby per­

m its th e shortening of th e beam. From these considerations it follows th a t b

u n it shortening = S a.

U ‘

0 where S av is th e average value of S.

The bending (warping) of th e beam can be considered as due to the th ird p a rt of th e stress since it is the addition of this p a rt th a t removes th e rem aining external restrain t and thereby perm its w arping of the beam. T he deflection v caused by th e m om ent — M is given by th e elemen­

ta ry theory of th e bending of beams as v = — — where I is the span and v is th e deflection of points w ithin th e span w ith respect to either of th e end points x = 0 and x = I. The maximum deflection vmax (warp­

ing) th a t occurs a t x = 1/2 is therefore 6 M

Eb3

r b b n

/ l \ 2 3Z2

1 C 1 c

\ 2 7 = 2 b - J S y d y - — J S d y

- 0 0

.(19)

By w riting the second p a rt of E r

- b j S dy, in term s of the unit 0

shortening S av it produces, and by w riting th e th ird p a rt of o„

r b b

E ( 6 “ 12 V) h f Sydy~ k f s dy

0 0

*The p a r t in th e b rack ets is num erically equal to one-sixth of th e u n it strain e th a t w ould b e produced in either th e to p or th e b o tto m of th e beam b y a m om ent ju s t sufficient to straig h ten it. T h a t is,

2bVmas

’’ ~w~

179 in term s of the deflection vmax it produces, the expressions for the stresses are p u t into more usable forms. W hen this is done, the following equations for stresses are obtained.

For complete longitudinal restraint (first p a rt of ax),

o j = E S ...(15) For restrain t against warping only (sum of first and second p arts of ox),

o j' = E (S — S av) ... (20) For no external restraint (sum of all three p arts of <rx),

2 bv„

= E S — S av +

3Z2 ⁽21⁾

Evaluation of the parameters — and ^ n ’maz

S co w s * . W hen E quation 5 for S / S oo is substituted in E quation 18 for shortening and in E quation 19 for warping and the indicated integrations are performed, the result is

Sax i r S ™ - T f r

Sc ~ b J Sc dy = 1 " Z _ e Hn ... (22)

0 1

2f>ymai 1 f S S av ~ ^

= b*J SZ y dy ~ 2SZ ^{= Z .} e °n ⁽²³⁾

0 1

W S c where

and

H n = 2 B 2

f t (B* + B + ft)

Gn = ( — * - v cos f t 2 / f t

If T, the non-dimensional tim e-factor, is small, the series in Equations 22 and 23 converge rath er slowly, and in th a t case it is convenient to use the following equations obtained by substituting E quation 7 into E qua­

tions 18 and 19, respectively.*

Sa

Sc eB2T <KB V T ) - 1 + ^ V 7T

• (24)

*The lower integration lim it for each integral is decreased from 0 to — » ,

180

2 bvm - ..( 2 5 )

JO U RN A L OF THE AMERICAN CONCRETE INSTITUTE January 1946

1_

2B + B 2

R2rr --- 2B V T

e 1 <KB V T ) - 1 ' V 7T

F u rth e rm o re , if th e p a ra m e te r B ^ T is v e ry sm all, it is still b e tte r to use th e follow ing e q u atio n s o b ta in e d b y ex p a n d in g th e expressions in th e b ra c k e ts of E q u a tio n s 24 an d 25.

= B T 1 - - A = B ^ T + ~ ( B V T ) 2

2 bv„

3 V T ' ' 2 ' ’ 15 V 7T

( 5 V r ) 3+ . . (24a)

B T 2

■)

. (25a)

In general th e follow ing rules will be found ap p licab le fo r ra p id e v a lu a -

S 2bv

tio n of th e p a ra m e te rs — an d to a fair degree of accu racy .

CO o l cx>

If r is m ore th a n a b o u t 0.05, use E q u a tio n s 22 a n d 23.

If T is less th a n a b o u t 0.05 an d B is m ore th a n a b o u t 5, use E q u a tio n s 24 an d 25.

If T is less th a n a b o u t 0.05 an d B is less th a n a b o u t 5, use E q u a tio n s 24a a n d 25a.

Forces and m om ents necessary fo r complete restraint. T h e force n eces­

sa ry for lo n g itu d in al r e s tra in t is f<rx' d A . T herefore, th e a v era g e force b

1 C

per u n it area is ~ I o f dy. F ro m E q u a tio n s 15, 5, a n d 22 th is becom es 0

force p er u n it area = E S <

■I

- T f t

e H ,

• (26)

F ro m E q u a tio n s 17, 5, a n d 23, th e m o m e n t p er u n it w id th n ecessa ry for re s tra in t a g a in st w arping is fou n d to be

SHRINKAGE STRESSES IN CONCRETE

S im p lifica tio n by taking B as equal to in fin ity. T h e p rin cip al e q u a ­ tio n s d eriv ed ab o v e reduce to sim pler form s an d th e c o m p u ta tio n of n u m e ric a l v a lu es is less te d io u s if th e a ssu m p tio n is m ad e t h a t B , i.e., f b / k , eq u als in fin ity . If B is large, say 100 or m ore, th e e rro r in tro d u c e d b y assu m in g i t to be in fin ity is negligible. H ow ever, if B is less th a n a b o u t 5, th e e rro r in tro d u c e d b y considering it to be in fin ity m a y be ap p reciab le as is show n, for exam ple, b y Fig. 8 , 9, a n d 14. W h e th e r ju stifiab le or n o t, th e assu m p tio n t h a t B = is fre q u e n tly m ad e in a n a l­

ogous pro b lem s to w hich th e diffusion e q u a tio n applies. T h is assu m p ­ tio n w as m ad e b y T e rzag h i a n d F rö h lich 4 in dev elo p in g th e th e o ry of s e ttle m e n t of fo u n d a tio n s due to consolidation of u n d erly in g m a te ria l, by G lover6 in a s tu d y of d is trib u tio n of te m p e ra tu re in con crete dam s, a n d b y C a rlso n 1 in a s tu d y of d istrib u tio n of m o istu re an d sh rin k ag e in con­

crete. T h e m ore im p o rta n t of th e above e q u a tio n s for th e special case of B = oo are given below :

E q u a tio n 5 becom es

S ^ - 4 ( - l ) - ‘ - ( 2 » - 1 ) i t T

- 1 - 2. cosj(2n

1 E q u a tio n 7 becom es

s

V

_ _ 8 ^ 1 — (2n — l )2 (ir2/4 ) T

S c 1 7T2 2 - 1

S c * 7T 2 ( 2n - l ) 2 e

E q u a tio n 24 becom es

$av 2 ^ rp

S oo V 7T E q u a tio n 23 becom es

CD

2 bvmai KT- 4

312S c Tr2 (2n - l ) 2 1

E q u a tio n 25 becom es 2bvmax _ V T _ T 3l2S c

( _ l ) - i 4 tt (2tt 1) 1

(2n - I ) 2 — T

e 4*

182 JO U RN A L OF THE AMERICAN CONCRETE INSTITUTE January 1946

Tables, curves, and com putations.* T ab les a n d d ia g ra m s such as th o se b y N ew m an 6 are av ailab le from w hich valu es of S / Sco a n d S av/ S co m a y be d ete rm in e d . H ow ever, such p u blished ta b le s are in g en eral n o t a d e ­ q u a te for th e p re se n t problem . T h e sm allest v alu e of th e p a ra m e te r 7 used b y N ew m an in his c o m p u ta tio n s was 0.1, w hereas th e stresses in concrete m a y be desired for a m uch earlier period, t h e ta b le s g i\e n here w ere p re p a re d for T as low as 0.001. M o reo v er, so fa r as is kncm n, th e p a ra m e te r Vmax h a d n o t previously been e v a lu a te d for th is or a n y

3 B S oo

analogous p roblem an d its e v a lu atio n is necessary for th e p ro b lem h ere considered.

A ste p in th e ev a lu a tio n of E q u a tio n s 15, 20, a n d 21 for th e th e o r e ti­

cal stresses in a b eam d ry in g from one side u n d e r th e d ifferen t m odes of

i ~ . S S av i 2&Z1 mdx

re s tra in t is th e e v a lu a tio n of th e th re e q u a n titie s —— , —— a n d - •

O oo Uco co

T h e first q u a n tity S / S oo as a fu n ctio n of th e th re e p a ra m e te rs y /b , B a n d T, is given in T ab le 2 an d show n g rap h ically in Fig. 7. T h e second q u a n ­ tity Sav/S co as a fu n ctio n of B an d T is given in T a b le 3 a n d show n grap h icallv in Fig. 8 .f T h e th ird q u a n tity P ^ Vmax as a fu n c tio n of B an d

3 l2S oo

T is given in T ab le 4 a n d show n g rap h ically in Fig. 9. T a b le s 5 a n d 6 giving th e stresses a " a n d ax (E q u a tio n s 20 an d 21) as fu n c tio n s of th e th re e p a ra m e te rs y /b , B , a n d T , w ere read ily p re p a re d a fte r th e th re e p rim a ry q u a n titie s h a d been e v a lu a te d (T ables 2, 3, a n d 4). R e su lts for B = 5 are show n g rap h ically in Fig. 10, 11, 12, a n d 13. Fig. 14 show s m ax im u m values of an d a n d of <^ }Vmax versu s th e p a ra m e te r B .

E S » E S « 3l2S ^

T h e co m p u ta tio n s m ade for th e p re p a ra tio n of th e ta b le s a n d d iag ram s are explained in p a r t below.

If th e p a ra m e te r T is so sm all t h a t th e e q u a tio n s b ased u p o n th e a ssu m p tio n t h a t th e b o d y e x ten d s to in fin ity m a y be used in ste a d of th e th e o re tic a lly co rre ct e q u atio n s, no difficulty is en co u n tered . F o r exam ple, le t y /b = 0.8, B = 5, a n d T = 0.01. E q u a tio n 7 th e n becom es

A = J - * ( _ ° 4 _ + 5 x o . A „ 5 X 0.2 + 25 X 0.01

jS . \ 2 X 0 .1 / \ 2 X 0.1 /

1 o r

= d> (1) - </> (1.5) e

* T h e v alu es of /3n g iv e n in T a b le 1 differ s lig h tly fro m th o s e giv en b y N e w m a n in so m e in s ta n c e s . T h e v alu es in T a b le 1 a re b eliev ed to b e a c c u r a te to th e n u m b e r of places given. T h e ta b le s o th e r th a n T a b le 1 h a v e n o t b ee n ch e ck ed b y d u p lic a te co m p u ta tio n s . B u t, ex c e p t fo r th e la s t d ig it, w h ich m a y b e in a c c u r a te b y a p o in t or tw o , th e se ta b le s a r e b eliev ed to b e re a so n a b ly a c c u ra te .

t i n F ig . 8 a n d 9, w h ere s h o rte n in g a n d w a rp in g w ere th e d e p e n d e n t v a ria b le s , th e s q u a r e r o o t of T for th e ab scissas w as fo u n d to b e b e tte r th a n T , b ecau se a co n sid e rab le p a r t of su c h g ra p h s w ere a p p r o x im a te ly s tr a ig h t lines. F o r th is re a so n th e s q u a re ro o t of T r a th e r th a n T w as u sed fo r th e ab scissas in th e co n ­ s tr u c tio n of o th e r d ia g ram s.

SHRINKAGE STRESSES IN CONCRETE F ro m m a th e m a tic a l ta b le s

1 25

<*> (1) = 0.15730; <t> (1.5) = 0.03389; e ' = 3.4903 T h e refo re S / S » = 0.1573 - 0.03389 X 3.4903 = 0.0390.

N o te t h a t th is is th e v alu e given in T a b le 2 for th e abo v e valu es of B , T , an d y /b . Also n o te t h a t for th e sam e B a n d T th e ta b le gives zero for y /b = 0, show ing t h a t it w as perm issible to use E q u a tio n 7 in ste a d of E q u a tio n 5.

W hen th e th e o re tic a lly co rrec t e q u a tio n s a re used, th e co m p u ta tio n s are m ore involved. F o r in sta n ce, let T = 0.1 in ste a d of 0.01 in th e abo v e exam ple. T will th e n be so large t h a t S / S <*, will h av e an a p p reci­

ab le v a lu e a t y /b = 0. T herefore, E q u a tio n 7 will n o t be ap p licab le an d E q u a tio n 5, th e ex act eq u a tio n , m u st be used. A s u b s titu tio n of values for T a n d y /b in to E q u a tio n 5 gives

00 _ o i f l 2

A = i _ V C° S ° '8/3n

S® 2 . 6 Fn cos

1

T h e first ste p in e v a lu a tin g th e above expression is to d e te rm in e j3„

w hich E q u a tio n 2c show s to be a fu n ctio n of f b / k an d n, i.e., B , a n d th e in teg er n. T h e d e te rm in a tio n of j3„ b y in te rp o la tio n is sim plified b y th e in tro d u c tio n of a n w here a n dep en d s on B a n d n. T h e e q u a tio n for /3n is th e n w ritte n

j8„ = (n — 1 -\-an) i r ... (28) C u rv es of a n versu s B for th e first six valu es of n a n d for n = 21 are show n in Fig. 2. B y m eans of Fig. 2 a n d E q u a tio n 28 a n y desired /3„

m a y be fo u n d w ith reaso n ab le ac c u ra cy for a n y v alu e of B . T h e first six v alu es of /3n for several different valu es of B are given in T a b le 1.

A fte r finding /3„ for th e given v alu es of B a n d n, th e fa c to rs F n, cos pn, cos H ) a n d e T (in are d eterm in ed . F n a n d cos as fu n c­

tio n s of B are show n in F ig. 3 a n d 4, respectively, for th e first six values of n. T h e fu n ctio n s cos a n d e ^ @n are read ily o b ta in e d from m a th e m a tic a l ta b le s a fte r th e p ro d u c ts V- a n d T (31 h a v e been d eter-

b

m ined. W h en th e p ro p e r values of th e fo u r fac to rs listed above are sub- s titu te d , th e ab o v e e q u a tio n for — becom es

s

S ⁰⁰