l\UCHIGAN STATE fiiGlfiiAY Dl£PJ\RTMENT

Charles M. Ziegler State Highvray Conuniss:ione1·

SLAB AC'i'ION n1 CONGRET.S PAVEMENTS

UNDEP. STATIC LOADS

by

=~. D ~ Chilcts

A joint renearch proj oct betvreen Engineer:Lng Hesearch Laboratory, University of Michigrm

and the Michif;an State High·v1ay Dr-3phrtment

Engineering Hesenrch Project No. M-·561 Highway Department Proj uct No. 44 F'-11

Presented at the 27th Annual Meeting of the Highway Research Board, Washington, D. 9·

December 2 - 5, 194.7

Rosearch Laborcctory Testing and Research Division

Ropori; No. 90 November 1, 1947

'1

Pag!2_

Introduction l

Description of Equipment and Test Procedure 2

Materials and Equipment 2

Measuring Devices 5

Application of Loads 5

Measurement of Destructive Effect 5

Subgrade Modulus 5

The Loading Program 7

Freo Edge Load:ing 8

c~~L=~g e

Comparative TeGts 10

Comparison of Model Investigation with Ro:mlk1 of thifl Study 12

Results of Other Investigations 13

Limitations of this Study 15

Results of Theoretical Analysis 13

Observations and Conclusions 21

Acknowledement 24

Bibliography 25

Table

Table

Table

'l'able

Graph

Graph

Graph

Graph

Graph

Graph

Graph

Graph

Graph

Graph

Graph

Graph

,Graph

Graph

Graph

I. Characteristics of Sub grade Sand

Page

5

4

15

20

II.

III.

IV.

1..

2.

3.

4.

6.

7.

8.

9.

10.

ll.

12.

15.

14.

15o

16.

Mix and Test Date for Concrete

Stresses Under a Single Whoel

Stresses Under Multiple Wheels

Bearing Plate Tests

Data for Subgradc Modulus

Single Wheel Loading Data

Data from Test on One Axle at

Slab

Slab Edge

5

6

7

[l

\)

1'hree Axle Data for Loading at F'ree Edge 9

Cm·ves Resulting from Tc"sts on Corner viitb One Axle 9

Results of Two Axle Tests ut Corner Location 9

Data from Corner Tests on Throe Axles 10

Curves' for Edge Loading 11

Curves for Corner Loading 11

Effect of Axle Spacing ll

'l'ypical Data from the Model Study 12

Extnnsometor Readings at Slub Corn(~r 15

Westergaard 1 s Moment Curves for the Interior 16

Westorgao.rd' s Moment Curves for the Edge 16

:B"'igure L Form

Figure 2. Wire

Figure 5. Form

Figure 4. Plan

for Sub grade

Heating Grid

for Test Slab

of 1Test Slab

LIST OF' }'IGURE[i

Page

5

5

4

4

}'igure 5. Method of Grinding Suri'aco for Gage AppUcation 4

Figure 6. Length Change Comparator

Figure 7. ArrangGment for Bearing Plate Test

Figure 8. Slab Loading for Sub grade l!lodulus

Figure 9. Load Being Applied 'I'hrough a Single Tire

Figure 10. One Axle at E'r·ee Edge> of Slab

Figure H. Two Axles at fU.ab Edg0

Figure 12. Three Axles on Cornor of Slab

Figure 13. Load Spadng for l\1iuH:i.ple Axles

5

5

6

7

8

8

10

16

Early in Hl44 an imrorotigation was begun to determine the destructive

effect of axle loading:> upon concrete Blabs. The Department of Engineering

Research of the Uni versi.ty of Michigan anCI the JVlichigan State Highway

Department were the participants in this proj leCt. Pr8l.irninary tests were

made upon a small model and these X'e~?ults have be1:::n pub1isl'wd. (1)

Briefly, the.st: results indicated tlmt undHr ~:;tatic conditions the

addition of rfheels to .s.n H:z.J.e .is not l}.n e:~:pediont method of increasing the

loading capacity; that t1v0 a:"lor,J in t•:mdem 11rrangsment could carry a

standing load moro than twice t~1at of n tlingle c:.xle if a proper axle

spacing wore UG:3d; cmd, that a th:coe axl8 syf.:1t8m could be spaced f30 as to

support static .Loads three t.i.mef.: the rd.ng.Le r.\xle valutlFJ.

Although thr~ model :;ervcd very W<'Jll to indicate the relative: effects

of various loading arrangements £tnd locationD upon the slo.b, it. was neces-

sv.ry to repeat certain exper:Lmente upon :: full :3cnle slab in order to deter-

mine absolute valueB 'IJhich could be used in slab and vehicl8 dt.~s:qp.1. Wii'h

thiG purpose in mlnd a B :Lnch uniform ala.b 11 foet wide by 28 feet long was

cast and tested in th(.;: Highway H(~search Laboratory o.t East Lansing~

LoadB corroilponding to full. hi.ghwa;, loads vmre applied through actual

trailc"r axles 11ith dual 10.00-·~;o tir<)G at 70 p.B.i. air procsure. The

loading positions were midway botvmen the end.t1 of tho lont~i tudinal frGG edge

and [elsa at the corner of' tht:; r,lab. Single axles, t·v:.'o o.xleB spaced from

3-l/2 to 9 feet, and three axles ~--:pa.cod from 4 to 7 feot were loaded and the

slab strains and deflections me:o:mred.

Note: Numbers in pttrentheE:eo refer to b:ib1iograph;-v o

,j

Rosul ts of the tests on the full--size f!l.ab correlated quite Hell with

those of the model. At a full ltl,OOO pound J.oaC: pcrr axlo, greater stresses

\'Vert~ produced in the slab by a ningle axle th~m by tvw or three axle cornbi·­

nations with four foot to dght foot axle spacingL1. At the u;mD.l four foot

spacing bot1,;Ieen axle~' ttw throe axle eombino.tion C9.used considerably less

str1::;ss than either [', two axl::; system ox· a ~d.ngl0 o.xle when the loads were

t\Ppl.ied at the cornur of tho slub.

1J.lhi,'3 r(Jport includes a descri.ption of the materio.lr-1 and equipment

used for this study, n dJ.r;cur;sLm rJ:t' the md,iK>d of ccpplication of the loads

and e. graphical precentat1on of neJc~ctcd dat[L. A comparison is made between

the model study nnd the full ncale invest:i.gatlon. The results of other

lnvoatigntiono cmd theoretical computation~~ L:.r.? shov,n to corroborate certain

data, and a bibliography of those sources is inclwl.ud.

Jllln t<3ri.als 1_tgd E.9..l!iH.'!2llt

The luborat.:Jry at En.st L:~msi.ng w::~s chosen for the si tG of the test

slab because there 1,;vas sufficient f;po.ce for the eonstruction of a large

slab and fac:i.lities for applying the l.oaclG. Other lnvet,tigators had per-·

formed tests upon 13labs cast ·Jut of dours, but the reiJults ~;yere affected by

vmrping due to clnngGS ln tomperatu:ce and mo.isturo. It was hoped thD.t

laboratory control would mi.nimiz,'} thic unfavorable condition.

A vwocLm form 115 feet by 32 feet by 2 feot was built upon the concrete

floor of the lD.bora.tory. TiH bars ~;wre plnced ~J.t tbo curners and !:J.t inner

points to prevent spre!lding .,dwn the form W&.f:l f:i.l1ed with sub grade material 0

--2--

A systefil of p8rforrr ted pipes was laid upon th0 floor and ccmnected to the.

VJ&tHr supply for i:,h(:J purpot!O nf cont,ro11:Lng [:;ubgrc.de ,·aoistu:r.e,, thereby

giving some c,:)ntrol ov,~r subgr~:uh; ~nvdulns ~ This stnge of construction is

shown in F'igm'e 1.

Six inches of fST:lVGl were ther.t placed i:n the bottom of the form ~Lnd

the remainder filled to v;:ithin tVlo :Lnclms of tho t:Jp vdth a Gcl'.JCtr:~d bo.nk

run sand. Thin E~and \'.Jtt.S chosen becauso of its s.1.1~1ilari ty to that used as

TABLE I~ GHAHJ.\.CTJi:li.ISTICS O:F' SUBGRliDE; SAND

Sit3Ve No. Percent Passtng

Percent Tvioistu1.:-;;:) Density (p. c .f.)

Sinve ArwJ:ysis

10 40 99 .. 43

20 97.57

Ik;nsity Values .

fiO IJ.j~1G

100 ?.flO

1 :3 5 7 9 11 13 98 104 107 10·1 10:< 101 92

In ~)rd.e:c tr) IilD.intG.in a m:Ln:lmun terrq)t:;rc.tur~,,; gradient in the slab r::::m10

me'thoc1 of control had to bo duvis(~d ~ Prelirc::in1""J.ry investigcLtion showed that

the differ\;:~ncE.J in tem.pc~:rature betvrer~rJ tb.o t·)p and bottom of the .slab would

be small, but tlu1 t t h·c lovmr f3Ul'fD.c•:o of' i;he slab li':Juld likely be cooler than

the top. Of the vari,XlS meth(Jls proposed :for b2.:1ting this loFJer surface,

the one vvhich appeared to Ltffoct the subgrad'": bearing captlcit.y the least was

an elr;ctrieally hented wire' g:cJ.d. This w~1s insta11t~cl as sho¥rn in Figure 2

and it w.;).8 e-overcd with a tvm i:neh thicknens of subg:co.do t:iu.nd.

A wooden forr;, 11 feet by 2G feet by D inches ;;,ls built up:ln the pre--

VUlS planed and f.l.uxi.U.ury Elquipmen:t· :Lncid.ento.l tD the tests ·aas installed.

-3·-

Figure J, F'orm fo;: <mbgn:.<.ch;, f;llow:l.nf. ti.e rods) w:Jtcr·­prrJcf:i.ng ~;.nd woi:c;turu eontrul pip<c· l:J.yout,

ri.g:urc 2. \·i:rc· h:'~'t~::Ln~ .. i;ricJ l:!.l.ttCXJ J.n su·bgn;-:1:=, to c;on trol tcrn;·:cn:·:~t'tt·d'·= d.Li'f\;rul t.:i:.d. in ;->.1 n 1·).

For the purpose of measuring stral.ns on the bottom of the slab, SR-4

gs.ges were attached to mortar bloclw and thane blocks were placed on the

subgrade in such a way that the gages v;ould be in the plane of the lov.rer

surface o.f the slab. Unforturw:tely these gag eo Vi ere not sufficiently

insulated to give reli.:ible renults c,fter n fei·;· week.s time.

For c. sepc1rute study, incidental to the loading investigation, dowel

bars of various ::lizes and len;stht3 vvere installed o.t one foot intervaln on

all edges of the 0lab. These bars nncl the mortar blocks are seen in li'igure

The test slab was cnst using· a carefully designed transit mixed air

entral.necl concrete. Tabl'e II gives the mix and strength data. At this time

( ')'

five installations of thermocouples and Bou:roucos moisture cellG r.;.Jwere made.

for the purpose of keeping accurate record of t<empr-lrature and moisture dif-

ferential throughout the olab and to aiel in their control. A diagram of the

slab and the locn.tion of meaf~uring equipment is given in Figure 4.

Curing was accomplirJhed by applying a membrane curing compound to the

slab four hour:J 11fter pourl.ng. The' relcdoive humidity of th8 r'JOm was main-

tained at about '70 percent and the tempeJ.·ature held at 75° for tMenty-eight

days. During tllie period moirture f:J,nd tHnperatt1re measurements ·~·~ere made

D.nd comparat_or reading~: \';ere taken for length change and "~Harping. Flat eur-

faces were ground on the slab ,surfe .. ce according to the plan of F'iE,ure 4 and

1/2 inch by 1/16 circular brae'S discs ·,'lere cemented to the slab for eleva-

tion and deflection measurements. SR-t!, strain gages vverc applied and vdred

to ,junction boxee for fc.cility Ln reading. The method of grinding smooth

0urf~ces for thet>e installations i:::. shown i:G F'iguro 56

-4-

l~'il~llr"C; ) .

<lo\vel bru·s Jl<.Jr;n f.:_~~r t<-:':~;t.

i.n til..::, c r:> •

TABLE II

MIX AilD TEST DA1'A FOB. CONCRETI' SLAB

Material

Cement PeniJ;lsular V .n. (Raw) Boichot 2lJS

·,; ei!.;hts per 22l!__yd. concx·ete

Fine Aggregate Coarse Aggregate Water

American Aggregate Green Oak--lOA

517 lbs. 1182.5 189:2.0

?oll.S

Fine Aggregate Gradation:

Sieve Size No. Percent Passing

Coarse Aggregate:

Sieve Size, inche8 Percent Passing

Average slump = 7--1/8 inches

4 lOCI

l 100

Average air content o: 7.5 percemt

8 :-)4

1.6 75

1; 2 5:.1

50 51

'A' '0 l)/ u

34

GO 17

100 l

No. 4 1

200 0

Average 28 day compressive strength 9-6"xl2" test cylinders = 36!'0 p.s.i.

Average 28 day moduluB of ruptn1·e 6-6"x8 11 x56" beams= 5135 p.s.i.

Average 28 day modulns of elastici.ty of 6 cylinders at 400 p.s.i. = 5.25xlo6 p.s.i.

~~

A•TYPE AR-1 ROSETTE -=TYPE A-I GAGE

SECTION THROUGH A,C,D,E

?-.., .. · s -- --~- -~·-. :·_oz ___ . _·6: ... ~:_·: 'p· __ ( MOISTURE CELLS

a' tJ• o._o. o • c::::R:- • • - a -D-:. : ~ ~ : 0 -~a~

. c:::::F'; ' .

~BMI

)/;i)-· --- /p

~ ~~BM2

LONGITUDINAL SECTION THROUGH 8

.'C._-~-- . o ·o.

~/"::~~b-.~ . · -0~' ~ ~- cy::::~-~CARLSON

;~·.'o:"·~~' :,o-. :liill:

"-''' I ~- ·\\~

BRACKETS

c.__,_ <1i] BM 3

lr-;; lr_MON./

I I,

PLAN aj TEST SLAB Figure 4

Measur~ Devices

The apparatus necessary for the measurement of moisture in the sub-

grade and concrete is thoroughly drcscribed in the teclmical bulletin to

'Nhich reference vvc ... s made. Temperatures \Iere found by reading on a GtandcJ,rd

potentiometer the small eom.f~ generated by iron-conntantan thermocoupleB.

Strains 1ge~ce mea~~ured b;y rc~:.istance changes in bonded wire SR-4 type A-1 and

lUl-l strain gages. ~Chese rer.d.r:tance chang·es Hero read directly as unit

strain by a Balchv"in ~SoutlTdcrk SH--4 ntrEin indicator. Ft-.:deral one--thounandth

d:La1.s at the Dlab ed~:;es and corners indicated deflectionn, ,,:bile one-ten

thousandth dials Y<'ere used. in caJ.iln'atod Tine_;:: to determine the load inten-

nity. A special compare.tor was con;:3tructed t.o ueanux-e lenbth change and

vrarping. This ir, i.llustrected in Figun:; fL

_f:_P.plication .. of I:~oad:3

All loach:: vrere applied by mc::ans of hydxau:Lic jacks rer_.._ctinL at;ainst an

11 I 11 beam on the labor& tory ceilinz. Co..l.ibra ted dyn<.Lmometer rint:;r;; served to

indicatH the load intensity·. Although loads oJ:' an:;r value up to 20,000

pounds could be app.U.od, most of t11e tur:t:c: r:ere made o t 10,000, 13,000,

16,000, and 18,000 pound nxle loads.

§ubgrasJ.e Mo~~~~~"!:l~~

In o:cd.ei' to make 8. conrpa.:~·lson of test resul tt! VJi th theoretical values,

it v:ra.s nec8f3SCi.1 .. J' to detr:rm.ino the mocJu.lus of fmbt)T<ide stiffnef:1El. Thir3 vm .. s

done by two methodE. First, befor1;; "L.he Glab 'l'i"{lf' poured a number of loading

-5-

Fig1U'0 5 · (,>f strain

Methoc~ of g:cinc:ing r::urfn.c:e fur applJc<JtJ.c,n go.gGs end d!O:flr~cti()n ttTgetr-J,

r i.gurc;; c•. L/:'nzth cLo.nt:::: r;nd w.•).J:'[)ing fL•:Cf".811r'Cil1Cnt:c~ heiw:: f!lEU~(;; bJr [:1jJ(Jn:ia1 Ci'.;{L_ ~~~r'F~.t(1l',

t-ests vmre mta.de using a. 50 inch p1ate. I1'igure '1 exh.Lbit'~ the apparatus, and

data for two locations are sho,;;Tl in Gr.s.ph 1. It is apparent that the

modulus, k, is a.bout 110 p.c~i. under the existing conditions,

At the end of a ;..~0 day curtng period further tests J\•ere made by

loarUng the slab in four locations. Havin.,; found the moduJ.us of elasticity

of the concre~e, the :mbgrad,:o modulus ,,as computed from both load-deflection

data and load--stren.s data by .f.'ormula,s devr·.'loped by her::rt;e:cgaard ( 3) and by the·

modified equationf·J from thE· A:r.,l:Lngton Figure 8 1::: an illustration

of the appt;lratus used for thef30 te::rLt~. Thr:; data are compiled and preroentm1

in Graph 2., and the accom[X:=tn:ling tnble ~ Thc~re appear,s to be good corre1a-·

ti.on b~?tv\;een thef'C t·Ho method;_: of testing since the value 110 which -.nas

obtained by the bearing plato method also appeal's cevera:L timell in the table.

From these tests the subgrado modulus values for t1:-H~ tv11o soil conditionr~

vrhich prevailed vore chosen o For the fir[-:rt cond.i.t.:Lo.n the Vt~lue k=llO p. c ~ i.

was used, and for the: saturated conclition k=60 p.c.~i. neemed to be a fa:lr

value.

In spite of' ritid control of temper~Lttu·e and hwnidi ty tf.tere vJas some

upward -,Harping of' t.he concrete elab~ A continuoun record of comparator

had rais<:;d about two tentJ.H;; i.nches~ Since the temperature differential was

Bmall, the curll.nt, v:as attributed to moicture and fundamental differences in

the concrete caused by tho met bod of p1<:.t.C8.merrt ~ In an attempt to rectify

this eond.ition the uppe:c ,sux-face-of tht:1 slab waD flooded with water' and left

in that ::;.tute w:1til there r>Jan no further dusni;;;El.J:'d. Inove.ment of tbe r,-3lab. The

recove~cy vlaS about fi:ft:f porcenl~~ A heavy coat of wembrc..ne cur:i.ng compound

w-as applied as soon as the water wa::1 removed.

-fl-

8

iii ~· It

Q c 0 .J

I

LOCATION A / /

PEN£TftATION IN HUHOftEDTHS INCHES

8

-~-Q c 4 0 .J

2

LOCATION B /

10 PENETRATION IN HUNDftEDTHS INCHES

Modulus of suborade stiffness, k= unit lo~d = 110 p.c.i. approximately deflectaon

Graph 1. 1'hirty inch bearing plate t'::stu for two loco.tion:::; on Lmbgrade.

Figure 8, Nk;th.;d .:)f fJJ.cd'J l.xJUng :f:Jr tbd ·ktercinatLm of sulJgrade lli'Y:1ulu;.; by ti1c·:Jr::?tic:c~l forcm1.:::Ls.

.. "' z ::> 0 Q.

0

"' .. "' ::> 0 :1: ,_ ~ 0 .. 0 -'

DATA FROM LOADS ON DIAMETER

9'' SLAB PLATE

10

8

6

4

2

0 0 10 20 30 40 so 60

DfFLECTION IN THOUSANDTHS INCHES

IDENTIFICATION' SAND NORMAL

a' INTERIOR POINT

b• FREE EDGE

C < JOINT EDGE

d = CORNER

a bed

Ill 0 z B~---+--6-+--P~--~ ::> 0 ... ~ 6 -1t~--+---~ .. 1------+o----

"' :::1 0

~ 4 ll----o-4H~H-----+----i ~ 0

~ 2 -'

0 70 0 20 40 60

MAXIMUM UNIT STRAIN

SAND SATURATED

a:~ INTERIOR b

1= FREE EDGE

C: JOINT EDGE

50

TABULATED VALUES OF SUBGRADE MODU' US K IN PC I I- ..

LOCATION 1ST SUBGRADE CONDITION 2ND CONDITION FORMULAS

BY DEfLECTION BY STRESSES BY DEFLECTIONS USED

INTERIOR 100 110 65 WESTERGAARD

FREE EDGE 110 110 65 WESTERGAARD

JOINT EDGE 80 95 40 B. P.R.

CORNER 55 60 - B.P R.

Graph 2

An soon as a sc~ries of slab elevation readings had been completed the

loading program was begun~ A preliminary serieB of tests were made at

royrnmetric pointe: with a metal plate for bem·ing area to determine the loc<<l

differencefJ in the slab and to attempt to attain good bearing between EJlab

and subr:;rade.

A single 10.00--20 tix·e at 70 p.t•:.i. infl11,tion pressure ·,1as now located

at several points on the ,:1.l.ab a.nd dF:f1ection ctnd straln readings 'imre taken~

Figure 9 is c1n e:x:am.ple of thin te;3t. Tho d.:1t~;. curve6 are sho·~~'TI in Graph ~).

A comparison of the13e curves Hith thoGf.) of Graph 2 EJhorie thb .. t the strains

and deflections under the \iheel are comparable to tho:Je 1mder the metc,l

plate. Apparently the greater contact area under the tire and conr,equent

reduced unit presr;ure upon the '1lab does not causcc cuoy appreciable decrease

in slab ~rGressr:)e below· those produced undPr the metal plate~

This study was .follo\~ed by ;:d.mi1ar testr3 on a fJinglo axle equipped

with dual tireE. Due to the small strain magnitudes and the diff'Lculty in

obtaining reUable deflection measurements at interior points of the slab,

tests at theso locations were dir:Jcontinucd, and the only data presented for

these and subsequent tef3tto are those for the edgB and corner locationn.

Next, two axlef3 \;;;·ere plti..ced rdth outer v;rhr~els on the free edge of the

slab. One series of tG;3ts ~Jus r1u1 vd.t.h thB axlHs symmetrically placed about

the middle point of the ed.E.;;e ·' and another uerie-f3 ·:,·m.s made vrith one axle at

the slab corner~ A variety of axle- spacinf,G ·,vas uced in eaeh {:,roup of tests~

.Finally tJJree azles ·,-~:ere loaded in the }Jame t('St p<...~ttern as rvas used

for two axles~ The mr'""'ximum hxlo 13pacing was necessarily liilli ted. because of

the Jength of the te13t ~~lrlb. For large 8pacin.gr:::J tl'.te ter;;ts at the center

-7-

10 (I)

0 2: :> 0 8 a,.

0 z "" en 6 ;:)

0 % 1-

:! 4

0

"" 0 ..J 2 ..J ~ 1-0 1- 0

A 8 c ?

f ~

/ I , I / I I

\J

t 7 ~/'

{ /./ I I r/ I

cJ/ v 0 /)( I I

I

I I /,' v ,,

A -INTERIOR POSITION ,, B- FREE EDGE -I

~- C- JOINT EDGE

D- SLAB CORNER

IO 26 30 40 50 60 DEFLEOTION IN THOUSANDTHS INCHES

(I)

0 z g s 1111------J.----rl-Q,.

0 z "" en 61-----~~--~~~~---4----~ :> 0 % 1-

. z 411-------,f-#-/-'f-----f---4-----j

0

"" 0 ..J 2 1---1---#,J~-

SUBGRADE MODULUS

,.J

"' 1-0 1- 0

K = 110 P.C.I.

20 40 60 80 MAXIMUM UNIT STRAIN

100

DEFLECTIONS AND MAXIMUM STRAINS DUE TO LOADS ON A SINGLE 10.00-20 TIRE AT 70 P.S.I. INFLATION PRESSURE

were 11ffected by the ends, and the "Centro at th(J end were i.nfluenced by the

center. However, a fair comparison may be mt:.de between t-wo and three axle

systems for small axle spacings~

§_:L.n..g_le Ax~e.: An arrangement for mnasurint:: ''trains and deflection,s at

the edge of the slab due to a load on one axle may be c.een in Figure 10. A

total of fifteen teBtrJ were made at edge locationr.l for en.ch of two subgrade

conditions, In order to avoid ecc~:.ontric ronu1ts due to lclcal condi tlons the

axle was Bhifted to pos:Lt:Lon,s both t1:Lcles of the lateral center line of the

slab and all of these results Y.'c:ce nve:raged for tho pre~;enta tion in Graph 4 o

The f:4trainG from Ylhich th(~ stre~werJ were eompu'ted 1,~-ere mer:~.sured longi~

tuclinally i.n a line on tho top of the slab pilrlolleJ to tho ec!g,e and nine

inches in-Nard from the edge. This location ··,.vruJ chosen becauGe this l:Lne

pavement. Although this is not the line of ma:dmum c:train it is suffi-

ciently close for the purpo~388 of thcr3e test£\~ It is also true that the

longitudinal strains are n.ot necesGarily maximum, but calcu1ationf:3 from 45°

rosette readings gave va.luefi vd thin 10 percent of the lon[;i tudinal magni--

tudes and Yrithin a few degrefcr: of the longituO.inal direction.

strain and def.loction readings "~i?ure noted when these axles Y1ere loo.ded

simultaneously. 1The distance betV\(-H311. the axlefJ V.'[-':~n varied from the mechani--

cal minimum of 5-1/2 fr:;et to D. mo.::dmwn of D feet~ Fir;ure 11 picture~:l one of'

these arrangements.

-8--

fii_:UI'r 1;_, vrc):p•·:rc:~ T.;;r·y

T',ru .:.~.xi\::;:~ tJl.~·c,)ci (Jn ;JJJ::b c.nd lo~:tcle(! t,, ;;t.):: i':uc·:Lrl ;.:tr.·:~_ins <;no c.' ,,J.L::r.tion,~r,

STRESS AND DEFLECTION CURVES FOR ONE AXLE AT FREE EDGE OF SLAB

~ .. () --.... b -o-.-... .. - 5 ..... _, .

-!!£1()

_...,_ 0

.,;

..: !:

100 "' "' .... "' ... "' 200

K = 110

'\ I/ '~ I

10 8 6 4 2. 0 2 4 6 8 ro LONGITUI!INAL OI!ITANCE FROM LOAD- IN F n: T

'I' 0 2 I I I I I I I I

X

"' ' ~ 10 ... .. ... ....

.20 ... "' "' .,; 0 ..: !

"' 100

"' ... "' ... "' 200

I--. I ~ 10,000 LEIS,<._ -+- I ---I ---o....._ _...-

.,

" I.,.. .... I~ 18,~00 I..BS. ,...., _IV' ..... ....... _ --

0"""-o-~ ........ f- - .,_...._ o--· 'l>.. "' • I .... ...... -,, "..{

~~/ K=60 ·--~ I --·

\, I \ I

·' '@ I

20

., "' "" 16 :::> 0 ... <> z 1.2 "" "' :::> 0

"' 1- 8 :! Si <> .... 4 ... .... " "" 0

150 200 250

/. I I~/ K= 110-:>,/,

~~ K=60

/. v/ v v

50 100

20

"' ~ "' "' :::> 0 ... 0

"' .. 12 .. :::0 0

"' ... "' 8

"' "' 9 4 ... .... "' "' 0

5 15 10 20

I\IAl!lliiUiil STRESS IN P.S.I, MAXIMUM OEFLECTION-IN.Xi0-!1

Graph 4 i--,'

h

Since four feet is a standard spacing for axles on a hoavy trailer, a

number of tests were made at this Dpncing and the averages of these results

are shown in Graph f). The ma;d.mum Btr(jsser.t for this arrangement do not

differ significantly frorn tho:Je due to the si.ngle axle. Hov;ever, the

deflections are greater under the two axle rwst<3lll than under one axle.

1'hree Axl,es: A third axle wa:s added to the group and the loading

tests were repeatE~d for this f:Jystem. The Elpacings for thifi t;roup 'ilere from

four to seven feet~ Again for comparativ~J ptu·ponec the four foot .spacing

v-ras emphasized and averages of these test~: aro gi.ven :Ln Graph 6.

The deflectionr. increasod over thoPe of one a:;;le and the t\w axle

systems. The streE'S!Js, ho;;mver, v;ere only BlJ.ghtly 1Bss than the values

under the single axle t The diffe:eences a~ce not signifi9ant o

CoE .. Y!.~~d:!J.l:.g

.§ing±~-~~J...:.~-~ Etrains and defleetions made by an ax1e at a corner of

the pavement slab v1ere mea.Bured at two cornsrs ,9,t extreme ends of the Glab.

As in tbe case of edge loading, Grapt.\ 7 is a IJOrtrayal Of average values,

An inspection of these dat~;t 4nd a comparircon \rl th Graph 4 reveals that

the corner deflectionfl a:re much greater than the de.f1ectionr:J at the edge at

both bi&)1 and low loads" For the softer sub[;l'<:~de, the rnaximum stresses also

are greater at the e.orner the1.n at the edl;e for correspond:Ln~ load:::~. However,

very little d:Lfferenee i;::.; noted for maximum f.?.tressee. at the ty;o locations

when the subgrade modulus was 110 p.e.i.

Two 4~l~£! Loads were applif)d to a tho c:tx:le system VJi th one axle

remaining at the corner and the 13econd axle being :iJ11Xard from the first at

distc:.nces from fotu' to nine feet o Hepoated test13 were made at two corners

and averagt::s of these claitn for the four foot sp~:1.cing are &,iven in Graph 8&

"' 'o ~ z • !II 0 ;::: ... "' ... .. "' a

"' "' ... "' ... "'

.. ,. 0 ... u

"' ... .. ... .,

., on

"' .. ... .. "' ., 2: ::1> 0 ... "' "' .. "' :::> 0

"' .. !!: ... ... >< .. "' ... ... <> g _,

STRESS AND DEFLECTION CURVES FOR TWo AXLES ON SLAB EDGE-4FT. SPACING

0

......

0 -10

30 --40

0

100

""'

20

Ill

12

/.

v 4

0

DI!ITANCE 01\1 CI!NTII:R I I'II:I!:T I I I I I I

----- [....----0- ...... - --·------~ 10,000 LBS. EACH AXLE· -.....

- ..,¥---...... ~ ..... 0 " -..... -- K= €10 I&,OOOLBS.EAC;;U:r-..:s>-j ___ r----L--

'1----... I I I 1:-- --"" '-&-..... ~o-......:;:: _......,.....

" '" I/ 1 "\1 ,.,.....- v-o "' L"'' / ".o o.c / ' II ' I 9 ~, I \,/

"

/ .... I ~7

K=IIO~~Vo/

/ v~ 1- K=60

U) ., !!! 0 ... c

"' .. ., g X ... !!:

..

s•

16

12

I K=lry I)'

..,..::--?

v ,/

/tf--tc--x= /

// /"}>" ...

..J >< c I ol

/

0/ /

100 150 200 250

"' ... ... ~ 0 _,

6

// I/ 4

1l 1 v/

0 10 20 30 40

lii!Al!IIAUiil OI!:I'I.II!CTION -llii.XI0"3

Uraph !) (' , , :.' I

STRESSES AND DEFLECTIONS CAUSED BY LOADS ON 3 AXLES- SLAB EDGE

2 0

"' ~ "' S! 10 ... ., ... -' ... 20 ... "' .,; 0

"' ~ "' Ill 100 ... "" ... "' 2

200

"' "' ""

0 0 ;:: <> ...

20 -' ... "' ""

0 ~

.,;

"' ;;: 100

"' "' ... "' ... 200 "'

JL -!'-. _,_ /fz----- 10 000 LBS EACIT AXLE I ,--,..--,_ - / I I

',, rz--. 18 Onn I A~ I' AGH Al<l F) __ , \---' ...... ............

1-..JC -- K = 110 -- ---· )--- 1-- T

-0 I :::::.,::...

~~ _ .....

~' ..,..a·:;~ -. -"'-, ~~,7:/l'~, ./'~~\V:· I'

\ I \ 1/ \ p' \ I ll,/ \

I~ -t::: lz-IO~OOLB's I'ACi AXLE: ~: ~~

1::::::-- (..- K: 60 ---........ , --- f..-- f--i·-~IOOOLB5 EAC~ AXLE 1--- ----r-- ~ -=- ,__

10

"' 20 0

"' => 0

16 .. " "' .. "' :> 12 0 :r ,_ ;!';

"' 8 -'

"' "' 0:

,~ .. , .._'\L_,~; :7

:..--

''l'\ [I' .,\V.~ 1',''\. _,..1

~ ' l/;6 '~>.\ ,() ?1\ l' \ 1/ \ II I

8 6 -2 0 2 4 6 -i

/

DISTANCE IN FEET fROM CENTER AXLE ALONG SLAB EDGE

-- ----~-

17 //'

K = 110 fyo, V/ o' •

1//~· b /0 0 I"K=60

i/

V/ /

"' 020

"' :> 0 0.

0 16

"' "' "' :> 0 "' 12111-----+--1--+---r-­... ;!'; w -' 8 111----1+----/--1-><

"' "'

I

'

10

I 0

I

K= 60

I I

I

"' 4 .. 0

"' 0 -' 0

ll/ ~ 4111--+-~--+--~

" "' 0 -' o~--~~~~--~~~ so 100 150 2)0 25 0

MAXIMUM STRESS AT SLAB EDGE·P.S.L 10 0 30 40

MAXIMUM DEFLECTION· I N.X 10·3

Graph 6

STRESS AND DEFLECTfON CURVES FOR ONE AXLE AT SLAB CORNER

... 0 §! ..

! • It 20 !:! ... <> "' _.

40 ... "' 0

10,000 LEIS. o-"";.-~- -r -- --V::~1s ooo u:~s .

l~ i/'

/ p

K = 110

..: 200 .,; 0:

"' 100 "' "' "' "' ... "' 0

.£)..

/,I -~ }-o'-n.\

~ ·~-::&:::: ~n-

0 2 10 12

' "' 0

0

... 40 1:l -' ...

I I I I o/~ "

1-.o- --- --o-10,000 u~s.-= r!l 1-- ..... -

~A 0

,/ v" . ~ 18,000 LEIS. /

/1

-14 Ill 16

·- f-

/ K = 60 "' "' 60

300 .,; 200

"' "" "' ::l I 00 0: ... "'

., 0

"' ::> 0 ... 0

"' .. .. ::> 0

"' ... '!;. a

" a ..J

0

2 0

6

2

8

.

4

\(

""-"" / )., 01_7.~ """-.. / ~!'-----

I /c I'-.. ?"'--. ---.: r/ j--r- -~ h r ,..,._,~a

1 ... - 110

v -::> I

7.=60z.jy /

I /0

/ /

/ ./"

/ / . '

/ _,/

II ., "' "' "' 0 ...

0

"' 12 "' .. "' "' "' :>:: 8 ... "'

/ /"' 0 ---- 0 50 150 200 250 300 100

MAXIMUM STRESS 1111 P. 5.1.

--- ---' I

I J 'I 1 K= 110;: ,·

~ .a .5"""15...~ i

I I

// J'

VI v, 20 40 110 80

MAXIMUM DEFLECTION-IIII.lll0"3

100

Graph '7

STRESS AND DEFLECTION CURVES FOR TWO AXLES AT SLAB CORNER-4FT. SPACING

..:200

.,; "' !

"' "' I! ... "'

or., ~

"' • "' 2 ... <> ... -' ... "' <>

.,;

..:

"' "' "' ... "' ... "'

"' .... .. "' "' ... ... <> "' s

100

0

0

20

40

60

200

100

0

0

•

8

4

0

l

/? / /

K= 110~ 7 -~

/D rK=f>O /

/ / ,,/" /

I" I" v/ 50 100 150 200 250

IAAlUMUN STIII!SS IN P.S.I.

----

"' <> 2

"' 5 .. <> ,. "' "' , 0 X ... ,. ... -' .. .. Ill

I

0

2

!I

.

,

:Jj /11

K= 110 ~ // --17 /~- K= 60 ;t -VI ..

<> "' <>

Lf' l/" -' 0 20 40 60

Deflections for this case \{ere 1ar[ser than for the single axle.

Although the maximtun stressen ·~vere ~;_.,reHter than those caused by one axle

when the slab wu~1 supportc.~d by the stiffer subgrarle, the strerJseB uere con­

siderably less than those for one axle when the test vms made upon the soft

m.lb grade •

'l1 hr~_g_Ax~es: Finally, three axles Here .so .~Jlaced that the first vms

on a corner and the others were eqw:;.lly spaced :i.rruan.tJ.y at c1istaHces of four,

five and six feet~ The .:1rrangemont ma;;;· be:< clearly serm in Fit:;ure 12. Aver­

age data from loading tests .::.tt the four foot spaclnt, arB ~:;ho'>Rl in Graph B.

Although the deflections for tl:w three axle ;:;;y-stem are greater than

the corresponding deflect:LonD for the single axle n.nd tHo axl.e arrangements,

the stressen are less. Apparently the deflsct.Lon c1.1.rve 1;:.~ f1attent?-d to such

an extent that larger subgrade displacement 'is obtained with a Dmaller slab

curvature.

Comparative Tests

.Effect of iVl~:UipL:!~:le~: Althougb the prev:Lously described tests

provided average valuer~ for rrtre~::sf~S and deflections for the arrangements

specified, it vras noted that the d:i.ffcn·tf-:1ces in maximwn stresses as produced

by the three systems at the edge of the slab were not signifJ.cant. The

sever.al tests madE in euch group gave m~:-~ximu:m strain values which differed

substantially froJi1 a mean value~ Hov,:ever, repeatf.3d tests upon a system

whoBe position on the slab -,;as not disturbed usua1ly produced results in

close agreement. 'l'h:Ls fact led to -the conclusion that some local condition

in or beloYJ tht) sJ.ab, such as grouping or EJize of aggr(~&,ate or perhaps sub­

grade bE.~aring beneath the RJ..ab, influencet"l the straln reading::;~

--10-

STRESSES AND DEFLECTIONS CAUSED BY LOADS ON 3 AXLES- SLAB CORNER

"' !2 0 .. !! 2 10 0 .... .., ... 20 ..J ... "' "'

200

;;; 100 0:

!!

"' "' "' .. .... .,

"' 0

0

100

0

>< 20

"" z 2 40 .... <>

0 -~--10 oobLBS EACH Axu; .. """2b,..o --1--0 ~---- --r

cl--1-- ~-~ 16 OOOLBS EACH A~LE ,.,....o""' ..... .,.·

0 /

""' I \

If ,o~ \

w \: . ,' \

~-"t 0 ,y_:...o~ : ...... -- -~D4 D D~D

""' 1\ J 0 ..-o, o/ .-" n_._rr::,-1 2 4 6 6 10 12 14 16 18 20 22

I I LA ND IN I' ET

I ' I I I

DISTANCE FAOioll II II E I!

IOOOOLBS E~CH AXlE_--:?,. ~_.;;- --6-->- 0 --~

v"' v":; -v f2-1e OOOLBS EACH 0 ... AXLE .....

cv , .. /

. I -.-... _. l:i 60

"' 200

"' ... 0 ... "" .... "'

o-"

~

,o, I \

//0"' ,-hb

l\~11 °\ D

I I

-':.r;::;-1-.--

...-o _. ....

D /"I-"'"

/IJ ,.-<J I -_,a ,.o I I I .

~20.-----.----.----~--~ z ::> 0 ... "' 161-----~---4~~~~~ "' .. ... ::> 0 121-----~~?4--~~--~ "' .... !!: a

~ 811-----~---4;---~--~ .. .. 0:

I /

~ 4 11----t~l-::'"--4----~--~ 0 .. g o~--~~--~--~~~ 20 40 60 80

t.IAKIIMUM DEFLECTION- Ill!! to-ll

Graph 9 r'"'

With this thought in mind, and the; ;,bj ect a direct compariwm of the

maximum stresses under the one, tvro, and thr€":8 axJc arrangemeuts, the

systerns vrere tested in such an ordnr that an axle onee ple~crjd J:';a/3 not dis-

turbed. The results of this method apyJlied to the axlero located at t.he slab

edge Etre Ei ven :i.n Gra pt. l.U ~

A simila:r set. of' compEl.ratJ.ve tec;ts v;o.s made at one corner of the slab.

The curves are s11ovnl .in Graph. 11. It 1nay be seen ttw.t the stresses are

quite high for theee teste:. Thil; f11ct may be e.<.plained by the warped condj.­

tion of the slab at this time.. The i.r:regularity of the i3trerJs curve for one

axle is evidence that :,.;arping affGcted tJ:lr;; rf:std.ts ~

Loads of 13,000 pounds, 16,000 pounds and 1e,ooo povnds Here used i.n

this latter series of tests in order to make compariA011G among the~ legal

loading val.uero. These data bring out the fact that from the standpoint of

slab stresses, tt.te r.;ingJe axle lD ,000 pormd load ir1 morr_=J f:levere than anjl

other J.oacl.inr; system tested. The detrimental.. effect of large deflections

under mul ti.ple axle loads hao not been detEJrminc;cL

Effect of Axle SpacLr=J?;

Although the DtreBs and deflection curve's for bm ax1es shovm in this

report a.rl::~ drawn from data obtained vrhen the ELXles ·r.ere spaced four feet

apart, other npacingr' v:ore used in an effort to .find the dint&nces at which

the slab strains would be thn least. When tlk' axleD 1owre located along the

slab edge the minimum strb . .in ;,cas found to occur at a n:ix foot spaclng, while

a four foot distanc(~ bet.-~.-e;;-;n axles p:r·oduced the lea~1t f;t.raln in tho corner

reg:ion G These figu:ces ma;y be ea~3ily verified b;:r examination of Graph 12.

0

100

200

0

:r: 100 (.)

z 200

w a:: <(

::0 0 0

en 100

a:: w 200

"-

en 0

z 0

::> 0 100

a. 200

z

en <Jl

w 0

a:: 1- 100 en

1-200

z ::>

0

100

200

0

100

200

I AXLE 18,000

2 AXLES

2 " 6 8 '-----DISTANCE

10 12 14 16)

IN FEET----'

en w :X: (.)

;z

en I

1-0

;z <(

<Jl

::> 0 :X:

1-

z

z 0

1-(.)

w --' LL

w 0

I . '. I . . . . . ' . . . I. .. •I'

I AXLE '' i

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

4 6 8 10 12 14 16 J ·~~~~-DISTANCE IN FEET~~~~-~·

2

STRESS a.n.d DEFLECTION ai ihL EDGE

Graph 10

I

0

z

UJ

a: <(

:::>

a U)

a: w (L

(/)

0

z :0

0 (L

z

300

200

100

0

300

20 0

10 0

0

300

200

100

0

300

20 0

100

0

300

200

10 0

0

z 300 :0

200

100

0

300

200

100

'---0

c

I ;II· I I I

v. I AXL£ -18,000 LBS.

-

/ \ ._) "· "':- -

p._ I Z AXLES - 18,000 LBS.

\ EACH

i - "

• • / \ .. \ 11/' .

I -

/\ 2 AXLES -13}100 LBS.

EACH

r \ •

• \, :._/ ,. ,,_ \,/ I

-2 AXLES- 16p00 LBS.

i' EACH

/' I '

/ ,; \,

&/' \•. f-\./

--3 AXLES- 18,000U35

" ~~:l EACH

I • '. I .1-= ' /fil ....... ,

•-. • a /' '· v •

\I '

f"\ 3 AXLES- 13,000 LBS.-

/'. • EACH

I I

·~. \-1-. Ia 1-·-l / '• • \1 Vi

' 3AXLES-16,000 LBS-

" EACH -

i ' i I I '

·.a/ \.a ' Ia J..-o-1,

/ . ' v ./j

2 4 6 8 10 t;_: 14 16

'---~DISTANCE IN FEET _ _____)

I . . I

0 Ia '

I AXLE- I B,OOO LBS .

'

I

o a . I .• .-~ -·-" .- , .. • -

2511----'f--1::7""+--t--r--t--+

(f) _.,."'/ · 2 AXLES- 18,0001

LBS, 50 /. EACH . w

:r '-' z

(/)

z <(

(/)

:J

0

I ,_ z:

z 0

,_ 0

w _J

IL

w 0

1

0 • '.' ' ' ! ' .• '• I

I

' ' '

2 AXLES- 13,000 LBS 50 EACH --

'

' ·I -'

fl. 0

•

.. ~····

2511---1---cl-··""r-----r-r-_··_· '+--+·-•-.· +-.vv ' 50,jl--f--j---j-2 AXLES-16,000 LBS

EACH

I I ''

0·~·~+~-~~·~~--+, ''

• 251--+--+-~~~··~-----~--r

~~-·~-50•k-: 3 AXLES -m,ooo LBS

EACH

0 • • • - ~--·-25 • -.v 50 3 AXLES-13,000 LBS :._

EACH

0 •

25 ~---

- 50"it------J-----j----J-- 3 AXLES-16,000 LBSl E MC H

•

0 2 4 6 8 10 12 14

\_____DISTANCE IN FEET____)

STRESS ccnd DEFLECTION ai the. CORNER

Graph ll /;? I .''.

w :l;ICJ ::OQ :>w XO: 0 <(<( :l;w zZ

<f) -10 w<n ._,w z"' <(I- -20 I (f)

I.)_] <(

1-Z z- -30 WCl o"' a:!::

"'" a.z 0 --'

LONGITUDINAL DISTANCE IN FEET BETWEEN TWO AXLES WHICH ARE PLACED WITH OUTER WHEELS ON SLAB EDGE

\ 4 5 6 7 8 9 10

\ ~ v ,_ /

\ ~ // (~""

\ '\. ............ _,...._: ~,

'-, [ ---"'1 K = 60 - I I

BOTH AXLES INWARD AT SOME DISTANCE FROM THE CORNER

ONE AliLE AT CORNER, THE SECOND INWARD

EFFECT MAXIMUM

a-j AXLE SPACING STRESS NEAR SLAB EDGE

Graph 12

A group of curves representing data from the model study are repeated

here as Graph 13. Comparisons be't1veen these curves and the corresponding

curves for simi.lar loading on the full--size slab shcw1 a marked flimilari ty.

The strain curves for the model have several timefl the amplitude of, the

curves for the la.rge slabo This indicate:3 that i;,hB model was considerably

overloaded.. ~rhe>::Je excessive loads magn.ifir:-;d the differencec, in defleetions,

hov1ever, and brought oui> fl.uctuat-.Lonc: that are not apparent in the full-size

slab stw'l.y.

No conflicts ure ~:;een bet'.'ieOD data from the largt:: slab and data from

the modeL The prototype study 'ira.s necerJoary for the determinat:i.on of

working values for slab d'?.s:i.;;n, but the relati vee effects of different

loading arrangements are brought out clear.l~r in the model study and are cor-

roborated by thiE: later investigati.on.

Static load tests somev;hat similar to those made in this r-;tucly hHve

been conducted by other investigators o A six v,heel truck study v.ras made by

Teller(S) in 1925. A six inch p.la:Ln concrete slab v;as tested, and deflection

and strain curveo for one and tY,'O a:xle:-:1 Viere .found. These W8re similar to

those of the present study. In 1931 tho HHnois Divj_sion of Highways(6)

made tests on the odge of a 9····6-D pnvement Hhen it \,-as subjected to loading

by four and six wheel trLlCkf:. 1\ga:i.n the one axle and t1:o a.xle data compare

wol.l with the results of the present investigation. A thcn'oUt,h study of

stresses in the corner region of concrete pavements was mo.de by 8pangler(7)

in 1;142. These re~:;ult2 FJGrt::' ueed as a guide f'o:c gage placement in the

present test.

-12-

"4 . -- i!IJOOk i I L ·-·····

tilll:t+~~ in ··. 1 ~;vf 1 ! ~ 1 .. ' .~v • 1 i

0 <:0 .eo ~ 110 00

-,~-~-... -ONE AXLE AT SLAB EDGE

q---'"';r~···--· r.__.:_ ,. ~ .:::z. -~2 L ·-- -'-- L_ __ ----- t -·

• . . r ". tf ;.'1ft '. ---. . ~ ___ _____._ - ----+- --------

v; ; ' 0~0'101;080100

~dioklt•ec""""""

lff~ ·.='"·· .. ·-.• .; ·~!I ~¥1 ~ ;[:;11

I ~nvntmT 0 2040ISOBOIOO

L~di:llanCOOiindlel

TWO AXLES AT SLAB EDGE

ltfFfflJHtll 1£NtftHI

40<002040 LOfttlhllllilal dlo!GIICD from ~- looif

j~-. e:ac~~ __ ~ 2QOO!lla. 0'1 4 .. llao!ll

:r.tM±U .. jfmj ffifffiiffl

402002040 ~ ... ~- ........ -

THREE AXLES AT SLAB EDGE

• -;, r-. i " j 1'-

I ~ I/ 1\ l > v !

1 v i ~ _____ 40 • ~- •• L~•- • ''

ONE AXLE AT SLAB CORNER

i~ II!TYHUG

1l!! mmrn ~403:120100

I)W<!n<o """" fli'>d ol sleb· iN::boo

TWO AXLES AT SLAB CORNER

CHANGE IN STRESS OVER THE SINGLE AXLE VALUE CAUSED BY LOAD ON A SECOND AXLE

·~····· . . ~- ' ., _ _;_ __ 'o

~- . ---= OIC20l040

A>l•""""""'l"'-

TWO AXLES AT SLAB EDGE

~~ 01020:)040

~Vooo -~---

EFFECT OF SECOND AXLE WHEN FIRST IS AT SLAB

CORNER

Graph l3

Numerous other studies have been made where the in1restj.gators u~::ed

actual vehicles and also lo,~ding plc.tes to apply loads to the concrete

slabf:'. Tho strains have been nH:;asured by mechanicul gages of standard and

solf recording types. Investigations by 0. Graf(lOland G. Wcu(ll)are

parti cv .. laT ly thorough.

A1t~hough the concrete slab unc.~tor investigation wae. constructed in the

laboratory, t<::~st conditions did not pl"ove -to be as ideal as anttcipatedo

Lc.ck of room preclud,~.;d tlw poss1bilitJT of mnld.ng any study .. under moving

loads, and a curling phe:nomenon which mat;·:.ria1ly a.ff.'ected the stress values

WD.fj encountered~

The extensometr~r ~hovm in FigtJ.rl2 6 gnv<.:: .-:~ 1·.:~r:0~C(l of slab Vlr:.n·ping at

each corner 4 Gro.ph 14 provid~·)s o, typin-:i1 curve o TL.ir~ upwG.rd movemEmt of

the c.ornerr: and edg~~::.~ ·~7as vc.rifi.ud by a pl'\.":)Cise lE!V(:lo Th.o cr.:.usc of this

slab distortion eanJ"wt be nttrlbutod. to a t(~:mperature differential, for con-·

tinuous l'GCords showed littlG or no differences :i.n tcmpcrutur2 rnadings

botvveen tho top n.:n1 bottom of the~ r.~::.LD.b ~ Further (-:):Xperimentatlon and

behetvior C':tn he presented.

\ The~ foregoing DdCtion r-:v.gg~.:~st;.:_1 that -thl) strc..in~J meD .. fmrod in this

produced streSSf;G c:1us.ing tt.--msilE:-~ strt."d.ns on thE upper surface. The so

~-J 5·-

z 20 I

V)

·~ 10 z <( I u I 1-

0

L'J -10 z w _J

-20

z

~ 0: 1-z 8

" ,v I

z 0 v; z ~ X w

rs LAB FLOODED ON TH OAV 64

/ \ r~AINTED WI~H CURING C~MPOUND

~ I-"

~ Tob (xt) ····~

~ .. -.. -

/ ~ __ , .,.-- ------- -- ' -- ?>o, ... _____ -- \ I /

\ ... / ---7-- -------"

L BOTIOM {Xb)

20 40 60 80 TIME IN DAYS

100 120 140

VERTICAL MOVEMENT AT CORNER, ~=~(XC Xt)

WHERE l::: DISTANCE IN INCHES FROM CORNER TO START OF CURL. n= SLAB THICKNESS IN INCHES. XtAND Xb ARE DI5PLACEMENTS AT TOP AND BOTTOM OF SLAB.

EXTENSO METER RECORD a.l: SLAB CORNER

Graph 14

values should br.~ added t0 those nteasurecl under corne.r loading and subtracted

fr:Jm the edge loadlng value~~ Q 8uch ·i...-arping strc.inf. were not, measured

because of the failure of tho gr.~ge::.1 ilnbe;1cie:1 in the concrE~te.

J?or the purpoCJe of thin study 'Jf the rcJlative effects of static loads

it vms not necessary to know the total stresses.. HowGver, their magnib1der:J

are of interest and a thcwretic<:d. o:x:amination of' the sJ.ab stresnes is

pre sen tad.

The Westergaard. forr~ula~/ 3) provtde n t~tlO()I'etical check on s~)rne of the

results of this ,,xpor.lmenYtal study. It 'mst b,o remembered thD t those

formulas were dce;ve.lopecl under thD a.Bsumpt.ion tb8.t the slab '.'.'US of infinite

extent rmd that th<o c~ubgrade prossure was prupc:ll:tLmal tG the deflection.

The finite diGenoions ~Jf the tt::st slab and the cu .. i':'ling ·Jf the edgos noces-

sarily modifioc~- -GhdDC :ref3ult;::: 9

Stresse ~;; at the slab corner, intcr:ior, trrJ,nsverDe edge, and longi-

tudinal edge producud by n single loading area wer,;:; c!.)frrputed by the

follov;ing fonHulac:

1 ___ a,i2\0.c t i

Interior ~3t:co::>s ··· O.;JHi25 P hZ

\ 4 log ~ + 1.0693)

0.571851' (1

,4 h? lor; ,to ·+ 0. 5595\

1 tl

where P -·- loa:1.1 h _ .. slab thickness, o. and b are functions of the

loading arr;;a und

t; -- 4 i---_:e.~h3 --· -vl2(1--u2)k.

--14-

For these computations P ~' 10,000 pounds; h ~c 9 inchcr;; a and b aro

average values from Bradbury0.5); the modulus •:>f elasticity of concrC~te,

E -- 5.2f) x 106 p.s.iG; Polsson 1 s ratio, u =: 0.15, <.:tnd thE: subgrc~de modulus,

k ··- llO p.c.i. The tables presented by Kelley(l2)were used to fucil.itato

th(:; computation.

~rhe samo data Vtere used in tht.~ rnodified verGi.orw <)f these~ formulas

VIhich evolved from the Arlington testr;( 4). Ther;e equutions are listed

herewith:

al/~~­t

1.:2

Interior st:eess 0. 31625 p ( w! \

Edgo stress 0.57180 p liZ

\

4 log t + O.l7B8j b

The results of these computations together with the results •Jf experi-

ment are presented in the follov-iing table:

'rABLE III

Stresses Under a Single Wheel with 10,000 lb. Luad

Corner

Interior

Longitudinal Edge

Transversu Edge

Westergaard foruulan

222 pr:d. o

166

lD!J.

254

-1.5-

Arlington forr:iUlas

311 psi.

H2

258

~rest Slab (Experimental)

350 psi.

250

550

5~)0

It is readily seen that the mo.'3.uurod strosses exceeded the values

cor,lputed by· both methods, G.lth:mc;h the Arlington forHrulas givce a closer

approach to the experiJ;H:mtal values than the Westergaard results.

~ds on Two Wheels.

WhfjlJ the slab WD.S loaded through tvvo v;heels on a single axle, these

stro::3Ses were soraowlKJ.t modified. Computn,tions !1re l~hown herswi th for

principal stresses due to a 20,000 pound axle land at the int•orior, and at

longitudinal and lateral edges of thG slab.

to 10,000 potmd l<;ads at positionD l DJJ.d ;~ only. The stresses due to the

load at l \·Jere found. prevj_ously n.nd axe equal in a11 directions. Let us

cnll this valu0 s1 .. Strer:1seo at l c1uo to load :2 1o:rhich is 72 inches in the

y-direction froa 1 are found vdth t.he help of w~.~sttn·g:·lar·cl 1 s(B)noiJ.Gnt curves

shown in Graph li).

For thit3 slab tho radius of rela"ti ve nti:ffness t :: 41.5 inches. The

distance from 1 to 2 iB 72 :\.ncher'' '' L74t. Hence, Mr '' !Vlv o= -.02.1. and Mt =

rihe seetion modulus .for tlH3 rectangular

1'hen Sy =: .::-· 02]:.2.0_\l . .Q.OQ = -16 p.G.i. 10 .. 5 . R .. l 1 ._ J.na o. r y

Longi tud.inally: 81 + -- 81 + 15

S~T ::-.; +15 p~s.i., conse--

load at 2 ~Then point .1 :i.s on tho longitudirw). erlf:.,2 of 'the slab has been

-16-

LOAD SPACING joz MULTIPLE AXLES

Figure 13 ·· ·

0.20

0.18

0.18

U)

1-z 0.14 w ::; 0 0.12 ::;

... 0 0.10

::i'l"-o:oa

1-z w 0.08 \)

G: ... 0,04 w 0 u

0.02

-0,00

-0,02

-0.04

0.22

O.IB

t5 0. !6

8 !l! .../ U)

<.:)

g <(

U) 1-z

0,\4

0.12

0.10

w 0.08 ::; 0 ::; t5 0.06

-0.02

-0.04

1~ ., ' ' ' ' ' \ \ BENDING MOMENT COEFFICIENTS l jt!'t a'Jf I ~ INTERIOR LOAD • I

\ \ ( Wes fergaard)

t \ \ \ ' • ' ~ *T =TANGENTIAL COEFFICIENT I \

\ '· • \ '• ' • ' ' ·,.

~ .. A.

It

o.st

-

"' 'o""''-- ·-.. ---------... .. -- ... -p

' ... --- --j -~ ~"~ADIAL COEFFICIENT

2t 3t 4t st •t RADIAL DISTANCE FROM LOAD

Gre.ph 15

--,----· I

BENDING MOMENT COEFFICIENTS fiJ't. LOAD cd EDGE {)'f SLAB

-cr---------- ; __

.Mx =RADIAL COEFF!CI ENT ALONG EDGE p C WESTERGAARD)

0

I.Ot 2.5t

DISTANCES ALONG SLAB EDGE

------ -"'-zy--

3.0t

Gre.ph 16

··j

previously computed. The effect or 1ond :2 a.t position 1 is found in a

.manner siuiln.r t0 that for an :Lnturior location~ Yihf:ldl 2 is at an interior

point, r.md it::J effect r:m 1 yJill npproz-.i.J:in.te th~) vcclue found in tn(; interi·'Jr

case, :numoly 15 p .. s .. i... ~rhe longitudinal stres~~ will then be Sc + Sx ::.~ S8 +

15.

load at l produces strr:ws [\:.. The influence of load 2. is f.-Jund by use of

Graph 16, which :l.s dravrn by ir1torpolati':m frr;r1 Westergaard t s graphs in order

to 'Jbtain a direct rc'ading curvo for u "~ 0.1b. At <}istance L?,H, the coef-

f . Cl. lt M -­l.OI)p -0.05'7, hence

transverse stress at l is

Loads on Four \Vhoelr3

1'hen tho total

~t.. The acelUJ.\lln.ti.Jn of stress(:JS clue tv L)adn at 1 B.nri 2 hav(~ been f\mnd.

Wo now find tht~ effects ~;f l'Jo.ds n t 5 and 4_ ..

Figure 13 sl'FJws load. 3 to be 48 inches fr:"Jr.1 l. Since 48 inches ::::

'I' 'li ]'1 1.15Gt, we find froJi, Grap:·:; 15 that !!tj: =' !b-". "-' 0.044 ond """r"r' --p p Mr.: = -0.01. .P

Hence -7.4 p.s.i..

ThG ef'fGct of the lGarJ. o.t 4 upon p-:)sitior:. 1 is slie;htly 1:tore complex ..

In this case the tnngentinl .:;.,n~~ ra,.:tiD.l ::.~tresses are not in the x and y

directions but rnn.k8 angles of 34° with tl:1ose u.xes. Position 4 is 86.5

inches= 2.08t from 1 r,:o by Gr'lph H tt~;atn -~l< "' 0.012 and ~r:

St = +9 o.nd Sr :::: -16 ~ Since the principal crl;rl:;Ewes frolil p:Jf:~i tions 2 and 5

o.re on the x anft y tlirections, th•.::: ef/r~ctt:\ of' St, and Sr in these d.irections

-1'1--

must be computed before the longitudinal and lateral stresses can be

accumulated. The stresses in directions x and y due to load 4 are:

and shear T - 1/2 ( -16 -G) cos2 54° ... -·9

The accwnulo.ted stresses in the x a,nd. y directions are:

Sx = sl + F ·" - 7 ·f 1 - sl + 9

Sy .. sl - 16 + 3' ,, - '1 ·- 81 + 10

and T 9

The principal stress(:;;S are found by the formccla

c• + 81"1!: §x --~ 2

" ~.:z + rr2 0 max -· 2 " 2 +

Sx ,,

.ih. .. ::_§;y; .,

+ ,. ~r2 Smin = ~~ - + 2 2

Whence Smax "' sl + 19 and c ·- sl + 0 . .5 \-"ndn

Longitudinal EdE§_: 'rhE> longi tudino.l str0ss is necessarily maximum at

the edge of the slab. Tho load at 2. addfl 15 p.s.i. as was shown in the two

wheel case ..

·-0. 01.2. and

Load 3 is o:t clistancc0 l.l56t

0 0 2. • 104 Sx "' _:-_,__f.:: . ..?.:c...c .. "-· "" --8 • 9 p • S • i . . 13.5

·" l l f G ' 16 Yh:: -"rom ., 1ence . rom rapn , P -

The effE•ct o.f load 4 ic: not

determined, but from the computations abovu, for the interior, it is

-ltJ ..

apparent that it is small. An approximate value for stress at the odge is

then

In~_s:_rior.: When th(·? load. is distributed through six wheels, the

greatest strecoses are at positions 1 and 2 (F'igurre 13). Consider the

stresses at point l. Computations for s't~re~.;n~~s under fom· y.;heel loading m....'1y

be used for this calculation. The r>tresscs at 1 due to load l aro s1 in all

directions. Load 2 contributes +15 p.rJ.i. longitudinally and -16 p.s.i.

laterally. Loads :'1 and 5 each affect l bcr longitudinal stresses of -7 and

lateral stresses of +53. Loads at 4 and 6 produce tangNttial and radial

stresses of +9 and ·--16 respeeti vely, and the t and r axe[l make angles of

-34° and +34° reGpectively lvith x.. These st:cesnes, accTlJGuluted, are ~?quivu-

lent to Sx = 2, Sy = -14, and T c.c --18. Fi.nally, thc1 sum of all the stresses

at 1 is:

Sy -· c• -· 16 + 66 - 28 -·· SJ. + :<:2 ''1

T -· 1f:;

Th . . ] . e pr1nc1pa _ st.re:ssf~s t:n·t~:

~ongi tudina1 Edge: For thrue axle)£! at the 1ongitncUnnl edge it is

sufficient to corapute tbG longitudinal strc::w und8r l ~ The stress due to

load 1 is 38 • By Graph 15 that duo to 2 is +15. Loads nt :'5 and 5 each

cau~e ~x to b" --0.012 from Graph 16, whence &ach Sx = -O ·912 1~. ;o '000

-8.9. The effects of 4 and 6 must be approximated by the method used for

interior loads. From abov·e the effect of these two loads i.n the longi-

tudimcl direction waE only 2 p.s.i, Bence the tot<':tl longitudinal stress due

to all loadn is:

The foregoing eomputations were made in terms of a variable S1 or 88

in order that we migbt make a comparison bet~7een the forHlulns for strens

computation.. A tabulation of those results is given belov;:

TABLE IV

Stresses Under Multiple Whoel~; vri.tl1 10 JOOO lb. Lor:~d Per Wheel

Max.imwa Computed Stress Load Type Position

Maximum Experimental St1:'ess Westergaard Arlington

1 axle Intericr 182 157

1 axle long. edge 220 209 253

1 axle trans. edge 192 229

2 nxles Interior 185 161

2 axles long. codge 250 200 244

3 axleD Interior 198 1'73

5 axles long. edge 220 193 257

It i;; readily ro1een that the Public Roads formula yields greater

stresses thnn Westergo.ard 1 El except for the:~ interior location, .. ~n:J in all

--20--

cases of multiple wheel loading our experimental values lie between the

values computed by th:Jse formulas.

The greatGr part of th1s report has ben:n l.im:ited to the effects of one

axle, two axles at an axle spuc:Lng of four feet, and three axler, at u four

foot spacing. Except v;here othorvii.so stnted, o.l1 observcttions and conclu~

sions given here are rr~strictcd to o. discusnion o.f re1:>ults fuund under these

lim_i tations. Tho ~Jut standing l~eE;,ult~s are as follovl~l:

1 ~ A comparison of thr~ curv~~s presenting average vo..luos sho1Ns that

the maximun1 stresser:1 for. a conf:lt;:nt a::<le lo:-:td o..re pt·oduced by a

single axle on the corner of the slab~

2. Stresses due t8 corner loading by al.l throe systmns v~rere consider-

ably grer:~.ter 1rYher;. the ;.:;laJ~ :r-c·sted ·)n a subg1.'Ltdc: ·.vi th modulus k =

60 p. c. i. than they were whcm k = 110 p. c. L

o. At k "' 110 p.c.L :otresr;Gs due to corner loading did not greatly

exceed thot:H? duo to edgo loading for any of tht3 three systems

te:Jted, but at k - 60 p.c.i., corn<Or loading produced greater

stresses than edge loading~

4. The special comparat:i. vu tc1sts, the data for which were given in

Gro.ph 10, indicate the fo11m•:Lng rd.:tti.onshi.pG for edge loading:

(a) ~rhe mc~ximmn ntress for the tv10 axle system under loads

of 18,000 poundro per axle did not exceed that for the

single axle at 18,000 pounds.

(b) The threz:: n.:x1e sy~3tem at la,ooo pounds per axle caused

1efiE3 (3tress t,ha.n either of the other two S~fstoms.

-21-

(c) The; maximum deflection und<cr the two axle By stem was

t~;fice that under the single axle, but the three axle

sy:::tem did not cause a corresponding increase in

deflection.

5. The corner load.ing t2sts vil1ich furnh1hed the data for Graph 11

shovved the follovving:

(a) Th0 singlo o.xlo produced a grenter maximurn stresA than

(b) Three axles at 18,000 pmmds :.,OJach produced luss than 70

percent as great a maximum ntress value as the single

axle, and two D.xles at 18,000 poUI1ds e!lCh caused a mnxi-·

.murn stress which was more than 90 p.;rc.:3nt of the single

axle value.

( ., \ c., The deflections undc:r· tvvo tl::d.er:J ·wur(: tvvunt~~ percent

greater than under onv nxl:.:;, but the three axle

defl(:;ctions v1ere only tt~n percent greater than those for

a s:Lngle axle.

6. Upward v~arping of the r;lGb at the ends aff'Dcted thn total stresses,

but ffi!?JO.Sl..n·ed v-~-J.luc;c wore found to lio b<:JtWGE.m thone computod by

tho Public. llonclf.\ and WestDTg~n.rd formulaB.

7. The results of thi.s atud:y- arr;~ so n.ea:r:>Jy those that might have been

preC.lictecl" by the rJKH-:L;1 invGstigaticm thc.;.t r;m-lel studies arc racom-

mended for furthGr :L.1VGBt.igation in ~<Lab stresses ..

what due to local conditionn ~ Strain diff',~rences as high as ten

micro--inches per inch werco found, mld when the strains dut: to

loading wore small, t.he;3e local differences caused docid~d

irregulari tie e. in the curves ..

9.. 'l'he intensity of maximum stress ctt1.l.S0d by the two-axle nystems v1as

influGnced by the axle spacing.. 'rhe opt.imum dist;~mco botween

axles is about five feet.

Throughout this rJtudy ths refJetlrch std'f hcs been. guided b;y the

suggestions of W. 0. Fremont, <lnd hi.s hel-p in outlining procedures tend

cri.ticit·~m of result~ is grc.tefull;y ::.~cknov;lcdgod.

Special equipment for thiG investigation VR~s cunstructcd by F ~ C.

Filter and A. G .. Davis. These mon. participated in the entire te:3tin15·

program o.nd their nj_d 1n tLe s:Lmp]j_f:i.cnt:Lon of technique c.nd in the

comp:Llation of data is apcn'cciatod.

(1) Michigan State Highway Depnrtmcmt -- .<i Model Study of Slab Action in Concrote Pavements -Proceedings of the Highway Hesearch Board - 1945.

(2) Bouyoucos and Mick, iviichigan State Collega Agricultural Experiment Station, T"clmj_cal Bulletin No. 1?2 -·April, 1940.

( 3) H .. M .. Westergaard - Cor.-1putation o.f Strcsser:< in Concrete Hoads -Prc;ceodings of tho Highway H8cWD.rch Board -- 1925.

( 4) Teller E'.nd Sutherland P'1rt 5 - Publie R:oe.i!n

-· ~.1 fW Structurn.l Design of Concrete Pavements .. i\pril, 1:1~5.

(5) L. lV. Teller -· 'l'he i:i.i.x Vihe-:o1 Truck and the Pavl•nnent ·-Public Hoads -Octobor, 19:25.

(6) Illinois Divisi0n of lligh-.1ayG - Investi0;ati.on •.)f tbe Effect of Wheel Loads Applied to the Pavement by Six khoel 'Trucks -·· Mimeographed 1951.

(7) M. G. Spcmr;lur ... Stresse>s .i.n the Corner Regi.·")ll of Corw.rete Pavements. Io~sa Engineoring Exporir,wnt Station - Bu.11ctin 167 -- 19112 ~

( 8) Vi.. 0. Frr:;r:.10nt - Effect of Vo.ri("JUS Ax:le Loc..r~in E;"Q ·Jr, Hirhway Pavements~ Michigc;,n State High~vay Department - R.ee;,.~rJ..rc;b R.e}u:r·t F-25 - 1942, Mimeographed~

( 9) Spangler and Ligb.tburn ·-- St:res;:_h3G in Concreto Po.vmnent Slabs .. Proceedings of' Highwo.y Her:Jearch Board ·- 1937 o

(10) 0. Graf - Aus V•Jrcmchcn. uit BetondGclwn de:r Hdchskraftfahrbalmen, clurchgefiihrt in den .. JD.hron 19M und HJ35 ... zc,wentverhr; GmbH., Berlin·~ Charlottenburg 2.

(11) G. Weil -· Einrichtungen :.:,ur Messuns der BrJanspruchunr.-:: von Betun.­Fahrbahnp1nttfm. - 1:!06 ... Zemmrtv,·:r1ae; CdJFl., Bcrli.n-Charlottenburg 2.

(12) E. F. KeJJ_,,:y - ApplicntLE of the HNmltn of Resoarch to the Structural Design of ConcrutQ Pa venwn.ts - Public ReaCts - July, 1959 ~

(13) R. D. BracJbur;y ... Evaluation of Whce<el LoQd DirJtribution .for the Purpose of Comput:lng StrestJes :Ln Concrete Pavements -· ProcecdingB of t.he Highway Hosearch B:xtrd -· l93LJ: o