Prestresse4 ConcreteBridge Members

Progress Report 15

FATIGUE ~ES1STANCE OF r~EST~ESSED

CONC~ETE ~EAMS IN ~ENDING

by

Co Eo Ekberg, Jr.Ro Eo WaltherR o GIJ Slutter

Lehigh University

Fritz Laboratory Report 223.15

April, 1957

c. E. Ekberg, Jr.,l A.M. ASeE, R. E. Walther,2

3and Ro G. Slutter, J.Mo ASeE

SYNOPSIS

A method is presented to predict the fatigue strength

of prestressed concrete based on the failure envelope of the

materials involved. A discussion of factors influencing the

fatigue strength, such as percentage of steel, level of pre-

stress, and ratio of dead to live load is also included.

Recomnlendations are given all. trle liotioll of safety and on the

safety factors for prestressed concrete under fatigue loading.

1. Assoc. Prof. of Civil Engineering, Lehigh University,Bethlehem, Pennsylvania.

2. Research Associate in Civil Engineering, Lehigh University,"

30 Research Instructor in Civil Engineering, Lehigh University,

1THE FATIGUE RESISTANCE

OF PR~STRESSED CONCRETE BEAMS IN BENDING

INTRODUCTIOl'l

A survey of the present status of research in,

fatigue of prestressed cqncrete members leaves a rather per­

plexing impression. Various authors have conducted and reported

numerous tests and dravID con81usions which are difficult to

compare, because of the complexity of the problem 0

A wide-spread opinion that prestressed concrete is

safe against fatigue failure, a belief probably stemming from

the early tests of Freyssinet~l) is confronted by a reluctance

on tIle part of many e11gineers to use prestressed concrete where

fatigue loading exists 0 The existence of these two notions

side by side is no·t su,rprising wl1.en one considers the large

scatter of dynamic safety factors from as low as 1.15 reported

by Leonhardt(2) to as high as 2.60 reported by Lin. (3) Other

authors mention ratios of dynamic t1l1tj_mate strength to static

ultimate strength of 0.40 (Inomata)(4) to 1.00 (Abeles).(5)

On the basis of the above facts, it seems imperative

that the causes for such wide variation should be investigated

tlleoretical1y ~ Tllis gives rise to the question of w11ether or

not it is possible to predict the dynamic ultimate strength

2

of a member with about the same degree of accuracy which can

be achieved in calculating the static ultimate strengthG

The literature offers little help in answering these

questions, and therefore it is the primary purpose of this

paper to present a method for the calculation of the dynamic

ultimate loads which permits an adequate consideration of the

safety of a member in bending 0 The method which is presented

here has been used successfully in checking the results of

several investigations, including thoseof Lehigh University.

PREDICTION OF THE BEHAVIOR

OF BEAMS UNDER DYNAMIC LOADS

The Stress-Moment Diagram

The first information required is the stress~

moment relationship for a critical cross-section of the beam

in question. In order to make this relationship as general

as possible, the ordinate and the abscissa are non-dimension­

alized by expressing the actual stresses and moments as a

percent of the corresponding ultimate stresses and moments

(Fig. l)e

The uppermost curve gives the theoretical relationship

between steel stress and total applied moment, from release of

prestress, through cracking, and up to the static ultimate

moment. In the elastic range, ioe. before cracking, the steel

3

stresses increase only very little and at cracking load there

is theoretically' a sudden break 0 In reality there is no such

discontinuity, since the cracks develop gradually 0 For theor­

etical considerations, however, it is convenient to assign

the total stress increase due to cracking to one moment ~- the

cracking moment. The curves in this example are drawn for a

beam having a percentage of steel required for balanced

design. A design is called nbalanced n if the stress in the

steel at the ultimate moment corresponds to the stress causing

0 0 2% permanent set in the steel o The stress~moment relationship

for the concrete top fiber starts out as a straight line, and

gradually becomes concave do~!ward as static ultimate load

is approached 0 The bottom fiber stresses, on the other hand,

increase more or less linearly to their terminal point at the

initial cracking moment 0 Such a diagram is characteristic for

a given cross-section and a given effec,tive prestressing force,

but it does not depend on the location of the cross-section,

because the dead load is included in the external bending moment.

Hence for a ·beam with straight tendons and constant moment of

inertia the above diagram is valid for any, cross-section. It

will be noted that the require~ents of the Bureau of public

Roads Criteria(6) were observed in laying out these curves, in

that, the steel ~tress under working load is at 60% of ultimate

4

t'~::'.I:'k.,':.~':l,l,e a,llCl cornpt~essi,"\"e concre·t~=

c,()rtc re, t:e. (j

8% and 40% of static ultimate for

,a.dxn:Lt:t:(?,dl'\T .. ll,ot {:tIl easy tasle f.,or,,J "

:Eo!:ml.JLl>~tt,:L{)Jj.> o~E a strairl com-

~p,\S\.t, ,fllJ :ll,., :L cross~section leads to

t'::LriC'

't:' ~2:.1. a t hip of steel and concrete cannot

1 :Ll1fO anr,1 ec~rnpa.>t'1,bi,.li ty condition

El met.hod. firs,t proposed by Colonetti: (7)

Ii t'.

T =: C

t'=, ~\rr -, AT ~

creep shr~nkage

lc)sses a,re 11,ot Sllbtracted 0)

a.c, (:iDr.tc:;cet~(3, st~CE",8.ses C'Etll, tl1uS be calculated

5

under the action of the eccentric compressive force C alone by

meal1.S c~,f tl1.~~, usual metll0cls of equilibri,um and cornpatibil:'!', :~yc (j

(Tl1e, fo~.rc,e C C'09.D. 1,ler'Jl "Ale-Ii lie olJ.tside tlie C170SS --sec 't1.011 ; if

M is large enough, the lever arm of the external forces

11 ::= M = !VIT C

can become larger than the effective depth of the beamG) The

l')"i~' t "C' ''l B -'l... ,,! '" ., ~.

1 a,D,(I (J (, l-lc:'vJ'e,\re:r;" _, sirlce, t[1i8 i11volves several steps

1,(~,d.ge~ of t1.1i2 st1:'f;S,~~,"""strai.rl re,lati,011srlips of COllcre.te and steel,

for Prestressing Steel and Concrete

1:'tru,s fax" ortl:y' t11,e static prope,rties of tIle critical

cor~"Ilectioll, tCJ tIle dYIlamical betlavi<..1r, two fundamental diagrams

'\A71"licl1 s110w tl1e, d.yrlamic, prope~... ties of stee,l a11d plai11 concrete

are introduced 0 It is customary in strength of materials to

present this information as a failure envelope for a given

fixed nQmber of cycles~ It is beyond the scope of this paper

to extensively deal with the question of how large to choose

t11is 'uuInber of cyc:le.s =0' a (Iuest,ion wl'1ich il1.Volves statistical

considerations of the expected load frequency as well as con-

sid,er'atic~n, {)£ tJ.:l'8 d.)rn,arn:Lc cllaracteristics of tIle material itself.

6

We only mention that most of these materials show a more-or­

less distinct fatigue limit, i~eo above a certain number of

cycles, 'VJhich can vary from one to several rnillions , the ultimate

capacity levels asymptotically to ~ fixed value. For our

combination of concrete and high tensile steel it seems to

be justified, although hot beyond critical consideration, to

restrict the discussion to a fatigue limit assumed at one

million cyclesd

The fatigue failure envelope for prestressing steel,

shown in Figo 2, is a modification of the well-known Goodman­

Johnson diagram 0 (8) This envelope indicates how much we can

increase the stress from a given lower level to obtain a failure

at about one million load~cycleso (Again the stresses are

expressed as a percentage of static ultimate strength.) The

higher this lower stress limit is chosen, the smaller becomes

the possible stress amplitude. Thus, the steel may resist a

repetitive range of stress amounting to 27% of static tensile

strength if the lower stress limit is zero, but only an 18%

range of stress if tIle lower stress limit is increased to 40% I)

The trend of possible stress ranges may be easily observed by

noting the vertical displacement and width of the shaded area.

At tile point w11ere trle abse.issa. is 40%, for example, it will

be seen from the vertical axis that the shaded area extends

7

from 40% to 58%0 Furthermore, it may be seen that if the

stress were increased so that the minimum stress limit was

80%, the shaded area extends to .only 85%, tru~s giving a possible

stress range of only 5%.

The failure envelope in Fig. 2 readily provides a

definition of the dynamic yield point for materials with no

distinct flow point 0 It can be shown by tests that there exists

a stress level smaller than the static ultimate strength above

which even an extremely small stress alternation will cause

dynamic fracture. This stress level, at about 95% of the

ultimate, shall henceforth be defined as the dynamic yield

A failure envelope analogous to the one for steel

can be found for concrete (Fig. 3). This curve 1s ,more complicated

because it must be drawn for a range covering both tensile

and compressive stresses.(9~or example, if a point is considered

~vherea tensile stress range occurs, the portion of the envel-

ope in the upper right quadrant adjacent point A will apply.

If a point, such as the bottom fiber of a prestressed beam is

being considered, it might readily be visualized that the

fiber may be stressed between, say, 20% compression and 10%

tension, for one million cycles before cracking took place.

The point on the envelope representing a possible range 'of

8

stress between 20% compression and 10% tension is denoted by

the point B. The third quadrant contains the portion of the

failure envelope representing compressive stress ranges o For

example, a compressive stress could be applied between the

limits of 40 and 80% (as denoted by point C) for one million

cycles before failure 0 The possible stress amplitudes are

again accentuated by the shaded area in the same way as for

the steel failure envelope 0 Apparently there exists also

a dynamic yield stress in concrete, approximately at 90% of

the static ultimate strength~

The Combined Diagram

Returning now to the problem of predicting the' dynamic

ultimate strength in bending of a critical cross-section, we

simply combine the three previous diagrams, with the stress­

moment relation as the central part, and the steel and concrete

failure envelopes plotted for reasons of clearness, to the left

and to the right 0 Figure 4 shows the resulting combined diagram,

which makes possible the deter~ination of the dynamic cracking

load, and the dynamic ultimate moment as limited by steel or

concrete.

Determination of the Dynamic Cracking Moment

Several simple steps are necessary in the determin­

ation of the dynamic cracking load. In Fige 4, point A on the

9

stress-moment diagram represents the bottom fiber stress at

dead load moment and is the starting point from which a horiz­

ontal projection is made ov~r to the failure envelope of concrete.

Having a lower stress limit at point B on the failure envelope,

it is only necessary to project vertically to point C and

establish the upper limit of the stress range which causes

cracks after.1 million cycles Q A horizontal projection from

point C back to an intersection with the bottom fiber curve ,on

the central diagram at point D results in the establishment of

the dynamic cracking moment at 40% of static ultimate moment.

In other words, in this particular example, 1 million load

cycles from the dead load moment to 40% of the statid ultimate

moment will induce cracks in the bottom fiber of concrete Q

Determination of the Dynamic Ultimate Moment

The ultimate moment under dynamic loading is found in

the same manner as the cracking load, i.e. by projecting in

turn from the moment~stress diagram to the failure envelope

and back. In order to obtain the minimum failure moment, how­

ever, we have to observe two failure conditions - one for steel

and one for concrete. The dynamic ultimate moment based on

the steel is 56% of static ultimate as found by projecting

from E to F to G and finally to HQ The dynamic ultimate based

on the top fiber of concrete is determined by projecting from

I to J to K to L and is 83% of the static ultimate moment.

10

Thus, the dynamic ultimate moment of the beam for which Fig o 4

was drawn is limited by the steel, and is 5 16% of static tl1timate

moment 0

DISCUSSION

The method described above provides an adequate means

for determining the dynamic capacity of a prestressed concrete

member in bending o It is therefore possible to discuss the

effect of the level of prestress, the effect of over- and

under-reinforcing, and the cracking characteristics.

Level of Prestress

In Figure 5 the stress moment relationships are plotted

for different levels of prestressing a Imagine three beams

which are identical except for the fact that the steel in the

first beam has no initial 'prestress, the steel in the second

beam ·has a prestress which is 30% of its static ultimate strengtl1.,

and the third beam has steel with 60% initial pr~stressQ The

moment-stress diagrams for the beam with zero initial prestress

is represented by the solid lines, for 30% initial prestress

by the dashed lines, and for 60% initial prestress by the

broken lines. For zero prestress the maximum possible stress

range in the steel is seen to be from zero to 30%, thus giving

a dynamic 'ultimate moment of only about 27% of static ultimate

11

moment by projecting from A to B to C to D. Correspondingly,

it can be found by projecting from E to F to G to H that the

beam with 30% initial prestress has a dynamic ultimate moment

based on the steel of 42% of static ultimate moment c The

highest dynamic ultimate moment results from the beam with the

highest initial prestress of 60% of its static tensile strength

(projecting from I to J to K to L) and this value is 52% of

static ultimate moment~

The' values of dynamic ultimate bending moment obtained

above for various levels of prestress indicate that under

otherwise -identical circumstances these values increase with

the increasing level of prestress. This statement has been

confirmed by many tests, the first having been performed by

Freyssinet(l) in 1934. It should be kept in mind, however,

that the concrete stresses are assumed to be not critical.

Over- and Under-reinforcing

Fig .-6 sl1.oWS the stress-moment relation for three

beams which are identical except for the amount and arrangement

of the prestressing steel. The first beam, for example, has

a small percentage of steel and is under-reinforced. The second

beam is assumed to be a balanced static design on the basis of

the definition previously stated in this paper 0 The third

beam has more steel than is needed for a balanced design and

is therefore over-reinforced.

12

The dynamic llltimat~3 moments for all tllree bealus

based on the steil may be found quite easilyo It will be

observed that the possible stress range, denoted by the line

B ~ C on the failure envelope for the steel is practically

common to all tl"1re,e, 'beams because of tILe close correlaticrn of

t11e stress-moment c'urVeSOVij3r the elastic rarlg~ ¢ The dYl1.amic

·ultimate moments based on the steel are observed to be the

abscissas of the points D, E, and F for the under-reinforced,

the balanced, and the over~reinforced designs, respectively 0

It is evident from an examination of the relationship

between the stress-moment curve G - J and the failure envelope

for the concrete, that the top fiber of the concrete does not

become critical unless there is a percentage of steel which is

larger than the balanced design percentage for static loads 0

In other words, there exists a balanced design percentage of

steel for dynamic loading which results in an optimum dynamic

'ultimate momen't 4 In F:lg 4 6, tl1is op"timum dynamic ultimate

moment is approximately 85% of static ultimate moment Q

It has been observed that the variation of the per­

centage of steel, within the range of practicable values, can

result in a wide range of variation of the ratio of dynamic

ultimate moment to static ultimate moment~ It follows that

probably the higher the percentage of steel in any given cross­

section, the higher will be the resistance to dynamic loading,

13

but the arrangement of steel must also be considered 0 In fact,

it appears to be advantageous from the point of view of fatigue

loading to use a percentage of steel higher than the percentage

for static balanced design, that is, over-reinforce the section.

l:-t is 1,J:1.o\)\7rl ttlat. in.. n10st (~ases "tl1.e steel, 110t the concre-te, is

the critical mater 1 in fatigue, even though statically the

section might fail by crushing of the concrete in the top flange.

1'1'16 aDJCiJ"llf}t,· ()If 1.1],;;,,:.reJ3.s,e. ill stee.l stress at; cracking of the bottom

ttl€ se.ct

proportional to the percentage of steel in

thermore, the slope of the curve becomes flatter

~!\Ji·tl:"h. i.l1C)::--e,ase:cl, ,B\te.e~l pe:rcen.tag1es, a11d 'lienee leads to an advan­

tage, ill, ()~(le,1""''"1~'€~i:n.:Ecrcc,irlg for fatigue conditions 0

1''l;~r2;r~e, 11-88 .al.~rt7ir:l)7S been -an a~vers:ion against IJsing Qver­

reinforced concrete members of any kind o The principal reason

for this seems to stem from the fact that over~reinforcedmembers

usually fail without warning, and before any large deflections

B.re obse,r'led. ') .t.lo\A}teve, 1"'" , in the case. of failures c.aused by

dy'namic loadJirlg; f~ail'ure, c.an occ\.1.r suddenly, whet"her the beam

is over~ or under-reinforced, and before deflections become

large, moreover) the chance that the strands or bars do not

fail sirnulta11eolls1y increases with tl1.e rlum"ber of tendons, thus

agairl fa'ToI~i_ng o\7er .... reinforc'in.g 0 Also, tIle failure can be

S·lLdd.en, flO fnatte-:t" w11ether i.t acc-urs in COllcrete or steel 0

14

Cracks

Olle fa'ct, whicl'1 is quite apparent from a st'udy of

Fig. 4 is that the dynamic cracking load will always be less

than the static cracking load. However, this should not be

regarded with too much concern by the engineer.

A reCel1.t iU\Testigation cond'ucted at the Fritz

Engineering Laboratory, consisting of static and dynamic test­

ing of two full scale pretensioned bridge members, tends to

substantiate the statements in the preceding paragraph. (10)

It was found that the stiffness and general behavior of the

dynamically tested beam compared quite favorably with the

second beam tested by a gradually applied load. Observations

revealed that whereas cracks in the dynamically loaded beam

occu1.4 red be,1.ow stati.c cracking momeo.t, their number and width

at higher moments compared very favorably with values for the

statically tested beam 0

Further study of Fig o 4 reveals that the ratio of

dead load moment to live load moment has very little effect

on the dynamic cracking moment (bottom fiber) a For example,

a ratio of dead to live load of zero will result in a theoretical

cracking moment of about 40% of static 'tlltimate moment, wl1.ereas

a ratio of five increases the cracking moment to only about

43% of static ultimate moment 0

15

j\.1 tho·ugh re.peti·tive load.ing n,ecessarily reduces trle

cracking load, the advantage of prestressing remains tt- -arne

in that it delays the occurence of cracks very substantially a

Notion of Safet~

l\1ow tha,-t a rneans of predic"t'irlg tIle dynamic strengtl1

of prestressed members is established, it seems justified to

discuss the notion of safety in order to arrive at a suitable

means of providing sufficient safety against the probability

of fa,ilure 0

Since the ,actual stre,sses due t.o prestressing do not

increase linearly with the applied moment, it was felt that

merely specifying allowable stresses is insufficientQ Thus

all ,a.ddition.\al restrict.ion i.s iruposed 011 t11e ex'ternal moment by

all the current codes~

Formiulas for lirnit,ing trle, maximum allowable bending

moments are as follows~

Mu1 t .:> K1 (D + L + I)

M"J1t .2 D + KZ (L + I)

Wllen ~ D = decl"cl loa,d mome:ntL = live load momentI = impact momentK1 ,K2 = load factors*

The nature of these formulas often leads to the idea that they

provide a margin of safety against overloading. Such an idea

*As discussed later~ K1 (but not K2) can be considered as asa,fety factor (,

16

is both erroneous and dangerous 0 The necessity for saflety

factors has quite a different reason, namely, it stems from

the fact that the prediction of the ultimate moment always

includes a number of uncertainties which can be summarized,(10)

according to RUsch, as follows:

1. Uncertainties in the calculation of the internal

forces and stress.

2. Uncertainties in the properties of the materials

(concrete and steel)o

3. Local weaknesses of the materials (segregations

in concrete and inclusions in ste~l).

The real purpose of the factor of safety should be to guarantee.

a sufficiently small probability of failure under working load

assumin.g that an 'unlucky coincidence of malpredictions due to

the above uncertainties might lead to a failure under working

loAd& Two points are significant, first, that a possible

failure is not excluded but only made highly improbable,' and

second, that this probability refers only to a given load -

usually taken to be the maximum working load~ The probability

itself can be obtained by a theoretical evaluation of statis-

tical surveys of simple tests.

Since these,tl.ncertainties are fully independent of

the nature of the load, it follows that dead and live loads

17

have to be dealt with in the same manner. A formula of tl1:e

kind

Mu1t ~ D + K2

(L + I)

is therefore notionally unsatisfactory although it is clear

in its intentions 0 It might be mentioned, incidentally,

that the notion of allowable stresses is clear in this respect,

because nobody ever suggested that one should distinguish

between allowable stresses due to live and due to dead loads.

If there are believed to be some uncertainties in

the live and impact load aSstlmptions, then it is necessary to

increase these load assumptions, but the safety factor should

not be used to take care of overloading since this would simply

mean that the required safety margin would no longer b~ guar­

anteed. It seems especially desirable for short bridges to

assume occasional overloading, thus eliminating the need for

any restrictions of the above type.

From all the foregoing it follows that the factor

of safety should clearly be separated from the overloading

capacity. The former is connected with the uncertainties of

our calc'111ation and assumpt:Lons while the latter is a definite

value and can only be determined by the actual destruction of

the structure~ Only in two cases does the safety factor

coincide with the overloading capacity; if the deviations

18

from our assumptions are all zero (which is never the case),

or if the underestimations are accidentally equalized by Qver-

estimations 0

If a given bridge should, during its life,-sustain

permanently larger loads than those for which it was designed,

then a careful consideration of the actual failure probability

can always decide whether or not such overloading can be toler-

ated t

All these remarks apply to dynamic as well as static

loading. Since the uncertai11ties ·'under the latter are not

smaller for one than for the other, the design moment should

be subjected to a restriction of the form

Md '1 t > K (D + L + I).yn-u --

As to the actual value of the safety factor K, this might- very

well be a matter for further discussion, but it is believed

t1:1at it should 'be somewhere around two.

Many engineers will argue that in an actual bridge

1 mill.ion cycles of tl1.e maximum load will be very unlikely;

this being true under certain conditions, we can only see ·one

way of taking into account such possibilities - that is, by a

statistical proof that the expected maximum load frequency

mimimizes the danger of fatigue failure. Such proof would cer-

tainly [)8 possible for many l1igl1.way bridges, but probably not

for railway bridges.

19

Another widespread argument against con.siderati,011S

of dynamic failures is based on t]J.e fact tllat the steel stress

variation is definitely small and that the corresponding failure

envelope leaves a margin of say three times as large a stress

variation ~ From all the foregoing." it follows that tllis factor

of three cannot be considered as a factor of safety. Certainly

no one would pretend that the safety factor of any structur~-is

two if only the allowable stresses are taken to half of the

ultimate stresses. The above argument, however, would amount

to an analogous, misleading contention Q

The foregoing discussion of the desirable requirements

on prestressed structures under repetitive loading may seem

rather pessimistic. To leave s'ucl"l an impression, however, is

quite opposite to the aim of this paper. An attempt was made

to show that prestressed concrete is very well suited to struc­

tures subjected to dynamic loads. The method described in this

paper is intended to give more certainty as to what is to be

expected under such conditions 0 Since it is believed that the

estimated dynamic capacity can be predicted, it seems only

fair to connect this also with sufficient safety requirements.

Although a dynamic safety factor of, say two, may

seem very severe, attention is drawn to the fact that fatigue

loading, especially if combined with severe exposure conditions,

20

involves more uncertainties than those mentioned previously

for static conditions.

Now investigate the possibility that the present

design requirements may provide s·ufficient safety against fat­

igure failure. If one considers the two formulas given by

the Bureau of Public Roads Criteria(9) for limiting the design

moment based on the ultimate strength of the member, it will

be seen readily by considering typical ratios of live load to

dead load that the ratio of dynamic ultimate strength to static

ultimate strength must be greater than about 0 0 8 if a factor

of safety of two against dynamic ·ultimate moment is achieved.

A 'beam must be carefully designed based on the method of

analysis given in this paper to have so high a ratio of dynamic

ultimate to static ultimate strength. Members designed on the

basis of only static considerations normally have ratios of

dynamic ultimate to static ultimate of 0.4 to 0.8e

The required dynamic factor of safety cannot be

achieved economically by making the required static ultimate

moment more ~onservative because designing for a higher static

ultimate moment would result in a larger cross-section of

member when all that may be required is a similar member with

a larger percentage of steel or a better arrangement of

st~el.

21

The majority of prestressed bridges will never be

in danger of fatigue failureQ But if a structure is subjected

to se,vere, repetiti'~Je, loa,dil1.g., tl1e probl,em of fail,ure de.ser\!es

,adequate, consirlerat,ion 0 For any. spe,cific, problem, the dynamic

ul:timate moment sl1.ot:.tld be c,alc'ulated by ,tl1.e procedure,s outlined

in tilis I)aper.

CONCLUDING REMARKS

Based on the knowledge of the fatigue properties

of concrete and prestressing steel and the determination of

the stress~moment relationships of prestressed beams, it has

been s'hown tha.i: :Lt is possible to l~:rr:'eclict t:he fa.tigtle properties

of members in bending revealing the following £acts~

l c The dynamic ultimate moment is always less than

the static ultimate moment and can vary over a very large range 0

2 0 The dynamic cracking moment is always smaller

than the static cx~ackillg mOKnent 0 Above stati..c cracking moment,

however, the number and width of cracks due to dynamic loading

correspond to tllose due to sta.tic loading l>

3. The ratio of the dynamic ultimate moment to

the static ultimate, moment is j,ncreased by ~

'a o Increasing the level of prestress 0

bo Increasing the percentage of steel ina beanlo

22

The .design of prestressed members under severe.

fatigue loading should be ~ased on the determination 6£ the

ul,timate dynamic rnoment ~ l:;onsidering the 'uncertainties always

present in the analysis and in the properties of materials,

adequate safety should be provided by a restriction of~ the

fol~low-ing kind~

Md .~ 1 t ., t > K (D + L + I)ynam~c U l,ma e-

The desirable value of the safety factor K (about 2) deserves

a careful consideration and should be specified according to

the particular type of structure and site conditions 0 In the

c,a,se of £at,igue load,ing, it is quite important that possible

o,rerloadi,ng be considered sepa17ately by inc.reas:lng the design

loads and using the above equation with the accepted value of K.

It should be emphasized that this paper is restricted

to pure bending of bonded prestressed beamsc If the future

will, as is hoped, provide more information about the stress

distribution under the combined action of moment and shear

or if more becomes kno~! about bond stresses, then a further

discu,ssion of suc.h failures under repetitive loading will

certainly be possible and follow the same pattern as suggested

in t11is paper t

23

ACKNOWLEDGMENTS

This study has been carried out in connection

with research sponsored by the following: Pennsylvania

Department of Highways; Uo So Bureau,of Public Roads; 'Concrete

Products of America, Division of American~MariettaCompany;

American Steel and Wire, Division of UQ So Steel Corporation;

John Ao Roebling!s Sons Corporation; ~einforced Concrete

Research Council; and Lehigh Uriiversityo

I00 ....------~-------.-------.,..--------,-.;.--------..

o~_....~--...------.

80

V)

PRESTRESSING STEEL I-Wen 60 z

I wV')

~W It- o0::: Iz ~J-- IWV') I~ w

1° o!W :J-J 40 ~ ...J

V') ltD ~Z IZw 1;;2...... IU

1<20

lID!lU

-20V)WV)V') Cow

'lvC~S~J-- -40 r~ r.V) Opw ~/8SR>en:i1 -600:::0...20()

-80

BENDING MOMENT eyo OF M ult )

-10020 40 60 80 100" . 0

..

NOTE: ALL STRESSES IN PERCENT OF CORRESPONDING STATIC ULTIMATE STRESS

STRESS - MOMENT DIAGRAM

FIGURE I

100 ~------r------.,..------~-----_- _

1008040 60

jLOWER STRESS LIMIT (jlo (J su)

20

(J su === STATIC ULTIMATE TENSILE STRENGTH

o

---~b

60 C!:::~:::;tilV)w0::::l-V)

400::::wc..c..::>

20 1--4-~-t---4-~~-A---_---:"'----4-------4------+---------i

.80 ~------+------+-------+----.JIC..I--+--I---..,~---"""'"------I

FATIGUE FAILURE ENVELOPE FOR

PRESTRESSING STEEL

(106 LOAD CYCLES)

FIGURE 2

-100 -80 -60 -40 -20 o 20

-20

-40

q

COMPRESSION -

LOWER STRESS LIMIT (1'0 (J ell)

~u == STATIC ULTIMATE COMPRESSIVE STRENGTHI I I

SIGN CONVENTION: TENSION +----If&--+----+----+---+--+-...--.-B- c

""'0""'Cm;::cV)--I:;:;0mtil

t-- +-- +-- ~I__+___I__4__-+--I___I__4___I__-I--~_c.n

r~=i

~---i----~--+-+--+--I--I--+---I---+--I---&----+----I-----f--..,.,.A----..!..-60

&.....- -'-- -I-- --'-- ......1.- -L... ---I-100

r-----------:~I--+--+---+--+--+---+--+---==hII~----4------I--------I-80

r------t------t------+----......---...I'-................---JO

FATIGUE FAILURE ENVELOPE FOR CONCRETE

(106 LOAD CYCLES)

FIGURE 3

FAILURE ENVELOPEFOR CONCRETE

NOTE: ALL STRESSES IN PERCENT OF

CORRESPONDING STATIC

ULTIMATE STRESS.

I ! I I =ei I I H Iv~:fJ"llJY I r;e

;;;; PRI:51R£S5\NG 511:::"" lI I' 60 i~, ~ .....----"Tl" I ~ I I

~' I .F JE I~~' I~ V) , ::E I 0 to-

~' ~ ~ I 0 ~ I ~ ~,~ I~ ~ I ~ ~ I ~ ~

I ~ ~ ~ ~l-::> 0

kr1111 ~ ...--40 I ~-::ll.-+= ~

~ V'). 0::: () 1< wI II ~ ~ () ~ I LL ~,I ~ Vi 0 ,Z <.) I ~ ::

I z....l ~~~ <I w c ~ 1< LL

I~IIIIIII~ 1--20-;:5 j _Vl-+~ ~ (1o : 10 t- .

iii ~ C"I ,~.._~ ---- --_: .....----~----~----'""'"--.-'""'".-.-~o 20 .40 60 80 0 ""J~ ----1--- ~ ~ ~ J,,( III'

"I l' I /'~II I1 I I

FAILURE ENVELOPE -20~A~ ._.~. __._. ._ ~ • ---- ._,..:.~ • ~.+++_ D

~~<y- '" I - II IFOR STEEL $" ~ I~ ti ~I I~u.~ C'o 1"3<3 I I

to -40 I((}O O~· ~C'L) I ::( 5 JI IV') R "~h' I LL

<.) I" It/) o"~ ~~ 0- I I~ <Q 1'0..0 I~ ~ 1/ I l~V) ~ Iz ~ I' 1~ -60 ~ -+~ 0 ~ IVi ~ II" J~1'1

~ ,I """,,~ K~ If( -..-- .......~ -,,-o -80 ~~_-..

() ,\~' II 11II ........ ,,-,,-JJ.......IIII ..

BENDING MOMENT (~o OF M UI; )

00. I I I I-I 0 20 40 60 80 100 -80 -60 -40 -20 0 20

STRESS - MOMENT DIAGRAM

FIGURE 4

FAILURE ENVELOPE

FOR CONCRETE

FAILURE ENVELOPE

FOR STEEL

--------r------,...-----r----IOO -----......-----.....-----,-----~----

~---~---~---~-~~~-~80 ~ DYNAMIC ~ILURE MOMENT

---_._---_... -V')wV')V')

I--------II------:l~--......----t--- ~ - 60'-1E..:-~--+-----+--o#--#ol-V)

-w .. -:::! EFFECTIVE PRESTRESSV)

-#--+----+--#-----;------t-- ~ _ 40 NO PRESTRESS __I-

e; eff = 0.3 cr su

c· ---- -----------~eff= 0.6 (J"su -100 -80 -60 -40 -20 0 20

¥----+-----t------1I---- 201--------.F------4-----4-----1-----..------i---..,.~-

-20 ~--_____,~~~--I~------lI__----J-___4....l--_-""""-+-----'"__+---_+_---__+_---_+---_+____t"

V)WV)V)w:= -40 ~----I-----+--~r:-c.,--+-_....:.---H-----"--I-------_+_---+_---_I_---__t__---_t___t"V)

W>ViV)wg: -601-----~---~---4_-...piii1~I++0---_I_f_--_+_+_---_#_---_t_---_+_--- ---'"""1

~ou

-80 .----- I I - --T .------ I

-1001 B~NDING M?MENT ("~ OF1Mu1f)1

NOTE: ALL STRESSES IN PERCENT OF

CORRESPONDING STATICULfiMATE STRESS.

o 20 40 60 80 loq

STRESS ~MOMENT DI4GRAM

FIGURE 5

FAILURE ENVELOPEFOR CONCRETE

o DYNAMIC FAILURE MOMENT

PRESTRESSING ST~ELA

I I I I I V I I I I I I-100 _o 20 40 60 80 100

-20

1VJWV').V)w0:::

t:iw~VlVl

~ Ic..

2 I8 -80 I

FAILURE ENVELOPEFOR STEEL

NOTE: ALL STRESSES IN PERCENT OF

CORRESPONDING STATIC

ULTIMATE STRESS.

V !!! 01 7'h i

I Y I I 20~ I ~ -~ I 1 I I I )1

--_..-::3 - 60V)V)w0:::.....V)r

I .£ I V I I ~ 401 I I I> i ~ V)

Zw.....

I I I i' 7Jv 80+1---+-1---1---

STRESS - MOMENT DIAGRAM

FIGURE 6

REFERENCES

1 0 Rowe, R. E.: "An i-\ppreciatio'n of the Work Carried onFatigue in Prestressed Concrete Struct'ures, Vi 'recl-;L­nical Report. Cement a'nd Conertate PJ.ssociation, LOl'"ldon <1

2 . Leonhard,t, E'.: Spannbeton f{ir die Praxis 0 Ernst and 8011,

Berlin, 19550

3. Lin, T. Y.: nS"treng"t11 of Con'tinuous Prestressed ConcreteBeams "under S'tatic and Repeaded Loads (t 11 Proceedingsof the Americarl COl1.Crete Institute, Vol. '51, po 1037,19S5:-

~. 0 Irlomata:- S.: "Fatigue Tests in Bending of PrestressedConcrete Beams e " Proceedings of the American Con­crete Institute, Vol. 49, po 766, 1952-1953c

5 . Abeles, P. ~N.: t'Sta-tic and Fatigu,e Tests on PartiallyPrestressed Concrete Construe·tions (l1f Proceedingsof the American Concrete Institute, Vola 51, PPo 361­376, 1955.

6. "Criteria for Prestressed Concrete Bridgeso" Departmentof Commerce, Bureau of publ.ic Roads, 1954· 0

7 0 Colonnetti, Go: nScienza dell,e Costrlliol1.i 0 n Einaudi, 19l~1 0

80 Ros, M.: Vorgespannte£ Beton (Bericht Nr o 155)0 Eidgen­tJssische MatialprHfungs- ~l1n,d Vers'uclisanstalt fUrIndustrie, Bauwesen un,d Gewerbe 0 S tl 79, Abb 0 78,ZHrich, 1946.

9. Ros, Me: Die materialtechnischen Grundlagen und Problemedes Eisenb~etoll.S im Hinbli1cl'C auf die zuktlnftigeGestaltung dar Stalhbeton- Bauweise (Bericht Nr o 162).Eidgen~ssische MaterialprUfungs- und VersuchsanstaltfUr Industrie, Bauwesen und Gewerbe. So 36, Abb o 55,Ztlric11, 1950.

10 0 Ekberg, Jr D' C 0 E /): "Dynamic Tests of a 55 Foo-t Pundel< BridgeMember \1 ,t Concrete Products Comparly of America, Potts-t OWI1, Pa 0' 1956 "

11 0 RUsch, H, ~ Schweizer Archiv, iVDer Eil1.fluss des Sicherheit­begriffes auf die technishen Regeln fUr vorgespanntenBetono" pp. 85-93, Heft 3, Mal."c11, 1954 0