A study on the fatigue strength behavior of reinforced concrete structures

Int. J. Pres. Ves. & Piping 40 (1989) 51-75

A Study on the Fatigue Strength Behavior of Reinforced Concrete Structures

Dong-I1 C h a n g & W o n - K y u Chai

Department of Civil Engineering, Hanyang University, Seoul, Korea

(Received 1 May 1989; accepted 23 May 1989)

.4 BS TR,4 C T

Fatigue tests were carried out in order to investigate the fatigue strength behavior of R/C (reinforced concrete) structures. Eleven R/C beams were used in the tests, the fracture mode, crack growth, relationships between loading and stirrup strains, and relationships between loading and mid-span deflections of the beams were observed under the fatigue loading. ,4 S - N curve was drawn to present the fatigue strength of the R/C beam.

According to the results, stirrup strains are closely related to the diagonal crack, not to the flexural one; the fatigue strength of the beam at 1 000 000 repeated loading cycles is about 65% of the static ultimate strength.

1 I N T R O D U C T I O N

From the results of previous studies, the behavior of R/C beams under static loading has been well known, but recently it has become necessary to know the characteristics of behavior under other loadings.

A fatigue fracture under repeated loading occurs in the weak parts of structural members in proportion to the number of repeated loading cycles. After such fatigue fracture reaches its last phase, structural members cannot function properly even under static loading. Because the R/C beams are composed of the brittle concrete and ductile steel, the residual micro-crack coalescence of concrete could occur by increasing the strain in the steel which decreases the adhesion of two materials. Therefore it is very important to develop an understanding of the fatigue crack growth behavior.13-15

51 hu. J. Pres. Ves. & Piph~g 0308-0161/89/$03.50 ~ 1989 Elsevier Science Publishers Ltd. England. Printed in Great Britain

52 Dong-ll Chang, Won-Kyu Chai

Recently, because of the increase of design load and traffic volume, the material strength (called the fatigue strength) versus the repeated loading of the R/C beam has become an important aspect so that many researches have been conducted about this.

Because these researches are mostly about the flexural fatigue behavior of the beams, not about the shear behavior, the shear behavior of the R/C beam under dynamic loading has not been thoroughly investigated. It was therefore considered necessary to study the shear behavior of the beam under fatigue loading.

The results of such fatigue tests are generally expressed in terms of S-N curves such as that shown in Fig. 1. If the S-N curve has an asymptotic line, the stress corresponding to this line is called the fatigue limit of the material. Most metals have fatigue limits like this, but because concrete does not have a clear fatigue limit, the fatigue strength under a specified repeated number of loading cycles (for example 1 000 000 cycles) is used as the fatigue limit of the concrete.11

H s u 9 proposed that the fatigue strengths of the R/C members under 500000 cycles corresponds with a value that is 60% or 65% of the static ultimate strengths. Generally, however, fatigue strengths at 1 000 000 cycles of R/C members are considered to be between 60% and 70% of static ultimate strengths. 1~'~? Chang 3'4'18 reported that the size of repeated loading determines the failure modes of beams. That is, low repeated loading causes flexural failure because of fatigue of the tension bars. On the other hand, high repeated loading causes shear failure. Hawkins 7 reported that the shear failure of R/C beams causing failure of stirrups can occur at 21% of the static shear strengths. Nordby ix suggested that the failure of R/C beams under fatigue loading results from the fatigue failure of tension bars and this failure makes sharp cracks in members, so accelerating stress concentration up to the rupture.

In fact, many formulae have been suggested to predict a fatigue life under

E

o

z w G: I - (n

~ T

- - I t I I i

NUMBER OF REPEATED LOADING CYQJE

Fig. 1. Typ ica l S-N curve .

Fatigue strength behavior of reinforced concrete structures 53

repeated loading, especially AASHTO, which specifies checking the stress range of the members under repeated loading. 1 Okamura et a[. 13-15 proposed an empirical formula about the relationship between the fatigue strength and stirrup strain followed by the increase of the repeated loading. On this basis, in this study, the behavior of R/C beams under fatigue loading has been investigated. In this work, crack growth, fracture mode and stirrup strains of the R/C beams have been observed, and the results compared with the existing empirical formula.

2 TEST PROGRAM

2.1 Outline of the test

The fatigue test in this study was performed in order to investigate the shear behavior of the R/C beams; the occurrence and progress of cracks, the relationships between loadings and stirrup strains and the relationships between loadings and mid-span deflections of the R/C beams were investigated. Eleven test specimens were used, which are classified in Table 1.

2.2 Dimensions and materials

The R / C beams were designed according to the ultimate strength design method of the Concrete Standard Specifications of Korea, 2° and they were made to have low web reinforcement in order to make it fracture by shear. The steel ratio was 1-492% which was 51"7% of the balanced steel ratio. The beams were of the same rectangular cross-section, of effective height 24 cm and width 24 cm and their span was 190 cm. Waterproof strain gauges were

TABLE 1 Classification of Specimens

Classification of Number of Name of specimens specimens specimens

Series I 4 4Fl-1, 4F1-2, 4F1-3, 4F1-4 Series II 2 4F2-1, 4F2-2 Series III 2 4F3-1, 4F3-2 Series" IV 3 6F4-1, 6F4-2, 6F4-3

Total 11

Note: In 4F1-2, 4 is the number of stirrups, F is fatigue test, 1 indicates specimen series, and 2 is the specimen number.


Z

3 5 , 2 0

P S T R A I N GAUGE

.., /

P Z

zo ~zo ~z 0 ~_zo~. zo_~s5 . [

1 9 0

(a)

J Z 4 ~

i lo! ,sL, s i s o -t

P

• I l

1 9 0 1

(b)

2 4

N

P

I 0 / 1 5 / 3 0 , ZO ~ 2 0 , 2 0 , Z O , 3 0 , 1 5 ~ 1 0 1 ~ t ÷-+. +o-+-~- (c)

Z 4 , - - - - J (

P

I - s5 _~ 2o_4_~q_~_~o ~_ ~ I 1 9 0 I

Fig. 2.

(d)

2 4 [ I

Typical test beam and strain gauges arrangement. (a) Specimen I series, (b) specimen II series, (c) specimen II1 series, (d) specimen IV series.

Fatigue strength behavior o f reinforced concrete structures

TABLE 2 Properties of Bars

55

Bar type d (cm) A~ (cm 2) try (MPa) au (MPa) E~ (10 3 MPa)

D 10 0'953 0'713 272"3 398-8 200.1 D 19 1-270 2"865 349" 1 492"3 200-1

d= diameter; A~ = cross-sectional area; O'y = yield strength; a u = ultimate strength; E~ = Young's moduli.

used to prevent mois ture effects. The posit ion o f strain gauges a t tached is shown in Fig. 2.

The deformed bars o f D19 specified in K S D 3504 were used and U-type vertical stirrups o f D 10 were established for shear reinforcement. The cross- sectional areas, yield strengths and Young 's modul i o f the bars are shown in Table 2, and the mix p ropor t ion o f concrete is given in Table 3. The typical compressive strength o f concrete was measured by 40 cylindrical specimens (10 x 20cm). It was 22.1 MPa.

2.3 Measur ing sy s t em

In order to measure the crack propaga t ion easily, pre-drawn grid lines o f 10 cm × 10 cm were used on bo th sides o f the test members. Three-point and four-point loading systems were used. 5 Shear span to depth ratio was 2.5 ( a / d = 2-5). 6'16'19'21

Fo r these fatigue tests a 245.2kN fatigue test machine was used. The m i n i m u m value o f repeated loading was fixed at 39.2 kN and the m a x i m u m value was different for each member in order to draw appropr ia te S - N curve graphs. Load ing was applied to m a x i m u m repeated loading statically and at the same time, the loading at which flexural and diagonal cracks initiate was investigated.

The loading was applied statically up to 100 repeated cycles. In the fatigue

TABLE 3 Mix Proportion of Concrete

sf s a u u s w/c a r a c (ram) (cm) (%) (N) (N) (%) (N) (N)

25 7 1 1672"0 3484.3 48 7387-7 112"8

sf = maximum size of coarse aggregate; s = slump value; a = air value; u = water per cubic centimetre; u~ = cement per cubic centimetre; w/c = water-cement ratio; af = fine aggregate per cubic centimetre; a~ = coarse aggregate per cubic centimetre.


test, loading speed was 5 Hz. During the test, stirrup strain, strain rate of tension bars, the crack growth procedures and mid-span deflection in the minimum or maximum loading were measured. The members which were not fractured, after receiving 1 300000 cycles of loading, were loaded statically up to fracture again.

3 THE RESULTS OF THE FATIGUE TESTS

3.1 Behavior of beams under fatigue loading

In these tests the fatigue loading coefficient (R) is defined as Pmax/P'u (where Pm,x is the maximum load, P'u is the static ultimate load). Table 4 shows the applied fatigue loading and the number of repeated loading cycles to fracture.

By increasing the number of repeated loading cycles, the strains of the stirrups were changed continuously as shown in Fig. 3(a)-(h). Generally, the strains of stirrups and tension bars are increasing, but the strains of stirrups tend to decrease temporarily as a result of the formation of diagonal cracks. The reason is that during fracture the presence of cracks inside the concrete brought about the redistribution of stress. 12'~4

The crack growth behavior caused by the increase of the number of repeated loading cycles, showed that the micro-cracks at the early stage have grown and diagonal cracks developed. The growing modes are shown in Fig. 4(a)-(n).

Mid-span deflections increased continuously with increasing numbers of cycles. In order to investigate the residual deformations of the members applied to an arbitrary repeated loading, the mid-span deflections after removing the loading are shown in Fig. 7(a)-(g), which shows distinct residual deformations.

In the case when the stress by fatigue loading is above the limit value for the R/C beams, then the beams are fractured. The fractures are divided into the compressive fracture of concrete and the fatigue fracture of the tension bar. In this test, specimens 4F1-2, 4F3,1 and 4F3-2 were fractured by fatigue fracture of the tension bars. The fatigue fractures of the tension bars occurred as follows: the steel receives excessive strain due to the stress concentration at cracked portions so that the steel breaks; finally the member is fractured. 2.8

In the 6F4 specimen series, all of the members were fractured diagonally following the diagonal cracks. Specimens 6F4-1 and 6F4-2 sustained more than 1 300 000 cycles of loading, and 6F4-3 was fractured at the moment the stirrup was fractured at 465 000 cycles.

TA

BL

E

4 Fa

tigue

T

est

Res

ults

Spec

imen

s P

Illa

, W

N)

R

NU

Fr

actu

re

mod

e (k

:) (k

?)

4Fl-

1 39

.2

132.

4 0.

50

264.

8 21

0.8

- 4F

l-2

39.2

16

67

0.63

26

4.8

- 11

1040

0

4Fl-

3 39

.2

1765

0.

67

264.

8 -

9550

00

4Fl-

4 39

.2

196.

1 0.

74

264.

8 -

448

220

Shea

r fr

actu

re

Fatig

ue

frac

ture

of

ten

sion

ba

rs

Shea

r fr

actu

re

Shea

r fr

actu

re

88.3

17

65

88.3

17

6.5

78.5

15

6.9

78.5

15

6.9

4F2-

1 39

.2

107.

9 0.

55

196.

1

4F2-

2 39

.2

156.

9 0.

80

196.

1 4F

3-1

39.2

19

6.1

0.74

26

4.8

186.

3

-

2000

000

or m

ore

1570

900

38

6 10

0

4F3-

2 39

.2

186.

3 0.

70

264.

8 40

7 60

0

6F4-

1

6F4-

2

39.2

39.2

1667

149.

6

0.65

0.58

256.

0

256.

0

210.

8

235.

4

1500

000

or m

ore

1330

000

or m

ore

Shea

r fr

actu

re

34.3

Shea

r fr

actu

re

44.1

Fa

tigue

fr

actu

re

58.8

of

ten

sion

ba

rs

Fatig

ue

frac

ture

44

.1

of t

ensi

on

bars

Sh

ear

frac

ture

73

.5

49.0

Shea

r fr

actu

re

63.7

39

.2

1 68.6

88.3

17

7

88~3

47.1

27.5

6F4-

3 39

.2

181.

4 0.

71

256.

0 -

465

CO

O

Shea

r fr

actu

re

73.5

39

.2

147.

1

Pm

in =

min

imum

re

peat

ed

load

; P

,,,

= m

axim

um

repe

ated

lo

ad;

R =

per

cent

age

of m

axim

um

repe

ated

lo

ad

over

st

atic

ul

timat

e lo

adin

g;

PA

= s

tatic

ulti

mat

e lo

ad;

P:’

= s

tatic

fra

ctur

ing

stre

ngth

in

the

cas

e of

loa

d cy

cles

of

mor

e th

an

1.3

mill

ion;

N

, =

num

ber

of r

epea

ted

load

ing

cycl

es

whe

n sp

ecim

en

was

fra

ctur

ed;

V,,

= P,

,/2;

P,,

= ap

plie

d lo

ad o

f be

ams

at w

hich

the

fir

st b

endi

ng

crac

k oc

curr

ed;

PdC

= a

pplie

d lo

ad o

f be

ams

at

whi

ch t

he s

hear

cra

ck

occu

rred

.

Y

SH E A R S T R A I N

( X 10 - 6 )

5OO

SHEAR

STR~N (XIO-6)

@

® i jooo

J ~ i J ,

I z 3 4 5 6 LOG N

(a)

1,000

®

®

5 0 0


t ~ J J J I z =3 4 5 6 tOO N

SHEAR

STR~ N ( X I O - 6 )

t, o o o

®

500

I Z 3 4 5 6

SHEAR ST RAI N (X 10 - 6 )

1,000

5 0 0

- ~ t i i , i

LO G N I Z 3 4 5 6 LOG N

(b) Fig. 3. The relationships between the repeated loading cycles and the stirrup strains. The numbers O - ® in (a)-(f) indicate the position of stirrups in (f). The mark (.) indicates the yield point. The numbers 0) @ in (g)-(h) indicate the position of stirrups in Fig, 4 (m), (n). (a)

Specimen 4F1-3, (b) specimen 4F1-4.


SHEAR STRAI N (X I0 - 6 )

I,OOO

700

500

300

SHEAR STRAI N ( X I O - 6 )

1,00 0

8 0 0

500

••®> 3 0 0

I Z 3 4 S 6 LOG N

LOG Nu > 6 . 3 l

I • 0 I Z 3 4 S 6 LOG N

(c)

SHEAR S T I l l N (X 10-6)

1,000

500

SHEAR STRAI N ( X l O - 6 )

Q

f I { I I I I 0 I Z 3 4 5 6 LOG N

I , O 0 0

5 0 0

(d)

LOG Nu : 6 . 196

®

®

I ~ I I , { -

I Z 3 4 5 6 LOG N

Fig. 3.--contd. (c) Specimen 4F2-1, (d) specimen 4F2-2.

60 Dong-II Chang, Won-Kyu Chai

SHEAR STRAIN ,x,o-~b o.

1,500

1,000

I 2 3 4 5 6 7 LDG N

SHEAR STRAI N I

( X l O - 6 ) | 2,000~

13500 1

l~O00 ~ 701

®

50(

LOG Nu = 5 .61

I Z 3 4 5 6 7 LOG N

(e)

SHEAR STRAIN

( X l 0 - 6 ) z,ooo

1,500

1,00 0

500

SHEAR sTR~J~ OqO-O)

Z,O00

I I I I ®i :® ~', ,,® I I I I

1,500

®

1,000 /

5OO d)

I ~ I I i , , I 0

I 2 3 4 5 6 7 LOG N

(0 Fig. 3--contd. (e) Specimen 4F3-1, (f) specimen 4F3-2.

,'o~ N


SHEAR STRAI N

( X l O - 6 )

1,000

500

®

®

1 2 3 4 5 6 ~ G N

SHIE& R SI'RGJ N

(X lO-6)

1,000

50O

(g)

I Z 3 4 5 6LOGN

SHEAR I STRAI N ]

(X I0- 6 )]~

1~500 I

i,ooo

5 0 C

_ 1 0 0 " - 1 2 3 4 5 6 LOG N

SHF.AR i STRAIN (Xl 0-6)

L500

I,OOC

0 , . =

- i o o | E z 3 , s • mo ,,,

(h)

Fig. 3--contd. (g) Specimen 6F4-3 (rear), (h) specimen 6F4-3 (front).


/

6 ~ 5 2 ;

(a)

P

LO G Nu

6.05 ' , 6., p5

LOG Nu ~ 6 . 3 9

6

\

= 6 . 0 5

(b)

4~-

@ ®

® @ 5.S5

~ f ,(/'( /

(c)

® l O G Nu = 5 . 9 6

®

4

LOG Nu = 5 . 6 5

(d)

Fig. 4. Diagrams showing the progress of crack propagation. The numbers in (a)-(d) indicate the value of log N. The marks I-XIV in (e)~h) mean the number of cycles as follows: I: 1 (cycle), Ii: 10, Ill: 32, IV: 100, V: 320, VI: 1000, VII: 3200, VIII: 10000, IX: 32000, X: 100000, XI: 320000, XII: 1 000000, XIII: 1 500000, XIV: 2000000. The numbers in (i)-(n) indicate the number of the repeated loading cycles. The numbers Q -@ indicate the position of stirrups. (a) Specimen 4Fl - I , (b) specimen 4F1-2, (c) specimen 4F1-3, (d) specimen 4F1-4.


P P z Z

i =x I I / Y ~ / . ' "m" i , , \ ~ , ~

. , d / S ; ( I 7 ~ , . (e)

z

I

f ~ I I ), ) "'~7' (f)

P P 2 2

i ~ !it

(g)

P P z z

_~x

(h) Fig. 4--contd. (e) Specimen 4F2-2 (front), (f) specimen 4F2-2 (rear), (g) specimen 4F3-2 (front),

(h) specimen 4F3-2 (rear).


_P £ 2 2

III I I0~ I-MI .UON : i I-MI UON

4o 'X ' '~ ~ ' °

(i)

P P 2 2

1 (j)

J . .~ J ON ,~JON

~ I IOC ~ . 0 31~ !30

7

/ JC

i - kill JO~,~

P p 2

~ 1.000 I 31,601 ;,

(k)

P P 2

l ~86,000 ,-Mll..I.I ~.~iOOC ~ O O l

,,"b~ N I-MI

3.11~ ?; OJ.kllllON "~

I-MI .UON 3 J6~30 0'1- Mil I 'ION . ~ : l ~ ~ 0 ~ 3Jm

"1'000 |

I "~" 320 I

(1)

LI_ION

Fig. 4--contd. (i) Specimen 6F4-1 (front), (j) specimen 6F4-1 (rear), (k) specimen 6F4-2 (front), (I) specimen 6F4-2 (rear).

Fatigue strength behavior o f reinforced concrete structures 6 5

O) NE

/ n f 7

2

® ®

~,ooo 31s,,~o

fo '° 1,

(m) ~ P 2 2

P 2

®

~/s( i o

®

\ \ ,

NW

~ 1 0 0

® ® ~) ® @ ,._.ca 0 .1 - ~3.1" Mll I ~q

SW 3,Jso ~ w ~ , z ~

,o ,o% \~,,~ ~ o I o ,

, o o / ~ . , tuo, " ' " " '

~ 6 0 DO I0( oJ-u ~ ~ ! ~,~o

..... (n) Fig. ~-contd. (m) Specimen 6F4-3 (front), (n) specimen 6F4-3 (rear).

@

S E

q

In specimen 6F4-3, diagonal cracks occurred at the first loading stage. The diagonal cracks were associated with strains over 250 x 10 - 6 at stirrup numbers 2, 5, 8, l0 and 11 and made strain at the other stirrups below 100 x 10 -6. After 10 cycles of loading, stirrup number 11 yielded and the strain of the E side increased greatly compared with the W side. In the stirrup number 2 (that was a pair with stirrup number 11) the strain increased quickly; stirrup number 2 also yielded at 1000 cycles. After 465 000 cycles of loading, stirrup number l l in the SE side was fractured; however the fractured side in this case was flat and little strain occurred. This was because the fatigue fracture occurred by widening of the diagonal cracks.

The cracks in the SE side were developed completely at the early stage of the repeated loading and no new cracks occurred, although the potential cracks were widened after 10 000 cycles. After loading to 465 000 cycles the rigidity of the beam was reduced by the fatigue failure of the stirrup, and then it was fractured without withstanding any further loading.

3.2 A comparison between the results of fatigue tests and the empirical formulae of Okamura

Okamura~3-15 proposed empirical formulae for predicting stirrup strain under repeated loading of concrete beams with web reinforcement. The


empirical formulae of Okamura are as follows:

where

~w

A t Vma x ~ ]/co

fiX{ Vma x - - Vco(1 - k l o g N ) }

A wE Z/S

~w = strain stirrup

At Vmax<V~o

fiX V¢ok log N e,, = AwE~Z/S

X

Vrllax

Vc k

Aw E, Z S

fix = coefficient determined by the relative positions of the stirrup to the supports or loading points, and fix = x/( l '5d): x < l '5d

= 1: x > l '5d -- distance from support or loading point = maximum shear force in the fatigue test = value of V c at l o g N of 0 = shear force carried by concrete = constant which indicates the rate of decreasing of Vc = area of pair of stirrups = Young's moduli of steel = distance from the centroid of compressive area to tensile steel = spacing of stirrups.

Okamura ' 3 - ' 5 also proposed an equation for predicting a range of stirrup stresses assuming that residual strain is zero after some cycles of loading.

Vma x - Vmi . E~¢w G r ~ O'ma x - - Gmi n - - Wmax

According to the test results, the average strain of the stirrups was proport ional to log N, and the result of the regression analysis by the Least Squares Method is as follows:

e W = a + b l o g N

where ew = average strain of stirrups a = average strain of stirrups at the number of load cycles of 1

obtained by the regression analysis b = rate of increase in stirrup strain due to increase of repeated

loading obtained by the regression analysis.

Figure 5 and Table 5 show the relationships between the empirical formulae and the regression analysis. The empirical formulae are in good accord with the regression analysis.


STRAIN

(XlO'6) ' 4 F I - 0 4 / S ~

I)000-- ; / ~ / /

/ J J/o"

50C / o

s'n~AIN (X 10-6)

4 0 0

3O0

200

I00

50"

Z 3 4 5 6 LOG N 0

/ 4 F 2 - - Z o /

3"/0 4F2- -1

/ o /

o~ °~ o / /

o/ o/

/ / ~,4 / / o

,~ ° O ¢1 I I l i : - -

I Z 3 4 5 6 LOG N

(a) (b)

STRAI N (X IO-6 )

))ooc

STRAIN (X I 0 - 6 )

70(]

400

200

5G:

400

300-

200

I00-

0 /

J / o J J

oy--

4 F 3 - - Z

4 F 3 - - 1

L o Z

~ o

/

o ~ z 3 4 5 6 t o o . o z a 4 LOa N

(c) (d)

Fig. 5. Comparisons between the regression analysis formula and the empirical formulae of Okamura. The solid lines indicate the empirical formulae of Okamura, and the dotted lines indicate the regression analysis formula. (a) Specimen 4F1-3, 4, (b) specimen 4F2-1, 2, (c)

specimen 4F3-1, 2, (d) specimen 6F4-I.


STRAIN ( X I 0 - 6 )

SO0

400"

300-

ZOO"

I 0 0

STRAI N ( X l 0 - 6 ) 2 / /

/ /

?

,-!

/

5 0 0

30 0 % / 0 , '

2001/ I 00 [

I I I I { I I ' ! t i t

0 I Z ~3 4 5 6 LOG N 0 I 2 3 4 5 6 LOG N

(e) (f) Fig. 5~contd. (e) Specimen 6F4-2, (f) specimen 6F4-3.

Figure 6 shows the relationship between stress range observed and that calculated by the empirical formulae of Okamura. Generally, the observed values had shorter ranges than the calculated ones. This is from Okamura's assumption that the residual strain is 0 after some load cycles. In Fig. 6, the observed value was expressed in terms of the average shear strain. 1°'15

TABLE 5 A Comparison Between the Regression Analysis Formula and the Empirical

Formulae of Okamura

Specimens Empirical formulae of Okamura Regression analysis formula

a b a b

4F l - ! 0 134 --- - - 4F1-2 0 134 104 34 4F1-3 229 128 271 121 4F1-4 457 128 476 137 4F3-1 370-5 38"9 515"63 56"73 4F3-2 651 19"8 544.2 96'6 6F4-1 155'5 81"6 83"6 98-2 6F4-2 174'9 70-8 127"4 113'9 6F4-3 272' 1 81"6 219"0 70"3

Fatigue strength behavior o f reinforced concrete structures 69

ESR.MEAS I X I 0 - 6 )

/

500 /

/ 400

/ 0 " 6 F 4 - I 3 0 ( /

[3" " 6 F 4 - 2

LX " 6 F 4 - 3 20 ( a e &

o A 0 El A • i o3 IOO" ta O I I • " I 0 4

: ', I ~ I I I - -

I o o 2(3o 30o 4 0 0 s o o 600 700 F . m . c ~ Ix 1 0 - 6 )

The relationships between stress range observed and that calculated by the empirical formulae of Okamura.

o

Fig. 6.

3.3 Fatigue strength of tension bars

The fat igue stress range o f de fo rmed bars was ob ta ined f rom the tension bars which were f rac tured dur ing the fat igue test. The fol lowing regression analysis fo rmula f rom Tab le 6 is ob ta ined

log O'er = 4"288 - 0"142 log N

3.4 Relationships between the number of repeated loading cycles and mid-span deflection

With the increase o f the n u m b e r o f repea ted cycles, mid-span deflect ion was increased as shown in Fig. 7.

TABLE 6 Beams in which Occurred Fracture of Tension Bars

Specimens tr' c Pmin Pm,x N, O'mi n ~rm~ Fracture mode (MPa) (kN) (kN) (MPa) (MPa)

4F1-2 22.9 39,2 166.7 1 110400 14-7 2 7 9 . 9 Fatigue fracture 4F3-1 24.0 39,2 196.1 386 100 14'7 329-2 of tension 4F3-2 24.0 39,2 186.3 407 600 14.7 312-8 bars

ar,~,, trine,, = stress of tension bars by linear elastic theory.


~ A D I T )

I I

9

5

4

1 0 0 , 0 0 0 CYCLE

3, 16 0 CYCLE 8 1 0 s O 0 0 CYCLE

I s'i" CYCLE / / ' /~ , f h / // / /i/,'z,ooo~ooc~cLE

/ ' I,'/," 1,'t / / I' / ' 1/11

I ; l ' 1" I,"1! _ _ : , . c . ~ . ~ , . ~ /," 11 /,' 1.11; /, ' l' l" l.'l" - - - : O E O . . . , . o /,, / / / , , / , / ,

O. 5 I. 0 Z.O Z .5 3.0

DEFLECTION ( MM )

(a)

LO'AO

LOAD

LOAD (T )

16

13

Fig. 7.

I 0

T

4

The relationships between repeated loading cycles and (a) Specimen 4F2-1, (b) Specimen 4F2-2.

~o ° ~oo, l,ooo, ooo CYCLE I ST CYCLE , , , -' , / / / / /

/,," / , / , , / ~,~o, ooo CYCLE / / / ;~,/" / / /,D,,'~ / '

/ / / ' / / / " / / / / ' / , ' t '

~ l ( t ~ , ~ ,._#~ ! ~ 1 1 ; : ; ; : / ' , , , , : : , , ; , , ; ; . : , : .

0 . 5 1.0 1.5 Z.O Z.5 3 .0 3 . 5 DEFLECTIONI MM )

(b)

mid-span deflections.

3.5 S - N curve

From the fatigue test, the relationships between fatigue life and strength could be expressed exponentially as follows:

R = Kexp ( - Clog N)

where coefficients K and C are determined by the test. The above formula can be rewritten as follows:

log R + C log e log N = log K where

l o g K = Y l o g R = ~ C l o g e = / 3


LOll) IT)

ZO

f7

14

1 0

I0 CYCLE 3~160CYCLE

,s, c,,c,, d. ~ 7 S..~o,ooo<-.<,=

S" / ' / x, "# / ' / " / / / / / / t I I / , /y/ ,

0 0.5 1.0 1,5 Z.O Z. 5 3.0 3.5 4,0 DEFLECTION(MM)

(c)

LOAD ( T )

19

15

I0

7

4

0

IO CYCLE I00 CYCLE I O~O00c't'CLE

IST CYCLE'C~//~///~ / / / ' 3Z31OOOCYCLE

,y

O. 5 I . . 0 I . 5 2 . 0 2 . 5 3.0 3.5 4 . 0

DEFLECTION (MM) (d)

Fig. 7--contd. (c) Specimen 4F3-1, (d) specimen 4F3-2.

then the above equation can be rewritten as:

Y= ~ + f l logN

From the test which was conducted in order to observe the change of stirrup strain, the relationship between fatigue life and strength of the member could be expressed as shown in Fig. 8 and by regression analysis the following formula was obtained:

R = 182.35 - 19"561 logN


o~/ ~,/ .~/ ~o,~/

' :/?/ )7 / / j /,'/ ~ /,'; 13 I l l / l / i /tll/i I

, / / i / / / / #

0 . 5 1 I. 5 2 2 . 5 3 3..~ 4

(e)

t I 5 5.5

DEFLECTIO N (MM)

LOAD (T)

1.5.25

13

I0

!

/, 7

Q.5 I 1 . 5 Z Z. 5

(f)

2// ,/'// ? /

3 3 . 5 4 4 . 5 5 DE FLECT~N(MM)

Fig. G--contd. (e) Specimen 6F4-1, (f) specimen 6F4-2.

1 8 . 5

16

13

I 0

%

L O ~ ( T )

0 . 5 I 1.5 Z Z. 5 3 5 . 5 4 4 .5 5 5 . 5 DEFLECTION |MM)

(g)

Fig. 7--contd. (g) Specimen 6F4-3.

,____

lOG z. 0

~ :~ 50 . u- U o c _

i,i a.

Fig. 8.

X 4 F I SERIES D 4 F Z , 3 SERIES

• 6 F 4 SERIES

,X •

4 5 6 7 @ LOG N

The relat ionship between fatigue life and s trength of the member.


From the above formula, the fatigue strength against 1 000 000 repeated loading cycles was 65% of the static ultimate strength. This value is very similar to the results of previous studies.

4 CONCLUSIONS

In general it must be stated that the behavior of R/C beams under fatigue loading is complicated, especially since each member shows to some extent a different reaction against fatigue loading owing to the non-homogeneity of the member itself. More specifically it was found that:

(1) Stirrup strain is closely related to diagonal crack formation, not to bending crack formation.

(2) With increase in the number of repeated cycles, the average stirrup strain was proportional to log N in the shear span. It follows that the shear strength of R/C members should be calculated considering the effects of the stirrups.

(3) The fatigue strength of the R/C beam at 1-million repeated loading cycles is about 65% of the static ultimate strength.

As a result of the present study, it is thought that the following further research is needed:

(1) More experimental values are necessary to define more precisely the S - N curve; close observations about stirrup strain and fatigue fracture of tension bars are also needed.

(2) A study of the redistribution of stress is required which has occurred in the fatigue problem.

(3) As an approach to the fracture mechanics of R/C members, the theoretical study of crack, fracture toughness and notch, etc., is needed. This would help to establish R/C fracture mechanics.

REFERENCES

1. American Association of State Highway and Transportation Officials, Specifications, Washington, USA, 1977.

2. Beardmore, P., Fatigue response of structural composites. Fatigue 84, Vol. H. Engineering Material Advisory Service Ltd, Warley, UK, 1984, pp. 1091-102.

3. Chang, T. S. & Kesler, C. E., Fatigue behavior of reinforced concrete beams. J. ACL 55 (Aug. 1958) 245-54.

4. Chang, T. S. & Kesler, C. E., Static and fatigue strength in shear of beams with tensile reinforcement. J. ACI, 54 (June, 1958) 1033-58.


5. Ferguson, P. M., Some implications of recent diagonal tension tests. J. ACI, 53 (Aug. 1956) 157-72.

6. Hassoun, M. N., Design of Reinforced Concrete Structures. PWS Publishers, 1985, pp. 217-31.

7. Hawkins, N. M., Fatigue characteristics in bond and shear of reinforced concrete beams. J. ACI, 71 (April 1974) 221-2.

8. Higai, T., Fundamental study on shear failure of reinforced concrete beams. Proc. JSCE, 279 (Nov. 1978) 113-26.

9. Hsu, T. C., Fatigue of plain concrete. J. ACI, 78 (July-August 1981) 292-305. 10. Matsumoto, Y., Nakamura, S., Kohno, K., Nakayama, N. & Okamura, H.,

Flexural behaviour of reinforced concrete beams with high strength deformed bars. Proc. JSCE, 122 (Oct. 1965) 1-28.

11. Nordby, G. M., Fatigue of concrete--a review of research. J. ACI, 55 (Aug. 1958) 191-219.

12. Ohta, A., Sasaki, E., Kosuge, M. & Nishijima, S., Significance of residual stresses in fatigue crack propagation behavior of welded joints, 4th International Conference on Structural Safety and Reliability, Vol. II, 1985. pp. 351-60.

13. Okamura, H. & Farghaly, S. A., Shear design of reinforced concrete beams for static and moving loads. Proc. JSCE, 287 (July 1979) 127-36.

14. Okamura, H. & Higai, T., Proposed design equation for shear strength of reinforced concrete beams without web reinforcement. Proc. JSCE. 300 (Aug. 1980) 131-41.

15. Okamura, H., Farghaly, S. A. & Ueda, T., Behaviors of reinforced concrete beams with stirrups failing in shear under fatigue loading. Proc. JSCE, 308 (April 1981) 109-22.

16. Park, R. & Paulay, T., Reinforced Concrete Structures. John Wiley, New York, 1975, pp. 254-68, 285-91,394-7.

17. Shah, S. P. & Chandra, S., Fracture of concrete subjected to cyclic and sustained loading. J. ACI, 67(10} (Oct. 1970) 816-25.

18. Taylor, R., Chang, T. S. & Kesler, C. E., Fatigue behavior of reinforced concrete beams. J, ACI, 56 (March 1959) 157-72.

19. Thaman, P. P., Reinforced Concrete Structural Elements. McGraw-Hill, New York, 1984, pp. 115-19.

20. The Minister of Construction in Korea, Concrete Standard Specifications, Seoul, Korea, 1984.

21. Wang, C. K. & Salmon, C. G., Reinforced Concrete Design. 4th edn. Harper & Row Publishers, New York, 1985, pp. 120-30, 170-6.

A study on the fatigue strength behavior of reinforced concrete structures

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