meets the fatigue requirements of structural reinforcement for bridge decks. ... test results were made for U.S. WWF, German WWF, and conventional.
W E L D E D W I R E FABRIC FOR B R I D G E S . Ih FATIGUE STRENGTH By Bilal M. Ayyub, ~ Member, ASCE, Peter C. Chang,2 Member, ASCE, and Naji A. AI-Mutairi,3 Associate Member, ASCE ABSTRACT" The purpose of the present paper is to provide experimental data and
analytical results for the fatigue strength of welded wire fabric (WWF), so that engineers can confidently use WWF as structural reinforcement. The material for the tested specimens was provided by manufacturers in the United States, Germany, and Canada. The specimens included steel from rods (steel before the cold-drawing process), plain wires, deformed wires, epoxy-coated deformed wires, tempered and nontempered WWF, and nontempered wires without welds. The experiments conducted show that the WWF made in the United States can be considered a ductile steel based on the relevant ASTM specifications. In addition, the fatigue life of the epoxy-coated WWF made in the United States exceeds those for reinforcement bars as tested by the Portland Cement Association. The results indicate that WWF meets the fatigue requirements of structural reinforcement for bridge decks. INTRODUCTION
Welded wire fabric ( W W F ) consists of p r e f a b r i c a t e d sheets with parallel longitudinal wires that are welded at regular intervals to transverse wires (ASTM A185 and A497). T h e wielding process is based on the electrical resistance and p r o p e r t i e s of the wires. T h e W W F studied in the present paper is different from the lighter wire-mesh styles traditionally used to control thermal and drying shrinkage of concrete, in that the wires used in this W W F are normally larger in diameter. These newer fabrics are referred to as W W F in the present paper. W W F has higher yield, ultimate tensile strength, and bonding strength than commonly used steel reinforcing bars (rebars) in current practice. In addition to these benefits, the use of W W F has been shown to reduce construction time and labor. The p r e f a b r i c a t e d nature of W W F is believed to reduce construction errors and facilitate quality control in the field. (Bern o l d e t al. 1989). To understand the strength p r o p e r t i e s of W W F , an experimental p r o g r a m was conducted. In the present p a p e r , fatigue tests of W W F are described and discussed. In the fatigue tests, the effects of tempering, e p o x y coating, and welded intersections of W W F were studied. Comparisons of the fatiguetest results were m a d e for U.S. W W F , G e r m a n W W F , and conventional reinforcement. In A y y u b et al. (1994), the tensile strength, chemical-composition analyses, and ductility tests of W W F are described to establish its stress-strain and ductility characteristics. 1Prof. of Civ. Engrg., Univ. of Maryland, College Park, MD 20742. 2Assoc. Prof. of Civ. Engrg., Univ. of Maryland, College Park, MD. 3Dept. Head, Kuwait Inst. for Scientific Res., Engrg. Div., Civ. & Build. Dept. Safat 13109, Kuwait. Note. Discussion open until November 1, 1994. Separate discussions should be submitted for the individual papers in this symposium. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on July 2, 1992. This paper is part of the Journal of Structural Engineering, Vol. 120, No. 6, June, 1994. 9 ISSN 0733-9445/94/0006-1882/$2.00 + $.25 per page. Paper No. 2856. 1882
FATIGUE STRENGTH
Factors Affecting Fatigue Strength To understand the different factors affecting the fatigue strength of WWF, previous research on WWF and conventional reinforcement bars was reviewed. The main factors include stress range, minimum stress, bar diameter, strength or grade of steel, bar geometry, chemical composition, surface deformations, residual stresses, bar coatings, welds at intersections, wire penetration, welding process, and tempering. In this section, the main factors that affect the fatigue strength of reinforcing bars are discussed. The stress range to which a specimen is subjected is the primary factor in determining its fatigue life. The fatigue limit is defined as the stress range in which the reinforcement would have a long fatigue life and can sustain a virtually unlimited number of stress cycles. The stress range is considered the predominant factor influencing fatigue strength in the finite-life region (Burton and Hognestad 1967; Martin and Schiessl 1982a, 1982b; Pfister and Hognestad 1964; Hanson et al. 1974; Helgason et al. 1976), but its effect on fatigue life greatly diminishes in the long-life region (Pfister and Hognestad 1974; Ihamb and MacGregor 1972). Although some investigators (Martin and Schiessl 1982a, 1982b; Rehm 1960; Fisher and Viest 1961) minimize the effect of minimum stress level on the fatigue strength, other observations from test data (Pfister and Hognestad 1964; Hanson et al. 1974) show that a decrease in fatigue strength is due to an increase in the tensile minimum stress, and an increase in the fatigue strength is related to an increase in the compressive minimum stress. Bar diameter was determined to have a nonlinear effect on the fatigue strength (Helgason et al. 1976; Weiman 1969; Osgood 1970; Kravshenko 1964; Tetelman and McEvily 1967; MacGregor et al. 1971; Kokubu and Okamura 1969). This effect was indicated by an increase in fatigue strength with a decrease in the diameter of the bar. In the finite-life region, the effect of decreasing the bar diameter was indicated by a shift in the S-N diagram towards an increase in fatigue strength (Helgason et al. 1976). Two different opinions were found in the reviewed literature regarding the effect of steel grade for conventional reinforcing bars on fatigue strength. Some investigators (MacGregor et al. 1971; Kokubu and Okamura 1969; Ihamb and MacGregor 1974) concluded that steel grade has only a slight or no effect on the fatigue strength, but others (Pfister and Hognestad !964; Helgason et al. 1976; Lash 1969; Gronqvist 1971) reported an increase in fatigue strength for higher steel grades, for example, grades 60 and higher. The geometry of conventional reinforcing bars results in stress concentration; however, it was found to have the least effort on the finite-life fatigue strength (Helgason et al. 1976). On the other hand, fatigue studies on identical bars (Pfister and Hognestad 1964;. Gronqvist 1971), except for their transverse deformation patterns, showed a large difference in fatigue strength. Due to the complex state of stress at the transverse deformation, the severity of the stress concentration is not well understood (Helgason et al. 1976). Another important factor that causes a considerable reduction in fatigue strength is the stress concentration at the welds of the welded joints (the welded intersections) in WWF (Burton and Hognestad 1967; Sanders et al. 1961). The fatigue life for hot-formed welds were found to be less than half of those for cold-formed welds (Hawkins and Heaton 1971). In addition, the fatigue characteristics of a plain WWF are dictated by the characteristics 1883
of the welded intersections, which in turn are dictated by geometrical stress concentrations caused by the welding process. Current ASTM specifications for WWF contain a requirement on weld shear strength, the strength against shear failure between the longitudinal and transverse wires. This strength is inversely proportional to the depth of weld penetration. Therefore, increasing the shear-weld strength of WWF can reduce the fatigue strength. Since shear-weld strength is of secondary importance in structural design, only the fatigue strength of wires with and without welds are studied in the present paper. In summary, the predominant factor affecting the fatigue life is the stress range; hence, it is the main test variable used in the present fatigue study. Other factors discussed include wire welding process, wire penetration, welds at intersection, and tempering. These factors are unique to WWF.
Experimental Program To achieve an understanding of the fatigue characteristics of WWF, U,S. epoxy-coated WWF, tempered and nontempered German mesh, and nontempered German bars were tested. The WWF specimens were cut from a D12 • D12 wire size. The shapes of these specimens are shown in Fig. 1, and a sample of the specimen dimensions for the U.S. epoxy-coated WWF is shown in Table 1. Based on the dimensions of specimens, the average values of penetration and percent reduction in diameter are 2.51 mm (0.0987 in.) and 12.5%, respectively. The coefficients of variations for both penetration and percent reduction in diameter is 0.15. The fatigue test was conducted on a total of 42 specimens. These tests included 12 specimens for each of the tempered German WWF and U.S. epoxy-coated wire fabric. The tests of these specimens were conducted at four stress ranges, namely 50, 40, 30, and 20 ksi (345, 276, 207, and 138 MPa). In addition, nine specimens for each of the nontempered German WWF and nontempered German bars (bars without weld) were conducted.
B
A
!
_1
nl
B
t
E
A
p
A
'_L_
I-
(a)
C
(b)
-!
(c)
FIG. 1. Specimens Usedin Fatigue Test: (a) U.S. Epoxy-Coated WWF; (b) German Tempered and Nontempered WWF; (c) German Nontempered Bars (A ~ 170.2 mm; B -~ 91.4 mm)
1884
TABLE 1.
Fatigue-Test Specimen Dimensions for U.S. Epoxy-Coated WWF
Length (in.) Calculated Thickness Specimen Weight A b B~ Total diameter at weldd no. a (g) (mm) (mm) (mm) (mm) (mm) (1) (2) (3) (4) (5) (6) (7) FL-1 FL-2 FL-3 FL-4 FL-5 FL-6 FL-7 FL-8 FL-9 FL-10 FL-11 FL-12
167.53 189.11 78.7 i267.8 163.31 188.8 73.7 262.4 164.41 187.7 77.7 265.4 158.60 188.5 66.8 255.3 166.90 186.8 80.8 267.6 179.39 193.2 92.3 285.4 160.56 184.3 73.3 1257.6 163.10 179.8 75.5 261.2 162.79 185.3 73.9 259.2 166.15 186.7 78.2 264.9 167.11 188.1 78.8 267.0 164.74 189.6 73.7 263.3
10.1 10.0 10.0 10.0 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.1
17.9 17.1 17.1 17.9 17.8 17.3 17.7 17.6 17.3 17.7 18.4 17.6
Penetra-
(8)
Percent reduction in diameterf (9)
2.26 2.99 2.92 2.14 2.30 2.84 2.43 2.57 2.88 2.44 1.73 2.59
11.21 14.90 14.59 10.64 11.46 14.06 12.07 12.77 14.28 12.08 8.59 12.86
tione
(mm)
Note: 1 g = (1/453.6) lb; 1 in. = 25.4 ram. "FL = Fatigue for epoxy-coated local (U.S.) WWF. bLongitudinal wire. cUpper transverse wire. dThickness at the weld is the average of four readings. ePenetration is the difference between the actual diameter of the bars before welding. fPercent reduction in diameter = [Penetration/(Transverse + Longitudinal) Diameter] x 100.
The stress ranges were 50, 40, and 20 ksi (345,276, and 138 MPa) for the former, and 50, 40, and 30 ksi (345,276, and 207 MPa) for the latter. The minimum load was held constant at one kip (4.448 KN), and, for the different specimens, the corresponding minimum stress was calculated, which depended on the cross-sectional area of the individual specimens. The maximum stress was varied depending on both the minimum stress and the stress range. Three tests were conducted at each stress range. Run out for this investigation was taken as at least 10,000,000 cycles. A typical fatigue-test result is shown in Table 2 for the U.S. epoxy-coated W W F . Similar tables were constructed for the t e m p e r e d and n o n t e m p e r e d German meshes and the n o n t e m p e r e d G e r m a n bars. In this test, the failure of a specimen is defined as reaching two completely s e p a r a t e d parts. The number of cycles to failure, the frequency used, and the position of fatiguecrack initiation were recorded as shown in Table 2.
Statistical Analysis of Fatigue Data No matter how carefully controlled a fatigue test is, a large variability in the results is expected. F o r this reason, a statistical approach to fatiguedata analysis can produce the largest possible amount of information with associated confidence levels. The fatigue specimens in this constant-load-cycle fatigue test were subjected to r e p e a t e d applications of a constant stress range Sn with a specified minimum stress, Stain. The n u m b e r of cycles to failure N is recorded as shown in Table 2. The resulting data can be presented in an S-N curve, as 1885
TABLE 2. Fatigue-Test Results for U.S. Epoxy-Coated WWF Specimen Diameter no.a (mm) (1) (2) FL-I FL-2 FL-3 FL-4 FL-5 FL-6 FL-7 FL-8 FL-9 FL-10 FL-11 FL-12
10.07 10.04 10_02 10.04 10.05 10.09 10.05 10.06 10.09 10.08 10.07 10.07
Area (mm 2) (3)
Minimum stress (MPa) (4)
79.61 79.23 78.84 79.10 79.35 80.00 79.35 79.48 80.00 79.87 79.68 79.68
55.8 56.1 56.4 56.3 56.1 55.6 56.1 56.0 55.6 55.7 55.8 55.8
Stress range Frequency (MPa) (Hz) (5) (6) 345 345 345 276 276 276 207 207 207 138 138 138
15 15 15 15 15 15 15 15 15 15 15 15
Cycles to failure (7)
Position ~ of fracture
114,500 119,100 157,310 325,620 381,240 343,430 1,288,250 1,857,950 1,537,010 10,000,000 5,728,460 10,000,000
W-L W-L PP-L W-U W-L W-U W-L G-U G-U RO G-U RO
(8)
aFL = Fatigue for local (U.S.) epoxy-coated WWF. bW = Weld-crack started at weld; PP = Pressure-point crack started at point where pressure was applied in electrical welding process; RO = Run out--specimen reached at least 10,000,000 without failure, so test was stopped; U = Fractured above weld intersection (upper); L = Fractured below weld intersection (lower); and G = Grip failure. Note: 1 in. = 25.4 ram; 1 in. 2 = 645.2 mm2; 1 ksi = 6.895 N/ram 2 = 6.895 MPa. shown in Fig. 2. T h e m e a n equation:
S-N
logN
line can b e r e p r e s e n t e d by the f o l l o w i n g = b -
mlogS
(1)
where b = i n t e r c e p t for the stress r a n g e ; and m = s l o p e of the S-N line. The values of b a n d m and the v a r i a n c e , which is a m e a s u r e o f dispersion from the m e a n S-N line, are s h o w n in T a b l e 3. Statistically, t h e variability of data can be d e s c r i b e d by a c o n t i n u o u s p r o b a b i l i t y density f u n c t i o n fN(n). By integrating the density f u n c t i o n , t h e distribution f u n c t i o n FN(n) (Wirsching and Y a o 1970) can be f o u n d . T h e r e f o r e , t h e s u r v i v o r s h i p f u n c t i o n Lu(n) can be d e t e r m i n e d as
LN(n) = 1 -- FN(n)
(2)
w h e r e N = r a n d o m v a r i a b l e d e n o t i n g fatigue life; and n = specific v a l u e of N. For the analysis of fatigue data, t h e W e i b u l l distribution is a widely u s e d m a t h e m a t i c a l m o d e l (Wirsching and Y a o 1970). It is also r e f e r r e d to as t h e third asymptotic distribution of t h e smallest e x t r e m e values. By substituting the c u m u l a t i v e W e i b u l l distribution into (2), the s u r v i v o r s h i p f u n c t i o n becomes
LN(n)
= exp
[ - ( n Y ~s] \ v,/ _1
w h e r e ~Xs = shape p a r a m e t e r o r W e i b u l l slope at stress level S; and scale p a r a m e t e r o r characteristic life at stress level S.
1886
(3) Vs =
60
-400
50
",,
N -300
4O .r v
gr "
~,~i\,
30
t~
\ '~,' .% ~-~ X
fl:
v
-200
t-
\
C o n v e ' r d i o n a , R e ",n, .
rrt~
"i"\,,-N...... \ . _%__..
r
(5
13.
.
2O Coated Local
. . . . . .
Epoxy
...........
Tempered German
. . . . . .
~,
Non-tempered German
WSM
Non-tempered German
Bars
.......
10 0.01
)
,
o~
WSM
WSM
.......
I
Category
B
-100
........
I
0.1 1.0 10.0 20.0 N, Number of C y c l e s (Million) FIG. 2. S-N Curve for U.S. Epoxy-Coated W W F T A B L E 3.
Regression Analysis of Fatigue Data of W W F
Regression Coefficient
(1)
Intercept (b) (2)
Slope (m) (3)
German tempered WWF German nontempered WWF German nontempered bars U.S. epoxy-coated WWF
11.077 11.285 10.084 12.746
-3.702 -3.814 -2.787 -4.486
Specimen type
Variance
(4) 2.052 1.216 4.187 3.220
x x • x
10 3 10.3 10.3 ]0 .3
In the Weibull distribution, the probability density function can assume various shapes depending on the shape factor as used. This shape is generally skewed resulting in a mean (or expected value) and a median that do not coincide. For the estimation of the shape factor e~s and the scale parameter V s , the method of moments can be used. Therefore, estimates of the mean Ix and the standard deviation ~ (denoted as Ix* and or*) can be determined from the data as follows:
Ix* = ~
n,
(4)
i=1
E*(N2) = ~
ni i=1
1887
(5)
TABLE 4. Estimates of Weibull Distribution Parameters and Fatigue Life Using Method of Moments for U,S. Epoxy-Coated WWF Stress range (MPa) (1)
i~* (cycles) (2)
E*(N 2)
r~*
• 108 (3)
(cycles) (4)
n at LN (cycles) 0.50 (9)
V~ •
COW (5)
c~* (6)
104 (7)
0.95 (8)
0.05 (10)
i
345 276 207
130,320 I73.5 19,175 0.1471 7.514 350,097 123.1 23,191 0.0662 13.528 1,561,070 24,913.2 233,201 0.1494 7.426
14.0 I 94,300 133,300' 162,000 38.0 305,100 369,900 412,100 170.0 1,139,600 1,618,100 1,970,700
Note: 1 ksi = 6.895 N/ram 2 = 6.895 MPa.
4(1t3
Probability .300
=~ 20o
r~
100
i , ,fJf.l
10
100
,
i
, ~r~11,l r i i*~flf@ 1,000 10,000 20,000
,
[ If@lIT(
10
Number of Cycles (Thousands)
i
1 11~f,l
100
1.000
*
, ,,,,-~
I0.000
2O,OOO
Number of Cycles (Thousands)
(a)
(b)
400
3O3
300
200
Q
r~ 00 100
100
10
lOO
1,000
10,000 20,000
I 10
Number of Cycles (Thousands)
i fltlHI 100
~ T ~$15111 1,000
q , ,,,t,d 10,OO(] 2 0 , 0 0 0
Number of Cycles (Thousands) (d)
(c)
FIG. 3. S-N-P Survival Curve: (a) German Tempered WWF; (b) German Nontempered WWF; (c) U.S. Epoxy-Coated WWF; (d) German Nontempered Bars (Bars without Weld) a * = x/-E-W(N 2) where k = total number of specimens v a r i a t i o n ( C O V ) is e s t i m a t e d a s 1888
(ix*) 2
in t h e g r o u p .
(6) The coefficient of
O-~
COV* = - ix*
(7)
Using (7) as the best estimate of COV, the estimate of the shape factor ~* can be found from the following equation using an iterative procedure:
E F
COY* =
~+1
.
1
(8)
F2 ~-~+1
Then an estimate of the scale parameter V~, can be !ound from Ix*
V~ =
r~+l
,
(9)
)
The estimates of Weibull distribution parameters using the method of moments and the estimate of the fatigue life are shown in Table 4 for the U.S. epoxy-coated WWF as typical results of the analysis. Similar values were found for the other specimens. The fatigue-life predictions using the parameters from the method of moments for the 95%, 50%, and 5% probability of survival are shown in columns 8, 9, and 10 of Table 4, respectively. The S-N-P (stress-cycle-probability) curves are shown in Fig. 3. Performance of WWF in Fatigue The data and observations presented in Table 2 as typical test results indicate that the fatigue crack initiates at the point where the pressure is applied in the electrical welding process in more than half the German tempered WWF. However, in most of the cases of the nontempered German and U.S. epoxy-coated WWF, the crack started in the weld. The effect of penetration on fatigue life should be a significant factor in the study of WWF, since as penetration is increased the fatigue life is expected to decrease. However, from studying the relationship between the penetration and fatigue life, it is noticed that the effect is not consistent especially for low stress ranges. Welded intersections are known to have stress concentrations, and are the most likely points of fatigue-crack initiation. The effect of welding is seen as a parallel shift in the S-N curve, where for the same stress range, the fatigue life decreases with welding. The effect of tempering on fatigue life is found to be a parallel shift of the S-N curve, where, for the same stress range, the fatigue life of a tempered specimen is less than a nontempered one. Thus, the advantage of increased ductility due to tempering is counteracted by a decrease in strength and fatigue life. Category B of the AASHTO specifications ("Standard" 1983) is used for the purp.ose of a comparative illustration. Based on the study of the S-N curves, it was determined that all specimens have a fatigue life in excess of category B of the AASHTO specifications, with the exception of the 50-ksi (345-MPa) stress range of the tempered German WWF. The nontempered German bars (bars without weld) and the U.S. epoxy-coated WWF have larger fatigue strength than conventional reinforcements tested by the Port1889
land Cement Association (MacGregor et al. 1971). However, conventional reinforcement had a longer fatigue life than the tempered and nontempered German WWF. The nontempered German bars (bars without weld) have the longest fatigue life at higher stress ranges and almost the same fatigue life as nontempered German WWF at low stress ranges. Although the U.S. epoxycoated wire fabric has higher penetration than the German steel, it still has a higher fatigue life. This unexpected result is believed to be due to the differences in diameters or cross-sectional areas. Another factor is the different deformation pattern used in the WWF. The German WWF has protruding deformations, whereas the epoxy-coated U.S. WWF has recessed (indented) deformations. Additional discussion and information about these results and conclusions are provided by AI-Mutairi (1989) and A1-Mutairi et al. (1989). CONCLUSIONS
The following conclusions are drawn from the results of the experimental program: 9 All tested specimens have a fatigue life in excess of category B of the A A S H T O specifications for bridges ("Standard" 1983). 9 The fatigue life for the epoxy-coated U.S. WWF exceeds those for conventional reinforcement bars tested by the Portland Cement Association. 9 The fatigue life of WWF is shortened by tempering and welding. 9 The advantage of increased ductility due to tempering is obtained at the expense of lower strength (Ayyub et al. 1994) and shorter fatigue life. 9 The fatigue life for a specimen with a welded intersection is lower than the fatigue life of a specimen without a weld. ~ Because the applied pressure in the electrical welding process is considered a critical factor affecting the fatigue life, a carefully selected and regularly monitored applied pressure should be used. 9 Although the epoxy-coated U.S. wire fabric has a larger weld penetration than the German steel, it has a higher fatigue life. ACKNOWLEDGMENT
This research was sponsored by the Maryland State Highway Administration, under Contract No. AW089-317-046. T h e writers also wish to thank the Kuwait Institute of Scientific Research for the support of a research assistant during the course of this research. APPENDIX I.
REFERENCES
AI-Mutairi, N. (1989). "The structural feasibility of using welded wire fabric in bridge decks," PhD thesis, University of Maryland, College Park, Md. Al-Mutairi, N., Ayyub, B., and Chang, P. (1989). "The feasibility of using welded steel mesh in bridge decks--structural evaluation and testing." Rep. Dept. of Civ. Engrg., Univ. of Maryland, College Park, Md. Ayyub, B. M., Chang, P., and AI-Mutairi, N. M. (1991). "Welded wire fabrics for bridges. I: ultimate strength and ductility." J. Struct. Engrg., ASCE, 120(6), 18661881. 1890
Bernold, L., Chang, P., Ayyub, B. M. (1989). "The feasibility of using welded wire fabric in bridge decks--construction & management assessment." Rep. Dept. of Civ. Engrg., Univ. of Maryland, College Park, Md. Burton, K. T., and Hognestad, E. (1967). "Fatigue test of reinforcing bars--tack welding of stirrups." J. ACI, 64(5), 244-252. Fisher, J. W., and Viest, I. M. (1961). "Fatigue tests of bridge materials of the AASHTO Road Test." HRB Spec. Rep. No. 66, Hwy. Res. Board, National Research Council, Washington, D.C., 132-147. Gronqvist, N. O. (1971). "Fatigue strength of reinforcing bars." ACI Publ. SP-26, American Concrete Institute (ACI), Detroit, Mich., 1011-1059. Hanson, J. M., Somes, N. F., and Helgasou, T. (1974). "Investigation of design factors affecting fatigue strength of reinforcing bars--test program." PCA Res. and Development Bull. RDO36.01D, Portland Cement Association, Skokie, 1|1. Hawkins, N. M., and Heaton, L. W. (1971). "The fatigue properties of welded wire fabric." Rep. SM71-3, Dept. of Civ. Engrg., Univ. of Washington, Seattle, Wash., 1-35. Helgason, T., Hanson, J. M., Somes, N. F., Corley, W. G., and Hognestad, E. (1976). "Fatigue strength of high-yield reinforcing bars." Nat. Cooperative Hwy. Res. Program Rep. 164, NCHRP, Washington, D.C. Ihamb, I. C., and MacGregor, J. G. (1972). "Fatigue of reinforcing bars." Struct. Engrg. Rep. No. 39, University of Alberta, Edmonton, Alberta, Canada. Ihamb, J. C., and MacGregor, J. G. (1974). "Effect of surface characteristics on fatigue strength of reinforcing steel." ACI Publ. SP-41, American Concrete Institute (ACI), Detroit, Mich., 139-167. Kokubu, M., and Okamura, H. (1969). "Fatigue behavior of bigh strength deformed bars in reinforced concrete bridge design." ACI Publ. SP-23, American Concrete Institute (ACI), Detroit, Mich.; 301-316. Kravshenko, P. Y. (1964). Fatigue Resistance, Pergamon Press, Elmsford, N.Y. Lash, S. D. (1969). "Can high-strength reinforcement be used for highway bridges." ACI Publ. SP-23, American Concrete Institute, Detroit, Mich., 283-299. MacGregor, J. G., Ihamb, I. C., and Nutall, N. (1971). "Fatigue strength of hotrolled deformed reinforcing bars." J. ACI, 68(3), 169-179. Martin, H., and Schiessl, P. (1982a). "Finite-life fatigue strength of welded fabric reinforcement." Betonwerk and Fertigteil-Tecknik (p), 47(12), 733-738 (in German). Martin, H., and Schiessl, P. (1982b). "Finite-life fatigue strength of welded fabric reinforcement." Betonwerk and Fertigteil-Tecknik (p), 48(1), 33-38 (in German). Osgood, C. (1970). Fatigue Design, Wiley-lnterscience, New York, N.Y. Pfister, J. F., and Hognestad, E. (1964). "High strength bars as concrete reinforcement, part 6, fatigue tests." J. PCA Res. and Development Lab., 6(1), 65-84. Rehm, G. (1960). "Contributions to the problem of the fatigue strength of steel bars for concrete reinforcement." 6th Congr. Preliminary Publ., International Association for Bridge and Structural Engineers, Stockholm, Sweden, 35-46. Sanders, W. W. Jr., Hoadley, P. G., and Munse, W. H. (1961). "Fatigue behavior of butt-welded reinforcing bars for concrete." Welding J., 40(12), 529-s-535-s. "Standard specifications for highway bridges." (1983). American Association of State Highway and Transportation Officials (AASHTO), 13th Ed., Washington, D.C. Steel welded wire fabric, plain, .for concrete reinforcement; A185-90. (1990). ASTM, Philadelphia, Pa. Steel welded wire fabric, deformed, for concrete reinforcement, A497-90. (1990). ASTM, Philadelphia, Pa. Steel wire, plain, for concrete reinforcement; A82-85. (1985). ASTM, Philadelphia, Pa. Tetelman, A. S., and McEvily, A. (1967). Fracture of structural materials, John Wiley and Sons, New York, N.Y. Weiman, M. H. (1969). "Detail design and manufacturing consideration." Metal fatigue: theory and design, A. F. Madayag, ed., John Wiley and Sons, New York, N.Y. 1891
Wirsching, P. H., and Yao, J. T. P. (1970). "Statistical,methods in structural fatigue." J. Struct. Div., ASCE, 96(6), 1201• A P P E N D I X II.
NOTATIONS
The following symbols are used in this paper: A, B, C = dimensions; b = intercept; C O V = coefficient of variation; COV* = estimated coefficient of variation; FN(n) = probability distribution function; k = total n u m b e r of specimens in the group; LN(n) = survivorship function; m = slope; N = r a n d o m variable denoting fatigue life; n = specific value of N; P = probability; S = stress range; Vs = scale parameter or characteristic life at stress level S; as = shape p a r a m e t e r of Weibull slope at stress level S; Ix* = estimate of the m e a n ; and tr* = estimate of the standard deviation.
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