Journal of Engineering Sciences, Assiut University, Vol. 35, No. 2, pp. 323-335 , March 2007
A MATHEMATICAL EXPRESSION FOR SPLIT TENSILE
STRENGTH OF STEEL FIBER REINFORCED CONCRETE
Mazen A. Musmar1 and Muhammad I. Rjoub2
Civil Engineering Department, Faculty of Engineering Technology, Al-Balqa'
Applied University. Amman 11134, P.O. Box 15008, Jordan
1
Email: [email protected]
(Received November 11, 2006 Accepted February 5, 2007)
Use of fiber reinforced concrete is increasing recently, brought forth by
the enhancements of concrete matrix engineering properties upon adding
fiber. The resulting material possesses higher tensile strength,
consolidated response and better ductility, which ultimately minimize
immature failures.
Thus it is beneficial to comprehend more the engineering properties of the
material. Accordingly, this study moves toward deriving a mathematical
expression that relates fiber reinforced concrete compressive strength to
split cylinder tensile strength.
Literature survey is carried out to collect data, pertinent to split cylinder
tensile strength versus compressive strength values, for fiber reinforced
concrete. Gathered data comprise compressive strengths from 20 MPa to
102 MPa.
Regression analysis is performed; a mathematical expression that predicts
split cylinder tensile strength of steel fiber reinforced concrete is
eventually derived. The predicted values fit well with experimental data.
The findings of this study shed more light on the tensile strength of fiber
reinforced concrete.
KEYWORDS: Steel Fiber Reinforced concrete, composite concrete
1- INTRODUCTION
The addition of steel fibers to concrete mixes improves the properties of the
obtained steel fiber reinforced concrete (SFRC). Symptoms of the improved
properties comprise tensile strength, shear strength as well as ductility and
toughness of the SFRC.
Such enhancements render using steel fibers attractive especially in high
strength concrete. The addition of steel fibers also reduces the concrete
brittleness, which increases with higher concrete strengths. Such improvements
in concrete properties support the use of SFRC in many structural applications
such as airports, highway paving layers, industrial floorings, and bridge decks.
Let alone its use as a repairing material.
323
Mazen A. Musmar and Muhammad I. Rjoub
324
Utilizing steel fibers as a technique for improving the properties of
concrete has been the issue of many studies on structural behavior of such
material, steered to acquire better understanding of the behavior of SFRC both
in linear and nonlinear stages, and to attain detailed authenticated provisions for
the design of such materials.
Few studies have been targeted towards investigating the relationship
between the split tensile strength and the compressive strength of SFRC. The
available relationships are either based on limited number of specimens or
narrow range of fiber content or fiber aspect ratio. Ashour et al. [1] suggested
the following equation for high strength concrete specimens of a single aspect
ratio, l/d of 75
(1)
f sp  4.95  2.13 v f
where v f is fiber content.
More parameters are presented within the expression addressed by Ashour
et al. [2], as follows:
f sp 
where f cuf
f sp
f cuf
(20  F )
 0.7  F
(2)
is the cube strength of fiber reinforced concrete [MPa].
is the splitting strength of fiber reinforced concrete [MPa].
is fiber reinforcement index = (l/d). v f .Df , where l and d are
steel fiber length and diameter respectively. Df is a bond factor
that equals 0.5 for round fibers, 0.75 for crimped fibers, and 1.0
for indented fibers.
In order to derive a credible mathematical relationship, based on a wide
range of fiber contents ( v f ), compressive strengths, and fiber aspect ratios (l/d),
this study is carried out; it embodies a large number of experimental data
collected from literature. The independent predictors comprise specimens of
different sizes, various concrete grades and aggregate types in addition to
F
multiple values of the fiber reinforcement index (v f *
l
).
d
2- RESEARCH SIGNIFICANCE
The importance of the study is that it employs a large number of experimental
data of SFRC, based on test specimens that cover a variety of factors of
significant effect on the SFRC split strength. The main goal is to present a
reliable mathematical expression that relates SFRC split strength with concrete
cylindrical compressive strength and fiber reinforcement index. It is hoped that
A MATHEMATICAL EXPRESSION FOR SPLIT TENSILE….
325
such expression could be ultimately employed in formulating appropriate
design provisions for fibrous concrete members, under shear, torsion and
tension.
3- DATA ANALYSIS
An experimental data pertinent to values of compressive strength f ' c , splitting
tensile strength, f sp , volumetric fiber content, v f , and fiber aspect ratio, l/ d of
358 SFRC cylindrical specimens are listed in Table (3). These data are gathered
from several research papers, (Batson [3], Craig et al. [4], Sharma [5], Robert
and Victor [6], El-Niema [7], Ashour et al. [2], Ashour and Wafa [8], Ghosheh
[9], Padmarajaiah [10], Marar and Celik [11], Kwak [12], Ayish [13], BaniYasin [14], Rjoub and Rasheed [15], and Aqaileh [16] ). Gathered data contain
compressive strength values from 20.65 MPa to 102 MPa. Also data include
concrete without fibers and concrete with fiber reinforcement indices. All the
compressive strength values presented in Table (3) are either for cylinders of
standard dimensions (150x300mm) or converted to standard cylindrical
strengths using conversion factors presented in Table (2). Regression analysis is
carried out with the split strength, f sp as the predicted dependent variable. The
scatter plot of experimental values of f sp versus f ' c indicates that the expected
relation could take the general expression
l
f sp      ((v f  )  ) 
d
f 'c
(3)
Parameters that are statistically insignificant are discarded, and the model
coefficients are eventually determined. The values of calculated regression
coefficients (,  and ) are found to be (0.614, 0.4 and 1.029) respectively.
Ultimately, the mathematical expression that predicts split cylinder tensile
strength of fiber reinforced concrete f sp is concluded as follows
l
f sp  (0.614  0.4 (v f  )1.029 ) 
d
f 'c
(4)
The P-values for the coefficients of regression analysis (,  and ) are
illustrated in Table (1).Their values are less than 0.001. Such low p-values
indicate that the predictor has a significant effect on the response variable. Thus
the predictors illustrated in Eq. (4) are statically significant, and have a definite
significant effect on the predicted values. Also, the adjusted coefficient of
determination, R2 is 0.840, implying that the regression predicted values are
acceptably close to the observed data.
Mazen A. Musmar and Muhammad I. Rjoub
326
f c'
Equation (4) can be normalized by dividing its two sides by the term
as follows
f sp
l
 (0.614  0.4 (v f  )1.029 )
d
f 'c
(5)
Equation (5) could be further simplified as follows
f sp
f 'c
 (0.6  0.4  (% FRI ))
(6)
Where FRI = v f .l/d
Figure (1) illustrates the scatter plot of %FRI versus the experimental split
strength divided by f c' , for the data listed in Table (3). The plot illustrates an
upper and lower bounds derived by regression analysis. Figure (2) also
illustrates the experimental split strength values versus the predicted values
according to Eq. (7). It indicates that the predicted values are close to test result
values. The plot of the data in both figures (Figs. 1 and 2) confirms the
reliability of the derived expression. Equation (6) may be written in the
following form
l
f sp  (0.6  0.4 (v f  ) ) 
d
f 'c
(7)
Table 1: Estimated parameters using regression analysis
P value
0.614
β = 0. 4
 = 1.029
< 0.001
< 0.001
< 0.001
R2 = 0.841
adjusted R2 = 0.840
Table 2: Conversion factors to standard cylindrical strength [15]
Cylinders
7.5x150 mm
0.95
Cylinders
100x200 mm
0.97
cylinders
150x300 mm
1
Cubes
100x100x100mm
0.78
Cubes
150x150x150mm
0.8
Cubes
200x200x200mm
0.83
A MATHEMATICAL EXPRESSION FOR SPLIT TENSILE….
327
Table 3: Compressive strength, fiber reinforcement index and split cylinder strength
Marar and Celic [11]
(Compression, splitting, Cylinders 150x300)
no
FRI
f'c (Mpa)
fsp
(MPa)
no
FRI
f'c (Mpa)
fsp (MPa)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
0
30
60
75
90
105
120
37.5
75
93.75
112.5
131.25
150
41.5
83
103.75
124.5
145.25
166
32.06
32.66
34.11
36.28
37.46
39.27
39.85
33.73
34.63
36.61
38.31
39.63
41.17
33.99
35.26
37.09
39.73
41.27
42.87
3.2
3.93
4.72
5.35
5.9
6.1
6.84
4.12
5.24
6.18
6.53
7.15
7.87
4.36
5.94
6.54
7.07
7.86
8.33
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
0
30
60
75
90
105
120
37.5
75
93.75
112.5
131.25
150
41.5
83
103.75
124.5
145.25
166
73.5
76.02
78.48
80.09
84.63
86.22
88.97
76.96
78.85
84.48
87.4
89.52
91.49
78.02
80.95
86.21
89.19
91.73
93.56
5.13
5.68
6.95
8.26
8.93
9.97
10.83
6.94
8.14
9.12
10.03
11.16
11.74
7.51
8.89
10.71
11.5
12.54
13.16
Craig et al [4]
no
FRI
f'c (Mpa)
fsp
(MPa)
no
FRI
f'c (Mpa)
fsp (MPa)
39
40
41
42
0.00
42.00
100.00
90.00
40.69
40.00
43.45
35.86
3.45
5.72
6.34
5.31
43
44
45
46
120.00
200.00
120.00
160.00
28.97
47.59
40.00
45.52
4.55
6.00
6.07
7.10
no
FRI
f'c (Mpa)
fsp (MPa)
51
52
53
72
67.5
67.5
48.6
47.7
43.2
7.16
6.96
6.62
Sharma [5]
no
FRI
f'c (Mpa)
47
48
49
50
0
0
0
0
42.3
43.2
47.7
46.8
fsp
(MPa)
4.55
4.6
4.83
4.79
Mazen A. Musmar and Muhammad I. Rjoub
328
Batson [3]
no
FRI
54
55
56
57
58
59
60
61
62
44
44
44
44
30.8
30.8
30.8
30.8
61.6
f'c
(Mpa)
40.19
40.19
40.19
40.19
40.19
40.19
40.19
40.19
39.71
fsp
(MPa)
5.71
5.71
5.71
5.71
5.71
5.71
5.71
5.71
6.18
no
FRI
f'c (Mpa)
fsp (MPa)
63
64
65
66
67
68
69
70
61.6
61.6
61.6
61.6
61.6
61.6
61.6
61.6
39.71
39.71
39.71
39.71
39.71
39.71
39.71
39.71
6.18
6.18
6.18
6.18
6.18
6.18
6.18
6.18
no
FRI
f'c (Mpa)
fsp (Mpa)
78
79
80
81
82
83
75
56.25
93.75
37.5
75
0
42.67
40.47
40.85
40.47
40.11
41.42
5.69
6.62
6.11
7.17
5.51
6.7
FRI
0.00
0.00
0.00
0.00
0.00
0.00
30.00
30.00
30.00
60.00
60.00
60.00
75.00
75.00
75.00
f'c (Mpa)
51.90
49.64
52.13
53.80
52.94
52.89
54.25
55.54
54.72
55.72
55.85
56.80
55.26
55.88
55.58
fsp (Mpa)
5.66
5.23
5.64
5.73
5.86
5.94
6.44
6.12
6.13
7.42
7.28
7.29
7.65
7.55
7.48
Ghosheh [9]
no
FRI
f'c (Mpa)
fsp
(MPa)
71
72
73
74
75
76
77
0
0
0
26.6
35
70
37.5
42.49
41.9
41.9
42.49
39.7
41.42
40.11
4.56
4.53
4.53
5.4
5.49
6.7
5.24
Bani-Yasin [14]
no
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
FRI
0.00
0.00
0.00
0.00
0.00
0.00
30.00
30.00
30.00
60.00
60.00
60.00
75.00
75.00
75.00
f'c (Mpa)
23.83
23.57
23.27
24.20
23.75
23.82
25.83
24.66
24.90
27.39
26.41
25.99
26.68
25.63
26.19
fsp
(Mpa)
2.62
2.49
2.54
2.49
2.46
2.51
2.86
3.02
2.97
3.44
3.15
3.01
3.88
3.64
3.58
no
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
A MATHEMATICAL EXPRESSION FOR SPLIT TENSILE….
329
Ayish[13]
no
FRI
f'c (Mpa)
fsp (Mpa)
no
FRI
f'c (Mpa)
fsp (Mpa)
114
115
116
117
118
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
0
30
60
60
0
60
0
30
60
60
0
60
30
30
30
60
60
60
30
30
30
20.1
21.37
22.753
22.91
20.65
22.15
32.76
34.48
35.72
36.43
32.60
35.96
21.73
22.07
21.55
22.33
22.08
22.16
22.72
22.32
22.25
3.1
3.23
3.67
3.53
3.14
4.08
3.84
4.15
4.66
4.36
3.74
4.64
2.48
2.31
2.47
2.49
2.46
2.51
2.6
2.66
2.54
119
120
121
122
123
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
60
60
60
90
90
90
30
30
30
60
60
60
30
30
30
60
60
60
90
90
90
22.78
24.65
23.02
24.65
25.45
24.73
53.52
52.03
53.66
51.74
52.74
53.35
54.52
53.66
54.29
54.85
56.68
54.14
56.64
56.03
56.72
2.94
2.76
2.79
3.58
3.63
3.68
5.96
6.08
5.91
5.73
5.86
5.94
6.26
6.57
6.45
6.89
7.3
7.13
8.56
8.34
8.63
Kwak [12]
no
FRI
f'c (Mpa)
fsp (Mpa)
no
FRI
f'c (Mpa)
fsp (Mpa)
156
157
0.00
31.25
60.72
61.89
4.32
5.88
158
159
46.88
31.25
66.54
29.88
6.08
3.83
Craig [4]
no
FRI
f'c (Mpa)
fsp (Mpa)
no
FRI
f'c (Mpa)
fsp (Mpa)
160
161
162
163
0
42
100
90
40.69
40.00
43.45
35.86
3.45
5.72
6.34
5.31
164
165
166
167
120
200
120
160
28.97
47.59
40.00
45.52
4.55
6.00
6.07
7.10
El-Neima [7] (Comp, splitting, cylinders 150x300mm)
no
168
169
FRI
0.00
51.08
f'c (Mpa)
22.34
26.19
fsp (Mpa)
1.96
4.50
no
186
187
FRI
25.00
25.00
f'c (Mpa)
61.70
39.90
fsp (Mpa)
6.39
5.14
Mazen A. Musmar and Muhammad I. Rjoub
330
(cont) El-Neima [7]
no
FRI
f'c (Mpa)
fsp (MPa)
no
FRI
f'c (Mpa)
fsp (MPa)
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
89.39
127.70
38.30
67.03
95.75
25.35
44.37
63.38
25.00
25.00
66.50
133.00
50.00
100.00
150.00
200.00
28.57
29.73
24.62
25.24
25.38
23.79
24.76
25.17
61.70
39.90
61.70
67.20
59.30
60.00
67.00
55.90
4.60
4.74
3.60
3.88
4.07
3.12
3.64
4.01
6.39
5.14
7.88
10.70
7.14
8.95
11.32
12.04
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
66.50
133.00
25.00
25.00
66.50
150.00
200.00
250.00
300.00
66.50
100.00
66.50
66.50
133.00
150.00
61.70
67.20
61.70
39.20
61.70
76.70
79.50
77.20
75.80
42.30
41.40
55.70
42.30
71.90
67.00
7.88
10.70
6.39
5.09
7.88
12.11
14.36
16.14
17.98
6.52
7.43
7.48
6.52
11.07
11.32
Robert [6]
no
FRI
f'c (Mpa)
fsp (MPa)
no
FRI
f'c (Mpa)
fsp (MPa)
203
204
205
0.00
14.25
28.50
54.15
55.96
59.47
2.90
5.60
6.00
206
207
208
42.75
57.00
57.00
56.72
54.15
51.40
5.90
5.60
6.20
Rjoub and Rasheed [15]
no
FRI
f'c (Mpa)
fsp (MPa)
no
FRI
f'c (Mpa)
fsp (MPa)
209
210
211
212
213
214
215
216
217
218
219
220
0
0
0
0
0
40
40
40
40
40
60
60
55.22
63.71
71.06
80.87
91.85
59.76
65.65
74.69
84.84
94.78
62.51
76.71
4.96
5.11
5.71
6.34
6.62
5.18
5.26
5.91
6.74
6.77
7.13
7.8
284
285
286
287
288
289
290
291
292
293
294
295
0
0
0
0
0
40
40
40
40
40
60
60
59.76
65.65
74.69
84.84
94.78
63.07
78.66
88.3
93.94
97.06
65.21
80.42
5.18
5.26
5.91
6.74
6.77
6.1
6.79
7.36
8.81
8.95
7.32
7.95
A MATHEMATICAL EXPRESSION FOR SPLIT TENSILE….
331
(cont) Rjoub and Rasheed [15]
no
FRI
f'c (Mpa)
fsp (MPa)
no
FRI
f'c (Mpa)
fsp (MPa)
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
60
60
60
80
80
80
80
80
120
120
120
120
120
0
0
0
0
0
40
40
40
40
40
60
60
60
60
60
80
80
80
80
80
120
120
120
120
120
0
0
85.62
92.98
97.03
64.73
78.91
88.01
94.67
99.21
66.81
80.82
91
96.73
100.18
65.25
70.21
79.51
89.27
98.92
66.23
79.42
90.79
94.13
99.94
67.37
82.66
91.89
95.62
99.98
69.71
84.31
93.76
96.2
100.12
71.23
85.21
94.78
98.71
101.3
48.74
58.81
8.62
9.64
9.83
8.12
8.93
9.69
10.96
11.47
9.22
9.98
10.73
11.68
12.27
5.28
5.47
6.12
6.58
6.93
6.32
6.97
7.92
8.92
9.02
7.8
8.37
9.2
9.95
10.31
8.81
9.42
10.72
11.63
11.92
9.85
10.84
11.93
12.78
13.08
3.8
4.01
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
60
60
60
80
80
80
80
80
120
120
120
120
120
0
0
0
0
0
40
40
40
40
40
60
60
60
60
60
80
80
80
80
80
120
120
120
120
120
0
0
90.17
94.97
98.11
67.31
81.14
92.93
96.62
99.98
69.08
82.29
94.17
96.35
100.2
51.22
60.85
69.39
80.11
86.48
54.71
67.91
79.31
89.76
93.62
55.84
69.42
81.61
90.31
94.28
57.88
73.07
82.60
92.61
95.31
59.12
74.24
84.63
93.11
96.72
57.87
64.08
8.9
9.75
9.92
8.41
9.15
10.04
11.1
11.5
9.71
10.74
11.23
12.04
12.63
3.95
4.1
4.42
4.75
4.98
4.94
5.94
6.28
6.63
6.91
6.41
7.04
7.68
8.2
8.57
7.23
7.92
8.45
9.57
9.92
8.21
9.00
9.64
10.36
10.74
4.08
4.25
Mazen A. Musmar and Muhammad I. Rjoub
332
(cont) Rjoub and Rasheed [15]
No
FRI
f'c (Mpa)
fsp (MPa)
No
FRI
f'c (Mpa)
fsp (MPa)
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
0
0
0
40
40
40
40
40
60
60
60
60
60
80
80
80
80
80
120
120
120
120
120
67.31
74.92
80.77
52.63
61.27
74.67
82.89
91.07
54.01
62.89
76.63
84.62
92.86
55.22
64.73
78.98
87.03
93.91
56.77
66.82
80.11
88.18
94.22
4.34
4.63
4.81
4.68
4.97
5.51
6.15
6.26
6.05
6.48
7.08
7.91
8.08
6.83
7.44
8.12
8.97
9.2
7.68
7.94
8.72
4.98
10.3
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
0
120
120
120
0
0
0
40
60
60
60
60
60
80
80
80
80
80
120
120
120
120
120
75.31
84.42
93.64
52.63
61.72
74.67
82.89
91.07
54.01
62.89
76.63
84.62
92.82
52.2
64.73
78.98
87.03
93.91
56.77
66.82
80.11
88.18
94.22
4.63
4.82
4.15
4.68
4.97
5.51
6.15
6.26
6.05
6.48
7.08
7.91
8.08
6.83
7.44
8.12
8.97
9.2
7.68
7.94
8.72
4.98
10.33
l
d
Note: FRI = (v f * ) s
4- CONCLUSIONS
The following conclusions can be drawn from this study
1- A suggested expression that predicts the split cylinder tensile strength of
steel fiber reinforced concrete is presented.
2- The outcomes of descriptive statistical analysis confirm the credibility
of the derived expression.
3- Concrete compressive strength, fiber content and the fiber aspect ratio
are the major effectual parameters in specifying the tensile strength of
fiber concrete.
A MATHEMATICAL EXPRESSION FOR SPLIT TENSILE….
333
2.5
f sp/(f'c) 0.5
2
Upper boundary
1.5
1
Lower boundary
0.5
Test result values
0
0
0.5
1
1.5
2
2.5
3
3.5
Hundreds
Fiber reinforcement Index,% FRI
Figure 1: Effect of steel fiber reinforcement index, % FRI versus fsp/(f'c)0.5
18.00
Predicted fsp (MPa)
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
0.00
5.00
10.00
15.00
20.00
Experimental fsp (MPa)
Figure 2: Experimental versus predicted split strength.
334
Mazen A. Musmar and Muhammad I. Rjoub
REFERENCES
[1] Ashour S. A. , Hasanian G. S. and Wafa, F. F." Flexural Behavior of High
Strength Fiber Reinforced Concrete Beams". ACI Structural Journal, V90,
No. 3 May-June 1993, pp279-287.
[2]
Ashour S. A. , Hasanian G. S. and Wafa, F. F."Shear Behavior of HighStrength Fiber Reinforced Concrete Beams", ACI Structural Journal,
March-April 1992, V. 89, No. 2, pp 176- 184.
[3]
Batson, G. Jenkins, E. and Spatney, R. "Steel Fibers as Shear
Reinforcement in Beams", ACI Journal, Oct 1972.
[4]
Craig, R. J. Parr, J. A. Germain, E. Mosquera, V. Kamilares, S. "Fiber
Reinforced Beams in Torsion", ACI Journal, Nov-Dec 1986.
[5] Sharma, A. K., "Shear Strength of Steel Fiber Reinforced Concrete Beams,
ACI Journal V. 83, No. 4, 1986, pp. 624-628.
[6] Robert, J. Ward. and Victor C. Li. " Dependence of Flexural Behavior of
Fiber Reinforced Mortar on Material Fracture Resistance and Beam Size",
ACI Materials Journal, Nov-Dec 1990, V87, No.6, pp627-637.
[7] EL-Niema, E. I. " Reinforced Concrete Beams with steel Fibers under
Shear", ACI Structural Journal, March-April 1991, V. 88, No. 2, pp 178183.
[8] Ashour S. A. , Hasanian G. S. and Wafa, F. F." Flexural Behavior of High
Strength Fiber Reinforced Concrete Beams". ACI Structural Journal, V90,
No. 3 May-June 1993, pp279-287
[9]
Ghosheh, N. "Reinforced Concrete Beams with steel Fibers ", M. Sc.
Thesis, Jordan University, 1999.
[10] Padmarajaiah, S. K.and Ramaswamy, A. " Crack-Width Prediction for
High-Strength Concrete Fully and Partially Prestressed Beam Specimens
Containing Stee Fibers", ACI Structural Journal, November-December
2001, V. 98, no.6, pp852-861
[11] Marar K.and Celik, T. "The Influence of FRI on the Relationship Between
Compressive and Tensile Strength of NSFRC and HSFRC" The sixth
International Conference on Concrete Technology, Amman-Jordan, Oct.
2002.
[12] Kwak, Y. K. Eberhard, M. O. Kim, W. S. and Kim, J. "Shear Strength of
Steel Fiber-Reinforced Concrete Beams without Stirrups", ACI Structural
Journal, July-August 2002, V.99, No. 4, pp530-538.
‫‪335‬‬
‫‪A MATHEMATICAL EXPRESSION FOR SPLIT TENSILE….‬‬
‫‪[13] Ayish, M. "Punching Shear Behavior of Flat Plates with Fiber Reinforced‬‬
‫‪Concrete", M. Sc. Thesis, June 2004, Jordan University of Sceince and‬‬
‫‪Technology.‬‬
‫‪Bani-Yasin, I. S. "Performance of High Strength Fibrous Concrete slab‬‬
‫‪concrete connections under gravity and lateral loads" M. Sc. thesis, Jordan‬‬
‫‪University of Sceince and technology", June, 2004.‬‬
‫]‪[14‬‬
‫‪[15] Rjoub, M. I. M., Rasheed, T.M. "Shear Strength of Steel Fiber High‬‬
‫‪Strength Concrete Beams", Seventh International Conference on Concrete‬‬
‫‪Technology, Oct. 2004.‬‬
‫‪[16] Aqaileh, Z. A. "The structural Behavior of High Strength Normal Two‬‬
‫‪Way Ribbed Slabs with Steel fibers", M. Sc. Thesis, Jordan University,‬‬
‫‪2004.‬‬
‫حساب مقاومة شد اانفاق للخرسانة باألياف‬
‫أص بباستخد بباألخاتخا م ببيةتخ لخلسم بببتاي أ د ببيسبتاا خلم ببلتلد ببال تك بببتخ حسب ب تخ ببي ب تحت ب ب تاد ببا تا دب ب ت‬
‫أصببي اتخ أ دببيسبتإسببلتاأببيكبتخا مببيةتخ لخلسمبببختحتالا ب تخ أ دببيسبتايا مببيةتخ لخلسمبببتلةيحلبببتأإ ب ت‬
‫ا جهببيلخ تخ ‪،‬ببل تاي أببيكبتا ب تخدبباجيابتحتلايسبببتأكأببمتحتا ب تا د ب تكبببتخ لل ح مبببتتللببيتم ب ل تكبببت‬
‫خ سهيمببتا ب تدب ح تحتخدباجيابتأكأببمتإسبلتخ اخب‬
‫ت ابى‪.‬م تخ ةبحأختحتإ مببضتأأب‬
‫تلب تخ أب ح تخ اخب ةت‬
‫إ ب تخ أصببي اتخ هسلدببمبتت أ دببيسبتايا مببيةتخ لخلسمبببختحتإ مببضتااجببضتإ ب دتخ ل خدبببتا ب تامجببيلتسلببح ت‬
‫ميأبتم اتإاسبتلةيحلبتخ أ ديسبت ‪،‬لتايدباألخاتك باتخاسقبامتالةيحلببتخ أ دبيسبت أبب ختحتاسبي ت‬
‫إ ت‬
‫تااتل خدبتاميسي تااأل تلةيحلبتخ أ ديسبت ‪،‬لتحتخ ةماتخ لسيظ ةت هيت لةيحلبتخ أ ديسبت أبب خت‬
‫‪2‬ت‬
‫حتسلت‪،‬ل تخ ل خدبتإ تأ ديسبتاإجهيلت د تل ت‪26.02‬تسمحا ‪/‬لا تتا تت‪262‬تسمحا ‪/‬لا ختحاباتإلبمت‬
‫‪2‬‬
‫ا مببمت ا ص ببي مبتت يكبببتخ اميس ببي تحتا بباتخد بباساي تإاسبببت ميأ ببمبتتااسببيحمتإاس بببتلةيحلبببتخ أ د ببيسبت ‪ ،‬ببلت‬
‫ايداألخاتك اتخاسقامتالةيحلبتخ أ ديسبت أب ت سقستخ أ ي خ ت‬
‫ت‬
‫خ اي ثتخاحم‪:‬تخ ل اح تليل تإ بتلدلي ‪:‬ت مبتخ هسلدبتخ ا سح حجمب\تسداتخ هسلدبتخ للسمب\تجيلخبتخ ا ةي ت‬
‫خ ا امةمب\تإلي ت\ت‪2231‬تاخ ختت‪22661‬ت\تخا ل‬
‫خ اي ببثتخ ‪.‬ببيسب‪:‬تخ ببل اح تل لببلتل ب م تخ جببح ‪:‬ت مبببتخ هسلدبببتخ ا سح حجمببب\تسدبباتخ هسلدبببتخ للسمببب\تجيلخبببت‬
‫خ ا ةي تخ ا امةمب\تإلي ت\ت‪2231‬تاخ ختت‪22661‬ت\تخا ل‬