DELFT UNIVERSITY OF TECHNOLOGY
DEPARTMENT OF CIVIL ENGINEERING
Report 5-85-6
Shear transfer across a single crack in reinforced
concrete under sustained loading
Part II. Appendices
Ir. J.W. Frénay
STEVIN LABORATORY
'^i.
-^:)>
>>^'t
1
-Stevin Laboratory
Department of Civil Engineering
Delft University of Technology
Report 5-85-6
Project 8209
June 1985
Shear transfer across a single crack in reinforced
concrete under sustained loading
Part II - Appendices
by
J.W. Frénay
t
^r
Mail ing address:
Delft University of Technology
Concrete Structures Group
Stevinlaboratory I I
Stevinweg 4
2628 CN D e l f t
The Netherlands
Technische Hogeschool
Of Bibliotheek
Aldelii^: Civiele Techniek
_ o / Stevinweg 1
S /l S^'dS'-ppostbus 5048
2600 GA Delit
^iiyÜ^^g
/
No part of this report may be published without written permission of the
author.
3
-CONTENTS
I. Mix proportions
II. Standard tests
III. Results of regression analysis for f; and
IV. Instrumentation
V. Accuracy of displacement measurements
VI. Correction of measured displacements
VII. Data of sustained tests and push-off tests
VIII. Results of regression analysis
4
-APPENDIX I. Mix proportions
Mix code B1632550 strength f' =51N/mm2
—
.
^ cc
(mix A)
Components [kg/m^l Sieve analysis of aggregate
Sieve opening
[mm] [ k g ]
sand 877.2
gravel 1065.0
cement-B 325.0
water 162.5
2429.7
8 - 16 623.7
4 - 8 441.3
2 - 4 312.1
1 - 2 220.9
0.5 - 1 156.2
0.25 - 0.5 110.3
0.10 - 0.25 77.7
1942.2 1
Mix code B1642037.5 strength f' =70N/mm2
(mix B)
sand
gravel
cement-B
water
superpl .2J%
857.3
1018.5
420.0
147.0
10.5
2453.3
8 - 16
4 - 8
2 - 4
1 - 2
0.5 - 1.0
0.25 - 0.50
0.10 - 0.25
596.5
421.9
298.3
212.0
148.6
105.0
93.5
1875.8 1
Sieve analysis of aggregate
[cum.X]
mix A
100.0
67.9
45.2
29.1
17.7
9.7
4.0
mix B
• 100.0
68.2
45.7
29.8
18.5
10.6
5.0
Fuller
100.0
70.7
50.0
35.4
25.0
17.7
12.5
sieve opening [mm]
8 - 1 6
4 - 8
2 - 4
1 - 2
0 . 5 - 1
0.25 - 0.5
0.1 - 0.25 1
5
-For a good mix the Netherlands concrete code recommends a minimum
quantity of fiiie material D < 250 '^m.
For a maximum particle size of 16mm at least 140 liters/m' of concrete
is specified.
0.10.25 051.0 2.0 40 8.0 16.0
Actual v a l u e s :
mix A : 3 2 5 / 3 . 1 + 7 7 . 7 / 2 . 5 5 = 134.2 1/m^
mix B . : 4 2 0 / 3 . 1 + 9 3 . 5 / 2 . 6 5 = 170.8 1/m^
6
-APPENDIX I I Standard t e s t s .
Tests on 150mm cubes
Mix
B
A
A
B
B
B
A
A
A
B
A
B
A
A
B
B
B
batch no.
and date
10
(250583)
11
(060683)
12
(200683)
14
(040783)
21
(220883)
23
(050983)
25
(190983)
27
(031083)
29A
(171083)
29B
(171083)
31
(311083)
34
(211183)
36
(051283)
37A
(121283)
378
(121283)
38
(191283)
41B
(090184)
age to
[days]
28
58
84
119
28
58
86
119
28
58
87
119
28
58
87
119
28
28
28
28
28
28
28
28
28
28
28
10
28
39
56
9
28
cc
number
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
moressive
f^^ [N/mm^J
70.77
76.67
77.85
74.37
53.20
60.15
61.67
65.01
53.69
54.96
60.95
62.74
68.67
80.90
75.82
79.56
67.30
73.79
56.66
48.38
53.47
73.22
47.60
73.58
49.98
49.47
70.34
59.19
67.15
70.62
69.88
57.91
68.63
v.c.[%]
4.3
4.9
3.8
5.0
3.2
6.9
3.6
3.5
4.2
8.7
4.4
3.3
3.7
0.7
5.2
6.8
4.9
7.6
6.5
4.9
1.1
2.3
4.7
3.8
2.8
3.3
2.4
5.7
12.3
13.4
7.2
7.9
5.1
ten
number
3
3
3
3
3
3
3
3
3
1
3
3
3
3
3
-s i l e S D l i t t i n a
^cspl t^/™^^]
4.04
3.59
3.28
3.89
T3.93
3.90
3.47
3.22
3.07
4.33
2.90
4.61
3.10
3.28
4.03
-v . c . [ . . ]
5.8
6.3
3.6
9.0
4.5
4.6
1.2
4.5
12.3
-6.5
0.3
7.3
5.5
2.6
7
-150mni cubes (continued)
A
A
A
B
B
41A
(090184)
47
(020484)
48
(090484)
49
(160484)
50
(240484)
9
21
28
39
29
7
28
27
5
5
5
5
6
6
6
6
43.55
45.79
52.64
51.65
51.07
38.83
69.24
67.05
2.6
8.8
3.7
7.8
3.2
2.0
3.8
5.3
-3
3
3
-2.30
3.42
3.66
-2.8
9.2
4.4
^150x400mm cylinders
A
B
12
(200683)
14
(040783)
28
58
87
149
28
56
87
135
2
2
2
2
2
2
2
2
34.75
38.54
39.98
44.57
53.96
64.54
57.64
65.08
5.8
9.3
10.7
4.4
5.0
0.9
15.1
4.4
150x150x600mm prisms
B
A
A
B
10
(250583)
11
(060683)
12
(200683)
64
(040783)
709
697
683
669
2
2
1
2
68.65
42.40
37.24
62.16
0.8
4.7
-3.9
ISOnim
batch
1 no.
10
11
12
14
21
23
25
27
29A
29B
31
34
36
37A
37B
Cubes
age to
[days]
28
58
84
119
28
58
86
119
28
58
87
119
28
58
87
119
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
67
72
74
71
51
58
59
63
51
50
58
60
67
80
71
73
60
68
63
76
51
56
45
48
52
53
71
73
43
49.
73.
71
50.
52.
46.
49.
70.
70.
.69
.62
.44
.64
.33
.89
.11
.56
.11
.84*
.53
.93
.11
.22
.51
96
84
.89
96
22
87
04
38
76
62
69
33
78
47
78
16
78
04
13
62
29
18
67
measured
val
70
77
79
72
53
59
62
63
54
53
60
62
67
81
76
79
66
68
70
78
54
59
46
50
53
53
71
73
47
49
69
76
48.
48.
50.
5 1 .
68.
68.
ues
.89
.38
.38
.84
.69
.82
.93
.87
.71
.87
.44
.27
.33
.16
.67
,96
.98
98
67
13
76
47
62
18
07
69
91
78
96
11
47
54
98
18
31
11
80
09
f ' [N/mm ]
ee73
80
79
78
54
61
62
67
55
60
63
65
71
81
79
84
68
69
75
78
55
62
47
51
53
54
72
75
47
48
73
76
50
49
50
48
72
72
.73
.00
.73
.62*
.58
.73
.98
.60*
.24
.18
.87
.02
.56
.33*
29
76*
62
47
16
62
51
31
56
82
38
36
58
96
02
27
69
84
80
73
58
93
04
27 j
150mm Cubes (continued]
38
41B
41A
47
48
49
60
10
10
28
28
39
39
56
56
9
9
28
28
9
9
21
21
28
28
39
39
29
29
7
7
28
28
27
27
61.42
59.02
52.62*
73.24
55.69*
79.07*
73.42
73.78
59.56
58.13
63.78
71.29
43.56
42.67
50.13
44.84*
49.91
54.04
56.13
47.295!^
49.24
53.28
39.87
39.42
65.42
69.42
69.33
69.82
59.91
53.56*
70.62
69.38
70.67
78.36
71.56
69.20
60.40
61.42
65.96
71.02
44.09
42.27
39.42*
47.42
51.51
53.02
47.47*
54.31
51.73
50.36
39.16
38.36
72.31
72.04
64.18
71.07
62.04
-69.87
-69.38
-61.42*
-50.04*
-71.11
-45.16*
-47.11
-54.71
-53.07
-49.56
52.13
37.73
38.44
68.49
6 7.78
65.91
62.03 1
0150x400mm cylinders
12
14
28
58°
87
149
28
56
87°
135
36.16
35.99*
36.95*
45.95
55.85
64.96*
51.50*
67.11
33.33
41.08
43.01
43.18
52.06
64.11
63.78
63.04
-_ 1
-1
150x150x600mm
1 10 708
11 697
12 683
14 669
prisms
69.02
40.98
37.24
60.44
68.27
43.82
-63.87
1
-eliminated to f i n d f ' ( t ) ace. to (3.10)
deformation c o n t r o l l e d .
150mm Cubes
9
-batch no.
10
11
12
14
21
23
25
27
29A
29B
31
34
36
37A
37B
48
49
50
age to
[days]
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
7
28
27
measured
^cspl
4.43
3.78
3.33
3.15
3.49
3.69
3.71
3.06
2.83
4.33
2.70
4.63
3.24
3.27
4.08
2.27
3.73
3.50
4.54
4.10
3.71
3.31
4.05
3.80
3.94
3.26
2.87
-2.94
4.60
3.22
3.20
3.91
2.27
3.20
3.65
values
[N/mni^]
4.54
4.24
3.73
3.38
4.13
4.05
4.06
3.34
3.50
-3.07
4.61
2.84
3.46
4.20
2.25
3.41
3.83
The next two plots show measured mean cube strengths approximated to a normal distri'
bution.
038 097 0S5 090 OflO 0.70 0.60 0.50 0.40 030 020 010 005 003 0.02 probability p(f>(cclh
V
A
^
\ D
\ o
Y
^
i 1 Mix A cast nr number O 11 3 A 12 3 D i l A 5 f j . ^ = 5308N/mm' v.t » 34%\
0.99 0.98 0.97 095 Q90 080 0.70 050 0.40 030 020 0.10 002 probobüity p l f ï f c c '\
\
^
.
\
\
\
\P
\
I Mix B 1 cost nr number [ O 10 3 A 14 3 0 3 8 4 V41B 5 •ccm' 69.64N/mm ' V.C.. 4.0% [| Q^
\
\
\
\
^
45 50 55 60 65 60 65 70 75 BO f' IN/mmM I'lN/mm']-11
APPENDIX I I I
Results of regression analysis f o r f ' and
•^ cc
cs
The regression coefficients a and 3 are given in accordance with
f' .t
f' (t )= - ^ ^ °
cc 0
CH. + Q.t
( I I I . 1 )
in which t^ is the age in days. The values of da and d3 represent h a l f of the
90%-confidence i n t e r v a l s of a and 3 r e s p e c t i v e l y .
150mm cubes
mix
batchno.
number
of
measur.
f'
cc
[N/mm^]
f '
cc
[N/mm^]
d f '
cc
d."
11
11
11
12
12
12
41A
41A
41A
12
11
10
12
11
10
20
18
17
11+12+41A 44
11+12+41A 40
11+12+41A 37
53.20
53.20
53.20
53.69
53.69
53.69
52.64
52.64
52.12
53.08
53.08
53.08
6.298
6.576
6.227
5.000
5.100
5.172
2.211
2.191
2.098
3.681
3.610
3.785
0.777
0.767
0.778
0.839
0.826
0.816
0.976
0.968
0.979
0.868
0.852
0.847
0.88
0.92
0.94
0.49
0.68
0.81
0.45
0.66
0.67
0.66
0.83
0.86
53.10 2.98 1.344
53.09 2.51 1.151
53.19 2.22 1.061
52.75
53.27
53.66
4 9 . 8 8
50.28
49.45
53.10
54.13
54.05
6.46 2.916
4.74 2.122
•3.77 1.664
7.64 1.032
5.07 0.692
4.75 0.671
9.74 0.709
5.07 0.455
4 . 3 8 0.434
0.028
0.025
0.024
0.061
0.045
0.036
0.066
0.046
0.045
0.033
0.021
0.020
10
10
10
14
14
14
38
38
38
41B
41B
41B
10 - 4 1 8
10 - 4 1 8
10 -.418
12
11
10
12
11
10
20
18
17
10
9
54
50
45
70.77
70.77
70.77
68.67
68.67
68.67
67.15
70.78
70.78
68.63
68.62
68.63
68.58
69.64
69.54
2.580
2.527
3.072
4.684
5.170
5.965
2.142
465
539
2.513
1.917
2.046
2.853
2.804
2.533
0.900
0.906
0.890
0.820
0.804
0.780
0.925
0.946
0.939
0.912
0.934
0.929
0.885
0.883
0.895
0.30
0.34
0.49
0.53
0.66
0.81
0.31
0.52
0.66
0.63
0.78
0.79
,57
7R0.80
71.33
71.00
70.81
71.65
67.85
69.19
67.06
68.45
68.77
68.49
68.49
68.49
69.50
70.84
70.65
6.82 2 . 2 4 ;
6.46 2.155
5.92 2.078
7.60 2.543
6.67 2.285
5.33 1.922
16.03 1.302
10.45 1.044
8.21 0.827
11.74 1.279
6.20 0.734
6.62 0.847
10.35 0.578
6.25 0.363
5.26 0.326
0.048
0.046
0.045
0.054
0.050
0.044
0.072
0.060
0.049
0.106
0.058
0.062
0.029
0.018
0.016
gl50x400inm cyl inders (continued)
A
B .
12
14
8
7
6
8
7
6
34.75
34.75
34.75
53.96
53.96
53.96
6.833
7.558
7.728
4.779
6.281
6.210
0.642
0.742
0.724
0.817
0.763
0.772
0.64
0.78
0.92
0.32
0.81
0.85
34.30
34.33
34.77
54.63
54.64
54.29
5.21
4.56
3.12
11.14
5.53
5.52
4.045
3.611
2.411
5.535
2.753
2.760
0.085
0.079
0.055
0.117
0.061
0.063
12
-Shrinkage deformations
c
have been calculated according to:
cs
-' .
c^^(t^«25 days, t) = ^ (HI.2)
c^3(t^«28 days, t) = -,+6.1n(t)
., . ,
(III.3)
in which
;;,3,Y
and
A
are regression coefficients and t is the duration of
the test in hrs. Test data have been analysed from t=240 hours onwards.
mix
A*
B*
A*
B*
A**
B**
A**
B**
spec.no.
IIA
11B
12A
12B
lOA
lOB
14A
14B
l l A
-
10A-11A
IIB
12A
12B
lOA
10B
14A
14B
11A-
10A-12B
14B
12B
14B
number of
observat.
17
17
17
17
17
17
13
13
20**
20**
17
17
17
17
17
17
13
13
20**
20**
[xlO^]
2.152
2.552
3.436
3.078
2.562
2.482
1.725
2.019
2.791
2.201
-404.5
-445.0
-476.8
-447.5
-448.75
-327.50
-368.50
-394.25
-439.75
-383.50
r^ 3,[xlO-^J
2.459
2.528
2.514
2.609
2.711
3.298
3.031
2.847
2.539
2.954
88.33
91.78
93.08
89.60
91.60
70.23
80.93
93.25
90.18
81.60
r'
0.989
0.993
0.996
0.999
0.989
0.974
0.969
0.992
0.955
0.960
0.974
0.908
0.984
0.997
0.971
0.933
0.973
0.968
0.967
0.958
da
[x10h
0.10
0.10
0.10
0.05
0.13
0.18
0.17
0.10
0.16
0.19
27.50
30.00
35.10
50.00
35.00
32.00
45.00
52.50
67.50
72.50
dB T
[xlO^
0.16 1
0.16
0.12
0.06
0.24
0.34
0.32
0.19
0.12
0.09
5.00
5.00
5.00
7.50
5.00
5.00
7.50
7.50
10.00
10.00
(111.2)
(111.3)
* equation
** equation
13
-s c a n n o .j 1
1 3
3 4 5 6 1 •'7 8 ? 1 0 1 1 1 3 1 3 1-1 I S 1 6 1 7 1 8 i V 2 0 2 1 2 2 2 3 2 4 2 S 2 61 27
2 B 2 9 3 0 3 1 3 2 3 3 3A 3 S 3 6 3 7 ! 3 8 j 3 9 j 4 0 t [ h r s ] 1 , 5 0 0 4 . S U 0 2 0 . S 0 0 4 S . 7 S 0 7 0 , S Ü 0 9 3 . S O 0 1 6 S . S O 0 2 3 6 . 7 S Ü 2 6 0 . S O U 2 6 7 . Ü O Ü 4 0 S , S Ü Ü 4 2 9 , 6 6 0 5 U 1 . 6 6 0 S S 3 , 5 Ü Ü 6 0 2 , S O 0 6 6 8 . 8 3 0 7 1 7 , 0 0 0 7 6 4 , 4 9 0 8 6 2 , 4 9 0 1 0 0 4 . 4 9 0 1 1 0 0 , 4 9 0 1 1 7 2 . 4 9 0 1 2 S 0 . 4 9 0 1 3 9 1 , 2 4 0 I S I O , 9 9 0 1 S 8 0 . 4 9 0 1 7 2 9 , 4 9 0 1 9 4 0 , 9 9 0 2 0 B 4 , 4 9 0 2 2 7 8 , 9 9 0 2 5 8 9 . 2 4 0 2 7 8 1 , 2 4 0 3 0 2 2 . 4 9 0 3 2 9 0 , 9 9 0 3 6 2 0 , 9 9 0 3 9 3 2 , 9 9 0 4 4 1 2 . 9 9 0 5 4 7 1 , 9 9 0 7 9 9 2 . 0 0 0 1 1 Ü 4 O.ÜÜ s p e c . n r . 1 0 . A filjl^ [mm] . 0 0 0 , 0 0 0 , 0 0 3 , 0 0 8 . 0 1 4 . O i S . 0 2 4 . 0 2 9 . 0 3 2 . 0 3 6 . 0 4 2 . 0 4 2 . 0 4 4 . 0 4 9 . O S O . Ü S 2 . 0 5 4 . 0 S 7 . 0 6 7 . 0 7 2 , 0 7 6 . 0 7 8 . 0 7 7 . 0 8 1 . 0 9 4 . 0 9 3 . 0 9 1 . 1 0 4 . 1 0 7 . 1 1 1 . 1 1 4 , 1 1 4 . 1 1 9 . 1 2 3 . 1 2 1 . 1 2 3 . 1 2 3 . 1 2 7 . 1 2 5 , 1 2 8 s p e c . n r . 1 0 . B A ' 5 ^ [mm] 0 . 0 0 0 U . OOO . 0 0 2 . 0 0 8 . 0 1 3 , 0 1 ; : . 0 2 3 . 0 2 8 . 0 3 2 . 0 3 6 . 0 3 9 . 0 4 1 . 0 4 S . 0 4 7 . 0 4 9 . 0 5 0 . 0 5 3 . 0 5 6 . 0 5 4 . 0 5 8 . 0 6 3 . 0 6 ^ . . 0 6 4 . 0 6 7 , 0 7 2 . 0 6 8 . 0 6 7 . 0 8 1 . 0 8 4 . 0 8 V . 0 8 9 . 0 9 1 . 0 9 7 , l O L . 1 0 1 . 1 1 (1 . 1 1 2 . 1 1 5 . 1 1 3 . 1 1 6 I s c a n n o .1 1
1 2
3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2Ü 2 3I 2.;!
2 j i^A Li b 2 6 d/ dU 2 9 3 0 3 1 3 2 3 3 3 4 3 5 3 6 t [ h r s ] s p e c •••SH 3 , 6 7 0 5 . 7 S 0 2 2 . 0 00 2 9 . 0 0 0 9 4 . 2 3 0 1 6 7 . 0 0 0 1 9 1 . 1 6 0 2 6 3 . 1 6 0 3 1 5 . 0 0 0 3 6 4 . 0 0 0 4 3 0 . 3 3 0 4 7 8 . 5 0 0 5 2 S . 9 9 0 6 2 3 . 9 9 0 7 6 5 . 9 9 0 8 6 1 . 9 9 0 9 3 3 . 9 9 0 1 0 1 1 . 9 9 0 1 1 5 2 , 7 4 0 1 2 7 2 , 4 9 0 1 3 4 1 , 9 9 0 1 4 9 0 . 9 9 0 1 7 U 2 . 4 9 0 1 Ü 4 5 . 9 V Ü 2 Ü 4 Ü . 4 9 0 2 3 5 0 . 7 4 0 2 5 4 2 . 7 4 0 2 y 8 3 . 9 9 U 3 0 5 2 . 4 9 0 3 3 8 2 . 4 9 0 3 6 9 4 . 4 9 0 4 1 7 4 . 4 9 0 5 2 3 3 . 4 9 U 7 3 2 0 . 0 0 0 8 1 6 0 . 0 0 0 1 1 2 0 a o o n r . l l A [mm] . 0 0 0 , 0 0 1 , 0 0 5 . 0 0 5 , 0 1 6 . 0 1 9 , 0 2 8 . 0 3 7 . 0 4 1 . Ü 5 2 . 0 5 3 . 0 6 1 , 0 6 2 , 0 6 4 , 0 7 2 , Ü 7 i • . 0 8 1 . 0 8 2 , 0 8 6 . Ü 8 8 . Ü 8 8 , Ü9Ü , 0 9 8 , l ü i . 1 0 5 . 1 1 3 , 1 1 0 , 1 2 1 , 1 3 0 . 1 2 6 , 1 3 0 , 1 3 7 , 1 4 3 . 1 4 3 , 1 4 5 , 1 4 8 s p e c . n r . 1 ! B A I J ^ [mm] . 0 0 1 j . 0 0 3 . 0 0 7 . 0 0 7 . 0 1 6 , 0 2 5 . 0 2 8 , 0 3 3 . 0 3 9 , 0 4 2 . 0 4 4 . 0 5 ; : . 0 5 3 , Ü5i.. . 0 6 2 . 0 6 6 . 0 7 0 , 0 7 1 , 0 7 6 . 0 8 1 . 0 8 0 . ü s ; : : . 0 9 5 . 0 9 8 , 1 0 2 , 1 0 7 , 1 U 9 , 1 1 6 , 1 2 6 , 1 2 1 . 1 2 7 , 1 3 2 , 1 3 7 . l 3 i l . 1 4 0 . 1 4 31
s c a n n o . 1 0 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 I S 1 6 1 7 1 8 1 9 2 0 2 1 1 0 0 2 3 t [ h r s ] 2 3 5 0 1 1 4 1 6 4 21 0 2 8 2 3 6 0 5 0 0 6 2 0 8 3 9 1 O 5 0 1 1 9 4 1 3 8 8 1 6 9 8 1 8 9 0 2 1 3 1 2 4 O 0 2 7 3 0 3 0 4 2 3 5 2 2 4 5 8 1 7 9 9 2 1 1 Ü 4 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 ( 0 00 7 5 0 5 0 0 5 0 0 7 5 0 ÜÜÜ 0 0 0 2 5 0 2 5 0 5 0 0 0 0 0 0 0 0 0 0 0 0 Ü ( 0 0 0 ÜÜO 0.0 0 s p e c . n r . 1 4 A . , I j ^ [mm] . 0 0 3 , 0 1 9 , 0 2 5 . 0 3 3 . 0 3 9 . 0 4 2 , 0 4 6 , 0 5 8 , 0 5 3 , 0 6 1 , 0 7 5 . 0 8 1 , 0 8 3 , 0 8 9 , 0 9 4 . 1 0 1 . 1 0 7 . 1 Ü 8 . 1 1 3 •. 1 1 8 . 1 2 4 , 1 2 5 . 1 2 Ö s p e c . n r . 14 E • • ' s h ' ' ™ l . 0 0 4 1 . 0 0 9 . 0 1 7 . 0 2 7 . 0 3 3 . 0 3 9 . 0 4 8 . 0 5 5 , 0 5 5 . 0 5 0 , 0 7 3 , 0 7 9 , 0 8 5 . 0 9 0, 095 1
. 100 1
. 1 0 7 . 1 0 9 , 1 1 4 . 1 2 0 , 1 2 6 . i 2 y , 1 3 U s c a n n o .1 1
1 2
3 4 5 6 '? S 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 ' j T 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 t [ h r s ] 1 6 . 7 5 0 4 2 . 7 5 0 1 1 2 . 5 9 0 1 3 6 . 5 0 0 1 6 0 . 7 5 0 2 0 8 . 3 5 0 3 0 6 , 2 5 0 4 4 8 . 2 5 0 5 4 4 , 2 5 0 6 1 6 , 2 5 0 6 9 4 , 2 5 0 8 3 5 . 0 0 0 9 5 9 . 7 5 0 1 0 2 4 . 2 5 0 1 1 7 3 . 7 5 0 1 3 8 5 . OOO 1 5 2 3 , 2 5 0 1 7 2 2 . 2 5 0 2 0 3 2 . 5 0 0 2 2 2 4 , 5 0 0 2 4 6 3 . 2 5 0 2 7 3 1 . 7 5 0 3 0 6 1 , 7 5 0 3i7i,7S(l 3 8 5 3 . 7 5 0 4 9 1 2 , 7 5 0 6 6 6 8 . 7 5 0 7 3 2 0 . 0 0 0 8 1 6 0 . 0 0 0 1 1 2 0 8,00 s p e c . n r . 1 2 A A I J J ^ [mm] . 0 0 0 . 0 0 2 , 0 1 2 , 0 1 3 , 0 1 8 , Ü 2 1 . 0 3 0 . 0 3 9 . 0 4 5 , 0 4 6 , 0 5 3 , 0 5 2 . 0 6 3 . 0 6 5 . 0 6 5 , 0 7 8 , 0 8 3 , 0 9 0 . 0 9 4 . 0 9 0 , 1 0 3 . 1 0 9 . 1 1 0 . 1 1 2 . 1 2 0 . 1 2 7 . 1 3 7 . 1 3 7 . 1 3 9 . 1 4 2 s p e c . n r . 1 2 B .AIJ,_I [mm] , U U .i , 0 0 Ö , 0 1 7 , 0 1 9 , 0 2 1 , 0 2 5 , 0 3 2 , 0 4 2 , 0 4 9 . 0 5 1 , 0 5 6 . 0 5 ! , 0 6 4 , 0 6 7 . 0 6 7 . 0 8 0 . 0 8 5 . 0 9 0 . 0 9 3 . 0 9 7 . 1 0 4 . 1 0 V , 1 1 0 , 1 1 0 , 1 2 0 , 1 2 r , 1 3 5 , 1 3 f c . 1 3 8 , 1 4 114
- 6 i 400,-CS
[10-"]
300SHRINKAGE
Specimennr.
mix B
tQ=26 d a y s
c ^ * ( l l n ( t ) '
t
« +(11
loA
200 100 m l O • 1 ^ 1 10 10 10* - 6 i 400ecs[10-"]
t [hrs)
300SHRINKAGE
Specimennr.
mix B
tp= 26 days
( x * ( i l n ( t )
• t«* fit
loB
200 100. . /
10b * ' ' * " . ' A *' "k
10 te't [hrs]
400
ecsHO-^l
300
200
100
SHRINKAGE
Specimennr.
mix A
tQ= 24 d a y s
o<.t(iln(t)
t
a * p t
: uA
0(9
I l i i i m l I i i i i i i i l I i ^ ^ i P i i i l I | > | | | 10 10'i,00
^csIlO 1
10"t [hrs]
300
200
100
SHRINKAGE
Specimennr.
mix A
to= 24 days
«•(!,.|n(t)
t
i l B
a* pt
i i I i I M l I I ^ 10t[hrs]
16
-400
erc;[10"'l
'CS
300
200
100
SHRINKAGE
S p e c i m e n n r .
m i x A
tQ= 23 days
o ^ - p l n ( t )
t
12A
::i>
10'
e i I P i I I 1 I I I M i l 1 * « * » • * j « * 10 10 10 10400
ecs[io-']
1010-t[hrs]
300
200
100
SHRINKAGE
S p e c i m e n n r .
mix A
1^= 23 days
p In(t)
t
1 2 B
«* ( ï t
o « ^
-10 •^^^
l u l I I I • I m l 10 10 10t [hrsl
17
400
^csHO''!
300
200
SHRINKAGE
Specimennr.
mix B
1^= 23 d a y s
« • ( l l n ( t )
t
1 4 A
« + (5t
100
I . — • — I — I I I 1 1 1 10 10 10400
EcsllO-^]
t [ h r s ]
300
200
SHRINKAGE
Specimennr.
mix B
tQ= 23 days
t
14B
100
d<r*
~ * i — -
10t [hrsl
Measured losses of weight of 150mm cubes and 150x150x500mm prisms,
The specimens were taken from the fog room at age t .
cast
no.
63
65
1
mix
B
A
toW
28
56
90
182
28
56
90
204
w(to) [gr]
76.8
36.0
84.2
87.2
87.6
83.4
72.0
76.3
79.0
90.1
76.4
81.1
95.0
96.6
112.2
111.6
103.5
120.9
126.6
91.3
107.5
94.4
/^w^^0O(.[gr]
2.9
1.4
3.1
1.5
1.4
1.2
0.5
0.7
0.7
0.4
2.9
3.2
2.6
1.8
2.1
1.7
1.4
1.6
2.3
1.0
1.5
1.1
'^^10+1050°c[9'•]
3.7
1.9
4 . 3
2.5
2.2
1.8
0.9
1.2
1.0
0.8
4.0
4.7
4.5
3.3
3.7
3.5
3.1
3.2
3.9
2.0
3.3
2.2
X eliminated values for eq. (3.27a) and (3.27b)
xx compressive strengths are reported in report 5-85-10.
batch no.
10
11
t
[hrs]
93.50
242.00
332.83
764.50
1511.00
2230.25
3932.50
5471.50
7227.50
168.00
863.00
1492.00
2041.00
3695.00
5234.00
6990.00
iw^grams]
34-30
*52-50
55-55
70-67
85-78*
110-110
150-150
195-195
225
-85-90
155-167
*185-200
230-250
295-320
365-390*
410-420
12
14
63
65
112.58
835.00
3367.25
1528.25
4915.25
6671.25
167.00
843.50
1559.00
2404.00
3046.00
4585.00
672.0
1488.0
3696.0
672.0
1488.0
4224.0
80-60 '.
172-150
'*325-285
130-210
350-300*
415-375*
33-27
63-43*
98-93
127-128
143-133
*195-190*
34.7
57.2
73.0
64.6
72.2
101.4
prisms
cubes
batch no.
10
n
12
14
63"'*
65X'*
specimen no
10A-10B
11A-11B
12A-12B
14A-14B
63A-63B-63C
65A-65B-65C
mix
B
A
A
B
B
A
t^Ldays]
26
24
23
23
28
28
PLkg/m^] [
of concrete
2383-2384
2367-2365
2366-2365
2395-2433 |
2401 (0.6% v . c . )
2381 (1.2% v . c . )
Measured average weight increase from demoulding
(t^=2 days) until removal from fog room (t <s»28 days):
mix A: 39 (cubes) and 208 (prisms) grams.
19
-APPENDIX IV Instrumentation
Specification of measuring instruments:
Computer
Floppy-disk
Printer
Amplifier
Displacement
trans-ducers
A/D converter
Hand-held device
Accumulators
Glue
Wheatstone bridge
SIiding layer
Jack
Load-cell
- CBM8032; 32
\(b
RAM; 28 Kb ROM
- CBM8050; 521 Kb
- CBM4022 : 80 characters/s
- HBM KWS/5T-5/GS-551
s u p p l y 4,
- H e w l e t t - P a c k a r d , 7ACD, T-lOO; ± 1/10 i n c h , 6 V power
- D a t e l ; HDAS-16ADC : ± 10 V o l t accuracy 4.884 mV
- Onno-Sokki : accuracy 0,001mm : ± 6.5mm
- OLAER : NPB-1 : PS = 330 bar PE=495 bar (1 l i t e r )
- T r i d o x : P h i l a d e l p h i a S.A. : F-88 epoxy
- Peekel : T200 : 0-10000 ym/m
- Rulon LD
- F r e y s s i n e t f l a t j a c k 220mm, 400 kN
- S t r a i n - g a u g e T o k y o ' s Sokki : P C - 5 - 1 1 ; 5mm; 120
±0.3-0-d isplacemen t - Irans±0.3-0-d ucers
-crock face
stejjl refereoce.
poiglue
'.%
(T
CROSS-SECIION A-A
1 .
-crock face
20
-isplacement
transducers
data-collector
•
9--f
^«
-P
-•
•
load<ell
amplifier
T= 0-24 HRS.
aid
con-verter
cbm 8032
micro-computer
^disk-drive
M printer
displacem. transducer
t=T^
updovi»
counter
calibrcrting-jjevice
Tneosuring pbirtS'
V
load-cell
P
whecfclone
^^^^
measuring -bridge
T= 2AHRS.-9(LDAYS
cbm 8032
micro-computer
1disk-drive
printer
shear creep 8209
electric sctiemes
- 21
/ ^
^ ^
<D
T= O -24 HRS.
1. flat hydraulic jack
2. electric oilpump
3. handforce oilpump
A. accumulator
shear creep 8209
hydraulic schemes
22
-APPENDIX V Accuracy of displacement measurements
»
a. Hand-held measuring device
A few analyses of the measuring system are presented. Specifications are given
in appendix IV.
a1. Choice of measuring system and reference points
The illustration below schematically shows the hand-held measuring-device.
The legs are placed in steel reference points stuck to the concrete surface.
The optimal position is disturbed by an inclination of these steel points
(..s) and by a difference in height of the layer of glue used (At). Hence:
AS = s(l-COSa) - R.COSO(l-COSa) ' (V.I)
lA = 2.AS+.-,t (V.2)
The maximum disturbance .'1 is l i m i t e d to h a l f of the measuring accuracy:
Al = y p +
(;:A)2'
- 1 •0.002mm (V.3)
The optimal position is found for a=0° and 0=90". The actual values are found
analytically;
.> x^ + y* = R2 (V.4)
dy/dx = -2x/
/ R ^ - X ^ '
= -6 (V.5)
Combining (V.4) and (V.5) gives X-+0.95R and 0=72°. If 1=100mm and R=2.5mm
are chosen, then the maximum difference in height can be calculated:
tanu = ZA/l s0.02-> o( = 1.14° (V.6)
After substituting (V.2) and (V.3) with o< = 1.14'
Al =
J]OQ.O'
+ (5.0.32 (1-cos1.14°))2' - 100.0 (V.7)
so that At • 0.60mm. This condition can be satisfied by a smooth concrete
sur-face and by use of a special steel jig for glueing. It can be concluded that:
23
-- large values for R and 1 should be chosen;
- angle of the cone is 1:6. Hence the contact area between the reference
point (see illustration below) and the measuring leg is relatively large.
The induced deformation will be negligible;
- it is important to reduce the difference in height between the reference
points. The screw thread is used to fix the steel jig temporarily.
a2. Accuracy
The actual displacements of the shear plane are found from both the zero
measurement before pre-cracking and the actual measurement. Possible errors
are:
- systematic e r r o r s : a wrong zero-measurement w i l l influence a l l additional
measurements in the same d i r e c t i o n ;
- stochastic errors;
- personal differences. It is advisable to have all measurements performed
by the same person (A1,=0.0010mm);
- deviations (height, inclination) from the ideal position of the steel
reference points (AI^^O.0005mm);
- deflection of measuring instrument and deformation of the legs (A1 o?»nought);
- electronic stability of instrumentation (A1.smought);
- digital signal output (A1.=0.0005mm);
-Fi
-
temperature fluctuations of the steel reference jig(u-.= 10 "K-l andAT=1''C)
and of the measuring device (o<j= lO" "K-l and
A T = 4 ° C ) .
The gauge length is assumed to be 100mm ( A 1 ^ = ( 1.0-1-4.0)x'lo"'^=0.0005mm).
By adding together all the errors the maximum deviation is found:
6 , •
A 1 = X Al'=0.0025mm (V.8)
i = 1
Assuming a normal error distribution, the final accuracy of the measured
dis-placements will be 0.0025x1.4=0.004mm.
24
-a3. Calibration
The displacements (y) of the movable leg of the hand-held device were
measured by means of imposed displacements (x) of a micrometer instrument.
The relation between x and y is assumed to be linear. Hence:
y 'a Ö
(V.9)
Y[mm
X l m m J
Neglecting adjustment errors both x and i t s variance can be c a l c u l a t e d :
X = ( y - Ü
y nt
- ' ' ' • = - , '
f U h . y )
^x
-~
(^) -"a " ^13) • "3 " % ^ • °y
Combining (V.IO) and (V.11) the approximate variance will be
2 ~ ' 2 _2 2 2
C R i t ) . ( S - H X . S - l - S ) :•
X ^ a
b
y
.
.
-
.
(V.IO)
(V.11)
(V.12)
Assuming a Student distribution of the observed phenomenon the total measuring
accuracy Af can be quantified. In order to reduce the error each result is the
average value of three measurements. Hence:
Af
1 .65
0.005mm
(V.13)
a4. Measurements in-situ
Measurements were performed on the surface of a concrete specimen (figs. 2.5
and 4.6 in report 5-85-5). The cracked as well as the uncracked situation
were simulated. For one and the same gauge length theoretically all values
should be identical. Systematic ( A X ) and stochastic (standard deviation s)
errors can be calculated with 90/ü confidence:
- 25
|AX| = |x.(cracked) - x (uncracked'
2.13.s = to_Q5 (4 meas.)-/s + s
1 o
(V.14)
(V.15)
error
gauge
length
1=70
[mm]
gauge
length
1 = 189
[mm]
Ax
2.13.S
added
0.001
0.002
0.003
0.002
0.003
0.005
As d e r i v e d e a r l i e r o n , t h e gauge l e n g t h w i l l i n f l u e n c e t h e e r r o r . I t can be
concluded from both i n v e s t i g a t i o n s t h a t the e r r o r i s Af=0.004mm (0.08% f u l l
-s c a l e ) .
a 5 . I n f l u e n c e o f measuring e r r o r on the c a l c u l a t e d d i s p l a c e m e n t s .
t = to: uncracked specimen
t = t i : cracked specimen underload
1 t:^ 70 mm
0
K(t ) =^175 mm
P o'
l,(t ) -188.5 mm
2^ 0
The actual displacements of the crack plane can be approximated by simple
formulae if w/l-(t ) <<1 and s/l.(t ) -•<1 :
w « y t ,
) -
i^(t^;
s^A%)
-l?(ti) - 1 ,
^. 2
(V.16)
(V.17)
The measuring error A1=0.004mm will correspond to displacement inaccuracies
(ï(w) and (f(s) :
26
<fw = i3w/^l^(t^)|.'5l^(t^) + !3w/9l^(t^)1.6l^(t^)
n^(t^) + 'li(t^)« 2.Af= 0.008mm (V.18)
is = |9s/ .<l^(t.,) j-Sl^(t^) + ldS/3l2(t^)|.Sl2(t^)
+
laS/.l^i-ól^
l(^o)^^2(^o) n .f
Y=\
+ 1} . Af
7ÏÏ
= {I^}^A + ]).Q^QOil = Q,Q]Qmm
(V.19)
Both separation and slip have been calculated iteratively by a computer
pro-gram using eq. (V.16) and (V.17) as initial values. The computation stops if
successive displacements change less than 10~ mn. Final values of w and s
agree with geometrical conditions. As an advantage, error propagation
can also be quantified in a more refined way. There is close agreement with
the results of eq. (V.18) and (V.19). The error of As is limited for "Ip^^o^'^
l^(tQ)>2. The final choice is 188.5/70.0=2.62).
b. Displacement transducer
b1 . Accuracy
i M
Electronically the accuracy of the system is limited by high-frequency waves
and by the resolution of the 12 bit A/D converter;
- sensitivity : 0.004 mm (5mV)
- binary resolution : 0.001 mm
Mechanical 1iniitations also contribute to the inaccuracies. The transducers
have been calibrated by a micrometer instrument analogous to the hand-held
measuring device. The final error amounts to 1 .65x0.006=0 .OlOmm (0.2%
full-scale).
b2. Influence of eccentricity . >.
In section 4.1 it is reported that the crack plane may experience bending
moments, viz.: .'J ' - . ,
iff I
-".
-y- M = 4.0 kNm at pre-cracking with Wn=0.01-0.03mm v'** *>-'^/'
yy ^^ ^ 0 .**^
- M = 2.4 kNm a t maximum shear l o a d i n g w i t h e=6mm.
XX
27
-In order to estimate the change in the displacements caused by M the
A X
following assumptions are made:
- the bending moment is transferred only by stirrups that cross the crack.
The reducing influence of the cracked concrete has been entirely neglected.
- the separation changes due to deformation of the reinforcement and to
in-duced si ip.
- both separation and slip equally change in proportion to the corresponding
axial steel stresses.
crack plane stimjps 08n\m M,j^Mspi = 4DkNm Myy;Mexc=2.4kNm
I =i.TT.4^ + TT-4^.(37.5^ + 112.5^) = 2B3.10^mm^
XX
I y=i-TT.4^ +TT.4^.41^.2 = 0.68.10^mm^
c (M )=4.10^.41/0.68.10^ = 242 N/mm^
max yy
o _ (M ) = 24.10^.112.5/2.83.10^= 95N/mm2
iTlaX X X
(V.20)
(V.21)
(V.22)
(V.23)
The order of change of both w and s is 0.03/2.8=0.01 mm;
Al^ = Al^.60/41 4 0.03mm and Al]= lj150/112.5
Al^/AlJ =60.112.5 . 242/(41.150.95)=2.8
(V.24)
(V.25)
A maximum shear load of 400 kN can be applied with four stirrups 08mm
(p =1.12%): the concrete grade f' =62 N/mm^ is necessary for a crack width
w=0.25mm according to eq. (2.2). It is concluded that the influence of the
eccentricity on w is less than 4'ó. For higher reinforcement ratios the change
28
-APPENDIX VI Corrections of measured displacements
Fig. 4.15 shows two corrections:
a. Corrections due to the use of two measuring systems. At the very moment
of change-over from one system to the other most specimens showed a small
discontinuity in the time-dependent displacements:
- increased displacements during the change-over period (At=l-4 hours) were
corrected via a simple regression analysis (w or s=a +i?-1 n(t)). Thus part of
the correction due to time-dependent displacement increases was considered;
- the remaining differences d(w) and d(s) in all the specimens show a more or
less normal distribution (see fig. 4.16) which points to stochastic measuring
errors. The fact that d(w) has a relatively large scatter is inherent in the
hand-held measuring device, as has been explained in appendix V. During
preliminary tests an 8 bit A/D converter was used. Logically it induced higher
values of d(w) and d ( s ) .
QQ^ifference d[mm]
006^
OOi
002.
0
-0D2
-00/.
-006
-0081
dlwl '« d i s l 90% anfKJence limit d l « L l90Xr; .
.LJ:
d(sl-.0O17mm dlv>>>0011mm ^ * * ' ^ t'~êm)
specimen ne,
Next, both d(w) and d(s) were spread over the two systems in proportion to
t h e i r respective accuracies (see also appendix V):
transducer:
hand-held instrument
^ " V a s ~ 0-56.d(w) (0.56 = 0.010/0.018)
S = s ^ . , , ± 0.50.d(s) (0.50=0.010/0.010)
^"%eas + 0 - 4 4 . d ( w ) (0.44=0.008/0.010)
^ = ^meas T 0 - 5 0 . d ( s ) (0.50=0.010/0.010)
( V I . i :
( V I . 2 ;
( V I . 3 ;
( V I . 4 )
29
-Corrections due to the deformation of the concrete near the crack between
the steel reference points
"c — m — I'l
H "
- * III — fill — 811— DiL
sew.,-I
\t,'H^J
Displ t>»tw*«n crock face ond ste«l rrf^rence points
Possible deformations are:
- effect of o'c on the crack width w:
ÖW
dir
(SW,
'sh
instantaneous deformation
time-dependent deformation
shrinkage
- effect of fc on the shear displacement s
4,
5. OS
OS,. : instantaneous shear deformation
d i r
time-dependent shear deformation
An example o f a c a l c u l a t i o n i s given b e l o w . I t r e l a t e s to a p u s h - o f f specimen
having a r e l a t i v e l y low r e i n f o r c e m e n t r a t i o o f the crack p l a n e . The s u s
-t a i n e d shear s -t r e s s l e v e l i s
T / T U = 0 . 8 0 .
Taking i n t o account the measuring
a c c u r a c y , o n l y t h e c o r r e c t i o n s 3 , 4 and 5 are i m p o r t a n t ;
Example o f p u s h - o f f specimen
f
cc
sy
A
c
E'
c
G'
c
1
= 6 7 . 3 N/mm2
= 460 N/mm2
= 36.10^ mm^
= 37.10^ N/mm^
= E ' / 2 . 4
c
= 50 mm
T / T =0.80
u
w =0.03mm
0
p = 1 . 1 2 %
0
i . e . 4 s t i r r u p s 0
5 mm
30
Results of computation:
Og = 80 N/mm2 i = 9.9 N/mm^
'cs
0.9 N/mm^ T ^ = 1 .S.S^f^^ .f^^/A^ = 3.3 N/r
s) = 0.19
24hrs) = 20.10'
tp (24hrs) = 0.19 T = T - T ,
C
r
c d
The corrections are:
'^^dir
^:w^
^^sh
^'^dir
OS
c
= l.r'/E'
c c
= ^w^.^.cp^(24hrs)
= l..^^(24hrs)
-
^-V^c
= .s,.^..p^(24hrs)
= 0.0010
= 0.0002
= 0.0010
= 0.0210
= 0.0039
mm
mm
mm
mm
mm
(VI.5)
(VI.6)
(VI.7)
(VI.8)
(VI.9)
The internal stresses on the crack plane o' and x have been calculated
^ c c
with a theoretical model of Pruijssers [27]. Creep and shrinkage deformations
have been obtained from the CEB-FIP model code [11].
Re 3.
Values of Sw , were measured using four unloaded push-off specimens. The
position of the steel reference points conformed to the loaded specimens.
Experiments started at t =23 days and lasted approximately six months.
Test data were analysed by non-linear regression analysis:
mix A : -W^^(t^,t) = { 6Ü3(-4- + 129)° "^^^ - 4592}.10"^ (VI,10)
mix B : 'iW3^(t^,t) = (1.3 . (t + 4i2)°-°°2 - 1.3} . 10° ' (VI.11)
in which t=duration of test in hours. An increase of the reinforcement ratio
will reduce shrinkage. For the relation between the embedded bars and the
concrete it is assumed [21] that:
£ ^ ( P Q > 0 ) = c^(p = 0)/(Un.p) (VI.12)
in which n=E /E' (mix A: n=6.2 and mix B: n=5.7) and ii=A /A . All the specimens
S c 5 c •
31
Combining ( V I . 1 0 ) - ( V I . 1 2 ) g i v e s :
mix A: 6 w ^ ( t , t ) = 1.069. (eq . ( V I . 10))/(1+6.2iO
mix B: 5w^^(t , t ) = 1.064. ( e q . ( V I . 11))/(1+5.7p)
( V I . 1 3 )
( V I . 1 4 )
20
15
10
ÖWrs [lO-^mm]
tQ=23days
mix A
fee'= 53.7 N/mm 2
• ^ stirrups 08mm 60,p[
measuring points:
O nr. 1 specimennr 27
A nr 3 specimennr 28
D nr U specimennr. 28
20
15
10
öWcs[lO"^mm]
t [hrs]
tQ= 23 days
mix B
fcc'= 67 2 N/mm
i stirrups 08mm
2 s .
-o
- o
60^measuring points :
O n r . r
• nr.2
A nr. 3]
D nr. L,
specimennr. 29
— specimennr. 30
equation tffltl
90% confidence
interval
10'
10^
10'
10-
10^
t [hrs]
- 32
Re 4.
The external shear force V='Z~'.A is transferred both by the opposing crack
faces C^c) and by the embedded bars (Td). The bars are subjected to a dowel
force F ,. The observed instantaneous crack widths w . vary between 0.1 and
d el -^
0.2mm, i.e. approximately 2-3 times the initial crack width. In that case
the dowel force on the bar will already have reached its maximum value
according to the model [20, 26, 27j :
T = T +•[ ,
c d
T , = n-1.3 . 0^ /f' • f ' /A
d ^ cc sy c
in which:
T , = shear stress component on dowels
i^ = shear stress component on concrete
n = number of dowels
0 = bar diameter
f' = cube compressive stress
cc ^
f = steel yield strength
sy
= reinf. ratio = 100 .n.n.02/(4A )
0
'^
' ' c
:VI.15)
; v i . i 6 )
T
i
tTc
c
(VI.17)
At a distance of approximately four times the bar diameter from the shear
plane the dowel force will be completely transferred to the surrounding
con-crete. If the shear stress increase in the concrete is simplified to develop
linearly in this area, the average shear stress will be:
T '^T -I- l,/2 = T - T , / 2
c c d d
;VI.18)
Now pure shear of the concrete is assumed between the steel reference points
(Ss ,.
dir
•^c -l/^c
(VI.19)
On substituting G^=
VJ{2+2.0^)=
0.42.E^ and 1= 60mm and combining (VI.
16)-(VI.18) the following equations are obtained:
^ix A: ^Sdi^.= 42.10"-^i'r'- 0 .1775 . p ^ / F ^ }
iix B:
k^^^-
3.9.10"3(r- 0.1775.p^yf^l
(VI.20)
33
-Re 5.
No experiments concerning time-dependent shear deformation of plain concrete
are reported in the literature. Ruetz [3l] derived a theoretical relation
between shear and normal deformation of concrete. Neglecting volumetric
change compared with deformation by transfiguration, it follows that:
S i = T e i / 2 " _ (VI.22)
de^/dt = è.dy^/dt (VI.23)
so that: dy /dt=2.de /dt=2.t: , .dcp /dt
'o c el c
and dy /dt=Y i -dip /dt (VI.24)
0 'el c
Integrating the last equation gives 70=~»el*tp^. By analogy it follows that:
6s^(t)=ós^.^.cpc(t)= Ö5j.^. (cp^(t+28)- U)^(28)} (VI.25)
in which:
6 s , . = d e f o r m a t i o n a c e . t o ( V I . 2 0 ) and ( V I . 2 1 )
d i r
cp = creep c o e f f i c i e n t of c o n c r e t e a c c o r d i n g t o [ 1 1 , 1 9 ]
t = d u r a t i o n o f l o a d a p p l i c a t i o n i n d a y s .
The creep c o e f f i c i e n t cit c o n s i s t s of a. number o f s i m p l e f u n c t i o n s :
t p ^ ( t ) = e g ( t ) + c p ^ . B ^ ( t - t ^ ) + c p ^ - { B ^ ( t ) - S ^ ( t ^ ) ( V I . 2 6 )
e ( t ) = 0 . 8 0 . d - f ( t ) / f ( t ^ ^ ) }
a L U L U
U)j = 0.40 tp^ = cp^^. ip^^^2.70 . 1.70 = 4.59
^ ^ ( t - t ^ ) = 0 . 2 4 + 0 . 0 9 . 1 n ( t - t Q ) p ^ ( t ) = - 0 . 1 2 + 0 . 1 5 . 1 n ( t )
For all cases it is assumed that the dowel force remains constant [28,29].
Hence with (VI .25)-(VI .26):
mix A: 6s^(tQ,t) = 6s^.^. {-2 .075+0.036-ln(t)+0.7071 n(t+28)} (VI.27)
34
-50
30
20
10
öS(jjp [lO"-^mm]
O
O
to= 28 days I I
— — mix A (eq.Tn.20)
mix B (eq,S3.21)
0 = 8mm
8 10
l [ N / m m ^ ]
ÖWrs o»" öSp[lO'^mm]
35
-APPENDIX VII Data of sustained tests and push-off tests
The measured displacements of specimensnos. 1-26 are presented on the
following page 36-62 . The data correspond to the plots in sections
4.4.2-4.4.6 (figs. 4.22-4.68). Three tables of the measured displacements
for each specimen are given:
- during load application and sustained shear loading (t «^ 90 days)
- during possible prolonged load application and unloading ( t > 9 0 days)
- on reloading to failure
All the tables show definite displacements as well as the necessary
correc-tions.
APPENDIX VIII Results of regression analysis
The regression coefficients *<i-'^io of eq. (5.2)-(5.4) are presented on
page 62-63,
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