Sam Eurocode UK Pretressed Beam Sample Report
March 15, 2017 | Author: coho_hc | Category: N/A
Short Description
Download Sam Eurocode UK Pretressed Beam Sample Report...
Description
Sample Report Precast Pre-tensioned Beam Example Eurocodes UK NA
Pre-tensioned Pre-stressed Beam Bridge Design Example 1.
Geometry & Basic Data ................................................................................................................... 5
2.
Carriageway Configuration .................................................................................................. ...........9
3. 4.
Global Analysis Model................................................................................................................... 13 Influence surfaces ......................................................................................................................... 17 a) Mid Span Sagging Moment ....................................................................................................... 19 b) Internal Support Hogging Moment ........................................................................................... 20 c) Internal Support Shear .............................................................................................................. 21
5.
Traffic Loading Configuration........................................................................................................ 23 a) Mid Span Sagging Moment ....................................................................................................... 25 b) Internal Support Hogging Moment ........................................................................................... 26 c) Internal Support Shear .............................................................................................................. 27
6.
Global Analysis Results.................................................................................................................. 29 a) Mid Span Sagging Moment ....................................................................................................... 31 b) Internal Support Hogging Moment ........................................................................................... 33 c) Internal Support Shear .............................................................................................................. 35
7.
Section Properties ......................................................................................................................... 37 a) Mid Span ................................................................................................................................... 39 b) Internal Support ........................................................................................................................ 41
8.
Data Summary after Tendon Design ............................................................................................. 43
9.
Temperature Gradient .................................................................................................................. 51
10. Shrinkage & Creep ........................................................................................................................ 55 11. Verification: Transfer Stresses ...................................................................................................... 63 12. Verification: SLS Bending - Mid Span ............................................................................................ 73 13. Verification: ULS Bending - Mid Span ........................................................................................... 93 14. Verification: SLS bending – Pier .................................................................................................... 99 15. Verification: SLS bending – Support ............................................................................................ 117 16. Verification: ULS Shear - Pier ...................................................................................................... 135 17. Verification: ULS Interface Shear ................................................................................................ 143 18. Verification: Web Shear Cracking ............................................................................................... 149 Appendix - National Annex NDP Values.............................................................................................. 157
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4
Pre-tensioned Pre-stressed Beam Bridge Design Example
1. Geometry & Basic Data
5
6
General Cross Section
Elevation
Plan
Grade C31/40 insitu concrete; Grade C50/60 precast concrete Grade B500B reinforcement steel Supports located 1m beneath soffit of slab Reinforced Concrete diaphragm over supports Cracked insitu concrete over central supports Slab reinforcement over internal supports (6m either side) Carriageway is 9.6 m wide with 1.2m footway on each side Designed for vertical highway loading groups Gr1a with French National Annex NDP values
7
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
USER NOTES
The design had been completed using the following process: 1) Four beams are created in SAM, two representing each span of the Y7 inner beams and the other two representing the edge beams of each span (with the upstand on the left hand side). At this stage all possible tendons are active. The differential temperatue profile is also determied and entered for each of the beams 2) A line beam analysis is carried out to determine the bending moments and shear forces atrributed to the dead load actions at each construction stage and the secondary moments and shears for differential temperatuire and differential shrinkage. Surfacing (SDL) actions are also established with the line beam analysis 3) A grillage model of the bridge deck is created using the beams prepared in 1) above. The grillage is to take account of the vertical level of each of the component beam elements by way of member eccentricities. This will give rise to a better distribution of effects but will intruduce (relatively small) axial forces into the beams. 4) Traffic load patterns are established for max sagging, hogging and shear for each node point along one of the central most beams, by using the load optimisation. This will give rise to three envelopes for sagging, hogging and shear. 5) The traffic live loads are transferred back to the table in the appropriate beam file. 6) An alaysis at transfer is carried out and some tendons are removed and debonded to reduce the compressive and tensile stresses to below limiting values. (This can be done with the tendon optimisation facility if required). Results output is produced for the mid span section 7) Other construction stages are checked at SLS Characteristic and ULS:STR to check compliance with stress limits and Bending capacity. Results output is produced for the mid span section. 8) Bending moments (sagging and Hogging) due to the full traffic action (plus other permanent and variable effects) ar checked for compliance at SLS and ULS. Results output is produced for the mid span section. 9) Transverse and Longitudinal shear reinforcement requirements are established and the results output for the most onerouse section as well as web shear cracking checks at SLS 10) Other reports of results, such as differential temp and shrinkage are produced and appended to the final report. 11) Time dependant creep effects are accounted for using the simplified method found in EN1992-2 Annex KK.7
SAM v6.50d
02/02/2012 10:57:11
© 2012 Bestech Systems Ltd
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Page: 1
Pre-tensioned Pre-stressed Beam Bridge Design Example
2. Carriageway Configuration
9
10
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Job No.: 6.5d Calc. By: DLG
Sample Reports
Structure: 2 span grillage prestresses beam deck Eurocodes + UK NA
Checked:
Data File: J:\...\6.50d Data Files\2 span prestress beam deck .sst 02/02/2012 09:29:44
Data Report
STRUCTURE
CARRIAGEWAYS CW1: Carriageway Carriageway is for road traffic loading. It is aligned to design line DL1 and is single. Primary carriageway has 2 lanes 4.0m wide.
Carriageway
Offset 1 (m)
Offset 2 (m)
Primary
-4.0
4.0
Footway 1
-5.5
-4.0
Footway 2
4.0
5.5
Loaded Widths for: CW1 CF1: Default Primary carriageway - Number of lanes: 2 Ref Offset Width Direction 1 0.0 3.0 with Chainage 2 3.0 3.0 against Chainage CF2: Load Opt. (created by load Primary carriageway - Number of Ref Offset Width 1 1.5 3.0 2 5.0 3.0
optimisation) lanes: 2 Direction with Chainage against Chainage
CF3: Load Opt. (created by load Primary carriageway - Number of Ref Offset Width 1 2.0 3.0 2 4.5 3.0
optimisation) lanes: 2 Direction with Chainage against Chainage
SAM v6.50d
02/02/2012 11:00:53
© 2012 Bestech Systems Ltd
11
Page: 1
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Job No.: 6.5d Calc. By: DLG
Sample Reports
Structure: 2 span grillage prestresses beam deck Eurocodes + UK NA
Checked:
Data File: J:\...\6.50d Data Files\2 span prestress beam deck .sst 02/02/2012 09:29:44
CF4: Load Opt. (created by load Primary carriageway - Number of Ref Offset Width 1 0.5 3.0 2 3.5 3.0
optimisation) lanes: 2 Direction with Chainage against Chainage
CF5: Load Opt. (created by load Primary carriageway - Number of Ref Offset Width 1 -0.25 2.5 2 3.5 3.0
optimisation) lanes: 2 Direction with Chainage against Chainage
CF6: Load Opt. (created by load Primary carriageway - Number of Ref Offset Width 1 0.0 3.0 2 5.0 3.0
optimisation) lanes: 2 Direction with Chainage against Chainage
CF7: Load Opt. (created by load Primary carriageway - Number of Ref Offset Width 1 0.0 3.0 2 4.5 3.0
optimisation) lanes: 2 Direction with Chainage against Chainage
CF8: Load Opt. (created by load Primary carriageway - Number of Ref Offset Width 1 0.0 3.0 2 3.5 3.0
optimisation) lanes: 2 Direction with Chainage against Chainage
SAM v6.50d
02/02/2012 11:00:53
© 2012 Bestech Systems Ltd
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Page: 2
Pre-tensioned Pre-stressed Beam Bridge Design Example 3. Global Analysis Model
13
14
This is a view of the structure that is modelled for the global analysis highlighting the beam considered for design
The beam in isolation indicates the cracked concrete slab over the central pier, shown dotted
15
16
Pre-tensioned Pre-stressed Beam Bridge Design Example
4. Influence surfaces a) Mid Span Sagging Moment b) Internal Support Hogging Moment c) Internal Support Shear
17
18
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job: Sample Reports Structure: 2 span grillage prestresses beam deck Eurocodes + UK NA Data File: J:\Marketing\Sample Reports\Sam Eurocode UK Prestressed Beam\6.50d Data Files\2 span prestress beam deck .sst 02/02/2012 09:29:44 Result Type: Influence Surface
Job No.: 6.5d Calc. By: DLG Checked:
Name: I7: BM55; My Sagging
Influence coefficients are expressed with respect to global axes. Analysis Run: 01/02/2012 14:39:12 Results shown for: Influence Coefficients - DZ (m)
SAM v6.50d Copyright © 2012 Bestech Systems Ltd
02/02/2012 11:14
19
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Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job: Sample Reports Structure: 2 span grillage prestresses beam deck Eurocodes + UK NA Data File: J:\Marketing\Sample Reports\Sam Eurocode UK Prestressed Beam\6.50d Data Files\2 span prestress beam deck .sst 02/02/2012 09:29:44 Result Type: Influence Surface
Job No.: 6.5d Calc. By: DLG Checked:
Name: I25: BM60; My Hogging
Influence coefficients are expressed with respect to global axes. Analysis Run: 06/02/2012 11:01:20 Results shown for: Influence Coefficients - DZ (m)
SAM v6.50d Copyright © 2012 Bestech Systems Ltd
06/02/2012 10:58
20
1
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job: Sample Reports Structure: 2 span grillage prestresses beam deck Eurocodes + UK NA Data File: J:\Marketing\Sample Reports\Sam Eurocode UK Prestressed Beam\6.50d Data Files\2 span prestress beam deck .sst 02/02/2012 09:29:44 Result Type: Influence Surface
Job No.: 6.5d Calc. By: DLG Checked:
Name: I26: BM60; Shear z-
Influence coefficients are expressed with respect to global axes. Analysis Run: 06/02/2012 11:01:20 Results shown for: Influence Coefficients - DZ (m)
SAM v6.50d Copyright © 2012 Bestech Systems Ltd
06/02/2012 10:59
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1
22
Pre-tensioned Pre-stressed Beam Bridge Design Example
5. Traffic Loading Configuration a) Mid Span Sagging Moment b) Internal Support Hogging Moment c) Internal Support Shear
23
24
25
26
27
28
Pre-tensioned Pre-stressed Beam Bridge Design Example
6. Global Analysis Results a) Mid Span Sagging Moment b) Internal Support Hogging Moment c) Internal Support Shear
29
30
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job No.: 6.5d Job: Sample Reports Calc. By: DLG Structure: 2 span grillage prestresses beam deck Checked: Eurocodes + UK NA Data File: J:\Marketing\Sample Reports\Sam Eurocode UK Prestressed Beam\6.50d D...\2 span prestress beam deck .sst 02/02/2012 09:29:44 Result Type: Envelope Result For: Beam
Name: E1: GR1A; ULS STR/GEO Mem 49-60: My+ Effect: Member End Actions
Moments at the member start end correspond with the local member axes directions. At the other end, moments are positive in the opposite direction to the local member axes. With this convention, a positive y or z moment at each end denotes sagging. The table displays the enveloped effect and associated values. The enveloped effect is Member End Moments - My (kN.m). Analysis Run: 01/02/2012 14:48:20 Results shown for: Member End Moments - My (kN.m)
New Selection Reference
Member End Forces Fx (kN)
Joint
49
53
-55.91424
47.25736
26.53076
25.44022
184.235
C1
182.2866
49
54
-35.84683
33.06124
271.1681
19.73986
335.773
C9
100.8609
50
54
-22.25784
22.6683
133.4978
23.78223
343.2836
C9
51.87468
50
55
54.62167
-2.692833
488.1098
-2.971709
1012.892
C17
-1.920841
51
55
20.41604
-2.863775
207.1184
1.268835
992.3741
C17
4.310352
51
56
28.14961
-3.108706
341.2957
0.4258882
1519.03
C25
7.980115
52
56
9.656585
-2.213639
12.84226
4.645401
1506.387
C25
8.499461
52
57
12.55014
-3.992453
271.0804
3.409645
1867.618
C33
11.38605
53
57
1.544886
-1.958322
-53.32979
7.16562
1859.024
C33
10.30689
53
58
2.67042
-3.684942
209.6595
5.880492
2039.232
C41
11.96043
54
58
0.7964923
-0.7758132
-115.5332
8.95645
2035.977
C41
11.81908
54
59
2.120291
-2.487774
138.1745
7.473753
2042.085
C49
10.77256
55
59
11.21513
0.7455683
-183.48
9.730765
2045.381
C49
12.77409
Load Ref
Fy (kN)
Member End Moments
Member
Fz (kN)
Mx (kN.m)
My (kN.m)
Unfactored
Type
Origin
Mz (kN.m)
Factors Gamma
Psi
Alpha
Factored gr
Lane
Other
Total
Compilation : C49: BM55; My Sagging; GR1A; ULS STR/GEO (SUM=2045.38) L57
LM1 UDL System
285.1668
1.35
1
0.61
1
1
1
234.8349
L59
LM1 UDL System
22.51277
1.35
1
2.2
1
1
1
66.86293
L121
Footway: UDL System (Footway)
9.465714
1.35
1
1
1
12.77871
L122
Footway: UDL System (Footway)
23.41097
1.35
1
1
1
31.60481
L123
LM1 UDL System
282.2928
1.35
1
2.2
1
0.2777778
1
232.8916
L124
LM1 Tandem System
649.1479
1.35
1
1
1
0.6666667
1
584.2331
L125
LM1 Tandem System
653.4628
1.35
1
1
1
1
1
1925.46
882.1748 My=2045.381
55
60
15.52078
0.798247
77.73588
7.514261
1895.5
C57
-17.79333
56
60
33.82052
-2.988016
-247.2278
2.884023
1907.151
C57
-16.04987
56
61
30.5347
0.8282612
-6.834788
5.464101
1621.272
C65
-15.22928
57
61
60.01271
-3.610685
-346.8961
0.3388024
1638.577
C65
-16.90568
57
62
48.37749
-4.715206
-110.8182
-0.6692659
1210.068
C73
-10.54859
58
62
82.86469
-10.92155
-465.8913
-5.560555
1230.844
C73
-17.32338
58
63
28.10417
-7.797547
-134.3832
-4.188907
651.3091
C81
-3.139501
59
63
52.21575
-12.04783
-505.8134
-5.459631
681.349
C81
-11.63125
59
64
5.630956
-9.655632
-151.3374
-7.84137
61.63301
C89
4.864394
60
64
11.72936
-13.57086
-406.9595
-8.120834
67.45666
C89
12.16619
60
65
-20.1954
-9.464386
-16.30567
-6.696705
7.586888
C97
20.18539
SAM v6.50d Copyright © 2012 Bestech Systems Ltd
02/02/2012 11:29
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Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job No.: 6.5d Job: Sample Reports Calc. By: DLG Structure: 2 span grillage prestresses beam deck Checked: Eurocodes + UK NA Data File: J:\Marketing\Sample Reports\Sam Eurocode UK Prestressed Beam\6.50d D...\2 span prestress beam deck .sst 02/02/2012 09:29:44 Result Type: Envelope Result For: Beam
Name: E2: GR1A; ULS STR/GEO Mem 49-60: MyEffect: Member End Actions
Moments at the member start end correspond with the local member axes directions. At the other end, moments are positive in the opposite direction to the local member axes. With this convention, a positive y or z moment at each end denotes sagging. The table displays the enveloped effect and associated values. The enveloped effect is Member End Moments - My (kN.m). Analysis Run: 01/02/2012 14:48:20 Results shown for: Member End Moments - My (kN.m)
New Selection Reference
Member End Forces Fx (kN)
Joint
49
53
157.8887
5.094982
427.0325
-2.139213
-540.7185
C2
49
54
137.6641
5.170721
207.8578
-4.97695
-203.0551
C10
-19.1692
50
54
99.32214
4.674887
210.546
-4.123625
-224.4301
C10
-1.098864
50
55
-2.612839
0.125463
-30.58831
-0.1607472
-70.21469
C26
-0.04289707
51
55
-0.8885847
0.4065817
-30.84254
-0.5053211
-69.16653
C26
0.2566332
51
56
-0.8885847
0.4065817
-30.84254
-0.5053211
-128.0478
C26
-0.5195683
52
56
1.513816
0.7835213
-31.26312
-0.8808424
-126.6603
C26
0.5482784
52
57
2.163933
1.11089
-33.71269
0.08934582
-189.3661
C42
-1.782311
53
57
7.540948
1.586132
-34.77629
-0.4991898
-186.3882
C42
0.3427496
53
58
7.498607
1.625769
-34.81647
-0.4645858
-252.8444
C58
-2.733323
54
58
14.86624
2.137992
-36.42122
-1.132049
-248.9295
C58
0.4624202
54
59
14.80931
2.121582
-36.50214
-1.130482
-318.4683
C74
-3.609668
55
59
24.70702
2.689951
-38.68139
-1.896179
-313.4238
C58
0.6104634
55
60
24.70534
2.664611
-38.80091
-1.896578
-387.4718
C74
-4.508079
56
60
37.70397
3.20602
-41.87492
-2.795453
-381.0251
C74
0.7231245
56
61
37.70397
3.20602
-41.87492
-2.795453
-460.9685
C74
-5.397487
57
61
54.36921
3.641991
-45.78775
-3.869674
-453.6171
C74
0.8188211
57
62
54.36921
3.641991
-45.78775
-3.869674
-541.0301
C74
-6.134068
58
62
74.1749
3.475636
-50.55485
-5.391563
-531.9109
C74
0.7229482
58
63
95.30983
2.989823
-90.50055
-4.804924
-656.2544
C82
-5.145004
59
63
117.8148
1.956679
-95.09197
-8.621091
-610.4323
C82
0.5380956
59
64
150.7005
-26.16291
-266.468
-23.02836
-888.0567
C90
24.02131
60
64
154.1473
-36.96983
-270.5991
-19.99131
-883.7671
C90
8.108403
60
65
190.1957
-15.75881
-469.0165
-9.477075
-1286.162
C98
13.65186
Load Ref
Fy (kN)
Member End Moments
Member
Fz (kN)
Mx (kN.m)
My (kN.m)
Unfactored
Type
Origin
Mz (kN.m) -0.3055404
Factors Gamma
Psi
Alpha
Factored gr
Lane
Other
Total
Compilation : C98: BM60; My Hogging; GR1A; ULS STR/GEO (SUM=-1286.16) L24
LM1 UDL System
-11.35887
1.35
1
1
-33.73584
L210
Footway: UDL System (Footway)
-0.2945362
1.35
1
1
1
-0.3976238
L211
Footway: UDL System (Footway)
-6.087952
1.35
1
1
1
-8.218735
L212
Footway: UDL System (Footway)
-4.697958
1.35
1
1
1
-6.342243
L213
Footway: UDL System (Footway)
-12.63796
1.35
1
1
1
-17.06124
L214
LM1 UDL System
-311.375
1.35
1
0.61
1
1
1
-256.4173
L215
LM1 Tandem System
-312.2353
1.35
1
1
1
1
1
-421.5176
L216
LM1 UDL System
-311.8849
1.35
1
2.2
1
0.2777778
1
-257.3051
L217
LM1 Tandem System
-298.1766
1.35
1
1
1
0.6666667
1
-268.3589
L218
LM1 UDL System
-5.659007
1.35
1
2.2
1
1
1
-1274.408
2.2
1
1
-16.80725 My=-1286.162
SAM v6.50d Copyright © 2012 Bestech Systems Ltd
02/02/2012 11:31
33
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Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job No.: 6.5d Job: Sample Reports Calc. By: DLG Structure: 2 span grillage prestresses beam deck Checked: Eurocodes + UK NA Data File: J:\Marketing\Sample Reports\Sam Eurocode UK Prestressed Beam\6.50d D...\2 span prestress beam deck .sst 02/02/2012 09:29:44 Result Type: Envelope Result For: Beam
Name: E9: GR1A; ULS STR/GEO Mem 50-60: Sh z Effect: Member End Actions
Forces at the member start end correspond with the local member axes directions. At the other end, forces are positive in the opposite direction to the local member axes. With this convention, a positive axial force at each end denotes compression. The table displays the enveloped effect and associated values. The enveloped effect is Member End Forces - Fz (kN). Analysis Run: 01/02/2012 14:48:20 Results shown for: Member End Forces - Fz (kN)
New Selection Reference
Member End Forces Fx (kN)
Joint
49
53
114.2916
-1.96579
525.8172
C105
-2.585628
-433.2182
-15.82357
49
54
114.2916
-1.96579
525.8172
C105
-2.585628
68.69836
-13.94714
50
54
60.94563
-1.625222
512.9767
C105
-1.151738
40.00968
-3.454434
50
55
60.94563
-1.625222
512.9767
C105
-1.151738
1019.329
-0.3517376
51
55
33.78914
-3.655797
426.0225
C113
2.064739
674.1985
-3.804586
51
56
33.78914
-3.655797
426.0225
C113
2.064739
1487.514
3.174661
52
56
14.18757
-4.877261
352.5722
C121
4.608999
1064.631
-4.992496
52
57
14.18757
-4.877261
352.5722
C121
4.608999
1737.724
4.318638
53
57
1.836164
-4.613297
289.4707
C129
5.606502
1238.806
-4.738815
53
58
1.836164
-4.613297
289.4707
C129
5.606502
1791.431
4.068386
54
58
-0.7389602
-3.216076
233.3123
C137
5.591374
1243.906
-3.223764
54
59
-0.7389602
-3.216076
233.3123
C137
5.591374
1689.32
2.91601
55
59
13.77622
-2.013512
-292.0244
C146
3.502448
1703.899
-10.56414
55
60
13.77622
-2.013512
-292.0244
C146
3.502448
1146.398
-6.720165
56
60
38.65846
5.128567
-363.1734
C154
6.038212
1668.201
11.65262
56
61
38.65846
5.128567
-363.1734
C154
6.038212
974.8671
1.861671
57
61
75.81142
5.968157
-428.0912
C162
4.055623
1443.039
11.47534
57
62
75.81142
5.968157
-428.0912
C162
4.055623
625.7743
0.08159752
58
62
110.6654
4.032943
-500.8601
C170
1.035365
1053.722
7.826455
58
63
110.6654
4.032943
-500.8601
C170
1.035365
97.5351
0.1272036
59
63
114.069
-0.4487549
-583.3914
C178
-1.563676
489.9939
1.894806
59
64
114.069
-0.4487549
-583.3914
C178
-1.563676
-623.7527
2.751519
60
64
86.18819
-2.104652
-655.6394
C186
-6.772206
-294.7813
4.316139
Load Ref
Fy (kN)
Member End Moments
Member
Fz (kN)
Origin
Mx (kN.m)
My (kN.m)
Unfactored
Mz (kN.m)
Factors
Type
Gamma
Psi
Alpha
Factored gr
Lane
Other
Total
Compilation : C186: BM60; Shear z-; GR1A; ULS STR/GEO (SUM=-655.64) L231
LM1 UDL System
-1.622997
1.35
1
2.2
1
1
1
-4.8203
L425
LM1 UDL System
-1.929261
1.35
1
2.2
1
1
1
-5.729905
L467
Footway: UDL System (Footway)
-0.7695409
1.35
1
1
1
-1.03888
L468
Footway: UDL System (Footway)
-2.989803
1.35
1
1
1
-4.036234
L469
Footway: UDL System (Footway)
-0.03466308
1.35
1
1
-0.04679515
L470
LM1 UDL System
-73.47056
1.35
1
2.2
1
0.2777778
1
-60.61321
L471
LM1 Tandem System
-143.3186
1.35
1
1
1
0.6666667
1
-128.9867
L472
LM1 UDL System
-118.1804
1.35
1
0.61
1
1
1
-97.32159
L473
LM1 Tandem System
-261.1537
1.35
1
1
1
1
1
-352.5574
L474
LM1 UDL System
-0.02488137
1.35
1
2.2
1
1
1
-0.07389768
L475
LM1 UDL System
-0.1395501
1.35
1
2.2
1
1
1
1
-603.6339
-0.4144637 Fz=-655.6394
60
65
86.18819
SAM v6.50d Copyright © 2012 Bestech Systems Ltd
-2.104652
-655.6394
C186
-6.772206
02/02/2012 11:35
35
-920.6156
6.325115
1
36
Pre-tensioned Pre-stressed Beam Bridge Design Example
7. Section Properties a) Mid Span b) Internal Support
37
38
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
Design code: Analysis:
EN 1992-2:2005 with UK National Annex (modified) Section Properties EN 1990 Equation 6.14 SLS Characteristic Exposure Class: XD1, XD2, XS1, XS2, XS3 Section Ref 1 at 10.5m from left end of beam
Section Ref: 1 "Section 1" depth of precast beam total depth of section
= 1300.0 mm = 1470.0 mm
Section properties are detailed below in the following sequence: PRECAST BEAM ALONE COMPOSITE BEAM TO STAGE 1
PRECAST BEAM ALONE Elastic section properties area, height to centroid, overall depth, 2nd moment of area, section modulus at bottom, section modulus at top,
Ac = 5.372E5 mm² za = 576.039 mm h = 1300.0 mm Iy y = 9.2977E10 mm⁴ Wb = 9.2977E10 / -576.04 -1.6141E8 mm³ Wt = 9.2977E10 / (1300.0-576.039) 1.28428E8 mm³
COMPOSITE BEAM COMPOSITE BEAM TO STAGE 1 Elastic section properties Area mm²
centroid mm
Sy mm³
Iy y mm⁴
Iy y (z=0) mm⁴
Precast beam
537225.68
Stage 1 i.s.
388869.57 1372.4329 1.057 5.04816E8 1.18215E9 6.9401E11
TOTAL
576.0392
α
1.0 3.09463E8 9.2977E10 2.7124E11
905051.14(transformed)
SAM v6.50d
8.14279E8
06/02/2012 10:28:43
© 2012 Bestech Systems Ltd
39
9.6525E11
Page: 1
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
height to centroid
= = Iy y = =
8.14279E8/9.051E5 899.705 mm 9.6525E11 - (9.051E5*899.705²) 2.3264E11 mm⁴
ELASTIC SECTION PROPERTIES SUMMARY TABLE Level mm
Iy y mm⁴
zn a mm
W mm³
Precast beam only Precast beam
B
0.0
T
1300.0
9.2977E10
576.0392
-1.6141E8 1.28428E8
In situ to stage 1 Precast beam In situ Stage 1
SAM v6.50d
B
0.0
T
1300.0
2.3264E11
899.70476
5.81164E8
B
1270.0
6.64191E8
T
1470.0
4.31262E8
06/02/2012 10:28:43
© 2012 Bestech Systems Ltd
40
-2.5857E8
Page: 2
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
Design code: Analysis:
EN 1992-2:2005 with UK National Annex (modified) Section Properties EN 1990 Equation 6.14 SLS Characteristic Exposure Class: XD1, XD2, XS1, XS2, XS3 Section Ref 2 at 21m from left end of beam
Section Ref: 2 "Section 2" depth of precast beam total depth of section
= 1300.0 mm = 1470.0 mm
Section properties are detailed below in the following sequence: PRECAST BEAM ALONE COMPOSITE BEAM TO STAGE 1
PRECAST BEAM ALONE Elastic section properties area, height to centroid, overall depth, 2nd moment of area, section modulus at bottom, section modulus at top,
Ac = 5.372E5 mm² za = 576.039 mm h = 1300.0 mm Iy y = 9.2977E10 mm⁴ Wb = 9.2977E10 / -576.04 -1.6141E8 mm³ Wt = 9.2977E10 / (1300.0-576.039) 1.28428E8 mm³
COMPOSITE BEAM COMPOSITE BEAM TO STAGE 1 Elastic section properties Area mm²
centroid mm
Sy mm³
Iy y mm⁴
Iy y (z=0) mm⁴
Precast beam
537225.68
Stage 1 i.s.
388869.57 1372.4329 1.057 5.04816E8 1.18215E9 6.9401E11
Rft in IS 1
4908.7385
TOTAL
576.0392
α
1407.5
1.0 3.09463E8 9.2977E10 2.7124E11 0.211
928282.4(transformed)
SAM v6.50d
3.2698E7 907471.06 8.46977E8
02/02/2012 11:41:05
© 2012 Bestech Systems Ltd
41
4.6E10 1.0113E12
Page: 1
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
height to centroid
= = Iy y = =
8.46977E8/9.283E5 912.413 mm 1.0113E12 - (9.283E5*912.413²) 2.3848E11 mm⁴
ELASTIC SECTION PROPERTIES SUMMARY TABLE Level mm
Iy y mm⁴
zn a mm
W mm³
Precast beam only Precast beam
B
0.0
T
1300.0
9.2977E10
576.0392
-1.6141E8 1.28428E8
In situ to stage 1 Precast beam In situ Stage 1
SAM v6.50d
B
0.0
T
1300.0
2.3848E11
912.41288
6.1529E8
B
1270.0
7.05066E8
T
1470.0
4.52167E8
02/02/2012 11:41:05
© 2012 Bestech Systems Ltd
42
-2.6137E8
Page: 2
Pre-tensioned Pre-stressed Beam Bridge Design Example
8. Data Summary after Tendon Design
43
44
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
DATA SUMMARY
ANALYSIS TYPE: EN 1992-2 Pre-tensioned Prestressed Beam With UK National Annex (modified) BEAM DETAILS Span:
Total length of pre-tensioned beam Distance from left support to beam end face Distance from right support to beam end face Total distance between supports
: : : :
21 m 0 m 0 m 21 m
Beam section varies along length of beam. Number of different sections No. of longitudinal construction stages No. of superimposed construction stages
: 2 : 2 : 1
Section 1
SAM v6.50d
02/02/2012 11:47:59
© 2012 Bestech Systems Ltd
45
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Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
Precast beam: Precast beam is standard section: Y7 Beam Property set: 2 "C40/50 Ecm 35.2 " Age of beam at transfer: 4.0 days Corresponding concrete strength at transfer: 23.8094 MPa In situ concrete - stage 1A: In situ is from standard section: - width : 2.0 m - depth : 0.2 m Property set: 1 "C31/40 Ecm 33.3 " Age of beam when stage 1A concrete is cast: 60 days Shear resistance width:
216.0 mm
Section 2
Precast beam: Precast beam is standard section: Y7 Beam Property set: 2 "C40/50 Ecm 35.2 " Age of beam at transfer: 4.0 days Corresponding concrete strength at transfer: 23.8094 MPa In situ concrete - stage 1B: In situ is from standard section: - dimensions (m) : 2.0 0.0 0.2 0.0 0.0 0.0 Property set: 1 "C31/40 Ecm 33.3 " Age of beam when stage 1B concrete is cast: 60 days Shear resistance width:
SAM v6.50d
216.0 mm
02/02/2012 11:47:59
© 2012 Bestech Systems Ltd
46
Page: 2
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
Tendons:
y-z coordinates mm
area transmission coeffients mm² α1 α2 ηp 1 η1 ηp 2
φ mm
draw-in mm/beam
property ref
-275.0
60.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
-225.0
60.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
-175.0
60.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
-125.0
60.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
-75.0
60.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Debonded
0.0
60.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Debonded
75.0
60.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Debonded
125.0
60.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
175.0
60.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
225.0
60.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
275.0
60.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
-75.0
110.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Debonded
-25.0
110.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Debonded
25.0
110.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Debonded
75.0
110.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Debonded
-25.0
210.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
25.0
210.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
-25.0
260.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
25.0
260.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
-80.0
1200.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
80.0
1200.0
150.0
1.0 0.19
3.2
1.0
1.2 16.0
3.0
4 Full stress
Debonded Tendons:
y-z coordinates mm
distance from left end (m) start end
-75.0
60.0
2.0
19.0
0.0
60.0
2.0
19.0
75.0
60.0
2.0
19.0
-75.0
110.0
2.5
18.5
-25.0
110.0
2.5
18.5
25.0
110.0
2.5
18.5
75.0
110.0
2.5
18.5
SAM v6.50d
02/02/2012 11:47:59
© 2012 Bestech Systems Ltd
47
Page: 3
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
Reinforcement:
y-z coordinates mm
diameter mm
Property ref
Start m
End m
Length m
900.0
1407.5
25.0
3
15.0
21.0
6.0
700.0
1407.5
25.0
3
15.0
21.0
6.0
500.0
1407.5
25.0
3
15.0
21.0
6.0
300.0
1407.5
25.0
3
15.0
21.0
6.0
100.0
1407.5
25.0
3
15.0
21.0
6.0
-100.0
1407.5
25.0
3
15.0
21.0
6.0
-300.0
1407.5
25.0
3
15.0
21.0
6.0
-500.0
1407.5
25.0
3
15.0
21.0
6.0
-700.0
1407.5
25.0
3
15.0
21.0
6.0
-900.0
1407.5
25.0
3
15.0
21.0
6.0
Location of sections
Position along span from left support: dimension (m) proportion
Section
0.0
0.0
1
"Section 1"
18.0
0.857
1
"Section 1"
18.0
0.857
2
"Section 2"
21.0
1.0
2
"Section 2"
PROPERTIES DETAILS ref: 1
Type: Concrete - Parabola-Rectangle Name: C31/40 Ecm 33.3
Design Code Part Characteristic strength
: fc k : fc k , c u b e : modulus of elasticity Ec m : Elastic modulus - long term : Ultimate compressive strain εc u : Tensile strength fc t m : Cement Class : Contains Silica Fume : Coefficient of thermal expansion: Density : Density increase for rft. :
SAM v6.50d
EN 1992-2 31.875 MPa 40.0 MPa 33.314469 GPa 13.325787 GPa 0.0035 -3.015931 MPa N - Normal and rapid hardening No 0.00001 /°C 24.0 kN/m³ 1.0 kN/m³
02/02/2012 11:47:59
© 2012 Bestech Systems Ltd
48
Page: 4
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
ref: 2
Type: Concrete - Parabola-Rectangle Name: C40/50 Ecm 35.2
Design Code Part Characteristic strength
: fc k : fc k , c u b e : modulus of elasticity Ec m : Elastic modulus - long term : Ultimate compressive strain εc u : Tensile strength fc t m : Cement Class : Contains Silica Fume : Coefficient of thermal expansion: Density : Density increase for rft. :
ref: 3
Type: Reinforcing Steel - Horizontal Name: Grade 500 Es 200.0
Yield strength modulus of elasticity Characteristic strain limit Density
ref: 4
EN 1992-2 40.0 MPa 50.0 MPa 35.220462 GPa 14.088185 GPa 0.0035 -3.508821 MPa N - Normal and rapid hardening No 0.00001 /°C 24.0 kN/m³ 1.0 kN/m³
fy k : Es : εu k : :
500.0 MPa 200.0 GPa 0.025 77.0 kN/m³
Type: Prestressing Steel - Horizontal Name: Grade 1600 Ep 195.0
tensile strength fp k : 0,1% proof stress fp 0 , 1 k : modulus of elasticity Ep : Relaxation loss after 1000 hours: Relaxation Class : Density :
1860.0 1600.0 195.0 8.0 1 77.0
MPa MPa GPa % kN/m³
ANALYSIS DATA Data for loss calculations:
%
Shrinkage strain is calculated from the data provided Creep coefficient is calculated from the data provided Differential shrinkage is calculated from the data provided Percentage of total long term loss which occurs before the section is made composite is 30.18 Age at start of drying shrinkage Ambient relative humidity Ambient temperature Maximum Curing temperature
= = = =
1.0 80.0 20.0 20.0
day % °C °C
Creep calculations are based upon EN 1992-1-1
SAM v6.50d
02/02/2012 11:47:59
© 2012 Bestech Systems Ltd
49
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Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
Data for shear calculations: Material property for transverse reinforcement: Grade 500 Es 200.0 Angle between concrete strut and beam axis, θ = 35.0° Angle between shear reinforcement and beam axis, α = 90.0° Enhancement close to supports is ignored Surface condition for precast / in-situ interface = Smooth Longitudinal force ratio β is calculated Angle for compression strut in slab, θf = 26.0°
SAM v6.50d
02/02/2012 11:47:59
© 2012 Bestech Systems Ltd
50
Page: 6
Pre-tensioned Pre-stressed Beam Bridge Design Example
9. Temperature Gradient
51
52
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
DIFFERENTIAL TEMPERATURE EN 1991-1-5:2003 Figure 6.2 non-linear Temperature Profile
EN 1991-1-5:2003 Figure 6.2 non-linear Temperature Profile Figure 6.2c: Type 3b. Concrete Beams Surfacing : surfaced Surfacing thickness : 0.1 m
Top warmer than bottom height m Temperature °C
Bottom warmer than top height m Temperature °C
0.0
13.5
0.0
-8.376
0.15
3.0
0.25
-0.56
0.4
0.0
0.45
0.0
1.27
0.0
1.02
0.0
1.47
2.5
1.22
-1.03
1.47
-6.488
Relaxing Forces Moment kN.m
Axial kN
Heating Temperature difference
-413.8371
-1015.993
Cooling Temperature difference
143.38408
976.55933
Note: The reinforcement has been ignored in the calculation of the above relaxing moments
Self Equilibrating Stresses
SAM v6.50d
02/02/2012 11:50:10
© 2012 Bestech Systems Ltd
53
Page: 1
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
Rectangle
Distance to top of section - m
Stress Heating
0.0
2.4760279
0.15
-0.769597
0.2
-0.885352
- MPa Cooling -1.437326 0.5291631
Y7 Beam
Distance to top of section - m
Stress Heating
0.17
-0.862578
0.25 0.4
0.2475882 1.0791869
-1.425518
0.45
1.1531531
1.02
0.8018382
1.22
SAM v6.50d
- MPa Cooling
0.3157990
1.27
0.1221205
1.47
1.358411
-1.760619
02/02/2012 11:50:10
© 2012 Bestech Systems Ltd
54
Page: 2
Pre-tensioned Pre-stressed Beam Bridge Design Example 10.
55
Shrinkage & Creep
56
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
DIFFERENTIAL SHRINKAGE MODIFIED BY CREEP - Primary Load effects
Section Reference: 2
"Section 2"
Evaluate the shrinkage strains using EN 1992-1-1 clause 3.1.4(6) Shrinkage in precast at time t = ∞ Age of concrete at time considered, t = ∞ Age of concrete at loading, t0 = 4.0 days Age of concrete at start of drying, ts = 1.0 days Relative humidity of enviroment, RH = 80.0 % Average temperature, Ta = 20.0 °C Type of cement = Class N for which, EN1992-1-1 Annex B.1(2) α = 0.0 Annex B.2(1) αd s 1 = 4.0 Annex B.2(1) αd s 2 = 0.12 3.1.2(6) s = 0.25 Characteristic strength of concrete, fc k = 40.0 MPa Mean compressive strength at 28 days (Table 3.1), fc m = fc k + 8.0 = 48.0 MPa Mean comp. strength at 4.0 days (3.1.2(6) ), fc m (t0 ) = fc m .exp[s.(1-√(28/t0 )] = 48.0*exp[0.25*(1-√(28/4.0)] = 31.809 MPa Constant value from Annex B.2(1) fc m 0 = 10.0 MPa Total Shrinkage: εc s = εc d + εc a
SAM v6.50d
(3.8)
02/02/2012 11:54:19
© 2012 Bestech Systems Ltd
57
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Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
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Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
Drying Shrinkage - Expression (3.9): εc d (t) = βd s (t,ts ).kh .εc d , 0 βd s (t,ts ) = 1.0 for t = ∞ From Table 3.3: kh = 0.79438 From Annex B, Expression (B.11):
-6
εc d , 0 = 0.85[(220+110.αd s 1 ).(exp(-αd s 2 .fc m /fc m 0 )].10 .βR H βR H = 1.55[1.0-(RH/100)³] (B.12) = 0.7564 For cement class N, αd s 1 = 4 αd s 2 = 0.12 hence, -6
εc d , 0 = 0.85[(220+110*4.0)*exp(-0.12*48.0/10.0)]*10 *0.7564 -6
and,
= 238.54*10
εc d (t) = 1.0*0.79438*238.54*10
-6
-6
= 189.491*10
Autogenous Shrinkage - Expression (3.11): εc a (t) = βa s (t).εc a (∞) βa s (t) = 1.0 for t = ∞ εc a (∞) = 2.5*(fc k -10.0)*10 -6 = 75.0*10 hence, εc a (t) = 1.0*75.0*10 = 75.0*10
-6
-6
-6
Total Shrinkage: εc s = εc d (t) + εc a (t) = 189.49131 + 75.0 = 264.49131*10
-6
Shrinkage in in-situ concrete at time t = ∞ Age of concrete at time considered, Age of concrete at loading, Age of concrete at start of drying, Relative humidity of enviroment, Average temperature, Type of cement for which, EN1992-1-1 Annex B.1(2) Annex B.2(1) Annex B.2(1) 3.1.2(6)
SAM v6.50d
t t0 ts RH Ta
α αd s 1 αd s 2 s
= ∞ = 4.0 days = 1.0 days = 80.0 % = 20.0 °C = Class N = 0.0 = 4.0 = 0.12 = 0.25
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Characteristic strength of concrete, fc k = 31.875 MPa Mean compressive strength at 28 days (Table 3.1), fc m = fc k + 8.0 = 39.875 MPa Mean comp. strength at 4.0 days (3.1.2(6) ), fc m (t0 ) = fc m .exp[s.(1-√(28/t0 )] = 39.875*exp[0.25*(1-√(28/4.0)] = 26.425 MPa Constant value from Annex B.2(1) fc m 0 = 10.0 MPa Total Shrinkage: εc s = εc d + εc a
(3.8)
Drying Shrinkage - Expression (3.9): εc d (t) = βd s (t,ts ).kh .εc d , 0 βd s (t,ts ) = 1.0 for t = ∞ From Table 3.3: kh = 0.79438 From Annex B, Expression (B.11):
-6
εc d , 0 = 0.85[(220+110.αd s 1 ).(exp(-αd s 2 .fc m /fc m 0 )].10 .βR H βR H = 1.55[1.0-(RH/100)³] (B.12) = 0.7564 For cement class N, αd s 1 = 4 αd s 2 = 0.12 hence, -6
εc d , 0 = 0.85[(220+110*4.0)*exp(-0.12*39.88/10.0)]*10 *0.7564 -6
and,
= 262.969*10
εc d (t) = 1.0*0.79438*262.969*10
-6
-6
= 208.897*10
Autogenous Shrinkage - Expression (3.11): εc a (t) = βa s (t).εc a (∞) βa s (t) = 1.0 for t = ∞ εc a (∞) = 2.5*(fc k -10.0)*10 -6 = 54.6875*10 hence, εc a (t) = 1.0*54.6875*10
-6
-6
-6
= 54.6875*10
Total Shrinkage: εc s = εc d (t) + εc a (t) = 208.89738 + 54.6875 = 263.58488*10
-6
Shrinkage in precast at time in-situ is placed (t = 60 days)
SAM v6.50d
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Age of concrete at time considered, t = 60.0 days Age of concrete at loading, t0 = 4.0 days Age of concrete at start of drying, ts = 1.0 days Relative humidity of enviroment, RH = 80.0 % Average temperature, Ta = 20.0 °C Type of cement = Class N for which, EN1992-1-1 Annex B.1(2) α = 0.0 Annex B.2(1) αd s 1 = 4.0 Annex B.2(1) αd s 2 = 0.12 3.1.2(6) s = 0.25 Characteristic strength of concrete, fc k = 40.0 MPa Mean compressive strength at 28 days (Table 3.1), fc m = fc k + 8.0 = 48.0 MPa Mean comp. strength at 4.0 days (3.1.2(6) ), fc m (t0 ) = fc m .exp[s.(1-√(28/t0 )] = 48.0*exp[0.25*(1-√(28/4.0)] = 31.809 MPa Constant value from Annex B.2(1) fc m 0 = 10.0 MPa Total Shrinkage: εc s = εc d + εc a
(3.8)
Drying Shrinkage - Expression (3.9): εc d (t) = βd s (t,ts ).kh .εc d , 0 βd s (t,ts ) = (t-ts )/[(t-ts )+0.04√h0 ³]
(3.10)
t-ts = 60.0-1.0 = 59.0 days βd s (t,ts ) = 59.0/(59.0+0.04√255.62³) = 0.26519 From Table 3.3: kh = 0.79438 From Annex B, Expression (B.11):
-6
εc d , 0 = 0.85[(220+110.αd s 1 ).(exp(-αd s 2 .fc m /fc m 0 )].10 .βR H βR H = 1.55[1.0-(RH/100)³] (B.12) = 0.7564 For cement class N, αd s 1 = 4 αd s 2 = 0.12 hence, -6
εc d , 0 = 0.85[(220+110*4.0)*exp(-0.12*48.0/10.0)]*10 *0.7564 -6
and,
= 238.54*10
εc d (t) = 0.26519*0.79438*238.54*10
-6
-6
= 50.2528*10
SAM v6.50d
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Autogenous Shrinkage - Expression (3.11): εc a (t) = βa s (t).εc a (∞) βa s (t) = 1-exp(-0.2√t) = 1.0-exp(-0.2*√60.0) = 0.78758 εc a (∞) = 2.5*(fc k -10.0)*10 -6 = 75.0*10 hence,
(3.13)
-6
-6
εc a (t) = 0.78758*75.0*10 -6
= 59.0686*10
Total Shrinkage: εc s = εc d (t) + εc a (t) = 50.252794 + 59.068556 = 109.32135*10
-6
Summary of data Section is composite from t = 60 days at time t = 60 days:
shrinkage strain in precast concrete, at time t = ∞
εa = 109.321 x10
shrinkage strain in in-situ concrete, differential shrinkage strain, εd i f f = εc - ( εb - εa )
εc = 263.585 x10
shrinkage strain in precast concrete,
εb = 264.491 x10
= 263.585 - (264.491-109.321) = 108.415 x10 creep coefficient, φ = 2.00881
-6
-6 -6
-6
(-φ)
1 - e creep reduction factor Φ = ——————————— φ 2nd moment of area of transformed section, Iy y height of centroid, za total transformed area, Ac elastic modulus of precast concrete, Ec , p elastic modulus of in situ concrete, Ec , i modular ratio
SAM v6.50d
= 0.43102 = = = = =
2.33E11 899.705 9.051E5 35.2205 33.3145
mm⁴ mm mm² GPa GPa
n0 = Ec , p / Ec , i = 35.2205/33.3145 = 1.05721
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Stage 1 In-situ area of concrete : 3.889E5 mm² height to centroid : 1372.43 mm force required to restrain shrinkage: Ac .Φ.Ec , i .εd i f f = = corresponding moment = =
-6
3.889E5*0.65982*33.3145*108.415 x10 605.383 kN 605.383*(1372.43-899.705) 286.182 kN.m (sagging)
self equilibrating stress in precast beam: top of beam = P/Ac + M/Wt = 605.38331/905051.14 + 286.18174/5.81164E8 = 1.1613224 MPa soffit of beam
= P/Ac + M/Wb = 605.38331/905051.14 + 286.18174/-2.5857E8 = -0.437889 MPa
self equilibrating stress in stage 1 concrete: at top = ( P/Ac + M.(zt -za )/Iy y + Φ.εd i f f .Ec , p )/α = (605.38331/905051.14 + 286.18174*570.29524/2.3264E11 + 0.4310270*-1.084E-4*35.220462 ) /1.0572122 = -0.260490 MPa at bottom
SAM v6.50d
= ( P/Ac + M.(zb -za )/Iy y + Φ.εd i f f .Ec , c )/α = (605.38331/905051.14 + 286.18174*370.29524/2.3264E11 + 0.4310270*-1.084E-4*35.220462) /1.0572122 = -0.493208 MPa
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Pre-tensioned Pre-stressed Beam Bridge Design Example
11.
Verification: Transfer Stresses
63
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Design code: EN 1992-2:2005 with UK National Annex (modified) Analysis: Stresses at Transfer EN 1990 Equation 6.14 SLS Characteristic Section Ref 1 at 10.5m from left end of beam
Section details:
Ref 1 "Section 1" at 0.5 x span = 10.5 m from left end of beam
Analysis:
Stresses at Transfer Serviceability Limit State: Characteristic
- EN 1990 Equation 6.14
ACTUAL STRESSES IN PRECAST BEAM No. of tendons fully bonded at this section: No. of tendons fully debonded at this section: No. of tendons deflected at this section:
SAM v6.50d
21 0 0
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Maximum Stressing Force - EN 1992-1-1 Clause 5.10.2.1(1)P For tendon property Grade 1600 Ep 195.0 k1 .fp k = 0.8*1860.0 = 1488.0 MPa k2 .fp 0 , 1 k = 0.9*1600.0 = 1440.0 MPa Wedge draw-in loss Clause 5.10.4(1)(i) draw-in strain = 0.003/21.0 = 1.43E-4 loss = Ep . strain = 195.0*1.43E-4 = 27.8571 MPa Heat Curing Clause 5.10.4(1)(ii)(Note) Concrete is cured at ambient temperature
Immediate Losses - EN 1992-1-1 Clause 5.10.4
height No of mm tendons 60.0
fp MPa
k1 /k2
draw-in MPa
heat cure MPa
area mm²
initial force kN
11
1600.0
0.9
27.8571
0.0
150.0
2330.0357
110.0
4
1600.0
0.9
27.8571
0.0
150.0
847.28571
210.0
2
1600.0
0.9
27.8571
0.0
150.0
423.64286
260.0
2
1600.0
0.9
27.8571
0.0
150.0
423.64286
1200.0
2
1600.0
0.9
27.8571
0.0
150.0
423.64286
TOTAL
21
4448.25
In accordance with clause 5.10.9(1), for SLS, the Characteristic value must be used. With ri n f = 1.0, Pk , i n f = 4448.25 kN
Friction Clause 5.10.4(1)(i)
All tendons are straight in this beam.
Initial Relaxation Clause 5.10.4(1)(ii)
Loss is calculated from clause 3.3.2(7) For tendon property Grade 1600 Ep 195.0 relaxation loss at 1000 hours, ρ1 0 0 0 = 8.0 % μ = σp i / fp k = 1440.0-27.8571-0.0/1860.0 = 0.75921 time after tensioning = 96.0 hours for Class 1 relaxation, use Expression (3.28) 6.7μ
0.75(1-μ)
5.39 . ρ1 0 0 0 . e . [t/1000] . 10 -5 = 5.39 * 8.0 * 161.863 * 0.65495 * 10 = 0.04571
SAM v6.50d
-5
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relaxation
height No of mm tendons 60.0
area x σp i
%
After relaxation loss kN
force kN
moment kN.m
11
2330.04 4.57 106.51239
2223.5233
133.4114
110.0
4
847.286 4.57 38.731779
808.55394
88.940933
210.0
2
423.643 4.57 19.365889
404.27697
84.898163
260.0
2
423.643 4.57 19.365889
404.27697
105.11201
1200.0
2
423.643 4.57 19.365889
404.27697
485.13236
4244.9082
897.49487
TOTAL
21
Moment about the centroid of the precast beam: Mr = 897.49487-(4244.9082*0.5760392) = -1547.739 kN.m Corresponding stresses: top stress = 4244.9082/537225.68+-1547.739/1.2843E8 = 7.9015362+-12.05139 = -4.149853 MPa bottom stress = 4244.9082/537225.68+-1547.739/-1.614E8 = 7.9015362+9.5890175 = 17.490554 MPa Self weight moment: c.s.a. = 5.372E5 mm²
[1]
density = 24.0 kN/m³ + 1.0 kN/m³ + 1.0 kN/m³ = 26.0 kN/m³ self weight = 5.372E5*26.0 = 13.9679 kN/m beam length = 21.0 m distance = 10.5 m Ms w = 0.5*13.9679*10.5*(21.0-10.5) = 769.979 kN.m Corresponding stresses: top stress = 769.979/1.2843E8 = 5.9954 MPa bottom stress = 769.979/-1.614E8 = -4.7704 MPa
Elastic Deformation - Clause 5.10.4(1)(iii) stress at top of precast beam stress at bottom of precast beam depth of precast beam elastic modulus of concrete at transfer
SAM v6.50d
= 1.84555 MPa = 12.7201 MPa = 1300.0 mm = 31.1307 GPa
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height
mm 60.0
No of tendons
conc stress MPa
conc strain
tendon force kN
tendon moment kN.m
11
12.21824
3.925E-4
126.28096
7.5768575
110.0
4
11.79999
3.79E-4
44.348407
4.8783248
210.0
2
10.96348
3.522E-4
20.602262
4.326475
260.0
2
10.54523
3.387E-4
19.816291
5.1522358
1200.0
2
2.682055
8.615E-5
5.0400412
6.0480495
216.08796
27.981943
TOTAL
21
Moment about the centroid of the precast beam: Me d = 27.981943-(216.08796*0.5760392) = -96.49319 kN.m hence, top stress = 1.8455-216.08796/537.22568--96.49319/1.2843E8 = 1.8455-0.4022294--0.751339 = 2.1946575 MPa bottom stress = 12.72-216.08796/537.22568--96.49319/-1.614E8 = 12.72-0.4022294-0.5978237 = 11.720096 MPa After a further 2 iterations of the above process, the top and bottom stresses are as follows: top stress = 2.16502461 MPa bottom stress = 11.7912468 MPa
Max Prestress Force after transfer - EN 1992-1-1 Clause 5.10.3.(2) For tendon property Grade 1600 Ep 195.0 k7 .fp k = 0.75*1860.0 = 1395.0 MPa k8 .fp 0 , 1 k = 0.85*1600.0 = 1360.0 MPa Maximum tendon stress after transfer = 1329.4 MPa which is not greater than 1360.0 and therefore OK.
TOTAL LOSS OF PRESTRESS SUMMARY Initial stressing force Prestress after all transfer losses
= 4448.25 kN = 4043.05 kN
Corresponding loss = 9.11 %
LIMITING STRESSES IN PRECAST BEAM Compression EN 1992-1-1 Clause 3.1.2(5) & 3.1.2(6) For transfer at t = 4.0 days fc k (t) = fc m (t) = βc c (t) = for Class N cement,
SAM v6.50d
fc m (t) - 8.0 βc c (t).fc m exp{s[1-√(28/t)]} s = 0.25
Equation 3.1 Equation 3.2
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hence
βc c (t) = = fc m = = fc m (t) = = and fc k (t) = =
exp{0.25[1.0-√28/4.0)]} 0.66269 fc k + 8.0 (from Table 3.1) 48.0 MPa 0.66269*48.0 31.8094 31.8094 - 8.0 MPa 23.8094 MPa
EN 1992-1-1 Clause 5.10.2.2(5) σc
35.0 MPa 1-RH/100 φR H = [ 1 + ——————————— . α1 ] .α2 0.33 0.1*h0 α1 = [35.0/48.0] α2 = [35.0/48.0] α3 = [35.0/48.0]
0.7 0.2 0.5
Expression (B.3b)
= 0.80163 = 0.93878 = 0.85391
φR H = [1.0 + (1.0-0.8) / (0.1*249.811 = 1.17777
0.33
) * 0.80163]*0.93878
β(fc m ) = 16.8/√fc m = 16.8/√48.0 = 2.42487
For Permanent Loads
In the absence of heat curing t0 , T =
SAM v6.50d
Expression (B.4)
4.0 days
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age is adjusted for expression (B.5) (for cement type and temperature) - for cement class N (α = 0) 9.0
t0 = t0 , T . [ —————————————— + 1.0 ]
= 4.0 =
2.0 + t0 , T 9.0
α
* [ —————————————— + 1.0 ]
2.0 + 4.0 4.0 day
>=0.5
1.2
Expression (B.9)
0
1.2
0.2
β(t0 ) = 1/(0.1+t0 ) 0.2 = 1/(0.1+4.0 ) = 0.70446 βc (t,t0 ) = 1.0 for time t = ∞ hence from (B.1) and (B.2): φ(t,t0 ) = 1.17777*2.42487*0.70446 = 2.01193
Expression (B.5)
Check for creep non-linearity EN 1992-1-1 clause 3.1.4(4) At the level of the centroid of the tendons, the compressive stress in the concrete at time t0 = 8.31165 MPa. This does not exceed 0.45*fck(t0), i.e. 18.0 MPa, so non-linear creep is not considered
Shrinkage Strain for concrete - EN 1992-1-1 clause 3.1.4(6) Total Shrinkage: εc s = εc d + εc a
(3.8)
Drying Shrinkage - Expression (3.9): εc d (t) = βd s (t,ts ).kh .εc d , 0 βd s (t,ts ) = 1.0 for t = ∞ From Table 3.3: kh = 0.80018 From Annex B, Expression (B.11):
-6
εc d , 0 = 0.85[(220+110.αd s 1 ).(exp(-αd s 2 .fc m /fc m 0 )].10 .βR H βR H = 1.55[1.0-(RH/100)³] (B.12) = 0.7564 For cement class N, αd s 1 = 4 αd s 2 = 0.12 hence, -6
εc d , 0 = 0.85[(220+110*4.0)*exp(-0.12*48.0/10.0)]*10 *0.7564 -6
= 238.54*10
SAM v6.50d
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and,
εc d (t) = 1.0*0.80018*238.54*10
-6
-6
= 190.877*10
Autogenous Shrinkage - Expression (3.11): εc a (t) = βa s (t).εc a (∞) βa s (t) = 1.0 for t = ∞ εc a (∞) = 2.5*(fc k -10.0)*10 -6 = 75.0*10 hence, εc a (t) = 1.0*75.0*10 = 75.0*10
-6
-6
-6
Total Shrinkage: εc s = εc d (t) + εc a (t) = 190.87688 + 75.0 = 265.87688*10
-6
Further Relaxation Clause 5.10.6(1)(b) Loss is calculated from clause 3.3.2(7) For tendon property Grade 1600 Ep 195.0 relaxation loss at 1000 hours, ρ1 0 0 0 = 8.0 % time after tensioning = 500000.0 hours μ = 0.75921 (as calculated for initial relaxation loss above) for Class 1 relaxation, use Expression (3.28) 6.7μ
0.75(1-μ)
-5
5.39 . ρ1 0 0 0 . e . [t/1000] . 10 -5 = 5.39 * 8.0 * 161.863 * 3.07185 * 10 = 0.21440 With the initial relaxation deducted, the variation in tendon stress from relaxation becomes: Δσp r / σp i = 0.21440 - 0.04571 = 0.16868 Summary of the above for Expression (5.46):
Estimated shrinkage strain εc s Creep coefficient at t for loading at t0 φ(t,t0 ) Relaxation, Δσp r Modulus of elasticity for prestressing steel Ep Modulus of Elasticity for concrete Ec m Area of all prestressing Ap Area of concrete section Ac Second moment of area of concrete section Ic
SAM v6.50d
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= = = = = = = =
265.877 2.01193 238.212 195.0 37.9636 3150.0 9.051E5 2.33E11
-6
x10 MPa GPa GPa mm² mm² mm⁴
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Ep /Ec m Ep /Ec m .Ap /Ac Ac /Ic
= 195.0/37.9636 = 5.1365*3150.0/9.051E5 = 9.051E5/2.33E11
= 5.1365 = 0.01788 = 3.8904
In the table below the following vary with tendon height: σc , Q P = Stress in concrete adjacent to tendons zc p = Section centre of gravity to tendons φ(t,t0 ) = Creep Coefficient (if non-linear creep is considered)
height
mm
Ap mm²
shrink εc s .Ep MPa
relax 0.8Δσp r MPa
φ(t,t0 )
51.846 190.57 2.012
σc , Q P MPa
creep Ep /Ec m .φ.σ MPa
denom ΔPc + s + r kN
zc p mm
60.0
1650.0
110.0
600.0
51.846 190.57 2.012 8.5109 87.954 789.705
8.605 88.927 839.705 1.175 465.43833
210.0
300.0
51.846 190.57 2.012 8.3226 86.008 689.705 1.133 86.962166
260.0
300.0
51.846 190.57 2.012 8.2284 85.035 639.705 1.121 87.637686
1200.0
300.0
51.846 190.57 2.012 6.4583 66.742
1.16 170.90467
-300.3 1.063 87.248992
Total force loss: Total moment loss:
898.19184 192.47246
Mc s r = 192.47246-(898.19184*0.8997047) = -615.635 kN.m
Corresponding stresses - before composite: top stress = ( 898.192/5.372E5+-615.64/1.284E8 )* 0.286 = ( 1.6719079+-4.793611 )* 0.286 = -0.893716 MPa bottom stress = ( 898.192/5.372E5+-615.64/-1.61E8 )* 0.2862 = ( 1.6719079+3.8141679 )* 0.286 = 1.5706162 MPa top stress bottom stress
= = = = = =
- after composite: ( 898.192/9.051E5+-615.64/5.812E8 )*(1.0- 0.286) ( 0.9924210+-1.059313 )*(1.0-0.286) -0.047742 MPa ( 898.192/9.051E5+-615.64/-2.59E8 )*(1.0-0.286 ) ( 0.9924210+2.3809161 )*(1.0-0.286) 2.4075798 MPa
Surfacing 1 Loading
MA p p l i e d = 99.65918 kN.m Corresponding stresses: top stress = 99.65918/5.8116E8 = 0.17148 MPa bottom stress = 99.65918/-2.586E8 = -0.3854 MPa
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Traffic gr1a TS - for Bending design 1 Loading
MA p p l i e d = 934.3025 kN.m PA p p l i e d = -43.8224 kN Corresponding stresses: top stress = -43.8224/905051.1 + 934.3025/5.8116E8 = -0.0484 + 1.60764 = 1.55922 MPa bottom stress = -43.8224/905051.1 + 934.3025/-2.586E8 = -0.0484 + -3.613 = -3.6618 MPa
Traffic gr1a UDL - for Bending design 1 Loading
MA p p l i e d = 324.4073 kN.m PA p p l i e d = -4.365749 kN Corresponding stresses: top stress = -4.365749/905051.1 + 324.4073/5.8116E8 = -0.0048 + 0.55820 = 0.55337 MPa bottom stress = -4.365749/905051.1 + 324.4073/-2.586E8 = -0.0048 + -1.255 = -1.2594 MPa
Traffic gr1a Footway - for Bending design 1 Loading
MA p p l i e d = 19.32731 kN.m PA p p l i e d = 1.418796 kN Corresponding stresses: top stress = 1.418796/905051.1 + 19.32731/5.8116E8 = 0.00157 + 0.03326 = 0.03482 MPa bottom stress = 1.418796/905051.1 + 19.32731/-2.586E8 = 0.00157 + -0.075 = -0.0732 MPa
TOTAL LOSS OF PRESTRESS SUMMARY Initial stressing force Prestress after all losses at t = ∞
= 4448.25 kN = 3142.75 kN
Corresponding loss = 29.3 %
LIMITING STRESSES IN PRECAST BEAM Compression EN 1992-2 Clause 7.2(102) k1 .fc k = 0.6*40.0 = 24.0 MPa In the presence of confinement or increase in cover this may be increased by up to 10%, i.e to: = 26.4 MPa
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Tension Tension is governed by crack width considerations, and reinforcement provided for crack width control. Exposure Class is XD1, XD2, XD3, XS1, XS2, XS3 ... ... for which decompression is checked for the Frequent combination of loads. Decompression requires all of the tendon to be at least 65.0 mm above the level of the neutral axis.
LIMITING STRESSES FOR IN SITU CONCRETE Compression EN 1992-2-2 Clause 7.2(102) To avoid longitudinal cracking, compressive stress is limited to: σc = k1 .fc k = 0.6*31.875 = 19.125 MPa Tension Tension is governed by crack width considerations, and reinforcement provided for crack width control. EN 1992-1_1 Clause 7.3 However, no tensile stress is present at this section.
TRANSMISSION LENGTH Bond stress at release, EN 1992-1-1 Clause 8.10.2.2(1) fb p t = ηp 1 .η1 .fc t d (t) where fc t d (t) = αc t .0.7fc t m (t)/γc fc t m (t) = -2.3253 MPa αc t = 1.0
[2]
- from EN 1992-1-1/3.1.6(2)
tendon type coefficient, bond condition coefficient, hence
Expression (8.15)
ηp 1 = η1 =
3.2 1.0
fc t d (t) = 1.0*0.7*-2.3253/1.5 = -1.0851 MPa
and
fb p t = 3.2*1.0*-1.0851 = -3.4724 MPa
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Basic transmission length, EN 1992-1-1 Clause 8.10.2.2(2) lp t = α1 .α2 .φ.σp m 0 /fb p t where speed of release coefficient, tendon surface coefficient, nominal diameter of tendon, tendon stress after release,
hence
Expression (8.16)
α1 = α2 = φ = σp m 0 =
1.0 0.19 16.0 mm 1440.0 MPa
lp t = 1.0*0.19*16.0*1440.0 / 3.47242 = 1.26068 m
Design value of transmission length, EN 1992-1-1 Clause 8.10.2.2(3) lp t 1 = 0.8*lp t = 0.8*1.26068 = 1.00854 m
STRUCTURAL EFFECTS OF TIME DEPENDENT BEHAVIOUR EN 1992-2 Annex KK.7 Age of concrete at first loading, Age of concrete when first composite, Age of concrete at time considered, Creep coefficient when first composite, Final creep coefficient, Creep coefficient increment, Specified value of Ageing coefficient,
t0 = 4.0 days tc = 60.0 days t = ∞ φ(tc ,t0 ) = 0.89250 φ(∞,t0 ) = 2.00881 φ(∞,tc ) = 1.20422 χ = 0.8
From Expression (KK.119): φ(∞,t0 ) - φ(tc ,t0 ) 2.00881-0.89250 ————————————————— = ————————————————————— 1 + χ.φ(∞,tc ) 1.0 + 0.8*1.20422 = 0.56856
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Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
SLS STRESS SUMMARY TABLE Concrete Stresses (MPa)
force kN
moment kN.m
In situ top bottom
Precast top bottom
CHARACTERISTIC PERMANENT ACTIONS AND PRESTRESS [3]
Prestress
4244.91
Self Weight
-1547.7
-4.1499
17.4906
740.364
5.76481
-4.5869
—————————————————————————————————————————————————————— Prestress + Self Weight Elastic Def
-203.96
TRANSFER
90.6953
1.61496
12.9036
0.32653
-0.9415
4040.95
-716.68
1.94149
11.9621
-257.14
176.251
0.89371
-1.5706
Erection
-2.3584
-0.0184
0.01461
In situ 1A
512.315
3.98911
-3.174
Cr+Sh+Rlx
B
In situ 1B
21.8745 0.0
0.05072
0.03293
0.03764
-0.0846
0.0
0.0
0.0
0.0
6.8436
7.14742
TOTAL PERMANENT EFFECTS, S0
Cr+Sh+Rlx
A
-641.05
439.384
TOTAL PERMANENT EFFECTS, S0 , ∞
0.34886
-0.0084
0.04774
-2.4076
0.39958
0.0245
6.89134
4.73984
CREEP REDISTRIBUTION according to EN 1992-2 Annex KK.7 Construction On Centering, Sc = G + P1 + P2 Permanent G
606.626
1.40663
0.91333
1.04381
-2.3461
Prestress P1
-2584.1
-5.992
-3.8906
-4.4465
9.99388
3.95141
3.95141
4.17748
4.17748
5.40251
3.50787
4.00902
-9.0107
2.68239
2.52957
-1.1711
-2.4635
3.08198
2.55407
5.72024
2.27635
99.6592
0.23108
0.15004
0.17148
-0.3854
286.182
-0.2604
-0.4932
1.16132
-0.4378
-101.97
-0.2364
-0.1535
-0.1754
0.39436
5.89172 6.87758
1.89092 1.8474]
3780.83
Prestress P2
2329.9
(Sc - S0 )*0.56856 Hence from KK.119, TOTAL CONSTRUCTION EFFECTS, S∞
[4]
[5]
SDL Diff. Shr. 1
605.383
Diff. Shr. 2
[Differential shrinkage is included when adverse TOTAL PERMANENT EFFECTS [including diff. shrinkage
SAM v6.50d
3.31306 2.81612
2.70412 2.05738
]
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Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
VARIABLE ACTIONS - CHARACTERISTIC COMBINATION Traffic Selected case: Traffic gr1a TS
1
934.303
-43.822 Traffic gr1a UDL 1
324.407
-4.3657 Traffic gr1a FT
1
19.3273
1.4188
2.16644
1.40668
1.60764
-3.6133
-0.0458
-0.0458
-0.0484
-0.0484
0.75222
0.48842
0.55820
-1.2546
-0.0046
-0.0046
-0.0048
-0.0048
0.04482
0.0291
0.03326
-0.0747
0.00148
0.00148
0.00157
0.00157
Total (Leading)
:
2.9146
1.87532
2.14742
-4.9944
Total (in Combination)
: 2.16974
1.39579
1.59838
-3.7202
-0.2524
-0.2885
0.64854
ψ0
0.75 0.75 0.4
Other traffic cases for comparison: Traffic gr1a TS
2
-167.69
-0.3888 0.0147
0.0147
0.01554
0.01554
-89.68
-0.2079
-0.1350
-0.1543
0.34682
0.00546
0.00546
0.00577
0.00577
0.04482
0.0291
0.03326
-0.0747
0.00148
0.00148
0.00157
0.00157
Total (Leading)
: -0.5303
-0.3367
-0.3867
0.94350
Total (in Combination)
: -0.4139
-0.2632
-0.3022
0.73324
14.063 Traffic gr1a UDL 2 5.225 Traffic gr1a FT
2
19.3273
1.4188
0.75 0.75 0.4
Temperature Restraint None defined Differential Temperature - Heating Diff. Tmp H1
-1016.0
Diff. Tmp H2
-413.84
2.47603
-0.8853
-0.8625
1.35841
285.183
0.66127
0.42936
0.49071
-1.1029
0.6
Differential Temperature - Cooling Diff. Tmp C1
976.559
Diff. Tmp C2
143.384
-1.4373
0.52916
0.24758
-1.7606
-98.811
-0.2291
-0.1487
-0.1700
0.38214
0.6
Other Variable Action None defined The following combinations of variable actions are evaluated in accordance with EN 1990 Equation 6.14b: a) Traffic as leading action + ψ0(Thermal + Other) b) Thermal as leading action + ψ0(Traffic + Other) c) Other as leading action + ψ0(Traffic + Thermal) For thermal actions the following cases are considered: i) temperature restraint alone ii) differential temperature: Heating iii) differential temperature: Cooling iv) temperature restraint + differential temperature: Heating v) temperature restraint + differential temperature: Cooling
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Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
The most adverse case is with Traffic as leading action
Traffic ψ0 x Thermal
2.9146
1.87532
2.14742
-4.9944
1.88238
-0.2735
0.04654
-0.8270
0.0
0.0
0.0
0.0
ψ0 x Other
TOTAL VARIABLE ACTIONS TOTAL PERMANENT (from above)
TOTAL COMBINATION
4.79698 1.60173 2.19396 -5.8215 3.31306 2.70412 6.87758 1.8474 —————————————————————————————————— 8.11005 4.30585 9.07154 -3.9741
WARNING - The flexural tensile stress exceeds the value of fct,eff so the section cannot be assumed to be uncracked. (EN 1992-1-1/7.1(2)) A cracked section analysis must be performed to derive the true compression stress in the concrete.
VARIABLE ACTIONS - FREQUENT COMBINATION Traffic Selected case:
ψ2
ψ1 = 0.75
Traffic gr1a TS
1
700.727
-32.867 ψ1 = 0.75
Traffic gr1a UDL 1
243.305
-3.2743 ψ1 =
1.62483
1.05501
1.20573
-2.71
-0.0343
-0.0343
-0.0363
-0.0363
0.56417
0.36631
0.41865
-0.9409
-0.0034
-0.0034
-0.0036
-0.0036
0.0
0.0
0.4
however, EN 1991-2 Clause 4.5.2 excludes the footway loading from the frequent action of LM1 Traffic gr1a FT
1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Total (Leading)
: 2.15123
Total (in Combination)
:
0.0
1.38355
1.58445
-3.6909
0.0
0.0
0.0
0.0
-0.2916
-0.1893
-0.2164
0.48640
0.01102
0.01102
0.01165
0.01165
-0.1559
-0.1012
-0.1157
0.26012
0.0041
0.0041
0.00433
0.00433
0.0
Other traffic cases for comparison: ψ1 = 0.75
Traffic gr1a TS
2
-125.77
10.5473 ψ1 = 0.75
Traffic gr1a UDL 2
-67.26
3.91875 ψ1 =
0.0
0.0
0.4
however, EN 1991-2 Clause 4.5.2 excludes the footway loading from the frequent action of LM1 Traffic gr1a FT
2
0.0 0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Total (Leading)
: -0.4324
-0.2755
-0.3161
0.76251
Total (in Combination)
:
0.0
0.0
0.0
0.0
0.0
Temperature Restraint None defined SAM v6.50d
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Differential Temperature - Heating ψ1 =
0.6
Diff. Tmp H1
-1016.0
Diff. Tmp H2
-413.84
2.47603
-0.8853
-0.8625
1.35841
285.183
0.66127
0.42936
0.49071
-1.1029
0.5
Differential Temperature - Cooling ψ1 =
0.6
Diff. Tmp C1
976.559
Diff. Tmp C2
143.384
-1.4373
0.52916
0.24758
-1.7606
-98.811
-0.2291
-0.1487
-0.1700
0.38214
0.5
Other Variable Action None defined The following combinations of variable actions are evaluated in accordance with EN 1990 Equation 6.15b: a) ψ1(Traffic) as leading action + ψ2(Thermal + Other) b) ψ1(Thermal) as leading action + ψ2(Traffic + Other) c) ψ1(Other) as leading action + ψ2(Traffic + Thermal) For thermal actions the following cases are considered: i) temperature restraint alone ii) differential temperature: Heating iii) differential temperature: Cooling iv) temperature restraint + differential temperature: Heating v) temperature restraint + differential temperature: Cooling The most adverse case is with Traffic as leading action
ψ1 x Traffic ψ2 x Thermal ψ2 x Other
TOTAL VARIABLE ACTIONS TOTAL PERMANENT (from above)
TOTAL COMBINATION
2.15123
1.38355
1.58445
-3.6909
1.56865
-0.2279
0.03878
-0.6892
0.0
0.0
0.0
0.0
3.71988 1.15556 1.62323 -4.3801 3.31306 2.70412 6.87758 1.8474 —————————————————————————————————— 7.03294 3.85968 8.50081 -2.5327
SLS FLEXURE Precast
Stress
(MPa)
After Transfer
After Erection
SAM v6.50d
T B
T B
E
Strain -6
Curvature -6
(x10 )
Deflection (mm)
(x10 )
(rad/m)
Here
1.94149 ET 62.3657
-247.61
15.0561
15.0561
-314.42
19.6107
19.6107
11.9621
384.253
2.81685 EI 151.712 10.4061
Max.
560.458
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After in situ 1A After in situ 1B
Long-term Dead
Diff. Temp H1 Diff. Temp H2 Diff. Temp C1 Diff. Temp C2
Traffic gr1a 1
T B T B
T B
T B T B T B T B
T B
6.80596 EI 366.561 7.23201 6.8436 7.14742
EI 368.588
245.625
-4.8888
-4.8888
-29.104
1.59073
1.59073
20.8834
-1.1442
-1.1726
26.3161
-1.4413
-1.4413
-7.2357
0.39644
0.40625
155.98
-6.7548
-6.7548
420.685
-12.688
-12.688
-29.993 6.51001
2.14742 ES 60.9709 -4.9944
5.84195
-18.789
-0.1020 ES -2.8964 0.22928
5.84195
23.1413
0.14855 ES 4.2178 -1.0564
-12.587
368.27
0.29442 ES 8.35951 -0.6617
6.14087
384.951
-0.5175 ES -14.695 0.81504
6.14087
389.507
8.04868 EL 687.582 4.31089
-17.651
-141.8
Extreme in-service
Curvatures here are derived from precast section height: 1300.0mm ET = Elastic Modulus at Transfer = 31130.7MPa [EN1992-1-1 Clauses 3.1.3-(3) and 3.1.2-(6) with age 4 days] EI = Intermediate Term Elastic Modulus = 18567.1MPa [EN1992-1-1 Clause 7.4.3-(5) at 60.0 days (φ=0.89693)] EL = Long Term Elastic Modulus = 11705.8MPa [EN1992-1-1 Clause 7.4.3-(5) at infinite time (φ=2.00881)] ES = Short Term Elastic Modulus [Ecm]
= 35220.5MPa
________ [1] Refer to EN 1991-1-1 Table A.1 Note 1) [2] For the derivation of this value refer to the limiting stress calculations for transfer [3] includes draw-in and initial relaxation [4] With immediate losses and shrinkage / creep / relaxation losses until time at which insitu is cast. [5] Secondary effects arising from prestress in continuous section.
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92
Pre-tensioned Pre-stressed Beam Bridge Design Example
13.
Verification: ULS Bending - Mid Span
93
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Design code: EN 1992-2:2005 with UK National Annex (modified) EN 1990 Equation 6.10 ULS Persistent / Transient Load case: Traffic gr1a TS - for Bending design 1 Section Ref 1 at 10.5m from left end of beam
Section details:
Ref 1 "Section 1" at 0.5 x span = 10.5 m from left end of beam
Analysis:
Traffic Actions: Bending for gr1a, loading I.D. 1 Ultimate Limit State: Persistent / Transient - EN 1990 Equation 6.10
ULS Stress / Strain summary for section with HOGGING moment
Location
strain
Precast beam
height mm
-0.02234
0.0
0.0035
22.6667
0.0
-0.025719
0.0
1470.0
- bottom -0.021744
0.0
1270.0
-0.025441
-1391.3
1200.0
- bottom -0.002627
-512.29
60.0
- top - bottom
In situ stage 1 - top Tendon
stress MPa
- top
1300.0
Resistance of section = -551.202 kN.m In conjunction with axial load of -48.0244 kN
SAM v6.50d
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ULS Stress / Strain summary for section with SAGGING moment
Location
strain
Precast beam
- top
-5.156E-4
- bottom -0.031223
height mm
0.0
1300.0
0.0
0.0
18.0625
1470.0
- bottom -0.001224
0.0
1270.0
-0.007966
-1391.3
1200.0
-0.03474
-1391.3
60.0
In situ stage 1 - top Tendon
stress MPa
- top - bottom
0.0035
Resistance of section = 5221.3 kN.m In conjunction with axial load of -48.0244 kN
TRANSMISSION and ANCHORAGE Bond stress at release, EN 1992-1-1 Clause 8.10.2.2(1) fb p t = ηp 1 .η1 .fc t d (t) where fc t d (t) = αc t .0.7fc t m (t)/γc fc t m (t) = -2.3253 MPa αc t = 1.0
[1]
- from EN 1992-1-1/3.1.6(2)
tendon type coefficient, bond condition coefficient, hence
Expression (8.15)
ηp 1 = η1 =
3.2 1.0
fc t d (t) = 1.0*0.7*-2.3253/1.5 = -1.0851 MPa
and
fb p t = 3.2*1.0*-1.0851 = -3.4724 MPa
Basic transmission length, EN 1992-1-1 Clause 8.10.2.2(2) lp t = α1 .α2 .φ.σp m 0 /fb p t where speed of release coefficient, tendon surface coefficient, nominal diameter of tendon, tendon stress after release,
hence
Expression (8.16)
α1 = α2 = φ = σp m 0 =
1.0 0.19 16.0 mm 1440.0 MPa
lp t = 1.0*0.19*16.0*1440.0 / 3.47242 = 1.26068 m
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Design value of transmission length, EN 1992-1-1 Clause 8.10.2.2(3) lp t 2 = 1.2*lp t = 1.2*1.26068 = 1.51281 m
Anchorage of the tendons for the ultimate limit state: Bond stress for anchorage EN 1992-1-1 Clause 8.10.2.3(2) where
fb p d = ηp 2 .η1 .fc t d
Expression (8.20)
fc t d = αc t .0.7fc t m /γc = 1.0*0.7*-3.5088/1.5 = -1.6374 MPa
tendon type coefficient, hence
ηp 2 = 1.2
fb p d = 1.2*1.0*-1.6374 = -1.9649 MPa
Total anchorage length, EN 1992-1-1 Clause 8.10.2.3(4) lb p d = lp t 2 + α2 .φ.(σp d -σp m ∞ )/fb p d
For a tendon stressed to its limit, stress in tendon,
σp d = -1391.3 MPa
[2]
prestress after all losses, hence
Expression (8.21)
σp m ∞ = -1069.1 MPa
lb p d = 1.51281 + 0.19*16.0*(-1391.3--1069.1)/-1.9649 = 2.01123 m
SUMMARY OF ACTIONS PERMANENT ACTIONS
ACTION TYPE
MOMENT kN.m
AXIAL kN
Beam erection before composite
=
996.3078
0.0
Construction stage 1A
=
691.6251
0.0
Construction stage 1B
=
29.53059
0.0
Surfacing
=
119.591
0.0
Differential Shrinkage / creep
=
0.0
0.0
1837.0545
0.0
TOTAL PERMANENT ACTIONS,
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VARIABLE ACTIONS
ACTION TYPE
MOMENT kN.m
Differential Temperature - Heating =
442.0334
AXIAL kN 0.0
Differential Temperature - Cooling = -153.1568
0.0
ψ0
ψ1
ψ2
0.6
0.6
0.5
0.6
0.6
0.5
Traffic gr1a TS - for Bending desi =
1261.308
-59.16024
0.75 0.75
0.0
Traffic gr1a UDL - for Bending des =
437.9499
-5.893762
0.75 0.75
0.0
Traffic gr1a Footway - for Bending =
26.09187
1.915374
0.4
0.4
0.0
TOTAL VARIABLE ACTIONS, γQ , 1 x Qk , 1 "+" ΣγQ , i x ψ0 x Qk , i TOTAL VARIABLE ACTIONS, Qk , 1 "+" Σψ0 x Qk , i [4]
Thermal effects are set to zero when not adverse. Traffic leading:
Traffic
1725.3498
-63.13863
0.0
0.0
0.0
0.0
Total
1725.3498
-63.13863
ψ0 x Traffic
1284.8802
-48.02435
442.0334
0.0
0.0
0.0
ψ0 x Thermal ψ0 x Other
Thermal leading:
Thermal
ψ0 x Other
Other leading:
Total
1726.9136
-48.02435
ψ0 x Traffic
1284.8802
-48.02435
ψ0 x Thermal
0.0
0.0
Other
0.0
0.0
Total
1284.8802
-48.02435
Critical case is with thermal leading
TOTAL COMBINATION
———————————————————————— 3563.9681 -48.02435
________ [1] For the derivation of this value refer to the limiting stress calculations for transfer [2] Typical value used - varies for each tendon [3] Shrinkage effects are excluded at ULS in accordance with EN1992-1-1/2.3.2.2(2) [4] When thermal effects are not adverse, they are excluded at ULS according to EN 1992-1-1/2.3.1.2
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Pre-tensioned Pre-stressed Beam Bridge Design Example
14.
Verification: SLS bending – Pier
99
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Design code:
EN 1992-2:2005 with UK National Annex (modified) EN 1990 Equation 6.14 SLS Characteristic Exposure Class: XD1, XD2, XS1, XS2, XS3 Load case: Traffic gr1a TS - for Bending design 1 Section Ref 2 at 21m from left end of beam
WARNING - A reduction of flange width to allow for shear lag effects may be appropriate for this beam. SAM makes no allowance for this. Refer to EN 1992-1-1/5.3.2.1
Section details:
Ref 2 "Section 2" at 1 x span = 21 m from left end of beam
Analysis:
Traffic Actions: Bending for gr1a, loading I.D. 1 At time considered, t = ∞ Serviceability Limit State: Characteristic - EN 1990 Equation 6.14
ACTUAL STRESSES IN PRECAST BEAM No. No. No. No.
of tendons fully bonded at this section: 0 of tendons fully debonded at this section: 7 of tendons deflected at this section: 0 of tendons partially stressed: 14 (i.e. within the transmission length) The prestress force in these tendons is interpolated in accordance with EN 1992-1-1 clause 8.10.2.2.
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Maximum Stressing Force - EN 1992-1-1 Clause 5.10.2.1(1)P For tendon property Grade 1600 Ep 195.0 k1 .fp k = 0.8*1860.0 = 1488.0 MPa k2 .fp 0 , 1 k = 0.9*1600.0 = 1440.0 MPa Wedge draw-in loss Clause 5.10.4(1)(i) draw-in strain = 0.003/21.0 = 1.43E-4 loss = Ep . strain = 195.0*1.43E-4 = 27.8571 MPa Heat Curing Clause 5.10.4(1)(ii)(Note) Concrete is cured at ambient temperature
Immediate Losses - EN 1992-1-1 Clause 5.10.4
height No of mm tendons TOTAL
fp MPa
k1 /k2
draw-in MPa
heat cure MPa
area mm²
initial force kN
0
0.0
In accordance with clause 5.10.9(1), for SLS, the Characteristic value must be used. With ri n f = 1.0, Pk , i n f = 0.0 kN
Friction Clause 5.10.4(1)(i)
All tendons are straight in this beam.
Initial Relaxation Clause 5.10.4(1)(ii)
Loss is calculated from clause 3.3.2(7) For tendon property Grade 1600 Ep 195.0 relaxation loss at 1000 hours, ρ1 0 0 0 = 8.0 % μ = σp i / fp k = 0.0-0.0-0.0/1860.0 = 0.0 time after tensioning = 96.0 hours for Class 1 relaxation, use Expression (3.28) 6.7μ
5.39 . ρ1 0 0 0 . e
0.75(1-μ)
. [t/1000]
= 5.39 * 8.0 * 1.0 * 0.17246 * 10 = 7.44E-5 relaxation
-5
height No of mm tendons TOTAL
area x σp i
%
. 10
-5
After relaxation loss kN
force kN
0
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moment kN.m 0.0
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Moment about the centroid of the precast beam: Mr = 0.0-(0.0*0.5760392) = 0.0 kN.m Corresponding stresses: top stress = 0.0/537225.68+0.0/1.2843E8 = 0.0+0.0 = 0.0 MPa bottom stress = 0.0/537225.68+0.0/-1.614E8 = 0.0+0.0 = 0.0 MPa Self weight moment: c.s.a. = 5.372E5 mm²
[1]
density = 24.0 kN/m³ + 1.0 kN/m³ = 25.0 kN/m³ self weight = 5.372E5*25.0 = 13.4306 kN/m beam length = 21.0 m distance = 21.0 m Ms w = 0.5*13.4306*21.0*(21.0-21.0) = 0.0 kN.m Corresponding stresses: top stress = 0.0/1.2843E8 = 0.0 MPa bottom stress = 0.0/-1.614E8 = 0.0 MPa
Elastic Deformation - Clause 5.10.4(1)(iii) stress at top of precast beam stress at bottom of precast beam depth of precast beam elastic modulus of concrete at transfer
height
mm TOTAL
No of tendons 0
conc stress MPa
conc strain
= 0.0 = 0.0 = 1300.0 = 31.1307
tendon force kN
MPa MPa mm GPa
tendon moment kN.m 0.0
0.0
Moment about the centroid of the precast beam: Me d = 0.0-(0.0*0.5760392) = 0.0 kN.m hence, top stress = 0.0-0.0/537.22568-0.0/1.2843E8 = 0.0-0.0-0.0 = 0.0 MPa bottom stress = 0.0-0.0/537.22568-0.0/-1.614E8 = 0.0-0.0-0.0 = 0.0 MPa After a further 0 iterations of the above process, the top and bottom stresses are as follows: top stress = 0.0 MPa bottom stress = 0.0 MPa
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Max Prestress Force after transfer - EN 1992-1-1 Clause 5.10.3.(2) For tendon property Grade 1600 Ep 195.0 k7 .fp k = 0.75*1860.0 = 1395.0 MPa k8 .fp 0 , 1 k = 0.85*1600.0 = 1360.0 MPa Maximum tendon stress after transfer = 1272.24 MPa which is not greater than 1360.0 and therefore OK.
ACTIONS DURING EXECUTION Erection of beam Loading
Bending moment from erection loadcase at current span location: MA p p l i e d = 0.0 kN.m Corresponding stresses: top stress = 0.0/1.2843E8 = 0.0 MPa bottom stress = 0.0/-1.614E8 = 0.0 MPa Remove the dead load applied for transfer calculations Ms w = 0.0 kN.m Corresponding stresses: top stress = 0.0/1.2843E8 = 0.0 MPa bottom stress = 0.0/-1.614E8 = 0.0 MPa
Time Dependent Losses - EN 1992-1-1 Clause 5.10.6 Simplified method using Expression (5.46) ΔPc + s + r = Ap .Δσp , c + s + r εc s .Ep + 0.8Δσp r + Ep /Ec m .φ(t,t0 ).σc , Q P Δσp , c + s + r = —————————————————————————————————————————————— 1 + Ep /Ec m .Ap /Ac (1+Ac /Ic .zc p ²)[1+0.8φ(t,t0 )] The calculated loss is apportioned partly to the precast beam alone and partly to the full composite section. For in-situ cast at 60 days, the proportion of the loss occurring before the in-situ is cast is calculated to be 30.0 %
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Losses are calculated for time t = ∞ Age of concrete at end of curing, Age of concrete at transfer,
ts = t0 =
1.0 days 4.0 days
Age is adjusted for expression (B.5) (for cement type & temperature) - for cement class N (α = 0) 1.2
α
adjusted t0 = t0 , T . [(9/(2+t0 , T )+1) >=0.5 Expression (B.9) 1.2 0 = 4.0 * [(9/(2+4.0 )+1] = 4.0 days Age of concrete at time considered, t = ∞ EN 1992-1-1/3.3.2(8) for relaxation, t is taken as 500,000 hours Concrete age coefficient (Expression (3.2)), βcc: βc c ( t ) = fc m ( t ) /fc m Expression (3.1) = exp{s[1-√(28/t)]} Expression (3.2) Coefficient for Class N cement, s = 0.25 βc c ( t 0 ) = exp{0.25[1.0-√(28/4.0)]} = 0.66269 βc c ( t ) = exp{0.25} = 1.28403 Characteristic strength of concrete, fc k = 40.0 MPa Mean compressive strength of concrete, fc m = 40.0 + 8.0 (from Table 3.1) = 48.0 MPa fc m 0 = 10.0 MPa fc m ( t 0 ) = βc c ( t 0 ) . fc m = 31.8094 MPa Ambient relative humidity = 80.0 % Notional size of member, h0 = 2Ac /u = 2*9.283E5/7245.89 = 256.223 mm Modulus of elasticity of concrete at 28 days, Ec m = 35.2205 GPa Modulus of elasticity of concrete at time considered, Ec m ( t ) = βc c ( t )
0.3
Expressions (3.5) & (3.1)
. Ec m
0.3
= 1.28403 * 35.2205 = 37.9636 GPa
Area of concrete cross section, Ac = 9.28E5 mm² Perimeter of concrete cross section, u = 7245.9 mm Notional size, h0 = 2*Ac /u = 2*9.283E5/7245.89 = 256.22 mm
Creep coefficient for concrete - EN 1992-1-1 clause 3.1.4 and Annex B.1 φ(t,t0 ) = φ0 . βc (t,t0 ) = φR H . β(fc m ) . β(t0 ) . βc (t,t0 )
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Expression (B.1) Expression (B.2)
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for fc m >
35.0 MPa 1-RH/100 φR H = [ 1 + ——————————— . α1 ] .α2 0.33 0.1*h0 α1 = [35.0/48.0] α2 = [35.0/48.0] α3 = [35.0/48.0]
0.7 0.2 0.5
Expression (B.3b)
= 0.80163 = 0.93878 = 0.85391
φR H = [1.0 + (1.0-0.8) / (0.1*256.223 = 1.17576
0.33
) * 0.80163]*0.93878
β(fc m ) = 16.8/√fc m = 16.8/√48.0 = 2.42487
Expression (B.4)
For Permanent Loads
In the absence of heat curing t0 , T = 4.0 days age is adjusted for expression (B.5) (for cement type and temperature) - for cement class N (α = 0) 9.0
t0 = t0 , T . [ —————————————— + 1.0 ] 1.2 2.0 + t0 , T 9.0
= 4.0 =
α
* [ —————————————— + 1.0 ]
2.0 + 4.0 4.0 day
>=0.5
Expression (B.9)
0
1.2
0.2
β(t0 ) = 1/(0.1+t0 ) 0.2 = 1/(0.1+4.0 ) = 0.70446 βc (t,t0 ) = 1.0 for time t = ∞ hence from (B.1) and (B.2): φ(t,t0 ) = 1.17576*2.42487*0.70446 = 2.00849
Expression (B.5)
Check for creep non-linearity EN 1992-1-1 clause 3.1.4(4) At the level of the centroid of the tendons, the compressive stress in the concrete at time t0 = 0.0 MPa. This does not exceed 0.45*fck(t0), i.e. 18.0 MPa, so non-linear creep is not considered
Shrinkage Strain for concrete - EN 1992-1-1 clause 3.1.4(6) Total Shrinkage: εc s = εc d + εc a
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Drying Shrinkage - Expression (3.9): εc d (t) = βd s (t,ts ).kh .εc d , 0 βd s (t,ts ) = 1.0 for t = ∞ From Table 3.3: kh = 0.79377 From Annex B, Expression (B.11):
-6
εc d , 0 = 0.85[(220+110.αd s 1 ).(exp(-αd s 2 .fc m /fc m 0 )].10 .βR H βR H = 1.55[1.0-(RH/100)³] (B.12) = 0.7564 For cement class N, αd s 1 = 4 αd s 2 = 0.12 hence, -6
εc d , 0 = 0.85[(220+110*4.0)*exp(-0.12*48.0/10.0)]*10 *0.7564 -6
and,
= 238.54*10
εc d (t) = 1.0*0.79377*238.54*10
-6
-6
= 189.347*10
Autogenous Shrinkage - Expression (3.11): εc a (t) = βa s (t).εc a (∞) βa s (t) = 1.0 for t = ∞ εc a (∞) = 2.5*(fc k -10.0)*10 -6 = 75.0*10 hence, εc a (t) = 1.0*75.0*10 = 75.0*10
-6
-6
-6
Total Shrinkage: εc s = εc d (t) + εc a (t) = 189.3473 + 75.0 -6
= 264.3473*10
Further Relaxation Clause 5.10.6(1)(b) Loss is calculated from clause 3.3.2(7) For tendon property Grade 1600 Ep 195.0 relaxation loss at 1000 hours, ρ1 0 0 0 = 8.0 % time after tensioning = 500000.0 hours μ = 0.75921 (as calculated for initial relaxation loss above) for Class 1 relaxation, use Expression (3.28) 6.7μ
5.39 . ρ1 0 0 0 . e
0.75(1-μ)
. [t/1000]
= 5.39 * 8.0 * 161.863 * 3.07185 * 10 = 0.21440
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-5
. 10
-5
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With the initial relaxation deducted, the variation in tendon stress from relaxation becomes: Δσp r / σp i = 0.21440 - 0.04571 = 0.16868 Summary of the above for Expression (5.46):
Estimated shrinkage strain εc s Creep coefficient at t for loading at t0 φ(t,t0 ) Relaxation, Δσp r Modulus of elasticity for prestressing steel Ep Modulus of Elasticity for concrete Ec m Area of all prestressing Ap Area of concrete section Ac Second moment of area of concrete section Ic Ep /Ec m Ep /Ec m .Ap /Ac Ac /Ic
= 195.0/37.9636 = 5.1365*0.0/9.283E5 = 9.283E5/2.38E11
= = = = = = = =
264.347 2.00849 238.212 195.0 37.9636 0.0 9.283E5 2.38E11
-6
x10 MPa GPa GPa mm² mm² mm⁴
= 5.1365 = 0.0 = 3.89252
In the table below the following vary with tendon height: σc , Q P = Stress in concrete adjacent to tendons zc p = Section centre of gravity to tendons φ(t,t0 ) = Creep Coefficient (if non-linear creep is considered)
height
mm
Ap mm²
shrink εc s .Ep MPa
relax 0.8Δσp r MPa
φ(t,t0 )
σc , Q P MPa
creep Ep /Ec m .φ.σ MPa
denom zc p mm
Total force loss: Total moment loss:
ΔPc + s + r kN 0.0 0.0
Mc s r = 0.0-(0.0*0.9124128) = 0.0 kN.m
Corresponding stresses - before composite: top stress = ( 0.0/5.372E5+0.0/1.284E8 )* 0.3 = ( 0.0+0.0 )* 0.3 = 0.0 MPa bottom stress = ( 0.0/5.372E5+0.0/-1.61E8 )* 0.3 = ( 0.0+0.0 )* 0.3 = 0.0 MPa top stress bottom stress
SAM v6.50d
= = = = = =
- after composite: ( 0.0/9.283E5+0.0/6.153E8 )*(1.0- 0.3) ( 0.0+0.0 )*(1.0-0.3) 0.0 MPa ( 0.0/9.283E5+0.0/-2.61E8 )*(1.0-0.3 ) ( 0.0+0.0 )*(1.0-0.3) 0.0 MPa
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Surfacing 1 Loading
MA p p l i e d = -130.6559 kN.m Corresponding stresses: top stress = -130.6559/6.1529E8 = -0.2123 MPa bottom stress = -130.6559/-2.614E8 = 0.49988 MPa
TOTAL LOSS OF PRESTRESS SUMMARY Initial stressing force Prestress after all losses at t = ∞
= =
0.0 kN 0.0 kN
Corresponding loss = 100 %
LIMITING STRESSES IN PRECAST BEAM Compression EN 1992-2 Clause 7.2(102) k1 .fc k = 0.6*40.0 = 24.0 MPa In the presence of confinement or increase in cover this may be increased by up to 10%, i.e to: = 26.4 MPa Tension Tension is governed by crack width considerations, and reinforcement provided for crack width control. Exposure Class is XD1, XD2, XD3, XS1, XS2, XS3 ... ... for which decompression is checked for the Frequent combination of loads. Decompression requires all of the tendon to be at least 65.0 mm above the level of the neutral axis.
LIMITING STRESSES FOR IN SITU CONCRETE Compression EN 1992-2-2 Clause 7.2(102) To avoid longitudinal cracking, compressive stress is limited to: σc = k1 .fc k = 0.6*31.875 = 19.125 MPa Tension Tension is governed by crack width considerations, and reinforcement provided for crack width control. EN 1992-1_1 Clause 7.3 However, no tensile stress is present at this section.
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TRANSMISSION LENGTH Bond stress at release, EN 1992-1-1 Clause 8.10.2.2(1) fb p t = ηp 1 .η1 .fc t d (t) where fc t d (t) = αc t .0.7fc t m (t)/γc fc t m (t) = -2.3253 MPa αc t = 1.0
[2]
- from EN 1992-1-1/3.1.6(2)
tendon type coefficient, bond condition coefficient, hence
Expression (8.15)
ηp 1 = η1 =
3.2 1.0
fc t d (t) = 1.0*0.7*-2.3253/1.5 = -1.0851 MPa
and
fb p t = 3.2*1.0*-1.0851 = -3.4724 MPa
Basic transmission length, EN 1992-1-1 Clause 8.10.2.2(2) lp t = α1 .α2 .φ.σp m 0 /fb p t where speed of release coefficient, tendon surface coefficient, nominal diameter of tendon, tendon stress after release,
hence
Expression (8.16)
α1 = α2 = φ = σp m 0 =
1.0 0.19 16.0 mm 1440.0 MPa
lp t = 1.0*0.19*16.0*1440.0 / 3.47242 = 1.26068 m
Design value of transmission length, EN 1992-1-1 Clause 8.10.2.2(3) lp t 1 = 0.8*lp t = 0.8*1.26068 = 1.00854 m
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STRUCTURAL EFFECTS OF TIME DEPENDENT BEHAVIOUR EN 1992-2 Annex KK.7 Age of concrete at first loading, Age of concrete when first composite, Age of concrete at time considered, Creep coefficient when first composite, Final creep coefficient, Creep coefficient increment, Specified value of Ageing coefficient,
t0 = 4.0 days tc = 60.0 days t = ∞ φ(tc ,t0 ) = 0.89250 φ(∞,t0 ) = 2.00881 φ(∞,tc ) = 1.20422 χ = 0.8
From Expression (KK.119): φ(∞,t0 ) - φ(tc ,t0 ) 2.00881-0.89250 ————————————————— = ————————————————————— 1 + χ.φ(∞,tc ) 1.0 + 0.8*1.20422 = 0.56856
SLS STRESS SUMMARY TABLE Concrete Stresses (MPa)
force kN
moment kN.m
In situ top bottom
Precast top bottom
CHARACTERISTIC PERMANENT ACTIONS AND PRESTRESS [3]
Prestress
0.0
Self Weight
0.0
0.0
0.0
0.0
0.0
0.0
—————————————————————————————————————————————————————— Prestress + Self Weight
0.0
0.0
Elastic Def
0.0
0.0
0.0
0.0
TRANSFER
0.0
0.0
0.0
0.0
0.0
Cr+Sh+Rlx
0.0
0.0
0.0
Erection
B
0.0
0.0
0.0
In situ 1A
0.0
0.0
0.0
In situ 1B
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0 TOTAL PERMANENT EFFECTS, S0
Cr+Sh+Rlx
A
0.0
0.0
TOTAL PERMANENT EFFECTS, S0 , ∞
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
CREEP REDISTRIBUTION according to EN 1992-2 Annex KK.7 Construction On Centering, Sc = G + P1 + P2 Permanent G
-1332.6
-2.9471
-1.89
-2.1658
5.09841
Prestress P1
0.0
0.0
0.0
0.0
0.0
Prestress P2
4659.29
10.3044
6.6083
7.57251
-17.826
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(Sc - S0 )*0.56856
4.18309
2.68266
3.07408
-7.2366
4.18309
2.68266
3.07408
-7.2366
-130.66
-0.2889
-0.1853
-0.2123
0.49988
278.488
-0.3240
-0.5449
1.10477
-0.4133
-203.89
-0.4509
-0.2891
-0.3313
0.78006
2.86173 3.63513
-6.7368 -6.37]
Hence from KK.119, TOTAL CONSTRUCTION EFFECTS, S∞
SDL Diff. Shr. 1
605.383
Diff. Shr. 2
[Differential shrinkage is included when adverse TOTAL PERMANENT EFFECTS [including diff. shrinkage
3.89413 3.1192
2.49735 1.66324
]
VARIABLE ACTIONS - CHARACTERISTIC COMBINATION Traffic Selected case: Traffic gr1a TS
ψ0
1
0.0
0.0
0.0
0.0
0.0
0.75
Traffic gr1a UDL 1
0.0
0.0
0.0
0.0
0.0
0.75
Traffic gr1a FT
0.0
0.0
0.0
0.0
0.0
0.4
1
Total (Leading)
:
0.0
0.0
0.0
0.0
Total (in Combination)
:
0.0
0.0
0.0
0.0
-1.4342
-0.9197
-1.054
2.48113
0.13361
0.13361
0.14125
0.14125
-1.179
-0.7560
-0.8664
2.0396
0.07909
0.07909
0.08362
0.08362
-0.0769
-0.0493
-0.0565
0.13305
-0.0119
-0.0119
-0.0125
-0.0125
Total (Leading)
: -2.4892
-1.5243
-1.7646
4.86613
Total (in Combination)
: -1.8359
-1.1218
-1.2993
3.60741
Other traffic cases for comparison: Traffic gr1a TS
2
-648.5
131.125 Traffic gr1a UDL 2
-533.09
77.6189 Traffic gr1a FT
2
-34.778
-11.642
0.75 0.75 0.4
Temperature Restraint None defined Differential Temperature - Heating Diff. Tmp H1
-1016.0
Diff. Tmp H2
-400.93
2.57552
-0.8043
-0.7740
1.31996
570.322
1.26131
0.80889
0.92691
-2.182
0.6
Differential Temperature - Cooling Diff. Tmp C1
976.559
Diff. Tmp C2
130.974
-1.5057
0.47350
0.18673
-1.7342
-197.6
-0.4370
-0.2802
-0.3211
0.75602
0.6
Other Variable Action None defined
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The following combinations of variable actions are evaluated in accordance with EN 1990 Equation 6.14b: a) Traffic as leading action + ψ0(Thermal + Other) b) Thermal as leading action + ψ0(Traffic + Other) c) Other as leading action + ψ0(Traffic + Thermal) For thermal actions the following cases are considered: i) temperature restraint alone ii) differential temperature: Heating iii) differential temperature: Cooling iv) temperature restraint + differential temperature: Heating v) temperature restraint + differential temperature: Cooling The most adverse case is with Thermal (iii/v) as leading action
ψ0 x Traffic Thermal
0.0
0.0
0.0
0.0
-1.9427
0.19324
-0.1344
-0.9781
0.0
0.0
0.0
0.0
ψ0 x Other
TOTAL VARIABLE ACTIONS TOTAL PERMANENT (from above)
TOTAL COMBINATION
-1.9427 0.19324 -0.1344 -0.9781 3.1192 2.49735 3.63513 -6.37 —————————————————————————————————— 1.1765 2.69059 3.50071 -7.3482
WARNING - The flexural tensile stress exceeds the value of fct,eff so the section cannot be assumed to be uncracked. (EN 1992-1-1/7.1(2)) A cracked section analysis must be performed to derive the true compression stress in the concrete.
VARIABLE ACTIONS - FREQUENT COMBINATION Traffic Selected case:
ψ2
ψ1 = 0.75
Traffic gr1a TS
1
0.0
0.0
0.0
0.0
0.0
0.0
Traffic gr1a UDL 1
0.0
0.0
0.0
0.0
0.0
0.0
ψ1 = 0.75 ψ1 =
0.4
however, EN 1991-2 Clause 4.5.2 excludes the footway loading from the frequent action of LM1 Traffic gr1a FT
0.0
0.0
0.0
0.0
Total (Leading)
1
0.0 :
0.0
0.0
0.0
0.0
Total (in Combination)
:
0.0
0.0
0.0
0.0
-1.0756
-0.6898
-0.7904
1.86085
0.10020
0.10020
0.10594
0.10594
-0.8842
-0.5670
-0.6498
1.5297
0.05932
0.05932
0.06271
0.06271
0.0
Other traffic cases for comparison: ψ1 = 0.75
Traffic gr1a TS
2
-486.37
98.3438 ψ1 = 0.75
Traffic gr1a UDL 2 58.2142 SAM v6.50d
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0.0
0.0
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ψ1 =
0.4
however, EN 1991-2 Clause 4.5.2 excludes the footway loading from the frequent action of LM1 Traffic gr1a FT
2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Total (Leading)
: -1.8004
-1.0974
-1.2716
3.55921
Total (in Combination)
:
0.0
0.0
0.0
0.0
-400.93
2.57552
-0.8043
-0.7740
1.31996
570.322
1.26131
0.80889
0.92691
-2.182
0.0
0.0
Temperature Restraint None defined Differential Temperature - Heating ψ1 =
0.6
Diff. Tmp H1
-1016.0
Diff. Tmp H2
0.5
Differential Temperature - Cooling ψ1 =
0.6
Diff. Tmp C1
976.559
Diff. Tmp C2
130.974
-1.5057
0.47350
0.18673
-1.7342
-197.6
-0.4370
-0.2802
-0.3211
0.75602
0.5
Other Variable Action None defined The following combinations of variable actions are evaluated in accordance with EN 1990 Equation 6.15b: a) ψ1(Traffic) as leading action + ψ2(Thermal + Other) b) ψ1(Thermal) as leading action + ψ2(Traffic + Other) c) ψ1(Other) as leading action + ψ2(Traffic + Thermal) For thermal actions the following cases are considered: i) temperature restraint alone ii) differential temperature: Heating iii) differential temperature: Cooling iv) temperature restraint + differential temperature: Heating v) temperature restraint + differential temperature: Cooling The most adverse case is with Thermal (iii/v) as leading action
ψ2 x Traffic ψ1 x Thermal ψ2 x Other
TOTAL VARIABLE ACTIONS TOTAL PERMANENT (from above)
TOTAL COMBINATION
SAM v6.50d
0.0
0.0
0.0
0.0
-1.1656
0.11594
-0.0807
-0.5869
0.0
0.0
0.0
0.0
-1.1656 0.11594 -0.0807 -0.5869 3.1192 2.49735 3.63513 -6.7368 —————————————————————————————————— 1.95358 2.6133 3.55448 -7.3237
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SLS FLEXURE Precast
Stress
(MPa)
After Transfer
After Erection
After in situ 1A After in situ 1B
Long-term Dead
Diff. Temp H1 Diff. Temp H2 Diff. Temp C1 Diff. Temp C2
Traffic gr1a 1 Extreme in-service
T B
T B
T B T B
T B
T B T B T B T B
T B
E
Strain -6
Curvature -6
(x10 )
Deflection (mm)
(x10 )
(rad/m)
Here
0.04918 ET 1.57982
2.16063
-4.E-16
15.0561
-33.436
1.7E-15
19.6107
-33.436
2.2E-16
6.14087
-32.586
-1.E-15
5.84195
-20.08
-2.E-16
-4.8888
-27.44
4.9E-16
1.59073
40.7406
1.9E-16
-1.1726
25.1724
-2.E-16
-1.4413
-14.116
-4.E-17
0.40625
-20.798
-2.E-15
-6.7548
-29.821
-3.E-15
-12.688
-0.0383
-1.229
-0.2764 EI -14.887 0.53063
28.5795
-0.2764 EI -14.887 0.53064
28.5797
-0.2649 EI -14.271 0.52154
28.0899
0.56105 EL 47.9295 0.86661
74.0333
-0.4644 ES -13.186 0.79197
22.4862
0.55615 ES 15.7905 -1.3092
-37.172
0.11203 ES 3.18107 -1.0405
-29.543
-0.1926 ES -5.471 0.45361
12.8793
-0.3085 ES -8.7613 0.64368
Max.
18.2758
Curvatures here are derived from precast section height: 1300.0mm ET = Elastic Modulus at Transfer = 31130.7MPa [EN1992-1-1 Clauses 3.1.3-(3) and 3.1.2-(6) with age 4 days] EI = Intermediate Term Elastic Modulus = 18567.1MPa [EN1992-1-1 Clause 7.4.3-(5) at 60.0 days (φ=0.89693)] EL = Long Term Elastic Modulus = 11705.8MPa [EN1992-1-1 Clause 7.4.3-(5) at infinite time (φ=2.00881)]
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ES = Short Term Elastic Modulus [Ecm]
= 35220.5MPa
________ [1] Refer to EN 1991-1-1 Table A.1 Note 1) [2] For the derivation of this value refer to the limiting stress calculations for transfer [3] includes draw-in and initial relaxation [4] With immediate losses and shrinkage / creep / relaxation losses until time at which insitu is cast. [5] Secondary effects arising from prestress in continuous section.
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Pre-tensioned Pre-stressed Beam Bridge Design Example
15.
Verification: SLS bending – Support
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Design code:
EN 1992-2:2005 with UK National Annex (modified) EN 1990 Equation 6.14 SLS Characteristic Exposure Class: XD1, XD2, XS1, XS2, XS3 Load case: Traffic gr1a TS - for Bending design 1 Section Ref 1 at 0m from left end of beam
WARNING - A reduction of flange width to allow for shear lag effects may be appropriate for this beam. SAM makes no allowance for this. Refer to EN 1992-1-1/5.3.2.1
Section details:
Ref 1 "Section 1" at 0 x span = 0 m from left end of beam
Analysis:
Traffic Actions: Bending for gr1a, loading I.D. 1 At time considered, t = ∞ Serviceability Limit State: Characteristic - EN 1990 Equation 6.14
ACTUAL STRESSES IN PRECAST BEAM No. No. No. No.
of tendons fully bonded at this section: 0 of tendons fully debonded at this section: 7 of tendons deflected at this section: 0 of tendons partially stressed: 14 (i.e. within the transmission length) The prestress force in these tendons is interpolated in accordance with EN 1992-1-1 clause 8.10.2.2.
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Maximum Stressing Force - EN 1992-1-1 Clause 5.10.2.1(1)P For tendon property Grade 1600 Ep 195.0 k1 .fp k = 0.8*1860.0 = 1488.0 MPa k2 .fp 0 , 1 k = 0.9*1600.0 = 1440.0 MPa Wedge draw-in loss Clause 5.10.4(1)(i) draw-in strain = 0.003/21.0 = 1.43E-4 loss = Ep . strain = 195.0*1.43E-4 = 27.8571 MPa Heat Curing Clause 5.10.4(1)(ii)(Note) Concrete is cured at ambient temperature
Immediate Losses - EN 1992-1-1 Clause 5.10.4
height No of mm tendons TOTAL
fp MPa
k1 /k2
draw-in MPa
heat cure MPa
area mm²
initial force kN
0
0.0
In accordance with clause 5.10.9(1), for SLS, the Characteristic value must be used. With ri n f = 1.0, Pk , i n f = 0.0 kN
Friction Clause 5.10.4(1)(i)
All tendons are straight in this beam.
Initial Relaxation Clause 5.10.4(1)(ii)
Loss is calculated from clause 3.3.2(7) For tendon property Grade 1600 Ep 195.0 relaxation loss at 1000 hours, ρ1 0 0 0 = 8.0 % μ = σp i / fp k = 0.0-0.0-0.0/1860.0 = 0.0 time after tensioning = 96.0 hours for Class 1 relaxation, use Expression (3.28) 6.7μ
5.39 . ρ1 0 0 0 . e
0.75(1-μ)
. [t/1000]
= 5.39 * 8.0 * 1.0 * 0.17246 * 10 = 7.44E-5 relaxation
-5
height No of mm tendons TOTAL
area x σp i
%
. 10
-5
After relaxation loss kN
force kN
0
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moment kN.m 0.0
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Moment about the centroid of the precast beam: Mr = 0.0-(0.0*0.5760392) = 0.0 kN.m Corresponding stresses: top stress = 0.0/537225.68+0.0/1.2843E8 = 0.0+0.0 = 0.0 MPa bottom stress = 0.0/537225.68+0.0/-1.614E8 = 0.0+0.0 = 0.0 MPa Self weight moment: c.s.a. = 5.372E5 mm²
[1]
density = 24.0 kN/m³ + 1.0 kN/m³ = 25.0 kN/m³ self weight = 5.372E5*25.0 = 13.4306 kN/m beam length = 21.0 m distance = 0.0 m Ms w = 0.5*13.4306*0.0*(21.0-0.0) = 0.0 kN.m Corresponding stresses: top stress = 0.0/1.2843E8 = 0.0 MPa bottom stress = 0.0/-1.614E8 = 0.0 MPa
Elastic Deformation - Clause 5.10.4(1)(iii) stress at top of precast beam stress at bottom of precast beam depth of precast beam elastic modulus of concrete at transfer
height
mm TOTAL
No of tendons 0
conc stress MPa
conc strain
= 0.0 = 0.0 = 1300.0 = 31.1307
tendon force kN
MPa MPa mm GPa
tendon moment kN.m 0.0
0.0
Moment about the centroid of the precast beam: Me d = 0.0-(0.0*0.5760392) = 0.0 kN.m hence, top stress = 0.0-0.0/537.22568-0.0/1.2843E8 = 0.0-0.0-0.0 = 0.0 MPa bottom stress = 0.0-0.0/537.22568-0.0/-1.614E8 = 0.0-0.0-0.0 = 0.0 MPa After a further 0 iterations of the above process, the top and bottom stresses are as follows: top stress = 0.0 MPa bottom stress = 0.0 MPa
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Max Prestress Force after transfer - EN 1992-1-1 Clause 5.10.3.(2) For tendon property Grade 1600 Ep 195.0 k7 .fp k = 0.75*1860.0 = 1395.0 MPa k8 .fp 0 , 1 k = 0.85*1600.0 = 1360.0 MPa Maximum tendon stress after transfer = 1272.24 MPa which is not greater than 1360.0 and therefore OK.
ACTIONS DURING EXECUTION Erection of beam Loading
Bending moment from erection loadcase at current span location: MA p p l i e d = 0.0 kN.m Corresponding stresses: top stress = 0.0/1.2843E8 = 0.0 MPa bottom stress = 0.0/-1.614E8 = 0.0 MPa Remove the dead load applied for transfer calculations Ms w = 0.0 kN.m Corresponding stresses: top stress = 0.0/1.2843E8 = 0.0 MPa bottom stress = 0.0/-1.614E8 = 0.0 MPa
Time Dependent Losses - EN 1992-1-1 Clause 5.10.6 Simplified method using Expression (5.46) ΔPc + s + r = Ap .Δσp , c + s + r εc s .Ep + 0.8Δσp r + Ep /Ec m .φ(t,t0 ).σc , Q P Δσp , c + s + r = —————————————————————————————————————————————— 1 + Ep /Ec m .Ap /Ac (1+Ac /Ic .zc p ²)[1+0.8φ(t,t0 )] The calculated loss is apportioned partly to the precast beam alone and partly to the full composite section. For in-situ cast at 60 days, the proportion of the loss occurring before the in-situ is cast is calculated to be 30.0 %
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Losses are calculated for time t = ∞ Age of concrete at end of curing, Age of concrete at transfer,
ts = t0 =
1.0 days 4.0 days
Age is adjusted for expression (B.5) (for cement type & temperature) - for cement class N (α = 0) 1.2
α
adjusted t0 = t0 , T . [(9/(2+t0 , T )+1) >=0.5 Expression (B.9) 1.2 0 = 4.0 * [(9/(2+4.0 )+1] = 4.0 days Age of concrete at time considered, t = ∞ EN 1992-1-1/3.3.2(8) for relaxation, t is taken as 500,000 hours Concrete age coefficient (Expression (3.2)), βcc: βc c ( t ) = fc m ( t ) /fc m Expression (3.1) = exp{s[1-√(28/t)]} Expression (3.2) Coefficient for Class N cement, s = 0.25 βc c ( t 0 ) = exp{0.25[1.0-√(28/4.0)]} = 0.66269 βc c ( t ) = exp{0.25} = 1.28403 Characteristic strength of concrete, fc k = 40.0 MPa Mean compressive strength of concrete, fc m = 40.0 + 8.0 (from Table 3.1) = 48.0 MPa fc m 0 = 10.0 MPa fc m ( t 0 ) = βc c ( t 0 ) . fc m = 31.8094 MPa Ambient relative humidity = 80.0 % Notional size of member, h0 = 2Ac /u = 2*9.051E5/7245.89 = 249.811 mm Modulus of elasticity of concrete at 28 days, Ec m = 35.2205 GPa Modulus of elasticity of concrete at time considered, Ec m ( t ) = βc c ( t )
0.3
Expressions (3.5) & (3.1)
. Ec m
0.3
= 1.28403 * 35.2205 = 37.9636 GPa
Area of concrete cross section, Ac = 9.05E5 mm² Perimeter of concrete cross section, u = 7245.9 mm Notional size, h0 = 2*Ac /u = 2*9.051E5/7245.89 = 249.81 mm
Creep coefficient for concrete - EN 1992-1-1 clause 3.1.4 and Annex B.1 φ(t,t0 ) = φ0 . βc (t,t0 ) = φR H . β(fc m ) . β(t0 ) . βc (t,t0 )
SAM v6.50d
Expression (B.1) Expression (B.2)
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for fc m >
35.0 MPa 1-RH/100 φR H = [ 1 + ——————————— . α1 ] .α2 0.33 0.1*h0 α1 = [35.0/48.0] α2 = [35.0/48.0] α3 = [35.0/48.0]
0.7 0.2 0.5
Expression (B.3b)
= 0.80163 = 0.93878 = 0.85391
φR H = [1.0 + (1.0-0.8) / (0.1*249.811 = 1.17777
0.33
) * 0.80163]*0.93878
β(fc m ) = 16.8/√fc m = 16.8/√48.0 = 2.42487
Expression (B.4)
For Permanent Loads
In the absence of heat curing t0 , T = 4.0 days age is adjusted for expression (B.5) (for cement type and temperature) - for cement class N (α = 0) 9.0
t0 = t0 , T . [ —————————————— + 1.0 ] 1.2 2.0 + t0 , T 9.0
= 4.0 =
α
* [ —————————————— + 1.0 ]
2.0 + 4.0 4.0 day
>=0.5
Expression (B.9)
0
1.2
0.2
β(t0 ) = 1/(0.1+t0 ) 0.2 = 1/(0.1+4.0 ) = 0.70446 βc (t,t0 ) = 1.0 for time t = ∞ hence from (B.1) and (B.2): φ(t,t0 ) = 1.17777*2.42487*0.70446 = 2.01193
Expression (B.5)
Check for creep non-linearity EN 1992-1-1 clause 3.1.4(4) At the level of the centroid of the tendons, the compressive stress in the concrete at time t0 = 0.0 MPa. This does not exceed 0.45*fck(t0), i.e. 18.0 MPa, so non-linear creep is not considered
Shrinkage Strain for concrete - EN 1992-1-1 clause 3.1.4(6) Total Shrinkage: εc s = εc d + εc a
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(3.8)
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Drying Shrinkage - Expression (3.9): εc d (t) = βd s (t,ts ).kh .εc d , 0 βd s (t,ts ) = 1.0 for t = ∞ From Table 3.3: kh = 0.80018 From Annex B, Expression (B.11):
-6
εc d , 0 = 0.85[(220+110.αd s 1 ).(exp(-αd s 2 .fc m /fc m 0 )].10 .βR H βR H = 1.55[1.0-(RH/100)³] (B.12) = 0.7564 For cement class N, αd s 1 = 4 αd s 2 = 0.12 hence, -6
εc d , 0 = 0.85[(220+110*4.0)*exp(-0.12*48.0/10.0)]*10 *0.7564 -6
and,
= 238.54*10
εc d (t) = 1.0*0.80018*238.54*10
-6
-6
= 190.877*10
Autogenous Shrinkage - Expression (3.11): εc a (t) = βa s (t).εc a (∞) βa s (t) = 1.0 for t = ∞ εc a (∞) = 2.5*(fc k -10.0)*10 -6 = 75.0*10 hence, εc a (t) = 1.0*75.0*10 = 75.0*10
-6
-6
-6
Total Shrinkage: εc s = εc d (t) + εc a (t) = 190.87688 + 75.0 = 265.87688*10
-6
Further Relaxation Clause 5.10.6(1)(b) Loss is calculated from clause 3.3.2(7) For tendon property Grade 1600 Ep 195.0 relaxation loss at 1000 hours, ρ1 0 0 0 = 8.0 % time after tensioning = 500000.0 hours μ = 0.75921 (as calculated for initial relaxation loss above) for Class 1 relaxation, use Expression (3.28) 6.7μ
5.39 . ρ1 0 0 0 . e
0.75(1-μ)
. [t/1000]
= 5.39 * 8.0 * 161.863 * 3.07185 * 10 = 0.21440
SAM v6.50d
-5
. 10
-5
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With the initial relaxation deducted, the variation in tendon stress from relaxation becomes: Δσp r / σp i = 0.21440 - 0.04571 = 0.16868 Summary of the above for Expression (5.46):
Estimated shrinkage strain εc s Creep coefficient at t for loading at t0 φ(t,t0 ) Relaxation, Δσp r Modulus of elasticity for prestressing steel Ep Modulus of Elasticity for concrete Ec m Area of all prestressing Ap Area of concrete section Ac Second moment of area of concrete section Ic Ep /Ec m Ep /Ec m .Ap /Ac Ac /Ic
= 195.0/37.9636 = 5.1365*0.0/9.051E5 = 9.051E5/2.33E11
= = = = = = = =
265.877 2.01193 238.212 195.0 37.9636 0.0 9.051E5 2.33E11
= = =
5.1365 0.0 3.8904
-6
x10 MPa GPa GPa mm² mm² mm⁴
In the table below the following vary with tendon height: σc , Q P = Stress in concrete adjacent to tendons zc p = Section centre of gravity to tendons φ(t,t0 ) = Creep Coefficient (if non-linear creep is considered)
height
mm
Ap mm²
shrink εc s .Ep MPa
relax 0.8Δσp r MPa
φ(t,t0 )
σc , Q P MPa
creep Ep /Ec m .φ.σ MPa
denom zc p mm
Total force loss: Total moment loss:
ΔPc + s + r kN 0.0 0.0
Mc s r = 0.0-(0.0*0.8997047) = 0.0 kN.m
Corresponding stresses - before composite: top stress = ( 0.0/5.372E5+0.0/1.284E8 )* 0.3 = ( 0.0+0.0 )* 0.3 = 0.0 MPa bottom stress = ( 0.0/5.372E5+0.0/-1.61E8 )* 0.3 = ( 0.0+0.0 )* 0.3 = 0.0 MPa top stress bottom stress
SAM v6.50d
= = = = = =
- after composite: ( 0.0/9.051E5+0.0/5.812E8 )*(1.0- 0.3) ( 0.0+0.0 )*(1.0-0.3) 0.0 MPa ( 0.0/9.051E5+0.0/-2.59E8 )*(1.0-0.3 ) ( 0.0+0.0 )*(1.0-0.3) 0.0 MPa
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Traffic gr1a TS - for Bending design 1 Loading
MA p p l i e d = 78.74031 kN.m PA p p l i e d = -19.96643 kN Corresponding stresses: top stress = -19.96643/905051.1 + 78.74031/5.8116E8 = -0.0221 + 0.13548 = 0.11342 MPa bottom stress = -19.96643/905051.1 + 78.74031/-2.586E8 = -0.0221 + -0.304 = -0.3265 MPa
Traffic gr1a UDL - for Bending design 1 Loading
MA p p l i e d = 32.7097 kN.m PA p p l i e d = -10.9063 kN Corresponding stresses: top stress = -10.9063/905051.1 + 32.7097/5.8116E8 = -0.0121 + 0.05628 = 0.04423 MPa bottom stress = -10.9063/905051.1 + 32.7097/-2.586E8 = -0.0121 + -0.126 = -0.1385 MPa
Traffic gr1a Footway - for Bending design 1 Loading
MA p p l i e d = 25.29156 kN.m PA p p l i e d = -3.118579 kN Corresponding stresses: top stress = -3.118579/905051.1 + 25.29156/5.8116E8 = -0.0034 + 0.04352 = 0.04007 MPa bottom stress = -3.118579/905051.1 + 25.29156/-2.586E8 = -0.0034 + -0.098 = -0.1012 MPa
TOTAL LOSS OF PRESTRESS SUMMARY Initial stressing force Prestress after all losses at t = ∞
= =
0.0 kN 0.0 kN
Corresponding loss = 100 %
LIMITING STRESSES IN PRECAST BEAM Compression EN 1992-2 Clause 7.2(102) k1 .fc k = 0.6*40.0 = 24.0 MPa In the presence of confinement or increase in cover this may be increased by up to 10%, i.e to: = 26.4 MPa
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Tension Tension is governed by crack width considerations, and reinforcement provided for crack width control. Exposure Class is XD1, XD2, XD3, XS1, XS2, XS3 ... ... for which decompression is checked for the Frequent combination of loads. Decompression requires all of the tendon to be at least 65.0 mm above the level of the neutral axis.
LIMITING STRESSES FOR IN SITU CONCRETE Compression EN 1992-2-2 Clause 7.2(102) To avoid longitudinal cracking, compressive stress is limited to: σc = k1 .fc k = 0.6*31.875 = 19.125 MPa Tension Tension is governed by crack width considerations, and reinforcement provided for crack width control. EN 1992-1_1 Clause 7.3
TRANSMISSION LENGTH Bond stress at release, EN 1992-1-1 Clause 8.10.2.2(1) fb p t = ηp 1 .η1 .fc t d (t) where fc t d (t) = αc t .0.7fc t m (t)/γc fc t m (t) = -2.3253 MPa αc t = 1.0
[2]
- from EN 1992-1-1/3.1.6(2)
tendon type coefficient, bond condition coefficient, hence
Expression (8.15)
ηp 1 = η1 =
3.2 1.0
fc t d (t) = 1.0*0.7*-2.3253/1.5 = -1.0851 MPa
and
fb p t = 3.2*1.0*-1.0851 = -3.4724 MPa
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Basic transmission length, EN 1992-1-1 Clause 8.10.2.2(2) lp t = α1 .α2 .φ.σp m 0 /fb p t where speed of release coefficient, tendon surface coefficient, nominal diameter of tendon, tendon stress after release,
hence
Expression (8.16)
α1 = α2 = φ = σp m 0 =
1.0 0.19 16.0 mm 1440.0 MPa
lp t = 1.0*0.19*16.0*1440.0 / 3.47242 = 1.26068 m
Design value of transmission length, EN 1992-1-1 Clause 8.10.2.2(3) lp t 1 = 0.8*lp t = 0.8*1.26068 = 1.00854 m
STRUCTURAL EFFECTS OF TIME DEPENDENT BEHAVIOUR EN 1992-2 Annex KK.7 Age of concrete at first loading, Age of concrete when first composite, Age of concrete at time considered, Creep coefficient when first composite, Final creep coefficient, Creep coefficient increment, Specified value of Ageing coefficient,
t0 = 4.0 days tc = 60.0 days t = ∞ φ(tc ,t0 ) = 0.89250 φ(∞,t0 ) = 2.00881 φ(∞,tc ) = 1.20422 χ = 0.8
From Expression (KK.119): φ(∞,t0 ) - φ(tc ,t0 ) 2.00881-0.89250 ————————————————— = ————————————————————— 1 + χ.φ(∞,tc ) 1.0 + 0.8*1.20422 = 0.56856
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SLS STRESS SUMMARY TABLE Concrete Stresses (MPa)
force kN
moment kN.m
In situ top bottom
Precast top bottom
CHARACTERISTIC PERMANENT ACTIONS AND PRESTRESS [3]
Prestress
0.0
Self Weight
0.0
0.0
0.0
0.0
0.0
0.0
—————————————————————————————————————————————————————— Prestress + Self Weight Elastic Def
0.0
TRANSFER
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Erection
0.0
0.0
0.0
In situ 1A
0.0
0.0
0.0
Cr+Sh+Rlx
B
In situ 1B
0.0 0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
TOTAL PERMANENT EFFECTS, S0
Cr+Sh+Rlx
A
0.0
0.0
TOTAL PERMANENT EFFECTS, S0 , ∞
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
CREEP REDISTRIBUTION according to EN 1992-2 Annex KK.7 Construction On Centering, Sc = G + P1 + P2 Permanent G
0.0
0.0
0.0
0.0
0.0
Prestress P1
0.0
0.0
0.0
0.0
0.0
Prestress P2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
(Sc - S0 )*0.56856 Hence from KK.119, TOTAL CONSTRUCTION EFFECTS, S∞
[4] [5]
SDL Diff. Shr. 1
605.383
0.0
0.0
0.0
0.0
0.0
286.182
-0.2604
-0.4932
1.16132
-0.4378
0.0
0.0
0.0
0.0
0.0
Diff. Shr. 2
[Differential shrinkage is included when adverse TOTAL PERMANENT EFFECTS [including diff. shrinkage
SAM v6.50d
0.0 -0.2604
0.0 -0.4932
] 0.0 1.16132
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0.0 -0.4378]
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VARIABLE ACTIONS - CHARACTERISTIC COMBINATION Traffic Selected case: Traffic gr1a TS
1
78.7403
0.18258
0.11855
0.13548
-0.3045
-0.0209
-0.0209
-0.0221
-0.0221
0.07585
0.04925
0.05628
-0.1265
-0.0114
-0.0114
-0.0121
-0.0121
0.05865
0.03808
0.04352
-0.0978
-0.0033
-0.0033
-0.0034
-0.0034
Total (Leading)
: 0.28154
0.17035
0.19773
-0.5663
Total (in Combination)
: 0.19177
0.11557
0.13427
-0.3893
-0.7655
-0.4970
-0.5680
1.27682
0.05356
0.05356
0.05663
0.05663
-0.3161
-0.2052
-0.2345
0.52720
0.02425
0.02425
0.02564
0.02564
0.05865
0.03808
0.04352
-0.0978
-0.0033
-0.0033
-0.0034
-0.0034
Total (Leading)
: -0.9484
-0.5896
-0.6803
1.78504
Total (in Combination)
: -0.7307
-0.4544
-0.5242
1.37422
-19.966 Traffic gr1a UDL 1
32.7097
-10.906 Traffic gr1a FT
1
25.2916
-3.1186
ψ0
0.75 0.75 0.4
Other traffic cases for comparison: Traffic gr1a TS
2
-330.15
51.2513 Traffic gr1a UDL 2
-136.32
23.2038 Traffic gr1a FT
2
25.2916
-3.1186
0.75 0.75 0.4
Temperature Restraint None defined Differential Temperature - Heating Diff. Tmp H1
-1016.0
Diff. Tmp H2
-413.84
2.47603
-0.8853
-0.8625
1.35841
0.0
0.0
0.0
0.0
0.0
143.384
-1.4373
0.52916
0.24758
-1.7606
2.0E-4
4.63E-7
3.01E-7
3.44E-7
-7.7E-7
0.6
Differential Temperature - Cooling Diff. Tmp C1
976.559
Diff. Tmp C2
0.6
Other Variable Action None defined The following combinations of variable actions are evaluated in accordance with EN 1990 Equation 6.14b: a) Traffic as leading action + ψ0(Thermal + Other) b) Thermal as leading action + ψ0(Traffic + Other) c) Other as leading action + ψ0(Traffic + Thermal) For thermal actions the following cases are considered: i) temperature restraint alone ii) differential temperature: Heating iii) differential temperature: Cooling iv) temperature restraint + differential temperature: Heating v) temperature restraint + differential temperature: Cooling
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The most adverse case is with Traffic as leading action
Traffic
0.28154
0.17035
0.19773
-0.5663
ψ0 x Thermal
1.48562
-0.5312
0.14855
-1.0564
0.0
0.0
0.0
0.0
ψ0 x Other
TOTAL VARIABLE ACTIONS TOTAL PERMANENT (from above)
TOTAL COMBINATION
1.76716 -0.3608 0.34628 -1.6228 0.0 -0.4932 1.16132 -0.4378 —————————————————————————————————— 1.76716 -0.8540 1.50761 -2.0607
VARIABLE ACTIONS - FREQUENT COMBINATION Traffic Selected case:
ψ2
ψ1 = 0.75
Traffic gr1a TS
1
59.0552
-14.975 ψ1 = 0.75
Traffic gr1a UDL 1
24.5323
-8.1797 ψ1 =
0.13693
0.08891
0.10161
-0.2283
-0.0157
-0.0157
-0.0165
-0.0165
0.05688
0.03694
0.04221
-0.0949
-0.0085
-0.0085
-0.009
-0.009
0.0
0.0
0.4
however, EN 1991-2 Clause 4.5.2 excludes the footway loading from the frequent action of LM1 Traffic gr1a FT
1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Total (Leading)
: 0.16962
Total (in Combination)
:
0.0
0.10164
0.11824
-0.3488
0.0
0.0
0.0
0.0
-0.5741
-0.3728
-0.4260
0.95761
0.04017
0.04017
0.04247
0.04247
-0.2370
-0.1539
-0.1759
0.39540
0.01819
0.01819
0.01923
0.01923
0.0
Other traffic cases for comparison: ψ1 = 0.75
Traffic gr1a TS
2
-247.61
38.4385 ψ1 = 0.75
Traffic gr1a UDL 2
-102.24
17.4029 ψ1 =
0.0
0.0
0.4
however, EN 1991-2 Clause 4.5.2 excludes the footway loading from the frequent action of LM1 Traffic gr1a FT
2
0.0 0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Total (Leading)
: -0.7528
-0.4683
-0.5402
1.41472
Total (in Combination)
:
0.0
0.0
0.0
0.0
0.0
Temperature Restraint None defined
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Differential Temperature - Heating ψ1 =
0.6
Diff. Tmp H1
-1016.0
Diff. Tmp H2
-413.84
2.47603
-0.8853
-0.8625
1.35841
0.0
0.0
0.0
0.0
0.0
143.384
-1.4373
0.52916
0.24758
-1.7606
2.0E-4
4.63E-7
3.01E-7
3.44E-7
-7.7E-7
0.5
Differential Temperature - Cooling ψ1 =
0.6
Diff. Tmp C1
976.559
Diff. Tmp C2
0.5
Other Variable Action None defined The following combinations of variable actions are evaluated in accordance with EN 1990 Equation 6.15b: a) ψ1(Traffic) as leading action + ψ2(Thermal + Other) b) ψ1(Thermal) as leading action + ψ2(Traffic + Other) c) ψ1(Other) as leading action + ψ2(Traffic + Thermal) For thermal actions the following cases are considered: i) temperature restraint alone ii) differential temperature: Heating iii) differential temperature: Cooling iv) temperature restraint + differential temperature: Heating v) temperature restraint + differential temperature: Cooling The most adverse case is with Traffic as leading action
ψ1 x Traffic ψ2 x Thermal ψ2 x Other
TOTAL VARIABLE ACTIONS TOTAL PERMANENT (from above)
TOTAL COMBINATION
0.16962
0.10164
0.11824
-0.3488
1.23801
-0.4426
0.12379
-0.8803
0.0
0.0
0.0
0.0
1.40764 -0.3410 0.24203 -1.2292 0.0 -0.4932 1.16132 -0.4378 —————————————————————————————————— 1.40764 -0.8342 1.40336 -1.6671
SLS FLEXURE Precast
Stress
(MPa)
After Transfer
After Erection
SAM v6.50d
T B
T B
E
Strain -6
Curvature -6
(x10 )
Deflection
(rad/m)
0.04918 ET 1.57982
2.16062
0.0
15.0561
-32.784
0.0
19.6107
-0.0383
Max.
-1.229
-0.2676 EI -14.415 0.52367
Here
(mm)
(x10 )
28.2042
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After in situ 1A After in situ 1B
Long-term Dead
Diff. Temp H1 Diff. Temp H2 Diff. Temp C1 Diff. Temp C2
Traffic gr1a 1
T B T B
T B
T B T B T B T B
T B
-0.2562 EI -13.799 0.51456
1.1E-6 -2.5E-6
ES 3.12E-5
105.09
0.0
-4.8888
-29.104
0.0
1.59073
7.79E-5
0.0
-1.1726
26.3161
0.0
-1.4413
1.46E-5
0.0
0.40625
16.6888
0.0
-6.7548
92.6748
0.0
-12.688
-29.993 -1.3E-5
0.19773 ES 5.61411 -0.5663
5.84195
-7.0E-5
2.06E-7 ES 5.86E-6 -4.6E-7
0.0
23.1413
0.14855 ES 4.2178 -1.0564
-31.933
-37.408
-0.5175 ES -14.695 0.81504
6.14087
27.7139
1.16132 EL 99.2094 -0.4378
0.0
27.7139
-0.2562 EI -13.799 0.51456
-31.933
-16.081
Extreme in-service
Curvatures here are derived from precast section height: 1300.0mm ET = Elastic Modulus at Transfer = 31130.7MPa [EN1992-1-1 Clauses 3.1.3-(3) and 3.1.2-(6) with age 4 days] EI = Intermediate Term Elastic Modulus = 18567.1MPa [EN1992-1-1 Clause 7.4.3-(5) at 60.0 days (φ=0.89693)] EL = Long Term Elastic Modulus = 11705.8MPa [EN1992-1-1 Clause 7.4.3-(5) at infinite time (φ=2.00881)] ES = Short Term Elastic Modulus [Ecm]
= 35220.5MPa
________ [1] Refer to EN 1991-1-1 Table A.1 Note 1) [2] For the derivation of this value refer to the limiting stress calculations for transfer [3] includes draw-in and initial relaxation [4] With immediate losses and shrinkage / creep / relaxation losses until time at which insitu is cast. [5] Secondary effects arising from prestress in continuous section.
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Pre-tensioned Pre-stressed Beam Bridge Design Example
16.
Verification: ULS Shear - Pier
135
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Design code: EN 1992-2:2005 with UK National Annex (modified) Analysis: Shears EN 1990 Equation 6.10 ULS Persistent / Transient Load case: Traffic gr1a TS - for Shear design 1 Section Ref 2 at 21m from left end of beam
WARNING - This analysis assumes that all tension steel (As) is adequately anchored to resist the required tensile forces. (Refer to clause 6.2.1(7), Figure 6.3, and clause 6.2.3(7) of EN1992-1-1).
Section details:
Ref 2 "Section 2" at 1 x span = 21 m from left end of beam
Analysis:
Traffic Actions: Shear for gr1a, loading I.D. 1 Ultimate Limit State: Persistent / Transient - EN 1990 Equation 6.10
SUMMARY OF ACTIONS PERMANENT ACTIONS
ACTION TYPE
SHEAR kN
MOMENT kN.m
AXIAL kN
Beam erection before composite
= -190.3096
0.0
0.0
Construction stage 1A
= -101.1947
0.0
0.0
Construction stage 1B
= -36.56457
0.0
0.0
Surfacing
= -45.26953
-156.7871
0.0
Differential Shrinkage / creep
=
0.0
0.0
0.0
-373.3384
-156.7871
0.0
TOTAL PERMANENT ACTIONS,
SAM v6.50d
γG x Gk
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[1]
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VARIABLE ACTIONS
ACTION TYPE
[2]
SHEAR kN
MOMENT AXIAL kN.m kN
ψ0
ψ1
ψ2
Traffic gr1a TS - for Shear design = -479.47 -407.05 37.0081 0.75 0.75
0.0
Traffic gr1a UDL - for Shear desig = -193.03 -710.42
0.0
Traffic gr1a Footway - for Shear d = -6.0115 TOTAL VARIABLE ACTIONS, γQ , 1 x Qk , 1 "+" Traffic leading:
Traffic
ψ0 x Other
Other leading:
105.33 0.75 0.75
-46.95 -15.717
0.4
0.4
0.0
[3]
ΣγQ , i x ψ0 , i x Qk , i
-678.5098
-1164.427
126.62151
0.0
0.0
0.0
Total
-678.5098
-1164.427
126.62151
ψ0 x Traffic
-506.7783
-856.888
100.46704
Other
0.0
0.0
0.0
Total
-506.7783
-856.888
100.46704
-1051.848
-1321.214
126.62151
Critical case is with traffic leading
TOTAL COMBINATION
This section is within a distance d from the face of a support. The following clauses therefore apply:
Clause 6.2.1(8)
The design shear reinforcement is taken as that which is required at the section a distance d from the face of the support. A check is carried out below that VEd does not exceed VRd,max.
Clause 6.2.2(6) and 6.2.3(8)
The load reduction factor β is not readily applicable to typical bridge loading, and it is conservative to ignore this clause.
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Clause 6.2.3(103) and 6.2.3(4) Shear Resistance with design shear reinforcement characteristic strength of shear rft, fy w k = 500.0 MPa material partial factor γs = 1.15 design strength of shear rft fy w k /γs , fy w d = 434.783 MPa characteristic strength of concrete fc k = 40.0 MPa material partial factor γc = 1.5 design strength of concrete fc k /γc , fc d = 26.6667 MPa angle between compression strut & beam axis, θ = 35.0 ° cotθ = 1.42815 angle between shear rft and beam axis, α = π/2 rad axial force in cross section, NE d = 126.622 kN area of concrete cross section, Ac = 9.283E5 mm² concrete compressive stress NE d /Ac , σc p = 126.622/9.283E5 = 0.13640 MPa effective depth, d = 1407.5 mm distance from edge of support, a = 0.0 mm
Maximum Shear Force Value The maximum value of shear resistance is given by: VR d , m a x = αc w .bw .z.υ1 .fc d (cotθ + cotα)/(1 + cot²θ)
Expression (6.14)
compression chord stress coefficient, αc w : (Note 3) σc p /fc d = 0.13640 / 26.6667 = 0.00512 hence from Expression (6.11aN) αc w = 1.00512 minimum width between tension and compression chords, bw = 216.0 mm
[4]
inner lever arm, strength reduction factor, υ1 : υ = 0.6[1.0-fc k /250] = 0.6*(1.0-40.0/250.0) = 0.504 υ1 = υ(1-0.5cosα) = 0.504
z = 1231.81 mm
Expression (6.6N)
αc w .bw .z.υ1 .fc d = 1.00512*216.0*1231.81*0.504*26.6667 = 3594.3 kN since α = π/2, the expression (6.14) reduces to: VR d , m a x = αc w .bw .z.υ1 .fc d /(cotθ + tanθ) Expression (6.9) = 3594.3/(1.42815 + 0.70020) = 1680.17 kN which is greater than VE d - Vc c d - Vt d (1051.85 kN) and therefore OK.
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Summary of link requirements along the beam:
dimension from left beam end m
As w / s mm²/mm
MR d / ME d
[5]
Spacing in mm for link diameter [6]
10.0mm
0.0
0.21857
718.648
1.17
0.21857
718.648
1.35167
0.21857
718.648
1.51281
0.21857
718.648
2.0
0.21857
718.648
2.01204
0.21857
718.648
2.5
0.21857
718.648
2.86364
0.21857
718.648
3.51281
0.21857
718.648
4.01204
0.21857
718.648
4.51204
0.21857
718.648
4.77273
0.21857
718.648
6.68182
0.57756
8.59091
0.21857
1.7
271.969 718.648
10.5
0.21857
12.4091
0.59500
718.648
14.3182
0.21857
718.648
16.2273
0.21857
718.648
16.488
0.21857
718.648
16.9872
0.21857
718.648
17.4872
0.21857
718.648
18.0
0.21857
718.648
18.0
0.21857
718.648
18.1364
0.21857
718.648
1.69
263.998
18.5
0.21857
718.648
18.988
0.21857
718.648
19.0
0.21857
19.4872
1.30135
17.9
718.648 120.705
19.5925
1.31204
11.9
119.722
20.0454
1.31204
11.9
119.722
21.0
1.31204
11.9
119.722
The interface shear requirement may be critical. Warning: At the locations indicated in the table above, additional longitudinal reinforcement is required to carry the shear in the tension chord of the truss.
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location.
The reinforcement requirements are calculated in the results for the relevant design
________ [1] Shrinkage effects are excluded at ULS in accordance with EN1992-1-1/2.3.2.2(2) [2] Thermal effects are not considered for ULS Shear in accordance with EN 1992-1-1 clause 2.3.1.2(2) [3] EN 1991-2 Clause 4.5.2 excludes the footway loading from the frequent action of LM1 [4] The inner lever arm is derived by analysing the section for the resistance moment and dividing the moment by the reinforcement tension force. (Using the resistance moment gives a conservative result). [5] The reduced resistance moment MRd is calculated for locations where design shear reinforcement is required. MRd/MEd must be greater than 1.0. [6] Based on 2 legs (double this value if using 4 legs)
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Pre-tensioned Pre-stressed Beam Bridge Design Example
17.
Verification: ULS Interface Shear
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Design code: EN 1992-2:2005 with UK National Annex (modified) Analysis: Longitudinal Shear EN 1990 Equation 6.10 ULS Persistent / Transient Load case: Traffic gr1a TS - for Shear design 1 Section Ref 2 at 21m from left end of beam
Section details:
Ref 2 "Section 2" at 1 x span = 21 m from left end of beam
Analysis:
Interface and web/flange shear for gr1a, loading I.D. 1 Ultimate Limit State: Persistent / Transient - EN 1990 Equation 6.10
SUMMARY OF ACTIONS PERMANENT ACTIONS
ACTION TYPE
SHEAR kN
Beam erection before composite
= -190.3096
Construction stage 1A
= -101.1947
Construction stage 1B
= -36.56457
Surfacing
= -45.26953
Differential Shrinkage / creep
=
TOTAL PERMANENT ACTIONS,
SAM v6.50d
γG x Gk
0.0
[1]
-373.3384
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VARIABLE ACTIONS [2]
ACTION TYPE
SHEAR kN
Traffic gr1a TS - for Shear design =
ψ0
ψ1
ψ2
-479.468
0.75 0.75
0.0
Traffic gr1a UDL - for Shear desig = -193.0303
0.75 0.75
0.0
Traffic gr1a Footway - for Shear d = -6.011462 TOTAL VARIABLE ACTIONS, γQ , 1 x Qk , 1 "+" Traffic leading:
Traffic
Other leading:
0.4
[3]
0.0
ΣγQ , i x ψ0 x Qk , i
-678.5098
ψ0 x Other
0.4
0.0
Total
-678.5098
ψ0 x Traffic
-506.7783
Other
0.0
Total
-506.7783
Critical case is with traffic leading
TOTAL COMBINATION
-1051.848
Clause 6.2.5 INTERFACE SHEAR CALCULATIONS This section is within a distance d from the face of a support. The following clause therefore applies:
Clause 6.2.1(8)
The design shear is taken as that which is present at the section which is a distance d from the face of the support.
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Summary of link requirements along the beam:
dimension from left beam end m
Flexural Shear As w / s mm²/mm
Interface Shear As w / s 6.2.5(3) mm²/mm
suggestion
Spacing in mm for link diameter [4]
10.0mm
0.0
0.21857
2.5653
2.33209
67.3557
1.17
0.21857
2.5653
2.33209
67.3557
1.35167
0.21857
2.5653
2.33209
67.3557
1.51281
0.21857
2.54248
2.33209
67.3557
2.0
0.21857
2.47351
2.33209
67.3557
2.01204
0.21857
2.47181
2.33209
67.3557
2.5
0.21857
2.40275
2.33209
67.3557
2.86364
0.21857
2.3513
2.33209
67.3557
3.51281
0.21857
2.17865
2.33209
67.3557
4.01204
0.21857
2.04589
1.98059
79.3095
4.51204
0.21857
1.91325
1.98059
79.3095
4.77273
0.21857
1.84411
1.98059
79.3095
6.68182
0.57756
1.37475
1.67646
93.697
8.59091
0.21857
0.70216
1.24977
125.687
10.5
0.21857
0.96417
1.2875
122.003
12.4091
0.59500
1.41625
1.68008
93.4954
14.3182
0.21857
1.84809
2.07448
75.7201
16.2273
0.21857
2.28193
2.36788
66.3377
16.488
0.21857
2.34875
2.36788
66.3377
16.9872
0.21857
2.47674
2.36788
66.3377
17.4872
0.21857
2.60467
2.83907
55.3279
18.0
0.21857
2.73586
2.83907
55.3279
18.0
0.21857
2.73586
2.83907
55.3279
18.1364
0.21857
2.77085
2.83907
55.3279
18.5
0.21857
2.85884
2.83907
55.3279
18.988
0.21857
2.97693
2.83907
55.3279
19.0
0.21857
2.97984
2.83907
55.3279
19.4872
1.30135
3.09753
2.83907
55.3279
19.5925
1.31204
3.12298
2.83907
55.3279
20.0454
1.31204
3.12298
2.83907
55.3279
21.0
1.31204
3.12298
2.83907
55.3279
________ [1] Shrinkage effects are excluded at ULS in accordance with EN1992-1-1/2.3.2.2(2) [2] Thermal effects are not considered for ULS Shear in accordance with EN 1992-1-1 clause 2.3.1.2(2) [3] EN 1991-2 Clause 4.5.2 excludes the footway loading from the frequent action of LM1 [4] Based on 2 legs (double this value if using 4 legs)
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Pre-tensioned Pre-stressed Beam Bridge Design Example
18.
Verification: Web Shear Cracking
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Design code: Analysis:
EN 1992-2:2005 with UK National Annex (modified) Shears at time t=infinity EN 1990 Equation 6.15 SLS Frequent Exposure Class: XD1, XD2, XS1, XS2, XS3 Load case: Traffic gr1a TS - for Shear design 1 Section Ref 2 at 21m from left end of beam
WARNING - This analysis assumes that all tension steel (As) is adequately anchored to resist the required tensile forces. (Refer to clause 6.2.1(7), Figure 6.3, and clause 6.2.3(7) of EN1992-1-1).
Section details:
Ref 2 "Section 2" at 1 x span = 21 m from left end of beam
Analysis:
Traffic Actions: Shear for gr1a, loading I.D. 1 At time considered, t = ∞ Serviceability Limit State: Frequent - EN 1990 Equation 6.15
SUMMARY OF ACTIONS
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PERMANENT ACTIONS
ACTION TYPE
SHEAR kN
AXIAL kN
Beam erection before composite
=
-140.97
0.0
0.0
Construction stage 1A
= -74.95901
0.0
0.0
Construction stage 1B
= -27.08487
0.0
0.0
Surfacing
= -37.72461
-130.6559
0.0
=
9.715637
-203.8871
0.0
-271.0229
-334.543
0.0
Differential Shrinkage / creep
MOMENT kN.m
TOTAL PERMANENT ACTIONS,
Gk
VARIABLE ACTIONS
ACTION TYPE
SHEAR kN
MOMENT kN.m
AXIAL kN
ψ0
ψ1
ψ2
Traffic gr1a TS - for Shear design = -355.16 -301.52 27.4134 0.75 0.75
0.0
Traffic gr1a UDL - for Shear desig = -142.99 -526.24 78.0224 0.75 0.75
0.0
Traffic gr1a Footway - for Shear d = -4.4529 -34.778 -11.642
0.0
TOTAL VARIABLE ACTIONS, ψ1 , 1 x Qk , 1 "+" Traffic leading:
Other leading:
0.4
ψ1 x Traffic
-373.6102
-620.8208
79.076887
0.0
0.0
0.0
Total
ψ2 x Other
Σψ2 , i x Qk , i
0.4
-373.6102
-620.8208
79.076887
ψ2 x Traffic
0.0
0.0
0.0
ψ1 x Other
0.0
0.0
0.0
Total
0.0
0.0
0.0
-644.633
-955.3638
79.076887
[1]
Critical case is with traffic leading
TOTAL COMBINATION
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WEB SHEAR CRACKING EN 1992-2 Annex QQ Characteristic strength of concrete in web, fc k Characteristic tensile strength 5% fractile, fc t k ; 0 , 0 5 Design shear on precast section before composite, VE d , 1 Total design shear on precast section, VE d Design shear on precast section after composite, VE d , 2 Stress in precast from Prestress P and bending ME d : at the top of the precast section, at the bottom of the precast section,
= = = = =
40.0 MPa -2.4562 MPa 243.01388 kN 654.34866 kN 411.33479 kN
σa = -1.1362 MPa σb = 3.32705 MPa
Height of precast section,
h =
1300.0 mm
Principal tensile stress is checked at the level of the centroid, and in addition at 100 points through the depth of the section to find the critical level.
First check at the composite section centroid: At the composite section centroid,
zf , m a x = 912.41288 mm
At this height: width of precast section, b = 354.58722 mm direct stress, σc 1 = σb + zf , m a x /h*(σa -σb ) = 3.32705 + 912.413/1300.0*(-1.1362-3.32705) = 0.19452 MPa For precast section: area beyond level zf , m a x first moment of area second moment of area
A1 = 1.547E5 mm² (A.z)1 = 8.269E7 mm³ Iy y , 1 = 9.2977E10 mm⁴
Shear stress at height zf , m a x τy z , E d , 1 = VE d , 1 * (A.z)1 /(Iy y , 1 .b) = 243.014 * 8.269E7 / (9.2977E10*354.587) = 0.60953 MPa For composite section: area beyond level zf , m a x (transformed) A2 = 3.825E5 mm² first moment of area (A.z)2 = 2.113E8 mm³ second moment of area Iy y , 2 = 2.3848E11 mm⁴
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Shear stress at level zf , m a x τy z , E d , 2 = VE d , 2 * (A.z)2 /(Iy y , 2 .b) = 411.335 * 2.113E8 / (2.3848E11*354.587) = 1.02807 MPa total shear stress,
τy z , E d = 0.60953 + 1.02807 = 1.63761
From Mohr's circle: centre, σc 0 = 0.5*(0.19452+0.0) radius,
σr
=
√[τy z , E d
2
= 0.09726 MPa
2
+ (σc 1 -σc 0 ) ]
= 1.64049 MPa
σ3 = 0.09726 + 1.64049 = 1.73776 MPa σ1 = 0.09726 - 1.64049 = -1.5432 MPa From Expression QQ.101, and using Annex QQ sign convention, σ1 = 1.54323 MPa fc t b = [1 - 0,8.(σ3 /fc k )].fc t k ; 0 , 0 5 = 0.96524 * 2.45617 = 2.37081 MPa
Now checking through the depth of the section:
Level at which critical tension stress occurs, zf , m a x = 442.0 mm At this height: width of precast section, b = 218.2 mm direct stress, σc 1 = σb + zf , m a x /h*(σa -σb ) = 3.32705 + 442.0/1300.0*(-1.1362-3.32705) = 1.80956 MPa For precast section: area beyond level zf , m a x first moment of area second moment of area
[2]
A1 = 2.489E5 mm² (A.z)1 = 9.905E7 mm³ Iy y , 1 = 9.2977E10 mm⁴
Shear stress at height zf , m a x τy z , E d , 1 = VE d , 1 * (A.z)1 /(Iy y , 1 .b) = 243.014 * 9.905E7 / (9.2977E10*218.2) = 1.18646 MPa For composite section: area beyond level zf , m a x (transformed) A2 = 2.489E5 mm² first moment of area (A.z)2 = 1.828E8 mm³ second moment of area Iy y , 2 = 2.3848E11 mm⁴
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Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
Shear stress at level zf , m a x τy z , E d , 2 = VE d , 2 * (A.z)2 /(Iy y , 2 .b) = 411.335 * 1.828E8 / (2.3848E11*218.2) = 1.44478 MPa total shear stress,
τy z , E d = 1.18646 + 1.44478 = 2.63124
From Mohr's circle: centre, σc 0 = 0.5*(1.80956+0.0) radius,
σr
=
√[τy z , E d
2
= 0.90478 MPa
2
+ (σc 1 -σc 0 ) ]
= 2.78246 MPa
σ3 = 0.90478 + 2.78246 = 3.68724 MPa σ1 = 0.90478 - 2.78246 = -1.8777 MPa From Expression QQ.101, and using Annex QQ sign convention, σ1 = 1.87767 MPa fc t b = [1 - 0,8.(σ3 /fc k )].fc t k ; 0 , 0 5 = 0.92625 * 2.45617 = 2.27505 MPa which is greater than σ1, so minimum reinforcement in accordance with 7.3.2 should be provided. ________ [1] EN 1991-2 Clause 4.5.2 excludes the footway loading from the frequent action of LM1 [2] Nearest multiple of depth/100
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© 2012 Bestech Systems Ltd
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156
Pre-tensioned Pre-stressed Beam Bridge Design Example
Appendix - National Annex NDP Values
157
158
Bestech Systems Limited 2 Slaters Court Princess Street Knutsford WA16 6BW Job:
Sample Reports
Beam:
Prestress Beam - Inner span 1 Eurocode + UK NA
Job No.: 6.5d Calc. By: dlg Checked:
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
NATIONALLY DETERMINED PARAMETERS European Standard EN1990 Description
Current
United Kingdom NDPs
Category A: domestic; residential area
0.7
0.7
Category B: office areas
0.7
0.7
Category C: congregation areas
0.7
0.7
Category D: shopping areas
0.7
0.7
Category E: storage areas
1
1
Category F: traffic area; vehicle weight
View more...
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