Appendix to 'Residual life assessment for bridges' lecture
Table 1 Values of the Coefficient ρ' [2]
|
v [km/h] l [m] |
10 |
20 |
30 |
40 |
50 |
60 |
70 |
80 |
90 |
100 |
110 |
120 |
140 |
160 |
180 |
|
2 |
0,0177 |
0,0360 |
0,0549 |
0,0746 |
0,0951 |
0,1163 |
0,1383 |
0,1612 |
0,1851 |
0,2099 |
0,2357 |
0,2625 |
0,3196 |
0,3815 |
0,4483 |
|
5 |
0,0177 |
0,0360 |
0,0549 |
0,0746 |
0,0951 |
0,1163 |
0,1383 |
0,1612 |
0,1851 |
0,2099 |
0,2357 |
0,2625 |
0,3196 |
0,3815 |
0,4483 |
|
7 |
0,0177 |
0,0360 |
0,0549 |
0,0746 |
0,0951 |
0,1163 |
0,1383 |
0,1612 |
0,1851 |
0,2099 |
0,2357 |
0,2625 |
0,3196 |
0,3815 |
0,4483 |
|
10 |
0,0177 |
0,0360 |
0,0549 |
0,0746 |
0,0951 |
0,1163 |
0,1383 |
0,1612 |
0,1851 |
0,2099 |
0,2357 |
0,2625 |
0,3196 |
0,3815 |
0,4483 |
|
15 |
0,0177 |
0,0360 |
0,0549 |
0,0746 |
0,0951 |
0,1163 |
0,1383 |
0,1612 |
0,1851 |
0,2099 |
0,2357 |
0,2625 |
0,3196 |
0,3815 |
0,4483 |
|
20 |
0,0177 |
0,0360 |
0,0549 |
0,0746 |
0,0951 |
0,1163 |
0,1383 |
0,1612 |
0,1851 |
0,2099 |
0,2357 |
0,2625 |
0,3196 |
0,3815 |
0,4483 |
|
30 |
0,0149 |
0,0303 |
0,0462 |
0,0625 |
0,0794 |
0,0968 |
0,1148 |
0,1334 |
0,1526 |
0,1725 |
0,1930 |
0,2142 |
0,2589 |
0,3067 |
0,3579 |
|
40 |
0,0133 |
0,0269 |
0,0409 |
0,0552 |
0,0700 |
0,0852 |
0,1008 |
0,1169 |
0,1335 |
0,1505 |
0,1681 |
0,1862 |
0,2240 |
0,2642 |
0,3069 |
|
50 |
0,0121 |
0,0245 |
0,0372 |
0,0502 |
0,0635 |
0,0772 |
0,0913 |
0,1057 |
0,1205 |
0,1357 |
0,1513 |
0,1673 |
0,2007 |
0,2360 |
0,2733 |
|
70 |
0,0105 |
0,0213 |
0,0323 |
0,0435 |
0,0549 |
0,0667 |
0,0786 |
0,0909 |
0,1034 |
0,1163 |
0,1294 |
0,1428 |
0,1706 |
0,1998 |
0,2304 |
|
100 |
0,0091 |
0,0183 |
0,0278 |
0,0374 |
0,0472 |
0,0571 |
0,0673 |
0,0776 |
0,0882 |
0,0990 |
0,1100 |
0,1212 |
0,1442 |
0,1683 |
0,1933 |
|
120 |
0,0084 |
0,0170 |
0,0257 |
0,0346 |
0,0436 |
0,0528 |
0,0622 |
0,0717 |
0,0814 |
0,0912 |
0,1013 |
0,1115 |
0,1325 |
0,1544 |
0,1771 |
Table 2 Values of the Coefficient ρ'' [2]
|
v [km/h] l [m] |
10 |
20 |
30 |
40 |
50 |
60 |
70 |
80 |
90 |
100 |
110 |
120 |
140 |
160 |
180 |
|
2 |
0,0673 |
0,1345 |
0,2018 |
0,2690 |
0,3363 |
0,4055 |
0,4708 |
0,5380 |
0,5380 |
0,5380 |
0,5380 |
0,5380 |
0,5380 |
0,5380 |
0,5380 |
|
5 |
0,0545 |
0,1090 |
0,1635 |
0,2181 |
0,2726 |
0,3271 |
0,3816 |
0,4361 |
0,4361 |
0,4361 |
0,4361 |
0,4361 |
0,4361 |
0,4361 |
0,4361 |
|
7 |
0,0429 |
0,0858 |
0,1287 |
0,1715 |
0,2144 |
0,2573 |
0,3002 |
0,3431 |
0,3431 |
0,3431 |
0,3431 |
0,3431 |
0,3431 |
0,3431 |
0,3431 |
|
10 |
0,0258 |
0,0515 |
0,773 |
0,1030 |
0,1288 |
0,1545 |
0,1603 |
0,2060 |
0,2060 |
0,2060 |
0,2060 |
0,2060 |
0,2060 |
0,2060 |
0,2060 |
|
15 |
0,0074 |
0,0148 |
0,0221 |
0,0295 |
0,0369 |
0,0443 |
0,0516 |
0,0590 |
0,0590 |
0,0590 |
0,0590 |
0,0590 |
0,0590 |
0,0590 |
0,0590 |
|
20 |
0,0013 |
0,0026 |
0,0038 |
0,0051 |
0,0064 |
0,0077 |
0,0090 |
0,0103 |
0,0103 |
0,0103 |
0,0103 |
0,0103 |
0,0103 |
0,0103 |
0,0103 |
|
30 |
0,0012 |
0,0024 |
0,0036 |
0,0048 |
0,0060 |
0,0072 |
0,0084 |
0,0095 |
0,0095 |
0,0095 |
0,0095 |
0,0095 |
0,0095 |
0,0095 |
0,0095 |
|
40 |
0,0004 |
0,0007 |
0,0011 |
0,0015 |
0,0019 |
0,0022 |
0,0026 |
0,0030 |
0,0030 |
0,0030 |
0,0030 |
0,0030 |
0,0030 |
0,0030 |
0,0030 |
|
50 |
0,0001 |
0,0001 |
0,0002 |
0,0002 |
0,0003 |
0,0003 |
0,0004 |
0,0004 |
0,0004 |
0,0004 |
0,0004 |
0,0004 |
0,0004 |
0,0004 |
0,0004 |
|
70 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
|
100 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
|
120 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
0,0000 |
Table 3 Definitions of characteristic lengths for fatigue calculations
|
Case |
Structural Element |
Characteristic Length LΦ |
|
DECK PLATE (Steel) closed deck with ballast bed (orthotropic deck plate) (for local stresses) |
||
|
1
|
Deck with longitudinal and cross ribs |
|
|
1.1 Deck plate (for both directions) |
3 x cross girder spacing |
|
|
1.2 Longitudinal ribs (including small cantilevers up to 0.50m)(*) |
3 x cross girder spacing |
|
|
1.3 Cross girders, end cross girders |
2 x length of cross girders |
|
|
2
|
Deck plate with cross girders only |
|
|
2.1 Deck plate (for both directions) |
2 x cross girder spacing + 3 m |
|
|
2.2 Cross girders, end cross girders |
2 x length of cross girders |
|
|
DECK PLATE (Steel) open deck without ballast bed (for local stresses) |
||
|
3
|
3.1 Rail bearers - as an element of a grillage - simply supported |
3 x cross girder spacing cross girder spacing + 3 m |
|
3.2 Cantilever of rail bearer |
the characteristic length leads to Φ3 = 2.0 |
|
|
3.3 Cross girders, end cross girders |
2 x length of cross girders |
|
|
DECK PLATE WITH BALLAST BED (structural concrete) (for local and transverse stresses) |
||
|
4
|
4.1 Deck plates as part of box girders or upper flange of main beam |
|
|
- spanning transversely to the main girders |
3 x span of deck plate |
|
|
- spanning in the longitudinal direction |
3 x span of deck plate or characteristic length of main girder; whichever is the lesser |
|
|
- transverse cantilevers supporting railway loading |
see footnote (**) |
|
|
4.2 Deck plate continuous over cross girders (in main-girder direction) |
2 x span of deck plate in the longitudinal direction |
|
|
4.3 Deck plate for trough bridges: |
|
|
|
- spanning perpendicular to the main girders |
span of deck plate |
|
|
- spanning in the longitudinal direction |
2 x span of deck plate or characteristic length of main girders - whichever is the lesser |
|
|
4.4 Deck slabs spanning transversely between steel beams embedded in concrete |
2 x characteristic length in the longitudinal direction |
|
|
MAIN GIRDER ELEMENTS |
||
|
5
|
5.1 Simply supported girders and slabs (including steel girders embedded in concrete) |
Span in main girder direction |
|
5.2 Girders and slabs continuous over n spans with: |
LΦ = k . Lm, at least max. Li (i=1,....,n) |
|
|
Lm = 1/n(L1+L2+..+Ln) |
 |
|
* In general all cantilevers greater than 0.50 m and supporting railway loads need a special study.
(**) These cantilevers need a special study.
Table 4 Maximum bending moments produced in a simply supported girder by the UIC 71 loading
|
L (m) |
Mmax (kN.m) |
L (m) |
Mmax (kN.m) |
L (m) |
Mmax (kN.m) |
|
1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0 3,2 3,4 3,6 3,8 4,0 4,2 4,4 4,6 4,8 5,0 |
62,5 75,0 87,5 100,0 112,7 125,8 139,3 153,2 167,5 182,2 197,3 212,8 241,2 275,0 312,5 350,0 387,5 425,0 462,5 500,0 537,7 |
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 32 |
732,2 974,2 1251 1543 1855 2187 2539 2911 3303 3715 4147 4599 5071 5563 6075 7159 8323 9567 10890 12300 13780 |
34 36 38 40 42 44 46 48 50 52 54 56 58 60 65 70 75 80 85 90 100 |
15340 16990 18710 20520 22400 24360 26410 28530 30740 33020 35380 37830 40350 42960 49820 57180 65040 73400 82260 91620 111800 |
Table 5 Historical types of trains for fatigue analysis [1]
Type 2.1 (1876-1890) ∑ P = 126,5 t L = 64.04 m
Type 2.2 (1876-1890) ∑ P = 249 t L = 77.18 m
Type 3.1 (1891-1905) ∑ P = 166 t L = 76.15 m
Type 3.2 (1891-1905) ∑ P = 325 t L = 76.98 m
Type 4.1 (1906-1920) ∑ P = 205.1 t L = 95.02 m
Type 6.1 (1936-1950) ∑ P = 294 t L = 98.78 m
Type 6.2 (1936-1950) ∑ P = 478 t L = 171.14 m
Type 6.3 (1936-1950) ∑ P = 732 t L = 271.9 m
Type 7.1 (1951-1965) ∑ P = 52 t L = 46.5 m
Type 7.2 (1951-1965) ∑ P = 346 t L = 151.1 m
Type 7.3 (1951-1965) ∑ P = 406 t L = 177.5 m
Table 6 ΛTj - for fatigue trains-past (k = 5) [2]
|
Period
|
Train j
|
Span l [m] |
|||||||||
|
2 |
3 |
5 |
7 |
10 |
15 |
20 |
25 |
50 |
100 |
||
|
1 |
1 2 |
|
|
|
|
|
|
|
|
|
|
|
2 |
1 2 |
0,637 0,709 |
0,595 0,653 |
0,423 0,523 |
0,411 0,515 |
0,364 0,445 |
0,577 0,577 |
0,370 0,412 |
0,298 0,335 |
0,277 0,358 |
0,116 0,197 |
|
3 |
1 2 |
0,721 0,884 |
0,612 0,823 |
0,466 0,529 |
0,464 0,515 |
0,404 0,445 |
0,635 0,577 |
0,453 0,412 |
0,335 0,335 |
0,325 0,374 |
0,134 0,224 |
|
4 |
1 2 3 |
0,753 0,938 0,891 |
0,685 0,798 0,810 |
0,477 0,580 0,484 |
0,463 0,577 0,465 |
0,445 0,549 0,446 |
0,692 0,756 0,635 |
0,494 0,535 0,453 |
0,410 0,447 0,373 |
0,358 0,423 0,390 |
0,155 0,189 0,218 |
|
5 |
1 2 3 |
1,116 1,117 1,104 |
0,891 0,911 1,019 |
0,657 0,666 0,748 |
0,668 0,609 0,748 |
0,647 0,589 0,744 |
0,981 0,813 1,043 |
0,700 0,618 0,701 |
0,596 0,522 0,560 |
0,569 0,537 0,537 |
0,247 0,229 0,245 |
|
6 |
1 2 3 |
1,090 1,148 1,071 |
0,869 0,888 1,010 |
0,672 0,699 0,814 |
0,671 0,692 0,832 |
0,687 0,699 0,813 |
0,981 0,983 1,156 |
0,741 0,741 0,782 |
0,596 0,596 0,634 |
0,569 0,569 0,602 |
0,229 0,239 0,253 |
|
7 |
1 2 3 4 |
0,427 1,134 1,148 1,385 |
0,411 0,810 0,843 1,264 |
0,252 0,658 0,660 0,883 |
0,185 0,673 0,674 0,782 |
0,125 0,689 0,691 0,777 |
0,174 1,040 1,043 1,100 |
0,124 0,782 0,783 0,743 |
0,112 0,634 0,634 0,634 |
0,114 0,602 0,602 0,667 |
0,053 0,232 0,237 0,310 |
|
8 |
1 2 3 4 |
0,427 1,068 1,193 1,413 |
0,411 1,016 0,981 1,355 |
0,252 0,515 0,670 0,805 |
0,185 0,447 0,643 0,633 |
0,125 0,416 0,582 0,499 |
0,174 0,592 0,722 0,676 |
0,124 0,415 0,541 0,468 |
0,112 0,373 0,450 0,379 |
0,114 0,325 0,439 0,409 |
0,053 0,134 0,184 0,211 |
Table 6 (continued) ΛTj - for fatigue trains-past (k = 3,75) [2]
|
Period
|
Train j
|
Span l [m] |
|||||||||
|
2 |
3 |
5 |
7 |
10 |
15 |
20 |
25 |
50 |
100 |
||
|
1 |
1 2 |
|
|
|
|
|
|
|
|
|
|
|
2 |
1 2 |
0,713 0,846 |
0,648 0,731 |
0,435 0,541 |
0,414 0,520 |
0,365 0,445 |
0,577 0,577 |
0,371 0,412 |
0,298 0,336 |
0,277 0,358 |
0,116 0,197 |
|
3 |
1 2 |
0,813 1,089 |
0,678 1,004 |
0,471 0,565 |
0,468 0,522 |
0,405 0,449 |
0,635 0,578 |
0,453 0,412 |
0,336 0,336 |
0,325 0,374 |
0,134 0,224 |
|
4 |
1 2 3 |
0,880 1,158 1,109 |
0,791 0,962 1,000 |
0,511 0,625 0,526 |
0,467 0,608 0,475 |
0,445 0,591 0,450 |
0,693 0,777 0,637 |
0,494 0,538 0,454 |
0,410 0,448 0,373 |
0,358 0,423 0,391 |
0,155 0,189 0,218 |
|
5 |
1 2 3 |
1,318 1,396 1,420 |
1,034 1,089 1,287 |
0,679 0,760 0,845 |
0,674 0,685 0,811 |
0,648 0,635 0,782 |
0,981 0,835 1,064 |
0,700 0,620 0,705 |
0,595 0,522 0,563 |
0,569 0,537 0,537 |
0,247 0,229 0,245 |
|
6 |
1 2 3 |
1,289 1,423 1,371 |
0,999 1,049 1,275 |
0,719 0,706 0,886 |
0,686 0,751 0,870 |
0,688 0,731 0,830 |
0,981 0,997 1,166 |
0,741 0,742 0,786 |
0,596 0,597 0,635 |
0,569 0,569 0,602 |
0,229 0,239 0,253 |
|
7 |
1 2 3 4 |
0,480 1,336 1,374 1,804 |
0,464 0,914 0,980 1,611 |
0,283 0,686 0,692 1,065 |
0,205 0,694 0,699 0,876 |
0,130 0,697 0,707 0,807 |
0,177 1,048 1,062 1,117 |
0,126 0,785 0,789 0,750 |
0,114 0,635 0,635 0,638 |
0,114 0,602 0,602 0,669 |
0,053 0,232 0,237 0,310 |
|
8 |
1 2 3 4 |
0,480 1,221 1,445 1,814 |
0,464 1,151 1,183 1,716 |
0,283 0,577 0,760 1,013 |
0,205 0,495 0,694 0,740 |
0,130 0,442 0,621 0,560 |
0,177 0,622 0,779 0,728 |
0,126 0,426 0,562 0,493 |
0,114 0,377 0,460 0,394 |
0,114 0,325 0,439 0,416 |
0,053 0,134 0,184 0,211 |
EXAMPLE OF CALCULATION PROCEDURE
Railway bridge, L = 5 m, built in 1902, consisting of two riveted plate girders each with a section modulus, w, of 2890 cm3
From Section 2.4, the dynamic amplification factor is given by:
From Table 4, maximum bending moment = 537,7 kN.m
σuic = (1,53 × 537,7 × 106) / (2 × 2890 × 103) = 14,23 kN/cm2
ΔσR = 100 N/mm2 ΛR = 1,65 × 100/142,3 = 1,16
Δσuic = 142,3 N/mm2
The calculation of the total damage S is shown in Table 7.
Sp = {365 /(2.106 × 1,165)} x 747 = 649 . 10-4
Υt = 1,40 x 1,15 = 1,61
1,615 x 649 ´ 10-4 = 0,70 ≤ 1
=> Safety (first case)
Table 7 Calculation of total damage in Example of calculation procedure
|
Period |
Tn (years) |
Train (type) |
Speed km/h |
Trains/ day |
ΛTj |
(ΛTj)k |
∑NjN (ΛTj)k |
1 - φ |
[(1+φj)/φuci]k |
 |
|
(1) 1902 - 1908 |
7 |
3.1 3.2 |
60 40 |
20 20 |
0,466 0,529 |
0,021 0,041 |
0,42 0,82 |
1,513 1,337 |
0,97 0,52 |
2,85 2,98 |
|
(2) 1009 - 1923 |
15 |
4.1 4.2 4.3 |
60 80 40 |
20 15 20 |
0,477 0,580 0,484 |
0,024 0,065 0,026 |
0,493 0,984 0,531 |
1,513 1,69 1,337 |
0,97 1,69 0,52 |
7,17 24,94 4,14 |
|
(3) 1924 - 1938 |
15 |
4.3 5.1 5.2 5.3 |
40 80 100 40 |
20 20 15 15 |
0,484 0,657 0,667 0,748 |
0,026 0,122 0,131 0,234 |
0,531 2,44 1,96 3,51 |
1,337 1,69 1,9 1,337 |
0,52 1,69 3,05 0,52 |
4,14 61,95 89,37 27,37 |
|
(4) 1939 - 1953 |
15 |
5.3 6.1 6.2 6.3 |
40 80 100 40 |
20 15 15 15 |
0,748 0,672 0,699 0,814 |
0,234 0,137 0,166 0,357 |
468 2,05 2,49 5,35 |
1,337 1,69 1,9 1,337 |
0,52 1,69 3,05 0,52 |
36,50 51,96 113,9 41,73 |
|
(5) 1954 - 1968 |
15 |
7.1 7.2 7.3 7.4 8.2 8.4 |
80 80 100 40 80 40 |
15 10 10 20 15 15 |
0,252 0,658 0,66 0,883 0,515 0,805 |
0,001 0,123 0,125 0,536 0,036 0,338 |
0,015 1,23 1,25 10,7 0,54 5,07 |
1,69 1,69 1,9 1,337 1,69 1,337 |
1,69 1,69 3,05 0,52 1,69 0,52 |
0,38 31,18 57,18 5,56 0,38 15,75 |
|
(6) 1969 - 1983 |
15 |
8.1 8.2 8.3 8.4 |
80 80 100 60 |
20 25 25 20 |
0,252 0,515 0,67 0,805 |
0,001 0,036 0,135 0,338 |
0,02 0,9 3,37 6,76 |
1,59 1,59 1,64 1,44 |
1,25 1,25 1,46 0,76 |
5,62 16,87 63,15 76,95 |
|
|
|
|
|
|
|
|
|
|
|
S = 747 |