1. Poe C.C, Jr, “Stress intensità Factor for a cracker Sheet with Riveted and Uniformly Spaced Stringer”, Journal of Engineering Material and Tecnology, Trans. ASME, Vol. 100, pp. 10-15, 1978.

(1)

- 64 -

Ringraziamenti

(2)

Bibliografia ____________________________________________________________________________________

- 65 - BIBLIOGRAFIA

1. Poe C.C, Jr, “Stress intensità Factor for a cracker Sheet with Riveted and Uniformly Spaced Stringer”, Journal of Engineering Material and Tecnology, Trans. ASME, Vol. 100, pp. 10-15, 1978.

2. A. Salvetti, G. Cavallini, A. Frediani, “Fracture Mechanics”, AGARD-AG- 292, pp. 133-178, 1983.

3. Rooke, D.P., Cartwright, D.J., “Compendium of Stress Intensity Factor”, Her Majest’s Stationery Office, London, 1976.

4. Bloom J.M. and Sanders J.L.Jr, “The effect of riveted stringers on the stress in a cracked sheet”, J Appl. Mech., 33:561-570 (1966)

5. Nishimura, T., “Stress Intensity Factor of Multiple Cracker Sheet with Riveted Stiffners”, Journal of Engineering Material and Technology, Trans. ASME, Vol. 113, pp. 280 – 284, 1991.

6. Nishimura, T., “Stress Intensity Factor of Multiple Cracker Sheet with Cracked Stiffners”, Journal of Engineering Material and Technology, Trans. ASME, Vol. 113, pp. 119 – 124, 1991.

7. Westergaard, H.M., “Bearing Pressures and Cracks”, Journal of Applied Mechanics, Trans. ASME, 1939.

8. Irwin, G.R., “Analysis of Stresse and Strains Near the End of Crack Traversing a Plate”, Journal of Applied Mechanics, Trans. ASME, Vol.24, 1939

9. Rooke, D.P., Cartwright, D.J., “The Compounding Method applied to crack in stiffened sheets”, Engineering Fracture Mechanics, Vol. 8 pp. 567-573, Pergamon Press, 1976

10. N.K. Salgado, M.H. Aliabadi, “The application of the dual boundary element method to the analysis of cracked stiffened panels”, Engineering Fracture Mechanics, Vol. 54 pp. 91-105, Pergamon Press, 1996.

11. D. Broek, “Elementary engineering fracture mechanics”, Kluwer Academic

Publisher, Dordrecht, 1986.

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Irrigidimento integro: risultati FEM Appendix I

______________________________________________________________________

I

APPENDIX I – Irrigidimento integro: risultati FEM

Irrigidimento integro: valori del fattore di intensità degli sforzi agli apici A e B della fessura calcolati al variare dell’area equivalente di irrigidimenti di geometria reale, per differenti valori della dimensione della fessura stessa e per posizioni differenti del corrente rispetto alla mezzeria di fessura.

• Corrente a doppia striscia

D.S.

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=9

a K_A K_B K_A K_B K_A K_B

x = 0 1034.5 1034.5 1027.7 1027.7 1022.6 1022.6

x = l/3 1036.6 1033.3 1030.2 1026.2 1025.4 1021

x = 2 l / 3 1039.3 1033 1033.4 1025.8 1029 1020.5

x = l 1042.1 1033.4 1036.6 1026.1 1032.5 1020.8

x = 3/2 l 1046.9 1036.1 1042.3 1029.2 1038.8 1024.1

x = 2 l 1051.6 1040.6 1048 1034.7 1045.1 1030.2

x = 3 l 1058.6 1050.7 1056.4 1046.6 1054.2 1042.9

x = 4 l 1062.3 1058 1060.8 1055.3 1059.3 1052.8

x = 5 l 1065.8 1065.1 1065.3 1064.3 1064.8 1063.5

x =

∞

1067.2 1067.2 1067.2 1067.2 1067.2 1067.2

=12

x = 0 1176.5 1176.5 1164.6 1164.6 1155.7 1155.7

x = l/3 1181.7 1173 1170.8 1160.4 1162.6 1151.1

x = 2 l / 3 1188 1171.9 1178.3 1159 1171 1149.5

x = l 1194.4 1172.7 1185.8 1159.7 1179.2 1150.1

x = 3/2 l 1204.3 1179.2 1197.5 1167.3 1192.2 1158.2

x = 2 l 1212.9 1190.1 1207.9 1180.3 1203.9 1172.8

x = 3 l 1223.6 1211.5 1220.9 1206 1218.4 1200.8

x = 4 l 1228.1 1223.2 1226.5 1220.1 1224.9 1217.2

x = 5 l 1231 1230 1230.4 1228.9 1229.7 1227.9

x =

∞

1232.6 1232.6 1232.6 1232.6 1232.6 1232.6

=15

x = 0 1298 1298 1280.9 1280.9 1268 1268

x = l/3 1307.4 1291.3 1292.1 1272.8 1280.5 1259

x = 2 l / 3 1318.2 1288.3 1304.9 1269.1 1294.7 1254.9

x = l 1328.8 1289.2 1317.3 1269.7 1308.3 1255.1

x = 3/2 l 1344 1300.5 1335.4 1282.8 1328.5 1269.3

x = 2 l 1356.3 1320.2 1350.3 1306.4 1345.4 1295.6

x = 3 l 1369.7 1355.2 1366.8 1348.6 1364.1 1342.4

x = 4 l 1374.3 1369.5 1372.7 1366.4 1371.2 1363.3

x = 5 l 1376.8 1375.5 1376 1374.2 1375.3 1373

x =

∞

1378.6 1378.6 1378.6 1378.6 1378.6 1378.6

(4)

______________________________________________________________________

II

=18

x = 0 1411 1411 1384.3 1384.3 1367.5 1367.5

x = l/3 1424.3 1400.7 1401 1371.3 1386 1353

x = 2 l / 3 1438.9 1395.3 1419.2 1364.3 1406.3 1345

x = l 1453 1395.4 1436.6 1363.6 1425.4 1343.7

x = 3/2 l 1472.5 1411.1 1461 1382.5 1452.7 1364.1

x = 2 l 1487.6 1440.1 1480.1 1419 1474.4 1404.9

x = 3 l 1502.3 1487.4 1499.1 1479.2 1496.3 1472.3

x = 4 l 1506.8 1502.3 1505.2 1499 1503.7 1495.9

x = 5 l 1508.8 1507.3 1508 1505.9 1507.2 1504.5

x =

∞

1510.8 1510.8 1510.8 1510.8 1510.7 1510.7

=24

x = 0 1603.7 1603.7 1572.9 1572.9 1549.3 1549.3

x = l/3 1625.7 1584.8 1599.1 1550.2 1578.4 1523.9

x = 2 l / 3 1648.7 1571.8 1626.4 1534.6 1608.8 1506.3

x = l 1670.9 1569 1652.7 1530.6 1638.2 1501.4

x = 3/2 l 1700.2 1593.6 1687.9 1559 1677.7 1532.1

x = 2 l 1721.1 1647.7 1713.4 1623.2 1706.8 1603.6

x = 3 l 1738.2 1722.5 1735.1 1714.6 1732.2 1707

x = 4 l 1742.6 1738.5 1741.1 1735.4 1739.6 1732.4

x = 5 l 1744.1 1742.3 1743.2 1740.7 1742.3 1739.1

x =

∞

1746.2 1746.2 1746.2 1746.2 1746.2 1746.1 Tab 2.2: K per corrente a doppia striscia

(5)

______________________________________________________________________

III

• Corrente a Z

Z

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=9

x = 0 1038.1 1038.1 1031.7 1031.6 1027.1 1027

x = l/3 1039.7 1037.2 1033.7 1030.5 1029.3 1025.8

x = 2 l / 3 1041.8 1036.9 1036.3 1030.2 1032.2 1025.5

x = l 1044.1 1037.2 1038.9 1030.4 1035 1025.6

x = 3/2 l 1047.8 1039.3 1043.4 1032.9 1040 1028.3

x = 2 l 1051.5 1042.9 1048 1037.3 1045.1 1033.2

x = 3 l 1056.9 1050.4 1054 1045.3 1051.2 1040.7

x = 4 l 1059.9 1056.3 1057.9 1052.8 1055.7 1049.4

x = 5 l 1062.8 1062.1 1062.2 1061 1061.2 1059.5

x =

∞

1065.2 1065.2 1065.3 1065.2 1065.2 1065.1

=12

x = 0 1184.1 1184.1 1172.5 1172.4 1164.3 1164.2

x = l/3 1188.2 1181.4 1177.5 1169.1 1169.9 1160.6

x = 2 l / 3 1193.2 1180.6 1183.6 1168 1176.7 1159.4

x = l 1198.2 1181.3 1189.7 1168.7 1183.3 1159.9

x = 3/2 l 1206 1186.4 1199.2 1174.8 1193.8 1166.5

x = 2 l 1212.9 1195 1207.7 1185.4 1203.4 1178.2

x = 3 l 1221.2 1211.4 1217.8 1204.2 1214.2 1197.6

x = 4 l 1224.9 1220.7 1222.8 1216.6 1220.3 1212.4

x = 5 l 1227.5 1226.5 1226.7 1224.9 1225.3 1223

x =

∞

1230.4 1230.3 1230.5 1230.4 1230.4 1230.3

=15

x = 0 1309.6 1309.7 1292.3 1292.3 1280.1 1279.9

x = l/3 1317 1304.4 1301.4 1285.7 1290.3 1272.7

x = 2 l / 3 1325.6 1302.1 1311.9 1282.9 1301.9 1269.6

x = l 1334 1302.9 1322.1 1283.5 1313 1269.9

x = 3/2 l 1346.2 1312 1337 1294.2 1329.7 1281.6

x = 2 l 1356 1327.6 1349.5 1313.5 1343.8 1303

x = 3 l 1366.7 1354.6 1362.8 1345.8 1358.7 1337.6

x = 4 l 1370.6 1366.3 1368.4 1361.9 1365.7 1357.2

x = 5 l 1372.9 1371.5 1371.8 1369.6 1370.3 1367.3

x =

∞

1378.1 1378 1378.1 1377.9 1378.1 1377.9

(6)

______________________________________________________________________

IV

=18

x = 0 1432.3 1432.3 1398.7 1398.6 1386.1 1385.8

x = l/3 1433 1413.4 1412.4 1388 1397.4 1370.2

x = 2 l / 3 1445.2 1409 1427.5 1382.5 1414.2 1363.9

x = l 1457 1409.2 1441.8 1382.1 1430 1363.1

x = 3/2 l 1473.5 1422.4 1462.2 1397.8 1452.9 1380.1

x = 2 l 1486.2 1446.7 1478.4 1427.9 1471.5 1413.6

x = 3 l 1498.5 1485.3 1494.4 1475.3 1489.9 1465.7

x = 4 l 1502.6 1498.3 1500.4 1493.8 1497.6 1488.8

x = 5 l 1504.6 1503 1503.3 1500.7 1501.6 1498

x =

∞

1510.2 1510 1510.2 1510 1510.1 1509.9

=24

x = 0 1623.1 1623.2 1588.3 1588.2 1565.5 1565.3

x = l/3 1640.9 1608.1 1610.4 1569.3 1587.9 1541.6

x = 2 l / 3 1659.6 1598 1633.8 1556.6 1614.1 1527.2

x = l 1677.7 1596.1 1656.3 1553.6 1639.4 1523.5

x = 3/2 l 1701.8 1616.4 1686.7 1577.9 1674 1550.1

x = 2 l 1719.1 1659.9 1709.3 1632 1700.2 1610.6

x = 3 l 1733.5 1719.6 1729.1 1708.1 1724 1696.5

x = 4 l 1737.7 1733.6 1735.4 1728.9 1732.5 1723.6

x = 5 l 1739.2 1737.2 1737.8 1734.5 1735.8 1731.2

x =

∞

1743.5 1743.2 1743.7 1743.4 1743.3 1743 Tab 2.3: K per corrente a Z

(7)

______________________________________________________________________

V

• Corrente a C

C

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=9

x = 0 1038.7 1038.6 1032.3 1032.2 1027.6 1027.4

x = l/3 1040.3 1037.7 1034.3 1031.1 1029.8 1026.2

x = 2 l / 3 1042.4 1037.5 1036.9 1030.9 1032.6 1025.9

x = l 1044.6 1037.8 1039.4 1031.1 1035.4 1026.1

x = 3/2 l 1048.2 1039.9 1043.9 1033.6 1040.3 1028.8

x = 2 l 1051.8 1043.5 1048.4 1038 1045.4 1033.6

x = 3 l 1056.7 1050.1 1054 1045.3 1051.2 1040.7

x = 4 l 1059.7 1056 1057.8 1052.7 1055.7 1049.4

x = 5 l 1062.8 1062 1062.1 1060.9 1061.1 1059.5

x =

∞

1065.2 1065.2 1065.2 1065.2 1065.1 1065

=12

x = 0 1185 1184.9 1173.6 1173.4 1165.2 1164.9

x = l/3 1189.1 1182.3 1178.6 1170.2 1170.7 1161.4

x = 2 l / 3 1194 1181.5 1184.6 1169.2 1177.4 1160.2

x = l 1198.9 1182.2 1190.6 1169.9 1184 1160.8

x = 3/2 l 1206.5 1187.4 1199.9 1176 1194.3 1167.4

x = 2 l 1213.2 1195.8 1208.2 1186.4 1203.8 1179

x = 3 l 1221 1210.9 1217.7 1204.1 1214.2 1197.7

x = 4 l 1224.8 1220.5 1222.7 1216.5 1220.2 1212.4

x = 5 l 1227.5 1226.4 1226.6 1224.9 1225.2 1222.9

x =

∞

1230.5 1230.5 1230.4 1230.3 1230.2 1230.1

=15

x = 0 1310.9 1310.7 1294 1293.6 1281.5 1281

x = l/3 1318.2 1305.5 1303 1287.3 1291.5 1274

x = 2 l / 3 1326.6 1303.4 1313.3 1284.6 1303 1270.9

x = l 1334.8 1304.3 1323.2 1285.3 1314 1271.3

x = 3/2 l 1346.6 1313.3 1337.8 1295.9 1330.4 1282.9

x = 2 l 1356.3 1328.6 1350 1314.9 1344.3 1304.1

x = 3 l 1366.4 1354 1362.7 1345.7 1358.7 1337.8

x = 4 l 1370.5 1366.1 1368.3 1361.7 1365.7 1357.2

x = 5 l 1372.9 1371.5 1371.7 1369.5 1370.1 1367.2

x =

∞

1378.1 1377.9 1378 1377.9 1375.9 1375.8

(8)

______________________________________________________________________

VI

=18

x = 0 1423.5 1423.2 1400.8 1400.4 1383.9 1383.4

x = l/3 1434.3 1414.8 1414.2 1390.1 1399 1371.9

x = 2 l / 3 1446.3 1410.6 1429 1384.7 1415.6 1365.8

x = l 1457.9 1410.9 1443.1 1384.5 1431.2 1365.1

x = 3/2 l 1473.9 1424 1463 1400 1453.8 1382.1

x = 2 l 1486.3 1447.8 1478.9 1429.6 1472 1415.2

x = 3 l 1498.3 1484.6 1494.3 1475.1 1489.8 1465.9

x = 4 l 1502.5 1498.1 1500.2 1493.5 1497.4 1488.7

x = 5 l 1504.5 1502.9 1503.2 1500.6 1501.4 1497.9

x =

∞

1510.2 1510 1510.1 1509.9 1507.8 1507.6

=24

x = 0 1625 1624.8 1591.1 1590.8 1565.7 1565.3

x = l/3 1642.3 1609.9 1612.8 1572.2 1590.2 1544.3

x = 2 l / 3 1660.6 1600 1635.6 1559.7 1616 1530.1

x = l 1678.3 1598.1 1657.6 1556.8 1640.9 1526.5

x = 3/2 l 1702 1618.2 1687.5 1580.9 1675 1553

x = 2 l 1719 1661 1709.6 1634.1 1700.8 1612.8

x = 3 l 1733.3 1718.8 1728.8 1707.6 1723.8 1696.6

x = 4 l 1737.6 1733.4 1735.3 1728.7 1732.3 1723.5

x = 5 l 1739.2 1737.2 1737.7 1734.4 1735.6 1731

x =

∞

1743.6 1743.3 1743.6 1743.4 1743.1 1742.8 Tab 2.4: K per corrente a C

(9)

______________________________________________________________________

VII

• Corrente ad Ω

Ω A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=9

x = 0 1045.3 1045.3 1036.4 1036.4 1033.5 1033.5

x = l/3 1046.8 1044.4 1038.1 1035.4 1035.2 1032.4

x = 2 l / 3 1048.7 1044.2 1040.3 1035.2 1037.5 1032.2

x = l 1050.6 1044.4 1042.4 1035.2 1039.6 1032.1

x = 3/2 l 1054 1046.2 1046.2 1037.2 1043.6 1034.2

x = 2 l 1057.4 1049.5 1050.2 1041 1047.8 1038.1

x = 3 l 1061.2 1054.4 1054.3 1045.7 1051.9 1042.3

x = 4 l 1064.1 1060.2 1057.7 1052.4 1055.3 1049.3

x = 5 l 1067.4 1066.5 1062.2 1060.9 1060.7 1058.9

x =

∞

1068.3 1068.3 1065.5 1065.5 1065.1 1065

=12

x = 0 1193.3 1193.3 1179.9 1179.9 1174.8 1174.8

x = l/3 1197 1190.9 1184.2 1177.1 1179.3 1171.9

x = 2 l / 3 1201.6 1190.1 1189.5 1176.2 1184.8 1170.9

x = l 1206.1 1190.5 1194.5 1176.4 1189.9 1171

x = 3/2 l 1213.2 1195.1 1202.6 1181.5 1198.5 1176.1

x = 2 l 1219.5 1203.1 1210.1 1190.7 1206.5 1185.7

x = 3 l 1226.1 1215.7 1217.9 1204.4 1214.6 1199.4

x = 4 l 1229.8 1225.1 1222.4 1215.8 1219.6 1211.8

x = 5 l 1232.7 1231.5 1226.7 1224.7 1224.7 1222.1

x =

∞

1233.9 1233.9 1230.8 1230.7 1230.2 1230.1

=15

x = 0 1320.6 1320.6 1302.2 1302.2 1294.4 1294.4

x = l/3 1327.5 1315.8 1310.2 1296.6 1302.7 1288.5

x = 2 l / 3 1335.3 1313.7 1319.2 1294.2 1312.3 1285.9

x = l 1342.9 1314.2 1327.8 1294.2 1321.1 1285.7

x = 3/2 l 1354 1322.3 1340.7 1303.3 1334.9 1295

x = 2 l 1363.1 1336.8 1351.8 1320.1 1346.9 1312.7

x = 3 l 1372.1 1359.1 1362.7 1345.5 1358.7 1339

x = 4 l 1376.1 1371.3 1368 1360.9 1364.8 1356.1

x = 5 l 1378.6 1377.1 1371.8 1369.3 1369.5 1366.2

x =

∞

1378.9 1378.9 1378.2 1378 1375.9 1375.8

(10)

______________________________________________________________________

VIII

=18

x = 0 1429.3 1429.3 1410.4 1410.4 1399.5 1399.5

x = l/3 1439.5 1421.4 1422.4 1401.3 1412.2 1389.9

x = 2 l / 3 1450.9 1417.4 1435.7 1396.5 1426.3 1384.8

x = l 1461.5 1417.2 1447.9 1395.6 1439.1 1383.5

x = 3/2 l 1476.6 1429.2 1465.9 1409 1458.3 1397.3

x = 2 l 1488.5 1451.8 1480.5 1435.5 1474.4 1425.3

x = 3 l 1499.2 1484.8 1494 1474.5 1489.4 1466.5

x = 4 l 1503.5 1498.6 1500 1492.6 1496.5 1487.3

x = 5 l 1505.7 1503.9 1503.3 1500.3 1500.7 1496.7

x =

∞

1511.1 1511.1 1510.3 1510.1 1507.9 1507.6

=24

x = 0 1636 1636 1599.8 1599.8 1581.4 1581.4

x = l/3 1652.8 1621.8 1619.8 1582.9 1602.9 1563.3

x = 2 l / 3 1670.4 1612.4 1641 1671.6 1625.8 1551.3

x = l 1687.3 1610.3 1661.1 1568.5 1647.5 1547.7

x = 3/2 l 1710 1629.3 1689.1 1590.3 1678 1570.4

x = 2 l 1726.6 1670.3 1710.2 1639.1 1701.7 1622.6

x = 3 l 1740.1 1724.5 1728.3 1705.7 1722.7 1695.1

x = 4 l 1744.6 1740 1735.1 1727.8 1731.3 1721.7

x = 5 l 1746.2 1744.1 1737.8 1734.1 1734.8 1729.7

x =

∞

1747.7 1747.8 1744.3 1744.2 1743.2 1742.8 Tab 2.5: K per corrente a

Ω

(11)

Irrigidimento integro: valutazione dell’errore Appendix II

______________________________________________________________________

IX

APPENDIX II – Irrigidimento integro: valutazione dell’errore

• Corrente a Z

Z

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=9

x = 0 0.3468 0.3468 0.3877 0.3781 0.4381 0.4284

x = l/3 0.2982 0.3760 0.3386 0.4173 0.3789 0.4679

x = 2 l / 3 0.2400 0.3761 0.2798 0.4271 0.3100 0.4876

x = l 0.1916 0.3664 0.2214 0.4173 0.2415 0.4680

x = 3/2 l 0.0859 0.3079 0.1054 0.3582 0.1154 0.4084

x = 2 l 0.0095 0.2205 0 0.2507 0 0.2904

x = 3 l 0.1608 0.0286 0.2277 0.1244 0.2854 0.2114

x = 4 l 0.2264 0.1609 0.2741 0.2375 0.3410 0.3240

x = 5 l 0.2823 0.2825 0.2918 0.3110 0.3392 0.3775

x =

∞

0.1878 0.1878 0.1784 0.1878 0.1878 0.1972

=12

x = 0 0.6418 0.6418 0.6738 0.6653 0.7386 0.7301

x = l/3 0.5470 0.7110 0.5690 0.7442 0.6240 0.8185

x = 2 l / 3 0.4358 0.7369 0.4478 0.7705 0.4844 0.8539

x = l 0.3171 0.7280 0.3278 0.7701 0.3465 0.8449

x = 3/2 l 0.1410 0.6069 0.1418 0.6384 0.1340 0.7115

x = 2 l 0 0.4100 0.0166 0.4302 0.0415 0.4583

x = 3 l 0.1965 0.0083 0.2546 0.1495 0.3459 0.2672

x = 4 l 0.2612 0.2048 0.3026 0.2877 0.3770 0.3959

x = 5 l 0.2851 0.2854 0.3016 0.3266 0.3591 0.4007

x =

∞

0.1788 0.1869 0.1707 0.1788 0.1788 0.1869

=15

x = 0 0.8858 0.8933 0.8821 0.8821 0.9452 0.9298

x = l/3 0.7289 1.0043 0.7146 1.0033 0.7595 1.0765

x = 2 l / 3 0.5582 1.0598 0.5336 1.0757 0.5530 1.1578

x = l 0.3898 1.0515 0.3631 1.0752 0.3580 1.1654

x = 3/2 l 0.1634 0.8765 0.1197 0.8809 0.0902 0.9597

x = 2 l 0.0221 0.5574 0.0593 0.5405 0.1191 0.5679

x = 3 l 0.2195 0.0443 0.2935 0.2081 0.3974 0.3589

x = 4 l 0.2700 0.2342 0.3142 0.3304 0.4027 0.4495

x = 5 l 0.2841 0.2917 0.3062 0.3359 0.3649 0.4169

x =

∞

0.0363 0.0435 0.0363 0.0508 0.0363 0.0508

(12)

______________________________________________________________________

X

=18

x = 0 1.4871 1.4871 1.0295 1.0225 1.3419 1.3205

x = l/3 0.6071 0.8985 0.8071 1.2032 0.8158 1.2553

x = 2 l / 3 0.4359 0.9723 0.5814 1.3165 0.5586 1.3857

x = l 0.2745 0.9793 0.3607 1.3385 0.3217 1.4232

x = 3/2 l 0.0679 0.7944 0.0821 1.0946 0.0138 1.1593

x = 2 l 0.0942 0.4562 0.1150 0.6233 0.1971 0.6154

x = 3 l 0.2536 0.1414 0.3145 0.2644 0.4296 0.4503

x = 4 l 0.2795 0.2670 0.3199 0.3481 0.4073 0.4769

x = 5 l 0.2791 0.2861 0.3126 0.3465 0.3729 0.4339

x =

∞

0.0397 0.0530 0.0397 0.0530 0.0397 0.0530

=24

x = 0 1.1952 1.2013 0.9696 0.9634 1.0348 1.0222

x = l/3 0.9263 1.4489 0.7017 1.2171 0.5983 1.1482

x = 2 l / 3 0.6568 1.6395 0.4529 1.4133 0.3284 1.3685

x = l 0.4053 1.6979 0.2174 1.4804 0.0732 1.4506

x = 3/2 l 0.0940 1.4105 0.0711 1.1978 0.2210 1.1612

x = 2 l 0.1163 0.7350 0.2399 0.5392 0.3882 0.4346

x = 3 l 0.2711 0.1686 0.3470 0.3805 0.4756 0.6189

x = 4 l 0.2820 0.2826 0.3285 0.3760 0.4098 0.5106

x = 5 l 0.2817 0.2936 0.3107 0.3575 0.3745 0.4563

x =

∞

0.1549 0.1721 0.1434 0.1606 0.1664 0.1779 Tab 2.6:

ε

per corrente a Z

(13)

______________________________________________________________________

XI

• Corrente a C

C

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=9

x = 0 0.4044 0.3948 0.4456 0.4360 0.4866 0.4672

x = l/3 0.3557 0.4240 0.3964 0.4752 0.4273 0.5067

x = 2 l / 3 0.2974 0.4337 0.3375 0.4947 0.3486 0.5264

x = l 0.2393 0.4240 0.2694 0.4849 0.2801 0.5165

x = 3/2 l 0.1240 0.3654 0.1533 0.4257 0.1442 0.4568

x = 2 l 0.0190 0.2779 0.0382 0.3179 0.0287 0.3289

x = 3 l 0.1798 0.0571 0.2277 0.1244 0.2854 0.2114

x = 4 l 0.2454 0.1894 0.2836 0.2470 0.3410 0.3240

x = 5 l 0.2823 0.2919 0.3013 0.3205 0.3487 0.3775

x =

∞

0.1878 0.1878 0.1878 0.1878 0.1972 0.2066

=12

x = 0 0.7173 0.7089 0.7669 0.7500 0.8153 0.7898

x = l/3 0.6223 0.7866 0.6618 0.8375 0.6919 0.8869

x = 2 l / 3 0.5025 0.8125 0.5318 0.8724 0.5436 0.9223

x = l 0.3753 0.8036 0.4032 0.8719 0.4054 0.9218

x = 3/2 l 0.1823 0.6906 0.2000 0.7398 0.1758 0.7881

x = 2 l 0.0247 0.4767 0.0248 0.5142 0.0083 0.5259

x = 3 l 0.2129 0.0495 0.2628 0.1578 0.3459 0.2588

x = 4 l 0.2694 0.2212 0.3108 0.2959 0.3852 0.3959

x = 5 l 0.2851 0.2935 0.3098 0.3266 0.3673 0.4089

x =

∞

0.1707 0.1707 0.1788 0.1869 0.1951 0.2032

=15

x = 0 0.9841 0.9689 1.0124 0.9818 1.0535 1.0148

x = l/3 0.8193 1.0877 0.8365 1.1264 0.8517 1.1774

x = 2 l / 3 0.6332 1.1585 0.6396 1.2066 0.6370 1.2590

x = l 0.4495 1.1577 0.4459 1.2137 0.4338 1.2743

x = 3/2 l 0.1931 0.9746 0.1794 1.0109 0.1428 1.0601

x = 2 l 0 0.6322 0.0222 0.6464 0.0818 0.6518

x = 3 l 0.2415 0.0886 0.3009 0.2155 0.3974 0.3438

x = 4 l 0.2773 0.2489 0.3216 0.3452 0.4027 0.4495

x = 5 l 0.2841 0.2917 0.3135 0.3432 0.3795 0.4242

x =

∞

0.0363 0.0508 0.0435 0.0508 0.1962 0.2035

(14)

______________________________________________________________________

XII

=18

x = 0 0.8781 0.8572 1.1779 1.1497 1.1851 1.1493

x = l/3 0.6972 0.9966 0.9334 1.3524 0.9292 1.3777

x = 2 l / 3 0.5117 1.0846 0.6858 1.4732 0.6570 1.5229

x = l 0.3361 1.0986 0.4504 1.5096 0.4053 1.5677

x = 3/2 l 0.0950 0.9059 0.1367 1.2500 0.0757 1.3024

x = 2 l 0.0875 0.5318 0.0811 0.7415 0.1630 0.7278

x = 3 l 0.2670 0.1886 0.3212 0.2779 0.4363 0.4366

x = 4 l 0.2862 0.2804 0.3333 0.3683 0.4207 0.4836

x = 5 l 0.2858 0.2928 0.3193 0.3532 0.3863 0.4406

x =

∞

0.0397 0.0530 0.0464 0.0596 0.1923 0.2056

=24

x = 0 1.3108 1.2986 1.1439 1.1252 1.0475 1.0222

x = l/3 1.0108 1.5591 0.8495 1.3993 0.7420 1.3210

x = 2 l / 3 0.7166 1.7625 0.5625 1.6093 0.4455 1.5555

x = l 0.4409 1.8209 0.2956 1.6829 0.1645 1.6443

x = 3/2 l 0.1058 1.5202 0.0237 1.3853 0.1612 1.3458

x = 2 l 0.1222 0.8007 0.2223 0.6670 0.3528 0.5704

x = 3 l 0.2827 0.2153 0.3644 0.4099 0.4873 0.6130

x = 4 l 0.2878 0.2942 0.3342 0.3876 0.4214 0.5164

x = 5 l 0.2817 0.2936 0.3165 0.3632 0.3860 0.4679

x =

∞

0.1491 0.1664 0.1491 0.1606 0.1778 0.1894 Tab 2.7:

ε

per corrente a C

(15)

______________________________________________________________________

XIII

• Corrente ad Ω

Ω A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=9

x = 0 1.0332 1.0332 0.8394 0.8394 1.0547 1.0547

x = l/3 0.9744 1.0628 0.7610 0.8885 0.9467 1.1042

x = 2 l / 3 0.8963 1.0726 0.6633 0.9080 0.8193 1.1335

x = l 0.8091 1.0532 0.5564 0.8791 0.6830 1.0949

x = 3/2 l 0.6736 0.9654 0.3728 0.7713 0.4599 0.9766

x = 2 l 0.5485 0.8480 0.2095 0.6052 0.2577 0.7610

x = 3 l 0.2450 0.3509 0.1992 0.0861 0.2187 0.0576

x = 4 l 0.1692 0.2075 0.2931 0.2756 0.3790 0.3336

x = 5 l 0.1499 0.1313 0.2918 0.3205 0.3865 0.4344

x =

∞

0.1030 0.1030 0.1595 0.1595 0.1972 0.2066

=12

x = 0 1.4079 1.4079 1.2967 1.2967 1.6258 1.6258

x = l/3 1.2782 1.5031 1.1316 1.4187 1.4161 1.7749

x = 2 l / 3 1.1318 1.5293 0.9416 1.4623 1.1648 1.8277

x = l 0.9701 1.4952 0.7283 1.4196 0.8992 1.7848

x = 3/2 l 0.7336 1.3304 0.4241 1.2019 0.5257 1.5220

x = 2 l 0.5412 1.0805 0.1818 0.8734 0.2155 1.0880

x = 3 l 0.2039 0.3455 0.2463 0.1328 0.3129 0.1167

x = 4 l 0.1382 0.1551 0.3354 0.3537 0.4346 0.4456

x = 5 l 0.1379 0.1218 0.3016 0.3429 0.4083 0.4746

x =

∞

0.1054 0.1054 0.1462 0.1544 0.1951 0.2032

=15

x = 0 1.7113 1.7113 1.6357 1.6357 2.0396 2.0396

x = l/3 1.5141 1.8620 1.3815 1.8356 1.7042 2.2895

x = 2 l / 3 1.2806 1.9335 1.0840 1.9394 1.3412 2.4108

x = l 1.0500 1.9023 0.7908 1.8931 0.9689 2.3800

x = 3/2 l 0.7386 1.6486 0.3953 1.5729 0.4794 1.9846

x = 2 l 0.4989 1.2418 0.1110 1.0378 0.1114 1.3027

x = 3 l 0.1749 0.2870 0.3009 0.2304 0.3974 0.2539

x = 4 l 0.1308 0.1313 0.3436 0.4041 0.4689 0.5309

x = 5 l 0.1306 0.1162 0.3062 0.3578 0.4235 0.4977

x =

∞

0.0218 0.0218 0.0290 0.0435 0.1962 0.2035

(16)

______________________________________________________________________

XIV

=18

x = 0 1.2803 1.2803 1.8505 1.8505 2.2865 2.2865

x = l/3 1.0559 1.4563 1.5045 2.1409 1.8553 2.6549

x = 2 l / 3 0.8271 1.5592 1.1493 2.3058 1.4022 2.8741

x = l 0.5816 1.5382 0.7804 2.2929 0.9520 2.8768

x = 3/2 l 0.2777 1.2664 0.3343 1.8808 0.3840 2.3760

x = 2 l 0.0605 0.8059 0.0270 1.1494 0 1.4313

x = 3 l 0.2068 0.1751 0.3414 0.3188 0.4633 0.3955

x = 4 l 0.2195 0.2469 0.3467 0.4288 0.4811 0.5782

x = 5 l 0.2059 0.2261 0.3126 0.3733 0.4331 0.5211

x =

∞

0.0199 0.0199 0.0331 0.0464 0.1857 0.2056

=24

x = 0 1.9743 1.9743 1.6815 1.6815 2.0298 2.0298

x = l/3 1.6396 2.2814 1.2779 2.0658 1.5285 2.5203

x = 2 l / 3 1.2991 2.5180 0.8897 2.1957 1.0456 2.9008

x = l 0.9720 2.5647 0.5057 2.4163 0.5645 2.9915

x = 3/2 l 0.5731 2.1911 0.0710 1.9682 0.0179 2.4389

x = 2 l 0.3185 1.3531 0.1871 0.9700 0.2997 1.1710

x = 3 l 0.1092 0.1160 0.3935 0.5218 0.5515 0.7020

x = 4 l 0.1146 0.0862 0.3458 0.4399 0.4794 0.6215

x = 5 l 0.1203 0.1032 0.3107 0.3806 0.4323 0.5434

x =

∞

0.0858 0.0915 0.1089 0.1147 0.1721 0.1894 Tab 2.8:

ε

per corrente a

Ω

(17)

Irrigidimento integro: confronto tra risultato FEM e risultato analitico Appendix III

______________________________________________________________________

XV

APPENDIX III – Irrigidimento integro: confronto tra risultato FEM e risultato analitico

D.S.

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=9

a K_B calc K_B FEM K_B calc K_B FEM K_B calc K_B FEM 1038.2 1034.5 1023.4 1027.7 1017.6 1022.6 x = 0

0,3532 0,4199 0,4924

1037.6 1033.3 1022.7 1026.2 1016.8 1021 x = l/3

0,4154 0,3378 0,4091

1037.8 1033 1023 1025.8 1017.1 1020.5 x = 2 l / 3

0,4654 0,2746 0,3325

1038.8 1033.4 1024.1 1026.1 1018.4 1020.8 x = l

0,5224 0,1916 0,2334

1041.8 1036.1 1027.6 1029.2 1022.3 1024.1 x = 3/2 l

0,5864 0,1584 0,1728

1066.8 1050.7 1033.3 1034.7 1028.9 1030.2 x = 2 l

1,5347 0,1341 0,1291

1066.8 1050.7 1056.7 1046.6 1055.5 1042.9 x = 3 l

1,5347 0,9631 1,2066

1074.2 1058 1065.3 1055.3 1065.3 1052.8 x = 4 l

1,5358 0,9476 1,1873

1074.2 1065.1 1065.3 1064.3 1065.3 1063.5 x = 5 l

0,8589 0,094 0,1693

1074.2 1067.2 1065.3 1067.2 1065.3 1067.2 x =

∞

0,6605 0,178 0,1780

D.S.

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=12

a K_B calc K_B FEM K_B calc K_B FEM K_B calc K_B FEM 1189 1176.5 1169.9 1164.6 1161.2 1155.7 x = 0

1,0609 0,4558 0,4750

1185.2 1173 1165.5 1160.4 1156.2 1151.1 x = l/3

1,0396 0,4403 0,4413

1183.9 1171.9 1164 1159 1154.4 1149.5 x = 2 l / 3

1,0204 0,4278 0,4274

1185 1172.7 1165.3 1159.7 1155.9 1150.1 x = l

1,0466 0,4789 0,5033

1191.3 1179.2 1172.6 1167.3 1164.2 1158.2 x = 3/2 l

1,0237 0,4512 0,5196

1203.4 1190.1 1186.7 1180.3 1180.3 1172.8 x = 2 l

1,1198 0,5407 0,6388

1243.1 1211.5 1232.8 1206 1232.8 1200.8 x = 3 l

2,6089 2,219 2,6623

1243.1 1223.2 1232.7 1220.1 1232.7 1217.2 x = 4 l

1,6234 1,0329 1,2737

1243.1 1230 1232.7 1228.9 1232.7 1227.9 x = 5 l

1,0614 0,3092 0,3909

1243.1 1232.6 1232.7 1232.6 1232.7 1232.6 x =

∞

0,8482 0,0081 0,0081

(18)

______________________________________________________________________

XVI

D.S.

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=15

a K_B calc K_B FEM K_B calc K_B FEM K_B calc K_B FEM 1315.9 1298 1292.4 1280.9 1280.4 1268 x = 0

1,3825 0,8974 0,9798

1308.2 1291.3 1283.4 1272.8 1270.2 1259 x = l/3

1,3086 0,8327 0,888

1304.9 1288.3 1279.6 1269.1 1265.9 1254.9 x = 2 l / 3

1,2910 0,8278 0,8733

1306.1 1289.2 1281 1269.7 1267.5 1255.1 x = l

1,3144 0,8911 0,9851

1316.4 1300.5 1292.9 1282.8 1281 1269.3 x = 3/2 l

1,2229 0,7894 0,9242

1337.2 1320.2 1317.1 1306.4 1308.5 1295.6 x = 2 l

1,2870 0,8165 0,9974

1399.9 1355.2 1389.9 1348.6 1391.5 1342.4 x = 3 l

3,2987 3,0618 3,6546

1390.2 1369.5 1378.6 1366.4 1378.6 1363.3 x = 4 l

1,5104 0,8932 1,1227

1390.2 1375.5 1378.6 1374.2 1378.6 1373 x = 5 l

1,0673 0,3202 0,4079

1390.2 1378.6 1378.6 1378.6 1378.6 1378.6 x =

∞

0,84 0 0

D.S.

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=18

a K_B calc K_B FEM K_B calc K_B FEM K_B calc K_B FEM 1427.1 1411 1398.8 1384.3 1383.3 1367.5 x = 0

1,1378 1,0484 1,1527

1414.7 1400.7 1384.5 1371.3 1366.9 1353 x = l/3

1,0008 0,9617 1,0309

1409.2 1395.3 1378.1 1364.3 1359.6 1345 x = 2 l / 3

0,9958 1,0097 1,0889

1410.5 1395.4 1379.6 1363.3 1361.4 1343.7 x = l

1,0812 1,1938 1,3138

1425.3 1411.4 1396.7 1382.5 1380.9 1364.1 x = 3/2 l

0,9815 1,0287 1,2302

1455.9 1440.1 1432.3 1419 1421.4 1404.9 x = 2 l

1,0984 0,9395 1,1769

1544.7 1487.4 1535.5 1479.2 1538.9 1472.3 x = 3 l

3,8549 3,8043 4,5234

1523.5 1502.3 1510.8 1499 1510.8 1495.9 x = 4 l

1,4110 0,7878 0,9967

1523.5 1507.3 1510.8 1505.9 1510.8 1504.5 x = 5 l

1,0742 0,3254 0,4187

1523.5 1510.8 1510.8 1510.8 1510.8 1510.7 x =

∞

0,84 0 0,0066

(19)

______________________________________________________________________

XVII

D.S.

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=24

0,6817 0,2204 0,2241

1591.5 1584.8 1549.5 1550.2 1522.2 1523.9 x = l/3

0,4215 0,0463 0,1146

1580.6 1571.8 1536.9 1534.6 1507.8 1506.3 x = 2 l / 3

0,5603 0,1469 0,0978

1582 1569 1538.5 1530.6 1509.6 1501.4 x = l

0,8293 0,5153 0,5482

1607.2 1593.6 1567.8 1559 1543 1532.1 x = 3/2 l

0,8549 0,5622 0,7097

1660.9 1647.7 1630.1 1623.2 1614 1603.6 x = 2 l

0,8008 0,4245 0,6456

1810 1722.5 1803.3 1714.6 1811.2 1707 x = 3 l

5,0805 5,1704 6,1019

1761 1738.5 1746.3 1735.4 1746.3 1732.4 x = 4 l

1,2932 0,629 0,8034

1761 1742.3 1746.3 1740.7 1746.3 1739.1 x = 5 l

1,0715 0,3217 0,4140

1761 1746.2 1746.3 1746.2 1746.3 1746.1 x =

∞

0,8458 0,0057 0,0115

Tabella 2.12: valutazione dell’errore commesso sostituendo il valore di K calcolato

con la formula analitica al valore FEM

(20)

Irrigidimento rotto: confronto tra risultato FEM e risultato analitico per la DS Appendix IV

______________________________________________________________________

XVIII

APPENDIX IV – Irrigidimento rotto: confronto tra

risultato FEM e risultato analitico per la doppia striscia

Tabella 3.1: Valori K metodo FEM per corrente a doppia striscia rotto

= 6

A

eq

A

_eq

= 9 A

_eq

= 12

=9

x = 0 1188 1188 1238.8 1238.8 1288.3 1288.3

x = l/3 1182.5 1192.7 1231.1 1245.2 1278.7 1296.4

x = 2 l / 3 1176.2 1196.2 1222.6 1250 1267.9 1302.5

x = l 1169.7 1198.5 1213.6 1253.3 1256.7 1306.7

x = 3/2 l 1159.2 1198.8 1199.3 1253.6 1238.6 1307.2

x = 2 l 1148.5 1194.8 1184.5 1248.2 1219.9 1300.3

x = 3 l 1128.4 1176.3 1156.6 1222.7 1184.7 1268

x = 4 l 1112.6 1151.4 1134.9 1188.9 1157.4 1226.1

x = 5 l 1086.9 1104.3 1098.2 1123.4 1109.7 1142.7

x =

∞

1067.7 1067.8 1068.1 1068.2 1068.6 1068.7

=12

x = 0 1354.7 1354.7 1406.9 1406.9 1458.1 1458.1

x = l/3 1346.8 1361.8 1396.1 1416.8 1444.4 1470.6

x = 2 l / 3 1338.5 1367.9 1384.6 1425.1 1429.8 1481.1

x = l 1330 1372.5 1373 1431.4 1415.2 1489.2

x = 3/2 l 1317.1 1374.6 1355.1 1434.4 1392.5 1492.9

x = 2 l 1304.3 1369.6 1337.3 1427.4 1370 1484

x = 3 l 1281.6 1341.1 1305.7 1388 1329.8 1434.1

x = 4 l 1265.7 1306.4 1283.4 1340.6 1301.4 1374.7

x = 5 l 1247.8 1264.3 1257.2 1281.4 1266.8 1298.9

x =

∞

1233.3 1233.3 1233.7 1233.8 1234.2 1234.3

=15

x = 0 1497.2 1497.2 1549 1549 1599.9 1599.9

x = l/3 1488 1506.2 1536.2 1561.5 1583.6 1615.8

x = 2 l / 3 1478.7 1514.7 1523.2 1573.2 1567.1 1630.5

x = l 1469.4 1522 1510.4 1583.2 1550.8 1643.3

x = 3/2 l 1455.3 1527.3 1490.7 1590.5 1525.9 1652.3

x = 2 l 1441.3 1522.1 1471.3 1583 1501.1 1642.7

x = 3 l 1417.8 1482.8 1438.2 1528.7 1458.9 1574

x = 4 l 1403 1440.9 1417.1 1471.2 1431.6 1501.6

x = 5 l 1390.5 1405.4 1398.4 1420.7 1406.5 1436.3

x =

∞

1378.6 1378.6 1378.5 1378.6 1380.2 1380.5

(21)

______________________________________________________________________

XIX

=18

x = 0 1624.3 1624.3 1674.8 1674.8 1724.6 1724.6

x = l/3 1614.5 1634.4 1661.1 1689 1707.1 1742.7

x = 2 l / 3 1604.9 1644.8 1647.6 1703.3 1689.8 1760.9

x = l 1595.4 1654.9 1634.4 1717.2 1672.9 1778.4

x = 3/2 l 1580.7 1664.4 1613.8 1730 1646.8 1794.4

x = 2 l 1566.1 1659.5 1593.3 1722.9 1620.5 1785.1

x = 3 l 1542.2 1608.7 1559.7 1652.9 1577.5 1696.6

x = 4 l 1529 1562.5 1540.5 1588.9 1552.3 1615.6

x = 5 l 1520.3 1533.5 1527 1547.2 1533.9 1561.2

x =

∞

1510.7 1510.8 1510.7 1510.7 1512.5 1512.8

=24

x = 0 1849.1 1849.1 1896.3 1896.3 1943 1943

x = l/3 1839.4 1859.5 1882.5 1911.4 1925.3 1962.4

x = 2 l / 3 1830.2 1872.2 1869.4 1928.6 1908.3 1984.3

x = l 1820.9 1885.9 1856.2 1947.3 1891.3 2007.9

x = 3/2 l 1806 1903.9 1835.3 1971.4 1864.4 2037.7

x = 2 l 1790.5 1901.3 1813.5 1967.3 1836.6 2032

x = 3 l 1767.4 1829.3 1780.6 1868.7 1794.1 1907.9

x = 4 l 1757.6 1781.8 1765.9 1802.2 1774.5 1823.1

x = 5 l 1752.8 1763.4 1758 1774.7 1763.3 1786.4

x =

∞

1746.9 1747 1747.4 1747.6 1747.9 1748.2 Tabella 3.1

(22)

______________________________________________________________________

XX 24

, 6 =

= a

A

_eq

A

_eq

= a 6 , = 18 A

_eq

= a 6 , = 15 A

_eq

= a 6 , = 12 A

_eq

= a 6 , = 9

a1 -0.0731 -0.0504 -0.0332 -0.0186 -0.008

a2 0.0834 0.041 0.014 -0.0054 -0.0145

a

3 0.0195 0.0327 0.0372 0.0361 0.0289

a4 1.0591 1.0752 1.0861 1.099 1.1152

24 , 9 =

= a

A

_eq

A

_eq

= a 9 , = 18 A

_eq

= a 9 , = 15 A

_eq

= a 9 , = 12 A

_eq

= a 9 , = 9

a1 -0.0968 -0.0685 -0.0462 -0.0257 -0.0112

a2 0.1074 0.0538 0.0194 -0.0074 -0.0196

a

3 0.0299 0.0464 0.0515 0.0498 0.0396

a4 1.0862 1.1087 1.1236 1.1413 1.1629

24 , 12 =

= a

A

_eq

A

_eq

= 12 , a = 18 A

_eq

= 12 , a = 15 A

_eq

= 12 , a = 12 A

_eq

= 12 , a = 9

a1 -0.1208 -0.0862 -0.0583 -0.0332 -0.0152

a2 0.1317 0.0665 0.0239 -0.0087 -0.0233

a

3 0.0398 0.0597 0.0655 0.0629 0.0498

a4 1.1130 1.1417 1.1606 1.1829 1.2093

Tabella 3.2 – Coefficienti

a

i

(23)

______________________________________________________________________

XXI

Tabella 3.3:

Calcolo del

KB

applicando la formulazione analitica e confronto con il valore FEM per il corrente a doppia striscia

D.S.

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=9

a K_B calc K_B FEM K_B calc K_B FEM K_B calc K_B FEM 1193 1188 1238.2 1238.8 1292.7 1288.3 x = 0

0,4204 0.0473 0.339

1197.7 1192.7 1243.1 1245.2 1297.8 1294.6 x = l/3

0.4202 0.1678 0.2455

1200.9 1196.2 1246.5 1250 1301.3 1302.5 x = 2 l / 3

0.3965 0.2829 0.0937

1202.5 1198.5 1248 1253.3 1302.9 1306.7 x = l

0.3312 0,419 0.2882

1201.1 1198.8 1246.6 1253.6 1301.4 1307.2 x = 3/2 l

0.1899 0.5582 0.4418

1194.6 1194.8 1239.9 1248.2 1294.4 1300.3 x = 2 l

0.0163 0.6661 0.4528

1163.4 1176.3 1207.5 1222.7 1260.6 1268 x = 3 l

1.093 1.2396 0.5799

1103 1151.4 1144.8 1188.9 1195.2 1226.1 x = 4 l

4.2013 3.7063 2.5216

D.S.

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=12

0,8836 0.5335 1.0025

1373.9 1361.8 1421.9 1416.8 1480.5 1470.6 x = l/3

0,8866 0.3571 0.6717

1380.3 1367.9 1428.5 1425.1 1487.4 1481.1 x = 2 l / 3

0,9065 0.2393 0.4256

1385.2 1372.5 1433.6 1431.4 1492.7 1489.2 x = l

0,9273 0.1544 0.236

1388.2 1374.6 1436.7 1434.4 1496 1492.9 x = 3/2 l

0,9919 0.1619 0.2046

1383.8 1369.6 1432.1 1427.4 1491.2 1484 x = 2 l

1,0379 0.3326 0.4846

1343 1341.1 1389.9 1388 1447.2 1434.1 x = 3 l

0,1386 0.1343 0.911

1243.2 1306.4 1286.6 1340.6 1339.6 1374.7 x = 4 l

4,8389 4.0276 2.5497

(24)

______________________________________________________________________

XXII

D.S.

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=15

a K_B calc K_B FEM K_B calc K_B FEM K_B calc K_B FEM 1513 1497.2 1561.3 1549 1621.4 1599.9 x = 0

1,0557 0.7971 1.3438

1522.2 1506.2 1570.8 1561.5 1631.2 1615.8 x = l/3

1,0608 0.5964 0.9548

1531.5 1514.7 1580.5 1573.2 1641.3 1630.5 x = 2 l / 3

1.1115 0.4621 0.6597

1539.8 1522 1588.9 1583.2 1650.1 1643.3 x = l

1.1663 0.363 0.4114

1547 1527.3 1596.5 1590.5 1657.9 1652.3 x = 3/2 l

1.2916 0.3745 0.3364

1544.3 1522.1 1593.6 1583 1654.9 1642.7 x = 2 l

1.4563 0.67 0.7426

1490.7 1482.8 1538.3 1528.7 1597.5 1574 x = 3 l

0.5332 0.6303 1.4932

1343.2 1440.9 1386.1 1471.2 1439.5 1501.6 x = 4 l

6.7786 5.7814 4.1385

D.S.

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=18

a K_B calc K_B FEM K_B calc K_B FEM K_B calc K_B FEM 1641.2 1624.3 1688.7 1674.8 1749 1724.6 x = 0

1,0401 0.8327 1.4148

1651.6 1634.4 1699.4 1689 1760 1742.7 x = l/3

1,0497 0.6163 0.9952

1663.3 1644.8 1711.5 1703.3 1772.5 1760.9 x = 2 l / 3

1,1233 0.4796 0.6606

1674.3 1654.9 1722.8 1717.2 1784.3 1778.4 x = l

1,1740 0.3285 0.3324

1685.2 1664.4 1734.1 1730 1795.9 1794.4 x = 3/2 l

1,2510 0.2341 0.0847

1683.2 1659.5 1732 1722.9 1793.8 1785.1 x = 2 l

1,4292 0.5274 0.4864

1613.2 1608.7 1659.9 1652.9 1719.1 1696.6 x = 3 l

0,2772 0.4234 1.3276

1409.5 1562.5 1450.3 1588.9 1502 1615.6 x = 4 l

9,7948 8.7234 7.029

(25)

______________________________________________________________________

XXIII

D.S.

A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=24

a K_B calc K_B FEM K_B calc K_B FEM K_B calc K_B FEM 1857.9 1849.1 1900.7 1896.3 1957.9 1943 x = 0

0,4785 0.2331 0.7653

1867.5 1859.5 1910.5 1911.4 1967.9 1962.4 x = l/3

0,429 0.0488 0.2809

1881.2 1872.2 1924.6 1928.6 1982.4 1984.3 x = 2 l / 3

0,4831 0.2099 0.0948

1895.6 1885.9 1939.2 1947.3 1997.5 2007.9 x = l

0,5138 0.4146 0.5162

1910.2 1903.9 1954.2 1971.4 2012.9 2037.7 x = 3/2 l

0,3297 0.875 1.2167

1905.5 1901.3 1949.3 1967.3 2007.9 2032 x = 2 l

0,2191 0.9134 1.1839

1788.7 1829.3 1829.9 1868.7 1884.9 1907.9 x = 3 l

2,2185 2.0766 1.2047

1446.5 1781.8 1479.8 1802.2 1524.3 1823.1 x = 4 l

18,8192 17.89 16.39

Tabella 3.3

(26)

Irrigidimento rotto: risultati FEM e analitici per la geometria reale Appendix V

______________________________________________________________________

XXIV

APPENDIX V – Irrigidimento rotto: risultato FEM e risultato analitico per la geometria reale

Tabella 3.4: Valori K metodo FEM per corrente a sezione Z rotto

Z A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=9

x = 0 1364 1367 1606,8 1616,3 1883,4 1904,6

x = l/3 1352 1376,2 1584,4 1632,7 1847,7 1929,5

x = 2 l / 3 1338,5 1382,6 1559,3 1643,9 1808,2 1946,1

x = l 1324,6 1386,5 1534,7 1651,8 1770,7 1959

x = 3/2 l 1302 1385,1 1494,5 1649,1 1708,7 1954,3

x = 2 l 1279,3 1374,6 1452,8 1628,7 1644,2 1920,6

x = 3 l 1237 1333,7 1376,2 1551,1 1526,2 1795,5

x = 4 l 1205,7 1284,6 1324,5 1470,4 1452,3 1679,2

x = 5 l 1147,5 1189,3 1214,3 1293,3 1278,8 1401,9

x =

∞

1071 1071,3 1076,2 1076,9 1081 1082

=12

x = 0 1544,4 1547,8 1801,3 1812,4 2092 2116,7 x = l/3 1526,8 1562,8 1768,4 1839,6 2040,2 2158,9 x = 2 l / 3 1507,8 1574,3 1733,5 1860,2 1985,5 2190,3 x = l 1488,8 1582 1699,6 1874,9 1933,8 2213,7 x = 3/2 l 1459,8 1582 1647,4 1874,5 1853,9 2212,2 x = 2 l 1431,6 1566,9 1596,4 1844,7 1775,7 2163 x = 3 l 1383 1503,6 1509,3 1725,5 1643,8 1973,2 x = 4 l 1349,7 1435,5 1452,1 1608,7 1560,1 1799,3 x = 5 l 1304,6 1347,7 1365 1444,6 1421,6 1542,2 x =

∞

1236,8 1237,3 1242,7 1243,7 1248,1 1249,6

=15

x = 0 1637,7 1639 1955 1966,1 2248,4 2272,8 x = l/3 1672,2 1717,2 1914,9 2002,9 2185,5 2330,4 x = 2 l / 3 1649,8 1734 1873,8 2033,3 2121,6 2377,7 x = l 1627,8 1746,5 1834,5 2057 2062,1 2415,3 x = 3/2 l 1594,9 1750,3 1775,6 2063,2 1972,5 2423,9 x = 2 l 1563,6 1731,6 1719,3 2026 1887,1 2362,4 x = 3 l 1512,1 1645,6 1627,5 1865,5 1749,6 2108,8 x = 4 l 1478,8 1563,7 1568,9 1722,7 1662,9 1894,7 x = 5 l 1443,4 1486,4 1498,7 1576,8 1549,8 1665,5 x =

∞

1383,1 1383,8 1389,4 1390,8 1395,2 1397,4

(27)

______________________________________________________________________

XXV

=18

x = 0 1822,8 1825,8 2085,2 2094,4 2387,8 2408,9 x = l/3 1799,8 1850,8 2040,5 2139,1 2307 2467,6 x = 2 l / 3 1775,5 1872,3 1996 2178,9 2238,5 2530,1 x = l 1751,8 1890,1 1953,8 2212,7 2174,7 2584 x = 3/2 l 1716,5 1899,7 1890,6 2229,3 2079,2 2609,1 x = 2 l 1683,1 1878,4 1830,6 2186,7 1988,7 2538,5 x = 3 l 1630 1769,6 1736,1 1984,6 1847,9 2220,7 x = 4 l 1597,8 1678,6 1678 1824,2 1761 1979,7 x = 5 l 1569,3 1611,6 1620,2 1696,2 1666,7 1777,4 x =

∞

1515,4 1516,3 1522,1 1523,9 1528,3 1531

=24

x = 0 2047,1 2048,7 2298,9 2304,2 2580,4 2592,1 x = l/3 2021,3 2075,9 2252,6 2354,4 2503,6 2662,7 x = 2 l / 3 1996,1 2103,2 2207,4 2404,9 2436,3 2741,4 x = l 1971,2 2128,5 2163,2 2451,6 2370,7 2815 x = 3/2 l 1933,9 2150,5 2096,9 2489,6 2272,1 2872,4 x = 2 l 1898,4 2128,1 2033,6 2444,8 2178,5 2798,7 x = 3 l 1845 1983 1938,7 2183,1 2038,4 2399,2 x = 4 l 1816 1885,9 1885,7 2012,1 1959 2145 x = 5 l 1796,4 1837,6 1844,6 1918,9 1891,3 1998,2 x =

∞

1750,4 1751,4 1757,1 1759,3 1763,6 1767,1 Tabella 3.4

(28)

______________________________________________________________________

XXVI

Tabella 3.5: Valori K metodo FEM per corrente a sezione Ω rotto

Ω A

_eq

= 6 A

_eq

= 9 A

_eq

= 12

=9

x = 0 1428,7 1428,7 1663,1 1663,1 1995,5 1995,5 x = l/3 1414,6 1440,1 1638,2 1683,4 1954,9 2030,4 x = 2 l / 3 1398,6 1448,2 1609,9 1697,8 1906,8 2053,3 x = l 1382,7 1454,3 1582,8 1710 1863,5 2075,9 x = 3/2 l 1356,2 1453,9 1536,7 1710,2 1788,2 2077,9 x = 2 l 1328,6 1442 1488,1 1688,8 1707,6 2041,7 x = 3 l 1276,9 1392,9 1398 1600,6 1559,1 1892,4 x = 4 l 1241,4 1338,7 1340,8 1513,2 1472,1 1760,4 x = 5 l 1168,9 1220,6 1213,6 1302,9 1260,1 1406,2 x =

∞

1075,4 1075,9 1076,3 1077 1079,3 1080,3

=12

x = 0 1615,1 1615,1 1861 1861 2208,9 2208,9 x = l/3 1593,9 1633,4 1823,3 1893,6 2146 2263,4 x = 2 l / 3 1571 1647,7 1782,9 1919 2078,6 2305,8 x = l 1548,6 1658,2 1744,3 1939 2016,1 2341,5 x = 3/2 l 1513,6 1659,4 1683,7 1942,1 1916,9 2348 x = 2 l 1479,3 1641,7 1624,2 1910 1819,4 2293,3 x = 3 l 1420,1 1565,5 1523,1 1774 1656,4 2064,2 x = 4 l 1381 1486,9 1459,6 1642,2 1558,6 1855,7 x = 5 l 1324,9 1377,7 1362 1449 1397,2 1532 x =

∞

1241,8 1242,6 1242,8 1243,9 1246,2 1247,6

=15

x = 0 1767,4 1767,4 2013,9 2013,9 2360,6 2360,6 x = l/3 1741 1791,7 1967,1 2057,1 2282,6 2432,9 x = 2 l / 3 1713,6 1812,2 1919 2093,5 2202,8 2493,7 x = l 1687,3 1828,3 1873,9 2123,7 2129,9 2546,5 x = 3/2 l 1647,4 1833,9 1805,6 2134,4 2019,1 2565,4 x = 2 l 1609,3 1811,5 1740,7 2094 1914,5 2496,4 x = 3 l 1546,8 1708,1 1636 1910,5 1749,5 2189,3 x = 4 l 1507,1 1611,6 1571,5 1747,2 1650,5 1928,4 x = 5 l 1463,1 1515,2 1494,9 1577,4 1523,8 1645,1 x =

∞

1379,8 1380,5 1379,5 1380,4 1393,3 1395,1

(29)

______________________________________________________________________

XXVII

=18

x = 0 1898,8 1898,8 2141,3 2141,3 2480,1 2480,1 x = l/3 1869 1927,6 2088,8 2192,3 2393,1 2565,5 x = 2 l / 3 1839,1 1953,7 2036,7 2239 2307,4 2643,7 x = l 1810,4 1976,1 1988,2 2280,5 2229,5 2715,6 x = 3/2 l 1767,5 1988,3 1915,6 2303,1 2113,2 2754,8 x = 2 l 1727 1962,6 1847,4 2256,9 2004,7 2675,4 x = 3 l 1685,5 1776,5 1741,4 2025,5 1840,4 2289,9 x = 4 l 1623,9 1722,9 1677,9 1841 1743,2 1994,7 x = 5 l 1588,5 1639,3 1616,3 1694,1 1641 1749,9 x =

∞

1512 1513 1511,7 1512,8 1526,2 1528,5

=24

x = 0 2122,1 2122,1 2346,5 2346,5 2652,5 2652,5 x = l/3 2090,1 2154,8 2292,5 2402,5 2566,6 2743 x = 2 l / 3 2058,8 2187,7 2240,5 2459,7 2484,4 2836,3 x = l 2028,2 2217,9 2190,3 2513,7 2406,7 2927,1 x = 3/2 l 1982,9 2243,5 2115,9 2559,7 2291,5 3003,8 x = 2 l 1940 2216,2 2046 2512,2 2184,3 2924,2 x = 3 l 1875,7 2043 1942,1 2215,6 2028 2445,9 x = 4 l 1840,6 1925,8 1885,8 2020,1 1943,4 2138,8 x = 5 l 1815,3 1864,7 1841 1914,3 1868,6 1965,3 x =

∞

1756,2 1758 1757,4 1759,7 1761,5 1764,5 Tabella 3.5