trasarea liniilor de egal efort principal

Tema 1 CHE

OCHI George-Bogdan MS6

UNIVERSITATEA POLITEHNICA BUCURESTI

FACULTATEA DE ENERGETICA

TEMA 1 CHE

STUDENT

OCHI GEORGE-BOGDAN

MS6

Tema 1 CHE


Calculul eforturilor principale si trasarea liniilor de egal efort principal

Metoda elementara de calcul a eforturilor.

Pentru calculul eforturilor intr-un punct oarecare din corpul barajului s-a folosit teoria

elasticitatii. Aceasta teorie exprima eforturile normale σx , σz si efortul tangential τ, intr-un

punct oarecare P(x,z) din corpul barajului, ca depinzand linear de coordonatele acestuia, adica:

𝜎𝑥 = 𝑎1𝑥 + 𝑏1𝑧

𝜎𝑦 = 𝑎2𝑥 + 𝑏2𝑧

𝜏 = 𝑎3𝑥 + 𝑏3𝑧

Coeficientii numerici a1, a2…b3, se determina folosind ecuatiile lui Laplace pentru un element

plan infinit mic, din jurul punctului P si punand conditii limita la parametrii barajului. Se obtin

relatiile:

𝑎1 = 𝛾𝑏

𝜆𝜆1(𝜆 − 𝜆1)

(𝜆 + 𝜆1)²− 𝛾

𝜆𝜆1(2 − 𝜆𝜆1 + 𝜆2)

(𝜆 + 𝜆1)3+ 𝑚𝛾

𝜆12

𝜆 + 𝜆1

𝑏1 = 𝛾𝑏

2𝜆²𝜆12

(𝜆 + 𝜆1)²+ 𝛾

𝜆²(𝜆 + 3𝜆1 − 2𝜆𝜆12)

(𝜆 + 𝜆1)3− 𝑚𝛾

𝜆𝜆12

𝜆 + 𝜆1

𝑎2 = 𝛾𝑏

(𝜆1 − 𝜆)

(𝜆 + 𝜆1)²+ 𝛾

(2 − 3𝜆𝜆1 − 𝜆12)

(𝜆 + 𝜆1)3+ 𝑚𝛾

1

𝜆 + 𝜆1

𝑏2 = 𝛾𝑏

(𝜆2 + 𝜆12)

(𝜆 + 𝜆1)²− 𝛾

(𝜆 − 𝜆1 − 2𝜆²𝜆1)

(𝜆 + 𝜆1)3− 𝑚𝛾

𝜆1

𝜆 + 𝜆1

𝑎3 = 𝛾𝑏

2𝜆𝜆1

(𝜆 + 𝜆1)²+ 𝛾

(𝜆 − 𝜆1 − 2𝜆²𝜆1)

(𝜆 + 𝜆1)3− 𝑚𝛾

𝜆1

𝜆 + 𝜆1

𝑏3 = 𝛾𝑏

𝜆𝜆1(𝜆1 − 𝜆)

(𝜆 + 𝜆1)²+ 𝛾

𝜆𝜆1(2 − 𝜆𝜆1 + 𝜆2)

(𝜆 + 𝜆1)3+ 𝑚𝛾

𝜆𝜆1

𝜆 + 𝜆1

Aceste expresii sunt determinate pentru λ, λ1 diferite de 0 si cazul lacului plin. Daca lacul

de acumulare este gol, expresiile pentru acesti coeficienti sunt la fel dar = 0 . Practic din

relatiile de mai sus raman numai primii termeni.

Eforturile normale σx, σz, τ, orientate dupa directiile axelor de coordinate, nu reprezinta

valorile extreme care actioneaza intr-un punct din corpul barajului. Eforturile maxime sau

minime, denumite eforturi principale, actioneaza dupa directii in general diferite de cele ale

axelor, numite directii principale. Directiile principale ale eforturilor normale sunt ortogonale

intre ele, iar de-a lungul lor eforturile de alunecare sunt nule. Eforturile maxime de alunecare

Tema 1 CHE


sunt orientate cu un decalaj de 45° fata de directiile principale. Se calculeaza in orice punct

P(x,z) pe langa eforturile σx, σz, τ si eforturile principale σ1, σ2, τm cu relatiile:

𝜎1,2 =1

2(𝜎𝑥 + 𝜎𝑧) ±

1

2√(𝜎𝑥 − 𝜎𝑧)2 + 4𝜋²

𝜏𝑚 = ±1

2√(𝜎𝑥 − 𝜎𝑧)2 + 4𝜋²

Foarte suggestive in reprezentarea starii de efort din baraj sunt liniile de egal effort

principal, denumite si linii izostatice, care reprezinta locul geometric al punctelor in care

actioneaza aceleasi eforturi principale σ1, σ2, τm:

𝜎1,2 =1

2(𝜎𝑥 + 𝜎𝑧) ±

1

2√(𝜎𝑥 − 𝜎𝑧)2 + 4𝜋2 = 𝜎0 = 𝑐𝑜𝑛𝑠𝑡

𝜏𝑚 = ±1

2√(𝜎𝑥 − 𝜎𝑧)2 + 4𝜋² = 𝜏𝑚 = 𝑐𝑜𝑛𝑠𝑡

Trasarea liniilor de egal efort principal

In cazul de fata, pentru barajul cu ambii parametrii inclinati se reprezinta grafic starea

eforturilor atat in cazul in care lacul de acumulare este gol, cat si in cazul cand acesta este plin.

Deci se cere, ca in cele doua situatii, sa se traseze eforturile totale amonte si aval si liniile de

egal efort principal σ1=ct, σ2=ct, τm=ct.

Se observa ca raportul 𝑥

𝑧= 𝑡𝑔𝛽 este o valoare constanta pe o raza (a) care uneste originea cu

punctual P (raza OP). Rezulta ca si paranteza va fi o valoare constanta (𝜎𝑥0, 𝜎𝑧

0, 𝜏𝑚0 ) pe raza

respectiva:

𝜎1,2 = [1

2(𝜎𝑥

0 + 𝜎𝑧0) ±

1

2√(𝜎𝑥

0 − 𝜎𝑧0)2 + 4𝜋2] 𝑧 = 𝜎1,2

0 𝑧

𝜏𝑚 = (±1

2√(𝜎𝑥

0 − 𝜎𝑧0)2 + 4𝜋²) 𝑧 = ±𝜏𝑚

0 𝑧

Tinand seama de acestea, trasarea liniilor de egal efort principal necesita parcurgerea

etapelor:

-calculul coeficientilor a1….b3

-calculul valorilor constante 𝜎𝑥0, 𝜎𝑧

0, 𝜏𝑚0 pentru diferite raze duse din varful O al barajului

-trasarea liniilor de egal efort principal

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-valorile σ1j, σzj, τmj se aleg astfel incat liniile de egal efort sa acopere intreaga sectiune

transversala a barajului;

Calculul si trasarea eforturilor totale amonte si aval in cazul lacului de

acumulare plin

Tabelul 1. Valorile eforturilor total, lac de acumulare plin

z σamFG σam

FH σamFV σam

FS Σσam σavFG σav

FH σavFV σav

FS Σσav

[m] [tf/m²] [tf/m²] [tf/m²] [tf/m²] [tf/m²] [tf/m²] [tf/m²] [tf/m²] [tf/m²] [tf/m²]

0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

5 9.07 -7.47 2.14 -2.50 1.24 2.93 7.47 -0.92 0.00 9.48

10 18.14 -14.94 4.28 -5.00 2.48 5.86 14.94 -1.84 0.00 18.96

15 27.22 -22.42 6.43 -7.50 3.73 8.78 22.42 -2.77 0.00 28.43

20 36.29 -29.89 8.57 -10.00 4.97 11.71 29.89 -3.69 0.00 37.91

25 45.36 -37.36 10.71 -12.50 6.21 14.64 37.36 -4.61 0.00 47.39

30 54.43 -44.83 12.85 -15.00 7.45 17.57 44.83 -5.53 0.00 56.87

35 63.50 -52.31 15.00 -17.50 8.69 20.50 52.31 -6.46 0.00 66.35

40 72.58 -59.78 17.14 -20.00 9.93 23.42 59.78 -7.38 0.00 75.83

45 81.65 -67.25 19.28 -22.50 11.18 26.35 67.25 -8.30 0.00 85.30

50 90.72 -74.72 21.42 -25.00 12.42 29.28 74.72 -9.22 0.00 94.78

55 99.79 -82.20 23.57 -27.50 13.66 32.21 82.20 -10.15 0.00 104.26

60 108.86 -89.67 25.71 -30.00 14.90 35.14 89.67 -11.07 0.00 113.74

65 117.93 -97.14 27.85 -32.50 16.14 38.07 97.14 -11.99 0.00 123.22

70 127.01 -104.61 29.99 -35.00 17.39 40.99 104.61 -12.91 0.00 132.70

75 136.08 -112.09 32.14 -37.50 18.63 43.92 112.09 -13.84 0.00 142.17

80 145.15 -119.56 34.28 -40.00 19.87 46.85 119.56 -14.76 0.00 151.65

85 154.22 -127.03 36.42 -42.50 21.11 49.78 127.03 -15.68 0.00 161.13

90 163.29 -134.50 38.56 -45.00 22.35 52.71 134.50 -16.60 0.00 170.61

95 172.37 -141.98 40.71 -47.50 23.59 55.63 141.98 -17.52 0.00 180.09

100 181.44 -149.45 42.85 -50.00 24.84 58.56 149.45 -18.45 0.00 189.56

105 190.51 -156.92 44.99 -52.50 26.08 61.49 156.92 -19.37 0.00 199.04

110 199.58 -164.39 47.13 -55.00 27.32 64.42 164.39 -20.29 0.00 208.52

115 208.65 -171.87 49.28 -57.50 28.56 67.35 171.87 -21.21 0.00 218.00

120 217.73 -179.34 51.42 -60.00 29.80 70.27 179.34 -22.14 0.00 227.48

125 226.80 -186.81 53.56 -62.50 31.05 73.20 186.81 -23.06 0.00 236.96

130 235.87 -194.28 55.70 -65.00 32.29 76.13 194.28 -23.98 0.00 246.43

138 250.38 -206.24 59.13 -69.00 34.27 80.82 206.24 -25.46 0.00 261.60

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Figura 1. Trasarea eforturilor, lac de acumulare plin

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Tabelul 2. Valorile eforturilor total, lac de acumulare gol

z FG W A σamFG σav

FG

[m] [tf/m] [m³] [m²] [tf/m²] [tf/m²] 0 0.00 0.00 0.00 0.00 0.00

5 24.54 2.79 20.45 9.07 2.93

10 98.16 11.15 81.80 18.14 5.86

15 220.86 25.09 184.05 27.22 8.78

20 392.64 44.61 327.20 36.29 11.71

25 613.50 69.70 511.25 45.36 14.64

30 883.44 100.37 736.20 54.43 17.57

35 1202.46 136.61 1002.05 63.50 20.50

40 1570.56 178.43 1308.80 72.58 23.42

45 1987.74 225.83 1656.45 81.65 26.35

50 2454.00 278.80 2045.00 90.72 29.28

55 2969.34 337.35 2474.45 99.79 32.21

60 3533.76 401.47 2944.80 108.86 35.14

65 4147.26 471.17 3456.05 117.93 38.07

70 4809.84 546.45 4008.20 127.01 40.99

75 5521.50 627.30 4601.25 136.08 43.92

80 6282.24 713.73 5235.20 145.15 46.85

85 7092.06 805.74 5910.05 154.22 49.78

90 7950.96 903.32 6625.80 163.29 52.71

95 8858.94 1006.47 7382.45 172.37 55.63

100 9816.00 1115.21 8180.00 181.44 58.56

105 10822.14 1229.52 9018.45 190.51 61.49

110 11877.36 1349.40 9897.80 199.58 64.42

115 12981.66 1474.86 10818.05 208.65 67.35

120 14135.04 1605.90 11779.20 217.73 70.27

125 15337.50 1742.51 12781.25 226.80 73.20

130 16589.04 1884.70 13824.20 235.87 76.13

138 18693.5904 2123.8 15577.992 250.384 80.82

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Figura 2. Trasarea eforturilor, lac de acumulare gol

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Calculul si trasarea liniilor de egal efort principal in cazul in care lacul de

acumulare este plin

Valorile constantelor:

Hc λ λ1 f ɣ ɣb m [m] [-] [-] [-] [tf/m³] [tf/m³] [-]

138 0.6184 0.1996 0.6555 1 2.4 0.5

Valorile coeficientilor in tf/m³:

a1 b1 a2 b2 a3 b3

-0.29966 0.575378 2.013792 0.650317 1.627678 0.375111

Pentru valorile λ si λ1 corespund unghiurile β=11.29° si β1=31.73°

Tabelul 3. Valorile constante 𝜎𝑥0, 𝜎𝑧

0, 𝜏𝑚0 in [tf/m³]

β[°] -11.29 -10 -5 0 15 20 25 30 31.73

tgβ -0.200 -0.176 -0.087 0.000 0.268 0.364 0.466 0.577 0.618

σx° 0.635 0.628 0.602 0.575 0.495 0.466 0.436 0.402 0.390

σz° 0.248 0.295 0.474 0.650 1.190 1.383 1.589 1.813 1.896

τ° 0.050 0.088 0.233 0.375 0.811 0.968 1.134 1.315 1.382

σ1° 0.642 0.650 0.779 0.990 1.725 1.995 2.285 2.600 2.716

σ2° 0.242 0.273 0.297 0.236 -0.040 -0.146 -0.260 -0.384 -0.431

τm° 0.200 0.188 0.241 0.377 0.883 1.071 1.272 1.492 1.573

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Tabelul 4. Valorile z[m] pentru lac plin-σ1=ct, lac plin

σ1\β[°] -11.29 -10 -5 0 15 20 25 30 31.73

20 31.17 30.76 25.67 20.21 11.59 10.02 8.75 7.69 7.36

40 62.34 61.53 51.34 40.41 23.19 20.05 17.51 15.39 14.73

60 93.52 92.29 77.01 60.62 34.78 30.07 26.26 23.08 22.09

80 124.69 123.06 102.68 80.82 46.38 40.09 35.01 30.77 29.45

100 0.00 0.00 128.35 101.03 57.97 50.11 43.77 38.47 36.81

120 0.00 0.00 0.00 121.23 69.57 60.14 52.52 46.16 44.18

140 0.00 0.00 0.00 0.00 81.16 70.16 61.27 53.85 51.54

160 0.00 0.00 0.00 0.00 92.75 80.18 70.03 61.54 58.90

180 0.00 0.00 0.00 0.00 104.35 90.20 78.78 69.24 66.27

200 0.00 0.00 0.00 0.00 115.94 100.23 87.53 76.93 73.63

220 0.00 0.00 0.00 0.00 127.54 110.25 96.28 84.62 80.99

240 0.00 0.00 0.00 0.00 0.00 120.27 105.04 92.32 88.36

260 0.00 0.00 0.00 0.00 0.00 130.30 113.79 100.01 95.72

280 0.00 0.00 0.00 0.00 0.00 0.00 122.54 107.70 103.08

300 0.00 0.00 0.00 0.00 0.00 0.00 131.30 115.40 110.44

320 0.00 0.00 0.00 0.00 0.00 0.00 0.00 123.09 117.81

340 0.00 0.00 0.00 0.00 0.00 0.00 0.00 130.78 125.17

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Figura 3. Linii de egal efort principal σ1=ct, in cazul lacului de acumulare plin

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Tabelul 5. Valorile z[m] pentru lac plin-σ2=ct, lac plin

σ2\β[°] -11.29 -10 -5 0 15 20 25 30 31.73

2 8.268 7.316 6.743 8.479 0.000 0.000 0.000 0.000 0.000

4 16.537 14.633 13.487 16.959 0.000 0.000 0.000 0.000 0.000

6 24.805 21.949 20.230 25.438 0.000 0.000 0.000 0.000 0.000

8 33.073 29.266 26.974 33.917 0.000 0.000 0.000 0.000 0.000

10 41.341 36.582 33.717 42.396 0.000 0.000 0.000 0.000 0.000

12 49.610 43.899 40.460 50.876 0.000 0.000 0.000 0.000 0.000

14 57.878 51.215 47.204 59.355 0.000 0.000 0.000 0.000 0.000

16 66.146 58.532 53.947 67.834 0.000 0.000 0.000 0.000 0.000

Figura 4. Linii de egal efort principal σ2=ct, in cazul lacului de acumulare plin

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Tabelul 6. Valorile z[m] pentru lac plin-τm=ct

τm\β[°] -11.29 -10 -5 0 15 20 25 30 31.730

10 50.036 53.087 41.446 26.527 11.331 9.340 7.859 6.702 6.356

20 100.072 106.175 82.892 53.054 22.663 18.680 15.718 13.404 12.711

30 0.000 0.000 124.338 79.580 33.994 28.020 23.578 20.106 19.067

40 0.000 0.000 0.000 106.107 45.325 37.360 31.437 26.808 25.422

50 0.000 0.000 0.000 132.634 56.657 46.700 39.296 33.510 31.778

60 0.000 0.000 0.000 0.000 67.988 56.040 47.155 40.212 38.133

70 0.000 0.000 0.000 0.000 79.320 65.380 55.015 46.914 44.489

80 0.000 0.000 0.000 0.000 90.651 74.720 62.874 53.617 50.844

90 0.000 0.000 0.000 0.000 101.982 84.059 70.733 60.319 57.200

100 0.000 0.000 0.000 0.000 113.314 93.399 78.592 67.021 63.555

110 0.000 0.000 0.000 0.000 124.645 102.739 86.452 73.723 69.911

120 0.000 0.000 0.000 0.000 135.976 112.079 94.311 80.425 76.266

130 0.000 0.000 0.000 0.000 0.000 121.419 102.170 87.127 82.622

140 0.000 0.000 0.000 0.000 0.000 130.759 110.029 93.829 88.977

150 0.000 0.000 0.000 0.000 0.000 0.000 117.889 100.531 95.333

160 0.000 0.000 0.000 0.000 0.000 0.000 125.748 107.233 101.688

170 0.000 0.000 0.000 0.000 0.000 0.000 133.607 113.935 108.044

180 0.000 0.000 0.000 0.000 0.000 0.000 0.000 120.637 114.399

190 0.000 0.000 0.000 0.000 0.000 0.000 0.000 127.339 120.755

200 0.000 0.000 0.000 0.000 0.000 0.000 0.000 134.041 127.110

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Figura 5. Linii de egal efort principal τm=ct, in cazul lacului de acumulare plin

Tema 1 CHE


Calculul si trasarea liniilor de egal efort principal in cazul in care lacul de

acumulare este plin

Valorile constantelor:

Neexistand apa in lac, greutatea specifica a apei va fi practic nula.

Hc λ λ1 f ɣ ɣb m

[m] [-] [-] [-] [tf/m³] [tf/m³] [-]

138 0.6184 0.1996 0.6555 0 2.4 0.5

Valorile coeficientilor in tf/m³:

a1 b1 a2 b2 a3 b3

0.185413 0.109294 -1.50214 1.514549 0.885451 -0.18541

Tabelul 7. Valorile z[m] pentru lac gol-σ1=ct

σ1\β[°] -11.29 -10 -5 0 5 10 15 20 25 30 31.73

10 5.300 5.419 5.920 6.499 7.182 7.997 8.970 10.093 11.252 12.159 12.352

20 10.600 10.838 11.839 12.999 14.364 15.994 17.939 20.185 22.503 24.318 24.704

30 15.901 16.257 17.759 19.498 21.547 23.991 26.909 30.278 33.755 36.478 37.056

40 21.201 21.676 23.679 25.998 28.729 31.988 35.879 40.370 45.007 48.637 49.408

50 26.501 27.095 29.599 32.497 35.911 39.985 44.849 50.463 56.258 60.796 61.759

60 31.801 32.514 35.518 38.996 43.093 47.983 53.818 60.556 67.510 72.955 74.111

70 37.101 37.933 41.438 45.496 50.276 55.980 62.788 70.648 78.762 85.114 86.463

80 42.401 43.352 47.358 51.995 57.458 63.977 71.758 80.741 90.013 97.274 98.815

90 47.702 48.771 53.278 58.495 64.640 71.974 80.727 90.833 101.265 109.433 111.167

100 53.002 54.190 59.197 64.994 71.822 79.971 89.697 100.926 112.517 121.592 123.519

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Figura 6. Linii de egal efort principal σ1=ct, in cazul lacului de acumulare gol

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Tabelul 8. Valorile z[m] pentru lac gol-σ2=ct

σ2\β[°] -11.29 -10 -5 0 5 10 15 20 25 30 31.73

2 0.000 0.000 40.179 23.463 17.194 14.163 12.807 13.007 16.518 48.537 0.000

4 0.000 0.000 80.357 46.926 34.389 28.326 25.614 26.013 33.035 97.075 0.000

6 0.000 0.000 120.536 70.388 51.583 42.489 38.421 39.020 49.553 0.000 0.000

8 0.000 0.000 0.000 93.851 68.778 56.652 51.228 52.027 66.070 0.000 0.000

10 0.000 0.000 0.000 117.314 85.972 70.815 64.036 65.033 82.588 0.000 0.000

12 0.000 0.000 0.000 0.000 103.166 84.978 76.843 78.040 99.105 0.000 0.000

14 0.000 0.000 0.000 0.000 120.361 99.141 89.650 91.046 115.623 0.000 0.000

16 0.000 0.000 0.000 0.000 137.555 113.304 102.457 104.053 132.141 0.000 0.000

18 0.000 0.000 0.000 0.000 0.000 127.467 115.264 117.060 0.000 0.000 0.000

20 0.000 0.000 0.000 0.000 0.000 0.000 128.071 130.066 0.000 0.000 0.000

22 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

24 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

26 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

28 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

30 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

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Figura 7. Linii de egal efort principal σ2=ct, in cazul lacului de acumulare gol

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Tabelul 9. Valorile z[m] pentru lac gol-τm=ct

τm\β[°] -11.29 -10 -5 0 5 10 15 20 25 30 31.730

10 10.60 10.90 12.20 13.76 15.67 18.03 20.86 23.89 26.05 25.60 24.71

20 21.20 21.80 24.40 27.52 31.35 36.06 41.72 47.79 52.11 51.20 49.41

30 31.80 32.70 36.60 41.28 47.02 54.09 62.58 71.68 78.16 76.80 74.12

40 42.40 43.60 48.80 55.04 62.70 72.12 83.45 95.57 104.21 102.40 98.82

50 53.00 54.50 60.99 68.81 78.37 90.15 104.31 119.47 130.26 128.01 123.53

60 63.60 65.41 73.19 82.57 94.04 108.18 125.17 0.00 0.00 0.00 0.00

70 74.20 76.31 85.39 96.33 109.72 126.21 0.00 0.00 0.00 0.00 0.00

80 84.80 87.21 97.59 110.09 125.39 0.00 0.00 0.00 0.00 0.00 0.00

90 95.40 98.11 109.79 123.85 0.00 0.00 0.00 0.00 0.00 0.00 0.00

100 106.00 109.01 121.99 137.61 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Figura 8. Linii de egal efort principal τm=ct, in cazul lacului de acumulare gol

trasarea liniilor de egal efort principal

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