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Constructii din beton armat Dimensionare planseu
An IV CCIA
1/8
CALCULUL PLANSEULUI PESTE ETAJUL CURENT
Bazele teoretice si metodelel numerice ale calculului placilor plane respecta principiile Teoriei
Elasticitatii. In practica se utilizeaza tabele cu coeficienti (STAS 10107/2) pentru placi dreptunghiulare
cu o singura deschidere si cu o incarcare uniform distribuita. Se considera simple rezemari sau
incastrari perfecte in lungul celor patru laturi si pot aparea 6 cazuri distincte.
A. Placi cu un singur camp
unde,
1
2
l
l=
Momentele pozitive in camp la centru:
2
2122
2
1111
qlM
qlM
=
=
Momentele pozitive in camp maxime:
2213max
2 qlM =
Sageata la centru:
3
4
110Eh
qlw =
Placa
tip 1
10 11 12 13
1.00 0.0487 0.0368 0.0368 0.0368
1.05 0.0536 0.0408 0.0330 0.0330
1.10 0.0584 0.0446 0.0296 0.0296
1.15 0.0631 0.0483 0.0266 0.0266
1.20 0.0678 0.0524 0.0239 0.0239
1.25 0.0728 0.0562 0.0215 0.0215
1.30 0.0767 0.0595 0.0192 0.0192
1.35 0.0809 0.0633 0.0172 0.0172
1.40 0.0850 0.0667 0.0154 0.0155
1.45 0.0890 0.0699 0.0138 0.0141
1.50 0.0927 0.0730 0.0124 0.0128
1.55 0.0963 0.0758 0.0111 0.0117
1.60 0.0997 0.0787 0.0101 0.0108
1.65 0.1029 0.0813 0.0090 0.0100
1.70 0.1060 0.0840 0.0081 0.0093
1.75 0.1093 0.0870 0.0073 0.0086
1.80 0.1118 0.0885 0.0066 0.0080
1.85 0.1145 0.0909 0.0059 0.0075
1.90 0.1169 0.0926 0.0054 0.0070
1.95 0.1195 0.0943 0.0048 0.0066
2.00 0.1215 0.0962 0.0043 0.0062
l 2
M2
M1
l1
Constructii din beton armat Dimensionare planseu
An IV CCIA
2/8
unde,
1
2
l
l=
Momentele pozitive in camp la centru:
2
2222
2
1211
qlM
qlM
=
=
Momentele pozitive in camp maxime:
2223max
2 qlM =
Sageata la centru:
3
4
120Eh
qlw =
Momentul maxim negativ pe reazem:
2224
'
2 qlM =
Daca latura incastrata este paralela cu 1l , sunt valabile
titlurile inferioare cu:
1
2
1' ==l
l
2
123
max
1 qlM = 2
124
'
1 qlM =
Placa
tip 2
20 21 22 23 24
0.50 0.0037 0.0042 0.0585 0.0585 0.1205
0.55 0.0052 0.0052 0.0565 0.0565 0.1196
0.60 0.0071 0.0063 0.0541 0.0541 0.1153
0.65 0.0092 0.0075 0.0513 0.0513 0.1121
0.70 0.0118 0.0090 0.0485 0.0485 0.1089
0.75 0.0147 0.0107 0.0457 0.0457 0.1053
0.80 0.0179 0.0127 0.0427 0.0427 0.1010
0.85 0.0215 0.0151 0.0398 0.0398 0.0967
0.90 0.0253 0.0178 0.0372 0.0372 0.0926
0.95 0.0292 0.0208 0.0344 0.0344 0.0887
1.00 0.0334 0.0243 0.0318 0.0340 0.0840
1.10 0.0422 0.0313 0.0270 0.0287 0.0758
1.20 0.0512 0.386 0.0227 0.0240 0.0688
1.30 0.0602 0.0461 0.0191 0.0199 0.0616
1.40 0.0689 0.0532 0.0159 0.0166 0.0555
1.50 0.0773 0.0602 0.0131 0.0138 0.0499
1.60 0.0852 0.0667 0.0110 0.0116 0.0449
1.70 0.0926 0.0725 0.0091 0.0099 0.0407
1.80 0.0994 0.0781 0.0076 0.0085 0.0367
1.90 0.1056 0.0833 0.0063 0.0074 0.0334
2.00 0.1112 0.0877 0.0051 0.0064 0.0305
' 20 22 21 23 24
M1
l1
l 2
M2
M 2
l1
l 2
M2
M1
M1
Constructii din beton armat Dimensionare planseu
An IV CCIA
3/8
unde,
1
2
l
l=
Momentele pozitive in camp la centru:
2
2322
2
1311
qlM
qlM
=
=
Momentele pozitive in camp maxime:
2233
max
2 qlM =
Sageata la centru:
3
4
130Eh
qlw =
Momentul maxim negativ pe reazem:
2134'
2 qlM =
Daca latura incastrata este paralela cu 1l , sunt valabile
titlurile inferioare cu:
1
2
1' ==l
l
2
133
max
1 qlM = 2
234
'
1 qlM =
Placa
tip 3
30 31 32 33 34
0.50 0.0063 0.0059 0.0800 0.0800 0.0298
0.55 0.0082 0.0081 0.0699 0.0699 0.0349
0.60 0.0100 0.0105 0.0602 0.0602 0.0400
0.65 0.0120 0.0130 0.0518 0.0518 0.0452
0.70 0.0139 0.0155 0.0441 0.0441 0.0497
0.75 0.0157 0.0180 0.0375 0.0375 0.0541
0.80 0.0174 0.0204 0.0317 0.0317 0.0577
0.85 0.0190 0.0226 0.0267 0.0267 0.0618
0.90 0.0205 0.0248 0.0224 0.0224 0.0648
0.95 0.0218 0.0268 0.0193 0.0193 0.0674
1.00 0.0230 0.0285 0.0158 0.0162 0.0700
1.10 0.0251 0.0315 0.0112 0.0123 0.0741
1.20 0.0267 0.0340 0.0078 0.0097 0.0769
1.30 0.0280 0.0360 0.0056 0.0081 0.0794
1.40 0.0289 0.0376 0.0039 0.0068 0.0813
1.50 0.0297 0.0388 0.0027 0.0059 0.0820
1.60 0.0302 0.0397 0.0019 0.0051 0.0833
1.70 0.0307 0.0405 0.0013 0.0045 0.0833
1.80 0.0309 0.0410 0.0009 0.0040 0.0833
1.90 0.0311 0.0412 0.0006 0.0036 0.0833
2.00 0.0313 0.0415 0.0004 0.0032 0.0833
' 30 32 31 33 34
l1
l 2
M2
M1
M1
M1
l1
M1
l 2
M2
M 2
M
2
Constructii din beton armat Dimensionare planseu
An IV CCIA
4/8
unde,
1
2
l
l=
Momentele pozitive in camp la centru:
2
2422
2
1411
qlM
qlM
=
=
Momentele pozitive in camp maxime:
2243
max
2 qlM =
Sageata la centru:
3
4
140Eh
qlw =
Momentele maxime negative pe
reazeme: 2
144
'
1 qlM =
2245'
2 qlM =
Placa
tip 4
40 41 42 43 44 45
1.00 0.0252 0.0234 0.0234 0.0249 0.0699 0.0699
1.05 0.0281 0.0263 0.0209 0.0221 0.0752 0.0657
1.10 0.0302 0.0285 0.0187 0.0197 0.0787 0.0608
1.15 0.0329 0.0311 0.0166 0.0176 0.0833 0.0569
1.20 0.0348 0.0333 0.0147 0.0158 0.0870 0.0530
1.25 0.0369 0.0357 0.0130 0.0140 0.0901 0.0496
1.30 0.0389 0.0377 0.0115 0.0124 0.0935 0.0462
1.35 0.0408 0.0397 0.0102 0.0111 0.0971 0.0432
1.40 0.0426 0.0415 0.0090 0.0100 0.1000 0.0405
1.45 0.0443 0.0433 0.0079 0.0091 0.1020 0.0380
1.50 0.0459 0.0450 0.0069 0.0084 0.1042 0.0358
1.55 0.0472 0.0463 0.0061 0.0077 0.1064 0.0338
1.60 0.0484 0.0476 0.0054 0.0071 0.1087 0.0318
1.65 0.0496 0.0490 0.0047 0.0066 0.1099 0.0301
1.70 0.0508 0.0503 0.0041 0.0061 0.1124 0.0284
1.75 0.0519 0.0513 0.0036 0.0057 0.1136 0.0268
1.80 0.0529 0.0524 0.0032 0.0053 0.1149 0.0253
1.85 0.0538 0.0535 0.0028 0.0050 0.1163 0.0239
1.90 0.0547 0.0543 0.0025 0.0047 0.1176 0.0227
1.95 0.0554 0.0552 0.0022 0.0044 0.1190 0.0216
2.00 0.0562 0.0559 0.0019 0.0042 0.1190 0.0205
M2
l 2
M 2
l1
M1
M1
Constructii din beton armat Dimensionare planseu
An IV CCIA
5/8
unde,
1
2
l
l=
Momentele pozitive in camp la centru:
2
2522
2
1511
qlM
qlM
=
=
Momentele maxime negative pe reazeme:
2153
'
1 qlM =
2254'
2 qlM =
Sageata la centru:
3
4
150Eh
qlw =
Daca latura simplu rezemata este paralela cu 1l , sunt
valabile titlurile inferioare cu:
1
2
1' ==l
l
Placa
tip 5
50 51 52 53 54
0.50 0.0019 0.0026 0.0408 0.0143 0.0833
0.55 0.0028 0.0034 0.0400 0.0173 0.0829
0.60 0.0038 0.0044 0.0388 0.0206 0.0820
0.65 0.0051 0.0055 0.0369 0.0241 0.0806
0.70 0.0065 0.0067 0.0351 0.0280 0.0794
0.75 0.0082 0.0080 0.0330 0.0321 0.0769
0.80 0.0100 0.0095 0.0310 0.0366 0.0741
0.85 0.0121 0.0111 0.0289 0.0412 0.0714
0.90 0.0142 0.0132 0.0267 0.0458 0.0680
0.95 0.0165 0.0157 0.0248 0.0504 0.0654
1.00 0.0188 0.0168 0.0227 0.0546 0.0617
1.10 0.0236 0.0217 0.0189 0.0649 0.0558
1.20 0.0284 0.0267 0.0155 0.0741 0.0500
1.30 0.0329 0.0314 0.0126 0.0820 0.0445
1.40 0.0371 0.0357 0.0101 0.0893 0.0392
1.50 0.0409 0.0397 0.0081 0.0943 0.0350
1.60 0.0442 0.0429 0.0063 0.0990 0.0310
1.70 0.0471 0.0461 0.0049 0.1031 0.0277
1.80 0.0495 0.0488 0.0039 0.1064 0.0249
1.90 0.0518 0.0513 0.0031 0.1111 0.0225
2.00 0.0539 0.0535 0.0025 0.1136 0.0203
' 50 52 51 54 53
l 2
M2
M 2
M
2
l1
M1
M1
l 2
M2
M 2
l1
M1
M1
M1
Constructii din beton armat Dimensionare planseu
An IV CCIA
6/8
unde,
1
2
l
l=
Momentele pozitive in camp la centru:
2
2622
2
1611
qlM
qlM
=
=
Momentele pozitive in camp maxime:
2263
max
2 qlM =
Sageata la centru:
3
4
160Eh
qlw =
Momentele maxime negative pe reazeme:
2164'
1 qlM =
2265'
2 qlM =
Placa
tip 6
60 61 62 63 64 65
1.00 0.0152 0.0176 0.0176 0.0176 0.0515 0.0515
1.05 0.0167 0.0198 0.0155 0.0156 0.0549 0.0482
1.10 0.0181 0.0217 0.0137 0.0137 0.0585 0.0449
1.15 0.0195 0.0236 0.0120 0.0121 0.0613 0.0418
1.20 0.0207 0.0254 0.0105 0.0106 0.0645 0.0388
1.25 0.0219 0.0270 0.0092 0.0092 0.0671 0.0362
1.30 0.0230 0.0287 0.0080 0.0080 0.0690 0.0336
1.35 0.0240 0.0300 0.0070 0.0070 0.0714 0.0314
1.40 0.0248 0.0313 0.0060 0.0061 0.0730 0.0292
1.45 0.0257 0.0327 0.0052 0.0053 0.0746 0.0272
1.50 0.0264 0.0338 0.0044 0.0048 0.0758 0.0254
1.55 0.0271 0.0347 0.0038 0.0043 0.0769 0.0238
1.60 0.0277 0.0356 0.0033 0.0040 0.0781 0.0223
1.65 0.0282 0.0364 0.0028 0.0037 0.0787 0.0210
1.70 0.0287 0.0372 0.0024 0.0034 0.0800 0.0198
1.75 0.0291 0.0379 0.0020 0.0032 0.0806 0.0187
1.80 0.0294 0.0385 0.0017 0.0030 0.0813 0.0176
1.85 0.0297 0.0389 0.0015 0.0028 0.0820 0.0167
1.90 0.0300 0.0394 0.0013 0.0026 0.0826 0.0158
1.95 0.0302 0.0397 0.0011 0.0025 0.0833 0.0150
2.00 0.0304 0.0400 0.0009 0.0024 0.0833 0.0143
l1
M1
M1
M1
l 2
M2
M 2
M
2
Constructii din beton armat Dimensionare planseu
An IV CCIA
7/8
B. Placi continue rezemate pe tot conturul
In alcatuirea planseelor, placile se prezinta rareori sub forma de panouri izolate. In general,
placile sunt continue pe doua directii.
Incarcarea aplicata distribuit pe suprafata planseului are o componenta permanenta si alta
temporara. Incarcarea permanenta actioneaza dupa o singura schema si anume aceea cu distributie
uniform pe toata suprafata planseului. Incarcarea temporara poate ocupa orice pozitie pe suprafata.
Incarcarea temporara actioneaza de fiecare data pe panouri intregi si este dispusa in pozitiile
care determina momentele incovoietoare maxime in campurile sau reazemele considerate.
Incarcarile permanente la nivelul etajului curent, notate cu g, provin din greutatea proprie
placa, pardoseala si stratul suport, pereti despartitori care reazema direct pe placa, tencuiala si finisaje
pe plafon.
Incarcarile temporare la nivelul etajului curent, notate cu p sunt constituite preponderent din
incarcarile utile.
B.1 Calculul momentelor maxime in campuri
In practica se foloseste un procedeu simplu, care, pe baza unor artificii de calcul permite
utilizarea coeficientilor de la panourile izolate de placa si pentru determinarea momentelor de calcul la
placile continue.
La placile continue, rezemate pe tot conturul, solicitate de incarcari permanente si temporare,
aplicate uniform distribuit, la care pe fiecare directie deschiderile sunt egale sau difera intre ele cu cel
mult 30%, se admite ca momentele maxime si momentele minime din campuri sa se determine pe baza
a doua scheme conventionale de rezemare.
Schema conventionala nr.1
In prima schema conventionala de rezemare a placilor, panourile se considera incastrate perfect
pe reazemele intermediare si simplu rezemate pe conturul planseului. Pe suprafetele tuturor panourilor
acestei scheme se aplica o incarcare conventionala, gravitationala, cu intensitatea 2
' pgq += si vor
rezulta valorile momentelor incovoietoare pe cele doua directii.
2111_
12
lp
gM ischema
+= 222
1_
22
lp
gM ischema
+=
Schema conventionala nr.2
In a doua schema conventionala de rezemare a placilor, se considera simplu rezemare pe tot
conturul acestora. Pe suprafetele tuturor placilor se aplica o incarcare conventionala dirijata alternant
de sus in jos si respectiv de jos in sus a carei intensitate pe unitatea de suprafata este data de relatia
2
" pq = , rezultand in acest fel, valorile momentelor incovoietoare pe cele doua directii.
21112_
12
lp
M schema
= 2212
2_
22
lp
M schema
=
Constructii din beton armat Dimensionare planseu
An IV CCIA
8/8
Momentele maxime si respectiv minime in campuri se obtin prin adunarea momentelor
incovoietoare produse de incarcarea q pe prima schema conventionala de rezemare, cu momentele
incovoietoare produse de incarcarea q pe a doua schema conventionala de rezemare. In final, valorile
momentelor incovoietoare pe cele doua directii sunt:
2_11_
11
schemaschema MMM += si 2_21_
22
schemaschema MMM +=
B.2 Calculul momentelor maxime pe reazeme
La placile continue, rezemate pe tot conturul, solicitate de incarcari permanente si temporare,
aplicate uniform distribuit, la care pe fiecare directie deschiderile sunt egale sau difera intre ele cu cel
mult 30%, se admite ca momentul maxim(in valoare absoluta) pe fiecare reazem intermediar sa se
determine prin considerarea a doua scheme conventionale de rezemare pentru cele doua panouri
adiacente reazemului respectiv.
Schema conventionala nr.1
In prima schema conventionala de rezemare a placilor, panourile se considera incastrate perfect
pe reazemele intermediare si simplu rezemate pe conturul planseului. Pe suprafetele tuturor panourilor
acestei scheme se aplica o incarcare conventionala, gravitationala, cu intensitatea 2
' pgq += si vor
rezulta valorile momentelor incovoietoare pe cele doua directii.
21
1_ '
12
lp
gM corespschema
+= 22
1_ '
22
lp
gM corespschema
+=
Schema conventionala nr.2
In a doua schema conventionala de rezemare, panourile se considera incastrate perfect pe
reazemul comun(cel ce este supus calculului) si simplu rezemate pe toate celelalte laturi. Pe
suprafetele ambelor panouri se aplica o incarcare dirijata de sus in jos, a carei intensitate pe unitatea de
suprafata este data de relatia 2
" pq = rezultand in acest fel valorile momentului incovoietor.
212_ '
12
lp
M corespschema
= sau 22
2_ '
22
lp
M corespschema
=
Momentul maxim(in valoare absoluta) pe reazemul considerat se va calcula prin insumarea
valorilor obtinute pe reazemul respectiv in cele doua scheme conventionale.
Pe fiecare schema in parte, momentele incovoietoare de o parte si de cealalta a reazemului pot
fi diferite. Pentru calcul, se va considera valoarea medie.