Presas de Gravedad

26
(67¨7,&$ ’( )/8,’26 (V HO HVWXGLR GHO FRPSRUWDPLHQWR GH ORV IOXLGRV \ GH OD GLVWULEXFLyQ GH HVIXHU]RV VH HVWXGLDUi HQ HVWH FDStWXOR 5HFRUGDQGR GH ODV SURSLHGDGHV GH ORV IOXLGRV pVWRV QR UHVLVWHQ HVIXHU]RV GH FRUWDQWHV WDQJHQFLDOHV HQ ORV IOXLGRV HQ UHSRVR VyOR VH SUHVHQWDQ HVIXHU]RV GH FRPSUHVLyQ 35(6,ˇ1 \ VH WUDQVPLWHQ D ORV OtPLWHV VyOLGRV D WUDYpV GH VHFFLRQHV SHUSHQGLFXODUHV )LJXUD (VIXHU]RV GH FRPSUHVLyQ HQ IOXLGRV HQ UHSRVR (VFDODV GH PHGLFLyQ GH OD SUHVLyQ /DV PHGLGDV GH SUHVLyQ VH SXHGHQ GDU HQ WpUPLQRV DEVROXWRV \ WpUPLQRV PDQRPpWULFRV \ HVWDV GLILHUHQ H[FOXVLYDPHQWH GH OD UHIHUHQFLD TXH FRQVLGHUDQ /DV PHGLGDV GH SUHVLyQ HQ HVFDOD DEVROXWD HVWiQ UHIHULGDV DO FHUR DEVROXWR YDFtR WRWDO R PtQLPD SUHVLyQ SRVLEOH HQ HVWD HVFDOD QR SXHGH SUHVHQWDUVH YDORUHV QHJDWLYRV SXHVWR TXH LPSOLFDUtD YDORUHV SRU GHEDMR GHO YDFtR WRWDO ItVLFDPHQWH LPSRVLEOH /D HVFDOD GH PHGLFLyQ PDQRPpWULFD R UHODWLYD UHJLVWUD ORV YDORUHV UHIHULGRV D OD SUHVLyQ DWPRVIpULFD ORFDO SUHVLyQ GHELGD D OD DWPyVIHUD HV GHFLU HQ HO VLWLR GH LQWHUpV /D SUHVLyQ DWPRVIpULFD HVWiQGDU R QRUPDO HV OD UHJLVWUDGD DO QLYHO GHO PDU \D D XQD ODWLWXG GH WLHQH XQ YDORU GH N3D PP+J DWP 7DPELpQ GHQRPLQDGD SUHVLyQ EDURPpWULFD

Transcript of Presas de Gravedad

Page 1: Presas de Gravedad

(67È7,&$�'(�)/8,'26��(V�HO�HVWXGLR�GHO�FRPSRUWDPLHQWR�GH�ORV�IOXLGRV�\�GH�OD�GLVWULEXFLyQ�GH�HVIXHU]RV�VH�HVWXGLDUi�HQ�HVWH�FDStWXOR���5HFRUGDQGR� GH� ODV� SURSLHGDGHV� GH� ORV� IOXLGRV�� pVWRV�QR� UHVLVWHQ�HVIXHU]RV� GH�FRUWDQWHV� �WDQJHQFLDOHV���HQ� ORV� IOXLGRV�HQ� UHSRVR�VyOR� VH�SUHVHQWDQ�HVIXHU]RV�GH� FRPSUHVLyQ� �35(6,Ï1�� \� VH� WUDQVPLWHQ� D� ORV� OtPLWHV� VyOLGRV� D� WUDYpV� GH�VHFFLRQHV�SHUSHQGLFXODUHV��

�)LJXUD��� (VIXHU]RV�GH�FRPSUHVLyQ�HQ�IOXLGRV�HQ�UHSRVR�

��(VFDODV�GH�PHGLFLyQ�GH�OD�SUHVLyQ��/DV� PHGLGDV� GH� SUHVLyQ� VH� SXHGHQ� GDU� HQ� WpUPLQRV� DEVROXWRV� \� WpUPLQRV�PDQRPpWULFRV�\�HVWDV�GLILHUHQ�H[FOXVLYDPHQWH�GH�OD�UHIHUHQFLD�TXH�FRQVLGHUDQ���/DV� PHGLGDV� GH� SUHVLyQ� HQ� HVFDOD� DEVROXWD� HVWiQ� UHIHULGDV� DO� FHUR� DEVROXWR��YDFtR� WRWDO�� R� PtQLPD� SUHVLyQ� SRVLEOH�� HQ� HVWD� HVFDOD� QR� SXHGH� SUHVHQWDUVH�YDORUHV� QHJDWLYRV�� SXHVWR� TXH� LPSOLFDUtD� YDORUHV� SRU� GHEDMR� GHO� YDFtR� WRWDO��ItVLFDPHQWH�LPSRVLEOH���/D�HVFDOD�GH�PHGLFLyQ�PDQRPpWULFD�R�UHODWLYD�UHJLVWUD�ORV�YDORUHV�UHIHULGRV�D�OD�SUHVLyQ�DWPRVIpULFD�� ORFDO� �SUHVLyQ�GHELGD�D� OD�DWPyVIHUD���HV�GHFLU�HQ�HO�VLWLR�GH� LQWHUpV�� �/D�SUHVLyQ�DWPRVIpULFD�HVWiQGDU�R�QRUPDO�HV� OD�UHJLVWUDGD�DO�QLYHO�GHO�PDU� \D� D� XQD� ODWLWXG� GH� ����� WLHQH� XQ�YDORU�GH������� N3D������PP+J����DWP���

������������������������������������������7DPELpQ�GHQRPLQDGD�SUHVLyQ�EDURPpWULFD�

Page 2: Presas de Gravedad

�)LJXUD��� (VFDODV�GH�PHGLFLyQ�GH�OD�SUHVLyQ��

�3UHVLyQ�HQ�XQ�SXQWR��&RQVLGHUHPRV� XQ� HOHPHQWR� GLIHUHQFLDO� GH� IOXLGR� HQ� UHSRVR�� ODV� IXHU]DV�DFWXDQWHV�VHUiQ��

��

)LJXUD��� )XHU]DV�GH�SUHVLyQ�HQ�XQ�HOHPHQWR�IOXLGR�GLIHUHQFLDO��

$O�HVWDU�HO�HOHPHQWR�HQ�UHSRVR�QR�H[LVWH�DFHOHUDFLyQ�\�SRU�WDQWR�OD�DSOLFDFLyQ�GH�OD�VHJXQGD� OH\�GH�1HZWRQ� � F ma=∑

JG G

� LQGLFD�TXH� OD�VXPDWRULD�GH�IXHU]DV�VHUi�

LJXDO�D�FHUR�

1 3 1 3 1 3

2 3 2 3 2 3

* * 0

* * 02 2

x x

z z

F dF dF dF dF sen p dz p ds sen

dxdz dxdzF dF dF dW dF dF cos p dx p ds cos

θ θ

θ γ θ γ

= − = − = − =

= − − = − − = − − =

∑∑

����

Page 3: Presas de Gravedad

�$O�FRQVLGHUDU�OD�JHRPHWUtD�GHO�WULiQJXOR�VH�WLHQH�TXH��

*

*

dzsen dz ds sen

dsdx

cos dx ds cosds

θ θ

θ θ

= ⇒ =

= ⇒ =�� � � � � � � ����

�&RPELQDQGR�HO�FRQMXQWR�GH�HFXDFLRQHV�����\�����VH�WLHQH���

1 *p ds senθ 3 *p ds senθ− 1 3

2

0

*

p p

p ds cosθ

= ⇒ =

3 *p ds cosθ− *ds cosθγ− 2 3

2 3 1 3 1 2 3

*0

2 2En un punto 0

dz dzp p

dz p p p p p p p

γ= ⇒ = +

≈ ⇒ = ∴ = ⇒ = =

�(VWR�VLJQLILFD�TXH�OD�SUHVLyQ�HQ�XQ�SXQWR�HV�OD�PLVPD�SRU�WRGDV�SDUWHV��HV�GHFLU�TXH�OD�SUHVLyQ�QR�WLHQH�VHQWLGR�YHFWRULDO�SRU�OR�WDQWR�HV�XQD�FDQWLGDG�HVFDODU�\�HV� XQD� SURSLHGDG� GHO� SXQWR� IOXLGR�� TXH� GHSHQGH� H[FOXVLYDPHQWH� GH� VXV�FRRUGHQDGDV���(O�SULQFLSLR�GH�3DVFDO��/D� SUHVLyQ� DSOLFDGD� HQ� XQ� SXQWR� GH� XQ� OtTXLGR� FRQWHQLGR� HQ� XQ� UHFLSLHQWH� VH�WUDQVPLWH� FRQ� HO� PLVPR� YDORU� D� FDGD� XQD� GH� ODV� SDUWHV� GHO� PLVPR�� � (VWH�HQXQFLDGR�� REWHQLGR� D� SDUWLU� GH� REVHUYDFLRQHV� \� H[SHULPHQWRV� SRU� HO� ItVLFR� \�PDWHPiWLFR� IUDQFpV� %ODV� 3DVFDO� ������������� VH� FRQRFH� FRPR� SULQFLSLR� GH�3DVFDO���/D�SUHQVD�KLGUiXOLFD�FRQVWLWX\H�OD�DSOLFDFLyQ�IXQGDPHQWDO�GHO�SULQFLSLR�GH�3DVFDO�\� WDPELpQ�XQ�GLVSRVLWLYR�TXH�SHUPLWH�HQWHQGHU�PHMRU�VX�VLJQLILFDGR��&RQVLVWH��HQ�HVHQFLD��HQ�GRV�FLOLQGURV�GH�GLIHUHQWH�VHFFLyQ�FRPXQLFDGRV�HQWUH�Vt��\�FX\R�LQWHULRU�HVWi� FRPSOHWDPHQWH� OOHQR� GH�XQ� OtTXLGR�TXH�SXHGH�VHU�DJXD�R�DFHLWH��'RV�pPERORV�GH�VHFFLRQHV�GLIHUHQWHV�VH�DMXVWDQ��UHVSHFWLYDPHQWH��HQ�FDGD�XQR�GH� ORV� GRV� FLOLQGURV�� GH�PRGR� TXH� HVWpQ� HQ� FRQWDFWR� FRQ� HO� OtTXLGR�� &XDQGR�VREUH�HO�pPEROR�GH�PHQRU�VHFFLyQ� 1A �VH�HMHUFH�XQD�IXHU]D� 1F �OD�SUHVLyQ� 1p �TXH�VH�RULJLQD�HQ�HO�OtTXLGR�HQ�FRQWDFWR�FRQ�pO�VH�WUDQVPLWH�tQWHJUDPHQWH�\�GH�IRUPD�LQVWDQWiQHD�D�WRGR�HO�UHVWR�GHO�OtTXLGR��SRU�WDQWR��VHUi�LJXDO�D�OD�SUHVLyQ� 2p �TXH�HMHUFH�HO�OtTXLGR�VREUH�HO�pPEROR�GH�PD\RU�VHFFLyQ� 2A ��HV�GHFLU���

Page 4: Presas de Gravedad

�)LJXUD��� 3UHQVD�+LGUiXOLFD���$SOLFDFLyQ�GHO�SULQFLSLR�GH�3DVFDO��

�1 2 2

1 2 2 11 2 1

F F Ap p F F

A A A= ⇒ = ⇒ = �

�6L�OD�VHFFLyQ� 2A �HV�YHLQWH�YHFHV�PD\RU�TXH�OD� 1A ��OD�IXHU]D� 1F �DSOLFDGD�VREUH�HO�pPEROR�SHTXHxR�VH�YH�PXOWLSOLFDGD�SRU�YHLQWH�HQ�HO�pPEROR�JUDQGH� 2F ���/D� SUHQVD� KLGUiXOLFD� HV� XQD� PiTXLQD� VLPSOH� VHPHMDQWH� D� OD� SDODQFD� GH�$UTXtPHGHV��TXH�SHUPLWH�DPSOLILFDU�OD�LQWHQVLGDG�GH�ODV�IXHU]DV�\�FRQVWLWX\H�HO�IXQGDPHQWR� GH� HOHYDGRUHV�� SUHQVDV�� IUHQRV� \� PXFKRV� RWURV� GLVSRVLWLYRV�KLGUiXOLFRV�GH�PDTXLQDULD�LQGXVWULDO���9$5,$&,Ï1�'(�/$�35(6,Ï1�81�)/8,'2�(1�5(3262��&RQVLGpUHVH� XQ� HOHPHQWR� GLIHUHQFLDO� GH� IOXLGR� GH� GLPHQVLRQHV�G[�� G\� \� G]�� \�SHVR�HVSHFtILFR�γ���6XSRQLHQGR�FRQRFLGD�OD�SUHVLyQ�S�HQ�HO�FHQWUR�GHO�IOXLGR��OD�SUHVLyQ�HQ�ORV�OtPLWHV�GHO�HOHPHQWR�GH�IOXLGR�VH�SXHGH�H[SUHVDU�HQ�WpUPLQRV�GH�OD�SUHVLyQ�S�\�GH�OD�YDULDFLyQ�GH�OD�SUHVLyQ�HQ�OD�GLUHFFLyQ�LQGLFDGD���(O�VLVWHPD�GH� IXHU]DV� DO� TXH� HVWi� VRPHWLGR� HO� HOHPHQWR� GLIHUHQFLDO� VH� REWHQGUi�PXOWLSOLFDQGR� OD� SUHVLyQ� HQ� ORV� OtPLWHV� GHO� HOHPHQWR� GHO� IOXLGR� SRU� HO� iUHD� GH�LQWHUDFFLyQ��FRPR�VH�tQGLFD�HQ�OD�)LJXUD�����

Page 5: Presas de Gravedad

�)LJXUD��� )XHU]DV�HQ�XQ�HOHPHQWR�GLIHUHQFLDO�GH�IOXLGR�HQ�UHSRVR��

�$SOLFDQGR� OD� VHJXQGD� OH\�GH�1HZWRQ� F ma=∑

JG G

�\�FRQVLGHUDQGR�TXH�XQ�FXHUSR�

HQ�UHSRVR�QR�WLHQH�DFHOHUDFLyQ�VH�WLHQH�TXH���(TXLOLEULR�GH�IXHU]DV�HQ�GLUHFFLyQ�\��KRUL]RQWDO���

0y yF ma

p

= =∑* *

2

p dydxdz

y

∂− ∂ p− * *

2

p dydxdz

y

∂+ ∂ 0

* 0 0p p

dyy y

=

∂ ∂− = ⇒ =∂ ∂

8Q�DQiOLVLV�VLPLODU�HQ�GLUHFFLyQ�[�SURGXFH� 0p

x

∂ =∂

�(O�UHVXOWDGR�DQWHULRU�VLJQLILFD�ItVLFDPHQWH�TXH�QR�KD\�YDULDFLyQ�GH�OD�SUHVLyQ�HQ�XQ�SODQR�KRUL]RQWDO���(V�GHFLU�OD�SUHVLyQ�HV�OD�PLVPD�HQ�XQD�KRUL]RQWDO���(TXLOLEULR�GH�IXHU]DV�HQ�GLUHFFLyQ�]��YHUWLFDO���

0z zF ma

p

= =∑* *

2

p dzdxdy

z

∂ − ∂ p− * *

2

p dzdxdy

z

∂ + ∂ dxdyγ− 0

*

dz

pdz

z

=

∂−∂

dzγ− 0p

zγ∂= ⇒ = −

Page 6: Presas de Gravedad

�$QWHULRUPHQWH� VH�GHPRVWUy�TXH� OD�SUHVLyQ�QR�YDUtD�HQ� XQD�KRUL]RQWDO��SRU� WDO�UD]yQ�OD�GHULYDGD�SDUFLDO�HQ�OD�YHUWLFDO�VH�FRQYLHUWH�HQ�XQD�GHULYDGD�WRWDO���

p dp

z dzγ γ∂ = − ⇒ = −

∂�

�/D�H[SUHVLyQ�DQWHULRU�HV�OD�HFXDFLyQ�JHQHUDO�GH�OD�KLGURVWiWLFD�H�LQGLFD�TXH�OD�YDULDFLyQ�GH�OD�SUHVLyQ��JUDGLHQWH�GH�SUHVLyQ��HV�SURSRUFLRQDO�D�OD�DOWXUD�\�HV�YiOLGD� SDUD� IOXLGRV� LQFRPSUHVLEOHV� �OtTXLGRV� \� DOJXQRV� JDVHV�� \� FRPSUHVLEOHV��JDVHV����8Q�GHVDUUROOR�DGLFLRQDO�VH�UHDOL]D�HQ� ORV�IOXLGRV� LQFRPSUHVLEOHV�FXDQGR�HO�SHVR�HVSHFtILFR� HV� FRQVWDQWH� γγγγ FWH� VH� SXHGH� LQWHJUDU� OD� HFXDFLyQ� JHQHUDO� GH� OD�KLGURVWiWLFD�HQWUH�GRV�SXQWRV�GH�FRWD�FRQRFLGD��DVt�HO�FDPELR�GH�SUHVLyQ�HVWDUi�UHODFLRQDGR�FRQ�HO�FDPELR�GH�DOWXUD�DVt��

( ) ( )2 2

1 1

2 1 2 1 2 1 1 2

si p z

p z

cte

dp dz dp dz p p z z p p z z

pp z p h h

γ

γ γ γ γ

γ γγ

=

= − ⇒ = − ⇒ − = − − ⇒ − = −

∆∆ = ∆ ⇒ ∆ = ⇒ =

∫ ∫

6L� HO� IOXLGR� HV� FRPSUHVLEOH�� HV� GHFLU� FXDQGR� HO� SHVR� HVSHFtILFR� HV� YDULDEOH�γγγγ YEOH�VH�GHEH�FRQRFHU�DOJXQD�H[SUHVLyQ�TXH�UHODFLRQH�HO�SHVR�HVSHFtILFR�FRQ�OD�SUHVLyQ��SDUD�DVt�LQWHJUDU�OD�HFXDFLyQ�JHQHUDO�GH�OD�KLGURVWiWLFD���/D�3DUDGRMD�GH�3DVFDO�

�)LJXUD��� 3DUDGRMD�GH�3DVFDO��

�&RPR� VH� GHPRVWUy� FRQ� DQWHULRULGDG� VL� γ� HV� FRQVWDQWH� OD� YDULDFLyQ� GH� SUHVLyQ�VROR�GHSHQGH�GH�OD�DOWXUD�R�FROXPQD�GH�IOXLGR�GLVSRQLEOH��DVt�HQ�OD�)LJXUD���VH�WLHQH� ODV� SUHVLRQHV� WUDQVPLWLGDV� SRU� OD� PLVPD� FROXPQD� GH� IOXLGR� SHUR� HQ�UHFLSLHQWHV�GH�IRUPD�GLIHUHQWH�HV�OD�PLVPD��3DUDGRMD�GH�3DVFDO����

1 2 3 4 4 5 6p p p p p p p hγ= = = = = = = ��

Page 7: Presas de Gravedad

3ULQFLSLR�GH�9DVRV�&RPXQLFDQWHV��6L�VH�WLHQHQ�GRV�UHFLSLHQWHV�FRPXQLFDGRV�\�VH�YLHUWH�XQ�OtTXLGR�HQ�XQR�GH�HOORV��pVWH� VH�GLVWULEXLUi�HQWUH�DPERV�GH� WDO�PRGR� TXH�� LQGHSHQGLHQWHPHQWH�GH� VXV�FDSDFLGDGHV��HO�QLYHO�GH�OtTXLGR�HQ�XQR�\�RWUR�UHFLSLHQWH�VHD�HO�PLVPR��eVWH�HV�HO�OODPDGR�SULQFLSLR�GH�ORV�YDVRV�FRPXQLFDQWHV��TXH�HV�XQD�FRQVHFXHQFLD�GH�OD�HFXDFLyQ�IXQGDPHQWDO�GH�OD�KLGURVWiWLFD��

�)LJXUD��� 3ULQFLSLR�GH�YDVRV�FRPXQLFDQWHV��

�0$1Ï0(7526��/RV�PDQyPHWURV� VRQ�GLVSRVLWLYRV�TXH�VH�XWLOL]DQ� IOXLGRV� LQFRPSUHVLEOHV��γ�FWH��SDUD�GHWHUPLQDU�OD�SUHVLyQ�TXH�DFW~D�VREUH�XQ�IOXLGR�\�VH�EDVDQ�HQ���

• 3DUD�XQ�PLVPR�IOXLGR�OD�SUHVLyQ�QR�YDUtD�HQ�XQD�KRUL]RQWDO��• /D�GLVWULEXFLyQ�KLGURVWiWLFD�GH�SUHVLyQ�• /D� SURSLHGDG� GH� XQ� IOXLGR� SDUD� WUDQVPLWLU� SUHVLRQHV� SXQWR� D� SXQWR� VLQ�

DOWHUDUOD��

�)LJXUD��� 0DQyPHWUR�HQ�8�

Page 8: Presas de Gravedad

3DUD� GHWHUPLQDU� OD� SUHVLyQ� HQ� $� VH� XWLOL]DQ� ORV� SULQFLSLRV� GHVFULWRV� HQ� HO�DSDUWDGR�DQWHULRU��DVt��

• ,JXDOGDG�GH�SUHVLRQHV�HQ�XQD�KRUL]RQWDOÎ� B Cp p= �• 'LVWULEXFLyQ�KLGURVWiWLFD�GH�SUHVLRQHV� 1 2*B A C atmp H p p h pγ γ= + ∴ = + �

�/RV� PDQyPHWURV� UHJLVWUDQ� OD� SUHVLyQ� PDQRPpWULFD�� HV� GHFLU� VRODPHQWH� OD�UHODWLYD�DO� IHQyPHQR�GH� LQWHUpV�� �6L� ORV�PDQyPHWURV�HVWiQ�HQ�FRQWDFWR�FRQ� OD�DWPyVIHUD�OD�SUHVLyQ�DWPRVIpULFD�VH�KDFH�FHUR���$O� LJXDODU� ODV� H[SUHVLRQHV� DQWHULRUHV�� FRQVLGHUDQGR� TXH� SDWP �� \� GHVSHMDQGR�SDUD�S$�VH�WLHQH���

2 1 *Ap h Hγ γ= − ��$Vt� OD�SUHVLyQ� VH�SXHGH�PHGLU� D�SDUWLU� GH�DOWXUDV�GH� IOXLGRV� FRQRFLGR�VX�SHVR�HVSHFtILFR�� � �6H� UHJLVWUDQ�GLIHUHQWHV� WLSRV�GH� IOXLGRV�SDUD�UHJLVWUDU� OD�YDULDFLyQ�GH�SUHVLyQ�VHJ~Q�VHD�OD�PDJQLWXG�GH�OD�SUHVLyQ�GH�LQWHUpV���

• 0HUFXULR�Î�SUHVLRQHV�DOWDV��• $JXDÎ�SUHVLRQHV�PHGLDV���EDMDV�

�3DUD� UHJLVWUDU� GLIHUHQFLDV� GH� SUHVLyQ� EDMDV� VH� XWLOL]D� HO� PDQyPHWUR� HQ� 8�LQYHUWLGR��FRQ�XQ�OtTXLGR�GH�GHQVLGDG�SHTXHxD���(V�LPSRUWDQWH�WHQHU�HQ�FXHQWD�DOJXQDV�FRQVLGHUDFLRQHV�DGLFLRQDOHV��

• /RV� OtTXLGRV� PDQRPpWULFRV� HVWiQ� LQIOXHQFLDGRV� SRU� FDPELRV� HQ� OD�WHPSHUDWXUD��FDPELRV�GH�GHQVLGDG��\�SRU�WDQWR�SXHGHQ�LQGXFLU�HUURUHV��

• 6H�GLVPLQX\H�ORV�HIHFWRV�GH�FDSLODULGDG�DO�XWLOL]DU�WXERV�GH�LJXDO�GLiPHWUR��VH�FDQFHODQ�ORV�HIHFWRV��

�0DQyPHWUR�GH�WXER�LQFOLQDGR��&RQVLVWH� EiVLFDPHQWH� HQ� XQ�PDQyPHWUR� FRQ� HO� WXER� LQFOLQDGR� �)LJXUD� ��� FRQ�iQJXOR� FRQRFLGR�� OD� HVFDOD� GH� PHGLFLyQ� VH� ORFDOL]D� VREUH� HO� WXER� LQFOLQDGR��SHUPLWLHQGR� XQD� OHFWXUD� PD\RU� �/�� TXH� OD� OHFWXUD� YHUWLFDO� �+��� FRQ� OD� TXH�UHDOPHQWH� VH� REWLHQH� HO� FDPELR� GH� SUHVLyQ�� � /� \� +� HVWiQ� UHODFLRQDGRV� SRU�WULJRQRPHWUtD��

pero * *A Ap H H L sen p L senγ θ γ θ= = ∴ = ��0HGLGRUHV�GH�SUHVLyQ�PHFiQLFD�WLSR�%RXUGRQ��(O� PDQyPHWUR� GH� %RXUGRQ� GHSHQGH�� SUHFLVDPHQWH�� GH� OD� HODVWLFLGDG� GH� ORV�PDWHULDOHV�XWLOL]DGRV�HQ�VX�FRQVWUXFFLyQ��(VWH�PDQyPHWUR��WDO�YH]�HO�PiV�FRP~Q�HQ� SODQWDV� GH� SURFHVRV� TXH� UHTXLHUDQ�PHGLFLyQ� GH� SUHVLRQHV�� FRQVLVWH� GH� XQ�

Page 9: Presas de Gravedad

WXER�PHWiOLFR�DFKDWDGR�\�FXUYDGR�HQ�IRUPD�GH��&���DELHUWR�VyOR�HQ�XQ�H[WUHPR���)LJXUD������

�)LJXUD��� 0DQyPHWUR�GH�WXER�LQFOLQDGR�

�$O� DSOLFDU� XQD� SUHVLyQ� DO� LQWHULRU� GHO� WXER� �VH� OH� LQIOD�� SRU� HMHPSOR�� OD� IXHU]D�JHQHUDGD� HQ� OD� VXSHUILFLH� �iUHD�� H[WHULRU� GH� OD� �&�� HV� PD\RU� TXH� OD� IXHU]D�JHQHUDGD�HQ�OD�VXSHUILFLH�LQWHULRU��GH�PRGR�TXH�VH�JHQHUD�XQD�IXHU]D�QHWD�TXH�GHIRUPD�OD��&��KDFLD�XQD��&��PiV�DELHUWD��(VWD�GHIRUPDFLyQ�HV�XQD�PHGLFLyQ�GH�OD� SUHVLyQ� DSOLFDGD� \� SXHGH� WUDVODGDUVH� D� XQD� HVFDOD� LQGLFDGRUD� SUHYLDPHQWH�FDOLEUDGD� WDQWR� FRPR� D� XQ� VLVWHPD� GH� YDULDFLyQ� GH� UHVLVWHQFLD� R� FDPSRV�HOpFWULFRV�R�PDJQpWLFRV��

�)LJXUD���� 0DQyPHWUR�GH�%RUGRQ�

Page 10: Presas de Gravedad

��/RV�PDQyPHWURV� GH�%RXUGRQ� UHJLVWUDQ� OD� SUHVLyQ�PDQRPpWULFD� R� UHODWLYD� \� OD�HVFDOD�HVWi�FDOLEUDGD�SDUD� UHJLVWUDU�XQD�SUHVLyQ�QXOD�FXDQGR�HVWi�HQ�FRQWDFWR�FRQ� OD� DWPyVIHUD�� � 6LQ� HPEDUJR� XQ� PHFDQLVPR� VLPLODU� WDPELpQ� SXHGHQ� VHU�XWLOL]DGRV� SDUD� UHJLVWUDU� OD� SUHVLyQ� DWPRVIpULFD�� DO� HVWDU� FRQHFWDGR� GLDIUDJPD�HOiVWLFR�� VL� DO� LQWHULRU� GHO� GLDIUDJPD� VH�KD� JHQHUDGR�XQ�YDFtR� FHUFDQR� DO� FHUR�DEVROXWR��ODV�SUHVLRQHV�LPSXHVWDV�VREUH�HO�H[WHULRU�GHO�GLDIUDJPD�KDUiQ�TXH�VH�GHIRUPH�\�VHD�WUDQVPLWLGR�D�OD�HVFDOD�PHGLGRUD����%DUyPHWUR�GH�PHUFXULR��7RUULFHOOL���������6H�FRQVWUX\H� OOHQDQGR�XQ� WXER�FRQ�PHUFXULR�OLEUH�GH�DLUH�H�LQYLUWLpQGROR�VREUH�XQ�UHFLSLHQWH�OOHQR�GH�PHUFXULR���'HVSUHFLDQGR�OD�SHTXHxD�SUHVLyQ�GH�YDSRU�TXH�VH�JHQHUD�SRU�HO�YDFtR�GHO�PHUFXULR�OD�SUHVLyQ�DWPRVIpULFD�VHUi atmp Hγ= �

�)LJXUD���� %DUyPHWUR�GH�0HUFXULR�GH�7RUULFHOOL�

Page 11: Presas de Gravedad

�7UDQVGXFWRUHV�GH�SUHVLyQ��6RQ� LQVWUXPHQWRV� TXH� PLGH� XQD� VHxDO� ItVLFD� �SUHVLyQ�� \� WUDQVPLWH� XQD� VHxDO�HOpFWULFD�� � (VWD� VHxDO� VH� SXHGH� UHJLVWUDU� \� DOPDFHQDU� SRU� PHGLR� GH� XQ�FRPSXWDGRU���

���)8(5=$6�62%5(�683(5),&,(6�3/$1$6�680(5*,'$6��3RUTXH�VH�UHTXLHUH�FDOFXODU�ODV�IXHU]DV"�&RPR�SDUWH�GHO�GLVHxR�GH�DOJXQDV�VXSHUILFLHV�TXH�VH�HQFXHQWUDQ�VXPHUJLGDV�HV�QHFHVDULR� FRQRFHU� OD� UHVLVWHQFLD� D� ODV� IXHU]DV� DFWXDQWHV� �KLGURVWiWLFD��� � 6H�UHTXLHUH� HQWRQFHV� GHWHUPLQDU� OD� PDJQLWXG�� GLUHFFLyQ� \� ORFDOL]DFLyQ� GH� ODV�IXHU]DV�VREUH�HO�iUHD���/D�IXHU]D�GHELGD�D�OD�SUHVLyQ�KLGURVWiWLFD�HQ�UHDOLGDG�VH�GLVWULEX\H�VREUH�WRGD�HO�iUHD�SHUR�VH�GHWHUPLQD�XQD� IXHU]D�UHVXOWDQWH�\�VX� ORFDOL]DFLyQ���/D�GLUHFFLyQ�VLHPSUH�VHUi�SHUSHQGLFXODU�DO�iUHD��9HDPRV� HO� FDVR� PiV� VLPSOH� GH� XQD� VXSHUILFLH� SODQD� YHUWLFDO� VH� VHFFLyQ�WUDQVYHUVDO� UHFWDQJXODU� GH� DQFKR�%� FRPSOHWDPHQWH�HQ� FRQWDFWR� FRQ�HO� OtTXLGR�GH� SHVR� HVSHFtILFR� γγγγ� FRQVWDQWH�� � (O� VLVWHPD� GH� UHIHUHQFLD� \� � LQLFLD� HQ� OD�VXSHUILFLH�GH�DJXD�\�HVWi�GLULJLGR�KDFLD�HO�IRQGR��

�)LJXUD���� )XHU]D�VREUH�XQD�VXSHUILFLH�SODQD�YHUWLFDO�GH�VHFFLyQ�UHFWDQJXODU�

Page 12: Presas de Gravedad

�/D�SUHVLyQ�HMHUFLGD�D�XQD�SURIXQGLGDG�FXDOTXLHUD�VH�REWHQGUi�FRQ� OD�HFXDFLyQ�JHQHUDO�GH�OD�KLGURVWiWLFD�\�(O�GLIHUHQFLDO�GH�IXHU]D�HMHUFLGR�VREUH�XQ�HOHPHQWR�GH�iUHD�GLIHUHQFLD�G\ %�TXH�HVWi�ORFDOL]DGR�D�XQD�SURIXQGLGDG�\�GHSHQGHUi�GH�GLFKD� SUHVLyQ�� � /D� IXHU]D� WRWDO� R� UHVXOWDQWH� )5� VREUH� OD� VXSHUILFLH� VHUi� OD�VXPDWRULD�GH�WRGRV�ORV�GLIHUHQFLDOHV���

*p yγ= � � dF yBdyγ= � �0

h

RF B ydyγ= ∫ �

�2

02

h

R

yF Bγ= � �

2

2R

hF Bγ= � �

2 2R

h hF Bh Aγ γ= = �

�(O� SXQWR� GH� DSOLFDFLyQ� GH� OD� IXHU]D� HV� HO� FHQWURLGH� GHO� WULiQJXOR� GH� SUHVLyQ��ORFDOL]DGR�D�K���GHVGH�OD�EDVH��)LJXUD�������'H� IRUPD� PiV� JHQHUDO� VL� VH� WLHQH� XQ� VXSHUILFLH� SODQD� LQFOLQDGD� \� SURIXQGD�VRSRUWDQGR�XQ�IOXLGR�FRQ�SHVR�HVSHFtILFR�γγγγ�FRQVWDQWH���FRQVLGHUDQGR�XQ�HMH�\�FRLQFLGLHQGR� FRQ� OD� LQFOLQDFLyQ� GH� OD� VXSHUILFLH� SODQD� \� FRQ� RULJHQ� HQ� OD�VXSHUILFLH�GHO�OtTXLGR���

�)LJXUD���� )XHU]D�VREUH�XQD�VXSHUILFLH�SODQD�LQFOLQDGD�\�VXPHUJLGD��

5HDOL]DQGR�XQD�DQiOLVLV�VLPLODU�DO�GH�OD�VXSHUILFLH�SODQD�YHUWLFDO�\�FRQVLGHUDQGR�TXH� *h y senθ= �

Page 13: Presas de Gravedad

* * *p h y senγ γ θ= = � � * * *dF y sen dAγ θ= ��0

A

RF sen ydAγ θ= ∫ �

GRQGH�0

A

CydA y A=∫ �� VLHQGR� \&� OD� ORFDOL]DFLyQ� GHO� FHQWURLGH� GH� OD� VHFFLyQ�

WUDQVYHUVDO�GH�OD�VXSHUILFLH�SODQD��PHGLGR�HQ�HO�HMH�LQFOLQDGR�\��

R CF y sen Aγ θ= � Î� R CF h Aγ= � Î� R CF p A= ��(VWR� VLJQLILFD� TXH� OD� IXHU]D� WRWDO� VH� SXHGH� REWHQHU� FRPR� HO� SURGXFWR� GH� OD�SUHVLyQ�HMHUFLGD�HQ�HO�FHQWURLGH�\�HO�iUHD�GH�OD�VHFFLyQ�WUDQVYHUVDO�HQ�FRQWDFWR�FRQ�HO�IOXLGR���6LHQGR� HVWi� OD� H[SUHVLyQ� JHQHUDO� SDUD� FDOFXODU� OD� IXHU]D� VREUH� OD� VXSHUILFLH�SODQD�� �(Q�HO�FDVR�SDUWLFXODU�GH� OD�VXSHUILFLH�SODQD�YHUWLFDO��K&� �K���\�HO�iUHD�$ % K��

2 2R

h hF A Bhγ γ= = � �

2

2R

hF Bγ= �� LGpQWLFD� D� OD� H[SUHVLyQ� SUHVHQWDGD�

DQWHULRUPHQWH���3DUD� OD� ORFDOL]DFLyQ�\3� GH� OD� IXHU]D� WRWDO�)5� VH� LJXDOD�HO�PRPHQWR�� UHVSHFWR�DO�RULJHQ��VXSHUILFLH� OLEUH���GH� OD�IXHU]D�UHVXOWDQWH�FRQ� OD�UHVXOWDQWH�GHO�PRPHQWR�GH�OD�IXHU]D�GLVWULEXLGD�VREUH�OD�VXSHUILFLH���

2

0 0

* *A A

R PF y sen ydA y sen y dAγ θ γ θ= =∫ ∫ �

GRQGH� 2

0

A

xxy dA I=∫ � VLHQGR� HVWH� HO� VHJXQGR� PRPHQWR� GH� iUHD� R� PRPHQWR� GH�

LQHUFLD�GH� OD�SODFD�UHVSHFWR�DO�HMH�[�� �3DUD�UHIHULUORV�D�XQ�HMH�FHQWURLGDO�ξξ�VH�XWLOL]D�HO�7HRUHPD�GH�6WHLQHU�TXH�DILUPD�ORV�PRPHQWRV�GH�LQHUFLD�FRQ�UHVSHFWR�D�ORV�HMHV�SDUDOHORV�VH�SXHGHQ�REWHQHU�DO�VXPDU�OD�GLVWDQFLD�FXDGUDGD�HQWUH�ORV�HMHV�PXOWLSOLFDGD�SRU�HO�iUHD��DVt��

2 *xx CI I y Aξξ= + �

�5HHPSOD]DQGR�VH�REWLHQH��

*R P xxF y sen Iγ θ γ= ⇒ *Cy senθ * * PA y γ= senθ ( )2 *CI y Aξξ + �

�2

2 ** * C

C P C P P C

C C

I y A Iy A y I y A y y y

y A y Aξξ ξξ

ξξ

+= + ⇒ = ⇒ = + ��

Page 14: Presas de Gravedad

/D�ORFDOL]DFLyQ�GH�OD�IXHU]D�UHVXOWDQWH�VH�FRQRFH�FRPR�FHQWUR�GH�SUHVLRQHV�\S�\�VLHPSUH�HVWi�ORFDOL]DGR�SRU�GHEDMR�GHO�FHQWURLGH���)8(5=$6�62%5(�683(5),&,(6�&859$6�680(5*,'$6��6XSRQLHQGR� XQ� OtTXLGR� DO� LQWHULRU� GHO� WDQTXH� )LJXUD� ���� SDUD� GHWHUPLQDU� OD�IXHU]D� WRWDO� UHVXOWDQWH� VREUH� OD� VXSHUILFLH� FXUYD� VH� GHEH� REWHQHU� ODV�FRPSRQHQWHV�KRUL]RQWDO�\�YHUWLFDO���6HSDUDQGR� HO� OtTXLGR� FRQWHQLGR� HQWUH� OD� VXSHUILFLH� FXUYD� \� VHQGRV� SODQRV�KRUL]RQWDOHV�\�YHUWLFDOHV�VH�WLHQH�HO�GLDJUDPD�GH�FXHUSR� OLEUH�$%&�GH�OD�)LJXUD�����ODV�IXHU]DV�DFWXDQWHV�VHUtDQ��

• )��IXHU]D�GH�SUHVLyQ�GHELGD�DO�IOXLGR�VREUH�OD�VXSHUILFLH�$%�• )��IXHU]D�KLGURVWiWLFD�GHELGR�DO�IOXLGR�VREUH�OD�VXSHUILFLH�$&�• )+�5HDFFLyQ�KRUL]RQWDO�GH�OD�VXSHUILFLH�FXUYD�VREUH�HO�IOXLGR�• )9�5HDFFLyQ�YHUWLFDO�GH�OD�VXSHUILFLH�FXUYD�VREUH�HO�IOXLGR�• :�SHVR�GHO�IOXLGR�FRQVLGHUDGR�

�3RU�DFFLyQ�UHDFFLyQ�OD�IXHU]D�TXH�HO�IOXLGR�HMHUFH�VREUH�OD�VXSHUILFLH�FXUYD�VHUtD� OD�PLVPD� SHUR� HQ�PDJQLWXG� FRQWUDULD� TXH� OD� IXHU]D�HMHUFLGD�SRU� OD�VXSHUILFLH�FXUYD�VREUH�HO�IOXLGR���/D�IXHU]D�WRWDO�GHO�IOXLGR�VREUH�OD�VXSHUILFLH�FXUYD��VHUi�OD�UHVXOWDQWH�GH�ODV�FRPSRQHQWHV�KRUL]RQWDOHV�\�YHUWLFDOHV�REWHQLGDV�HQ�HO�GLDJUDPD�GH�FXHUSR�OLEUH��

)LJXUD���� )XHU]D�VREUH�XQD�VXSHUILFLH�FXUYD�VXPHUJLGD��

Page 15: Presas de Gravedad

3XHVWR�TXH�HO�IOXLGR�HVWi�HQ�UHSRVR�VH�WLHQH�TXH� F ma=∑JG G

�(TXLOLEULR�GH�IXHU]DV�HQ�GLUHFFLyQ�[��KRUL]RQWDO���

2 20 0x x H HF ma F F F F= = − = ⇒ =∑ �

�(TXLOLEULR�GH�IXHU]DV�HQ�GLUHFFLyQ�\��YHUWLFDO���

1 10 0y y V VF ma F F W F F W= = − − = = +∑ �

/D�IXHU]D�WRWDO�VHUi� 2 2T H VF F F= + �

�&RPR� HO� IOXLGR� QR� DFHSWD� HVIXHU]RV� FRUWDQWHV� OD� ORFDOL]DFLyQ� GHO� )+� GHEH� VHU�FROLQHDO� FRQ�)�� \� )9� GHEH� VHU� FROLQHDO� FRQ� OD� UHVXOWDQWH� GH� )�� ��:�� � (VWR� VH�REWLHQH� FXDQGR� OD� VXPDWRULD� GH� PRPHQWRV� UHVSHFWR� D� FXDOTXLHU� SXQWR� GHO�VLVWHPD�HV�QXOD���2WUD�IRUPD�PiV�VLPSOH�GH�FDOFXODU�OD�FRPSRQHQWH�YHUWLFDO�GH�OD�IXHU]D�VREUH�OD�VXSHUILFLH�FXUYD�HV�FRQVLGHUDQGR�HO�VLJQLILFDGR�ItVLFR�GHO�UHVXOWDGR�REWHQLGR�FRQ�DQWHULRULGDG���)��HV�OD�IXHU]D�GHELGD�D�OD�SUHVLyQ�KLGURVWiWLFD�SRU�OD�FROXPQD�GH�IOXLGR�KDVWD�OD�VXSHUILFLH� $%�� � 1 AB AB AB AB EncimaAB EncimaABF P A h A Wγ γ= = = ∀ = �� HV� GHFLU� HO� SHVR� GHO�IOXLGR� KDVWD� OD� VXSHUILFLH� $%�� DVt� OD� IXHU]D� YHUWLFDO� VHUi� HO� WRWDO� GHO� SHVR� GHO�IOXLGR�TXH�HVWi�SRU�HQFLPD�GH�OD�VXSHUILFLH�FXUYD���/D�UHVXOWDQWH�YHUWLFDO�HVWDUi�ORFDOL]DGD� HQ� HO� FHQWURLGH� GH� OD� ILJXUD� TXH� VH� IRUPD� HQFLPD� GH� OD� VXSHUILFLH�FXUYD��

�)LJXUD���� )XHU]D�YHUWLFDO�VREUH�XQD�VXSHUILFLH�FXUYD�VXPHUJLGD��&DVR���

6L�HO� IOXLGR�VH�HQFXHQWUD�GHEDMR�GH�OD�VXSHUILFLH�FXUYD�&%�HO�DQiOLVLV�GH�FXHUSR�OLEUH� HV� VLPLODU� DO� FDVR� DQWHULRU� \� OD� IXHU]D� YHUWLFDO� UHVXOWDQWH� VHUi� HO� SHVR�HTXLYDOHQWH�GH�IOXLGR�TXH�HVWDUtD�SRU�HQFLPD�GH�OD�VXSHUILFLH�&%�

Page 16: Presas de Gravedad

�)LJXUD���� )XHU]D�YHUWLFDO�VREUH�XQD�VXSHUILFLH�FXUYD�VXPHUJLGD��&DVR���

�(678',2�'(�)8(5=$6�62%5(�81$�35(6$��(Q�HO�DQiOLVLV�GH�HVWDELOLGDG�GH�SUHVDV�VH�UHTXLHUH�JDUDQWL]DU�TXH�ORV�HVIXHU]RV�Pi[LPRV�\�PtQLPRV�HQ�OD�EDVH�GH�OD�SUHVD�VHDQ�GH�FRPSUHVLyQ��SDUD�TXH�WRGD�OD�EDVH�GH�OD�SUHVD�HVWp�VRSRUWDQGR�ORV�HVIXHU]R���3DUD�HOOR�VH�GHEH�JDUDQWL]DU�TXH�OD�IXHU]D�UHVXOWDQWH�HQ�OD�EDVH�GH�OD�SUHVD�HVWH�ORFDOL]DGD�HQ�HO�WHUFLR�PHGLR�GH�OD�SUHVD���(O� FKHTXHR� GH� HVWDELOLGDG� GH� SUHVDV� VH� GHEH� YHULILFDU� SDUD� FRQGLFLRQHV� GH�HPEDOVH�OOHQR�\�HPEDOVH�YDFtR��\�DQDOL]DU�OD�VLWXDFLyQ�PiV�GHVIDYRUDEOH���(Q�FRQGLFLRQHV�GH�HTXLOLEULR�HVWiWLFR�VH�GHEH�YHULILFDU�TXH�OD�SUHVD�QR�SUHVHQWH�URWDFLyQ�R�YROFDPLHQWR�DOUHGHGRU�GH�QLQJXQR�GH�VXV�HMHV���(VWR�VLJQLILFD�TXH�ORV�PRPHQWRV�SURGXFLGRV�SRU�ODV�IXHU]DV�TXH�LQGXFHQ�DO�YROFDPLHQWR�GH�OD�SUHVD�\�ODV� IXHU]DV� UHVLVWHQWHV� HVWpQ� HTXLOLEUDGRV�� � 3DUD� HVWD� YHULILFDFLyQ� VH� UHTXLHUH�KDFHU�XQ�DQiOLVLV�HVWiWLFR�GH�ODV�IXHU]DV�DFWXDQWHV�GHWDOODGDV�D�FRQWLQXDFLyQ���/DV�IXHU]DV�D�FRQVLGHUDU��)LJXUD�����VREUH�OD�SUHVD�VRQ��

• )XHU]D�+LGURVWiWLFD��)���)���• 3HVR�:��• 6XESUHVLRQHV�)VXE�VH�SUHVHQWD�FXDQGR�H[LVWH�IOXMR�VXEWHUUiQHR��GHSHQGH�

GH�SHUPHDELOLGDG�.��• 5HDFFLyQ� GHO� VXHOR� )5�� GHEH� HVWDU� ORFDOL]DGD� HQ� HO� WHUFLR�PHGLR� GH� OD�

EDVH�SDUD�JDUDQWL]DU�HVWDELOLGDG��• 6tVPLFDV� \� 2OHDMHV�� � 1R� VRQ� FRQVLGHUDGDV� HQ� HO� SUHVHQWH� DQiOLVLV�� VLQ�

HPEDUJR�SDUD�XQ�HVWXGLR�GHWDOODGR�VH�GHEHQ�WHQHU�HQ�FXHQWD���$�FRQWLQXDFLyQ�VH�GHWDOOD�HO�FDOFXOR�GH�FDGD�XQD�GH�ODV�FRPSRQHQWHV�GH�IXHU]D�VHJ~Q�OD�)LJXUD�����

Page 17: Presas de Gravedad

H

H/3

d

e/3 hβ

e

b

XFRVFRV

B

XFsubFsub

FRH F2

ac

Xw1

Xw2

F1

W2

W1

L

Hγ hγ

γ

γ

�)LJXUD���� )XHU]DV�DFWXDQWHV�VREUH�XQD�SUHVD��

�/D�PDJQLWXG�\�ORFDOL]DFLyQ��UHVSHFWR�DO�SXQWR�GH�URWDFLyQ�%��GH�HVWiV�IXHU]DV�VH�FDOFXODQ�DVt���0DJQLWXG�)XHU]DV�KLGURVWiWLFDV��

2

1 2

HF Lγ= �

2 *2

hF e Lγ= ��)��HV�SHUSHQGLFXODU�D�OD�FDUD�H[SXHVWD�GH�OD�SUHVD��SRU�WDO�UD]yQ�

SDUD�HO�DQiOLVLV�HVWiWLFR�VH�GHVFRPSRQH�HQ�[�\�\��DVt���

2 2 2 2*cos *H VF F F F senβ β= = ��/RFDOL]DFLyQ�)XHU]DV�KLGURVWiWLFDV��

• )���ORFDOL]DGD�D�+���GH�OD�EDVH��• )���ORFDOL]DGD�D�H���GHO�SXQWR�%��GRQGH� he senβ= �

�0DJQLWXG�\�ORFDOL]DFLyQ�GH�ORV�GH�ORV�3HVRV�:�GH�OD�SUHVD�GH�FRQFUHWR���

1 * *cW a d Lγ= ���� ( )2 * *cW b a c Lγ= − �

1 2WaX b= − ���� ( )2

2

3WX b a= − ��

Page 18: Presas de Gravedad

0DJQLWXG�GH�OD�6XESUHVLyQ��( )

* *2sub

H hF K l bγ

+= �

/D�VXESUHVLyQ�HVWi� ORFDOL]DGD�HQ�HO�FHQWURLGH�GHO�SULVPD�GH�VXESUHVLRQHV��TXH�SDUD�HO�FDVR�DQDOL]DGR�FRUUHVSRQGH�D�XQ�WUDSHFLR��DVt���

( )( )2

3sub

H hbX

H h

+=

+��GHVGH�HO�SXQWR�%��

�3DUD�GHWHUPLQDU�OD�UHDFFLyQ�GHO�VXHOR�VH�UHDOL]D�VXPDWRULD�GH�IXHU]D���

1 2 20 0V sub RV VF W W F F F= ⇒ − − + + − =∑ �

1 2 2RV sub VF W W F F= + − + ��

1 20 0H RH H fF F F F F= ⇒ + − − =∑ �

2 1 0RH H fF F F F= − + = �

/D�UHVXOWDQWH�WRWDO�VH�VHUi� 2 2RT RH RVF F F= + �

�3DUD�GHWHUPLQDU�OD�ORFDOL]DFLyQ�GH�OD�IXHU]D�UHVXOWDQWH�YHUWLFDO�\�YHULILFDU�TXH�VH�HQFXHQWUH�HQ�HO�WHUFLR�PHGLR�VH�XWLOL]D�HO�HTXLOLEULR�GH�PRPHQWRV�DOUHGHGRU�GH�FXDOTXLHU�SXQWR��VH�VLJXLHUH�VH�UHDOLFH�UHVSHFWR�DO�SXQWR�%��

1 1 1 2 2 20 * * * * * * 03 3 RVB W W sub sub RV F

H eM F W X W X F X F F X= ⇒ − + + − + − =∑ �

1 1 1 2 2 2* * * * *3 3

RV

W W sub sub

FRV

H eF W X W X F X F

XF

− + + − += �

�9HULILFDFLyQ�GH�HVIXHU]RV�HQ�OD�IXQGDFLyQ�

b

τ max min

X . FRV

FRV

B

�)LJXUD���� 'LVWULEXFLyQ�GH�HVIXHU]RV�UHVXOWDQWHV�VREUH�OD�EDVH�GH�OD�SUHVD��

Page 19: Presas de Gravedad

/D� IXHU]D� UHVXOWDQWH� )59� VH� GLVWULEX\H� D� OR� ODUJR� GH� OD� EDVH� GH� OD� SUHVD��XVXDOPHQWH�VH�FRQVLGHUD�XQD�GLVWULEXFLyQ�OLQHDO�GH�OD�IXHU]D��HQWUH�XQ�HVIXHU]R�Pi[LPR�τPD[�\�XQ�HVIXHU]R�PtQLPR�τPLQ�� �/D�IXHU]D�UHVXOWDQWH�GH�ORV�HVIXHU]RV�HV� HTXLYDOHQWH� D� OD� IXHU]D� UHVXOWDQWH� YHUWLFDO�� \� HO� FHQWURLGH� GHO� SULVPD� GH�HVIXHU]RV�GHEH�FRLQFLGLU�FRQ� OD� ORFDOL]DFLyQ�GH�OD�IXHU]D�UHVXOWDQWH�YHUWLFDO��$Vt�ODV�HFXDFLRQHV�UHVXOWDQWHV�VHUiQ���

min max *2

+ = VR b L

τ τ�

min max

min max

2*

3

+= + RV

bX b L

τ ττ τ

�� VLHPSUH� \� FXDQGR� OD� ORFDOL]DFLyQ� GH� OD� IXHU]D� HVWp�

UHIHULGD�DO�SXQWR�%���(VWH�VLVWHPD�GH�HFXDFLRQHV�VH�UHVXHOYH�SDUD�τPD[�\�XQ�HVIXHU]R�PtQLPR�τPLQ��)DFWRUHV�GH�VHJXULGDG���)LQDOPHQWH��VH�GHEH�FRQVLGHUDU�TXH�VH�FXPSODQ�FLHUWRV�IDFWRUHV�GH�VHJXULGDG���

• 6HJXULGDG��D�YROFDPLHQWR�)6�YROFDPLHQWR��!�����

Volcamiento

MomentosestabilizantesFS

Momentosvolcadores= ∑

∑�

1 1 2 2 2

1

* * *3

* * *3 RV

W W

Volcamiento

sub sub RV F

eW X W X F

FSH

F F X F X

+ +=

+ +�

• 6HJXULGDG��D�GHVOL]DPLHQWR�)6�GHVOL]DPLHQWR��!������

Vdeslizamiento

H

f FFS

F= ∑

∑�

1 2

* RVdeslizamiento

H

f FFS

F F=

−�

GRQGH�I�HV�HO�IDFWRU�GH�IULFFLyQ���)/27$%,/,'$'�<�(67$%,/,'$'��/D�IORWDELOLGDG�HV�OD�WHQGHQFLD�GH�XQ�IOXLGR�SDUD�HMHUFHU�XQD�IXHU]D�GH�DSR\R�VREUH�XQ�FXHUSR�FRORFDGR�HQ�pO���3ULQFLSLR�GH�$UTXtPHGHV���

Page 20: Presas de Gravedad

7DPELpQ� FRQRFLGR� FRPR� /H\� GH� %R\DQWH]� R� (PSXMH�� LQGLFD� TXH� XQ� FXHUSR�VXPHUJLGR�HQ�XQ� IOXLGR� UHFLEH�XQ�HPSXMH�KDFLD�DUULED�FRQ�XQD�IXHU]D� LJXDO�DO�SHVR�GHO�OtTXLGR�GHVSOD]DGR��/H\�GH�)ORWDFLyQ��8Q�FXHUSR�IORWDQWH�GHVSOD]D�VX�SURSLR�SHVR�GHO�OtTXLGR�HQ�HO�FXDO�IORWD���

B

C

D

E

V

C

B V D

E

DIAGRAMA DE CUERPO LIBRE

FEmpuje

γf

h

A1

2

1

2

W

W1

2

p *A

p *A�

)LJXUD���� $QiOLVLV�GH�IXHU]DV�VREUH�XQ�HOHPHQWR�FRPSOHWDPHQWH�VXPHUJLGR��

�3DUD� GHWHUPLQDU� OD� UHDFFLyQ� TXH� HO� IOXLGR� HMHUFH� VREUH� XQ� FXHUSR� WRWDOPHQWH�VXPHUJLGR� �YROXPHQ� FRQWHQLGR� HQWUH� %&'(�� VH� VHOHFFLRQDUi� XQ� YROXPHQ� GH�IOXLGR� TXH� FLUFXQVFULED� HO� FXHUSR�%&'(� FRQ� DOWXUD�K� \� iUHD� WUDQVYHUVDO�$�� � (O�DQiOLVLV�GHO�GLDJUDPD�GH�FXHUSR�OLEUH�LQGLFD�ODV�VLJXLHQWHV�IXHU]DV���

• 3UHVLyQ� KLGURVWiWLFD� HQ� ODV� IURQWHUDV� GHO� YROXPHQ� VHOHFFLRQDGR�� 1p A� \�

2p A�• 3HVR�GHO�IOXLGR�H[WHULRU�DO�FXHUSR�%&'(�� 1W �\� 2W �• 5HDFFLyQ�GHO�FXHUSR�VREUH�HO�IOXLGR��TXH�SRU�DFFLyQ�UHDFFLyQ�HV�OD�PLVPD�

TXH� OD�IXHU]D�GHO� IOXLGR�VREUH�HO�FXHUSR�%&'(��XVXDOPHQWH�GHQRPLQDGD�IXHU]D� GH� HPSXMH��)(�� � /D� UHDFFLyQ� GHO� FXHUSR� VREUH� HO� IOXLGR�HV�KDFLD�DEDMR��SHUR�OD�IXHU]D�GH�HPSXMH�HMHUFLGD�SRU�HO�IOXLGR�VREUH�HO�FXHUSR�HV�HQ�GLUHFFLyQ�FRQWUDULD��HV�GHFLU�YHUWLFDO�KDFLD�DUULED��

�(Q�FRQGLFLRQHV�GH�HTXLOLEULR�HVWiWLFR��

1 2 1 20 0y RF p A p A W W F= ⇒ − + − − − =∑ �

Page 21: Presas de Gravedad

( ) ( )2 1 1 2EF p p A W W= − − + �

�$SOLFDQGR� OD� GLVWULEXFLyQ� KLGURVWDWLFD� GH� SUHVLRQHV�� OD� GLIHUHQFLD� GH� SUHVLyQ�HQWUH�ORV�H[WUHPRV�GHO�FXHUSR�OLEUH�HV� 2 1 fp p hγ− = ��

�( )1 2E fF hA W Wγ= − + �

�GRQGH� f hAγ �HV�HO�SHVR�GHO�YROXPHQ�FRQWHQLGR�HQWUH�ODV�IURQWHUDV�����

�( )1 2 1 2E E cuerpo f cuerpoF W W W F W γ∀ −= − + ⇒ = = ∀ �

�&RQ�XQ�DQiOLVLV�VLPLODU�FXDQGR�HO�FXHUSR�HVWi�SDUFLDOPHQWH�VXPHUJLGR��HV�GHFLU�IORWDQGR��VH�WLHQH���

E f desplazadoF γ= ∀ �

�(Q�FRQGLFLRQHV�GH�HTXLOLEULR�HVWiWLFR�OD�IXHU]D�GH�HPSXMH�VHUi�LJXDO�DO�SHVR�GHO�FXHUSR�� � /D� IXHU]D� GH� HPSXMH� )(� DFW~D� YHUWLFDO� KDFLD� DUULED� D� WUDYpV� GHO�FHQWURLGH�GHO�YROXPHQ�GHO�FXHUSR�R�GHO�YROXPHQ�GHVSOD]DGR��VHJ~Q�VHD�WRWDO�R�SDUFLDOPHQWH� VXPHUJLGR�� � (O� VLWLR� GH� DSOLFDFLyQ� GH� OD� IXHU]D� GH� HPSXMH� VH�GHQRPLQD�FHQWUR�GH�HPSXMH�R�FHQWUR�GH�ER\DQWH]��(VWDELOLGDG���6H� FRQVLGHUD� XQ� FXHUSR� HQ� XQ� IOXLGR� HVWDEOH� VL� UHJUHVD� D� VX� SRVLFLyQ�RULJLQDO�GHVSXpV�GH�SUHVHQWDU�XQD�URWDFLyQ�DOUHGHGRU�GH�XQ�HMH�KRUL]RQWDO���/D�HVWDELOLGDG�GHSHQGH�GH�OD�SRVLFLyQ�UHODWLYD�HQWUH�OD�ORFDOL]DFLyQ�GH�OD�IXHU]D�GH�HPSXMH�\�HO�FHQWUR�GH�JUDYHGDG���(V�GLIHUHQWH�GHSHQGLHQWH�VL�HO�FXHUSR�HVWi�727$/0(17(�VXPHUJLGR�R�3$5&,$/0(17(�VXPHUJLGR���(VWDELOLGDG�GH�FXHUSRV�WRWDOPHQWH�VXPHUJLGRV���8Q�FXHUSR�727$/0(17(�VXPHUJLGR�HV�HVWDEOH�VL�HO�FHQWUR�GH�JUDYHGDG��SXQWR�GH� DSOLFDFLyQ� GHO� SHVR�� HVWi� SRU� GHEDMR� GHO� FHQWUR� GH� ER\DQWH]� �SXQWR� GH�DSOLFDFLyQ�GH�OD�IXHU]D�GH�HPSXMH���YpDVH�)LJXUD����

Page 22: Presas de Gravedad

EF

W

FE

W

EF

EF

W W

a) Estable. b) Inestable

�)LJXUD���� (VWDELOLGDG�GH�FXHUSRV�FRPSOHWDPHQWH�VXPHUJLGRV��

�/D�VLWXDFLyQ�D��GH�OD�)LJXUD����GRQGH�HO�FHQWUR�GH�JUDYHGDG�HVWi�SRU�GHEDMR�GHO�FHQWUR� GH� IORWDFLyQ� LQGLFD� TXH� DQWH� XQD� URWDFLyQ� GHO� FXHUSR�� VH� JHQHUDUi� XQ�PRPHQWR� UHFXSHUDGRU�GH� OD�SRVLFLyQ� LQLFLDO�� �(Q�HVWH�FDVR�HO�FXHUSR�VH�SXHGH�FRQVLGHUDU�HVWDEOH���3RU� HO� FRQWUDULR� HQ� OD� VLWXDFLyQ� E�� GH� OD�)LJXUD� ���� HO� FHQWUR� GH� JUDYHGDG� VH�HQFXHQWUD� SRU� HQFLPD� GHO� FHQWUR� GH� IORWDFLyQ� \� FXDOTXLHU� URWDFLyQ� LPSXHVWD�VREUH� HO� FXHUSR� JHQHUDUi� XQ� SDU� R� PRPHQWR� LQHVWDEOH� FRQWLQXDQGR� FRQ� OD�URWDFLyQ�LQLFLDO���(Q�HVWH�FDVR�HQ�FXHUSR�VH�FRQVLGHUD�LQHVWDEOH���&XDQGR�HO�FHQWUR�GH�JUDYHGDG�\�HO� FHQWUR�GH� IORWDFLyQ�FRLQFLGHQ��HO�SHVR�\� OD�IXHU]D�GH�HPSXMH�DFW~DQ�D�WUDYpV�GHO�PLVPR�SXQWR�VL�TXH�VH�SURGX]FD�XQ�SDU���(Q� HVWH� FDVR� HO� FXHUSR� WHQGUtD� XQD� HVWDELOLGDG� QHXWUD� \� SHUPDQHFHUtD� HQ�FXDOTXLHU�RULHQWDFLyQ�HQ�OD�TXH�VH�FRORTXH�UHVSHFWR�DO�HMH�KRUL]RQWDO���(VWDELOLGDG�GH�FXHUSRV�SDUFLDOPHQWH�VXPHUJLGRV���3DUD� OD� HVWDELOLGDG� GH� FXHUSRV� SDUFLDOPHQWH� VXPHUJLGRV� R� IORWDQWHV� VH� GHEH�DQDOL]DU�OD�SRVLFLyQ�UHODWLYD�GHO�PHWDFHQWUR�0��FRPR�VH�LQGLFD�D�FRQWLQXDFLyQ��

Page 23: Presas de Gravedad

CECG

CE*

M

dV

EdF

d

EF

FE*

W x x

y

dA

A

Par

Corte transversal Sección en planta

dFE

�)LJXUD���� (VWDELOLGDG�GH�FXHUSRV�FRPSOHWDPHQWH�VXPHUJLGRV�

0��� 0HWDFHQWUR�� LQWHUVHFFLyQ�GH� OD� OtQHD�GH�DFFLyQ�GH� *EF �FRQ�HO� HMH�FHQWUDO�

GHO�FRUWH�WUDQVYHUVDO�CG ��� /RFDOL]DFLyQ�GHO�FHQWUR�GH�JUDYHGDG��SXQWR�GH�DSOLFDFLyQ�GHO�SHVR��W ���� 3HVR�CE��� 3RVLFLyQ�GHO�FHQWUR�GH�HPSXMH�LQLFLDO��HV�GHFLU�DQWHV�GH�OD�URWDFLyQ��

EF ���� )XHU]D�GH�HPSXMH�LQLFLDO��DQWHV�GH�OD�URWDFLyQ��*CE �� 3RVLFLyQ�GHO�FHQWUR�GH�HPSXMH�ILQDO��HV�GHFLU�GHVSXpV�GH�OD�URWDFLyQ��

*EF ���� )XHU]D�GH�HPSXMH�ILQDO��GHVSXpV�GH�OD�URWDFLyQ��&XDQGR�VH�SUHVHQWD�XQD�URWDFLyQ�HQ�XQ�FXHUSR�SDUFLDOPHQWH�VXPHUJLGR��)LJXUD����� VH� SUHVHQWD� XQ� GHVSOD]DPLHQWR� DGLFLRQDO� GH� IOXLGR� SUHVHQWiQGRVH� XQD�IXHU]D�GH�HPSXMH� *

EF �HVWi�ORFDOL]DGD�HQ�HO� *CE D�XQD�GLVWDQFLD�δ�GHO�FHQWUR�GH�HPSXMH�LQLFLDO���(O� GHVSOD]DPLHQWR� GH� IOXLGR� KDFLD� HO� ODGR� L]TXLHUGR� RFDVLRQD� XQD� IXHU]D� GH�HPSXMH� EdF �� DO� ODGR� GHUHFKR� SDVD� OD� VLWXDFLyQ� FRQWUDULD�� HV� GHFLU� VH�GHMD�GH�SUHVHQWDU�XQD�IXHU]D�SRU�HO�YROXPHQ�TXH�DQWHULRUPHQWH�VH�GHVSOD]DED��HVWR�VH�SXHGH�UHSUHVHQWDU�SRU�XQD�IXHU]D�GH�HPSXMH�QHJDWLYD� EdF− ��HVWDV�GRV�IXHU]DV�IRUPDQ�XQ�PRPHQWR�R�SDU���3DUD�HYDOXDU�HO�SDU�VH�WLHQH���

2E

A

dV xd dA dF dV xd dA dPar x xd dA Par d x dAθ γ γ θ γ θ γ θ= ⇒ = = ⇒ = ⇒ = ∫ �

yyPar d Iγ θ= �

�/D� PDJQLWXG� GH� OD� IXHU]D� GH� HPSXMH� HV� OD� PLVPD� DQWHV� \� GHVSXpV� GH� OD�URWDFLyQ�� VLQ� HPEDUJR� DQWH� OD� URWDFLyQ� VH� SUHVHQWD� XQ� GHVSOD]DPLHQWR� GHO�FHQWUR�GH�HPSXMH�� � /D� FRQILJXUDFLyQ� LQFOLQDGD� GHO� VLVWHPD�VHUi� OD�GHELGD�D� OD�IXHU]D�GH�HPSXMH�ILQDO� *

EF �XELFDGD�HQ� *CE ��HVWH�HIHFWR�VHUi�FRQVLGHUDGR�FRQ�HO�

Page 24: Presas de Gravedad

SDU���(O�HIHFWR�GH� *EF �DSOLFDGD�HQ� *CE �HV�HVWiWLFDPHQWH�HTXLYDOHQWH�D�OD�IXHU]D�

GH�HPSXMH�LQLFLDO� EF �DSOLFDGD�HQ�CE�PiV�HO�SDU���/D�LQTXLHWXG�VHUi�FXDO�HV�OD�

GLVWDQFLD�δ�HQWUH� *CE \�CE��HV�GHFLU�GHVSOD]DPLHQWR�GH�OD�OtQHD�GH�DFFLyQ�GH�OD�IXHU]D�GH�HPSXMH�� �$O� LJXDODU�PRPHQWRV�UHVSHFWR�D�XQ�HMH�SDUDOHOR�D�\�\�TXH�SDVp�SRU� *CE ��

* *0 * 0 yyE E

E

d IPar ParF F Par

F W W

γ θδ δ+ − = ⇒ = = = �

�3RU� WULJRQRPHWUtD� GHO� WULiQJXOR� IRUPDGR� HQWUH�0� *CE �CE� \� UHHPSOD]DQGR� OD�H[SUHVLyQ�DQWHULRU�VH�WLHQH���

0lim 1

*

yy

yy

d

yy yy

d II sendWsend MCEsendMCE send send dW

dI I

MCEW Vd

θ

γ θγδ δ θθ θθ θ θ

θγ

→= ⇒ = = = ⇒ =

= =

��

�GH� GRQGH� VH� SXHGH� REWHQHU� OD� ORFDOL]DFLyQ� GHO� PHWDFHQWUR� FRPR� OD� GLVWDQFLD�GHVGH� HO� FHQWUR� GH� HPSXMH�� � &RQRFLGD� OD� SRVLFLyQ� GHO� PHWDFHQWUR� VH� SXHGH�FDOFXODU�OD�DOWXUD�PHWDFpQWULFD�0*��TXH�HV�OD�GLVWDQFLD�HQWUH�HO�PHWDFHQWUR��0��\�HO�FHQWUR�GH�JUDYHGDG��&*���DVt��

lMCEMG −= �'RQGH�0&(�HV�OD�GLVWDQFLD�HQWUH�HO�PHWDFHQWUR�\�HO�FHQWUR�GH�HPSXMH��\�l�HV�OD�GLVWDQFLD�HQWUH�HO�FHQWUR�GH�HPSXMH��&(��\�HO�FHQWUR�GH�JUDYHGDG��&*����/D� HVWDELOLGDG� GH� XQ� FXHUSR� SDUFLDOPHQWH� VXPHUJLGR� GHSHQGH� GH� OD� SRVLFLyQ�UHODWLYD�GHO�PHWDFHQWUR�\�HO�FHQWUR�GH�JUDYHGDG���

• 8Q� FXHUSR� VH� FRQVLGHUD� HVWDEOH� VL� HO� PHWDFHQWUR� HVWi� ORFDOL]DGR� SRU�HQFLPD� GHO� FHQWUR� GH� JUDYHGDG�� � (QWUH�PD\RU� VHD� OD�GLVWDQFLD�HQWUH�HO�PHWDFHQWUR� \� HO� FHQWUR� GH� JUDYHGDG�� HV� GHFLU� OD� DOWXUD� PHWDFpQWULFD��PD\RU�VHUi�OD�HVWDELOLGDG�GHO�FXHUSR���6L� OD�DOWXUD�PHWDFpQWULFD��0*��HV�SRVLWLYD�HO�FXHUSR�HV�HVWDEOH��

• 8Q� FXHUSR� HV� LQHVWDEOH� VL� HO�PHWDFHQWUR� HVWi� ORFDOL]DGR�SRU�GHEDMR�GHO�FHQWUR�GH�JUDYHGDG���6L� OD�DOWXUD�PHWDFpQWULFD�HV�QHJDWLYD�HO�FXHUSR�HV�LQHVWDEOH��

��(6)8(5=26�(1�78%(5Ë$6��6H�DQDOL]DUi�D�FRQWLQXDFLyQ�OD�VLWXDFLyQ�GH� ORV�HVIXHU]RV�D�ORV�TXH�VH�VRPHWHQ�ODV� WXEHUtDV� FXDQGR� XQ� IOXLGR� VH� HQFXHQWUD� HQ� UHSRVR�� HV� GHFLU� HQ� FRQGLFLyQ�HVWiWLFD�� � 3DUD� HVWD� VLWXDFLyQ� QR� H[LVWH� WUDFFLyQ� ORQJLWXGLQDO� SRU� OR� TXH�

Page 25: Presas de Gravedad

VRODPHQWH� H[LVWHQ� IXHU]DV� GH� WHQVLyQ� �7�� HQ� ORV� H[WUHPRV�� TXH� WUDWDUtDQ� GH�UHYHQWDU�OD�WXEHUtD����3DUD�HO�DQiOLVLV�GH�HVIXHU]RV�VH�FRQVLGHUD�XQ�FRUWH� ORQJLWXGLQDO�HQ� OD�WXEHUtD�\�XQ� WUDPR� GH� ORQJLWXG� �l��� FRPR� VH� LQGLFD� HQ� OD� )LJXUD� ���� � /DV� IXHU]DV�UHVXOWDQWHV�VHUiQ���)+� ���IXHU]D�KLGURVWiWLFD�GHELGD�D�OD�SUHVLyQ�3�HQ�OD�WXEHUtD��7� ���IXHU]D�GH�WHQVLyQ�JHQHUDGDV�DO�LQWHULRU�GH�OD�WXEHUtD��H� ���HVSHVRU�GH�OD�WXEHUtD��

3 3YEOH

�7

�7 ���H�������

���H�������

)+

�7

�7

+)

<)+

��

D�� � � � � � � E��

)LJXUD���� 'LVWULEXFLyQ�GH�HVIXHU]RV�HQ�XQD�WXEHUtD��D��FXDQGR�OD�SUHVLyQ�HV�OD�PLVPD�HQ�OD�WRWDOLGDG�GH�OD�WXEHUtD��E��FXDQGR�OD�SUHVLyQ�YDUtD�D�WUDYpV�GH�OD�WXEHUtD�

��<D� TXH� HO� IOXMR�DO� LQWHULRU�GH� OD� WXEHUtD� VH�HQFXHQWUD�HQ� FRQGLFLRQHV� HVWiWLFDV�SDUD� HO� DQiOLVLV� GH� ORV� HVIXHU]RV� VH� XWLOL]DQ� ODV� GRV� HFXDFLRQHV� EiVLFDV� GH�HTXLOLEULR�HVWiWLFR��VXPDWRULD�GH�IXHU]D�\�VXPDWRULD�GH�PRPHQWRV�LJXDO�D�FHUR��HV�GHFLU��

00 ==∑ ∑ OH MF �

�3UHVLyQ�FRQVWDQWH��6L� OD� SUHVLyQ� HV� FRQVWDQWH� HQ� WRGRV� ORV� SXQWRV� GH� OD� WXEHUtD� ORV� HVIXHU]RV�UHVXOWDQWHV�VREUH�OD�SDUHG�GH�OD�WXEHUtD�VHUiQ�ORV�PLVPDV��SRU�WDQWR�7� 7��\�HO�DQiOLVLV�GH�HTXLOLEULR�HVWiWLFR�SDUD�OD�VLWXDFLyQ�D��GH�OD�)LJXUD����VH�UHGXFH�D�OD�VXPDWRULD�GH�IXHU]DV���

22

0200 21

HH

HHH

FTTF

TFTTFF

=⇒=

=−⇒=−−⇒=∑�

/D�IXHU]D�KRUL]RQWDO�VH�REWLHQH�GH�OD�PXOWLSOLFDFLyQ�GH�OD�SUHVLyQ�SRU�HO�iUHD�GH�DFFLyQ��GH�DTXt�TXH���

lRpFH *2*= ��GRQGH�5�HV�HO�UDGLR�GH�OD�WXEHUtD���

Page 26: Presas de Gravedad

)LQDOPHQWH��OD�IXHU]D�GH�WHQVLyQ�VHUi���

lRplRpF

T H **2

*2*2

=== �

�(VWD�IXHU]D�GH�WHQVLyQ�HV�VRSRUWDGD�SRU�HO�iUHD�e*l��SRU�OR�WDQWR�HO�HVIXHU]R�GH�WHQVLyQ�HQ�OD�WXEHUtD���HVWi�GDGR�SRU���

e

Rp

le

lRp

A

T **

** =⇒== σσ �

�3UHVLyQ�9DULDEOH��6L�OD�SUHVLyQ�YDUtD�D�WUDYpV�GH�OD�WXEHUtD�ORV�HVIXHU]RV�UHVXOWDQWHV�VREUH�OD�SDUHG�GH�OD�WXEHUtD�GLIHUHQWHV�\�SRU�OR�WDQWR�HQ�HO�DQiOLVLV�GH�HTXLOLEULR�HVWiWLFR�SDUD�OD�VLWXDFLyQ�E��GH�OD�)LJXUD����GHEH�LQFOXLU�OD�VXPDWRULD�GH�IXHU]DV�\�PRPHQWRV���

2121 0200 TTFTFTTFF HHHH +=⇒=−⇒=−−⇒=∑ �

lypTR

ylRpTyFRTM

H

H

H

F

FFHO

**2

****20*2*0

1

11

=

=⇒=−=∑ �

( )H

H

F

FH

yRlpT

lyplRpTTFT

−=

−=⇒−=

2*

*****2

2

212�

�(V�GHFLU�TXH�H[LVWHQ�GLIHUHQWHV�IXHU]DV�GH�WHQVLyQ��HQ�OD�SRUFLyQ�VXSHULRU�\�HQ�OD�SRUFLyQ�LQIHULRU���3RU�WDQWR�KDEUiQ�GRV�HVIXHU]RV�GH�WHQVLyQ�D�YHULILFDU���

( )e

yRpl

le

lyp

A

T

e

yp

le

lyp

A

T

HH

HH

FF

FF

−=⇒==

=⇒==

2

*

**

*

*

**

22

2

12

1

σσ

σσ�

�9HULILFDFLyQ�GLVHxR�GH�WXEHUtDV���3DUD�FRQGLFLRQHV�GH�GLVHxR�VH�YHULILFD�TXH�HO�HVIXHU]R�TXH�UHVLVWH�OD�WXEHUtD�VHD�VXSHULRU� DO� HVIXHU]R� GH� WHQVLyQ� � diseño resistenteσ σ< ��SURGXFLGR� SRU� HO� GREOH� GH� OD�SUHVLyQ� GH� VHUYLFLR� � 2*diseño servicioP P= ��� VLHQGR� HVWD� ~OWLPD� OD� SUHVLyQ� Pi[LPD�EDMR� OD� FXDO� HVWDUi� H[SXHVWD� OD� WXEHUtD�� � (O� HVIXHU]R� TXH� UHVLVWH� OD� WXEHUtD�GHSHQGH� GHO� PDWHULDO� \� HVWi� LQFOXLGD� HQ� ODV� HVSHFLILFDFLRQHV� R� FDWiORJRV� TXH�HQWUHJD�FDGD�FRPSDxtD�SURGXFWRUD�GH�WXEHUtD�����KWWS���ILVLFD�XVDFK�FO�aOKURGULJ�ILVLFD��FKWH[WR�KWPO���