SHIP S STABILITY

Document Sample
SHIP S STABILITY Powered By Docstoc
					   Principles of Ship’s
        Stability



PETRAS PIKSRYS
    SHIP’S STABILITY

• SHIP’S STABILITY IS
  THE TENDENCY OF
  SHIP TO ROTARE ONE
  WAY OR THE OTHER
  WHEN FORCIBLY
  INCLINED
    WHAY IS STABILITY IS SO

         IMPORTENT ?

IF THE SHIP LOST STABILITY WHAT
  WILL BE HAPPENED:
 1. LOST OF MOBILE
 2. LOST THE HUMANS LIFES
 3. LOST THE SHIP
 4. LOST THE CARGO
 5. OIL POLLUTION
  FUNDAMENTALS OF STABILITY

STABILITY is the tendency of vessel to rotate one way or the
              other when forcibly inclined.



IMPORTENT !!

Ship’s stability can’t catch directly

Stability can define only by calculating
  HOW CALCULATING SHIP‖S
 STABILITY AND CARCO PLAN ?

• 1.By previous similar cargo plan.
• 2.By standard cargo plan according
  ―STABILITY BOOKLET‖
• 3.By standard cargo plan forms
• 4.By special cargo plan computer
• 5.By standard PC with special cargo
  plan program
• 6.By special or standard hand
  calculator
SHIP’S STABILITY CRITERIAS

• THERE ARE TWO SHIP’S STABILITY
  CRITERIAS:
• 1 h>0 ship’s metacenter height always
  positive.
• 2 Zg < Zcritical
•   h = Zm – Zg
•   Zg defined by calculating
•   Zm define according hydrostatic curves
• Zg critical define according special
    diagram.
SHIP’S STABILITY CALCULATING
• SHIP’S STABILITY CALCULATING BY
  MOMENT FORMULAS.
• MAIN OBJECT OF CALCULATING TO
  DEFINE SHIP’S STABILITY CRITERIAS:
• GM=h METACENTER HEIGHT
• Zg    SHIP’S GRAVITY HEIGHT
• MOMENT FORMULA:


•      D0Z0+P1Z1+P2Z2+…….+PnZn
• Zg =
•       D0 + P1 +P2 + …….. + Pn
SHIP’S STABILITY CALCULATING

• Zg critical CURVE
 6.60
 6.50

 6.40

                      Zg critical
6.30

6.20

6.10


   8000 10000 12000   14000   16000 18000   20000
   WHO CALCULATING SHIP’S
 CARGO PLAN AND STABILITY?
• 1.CARGO OFFICER (ch.mate)
• 2.PORT CARGO OFFICER (supercargo)
• 3.SHIP’S MASTER
 SHIP’S STABILITY

          STABILITY




INITIAL   OVERALL     DYNAMIC
            STABILITY
INITIAL STABILITY - The stability of a ship
in the range from 0 to 7/10 of
inclination.

OVERALL STABILITY - A general measure of a
ship's ability to resist capsizing in a
given condition of loading.

DYNAMIC STABILITY - The work done in heeling
a ship to a given angle of heel.
          INITIAL SHIP’S STABILITY

     • Initial ship’s stability when ship inclinating
       from 7 till12 degrees. Ship’s underwater
       body did not change volume
     •         V0=V1                          V1
                          m                        L1
     V0
w                       G                          L

W1                                  C1
                         C
   INITIAL METACENTRIC
         FORMULA
M=D lst                 Qst
              m
lst=hsinQ                           h
M=D h sin Q       lst

              G    D
                          Vg

                               C1

              C
        SHIP’S STABILITY
         CALCULATING
• Initial stability calculating by ship’s
  stability triangle
• Calculating formula lst= h sinQ
• Overall stability calculating by
  hydrostatic ship’s body formula lf
• Dynamic stability is the area under
  the static stability curve
• Dynamic stability also potential
  energy available to return the ship to
  the upringing
        STABILITY TRIANGLE
                      m
lst =hsin Q
l   f
              h           Q
                          l   st

                      G                 Vg
                              D
                                             C1
                                   lf

                  C
                    PHANTACORENS

           SHIP’S BADY FORM STABILITY ARMS lf

lf
     2.8
               80    90
     2.4       70
               60
     1.6
               50
               40
     1.2
               30
     0.8
               20

     0.4       10


       4000 6000    8000   10000   12000 14000 16000       20000
                                                     18000
                                   DISPLACEMENT
                METACENTRIC HEIGHT
Metacentric height GM is calculated by subtracting KG
From KM (GM=KM-KG), GM is a measure of the ship.s
stability. KM=h.
With initial stability(0 – 10 deg.) the metacenter does not
move, and Sine function is almost linear(a straight line).
Therefore, the size of the ship,s Righting Arm, GZ, is
directly prportional to the size of the ship’s Metacentric
Height, GM.

 IMPORTENT !

Thus , GM is a good measure of the ship’s
initial stability.
    METACENTRIC HEIGHT

            m
                   h     L
W           G
                   a
                   a
            C
  MAIN STABILITY POINTS

• There are three main stability
  points:
• m- metacenter is the end of
  hydrostatic force when ship
  listing.
• G- centre of ship gravity
• C- centre of ship underwater
  body.
           SHIP’S STABILITY

     • STABILITY REFERENCE POINTS
                     m
                                    h
      Zm
                     G
WO                              r   Lo
           ZG
                                    a
                Zc   C
      MAIN STABILITY POINTS
 •         m metacenter
 •         G center of gravity
 •         C center of buoyancy
            m
                       Q
                                       L1
       h

                                            Q
Wo                                LO
       a     G

 W1          C             C1
SHIP’S STABILITY

 METACENTER


     m




    C0
        SHIP’S STABILITY

• METACENTRIC HEIGHT FORMULAS
• h=r-a
• h=zm –   zG
• h=zc - ro -   zG
    METACENTRIC HEIGHT
•   METACENTRIC HEIGHT MEENS SHIP’S INITIAL STABILITY



                         m


                                    h
W                         G
                                         r0
                                    a

                         C
          Three states of static equilibrium


         (a) Positive stability - m above G
          (b) Neutral stability – m and   G in
              the same position
        ( c )Negative stability –m below G


m                                G

        h>O     m       G h=O          h<O

G                                 m
    a               b                 c
      POSITIVE SHIP’S STABILITY

     • Positive ship’s stability when m above G
     •                    h>0

                  h
                       m                          L1

     W                  G                              L

W1
                       C          C1
           SHIP’S STABILITY CURVE

                POSITIVE SHIP’S STABILITY
L
    l st
                       h>0




     h
                                    57, 3
                                            Q   Q
 NEUTRAL SHIP’S STABILITY

• Neutral ship’s stability when m and
  G in the same position
•                  h=0

               Gm                  L
  W

               C          C1
       SHIP’S STABILITY

• NEUTRAL SHIP’S STABILITY
 lst
            h=0




                             Q
     NEGATIVE SHIP’S STABILITY

• Negative ship’s stability when m
  below G
•                  h<0


               G                 L1
                        h
W              m                      L
                            C1
W1             C
 NEGATIVE   SHIP’S    STABILITY

             h=-0
Mst



                                   Qst
                     57.3




                                  -h
STABILITY CONDITIONS
The positions of Gravity and the Metacenter will indicate the initial stability
 of a ship.
  Following damage, the ship will assume one of the following three stability
  conditions:
             1. POSITIVE STABILITY. The metacenter is located above
               the ship’s center of gravity.
            As the ship is inclined, Righting Arm are created which tend
              to return the ship to it’s original, vertical position.
             2. NEUTRAL STABILITY. The metacenter and the ship’s
             center of gravity are in the same location. As the ship is inclined,
.             there are no returing moment.
             3. NEGATIVE STABILITY. The ship,s center of gravity is
             above the metacenter.
            As the ship is inclined, negative Righting Arms (called upsetting
            arms) are created which tend to capsize the ship.
                   OVERALL
       METACENTRIC FORMULA
                      M
                                M=( lf —lst)D
     • h=Zm - ZG      m                         L1

            h

                      G   lst
W0                                              L0
                                  Vg

                                       C1       Zm
                          lf                         ZG
W1                   C


                M- UPSERTING MOMENT
        METACENTRIC HIGHT
METACENTRIC HIGHT IS FIRST DERIVATIVE SHIP‖S
  STABILITY CURVE
       Mst
lst




 h
                                               Q
                             57,3
        METACENTER HEIGHT
Metacenter height GM is a determine of ship’s
stability curve
    •                    m                  L1



     W            h                             L
                        G

                                      C1
     W1
                         C

        METACENTER MOMENT IS UPSERTING MOMENT
                      M= D h sin Q
    DYNAMIC STABILITY




W                       L
    SHIP’S DYNAMIC STABILITY

    • DYMAMIC MOMENT
M




                       MDYNAMIC
                       MOMENT




                                  Q
       SHIP’S STABILITY

• STATIC MOMENT CURVE

 M




                          Q
    SHIP’S DYNAMIC STABILITY

    • MAXIMUM DYNAMIC ANGLE
M                                Qdyn WHEN S1= S2
               S2

     S1




          Q   static   Q   dyn       Q dyn max Q
    SHIP’S DYNAMIC CURVE

• SHIP’S DYNAMIC STABILITY CURVES APPLICATES
  IS EQUVALENT STATIC CURVES AREA

                   S=Mdyn
  Mdyn




                        Mdyn

                                               Q
DYNAMIC STABILITY
The dynamic stability is the area under the curve in metre-radians
Multiplated by the ship,s displacement in tonnes. It is areas under
 the GZ
Curve which are required for checking stability criteria which
 depending
Upon the ship,s data may be expressed in metre-degrees or
 metre-radians.
The area unde GZ curve also the potential energy available to
return the
Ship to the upringht.
Principle of conservation of energy, the potential energy
 in converted into
Rotation energy as the ship moves towards the upright.
      DYNAMIC STABILITY
Mst         CURVE



                   Mst             Mdin



Md




                         Q                Q
                             max
    STABILITY   ELEMENTS
THE LAW OF BUOYANCY
THE LAW OF GRAVITY
STABILITY REFERENCE POINTS
LINEAR MESURMENTS IN STABILITY
THE STABILITY TRIANGLE
RIGHTING MOMENT
STATIC STABILITY CURVE
DYNAMIC STABILITY CURVE
ROLLING PERIOD
          Learning Objectives

• Comprehend the concepts of hydrostatics, buoyancy,
  and Archimedes' principle
• Comprehend static equilibrium of a floating vessel and
  the relationship of the centers of gravity and buoyancy
  to righting arms and stability
• Comprehend and identify positive, negative and
  neutral conditions of stability
• Comprehend the effects of movements of the centers of
  gravity and buoyancy on vessel stability
• Know how ship's stability curves are derived and
  comprehend their use in determining stability condition
           Definitions

•   Draft
•   Freeboard
•   Depth of hull
•   Reserve buoyancy
•   List / Trim
     SHIP’S HULL    MARKINGS


At XVIII hundred one Englishman called
PLIMSOL in Great Britan Parlament filds
for marcks on the hull to for Safe shipping.
Now thats marks called PLIMSOL MARKS.
             PLIMSOL DISC

    • PLIMSOL DISC DIVAIDING SHIP‖S
      BODY IN TWO PARTS:
    • 1. RESERVE BUOYANCY
    • 2. DISPLACEMENT



W        L        RESERVE BOYANCY

                  DISPLACEMENT
            FREE BOARD


• SHIP’S MAIN FREE BOARD MEENS SHIP’S
  RESERVE BUOYANCY


•          DRAFT

• SHIP’S MAIN DRAFT MEENS SHIP’S
  DISPLACEMENT
     RESERVE BUOYANCY


• MAINTAIN FREEBOARD – RASERVE
  BUOYANCY PRIOR TO PREVENT
  LIMITING DRAFTS ARE ASSIGNED
  TO EXCESIVE HULL STRESS AS A
  RESULT OF OVERLOADING
     FREE BOARD
FREE BOARD MEENS RESERVE BUOYANCY



                          TF
             FREE BOARD
                          F         S
WL

                                    W

                                    WNA
                DRAFT

• MAIN DRAFT MEENS SHIP‖S DISPLACEMENT




W                                 L

                                      DRAFT
             Buoyancy

• Archimedes' principle
• Calculations of displacement (W)
• The effect of salt water and fresh water
  on displacement (relate to draft)
  [1/35 vs 1/36]
         Archimede’s principle

                   BOYAD
A body immersed (or floating) in water will
buoyed


           ARCHIMEDE’S FORCE

By a force equal to the weight of the water
displaced.
THE LAWS OF BUOYANCY

1. Floatating objects posses the property of buoyancy.
2. A floatating body displaces a volume of water equal in
   a body immersed (or floating) in water will be duoyed
   up by a force equal to the weight of the water displaced
                                           D=Vg

                                 D
                W         G
                                          L

                          C

                                 Vg
    SHIP’S BUOYANCY
• D=V*g


                             L
W             G
                  D


                  V*g
              C



          ARCHIMEDES FORCE
PLIMSOL       MARKS (Load lines)

Markings of minimum allowable freeboard for registred cargo-
Carryng ships.Located amidships on both the port and starboard
sides the ship.
Since the required minimum freeboard varies with water density
 and severity of weather, different markings are used for:

 - TF – Tropical Fresh Water      TF
  - F - Fresh Water                F
  - T - Tropical Water (sea water) T
  - S - Standard Summer                               S
  - W - Winter
                                                       W
  - WNA-Winter North Atlantic
                                                       WNA
SHIP’S HULL MARKINGS
Calculative Draft Marks

Used for determining displacement and other properties
of the ship for stability and damage control.
Those draft marks indicate the depth of the keel (baseline)
below the waterline.


TWO POSIBLE MARKING SYSTEMS:

1. Roman numerals in height
2. Arabic numerals in height
                DRAFT IN FEETS
• 1 ft = 0.3048 m



                 XVII
                 XVI
                 XV
                 XIV
                 XIII
            DRAFT IN METRES

• 1 ft = 0.3048 m
                    44

                    42

                    40
                    38

                    36
SHIP’S HULL MARKINGS

Navigational Draft Marks
Ship’s operational drafts.
These draft marks include the depth of any
projections below the keel of the ship.
Limiting Draft Marks
Limiting drafts are assigned to maintain
reserve buoyancy (freeboard) prior to
damage, and to prevent excessive hull stresses
as a result of overloading.
DISPLACEMENT
The weight of the volume of water that is displaced by the
underwater portion of the hull is equal to the
weight of the ships



GRAVITY
The force of gravity acts vertically downward through the ship’s center
Of gravity. The magnitude of the force depends on the ship’s total weight.



MOMENT
The endency of a force to produce a rotation about a pivot point.
This works like a torque wrench acting on a bolt.
               DISPLACEMENT



•   D=DLS + DS + DC
•   D – Displacement
•   DLS – Weight light ship
•   DS - Weight supply
•   DC - Weight cargo
               GRAVITY
• THE FORCE OF GRAVITY ACTS VERTICALY
  DOWNWARD THROUGHT THE SHIP‖S CENTER OF
 GRAVITY

  W                 G
                        D= DL+DC+DS   L
     SHIP’S STABILITY




• METACENTER MOMENT
  =UPSERTING MOMENT
 M = D h sin O
    RIGHTING MOMENT

• THE TENDENY OF A FORCE TO
  PRODUCE A ROTATION ABOUT
  A PIVOT POINT

         m
         h         M=Dh   sinQ
         G        Vg
              D
                  C1
         C0
                     GRAVITY

• The force of gravity acts vertically downward throught
  the ship’s center of gravity.


•                        D=Vg
                               D
         W                                      L
                         G


                          C

                               Vg
Application of following terms to
 overall stability


(a)Couple

(b)Righting arm (GZ)

(c)Righting moment (RM) - RM= GZ (W)

(d)Upsertting moment
                DEFINITIONS
Couple. Since the forces of buoyancy and gravity are equal and act
along parallel lines, but in opposite directions, a rotation is developed

Righting arm. The distance between the forces of buoyancy and
gravity is know as the ship’s righting arm.


 Righting moment. The righting moment is equal to the ship’s
 Righting arm multiplied by the ship’s displacement.

Metacentric height. The distance between center of gravity G and
Metacener M .
The development of the static stability curve from the
cross curves of stability


Foctors involed:

        - G does not change position as heeling angle
                changes

         - C is always at the geometric center of the volume
              of the underwater hull

         - the shape of the underwater hull changes as
                heeling angle changes
SHIP’S STABILITY CURVE

Using curves,find
        (a) Maximum rigting
arm (GZ) GZ=h
        (b) Angle of heel where
maximum GZ arm ocurs
l static maximum
        (c) Range of critical
stability Q critical
             SHIP’S STABILITY

    • STABILITY CURVES ELEMENTS
     lst



      l static max



h                                      Q

                     57.3
                            Q   critical
           STATIC STABILITY CURVE
 When a ship is inclined through all angles of
                       heel,and the
 righting arm for each angle is measured, the
      statical stability curve is produced. This
    curve is a “snapshot‖of the ship’s stability at
       that particular loading condition.Much
    information can be obtained from this curve,
                         including:
1. Range of Stability: This ship will generate Righting
   Arms when inclined from 0 deg. Till to approximately 74 dg.
2. Maximum Righting Arm: The angle of inclination
           where the maximum Righting Arm occurs
 3. Danger Angle:One half the angle of the maximum
                        Righting Arms.
DRAFT DIAGRAM AND FUNCTIONS
         OF FORM

The Draft Diagram is a nomogram located in
Section II(a) of the Damage Control Book.
It is used for determining the ship’s displacement, as well as other
 properties of the ship, including:
    - Moment to Trim One Inch (MT1);
    - Tons per Inch Immersion (TPI);
    - Height of Metacenter       (KM);
    - Longitudinal Center of Flotation (LCF)
    - Longitudinal Center of Buoyancy(LCB)
  -Displacement (D)
  -VOLUME V m

   -Weight, drafting per 1 cm
      DRAFT NOMOGRAM
8.2     18000   17900   19900    26.5
7.8     17000   16860   18800     26
7.2     16000   15845   17600    25.5
6.8     15000   14840   16600     25
6.4     14000   13840   15500    24.5
6.0     13000   12820   15000     24
5.6     12000   11820   14600    23.5
5.2     11000   10820   14400     23
4.8     10000   9820    14200    22.5
4.4     9000    8820    14000     22
Tm      Dt      Vm3     M t/cm   P t/cm
         HYDROSTATIC CURVES

•   SHIP’S FLOATING BODY FUNCTIONS CAN CALCULATING
    BY HYDROSTATIC CURVES. THIS CURVES IS FUNCTIONS
    FLOATING SHIP’S BODY STABILITY AND UNDERSEA
    SHIP’S BODY CAPITICY.
•   ARGUMENT FOR CALCULATING IS SHIP’S DRAFT
•   FUNCTIONS FOR CALCULATING:
•      a) DISPLACEMENT D
•      b) VOLUME V
•      c) FLOATING CENTER Xf
•      d) BOYAD CENTER     XC Zc
•      e METACENTER RADIUS r
       f) SQUERE OF WATERLINE S
              HYDROSTATIC CURVES

        • SHIP’S FLOATING BODY FUNCTION CURVES
DRAFT


                                                    V
               Zc           r

        Xf
                                                    D
                                     S



                                                 FUNCTIONS
                  COUPLE
                      m
M=D h sin Q
              h           Q
                          l   st

                      G            Vg
                              D
                                        C1


                  C
PLIMSOL DISC
        TF
        F      T


               S

               W

               WNA
     LIST

     Q      L1

             Q Lo
WO
W1
      ROLLING PERIOD
                                             CB
• SHIP‖S STABILITY AND ROLLING PERIOD
                                        T=
                                              h



W                                       L
ROLLING PERIOD
The rolling period of the ship’s dependenced from ship’s stability. The formula
Between ship,s stability and rolling :

       T = c*B/sqr GM

In this formula:
            T – rolling period in sec.
            c - constanta
            B – the ship’s beam to outside of hull.

  Note: the constanta c dependenced from ship’s displacements.
There are the followings meanings:
   c=0.88 – when ship is empty or ballast;
   c=0.78 - when the ship has on board amout 20 %
   c=0.75 – when liquids on board 10%
   c=0.73 – when all liquids on board amout 5%

HOWEVER, for all lagers ships Lloyd’s Register of shipping and the 1991 HMSO
         Code of Practice for Ro-Ro ships use c= 0.7
 SHIP’S STABILITY VARIATIONS

• LOADING CARGO
                m0

                                      h0
                  G0


                  C0




STABILITY REFERENCES POINTS BEFORE LOADING
 SHIP’S STABILITY VARIATIONS
                 h0 < h1
• LOADING CARGO IN HOLD
                   m1
                m0
   h1
                                       h0
                  G0
                   G1
                  C1
                  C0
                           p


 STABILITY REFERENCES POINTS AFTER LOADING
 SHIP’S STABILITY VARIATIONS
                       h0 >h1
• LOADING CARGO AT DECK
                            m1
             P1   m0             P2
   h1
                            G1         h0

                       G0
                            C1
                   C0




 STABILITY REFERENCES POINTS AFTER LOADING
 SHIP’S STABILITY VARIATIONS

• MOVING CARGO
               m0

                                      h0
                 G0


                 C0




STABILITY REFERENCES POINTS BEFORE MOVING
   SHIP’S STABILITY VARIATIONS

 • MOVING CARGO
                                     P2
           P1     m0

                                          h0
                    G0


                    C0




STABILITY REFERENCES POINTS BEFORE MOVING DOWN
  SHIP’S STABILITY VARIATIONS
                                   h1 > h0
• MOVING CARGO
                 m0

                                        h0      h1
                   G0

                      G1
                   C0
            P1                    P2



STABILITY REFERENCES POINTS AFTER MOVING DOWN
    SHIP’S STABILITY VARIATIONS

  • MOVING CARGO
                   m0

                                         h0
                     G0


                     C0
             P1           P2



STABILITY REFERENCES POINTS BEFORE MOVING UPWARD
   SHIP’S STABILITY VARIATIONS
                    h0 > h1

 • MOVING CARGO                      P2
          P1      m0
                                                  h1
                       G1                 h0
                       G0

                    C0




STABILITY REFERENCES POINTS AFTER MOVING UPVARD
     SHIP’S STABILITY VARIATIONS

• LOADING CARGO

                   m
                                   h0
                                        h1
                   G0   G1     L0
W0

                  C0
    SHIP’S STABILITY VARIATIONS

•         FREE LIQUID AREA




                  G0
W0                           L0


              C0


             P0
                            M Moment liquid

    SHIP’S STABILITY VARIATIONS
                            M Moment upserting
•         FREE LIQUID AREA
              m


                                           L1
                  G0
                                                Q
W0                                    L0
                             C1
W1            C0

                             P1
                       P1
 SHIP’S STABILITY VARIATIONS
                            M1
• FREE LIQUID AREA

               Y1


   Q1                P1          M2



                     Y2


                            P2
 M2>M1              Q2>Q1
                               Mcargo


     SHIP’S STABILITY VARIATIONS

 •         HANGING CARGO                     Q

                                             lz




 W0                             L1
                                        L0
                                             P


W1

           Mcargo= Pcargo lz sin Q
                           TRIM
Trim means different between draft fore TF and draft aft TAF




W1

W                                                               L

       TAF                                                     L1
                                                        TF
    TAf
              SHIP’S TRIM DIAGRAM
 m
9
8

7

6

5

4

3

2

1                                         Tf
          2    3   4   5   6   7    8   9 m
             SHIP’S TRIM DIAGRAM
  Dt


4000

3600

3200
2800
2400

1600
1200

                                                 3
   -5   -4    -3   -2   -1    0   1   2   Xc m
                             0
                    SHIP’S STABILITY VARIATIONS


                           TRIM
Trim means different between draft fore TF and draft aft TAF




W1

W                                                               L
                                    lx
       TAF                                                     L1
                P
                                                        TF

      SHIP’S TRIM BEFORE SHIFTING CARGO
                             Mdif

                              DH
                   SHIP’S STABILITY VARIATIONS


                             TRIM
  Trim means different between draft fore TF and draft aft TAF

                                            P lx
                                      d=            L
                                            DH
  W1

  W                                                               L

         TAF0      P                    P                        L1
TAF1                             lx                                   d

                                                   TF0     TF1
                             L

         SHIP’S TRIM AFTER SHIFTING CARGO
     LIST

     Q      L1

             Q Lo
WO
W1
     SHIP’S STABILITY VARIATIONS


                   LIST

               P


                                         Lo
WO




     SHIP’S LIST BEFORE SHIFTING CARGO
     SHIP’S STABILITY VARIATIONS

                   LIST

                            ly
               P                        P
                                            L1


WO                                 Q         Lo

W1



                   tg Q =   P ly
                            Dh
     SHIP’S LIST AFTER SHIFTING CARGO

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:477
posted:11/29/2011
language:English
pages:102