Hydrostatic Forces and Moments; Archimedes’ Principle. Numerical Integration Areas, Volumes, Moments, Centroids, and Moments of Inertia . Weight Estimates, Weight Schedule . Hydrostatic Stability

1. Section 9
Displacement and Weight
REPRESENTED BY :
• AMR KUNPER
• AHMED AMR
• ABDALARHMAN MAHER
• AHMED TAHA
• MAYAR MOHAMED
• NOURAN MAGED

2. Contents
• Hydrostatic Forces and Moments; Archimedes’ Principle.
• Numerical Integration
• Areas, Volumes, Moments, Centroids, and Moments of Inertia .
• Weight Estimates, Weight Schedule .
• Hydrostatic Stability .

3. Hydrostatic Forces
and
Moments; Archimedes’ Principle.

4. Archimedes Principle

5. P= Po – ρgh
Where:
Po : is the atmospheric pressure acting
on the Surface.

6. Hydrostatic Force
• A solid boundary in the fluid is subject to a force on any differential area
element ds equal to the static pressure p times the area of element .
• The contribution to force is:
where n is the unit normal vector.
• The contribution to moment about the origin is:
where r is the radius vector from the origin to the surface element.

7. By application of Gauss’ theorem the surface integrals are converted to
volume integrals, so
Where:
• kˆ is the unit vector in the vertical upward direction
• V is the displaced volume.
Because this force is vertically upward, it is called the “buoyant force”
Its moment about the origin is;
This Equations are the twin statements of “Archimedes’ principle”
Hydrostatic Force

8. Numerical Integration

9. Numerical Integration
• Many of the formulas involved in calculation of hydrostatic and mass
properties are expressed in terms of single or multiple integrals. (Multiple
integrals are computed as a series of single integrals )
The integral expression :
only meaningful if y is defined at all values of x in the range of integration,
a to b. ( main condition )
representing the area between the curve
y-axis and the x-axis

10. Two fundamental steps:
1. Adoption of some continuous function of x (the interpolate that
matches the given ordinates at the given abscissae)
2. Integration of the continuous interpolant over the given interval.
Methods of interpolations
1. Sum of Trapezoids ( general )
2. Trapezoidal Rule
3. Simpson’s First Rule
Numerical Integration

11. Sum of Trapezoids: ( general )
• The simplest interpolant is a piecewise linear function joining the
tabulated points (xi, yi) with straight lines.
• When the xi are irregularly spaced
The area from xi-1 to xi is (yi-1 + yi) (xi - xi-1)/2, so the integral is
approximated by

13. Trapezoidal Rule :
When the tabulation is at uniformly spaced abscissae , then the intervals in
equation are constant, xi – xi-1 = Δx =h

14. Trapezoidal Rule :

15. Simpson’s First Rule
• The tabulation is at uniformly spaced abscissae the number of intervals is even
• The function is known to be free of discontinuities in both value and slope.

16. Areas, Volumes, Moments, Centroids,
and
Moments of Inertia .

17. Areas, Volumes, Moments, Centroids,
and Moments of Inertia .
ⅆ𝐴 = ⅆ𝑥 ⅆ𝑦
𝐴 =
𝑅
ⅆ𝑥 ⅆ𝑦
𝑉 = ⅆ𝑥 ⅆ𝑦 ⅆ𝑧 = ⅆ𝑉
𝑆 𝑥 = ⅆ𝑦 ⅆ𝑧
is the area of a plane section normal to
the x axis at location x, the so-called
section area curve .
𝑉 = 𝑆 𝑥 ⅆ𝑥

18. R
contour C
x
y
𝐶 = 𝜕𝑅
Green’s theorem allows some area integrals to be expressed
as line integrals around the boundary 𝜕𝑅. In general 2-D form,
𝑅
𝜕𝑄
𝜕𝑥
−
𝜕𝑃
𝜕𝑌
ⅆ𝑥ⅆ𝑦
=
𝜕𝑅
𝑃ⅆ𝑥 + 𝑄ⅆ𝑦
where P and Q are arbitrary differentiable functions of x and y.
• P = -y , Q = 0
𝐴 =
𝑅
ⅆ𝑥 ⅆ𝑦 = −
𝜕𝑅
𝑦ⅆ𝑥
• P = 0 , Q = x
𝐴 =
𝑅
ⅆ𝑥 ⅆ𝑦 =
𝜕𝑅
𝑥ⅆ𝑦
𝐴 = 𝑖=0
𝑁
𝐴𝑖, where 𝐴𝑖= 𝑐𝑖
𝑥 ⅆ𝑦
𝐶𝑖, 𝑖 = 0 , … . , 𝑁−1
𝑋𝑖, 𝑖 = 0 , … . , 𝑁−1
𝑋𝑖 𝑋𝑖, 𝑦𝑖
𝑋0= 𝑋 𝑁−1

19. Segment 𝑖 runs from 𝑋𝑖−1 to 𝑋𝑖; it can be parameterized as :
𝑋𝑖−1 1 − 𝑡 + 𝑋𝑖t , with 0 ≤ 𝑡 ≤ 1
𝑋 = 𝑋𝑖−1 1 − 𝑡 + 𝑋𝑖t
y = 𝑦𝑖−1 1 − 𝑡 + 𝑦𝑖t
So ⅆ𝑦 = 𝑦𝑖 − 𝑦𝑖−1 ⅆ𝑡
𝐴𝑖=
𝑋𝑖−1+ 𝑋𝑖 𝑦𝑖 − 𝑦𝑖−1
2

20. The first moments of area
This same general scheme can be applied to compute integrals of other
polynomial quantities over arbitrary polygonal regions.
The first moments of area of R with respect to x and y are defined as :
𝑚 𝑥 , 𝑚 𝑦 =
𝑅
𝑥 , 𝑦 ⅆ𝑥 ⅆ𝑦
𝑚 𝑥 = 𝜕𝑅
−𝑥𝑦ⅆ𝑥 , 𝑚 𝑦 = 𝜕𝑅
−𝑦2/2ⅆ𝑥

21. Moments of inertia ∶
𝐼 𝑥𝑥 , 𝐼 𝑥𝑦 , 𝐼 𝑦𝑦 =
𝑅
𝑦2
, 𝑥𝑦, 𝑥2
ⅆ𝑥 ⅆ𝑦
The centroid (center of area) is the point with coordinates :
𝑚 𝑥
𝐴
,
𝑚 𝑦
𝐴
The 3-D moments of displaced volume for a ship are :
• 𝑀 𝑥 = 𝑥 ⅆ𝑥 ⅆ𝑦 ⅆ𝑧 = 𝑥 𝑆 𝑥 ⅆ𝑥 = 𝑚 𝑥 𝑥 ⅆ𝑥
• 𝑀 𝑦 = 𝑦 ⅆ𝑥 ⅆ𝑦 ⅆ𝑧 = 𝑦 𝑆 𝑥 ⅆ𝑥 = 𝑚 𝑦 𝑥 ⅆ𝑥
• 𝑀 𝑦 = 𝑧 ⅆ𝑥 ⅆ𝑦 ⅆ𝑧 = 𝑧 𝑆 𝑥 ⅆ𝑥 = 𝑚 𝑧 𝑥 ⅆ𝑥
The 3-D centroid (CB, center of buoyancy) is the point with coordinates :
𝑀 𝑥
𝑉
,
𝑀 𝑦
𝑉
,
𝑀 𝑧
𝑉

22. Offshore Structures !

23. 𝒂 = (𝒙 𝟐 − 𝒙 𝟏 ) ∗ ( 𝒙 𝟑 − 𝒙 𝟏 )/𝟐
• Points on the panel are parameterized with parameters 𝑢, 𝑣 as :
𝑥 𝑢 , 𝑣 = 𝑥1 + 𝑥2 − 𝑥1 𝑢 + 𝑥3− 𝑥1 𝑣
• Integrations over the triangle that is the vertical projection of the
panel onto the z = 0 plane :
𝑄 𝑥, 𝑦 ⅆ𝑥 ⅆ𝑦 = 2𝐴
0
1
0
1−𝑣
𝑄 𝑥, 𝑦 ⅆ𝑢 ⅆ𝑣
• The volume of the prism are evaluated as :
𝑉 =
𝐴 𝑧1 + 𝑧2 + 𝑧3
3
x
y
z
𝒙 𝟏
𝒙 𝟐𝒙 𝟑

24. • The moments of volume of the prism are evaluated as :
𝑀 𝑥= 𝐴[ 𝑥1 + 𝑥2 + 𝑥3 𝑧1 + 𝑧2 + 𝑧3 + 𝑥1 𝑧1 + 𝑥2 𝑧2 + 𝑥3 𝑧3 ]/12
𝑀 𝑦= 𝐴[ 𝑦1 + 𝑦2 + 𝑦3 𝑧1 + 𝑧2 + 𝑧3 + 𝑦1 𝑧1 + 𝑦2 𝑧2 + 𝑦3 𝑧3 ]/12
𝑀𝑧= 𝐴[ 𝑧1 + 𝑧2 + 𝑧3
2
+ 𝑧1
2
+𝑧2
2
+ 𝑧3
2
]/24
• The waterplane area of this prism is A, with its centroid at :
( 𝑥1 + 𝑥2 + 𝑥3 )/3, ( 𝑦1 + 𝑦2 + 𝑦3 )/3,0
• Its contributions to the waterplane moments of inertia are:
𝐼 𝑥𝑥= 𝐴 𝑦1
2
+𝑦2
2
+ 𝑦3
2
+ 𝑦1 𝑦2 + 𝑦2 𝑦3 + 𝑦3 𝑦1 /6
𝐼 𝑥𝑦 = 𝐴 𝑥1 + 𝑥2 + 𝑥3 𝑦1 + 𝑦2 + 𝑦3 + 𝑥1 𝑦1+ 𝑥2 𝑦2 + 𝑥3 𝑦3 /12
𝐼 𝑦𝑦 = 𝐴 𝑥1
2 +𝑥2
2 + 𝑥3
2 + 𝑥1 𝑥2+ 𝑥2 𝑥3 + 𝑥3 𝑥1 /12

25. Weight Estimates
&
Weight Schedule

26. Weight Estimates
Archimedes’ principle
states the conditions for a body to float in equilibrium:
its weight must be equal to that of the displaced fluid and
its center of mass must be on the same vertical line as the center of
buoyancy.

27. Weight Estimates
• The intended equilibrium will only be obtained if ..
The vessel is actually built
& loaded with the correct weight and weight distribution.
• Preparation of a reasonably accurate weight estimate is therefore a
critical step in the design of essentially any vessel, regardless of size.

28. Weight Estimates
• Weight is :
the product of Mass times acceleration due to gravity, g.
• The total mass:
the sum of all component masses,
the center of mass (or center of gravity) can be figured by accumulating x, y, z
moments:
where mi is a component mass ,{xi, yi, zi} is the location of its center of mass.

29. • The mass moments of inertia of the complete ship about its
center of mass are obtained from the parallel axis theorem.
• The parallel axis theorem :
used to determine the mass moment of inertia or the second
moment of area of a rigid body about any axis..
Weight Estimates

30. Weight Schedule
The weight schedule is a table of ..
• weights
• centroids
• moments
arranged to facilitate the above calculations.

31. Today it is most commonly maintained as a spreadsheet.
Advantage:
that its totals can be updated continuously as component weights are
added and revised.
Weight Schedule

32. • Often it is useful to categorize weight components into groups..
e.g. Hull , propulsion , tanks , cargo
• Some component weights can be treated as points
e.g. an engine or an item of hardware
• Some weights are distributed over curves and surfaces;
their mass calculation has been outlined in Sections 3 and 4
Weight Schedule

33. Weight Schedule
Complex-Shaped volumes or solids
Weights that are complex-shaped volumes or solids are generally the
most difficult to evaluate..
for example : ballast castings and tank contents
In this case the general techniques of volume and centroid computation
developed for hydrostatics could be used .

34. Weight Schedule
monitoring weights and center of gravity
The architect, builder, and owner/operator all have an interest in monitoring
weights and center of gravity throughout construction and outfitting ,
so the.. flotation, stability, capacity , performance requirements, and
objectives are met when the vessel is placed in service.

35. Weight Schedule
Weight analysis and flotation calculations
Weight analysis and flotation calculations are an ongoing concern during
operation of the vessel,
as .. cargo and stores are loaded and unloaded.
Often this is performed by on-board computer programs which contain
a geometric description of the ship and its partitioning into cargo spaces and
tanks.

36. Hydrostatic Stability

37. Hydrostatic Stability
• Archimedes’ principle provides necessary and sufficient conditions for a
floating object to be in equilibrium.
• However, further analysis is required to determine whether such an
equilibrium is stable.
• The general topic of stability of equilibrium examines whether, following a
small disturbance that moves a given system away from equilibrium, the
system tends to restore itself to equilibrium, or to move farther away from
it

38. 1-A ball resting at the low point of a concave surface is a
prototype of stable equilibrium
If the ball is pushed a little away from center, it tends to roll back

39. 2-The same ball resting at a Maximum of a convex surface is a typical
unstable equilibrium
Following a small displacement in any direction the ball tends to accelerate
away from its initial position

40. 3-On the boundary between stable and unstable behavior, there is neutral
stability represented by a ball on a level plane.
• There is no tendency either to return to an initial equilibrium, or to
accelerate away from it Stability can depend on the nature of the
disturbance .

41. 4-the ball resting at the saddle point on a saddle-shaped surface
• In this situation, the system is stable with respect to disturbances in one
direction and simultaneously unstable with respect to disturbances in
other directions
• A ship can be stable with respect to a change of pitch and unstable with
respect to a change in roll, or (less likely) vice versa

42. • In order to be globally stable, the system must be stable with respect to all
possible “directions” of disturbance, or degrees of freedom
A 3-D rigid body has in general six degrees of freedom: linear displacement
along three axes and rotations with respect to three axes.

43. • Let us first examine hydrostatic stability with respect to linear
displacements.
• When a floating body is displaced horizontally, there is no restoring
force arising from hydrostatics.
This results in neutral stability for these two degrees of freedom
rotation about a vertical axis results in no change in volume or
restoring moment, so is a neutrally stable degree of freedom.

44. • The vertical direction is more interesting
• A rigid body floating in equilibrium with positive water plane area Awp
• Is always stable with respect to vertical displacement. If the disturbance is a
small positive (upward) displacement in z, say dz, the displaced volume
decreases (by -Awpdz), decreasing buoyancy relative to the fixed weight, so
the imbalance of forces will tend to return the body to its equilibrium
flotation.

45. The two remaining degrees of freedom are rotations about horizontal axes; for
example, for a ship, trim (rotation about a transverse axis) and heel (rotation
about a longitudinal axis).
the centroid of waterplane area, also known as the center of flotation (CF), is a
pivot point about which small rotations can take place with zero change of
displacement; and the stability of these degrees of freedom depends on the
moments of inertia of the waterplane area about axes through the CF

47. • Because these coefficients pertain to small displacements from an
equilibrium floating attitude, they are called transverse and
longitudinal initial stabilities.
• Their dimensions are moment/radian (i.e., force × length / radian).
They are usually expressed in units of moment per degree.

48. • Initial stability is increased by increased moment of inertia of the
waterplane, increased displacement, a higher center of buoyancy, and a
lower center of gravity.
• Because of the elongated form of a typical ship, the longitudinal initial
stability is ordinarily many times greater than the transverse initial stability

49. It is common to break these formulas in two, stating initial stabilities in
terms of the heights of transverse and longitudinal metacenters Mt and
Ml above the center of gravity G:
In terms of metacentric heights, in this notation, the initial stabilities
become simply:

50. REPRESENTED BY :
• AMR KUNPER
• AHMED AMR
• ABDALARHMAN MAHER
• AHMED TAHA
• MAYAR MOHAMED
• NOURAN MAGED