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Section 9

Displacement and WeightRepresented by :Amr kunperAhmed amrAbdalarhman maherAhmed TahaMayar MohamedNouran Maged

ContentsHydrostatic Forces and Moments; Archimedes Principle.Numerical IntegrationAreas, Volumes, Moments, Centroids, and Moments of Inertia .Weight Estimates, Weight Schedule .Hydrostatic Stability .

Hydrostatic Forces and Moments; Archimedes Principle.

Archimedes Principle

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P= Po gh Where:Po : is the atmospheric pressure acting on the Surface.

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Hydrostatic ForceA 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.

By application of Gauss theorem the surface integrals are converted to volume integrals, so

Where: k 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 forceIts moment about the origin is;

This Equations are the twin statements of Archimedes principle

Hydrostatic Force

Numerical Integration

Numerical IntegrationMany 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

Two fundamental steps:Adoption of some continuous function of x (the interpolate that matches the given ordinates at the given abscissae)Integration of the continuous interpolant over the given interval.Methods of interpolationsSum of Trapezoids ( general )Trapezoidal RuleSimpsons First Rule

Numerical Integration

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 spacedThe area from xi-1 to xi is (yi-1 + yi) (xi - xi-1)/2, so the integral is approximated by

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

Trapezoidal Rule :

Simpsons First RuleThe tabulation is at uniformly spaced abscissae the number of intervals is evenThe function is known to be free of discontinuities in both value and slope.

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

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

Rcontour Cxy

The first moments of area

Offshore Structures !

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xyz

Weight Estimates & Weight Schedule

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 andits center of mass must be on the same vertical line as the center of buoyancy.

Weight EstimatesThe 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.

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.

The mass moments of inertia of the complete ship about its center of mass are obtained from the parallel axis theorem.

Theparallel axis theorem :used to determine themass moment of inertiaor thesecond moment of areaof arigid bodyabout any axis..

Weight Estimates

Weight ScheduleThe weight schedule is a table of ..

weights centroids moments

arranged to facilitate the above calculations.

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

Often it is useful to categorize weight components into groups.. e.g. Hull , propulsion , tanks , cargoSome component weights can be treated as pointse.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

Weight ScheduleComplex-Shaped volumes or solidsWeights 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 .

Weight Schedulemonitoring weights and center of gravityThe 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.

Weight ScheduleWeight analysis and flotation calculationsWeight 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 containa geometric description of the ship and its partitioning into cargo spaces and tanks.

Hydrostatic Stability

Hydrostatic StabilityArchimedes 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

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

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

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 .

4-the ball resting at the saddle point on a saddle-shaped surfaceIn 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

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

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 freedomrotation about a vertical axis results in no change in volume or restoring moment, so is a neutrally stable degree of freedom.

The vertical direction is more interestingA rigid body floating in equilibrium with positive water plane area AwpIs 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.

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

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.

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

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:

Represented by :Amr kunperAhmed amrAbdalarhman maherAhmed TahaMayar MohamedNouran Maged