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1.2.4. H.f. on curvilinear surfaces

We note that dF is no longer a parallel system of forces! Therefore they (dF) no longer reduce to

a unique force (resultant force), but to a resultant force F and a resultant moment M o.

In Civil Engineering the curvilinear surfaces used in all cases are regular surfaces: cylindrical,

spherical etc. or their combinations. In these cases the distributed forces F d on the regular

surface reduce again to a unique force F applied in the center of mass of the body defined by

that surface. To compute the resultant force F we proceed as follows:

- the surface A is projected on the 3 planes Oxy, Oyz and Ozx and we get 3 plane surfaces

we know already to compute the resultant forces zF , xF and yF .

- the resultant force F is given by

z y x F F F F (vectorally)

or in the elementary form: z y x F d F d F d F d

On the other hand we have:

x x x x zdAdA pdF dF

where: x x x OF ,cos

Similarly: z z

y y

dA zdF

dA zdF

And

)( )(

)( )(

Az Az

z z z

y y

Ax Ax

x x x x

GV dA zdA zF

SF

S zdAdF F

V

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V- the volume contained by the same surface and its projection on the free (horizontal) water

surface.

Synthesis:

-

whatever the surface of the element is, the hydrostatic force is the resultant on that surfaceof the water pressure;

- the water pressure is computed fro the fundamental equation of hydrostatics;

- the fundamental equation of hydrostatics is defined versus an xOy plan, which is the plan

of the water free surface;

- the hydrostatic force depends on the water depth and on the magnitude of the water

bottom surface (next figure);

The floating of bodies

Law of Archimedes –

on a body a liquid in rest acts on a body sunk in that liquid upwards with aforce equal to the weight of the volume of the displaced liquid. Aiming at getting the

mathematical form of this la, we take a sunk body with a paralelipipedical form.

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body

a

V zba

z zba zba zbaF

zbaF

zbaF

1212

22

11

Depending on the ratio G/Fa, we have 3 different situations:

- G<Fa – the body floats on the liquid while a certain part of it is above the liquid;

- G=Fa – the body is floating inside the liquid;

- G>Fa – the body goes to the water bottom.

Every case has application in engineering. For example, the submarine may be found in any of

the above cases. In Civil Engineering (floating bridge construction) the most common is case 1.

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HYDRODYNAMICS

2.1. Introductions to HydrodynamicsThe mathematical model associated to a liquid in motions is much more complicated than the

model associated with a liquid in rest. This is because every liquid particle has its own motions,

quite different from the motions of the particles of a rigid body.

The hydrodynamic analysis requires several introductory notions:

- Flow line (linie de curent). It is an imaginary line which has the property that in any of its points

the speed vector V is tangent to it.

- Flow stream ( firul de current ). It is a material line containing the liquid particles in the flow

line.

- Flow tube (tub de current ). It is a whole number of flow lines that have a closed line has their

directions. It is not necessarily material.

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- Liquid flow (curent de lichid). It is the whole number of flow lines contained in flow tube.

- Yield flow (flux sau debit lichid). It is the volume of liquid that passes through a unit surface in

the unit of time.

- Live area A (area vie). I is the cross sectional area of the liquid flow.

- Wetted perimeter P ( perimetru udat). This is the length of the contact line of the liquid with the

tube.

- Hydraulic radiusP

A R

- Hydraulic gradient I ( panta hidraulica). In the case of free surface streams, or unrestricted flow,

I is the natural gradient of the ideal liquid surface.

l

h I

In case of pressure flow

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l

h I

- Roughness (rugozitate). It is the degree of finishing of the surface of the hydraulics

transportation means.

- Classification of liquid flows (clasificarea curentilor de lichid):

According to flow location:

o one-dimensional flows M(z)

o two-dimensional flows M (y, z)

o free-dimensional flows M(x, y, z)

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