# eccentricity

## Analysis of Gravity Dam

Dams are structures whose purpose is to raise the water level on the upstream side of river, stream, or other waterway. The rising water will cause hydrostatic force which will tend the dam to slide horizontally and overturn about its downstream edge or toe. The raised water level on the upstream edge or heel will also cause the water to seep under the dam. The pressure due to this seepage is commonly called hydrostatic uplift and will reduce the stability of the dam against sliding and against overturning.

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## Total Hydrostatic Force on Surfaces

**Total Hydrostatic Force on Plane Surfaces**

For horizontal plane surface submerged in liquid, or plane surface inside a gas chamber, or any plane surface under the action of uniform hydrostatic pressure, the total hydrostatic force is given by

where *p* is the uniform pressure and *A* is the area.

In general, the total hydrostatic pressure on any plane surface is equal to the product of the area of the surface and the unit pressure at its center of gravity.

where *p _{cg}* is the pressure at the center of gravity. For homogeneous free liquid at rest, the equation can be expressed in terms of unit weight

*γ*of the liquid.

where $\bar{h}$ is the depth of liquid above the centroid of the submerged area.

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## 255 Equivalent loads to a compression member with eccentric load

**Problem 255**

A short compression member carries an eccentric load P = 200 lb situated 2 in. from the axis of the member, as shown in Fig. P-225. In strength of materials it is learned that the internal stresses are determined from the equivalent axial load and couple into which P may be resolved. Determine the equivalent axial load and couple.

## The Parabola

### Definition of Parabola

Parabola is the locus of point that moves such that it is always equidistant from a fixed point and a fixed line. The fixed point is called focus and the fixed line is called directrix.

### General Equations of Parabola

From the general equation of all conic sections, either $A$ or $C$ is zero to form a parabolic section.

For $A = 0$, the equation will reduce to $Cy^2 + Dx + Ey + F = 0$ or

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## The Hyperbola

## Definition

Hyperbola can be defined as the locus of point that moves such that the difference of its distances from two fixed points called the foci is constant. The constant difference is the length of the transverse axis, 2a.

## General Equation

From the general equation of any conic (A and C have opposite sign, and can be A > C, A = C, or A < C.)

$Ax^2 - Cy^2 + Dx + Ey + F = 0 \,$ or

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## Elements of Ellipse

Elements of the ellipse are shown in the figure below.

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## The Ellipse

**Definition of Ellipse**

Ellipse is the locus of point that moves such that the sum of its distances from two fixed points called the foci is constant. The constant sum is the length of the major axis, 2*a*.

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## Conic Sections

**Definition**

Conic sections can be defined as the locus of point that moves so that the ratio of its distance from a fixed point called the focus to its distance from a fixed line called the directrix is constant. The constant ratio is called the eccentricity of the conic.

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