## Length of Arc

## Length of Arc in Rectangular Plane

Apply Pythagorean theorem to the triangular strip shown in the figure:

$(ds)^2 = (dx)^2 + (dy)^2$ ← Equation (1)

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Physical Properties of Soil
## Phase Diagram of Soil

Soil is composed of solids, liquids, and gases. Liquids and gases are mostly water and air, respectively. These two (water and air) are called voids which occupy between soil particles. The figure shown below is an idealized soil drawn into phases of solids, water, and air.

**Weight-Volume Relationship from the Phase Diagram of Soil**

total volume = volume of soilds + volume of voids

$V = V_s + V_v$

volume of voids = volume of water + volume of air

$V_v = V_w + V_a$

total weight = weight of solids + weight of water

$W = W_s + W_w$

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Apply Pythagorean theorem to the triangular strip shown in the figure:

$(ds)^2 = (dx)^2 + (dy)^2$ ← Equation (1)

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The Cone
**Cone**
The surface generated by a moving straight line (generator) which always passes through a fixed point (vertex) and always intersects a fixed plane curve (directrix) is called conical surface. Cone is a solid bounded by a conical surface whose directrix is a closed curve, and a plane which cuts all the elements. The conical surface is the lateral area of the cone and the plane which cuts all the elements is the base of the cone.

Like pyramids, cones are generally classified according to their bases.

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Solids for which V = (mean B)h

* Frustum* of a pyramid (or cone) is a portion of pyramid (or cone) included between the base and the section parallel to the base not passing through the vertex.

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The Quadrilateral

**Quadrilateral** is a polygon of four sides and four vertices. It is also called *tetragon* and *quadrangle*. For triangles, the sum of the interior angles is 180°, for quadrilaterals the sum of the interior angles is always equal to 360°

$A + B + C + D = 360^\circ$

**Classifications of Quadrilaterals**

There are two broad classifications of quadrilaterals; *simple* and *complex*. The sides of simple quadrilaterals do not cross each other while two sides of complex quadrilaterals cross each other.

Simple quadrilaterals are further classified into two: *convex* and *concave*. Convex if none of the sides pass through the quadrilateral when prolonged while concave if the prolongation of any one side will pass inside the quadrilateral.

The following formulas are applicable only to convex quadrilaterals.

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The Triangle

**Definition of a Triangle**

Triangle is a closed figure bounded by three straight lines called sides. It can also be defined as polygon of three sides.

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The Three-Moment Equation

The three-moment equation gives us the relation between the moments between any three points in a beam and their relative vertical distances or deviations. This method is widely used in finding the reactions in a continuous beam.

Consider three points on the beam loaded as shown.

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Solution to Problem 341 | Torsion of thin-walled tube

**Problem 341**

Derive the torsion formula τ = Tρ / J for a solid circular section by assuming the section is composed of a series of concentric thin circular tubes. Assume that the shearing stress at any point is proportional to its radial distance.

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Sum and Product of Roots

The quadratic formula

$x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

give the roots of a quadratic equation which may be real or imaginary. The ± sign in the radical indicates that

$x_1 = \dfrac{-b + \sqrt{b^2-4ac}}{2a}$ and $x_2 = \dfrac{-b - \sqrt{b^2-4ac}}{2a}$

where x_{1} and x_{2} are the roots of the quadratic equation ax^{2} + bx + c = 0. The sum of roots x_{1} + x_{2} and the product of roots x_{1}·x_{2} are common to problems involving quadratic equation.

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Derivation of Quadratic Formula

The roots of a quadratic equation *ax*^{2} + *bx* + *c* = 0 is given by the quadratic formula

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The derivation of this formula can be outlined as follows:

- Divide both sides of the equation
*ax*^{2}+*bx*+*c*= 0 by*a*. - Transpose the quantity
*c*/*a*to the right side of the equation. - Complete the square by adding
*b*^{2}/ 4*a*^{2}to both sides of the equation. - Factor the left side and combine the right side.
- Extract the square-root of both sides of the equation.
- Solve for
*x*by transporting the quantity*b*/ 2*a*to the right side of the equation. - Combine the right side of the equation to get the quadratic formula.

See the derivation below.

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