Strength of Materials I: Moment of Area

ME 255 Strength of Materials I Lecture 1: Review of Moment of Area

A moment about a given axis is defined as something multiplied by the distance from that axis measured at 90° to the axis.
The moment of force can be expressed as:
- Moment of Force = Force $ imes$ Distance from an Axis
The moment of mass is defined in a similar manner:
- Moment of Mass = Mass $ imes$ Distance from an Axis
The moment of area is calculated as:
- Moment of Area = Area $ imes$ Distance from an Axis

(Figure illustrating moments could be included here)

For mass and area, the challenge lies in determining the correct distance, given that mass and area are not concentrated at a singular point.
The point where we can assume mass is concentrated is termed the centre of gravity.
Conversely, the point of assumption for area concentration is known as the centroid.
Visualizing area as a thin sheet, the centroid coincides with the center of gravity.
- This is the point on which the thin sheet can balance perfectly on a sharp point without tipping in any direction.

Considering a flat area positioned at some distance from an axis (denoted as G):
- The centroid's distance from the axis s-s is denoted as $ar{y}$.
- The axis drawn through G parallel to s-s is illustrated as the axis g-g.
- The first moment of area about the axis (s-s) is computed as the product of the area (A) and distance ($ar{y}$).

Upon comparing the equations, if an area possesses an axis of symmetry, then:
- First moment of area with respect to that axis is zero, leading to:
- Q_y = 0
- It follows that the centroid C lies on that axis.

Determine the first moments of area for a rectangular area.
- Given:
- Area of Rectangle A = b imes h
- First Moment about x-axis:
  Q_y = A ar{y} = b h rac{1}{2} h
- Thus, Q_y = rac{1}{2} b h^2
- First Moment about y-axis:
  Q_x = A ar{x} = b h rac{1}{2} b
- Leading to Q_x = rac{1}{2} b^2h

Finding the first moment of area of a circular area touching its edge in terms of diameter (d).
- For a circle, the centroid is at the geometric center, at a distance of $rac{D}{2}$ from the edge.
- Area:
  A = rac{eta D^2}{4}
- First moment about x-axis:
  Q_x = A ar{y} = rac{eta D^2}{4} rac{D}{2} = rac{eta D^3}{8}

For a triangular area, calculate:
1. The first moment $Q_x$ of the area.
2. The ordinate of the centroid of the area.
From similar triangles:
ext{Area} = rac{1}{2} b h
First moment equation; integrating to find $Qx$: Qx = ext{Integral}(y dA)
- The expression needs careful analysis of the limits involved for optimal integration application.

A composite area A can be divided into simpler geometric shapes; thus, the first moments can be treated individually and combined:
- Qx = A1 ar{y1} + A2 ar{y2} + … + An ar{y_n}
- Total first moment $Qx$: Qx = ext{sum of all first moments}
- Centroid C can also be computed using:
  ar{y} = rac{Q_x}{A}

Determine the first moments through the centroidal axes and locate centroid C of Area A.
Using summations of area moments gives the total centroid coordinates.
- Suppose:
  A = 4000 mm^2, Y = rac{184000}{4000} = 46 mm

The moment of area, when multiplied by the perpendicular distance between the centroid of the area and an axis, results in a quantity known as the moment of inertia.
It is essential for analyzing the properties of beams and structural elements under loading.
The second moment about the x-axis is defined as:
I_x = ext{Integral}(y^2 dA)
About the y-axis:
I_y = ext{Integral}(x^2 dA)

Defined as the sum of the area multiplied by the square of the distance from an axis 90° to the area plane:
J = ext{Integral}(r^2 dA) where r denotes the radial distance from the axis.
The formula can also be expressed in terms of Cartesian coordinates:
J_0 = ext{Integral}(x^2 + y^2 dA)

Denotes the distance, when squared and multiplied by the area, yields the moment of inertia about that axis.
- Radius of gyration relative to the x-axis is denoted as:
  rx = rac{Ix}{A}
Similar expressions hold true for y-axis and polar coordinates:
ry = rac{Iy}{A}
r_0 = rac{J}{A}

For a rectangular area:
a. Moment of inertia $Ix$ about the centroidal x-axis:
Ix = ext{Integral}(y^2 dA) involving integration across the dimensional limits of the rectangle.
b. Radius of Gyration $rx$:
By utilizing our earlier discussions, we derive:
rx = rac{I_x}{A}

For a circular area, determine:
a. Polar Moment of Inertia $J0$:
b. Rectangular Moments of Inertia $Ix$ and $I_y$; symmetrical properties expedite calculations through equal expressions.

Determine moment of inertia $I_x$ for the stated area considering the centroidal coordinate established from example 1-3 for placement of the axes.
Compute the individual moments from the segmented parts of the total area and then aggregate these values for the final sum, applying parallel axis correlations where necessary to enhance accuracy.

Calculating centroid and moment of inertia of the composite area in Figure E1-8.