Area Moment of Inertia: Manual Calculation vs CAD

Area Moment of Inertia: Manual Calculations Validated in CAD (SolidWorks)

  1. Step 1: Area M of I by Integration

    First, let's calculate the Area underneath the curve using Integration: A = ∫da; which for this area underneath the curve gives us 2.0 in^2.

    Second, calculate the X-bar and y-bar or the centroid of the area.

     x = ∫xda/∫da = 3.2 in.

    y = ∫yda/∫da = .571 in.

  2. Step 2: M of I about X-axis

    We first calculate the Moment of Inertia about the X and Y axes with the equations shown using a vertical element. Then the Parallel Axis Theorem is used to move the Moment of Inertia to the Centrodial X-axis and Centrodial Y-axis.

    Notice: that in the manual method for a function like this, for the Moment of Inertial, the value was first calculated with respect to the X-axis using Ixx = 1/3 bh^3. This is because the vertical element in Integration would have varying locations for the Centrodial x-axis but a constant x-axis. If a horizontal element was used, then we could use the standard Ixx= y^2da

  3. Step 3: Sketch in function in CAD

    In this case we are suing SolidWorks to create the curve.

    a- Sketch on the X-Y Plane

    b- Equation Driven Curve

    c-For this function, notice the value of y = 1/32 x^3, so in SolidWorks we input these values as shown:

    Which gives us the function curve:

    The enclose the curve with vertical and horizontal lines

  4. Step 4:

    Now, calculate the Section Properties of the Area;

    The values match our manual calculations.

    Area = 2.00 in^2

    x-bar = 3.2 in

    y-bar = .571 in

    Ixx = .414 in^4

    Iyy = .853 in^4

    Izz = 1.267 in^4