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