Find the Principal Axes of the Section Shown

[dbaliki918]

Homework Statement

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Statement: Find the principal axes of the section shown:

The origin is on the top left corner.

Homework Equations

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Centroid equations:

Second moments of area:

Mohr's Circle for I equations:
Coordinates

Centre

Angle from principal axes:

The Attempt at a Solution

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Finding the centroids:

Finding I_y, I_z, and I_yz (Finding Iyz is where I'm having some difficulties):

THE PROFESSOR-PROVIDED SOLUTION FOR I_yz (I don't know how to determine whether or not y₁, z₁, y₂, z₂ are positive or negative) (My prof's answer was I_yz = -1.188×10⁶mm⁴) Here is his provided solution:

I know where his values come from, I just need a surefire trick on determining the sign of y₁, z₁, y₂, z₂.

Mohr's Circle for I (I know I'm missing the axis labels. x-axis is I_s, y-axis is I_st) Each point is multiplied by 10^-6:

Centre:

Angle of Principal axes:

TL;DR Need a trick on determining the sign of y₁, z₁, y₂, z₂ from the I_yz equation.

 

[SteamKing]

The Attempt at a Solution

[/B]
Finding the centroids:

Finding I_y, I_z, and I_yz (Finding Iyz is where I'm having some difficulties):

THE PROFESSOR-PROVIDED SOLUTION FOR I_yz (I don't know how to determine whether or not y₁, z₁, y₂, z₂ are positive or negative) (My prof's answer was I_yz = -1.188×10⁶mm⁴) Here is his provided solution:

I know where his values come from, I just need a surefire trick on determining the sign of y₁, z₁, y₂, z₂.

Need a trick on determining the sign of y₁, z₁, y₂, z₂ from the I_yz equation.

All of this laborious calculation can be replaced by a simple tabular form calculation as attached in CrossProduct.pdf below.

Your calculation of Iy and Iz values for the angle section is slightly incorrect. You must calculate the inertia for each leg of the angle about its own centroid and then transfer it to the y-axis or z-axis, before you transfer the inertia back to the centroid of the angle as a whole.

The cross product of inertia of each piece of the angle is zero about its own centroidal axes. The tabular form first calculates the Ayz transfer values for each piece to obtain the inertia about the y-z axes. Then, the centroidal location for the angle and the Iyz about the centroid are calculated below the table.

You don't need to worry about learning any tricks or memorizing what is positive and what is negative for the y-bar and z-bar values.