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Find the Principal Axes of the Section Shown

  1. Apr 4, 2015 #1
    1. The problem statement, all variables and given/known data

    Statement: Find the principal axes of the section shown:
    ATIv6jH.png
    The origin is on the top left corner.

    2. Relevant equations

    Centroid equations:
    aHUyIgs.png
    kDNlmDV.png

    Second moments of area:
    NiXr9bt.png
    Hw4vfCi.png
    wuSfKn1.png

    Mohr's Circle for I equations:
    Coordinates
    ktfPEuc.png

    Centre
    Awo2Bri.png

    Angle from principal axes:
    czIA2vx.png

    3. The attempt at a solution

    Finding the centroids:
    WRcThjc.png

    Finding Iy, Iz, and Iyz (Finding Iyz is where I'm having some difficulties):
    cE8n6Ca.png

    gohBb4z.png

    THE PROFESSOR-PROVIDED SOLUTION FOR Iyz (I don't know how to determine whether or not y1, z1, y2, z2 are positive or negative) (My prof's answer was Iyz = -1.188×106mm4) Here is his provided solution:
    9o4ouXM.png
    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 Is, y axis is Ist) Each point is multiplied by 10-6:
    4eKkZzL.png

    Centre:
    hop1JJJ.png

    Angle of Principal axes:
    Op2UIe7.png

    TL;DR Need a trick on determining the sign of y1, z1, y2, z2 from the Iyz equation.
     
  2. jcsd
  3. Apr 5, 2015 #2

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    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. :wink:
     

    Attached Files:

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