# Question C.5: (a) Draw the Mohr’s circle for the centroidal moments and pr......

(a) Draw the Mohr’s circle for the centroidal moments and products of inertia for the L-shaped area in Fig. 1 of Example C-2, given that:

$I_y = \frac{5894}{147}t^4 = 40.10t^4, I_z = \frac{33,103}{294}t^4 = 112.60t^4$

$I_{yz} = \frac{-270}{7}t^4 = -38.57t^4$

(b) Use the Mohr’s circle constructed in Part (a) to compute the principal moments of inertia $I_{p_1}$ and $I_{p_2}$ and to locate the principal axes. Show the orientation of the principal axes on a sketch.

Step-by-Step
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(a) Sketch Mohr’s circle and calculate the principal moments of inertia. Points Y and Z are plotted and Mohr’s circle is then drawn (Fig. 1). From the circle,

$I_{avg.} = \frac{40.10t^4 + 112.60t^4}{2} = 76.35t^4$

$R = \sqrt{\left(\frac{112.60t^4 – 40.10t^4}{2}\right)^2+ (38.57t^4)^2} = 52.93t^4$

$I_{p_1} = I_{avg.} + R = 129.3t^4$

$I_{p_2} = I_{avg.} – R = 23.4t^4$            (a)

(b) Determine the orientation of the principal axes and show them on a sketch.

$\tan |2θ_{zp_1}| = \frac{38.57t^4}{\left(\frac{112.60t^4 – 40.10t^4}{2}\right)} = 1.064$

Therefore, $2θ_{zp}$ = 46.78° (clockwise), so

$θ_{zp_1} = θ_{yp_2}$ = 23.4°  clockwise         (b)

Note that the orientations of the principal axes in Fig. 2 are such that the contributions to $I_{p_1p_2}$ of the areas in the four quadrants cancel out, giving $I_{p_1p_2}$ = 0.

Question: C.4

## For the L-shaped area in Fig. 1 of Example C-2, (a) determine the orientation of the centroidal principal axes and show the orientation on a sketch. (b) Determine the principal moments of inertia. ...

(a) From Eq. (C-23), \tan 2θ_p = \frac{- I_...
Question: C.2

## Determine the centroidal moment of inertia Iy for the L-shaped section in Example C-1. (Here, in Fig. 1, the origin of the (y, z) reference frame is at the centroid of the composite area. The centroidal reference axes for the rectangular “legs” of the L-shaped area are (y1, z1) and ...

We can combine Eqs. (C-14) with the parallel axis ...
Question: C.1