Buckling in various planes, finding moment of inertia

[member 392791]

Homework Statement

Homework Equations

The Attempt at a Solution

With this problem and in general, I am having difficulties knowing what should be the cubic and what shouldn't be from visual inspection, so in this case I can't tell why I_x is 1/12ba^3, as opposed to 1/12ab^3. How can I tell from looking at this which one is right? Also, the same goes for I_y, is it just the opposite with the b cubed?The solution is also saying Iz = Ix...what??

When it says buckling in the xy plane or xz plane, I am wondering, does that mean the column bulges out normal to the plane, or does it bulge out in the same direction as the plane? Does being braced in a plane imply that its fixed, or does it mean its pinned?Honestly, I am wondering why its not Ix = 1/12aL^3...it looks like L is the height and a is the base with regards to the x-axis.A similar idea with problem 10.25. I don't understand why K is 1 in the xz-plane, but 2 in the yz-plane for the effective length. If what I'm thinking is right about how buckling in a plane is in the direction of the plane where the buckling occurs, not normal to it, then for the yz-plane the bar going through the middle impedes the ability to buckle in the y direction, so I guess that end is fixed, but for K=2 that means one end is fixed and one end is free to move, and I don't see which end is free to move here.

 

[SteamKing]

For buckling in general, the column is going to deflect about the axis which produces the minimum gyradius. Since the gyradius r = SQRT (I/A), this means you want to find the axes about which I is a minimum value. For a rectangular cross section, bending will occur with respect to whichever dimension, the width or the height (thickness) of the cross section is less.

In the sample problem, the base of the column is assumed to be fixed. The plates at the top do not allow a deflection to develop there, but the column may still rotate, which is why the effective length of the column is 0.7 times the distance between the base and the location of the plates.

In Prob. 10.25, the bracing bars are placed such that no deflection along the x-axis can develop; the column is constrained to buckle in the other direction, with rotation occurring about the x-axis. Due to the geometry of the bracing, the effective lengths differ as shown in the small diagrams in the solution. Since you are trying to determine the dimensions of the cross section, you must investigate buckling about each possible axis.

 

[Icy25]

Homework Statement

Homework Equations

The Attempt at a Solution

With this problem and in general, I am having difficulties knowing what should be the cubic and what shouldn't be from visual inspection, so in this case I can't tell why I_x is 1/12ba^3, as opposed to 1/12ab^3. How can I tell from looking at this which one is right? Also, the same goes for I_y, is it just the opposite with the b cubed?The solution is also saying Iz = Ix...what??

When it says buckling in the xy plane or xz plane, I am wondering, does that mean the column bulges out normal to the plane, or does it bulge out in the same direction as the plane? Does being braced in a plane imply that its fixed, or does it mean its pinned?Honestly, I am wondering why its not Ix = 1/12aL^3...it looks like L is the height and a is the base with regards to the x-axis.A similar idea with problem 10.25. I don't understand why K is 1 in the xz-plane, but 2 in the yz-plane for the effective length. If what I'm thinking is right about how buckling in a plane is in the direction of the plane where the buckling occurs, not normal to it, then for the yz-plane the bar going through the middle impedes the ability to buckle in the y direction, so I guess that end is fixed, but for K=2 that means one end is fixed and one end is free to move, and I don't see which end is free to move here.

Your issue was exactly the same as mine, down to the sample problem in questions. I made an account just to respond, I believe it is due to the N.A. with respect to the bending. For instance, if buckling in the XZ plane, the cross section bends about the Y axis. Making the Y axis the N.A. If you were to re-orient this cross section such that the Y axis is horizontal, I think you would then understand. Cheers (11 years later)