Can anyone explain to me how and what flexural stiffness as used in the AISC 13th edition on page 16.1-426 under torsional bracing is? From my understanding, flexural stiffness is just EI, how is it that they get 2EI/L for single curvature and 6EI/L for double curvature? And also, if possible, how is it that the plate attached to bottom flanges will bend in single curvature and plate attached to the top flange bend in double curvature? My problem is I'm attaching a plate to the web of a beam and it doesnt fall into either case
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Flexural Stiffness 1
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I don't know AISC, any edition, so I'm not sure I can help you. As I recall, the flexural stiffness of a beam is the moment required to produce unit rotation at the point of application of the moment.
Consider a simple beam A-B of span L. Apply a moment at point B. What is the rotation at B? Call it [θ][sub]B[/sub].
The Moment diagram is triangular, 0 at A and M at B. The area under the M/EI diagram is ML/2EI, so the rotation at A is ML/6EI and the rotation at B is ML/3EI.
[θ][sub]B[/sub] = ML/3EI
or M = [θ][sub]B[/sub]*3EI/L.
By definition, Stiffness = M when [θ][sub]B[/sub] =1.
So stiffness = 3EI/L when A is hinged.
If A is fixed against rotation, the beam is stiffer. A larger moment is required at point B in order to produce unit rotation at B. In fact, the stiffness of the beam with point A fixed against rotation is 4EI/L. Check it out.
BA
Consider a simple beam A-B of span L. Apply a moment at point B. What is the rotation at B? Call it [θ][sub]B[/sub].
The Moment diagram is triangular, 0 at A and M at B. The area under the M/EI diagram is ML/2EI, so the rotation at A is ML/6EI and the rotation at B is ML/3EI.
[θ][sub]B[/sub] = ML/3EI
or M = [θ][sub]B[/sub]*3EI/L.
By definition, Stiffness = M when [θ][sub]B[/sub] =1.
So stiffness = 3EI/L when A is hinged.
If A is fixed against rotation, the beam is stiffer. A larger moment is required at point B in order to produce unit rotation at B. In fact, the stiffness of the beam with point A fixed against rotation is 4EI/L. Check it out.
BA
I don't know AISC, any edition, so I'm not sure I can help you. As I recall, the flexural stiffness of a beam is the moment required to produce unit rotation at the point of application of the moment.
Consider a simple beam A-B of span L. Apply a moment at point B. What is the rotation at B? Call it [θ][sub]B[/sub].
The Moment diagram is triangular, 0 at A and M at B. The area under the M/EI diagram is ML/2EI, so the rotation at A is ML/6EI and the rotation at B is ML/3EI.
[θ][sub]B[/sub] = ML/3EI
or M = [θ][sub]B[/sub]*3EI/L.
By definition, Stiffness = M when [θ][sub]B[/sub] =1.
So stiffness = 3EI/L when A is hinged.
If A is fixed against rotation, the beam is stiffer. A larger moment is required at point B in order to produce unit rotation at B. In fact, the stiffness of the beam with point A fixed against rotation is 4EI/L. Check it out.
BA
Consider a simple beam A-B of span L. Apply a moment at point B. What is the rotation at B? Call it [θ][sub]B[/sub].
The Moment diagram is triangular, 0 at A and M at B. The area under the M/EI diagram is ML/2EI, so the rotation at A is ML/6EI and the rotation at B is ML/3EI.
[θ][sub]B[/sub] = ML/3EI
or M = [θ][sub]B[/sub]*3EI/L.
By definition, Stiffness = M when [θ][sub]B[/sub] =1.
So stiffness = 3EI/L when A is hinged.
If A is fixed against rotation, the beam is stiffer. A larger moment is required at point B in order to produce unit rotation at B. In fact, the stiffness of the beam with point A fixed against rotation is 4EI/L. Check it out.
BA
paddingtongreen
Structural
BA should get an "E" for effort.
Michael.
Timing has a lot to do with the outcome of a rain dance.
Michael.
Timing has a lot to do with the outcome of a rain dance.
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I believe that the discusion in the Article on Torsional Bracing Stiffness accounts for the rotation at the far end of the stiffener. If the rotation at the far end of the brace were zero, then the stiffness would be 4EI/L, as shown by BA above. However, if the far end of the brace rotates the same amount as the near end, as often happens when the bracing is attached to the top flange, the stiffener requires more moment to rotate it to the unit angle. The additional moment is 2EI/L, giving an apparent stiffness of 6EI/L. If the beam at the far end rotates in the opposite direction (single curvature), then the required moment to rotate a unit angle is reduced by the 2EI/L, so the apparent stiffness is 2EI/L (4EI/L minus 2EI/L).
Stiffness in general is an extensive property, meaning it is directly dependent on the system one is investigating.
That is to say that it is only useful on a case-by-case basis.
"I have situation 'x' with such n' such end conditions, then beam "A" is stiffer than beam "B"" In this case beams "A" and "B" differ by EI.
Remove the system constraints and the term is relatively meaningless.
wow, this would make a really good Critical Thinking term paper, huh?![[2thumbsup]](/data/assets/smilies/2thumbsup.gif "[2thumbsup] [2thumbsup]")
That is to say that it is only useful on a case-by-case basis.
"I have situation 'x' with such n' such end conditions, then beam "A" is stiffer than beam "B"" In this case beams "A" and "B" differ by EI.
Remove the system constraints and the term is relatively meaningless.
wow, this would make a really good Critical Thinking term paper, huh?
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