Also, How to convert the moment-curvature which is calculated from the sectional analysis to moment-rotation relationship?
Thank you in advance for your kind help and cooperation.
Dear Moussa;
In geometry, curvature is rate of change of rotation. For a cross-section undergoing flexural deformation, it can computed as the ratio of the strain to the depth of neutral axis.
Thank you Ghavami. Is there a formula to convert the curvature to rotation?
Hello Ahmed,
If you consider a beam with axis x, the rotation is the angular displacement of a single section with respect to the initial position. If w(x) is the vertical displacement of the beam, the rotation will be the derivative of w with resoect to x:
theta = dw/dx.
The curvature is the variation in rotation along the axis (the derivative of the rotation with respect to x), i.e. the second derivative of w with respect to x:
curvature = d2w/dx2
The bending moment in the beam can be expressed as (Euler-Bernoulli theory):
M = EI d2w/dx2
where E is the Young modulus and I the moment of inertia of the section, from which you can derive the moment-curvature / moment-rotation relationship.
Hope this helps.
Best,
G.
briefly,
chi= M/EI= 1/r (curvature)
teta= chi . d (Finite rotation)
Where,
r is the curvature radius
d is the lenght of plastic hinge
All good and comprehensive answers from colleagues. Nothing to add.
In short, moment-curvature is at the section level 'force-deformation' relationship that needs only the sectional properties, ie flexural rigidity. On the other hand moment-rotation is at the member level 'force-deformation' relationship that needs member properties as well. So, L must show up in moment-rotation relation while only EI will come in moment-curvature relation.
Hello all, a small follow up question...
To convert between a rotation and curvature a length is required (given the definition of a curvature). For design this is typically the plastic hinge length for which their is an abundance of resources to assist.
This however suggests the conversion is only applicable once the section reaches a plastic state. If we want to find out the rotation at yield (or just before yield) how would one select a relevant length to use? Would a relationship between displacement and depth be required, to which we can manipulate to find the rotation? Or is the section depth more suitable?
Thanks
A
Let us assume that one is interested to find out the rotation at a point, L distance away from a fixed support (which does not rotate). So, he/she needs to integrate the curvature from the support to that point to get the rotation there. When things are within elastic limit the exact curvature variation is known (from the elastic theory) and the integration can be performed in exact sense. Hence, there is no need to have any empirical formula with some associated length. If the support is anything other than a fixed one, the same quantity will give the rotation w.r.t. the support rotation.
Hello all
The yield rotation can be determined using area of yield curvature diagram. As we know that, the idealized yield curvature diagram is triangular in shape, the integration of this diagram gives yield rotation. Similarly, the idealized plastic curvature diagram is rectangle in shape, the integration of this diagram gives plastic rotation. The addition of yield rotation and plastic rotation gives ultimate rotation.
Cheers.
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