Moment-Rotation and Moment-Curvature

[amousa]

What is the difference between moment-rotation relationship and moment-curvature relationship for a structural element (e.g. beam, column)?
Also, How to convert the moment-curvature which is calculated from the sectional analysis to moment-rotation relationship or the other way around?

Thank you in advance for your kind help and cooperation.

 

[KootK]

Curvature is an instantaneous phenomenon along a member's length (calculus). Usually, curvature has to be integrated over some finite length to generate rotation.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[amousa]

Thank you very much for you help. Could you please provide me with an example for such integration?. What I do not get is how I should integrate the whole moment-curvature curve over the plastic hinge length to get the moment-rotation curve.

 

[dik]

It's best to consider moment rotation for elastic conditions only. In idealised areas of plastic hinges, rotation continues without additional restraint once Mp has been obtained.

Dik

 

[KootK]

I don't have an example ready that I can post. If you look into the seismic texts by either Priestly or Mohle, you'll find oodles of examples. Is it concrete/masonry that you're interested in here?

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[wannabeSE]

dik,
We must be thinking of two separate issues. Non-linear analysis considers the hinges in the plastic region so the load redistributes to elements that haven't yielded. I typically neglect the moment rotation of hinges in elastic analysis.

 

[amousa]

Actually I am trying to model a deteriorated 2D RC frame structure under seismic loads and I am taking the joint modeling into consideration. Therefore, at the interface of the beam\column-joint, there are two rotational springs in series. One to account for the corrosion effect on the beam/column (i.e. bond-slip) and the other one to account for the joint shear. Each spring has its own Moment-Rotation curve. Thus, I have to obtain first the Moment-Curvature curves of the beam/column then I need to transfer them to Moment-Rotation curves to use them in my model. I am very sorry for making this long and I am really grateful for you kind help and replies. Thank you very much.

 

[BAretired]

Curvature is the reciprocal of radius of curvature. The units are 1/L or L[sup]-1[/sup] where L represents length. In the imperial system, curvature would be measured in ft[sup]-1[/sup] or in[sup]-1[/sup].

Curvature of a structural member is closely approximated by the expression M/EI. A dimensional analysis shows that the units turn out to be "#/(#/in[sup]2[/sup]*in[sup]4[/sup]) or in[sup]-1[/sup].

If you plot the M/EI diagram for a structural beam, you are plotting curvature from end to end of beam. In the case of a uniformly loaded beam, moment is a parabola. If EI is constant over the span, then M/EI is also a parabola. The rotation between any two points on the beam is the area under the M/EI curve between the two points.

A simple beam loaded with an equal and opposite moment at each end has a constant moment across the span, hence constant curvature. That is cylindrical bending. The change in rotation from one end to the other is the area under the M/EI curve, namely ML/EI where L is the span. By symmetry, the rotation is equal in magnitude at each support, so the rotation at each end is ML/2EI.

Rotations for other types of loading can be worked out in similar fashion. Sometimes the geometry gets a bit messy, but the concept is straight forward. The conjugate beam method is an easy way to determine slopes and deflections of simple span beams and I recommend you check that out using Google.

BA

 

[IDS]

What BARetired says is all correct, but if you have a local hinge of finite length with constant moment life becomes much simpler:

Rotation = Curvature x Hinge Length

If the hinge has zero length it's even simpler:

Curvature = 0 up to the yield moment, then infinite.
Rotation = 0 up to the yield moment, then rotational stiffness = 0, so you effectively have a pinned joint with a point moment.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[JasonMcKee71]

Curvature is expressed in Length^-1 units while rotation in degrees or rads.

Jason McKee
proud R&D Manager of
Cross Section Analysis & Design
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