Moment-Rotation Relationship

[PrabakaranKesavan]

Dear Sir/Madam
I would like to know how to develop the moment-rotation relationship from the moment-curvature curve which we usually obtain from section analysis. I need to model an unreinforced masonry pier following a lumped plasticity approach (no fiber section, only phenomenological laws), so I model the pier as a beam-column element with two flexural hinges assigned at the top and bottom and shear-hinge at mid-height. I could develop the moment-curvature curve from the STKO section analysis. However, as an input to the flexural hinge, I need the moment-rotation relationship. It is not clear to me how to obtain the moment-rotation curve. Kindly guide me through this topic. Thanks.

[STKO Team]

Dear user it is pretty simple, you just need to follow these steps:

1. obtain the moment-curvature response
2. choose any uniaxial material you want, and calibrate its parameters to match the moment-curvature response
3. now you have a moment-curvature phenomenological law.
4. you need to convert it to a moment-plastic_rotation, for this you need the plastic hinge length
5. now the moment-plastic_rotation is a rigid-plastic material, and it makes sense because the hinge is a zero-length and the elastic deformation is already accounted for by the inner elastic beam
6. however you cannot input a rigid-plastic hinge, so you need to use a regularized (sufficiently high) stiffness, you can take it, for example, 2 or 3 orders of magnitude larger than the elastic stiffness

Here is a sample file where you have a beam-with-hinges using a finite length plastic hinge that uses a fiber cross section. It is compared with a lumped-plasticity beam with zero-length hinges using ModIKM uniaxial materials for moment rotations.

plastic_hinge_from_MK.zip

(165.46 KiB) Downloaded 396 times

I used this Excel file to calibrate the moment-rotation response and the main material parameters of the ModIKM law starting from the moment-curvature response obtained from the bottom-section of the first beam

calculation.zip

(52.96 KiB) Downloaded 424 times

Immagine 2020-12-03 154137.png (129.11 KiB) Viewed 6049 times

And this is the Ux-Fx curve of the 2 columns compared, as you can see they match really well:

Immagine 2020-12-03 154138.png (158.22 KiB) Viewed 6049 times

[PrabakaranKesavan]

Dear Sir/Madam
Thanks a lot for a really nice and detailed explanation. It gives clarity. Greatly appreciate your help.

[ankurjain1992]

I have different size columns in my model and so I need to calibrate ModiMKPinching parameters as I am modelling it for shear failure.

How to determine moment-curvature analysis I am not getting ?

What amount of load P and Moment should be applied ?

[Clarabella]

Please check the literature, several papers have been published on this topic

[ankurjain1992]

ok i got how to do the Moment-curvature analysis. But at least can you tell me that in the picture below for that column what axial force and moment shall I use to get Moment-curvature for this column so that I use it to determine other parameters of ModiMKPinching.

[Lucy2012]

Dear ankurjain1992,

I suggest to use the axial load of the columns at the end of the gravity load, just before applying the horizontal load.

Regards

[ankurjain1992]

Thank you for the reply .

What about the Moment ?

Shall I apply unit moment or moment also we take from the end of gravity load?

And that moment should be Mx, My or Mz ?

As we change the moment value, moment-curvature curve values change drastically.

[Lucy2012]

Moment is calculated with the axial load using the interaction diagram. The axes are those of the element as defined in the manual

[ankurjain1992]

Dear STKO Team,

I am able to evaluate most of the parameters of ModiMKPinching material that I want to use for model Shear failure in my Reinforced Concrete Building Model. But I am not able to find any data for some of the parameters and in literature, they have considered steel structure but mine is Reinforced Concrete structure. I am listing those parameters which I am not able to calculate but you have used those in one of your e-learning videos. If possible kindly give some idea how to calculate them.

$FprPos (You have considered 0.4 in your e-learning tutorial)
$A_Pinch (You have considered 0.1 in your e-learning tutorial)
$Lamda_S (You have considered 0 in your e-learning tutorial)
$Lamda_C (You have considered 0 in your e-learning tutorial)
$Lamda_A (You have considered 0 in your e-learning tutorial)
$Lamda_K (You have considered 0 in your e-learning tutorial)
$Res_Pos (You have considered 1.05 in your e-learning tutorial)
$theta_u_Plus (You have considered 100 in your e-learning tutorial)
$D_Plus (You have considered 0 in your e-learning tutorial)

Thank you in advance.