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Definition of Displaced Axis Rotations for Column, Shearwall and Beam Elements

The displaced axis rotation values ​​are used in linear performance analysis to calculate the strain and plastic rotation demands of columns, beams and shear elements.


E = Concrete modulus of elasticity
h = Section height
I = Moment of inertia
l c = Element net opening
l s = Shear opening
M y = Effective yield moment
M yi = Effective yield momentat i end
M yj = Effective yield moment at end j
Δ = Element node points to shift
φ y = Yield curvature
φ t = total curvature
θ p =Plastic rotation demand
θ i = i joint rotation
θ j = j node rotation
θ y = flow rotation
θ yi = flow rotation at end i
θ yj = flow rotation at end j
θ k = displaced axis rotation
θ ki = displaced axis rotation at end i


The deformation properties of a typical bending element under double curvature bending are shown in Figure 15A.1 . Here, l is the total length of the element, l c is the net span, Δ is the displacement between floors, θ i and θ j are the rotations of joints i and j, respectively, θ ki and θ kj are the displaced axis rotations at the ends i and j, respectively.


When the bending element is in the linear elastic deformation state, the relation of displaced axis and joint rotations at the i end and the displacement between floors is defined in Equation (15A.1) .

In beam elements, the translational value between floors can generally be taken as zero (= 0). When flow occurs at the i end of the element, the total displaced axis rotation at the i end is equal to the sum of the flow rotation and plastic rotation at this end.


The relations between the tip yield rotations and end moments at the i and j ends of a bending element that has become flowing at both ends are given in Equation (15A.3) . The definition of yield rotations for elements with both ends in flowing state corresponds to the most unfavorable situation in the calculation of unit deformation demands according to 15.5.4 .

In Eq. (15A.3) , EI is the bending stiffness of the uncracked section , M yi and M yj are the effective yield moments at the i and j ends , respectively. The directions of the yield moments are counterclockwise plus, and minus clockwise. Therefore, Equation (15A.3) includes both double and single curvature bending cases. The effective yield moment M y , Annex 5A.1 'will be obtained according to the definition given.


The relationship between the yield rotation and the yield moment of a bending element defined as a curtain according to at the lower end of any floor of the building is given in Equation (15A.4) .

Where l c is the shear gap (ratio of moment / shear force in cross section). It can be taken as approximately half the distance from the base of each floor to the top of the curtain.

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