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

**ICONS**

* E = *Concrete modulus of elasticity

*Section height*

**h =***Moment of inertia*

**I =***Element net opening*

**l**_{c}=*Shear opening*

**l**_{s}=**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*

**=****rotation**

_{joint}**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*

**=****15A.1. DEFINITIONS**

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.

**15A.2. REPLACED AXIS ROTATION**

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.

**15A.3. FLOWING ROTATION IN FRAME ELEMENTS**

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.

**15A.4. FLOWING ROTATION IN CURTAIN ELEMENTS**

**The** relationship between the yield rotation and the yield moment of a bending element defined as a curtain according to **4.5.3.2** 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.