3.1.2. State Determination

State determination for using the stiffness-based beam-column element with fiber sections includes the following steps:


displacement Δ **d***<sup>e</sup>*,*i*+<sup>1</sup> = *<sup>T</sup>e*Δ**d***<sup>e</sup>*,*i*+<sup>1</sup> in the local coordinate system.

3.Section level: The incremental section deformation <sup>Δ</sup><sup>ν</sup>*S*,*<sup>i</sup>*(*xk*) at the integration points beobtained¯ <sup>Δ</sup>*φ*(*xk*)]*<sup>T</sup>*,where

can as <sup>Δ</sup><sup>ν</sup>*S*,*i*+<sup>1</sup>(*xk*) = *Bd*(*xk*)<sup>Δ</sup> **d***<sup>e</sup>*,*i*+<sup>1</sup> = [<sup>Δ</sup>*ε*(*xk*), <sup>Δ</sup>*ε*(*xk*) and <sup>Δ</sup>*φ*(*xk*) are the incremental axial strain and curvature, respectively.

4. Fiber level: The incremental strain of the *j*th fiber is <sup>Δ</sup>*<sup>ε</sup>j*,*i*+<sup>1</sup> = −Δ*ε*(*xk*) + <sup>Δ</sup>*φ*(*xk*)*yj*. Then, the linear or nonlinear constitutive relationships of the material can be applied to obtain the stress *<sup>σ</sup>j*,*i*+<sup>1</sup> of the *j*th fiber.

5. Section level: The section internal force **<sup>S</sup>***i*+<sup>1</sup>(*xk*) at the integration points can be assembled as **<sup>S</sup>***i*+<sup>1</sup>(*xk*) = [*Ni*+<sup>1</sup>(*xk*), *Mi*+<sup>1</sup>(*xk*)]*<sup>T</sup>* = %∑*Nf j*=1 *<sup>σ</sup>j*,*i*+<sup>1</sup>*Aj*, ∑*Nf j*=1 *<sup>σ</sup>j*,*i*+<sup>1</sup>*Ajyj*&*<sup>T</sup>*.

¯

6. Element level: The element force **<sup>f</sup>***<sup>e</sup>*,*i*+<sup>1</sup> in the local coordinate system can be integrated by the section forces at the integration points as ¯ **<sup>f</sup>***<sup>e</sup>*,*i*+<sup>1</sup>= ∑<sup>5</sup>*k*=<sup>1</sup>*Bd*(*xk*)*<sup>T</sup>***S***i*+<sup>1</sup>(*xk*)*<sup>ω</sup>kLe*.

Then, the element force **<sup>f</sup>***<sup>e</sup>*,*i*+<sup>1</sup> = *TTe* ¯ **<sup>f</sup>***<sup>e</sup>*,*i*+<sup>1</sup> in the global coordinate system can be obtained.

7. Structure level: The element forces of all elements can be assembled as the first-order restoring force **R***i*+1.
