Direct evaluation method for load-deformation curve of precast prestressed concrete frame with different tendon forces

Abstract

To establish a more accurate, flexible, and practical performance evaluation method for unbonded precast prestressed concrete (PCaPC) frames, a macro model was illustrated based on the exterior beam–column sub-assemblages of unbonded PCaPC frames, which considers different tensile forces of the post-tensioned tendons (i.e. unsymmetrical tendon forces) and high-strength concrete. Based on the macro model, closed-form equations of beam axial shortening were provided by precisely considering the deformation compatibility and force equilibrium conditions along the beam. In this study, the iteration method for the comprehensive evaluation of different performance limit states, as well as the load–deformation relationship of the beam–column sub-assemblages, were constructed and simplified. Accuracy and superiority of the proposed evaluation methods were verified by comparison with experimental data and evaluation results of previous methods. Both damage tolerance and self-centering performance of unbonded PCaPC frames can be more accurately measured by the proposed methods. Meanwhile, the consideration of unsymmetrical tendon forces makes the methods more flexible for different types of unbonded PCaPC frames. In addition, parametric study was conducted to understand the effects of critical design parameters on the performance limits. Preliminary design advices were given to show the application potential of this study.

Appendices

Appendix 1

Based on the deformation compatibility shown in Fig. 2b, the concrete axial shortening Δ_c,cp at the tendon position on the compression side and the opening distance δ_d,tp at the tendon position on the tension side can be expressed as follows:

$$ \Delta_{c,cp} = \Delta_{c,ex} \frac{{x_{n} - \left( {D - d_{p} } \right)}}{{x_{n} }} - l_{c0} = \left( {\varepsilon_{p0} - \varepsilon_{p,c} } \right)l_{p} ,$$

(17)

$$ \delta_{d,tp} = \Delta_{c,ex} \frac{{d_{p} - x_{n} }}{{x_{n} }} + l_{c0} = \left( {\varepsilon_{p,t} - \varepsilon_{p0} } \right)l_{p} , $$

(18)

where the tendon deformations are given as products of the strains and lengths of the tendons. By solving Eqs. (17) and (18), the strain of the tendons on the compression and tension sides can be expressed as Eqs. (1) and (2).

Appendix 2

As shown in Fig. 6, when the concentrated force V is applied at the inflection point, the distribution of the sectional bending moment from the beam–column interface to the inflection point is triangular. Because the tendons are unbonded, their axial forces and the position of resultant force d_pt,t are constant along the beam. Therefore, the position of the concrete compressive resultant force changes linearly along the beam. Meanwhile, the compressive and tensile resultant forces meet at the inflection point. Therefore, at an arbitrary beam section from the beam end to the inflection point, distance C_cx from the concrete compressive resultant force to the extreme compression fiber can be expressed as follows:

$$ C_{cx} \left( X \right) = \frac{{3\left( {d_{p} - \beta d} \right) - x_{n} }}{3l}X + \frac{{x_{n} }}{3}\quad {\text{for}}\quad 0 \le X \le l,$$

(19)

where X is the horizontal distance from the beam–column interface to an arbitrary point between the beam–column interface and inflection point.

Figure 6 shows that the concrete compressive strain has a triangular distribution from the beam end to point A, where the neutral axis position coincides just with the extreme tension fiber of the beam section. If the distance from the beam end to point A is represented by L₁, and the distance to an arbitrary point within point A is represented by X₁, then the neutral axis depth at an arbitrary beam section x_nx equals three times the value of C_cx in the range of L₁. Furthermore, the concrete compressive resultant force is constant at an arbitrary beam section in the range of L₁, and can be expressed as follows:

$$ \frac{1}{2}x_{n} \varepsilon_{n} E_{c} b = \frac{1}{2}\left[ {\frac{{3\left( {d_{p} - \beta d} \right) - x_{n} }}{l}X_{1} + x_{n} } \right]\varepsilon_{nx} E_{c} b\quad {\text{for}}\quad 0 \le X_{1} \le L_{1}. $$

(20)

Then, in the range of L₁, the concrete compressive strain at the extreme compression fiber ε_nx can be obtained.

$$ \varepsilon_{nx} \left( {X_{1} } \right) = \frac{{x_{n} \varepsilon_{n} }}{{{{\left( {3d_{p} - 3\beta d - x_{n} } \right)X_{1} } \mathord{\left/ {\vphantom {{\left( {3d_{p} - 3\beta d - x_{n} } \right)X_{1} } l}} \right. \kern-\nulldelimiterspace} l} + x_{n} }}\quad {\text{for}}\quad 0 \le X_{1} \le L_{1}. $$

(21)

At point A, x_nx equals D, and ε_nx is defined as ε_nL1 = x_nε_n/D, where ε_n represents ε_nx at the beam end. The concrete compressive strain has a triangular distribution, and C_cx equals D/3. The value of C_cx also changes linearly from point A to the inflection point. At the inflection point, the compressive and tensile resultant forces meet each other, and C_cx equals (d_p—βd). If the distance from point A to the inflection point is represented by L₂, and the distance to an arbitrary point between point A and the inflection point by X₂, C_cx can then be expressed in the range of L₂ as follows:

$$ C_{cx} \left( {X_{2} } \right) = \frac{{3\left( {d_{p} - \beta d} \right) - D}}{{3L_{2} }}X_{2} + \frac{D}{3}\quad {\text{for}}\quad 0 \le X_{2} \le L_{2}. $$

(22)

As shown in Figs. 6 and

Fig. 20

Distribution of concrete compressive strain in the range of L₂

20, the distribution of the concrete compressive strain becomes trapezoidal in the range of L₂. Considering that the compressive resultant force remains constant at an arbitrary beam section, and also the trapezoid calculation formula, if the concrete compressive strain at the extreme tension fiber of the beam section is e_nx, and ε_nx at point A is ε_nL1, ε_nx equals (ε_nL1− e_nx) in the range of L₂.

Considering the definition of the resultant force, as shown in Fig. 20, the trapezoidal distribution of the concrete compressive strain is divided into the triangular and rectangular parts. By taking the bending moment about the extreme compression fiber, the sum of the bending moments contributed by the rectangular and triangular parts equal the bending moment of the compressive resultant force as follows:

$$ e_{nx} \left( {X_{2} } \right)D\frac{D}{2} + \frac{1}{2}\left( {\varepsilon_{nL1} - 2e_{nx} \left( {X_{2} } \right)} \right)D\frac{D}{3} = \frac{{\varepsilon_{nL1} D}}{2}C_{cx} \quad {\text{for}}\quad 0 \le X_{2} \le L_{2} . $$

(23)

By substituting ε_nL1 = x_nε_n/D, and Eq. (22) into Eq. (23), e_nx can be obtained; and ε_nx in the range of L₂ can be expressed as follows:

$$ \varepsilon_{nx} \left( {X_{2} } \right) = \varepsilon_{nL1} - e_{nx} \left( {X_{2} } \right) = \frac{{x_{n} \varepsilon_{n} }}{D} - \frac{{x_{n} \varepsilon_{n} \left( {3d_{p} - 3\beta d - D} \right)}}{{D^{2} L_{2} }}X_{2} \quad {\text{for}}\quad 0 \le X_{2} \le L_{2} .$$

(24)

The concrete compressive strain ε_nx at the extreme compression fiber of an arbitrary section thus has been determined along the beam length by Eqs. (21) and (24). Because the concrete axial shortening Δ_c,ex at the beam–column interface is considered as an accumulation of ε_nx along the beam length, Δ_c,ex can be calculated by the integral of ε_nx as follows:

$$ \Delta_{c,ex} = \int_{0}^{{L_{1} }} {\frac{{x_{n} \varepsilon_{n} }}{{\left( {3d_{p} - 3\beta d - x_{n} } \right)X_{1} {/}l + x_{n} }}dX_{1} } + \int_{0}^{{L_{2} }} {\left[ {\frac{{x_{n} \varepsilon_{n} }}{D} - \frac{{x_{n} \varepsilon_{n} \left( {3d_{p} - 3\beta d - D} \right)}}{{D^{2} L_{2} }}X_{2} } \right]dX_{2} } . $$

(25)

The integral boundary L₁ can be obtained by substituting C_cx = D/3 into Eq. (21), and solving for X. The integral boundary L₂ equals (l – L₁). After L₁ and L₂ are determined, Δ_c,ex can be calculated and expressed as Eq. (5).