24.1 Introduction

Engineers are facing the challenging situation in the analysis of RC structures while providing safe and economical buildings to the society. Nonlinear analysis has become one of the important means of achieving the above goal. RC structures are made out of structural elements such as column, beam and slab; also their elastic and inelastic behaviour determines the load response behaviour of the structure. Rotation capacity and formation of plastic hinge in the structural members is one of the main issues in understanding the inelastic behaviour of the structure. More specifically in the earthquake design of structures, ductility plays a key role in the design criteria set by many countries’ code provisions.

When the load applied is amplified, the hinge begins to form at locations where the moment is equal to the plastic moment, if further increase in the applied load then hinges transforming into plastic hinges and converting into mechanism followed with failure. Framed structures are designed to achieve the above by proportioning the beams and columns so that the majority of the plastic hinges are formed in the beams not in the column. For this reason, an important parameter is the plastic hinge, which controls the rotation capacity of the structural members. There are two different types of plastic hinges formed in RC structure, such as reversing plastic hinges and unidirectional plastic hinges. The reversing plastic hinge sustains both positive and negative inelastic rotations in the same region, and unidirectional plastic hinge sustains both in different regions of the beam element during earthquake [1]. It was observed that plastic hinge length lp in RC beams influenced by a number of parameters such as cross-sectional dimensions, yield and ultimate curvature, material characteristics, tension and compression reinforcement ratios, support type, intensity and nature of loading [2, 3].

The main objective of the present study is to propose a soft computing method that can estimate the rotation capacity and length of plastic hinge by using the available data from experimental studies conducted on RC beams [4,5,6,7,8,9]. The regression analysis was carried out to find out the influencing parameters on which the rotation capacity and plastic hinge length and also to demonstrate the relationship between the plastic hinge length and rotation capacity.

24.2 Rotation Capacity and Plastic Hinge Length

In the present investigation, the rotation capacity is defined as the rotation of critical section of a beam from yield stage to the ultimate stage, and the plastic hinge length is defined as the length of inelastic zone over which bending moment will be larger than or equal to yield moment over the length of the beam. It is generally agreed that the inelastic rotations are concentrated over a length called “plastic hinge length” where M ≥ My as shown in Fig. 24.1. The length of plastic hinge is dependent on many parameters, such as the shape of the ultimate bending moment diagram, and is also affected by the length between zero moment points and distance between support, reinforcement ratio, the characteristic strength of concrete and depth of section [2]. The corresponding plastic hinge length is determined with respect to integration of the curvature distribution for distinctive members.

Fig. 24.1
figure 1

Length of plastic hinge

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The correlation between the cross section, ductility and the length of plastic hinge needs to be accurately determined. It has been observed there is a large variation in the plastic hinge length formulae proposed by the researchers as shown in Table 24.1. Sawyer, Corley and Mattock, for example, considered only the member’s length and depth. Priestley and Park consider the length and diameter of reinforcement of the beam.

Table 24.1 Existing length of plastic hinge (lp) models [8]
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To be able to demonstrate the variations in the length of plastic hinge, the geometric and material properties were taken as an example from [6] l = 6000 mm, b = 300 mm, d = 540 mm, f’c = 30.9 MPa, z = 3000 mm and l/d = 11.1. For the above data, plastic hinge lengths were obtained using the above equations and they are shown in Fig. 24.2. It was observed a large variation in the plastic hinge length which encouraged us to study on the parameters effect and regression analysis.

Fig. 24.2
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Variation of plastic hinge length (lp)

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24.3 Parametric Study

In the present study, the central point loading configuration and different equivalent length of beams were selected from the literature [4,5,6,7,8,9]. The range of the parameters such as characteristics strength of concrete and steel, width and depth of beams, tension and compression steel percentages and all other essential parameters that affect the beam behaviour under central point loading is shown in Table 24.2.

Table 24.2 Range of parameters used
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24.4 Determination of Length of Plastic Hinge

The length of the plastic hinge lp was determined based on different parameters obtained from the previous studies. The beam plastic rotation (θp) was calculated from Eq. (1) [10].

$$\theta_{\text{p}} = \frac{0.004}{{\left( {x_{\text{u}} /d} \right)}}$$
(1)

The plastic hinge length lp was calculated based on Eq. (2) [2], and curvature at yielding moment (φy) and the curvature at ultimate moment (φu) and plastic hinge length values were analytically calculated.

$$l_{\text{p}} = \frac{{\theta_{\text{p}} }}{{(\varphi_{\text{u}} - \varphi_{\text{y}} )}} = \frac{{\theta_{\text{p}} }}{{\varphi_{\text{p}} }}$$
(2)

Illustrative graphs have been plotted to observe the variation of lp against the parameters which are shown in Fig. 24.3a–f.

Fig. 24.3
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Plastic hinge length (lp) versus various parameters

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It has been observed from Fig. 24.3a–f that the length of plastic hinge has not shown much variation with grade of concrete, and however, reduction in lp was observed with increase in reinforcement ratio and further it was observed that lp increased when the depth of the beam increased, l/d ratio, z and z/d ratio.

24.5 Nonlinear Regression Analysis

Nonlinear regression analysis is a method of mathematical modelling. In this study, cross-sectional dimensions, material characteristics, yielding and ultimate deflections, tension and compression reinforcement ratios were observed and parameters that affect on length of plastic hinge are analysed by using the nonlinear regression analysis. Formulated our regression equation for plastic hinge length (lp) as follows.

$$l_{p} = 0.37 \times b^{0.00896} \times d^{0.667} \times f^{\prime } c^{ - 0.0016} \times z^{0.3264} \times \rho^{ - 0.00207} \times \left( {\frac{l}{d}} \right)^{0.059266}$$
(3)

It has been observed from Fig. 24.4 that the comparison between regression plastic hinge length lp reg versus existing models lp trend lines shows that the proposed model is predicted well with Mattock, Sawyer, Corely and Baker’s models and R2 values are almost equal to unity.

Fig. 24.4
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Comparison between regression lp reg and existing models lp

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It has been observed from Fig. 24.5 that the comparison between regression plastic hinge length lp reg versus θp trend lines shows that there is no significant relationship between lp and θp.

Fig. 24.5
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Comparison between regression lp and θp

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24.6 Conclusions

Based on the parametric study and proposed nonlinear regression analysis, the following conclusions are drawn.

  • The plastic hinge length results are obtained by using plastic rotation and curvatures at ultimate and yield load (vide Eqs. 1 and 2).

  • Demonstrative graphs have been plotted to show the variation of influencing parameters. It was observed from Fig. 24.3 that the length of plastic hinge has not shown much variation with grade of concrete, and however, reduction in lp was observed with increase in reinforcement ratio.

  • Further, it was observed that lp amplified with increase in depth, l/d ratio, z and z/d ratio.

  • Analytical expressions given in the literature indicated that only the cross-sectional dimensions are under the bending effect of beam and other influencing parameters were not considered.

  • The plastic hinge length (lp) obtained by expressions given by the researchers gave different values and are not comparable (Fig. 24.2).

  • It has been observed that analytical plastic hinge expression and proposed nonlinear regression expression yielded results that are comparable.

  • Based on the comparison between lp calculated by existing model and lp.reg calculated by nonlinear regression model, the results show that the nonlinear regression plastic hinge model (lp.reg) is versatile and can be used all types RC beams.

  • Based on comparison between rotation capacity and length of plastic hinge, there was no significant relation between them and hence, further investigation has to be done over large number of data.