Elemental damping formulation: an alternative modelling of inherent damping in nonlinear dynamic analysis

Abstract

To date, nonlinear dynamic analysis for seismic engineering predominantly employs the classical Rayleigh damping model and its variations. Though earlier studies have identified issues with the use of this model in nonlinear seismic analysis, it still remains the popular choice for engineers as well as for software providers. In this paper a new approach to modelling damping is initiated by formulating the damping matrix at an elemental level. To this regard, two new elemental level discrete damping models adapted from their global counterparts are proposed for its application in nonlinear dynamic analysis. Implementation schemes for these newly proposed models using Newmark incremental method and revised Newmark total equilibrium method is outlined. The performance of these proposed models, compared to existing models, is illustrated by conducting nonlinear dynamic analyses on a four story RC frame designed to Eurocodes. The incremental dynamic analysis study presented in the paper illustrates the fact that both the proposed models seem to produce more reliable results from an engineering perspective in comparison to the global models. It is also shown that the proposed elemental damping models lead to smaller and more realistic damping moments in the plastic hinges. Furthermore, these models could be easily included in existing software frameworks without adding noticeably to the computational effort. The computation time required for these models is approximately equivalent to the one required when using the tangent Rayleigh damping matrix with constant coefficients.

Appendices

Appendix 1: Computation of elemental frequencies using consistent mass matrix

1.1 Elemental Rayleigh damping

Elemental frequencies of beam elements are obtained by assuming free–free boundary condition. Using the consistent mass formulation, the mass matrix is given as,

$${\mathbf{M}} = \frac{{\rho A_{e} l_{e} }}{420}\left[ {\begin{array}{*{20}c} {4l_{e}^{2} } & { - 3l_{e}^{2} } \\ { - 3l_{e}^{2} } & {4l_{e}^{2} } \\ \end{array} } \right]$$

(42)

and the flexibility matrix with plastic hinge spring flexibility $f_{s}$ in series is given as,

$${\mathbf{F}} = \left[ {\begin{array}{*{20}c} {\frac{{2l_{e} }}{{6EI_{e} }} + f_{s} } & {\frac{{l_{e} }}{{6EI_{e} }}} \\ {\frac{{l_{e} }}{{6EI_{e} }}} & {\frac{{2l_{e} }}{{6EI_{e} }} + f_{s} } \\ \end{array} } \right]$$

(43)

Now elemental frequencies can be computed by solving Eq. 44 given below as,

$$\left[ {\begin{array}{*{20}c} {\frac{{2l_{e} }}{{6EI_{e} }} + f_{s} } & {\frac{{l_{e} }}{{6EI_{e} }}} \\ {\frac{{l_{e} }}{{6EI_{e} }}} & {\frac{{2l_{e} }}{{6EI_{e} }} + f_{s} } \\ \end{array} } \right]\frac{{\rho A_{e} l_{e}^{3} }}{420}\left[ {\begin{array}{*{20}c} 4 & { - 3} \\ { - 3} & 4 \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varphi_{e,1} } \\ {\varphi_{e,2} } \\ \end{array} } \right\} = \frac{1}{{\omega_{e}^{2} }}\left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right]$$

(44)

Solving Eq. 44 gives the elemental frequencies and elemental mode shapes for the element under consideration. If the state of the element is elastic the plastic hinge spring flexibility $f_{s}$ would be zero otherwise the spring flexibility exists. It can be clearly seen that Eq. 44 can also be solved analytically if required. The elemental frequencies thus obtained based on the state of the element (elastic or inelastic) is used in Eq. (7) to compute the coefficients and the elemental matrix is obtained by substituting these coefficients in Eqs. (8) or (9). The elemental damping matrix thus obtained is transformed back to the system level using the standard transformation matrices.

1.2 Elemental Wilson Penzien damping

Frequencies and elemental mode shapes are computed as per Eq. (44) and is substituted back in Eq. (17) to compute elemental Wilson Penzien damping matrix.

Appendix 2: Description of the four story frame

The four story structure used in the present study is a test frame tested in European laboratory for structural assessment (ELSA) designed according to previous versions of Eurocodes 2 and 8 (Arede 1997). As already stated the structure falls in the high ductility class and is designed for a PGA of 0.3 g with soil type B and behavioral factor of 5. For full reinforcement details refer (Arede 1997). Figure 9 represents the geometric dimensions of the frame elevation. Floor slab masses are assumed to be lumped at the beam column joints. C25/30 grade concrete with B500 Tempcore reinforcing steel with yield strength of 500 MPa was used for the actual construction. As the frame is already experimentally tested more information regarding the material characteristics of the frame is available in (Arede 1997). As our focus is to qualitatively illustrate the performance of the proposed damping models, following simplified structural data is used for the present study.

Fig. 9

Four story frame

Material property (Dolsek 2010)

Considerable discrepancies in the material characteristics were observed in the testing in comparison to the nominal value as given by Eurocode 2. Adopted value for the present study is given as below:

$${\text{Mean concrete strength}} = 33 \;{\text{MPa}}$$

$$\text{Modulus}{\text{ of elasticity}} = 3.1 \times 10^{10} \;{\raise0.7ex\hbox{$\text{N}$} \!\mathord{\left/ {\vphantom {\text{N} {\text{m}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\text{m}^{2} }$}}$$

Geometric properties (Arede 1997)

Member number	Width of the member (mm)	Depth of the member (mm)
1,6,11,16,17,12,7,2,3,8,13,18	450	450
4,5,9,10,14,15,19,20	300	450

Nodal mass (Arede 1997)

Floor level	Mass per node (kg)
1st floor	29,800
2nd–4th floor	29,500

Yield rotations

Yield rotations are computed as per Fardis (2007). If section yielding is identified as yielding of steel reinforcement, yield curvature is given as,

$$\phi_{y} = \frac{{f_{y} }}{{E_{s} \left( {1 - \xi_{y} } \right)d}}$$

(45)

In Eq. (45),

$$\xi_{y} = \left( {\alpha^{2} A^{2} + 2\alpha B} \right)^{{\frac{1}{2}}} - \alpha A$$

(46)

where

$$\alpha = \frac{{E_{s} }}{{E_{c} }},\quad A = \rho_{1} + \rho_{2} + \rho {}_{v} + \frac{N}{{bdf_{y} }},\quad B = \rho_{1} + \rho_{2} \delta_{1} + 0.5\rho_{v} \left( {1 + \delta_{1} } \right) + \frac{N}{{bdf_{y} }}$$

(47)

$b$ is the width of the compression zone, $d$ is the effective depth of the section, $\rho_{1}$ and $\rho_{2}$ are the ratio of the tension and compression reinforcements normalized on $bd$, $\rho_{v}$ is the ratio of the web reinforcement, $\delta_{1}$ is the ratio of the distance of center of compression reinforcement from the extreme compression fibers to the effective depth of the section, $E_{s}$ and $E_{c}$ are elastic modulus for steel and concrete. Only yield rotation corresponding to flexural yielding is considered in the present study and is given as,

$$\theta_{y} = \frac{{\varphi_{y} L_{s} }}{3}$$

(48)

where $L_{s}$ is the shear span.

Member number	Yield rotation (ith node) (positive/negative) (rad)	Yield rotation (jth node) (positive/negative) (rad)
1, 2, 3	0.04	0.0074
6, 8, 11, 13, 16, 18	0.0064	0.0064
7	0.0054	0.0061
12, 17	0.0061	0.0061
4, 9	0.0093	0.0093
5, 10	0.0062	0.0062
14, 19	0.009	0.009
15, 20	0.006	0.006