Nonlinear Time History Analyses (NLTHA)

In practice, structural engineers typically design considering an individual earthquake signal in a particular direction of interest. However, this does not represent reality since earthquakes can cause ground shaking in any direction. Fortunately, with access to accelerograms the response of systems under these records can be simulated. In this manner, a SDOF column can be subjected to the two horizontal acceleration records at the same time, and its 3D response through the duration of the loading can be obtained. In this study, the program RUAUMOKO developed by Athol Carr was used to implement the nonlinear analyses. This program is able to produce a piece-wise time history response of three-dimensional structures to ground accelerations (Carr, 2017). Figure 4.1 illustrates this with an idealized column with lumped mass at the top node (Y= 𝐻_𝑐𝑜𝑙), which is free while the bottom node (Y=0) is fixed. The structure is then subjected to the pair of as-recorded motions and the top node is free to displace in any direction. The maximum displacement of the RC column is identified as the maximum distance between the top node and the point of origin (X=0, Z=0).

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Figure 4.1: Schematic of idealized RC column design under corresponding pair of as-recorded motions, where Δ_max is the maximum radial displacement from the origin.

It is essential to recognize the importance of the decisions made in modeling before performing the nonlinear analyses, even for simple systems. While the RC column designs have previously been performed in Chapter 3 for the two response spectra definitions, RotD50 and RotD100, it is necessary to identify other properties of the systems that are needed as input for the nonlinear analyses. Following Equation 3.16 and Equation 3.17, the moment capacity at yield (𝑀𝑦)

and cracked section moment of inertia (𝐼_𝑐𝑟) can be found for each of the 45,500 designs (22,750 for each spectra definition). In addition, it is necessary to provide the plastic hinge length (𝐿_𝑝),

over which strain and curvature are maxima at the base of each column design.

𝑳_𝒑= 𝒌𝑳_𝒄+ 𝑳_𝑺𝑷 ≥ 𝟐𝑳_{𝑺𝑷 Equation 4.1} where 𝒌 = 𝟎. 𝟐 (𝑭𝒖 𝑭𝒚− 𝟏) ≤ 𝟎. 𝟎𝟖 Equation 4.2 𝑳𝑺𝑷 = 𝟎. 𝟎𝟐𝟐𝒇𝒚𝒆𝒅𝒃𝒍 (𝒇𝒚𝒆 𝒊𝒏 𝑴𝑷𝒂) 𝑜𝑟 𝑳_𝑺𝑷 = 𝟎. 𝟏𝟓𝒇_𝒚𝒆𝒅_𝒃𝒍(𝒇_𝒚𝒆𝒊𝒏 𝒌𝒔𝒊) Equation 4.3

The calculation of 𝐿𝑝 (Equation 4.1) takes into account the strain penetration length (𝐿𝑆𝑃),

63 shown in Equation 4.3. For this study, these two parameters were kept constant for all designs with

𝑓_𝑦𝑒 = 𝑓_𝑦 = 400 𝑀𝑃𝑎 and 𝑑_𝑏𝑙 = 25 𝑚𝑚.

Moreover, another input parameter that needs to be provided for the NLTHA is the bilinear factor for the moment-curvature response. This is calculated following Equation 4.4 as a function of the plastic hinge length (𝐿𝑝), the column height (H), and the bilinear factor for the force-

displacement response, which was assumed as 𝑟_Δ= 0.05 for all column designs.

𝒓_𝝓= 𝟑 𝑳𝒑 𝑯 (_𝒓 𝟏

𝚫 − 𝟏)

Equation 4.4

The RUAUMOKO program provides a wide range of modeling options. In this research, RC columns were represented using a one component Giberson Beam frame element with lumped mass, while the damping was modelled using tangent stiffness proportional Rayleigh damping where the damping matrix is established using secant stiffness (ICTYPE=6). More details on the damping model can be found in Chapter 6.

Figure 4.2: Modified Takeda Hysteresis rule with Emori Unloading (Carr, 2017).

For all column designs, the hysteresis rule used was the modified Takeda degrading stiffness with Emori unloading (Carr, 2017). Figure 4.2 shows how RUAUMOKO defines this

64 hysteresis rule. First, the initial loading slope of the hysteresis is defined with the initial stiffness (𝑘_𝑜), and following slopes after yield are defined with the bilinear factor (𝑟 = 𝑟_Δ) with constant 𝛽 = 0, and constant 𝛼 = 0.5, which defines the unloading stiffness as a function of the initial stiffness (𝑘₀). These parameters result in the thinnest possible Takeda hysteretic model, which is typically used for the modeling of bridge columns, as well as being consistent with the damping-ductility relationship used in the designs discussed in Chapter 3.

The input file for each design also includes the callout for the corresponding pair of as- recorded ground acceleration records, for which the corresponding response spectrum was used to design the particular design. In this manner, the RC circular column is subjected to the actual ground motion record pair (see Figure 4.1) for which it was designed (as discussed in Chapter 3). This allows the direct comparison of the expected maximum displacement (Δ𝑡) and the maximum displacement given by the NLTHA (Δ_𝑚𝑎𝑥) as a displacement ratio of the maximum to the target displacement (Δ_𝑚𝑎𝑥/Δ_𝑇). If this ratio is equal to 1, it can be said that the structure achieved the performance it was designed to. An example of an input and output text file for one NLTHA of a column in provided in Appendix C.

4.1.1 Summary of Design Verification Assumptions

A nonlinear time history analysis was performed for each of the 45,500 column designs, (22,750 for each RotD50 and RotD100) using the corresponding pair of as-recorded ground motions. In summary:

 SDOF circular reinforced concrete bridge columns idealized as frame elements with two nodes, top node free (Y = 10 m) and bottom node fixed (Y = 0).

 Force – Displacement behavior characterized by the Takeda degrading stiffness hysteretic rule with 𝛼 = 0.5 and 𝛽 = 0.

65  Tangent stiffness proportional Rayleigh damping where the damping matrix is

established using secant stiffness (ICTYPE=6 in Ruaumoko).

 Plastic hinge length and curvature bilinear factor were calculated based on constant properties (i.e., 𝑑_𝑏𝑙 = 25𝑚𝑚, 𝑟_Δ= 0.05).