# Evaluation of DSSI Effects on the Dynamic Response of Bridges to Traffic Loads

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Factors Controlling DSSI

_{0}is introduced as ${a}_{0}=\frac{\omega B}{{V}_{s}}$ where ω is the excitation frequency, B is the half-width of a footing (or radius if circular), and V

_{s}is the effective shear wave velocity of the underlying soil. The effect of the following factors will be examined in the follow-up sections.

- Structure-to-Soil Stiffness Ratio (Rigidity Ratio):

- Structure-to-Soil Slenderness Ratio:

- Structure-to-Soil Mass Ratio:

_{0}> 1.5) [21]. In general, increasing the mass ratio leads to an increase in impedance, owing to a more meaningful increase in the real part (stiffness) than the imaginary part (damping) [22,23]. This is expected since hysteretic (material) damping is frequency-independent at low strains since it is controlled by the frictional forces between the soil and foundation at the interface. The dependence of the imaginary part of impedance on frequency would be driven by radiation damping, which can vary in complexity. It can be neglected in some sites, while other sites may exhibit significant viscous damping [14].

- Embedment Ratio:

- Other Factors:

## 3. Experimental Program

_{s}) profile was obtained through Multichannel Analysis of Surface Waves (MASW) testing before bridge testing. Various inversion parameterizations for the theoretical fundamental mode Rayleigh wave dispersion, considering several layering ratios, were evaluated to match the experimental data and to obtain the V

_{s}profile of the site. An average shear wave velocity of 200 m/s was deemed appropriate for the depth down to about 15 m.

## 4. Description of Numerical Model

^{−5}. Therefore, linear elastic material properties were assigned in both models to all domains, since the response was in the elastic range due to the controlled vibration levels. Hence, soil properties remain constant and they represent low-strain moduli and damping conditions.

_{dM}+ Kβ

_{dk}, where α

_{dkM}is the mass damping parameter, and β

_{dk}is the stiffness damping parameter. A value of β

_{dk}= 0.0065 was selected to represent the structural damping. α

_{dM}= 0 was set in the study since inertial effects are low in low-frequency ranges, and the behavior is more stiffness controlled. In addition, β

_{dk}was fixed in both time and frequency domains to maintain compatibility and modeling simplicity. It is a theoretical limitation that the damping loss factor cannot be set as a function of frequency in time domain studies. However, multiple Rayleigh damping ratios at multiple frequencies can lead to more accurate results in the frequency domain. Free tetrahedral elements with adaptive sizing were used to mesh the entire domain. Figure 9 illustrates the overall meshed model, in which concrete elements had coarser meshes, while steel elements (girders and bracing) were fine-meshed. This meshing resulted in 1,007,262 degrees of freedom (DoFs).

## 5. Experimental Results

_{xx}= Re(φ)

^{T}× Re(φ), L

_{yy}= Im(φ)

^{T}× Im(φ), and L

_{xy}= Re(φ)

^{T}× Im(φ) are scalar products from the mode shape φ.

## 6. Results from Numerical Simulation and Model Validation

_{i}) can be defined through (2).

_{i}} is the ith mode shape vector, [M] is the mass matrix, and {D} is a unit displacement/rotation vector in the direction of excitation for global Cartesian coordinates and rotations about their axes. The effective mass M

_{eff,i}is defined as ${\gamma}_{i}^{2}$. Subsequently, the ratio of effective mass to total mass indicates the contribution of the ith mode to the dynamic response of the bridge (MPF). The estimated total mass of the bridge is 1,029,400 kg, including the weight of the T-Rex. The mass moment of inertia was used instead of the mass to calculate MPFs for rotational DOFs. This is not as straightforward as determining translational MPFs, since this calculation requires knowledge of the center of mass of individual components of the SFS system. However, as an approximation, (3) provides an estimate for the rotational MPF of typical/notional bridges.

_{i,r}is the mass moment of inertia about the rth axis for the ith (e.g., J

_{1,x}= m[Y

^{2}+ Z

^{2}]), (X, Y, Z) is the center of mass of the SFS system, and n is the modifier between 3–5 to account for the overall off-center estimation in lieu of individual bridge components estimation. An average value of n = 4 was used, and the center of mass of the SFS system was found to be at (6.05, 33.7, −1.19) m from the origin indicated in Figure 11. Table 3 summarizes the estimated MPFs for each DOF. It can be observed that the presented modes capture most of the responses fairly, with the lowest MPF of ~0.65 for the significant motions. As expected, MPF

_{y}and MPF

_{z-z}, which correspond to translation along Y (longitudinal swaying) and rotation about Z (global torsion), are approximately 0. This confirms a proper assignment of boundary conditions. Furthermore, Mode#5 contributes as high as ~0.72 of MPF

_{y-y}(rocking), while Mode#3 contributes ~0.18 (both modes represent 0.9 of this motion). This indicates that rocking is primarily the vibration mode of Mode#5, while also exhibiting some swaying. On the other hand, Mode#3 has predominantly translation motion (swaying) compared to Mode#5. Nevertheless, swaying–rocking is coupled to some extent in both modes.

_{ini}− δ

_{f}× t

_{ini}= 15 Hz, t is time, and δ

_{f}is the frequency gradient (0.4375 Hz/s). Figure 12 shows response time histories caused by the lateral shaking for both the experimental and numerical results. The DSSI-incorporating model was more accurate than the fixed-base model when compared to the experimental results. Around resonance, the DSSI model showed a better match with the experimental results, supplemented by a lower mean absolute error (MAE). Table 4 shows a comparison of measured response and corresponding frequencies as obtained from the experimental results and both FE models. The maximum response from the experiment, 2.62 cm/s, occurred at a frequency of 4.16 Hz, as shown in Table 4. The maximum amplitude at the same frequency of the DSSI model was 2.53 cm/s, with a relative amplitude error of −3.5%. Meanwhile, the response at the same frequency from the fixed-base model was 2.41 cm/s with a relative error of −8.1%. Therefore, the fixed-base assumption led to a higher error in the response amplitude at resonance. The decrease in resonant frequencies due to DSSI, compared to a fixed base model, is in agreement with experimental results from other studies [39,40]. This is important since, despite the error in estimating the peak response being lower, the fixed-base assumption may lead to the omission of a vibration mode. This lateral mode is caused by soil flexibility, and its omission could lead to analysis and design errors [41,42,43]. The validated models were used to calculate displacements. Figure 13 shows a comparison of the deck’s response to horizontal excitation at a magnitude of 93.4 kN between the fixed-base and model the model including DSSI effects. It can be observed that the fixed-base model does not exhibit the multiple modes determined from the experimental study, hence skipping some modes, although they exist from the eigenfrequency analysis. Therefore, DSSI incorporation overall leads to a better match with amplitude and dynamic behavior when compared to experimental results. For the tested bridge, the incorporation of DSSI effects led to an increase in lateral displacement relative to the fixed-base model. For the mode exhibited by both models (~4.2 Hz), this increase is approximately 14%.

## 7. Parametric Study of Factors Influencing Dynamic Response

_{s}), structural height (H

_{tot}), foundation half width (B), and foundation depth (D

_{f}). Specific combinations are shown herein rather than all possible combinations while holding other parameters constant.

- Embedment ratio
- Structure-to-Soil Stiffness Ratio (Rigidity Ratio)
- Structure-to-Foundation Slenderness Ratio

_{0}= ωB/V

_{s}). For the second peak which describes rocking, it is evident that increasing $\overline{h}$ led to a reduction in DSSI alteration of response relative to fixed-base. This means that this effect diminishes for more slender structures. Rotations dictate the response, and they become more controlled by the moment arm rather than the boundary condition, i.e., fixed vs DSSI. This effect also depends on the stiffness ratio, which is discussed next. For flexible structures, the rocking of the foundation is less important than for rigid structures. As for the first peak, increasing $\overline{h}$ led to increasing the DSSI alteration of response. The bridge experiences stronger coupling between lateral, rocking, and vertical modes. Therefore, further investigation is required to determine this effect on rotations independently. However, overall softening was observed from the first peak relative to a fixed base, where the resonant frequency was reduced further than the computational frequency domain considered (around detected resonant frequencies from the experimental program). Away from the lateral resonant frequencies, the ratio approaches 1 as $\overline{h}$ increases in the interval a

_{0}> 0.9. On the other hand, the response is reduced relative to a fixed base at the anti-resonance down to a ratio of 0.75. The valley between the two peaks corresponds to a vertical mode shape as determined in experimental results, which again suggests some coupling in the bridge dynamic response that counteracts the rocking behavior. Figure 23 shows the effect of the $\overline{h}$ on the peak-amplitude frequency. As discussed, the effect of DSSI decreased with increasing $\overline{h},$ which also means structural softening due to DSSI effects diminishes as slenderness increases when examined at the same rigidity ratio.

_{s}and h

_{tot}. Figure 24 shows the effect of the rigidity ratio $\overline{s}$on the peak amplitude ratio of DSSI/fixed lateral deck response. Increasing $\overline{s}$led to a decrease in the maximum amplitude since the rigidity of the SFS is increasing. This is in agreement with the literature, since the system becomes closer to the fixed-base assumption [14]. Figure 25 shows the reduction of the second peak amplitude ratio and the frequency of the peak as a function of dimensionless frequency with increasing $\overline{s}$. This follows the expected behavior that increasing $\overline{s}$would eventually lead to a fixed-base condition.

## 8. Conclusions

- Large-amplitude mobile shakers are effective in exciting the entire bridge-foundation-soil system and they facilitate the accurate determination of global dynamic characteristics in a non-destructive manner. This was proven through the experimental program reported in this study. Therefore, this study opens the possibility for the use of such shakers in future testing of in-service bridges.
- Shaking in multiple directions using linear chirp functions enabled the identification of natural frequencies (modes) of the tested bridge. It was demonstrated that shaking the bridge at multiple locations can facilitate the detection of more natural modes/frequencies. Furthermore, ignoring DSSI effects in bridge models can lead to inaccurate dynamic response and omission of some modes of vibration.
- The rocking behavior of the foundation was captured from the experimental program, which served as a basis for the comparison of different numerical models established in this study. Coupling between vibration modes or the presence of closely spaced modes can pose a challenge to the intuitive understanding of the dynamic behavior of an excited bridge. Therefore, examining pure motion modes (e.g., rocking, vertical, etc.) assists in understanding dynamic characteristics more easily.
- Peak amplitude, resonant frequency, and the overall time history were shown to be unique for each combination, and this allows for discerning which set of parameters fits the experimental results.
- The results obtained from evaluating the rigidity, embedment, and slenderness ratios in this study agreed with the typical dynamic behavior reported in the literature. The rigidity and slenderness ratios are the most influential parameters in controlling the extent of DSSI effects on the rocking response of bridges, which is similar to the one evaluated in the current study.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Typical cross-section at notional bridge piers, showing different components of a structure-foundation soil system.

**Figure 7.**Overall sensor layout and T-rex loading positions. Reprinted with permission from Ref. [34].

**Figure 10.**Transverse response time histories from sensor locations mentioned in the legend to transverse shaking at 18 kN with T-Rex centered above the pier.

**Figure 12.**Lateral velocity from west geophone at the deck from the test, fixed-base model, and DSSI model due to shaking at a 93.4 kN lateral load magnitude.

**Figure 14.**Results from frequency-domain models compared to experimental results for the lateral response of the deck to lateral shaking at 93.4 kN.

**Figure 15.**Results from frequency-domain models compared to experimental results for the vertical response of the midspan to lateral shaking at 93.4 kN.

**Figure 16.**Effect of varying H

_{to}

_{t}on lateral deck response to lateral shaking (Shown for V

_{s}= 200 m/s, B = 5.7 m, D

_{f}= 2 m).

**Figure 17.**Effect of varying V

_{s}(m/s) on lateral deck response to lateral shaking (Shown for H

_{tot}= 5 m, B = 5.7 m, D

_{f}= 2 m).

**Figure 18.**Effect of varying D

_{f}(m) on the lateral deck response to lateral shaking (Shown for H

_{tot}= 5 m, B = 5.7 m, V

_{s}= 200 m/s).

**Figure 19.**Effect of varying D

_{f}(m) on lateral footing response to lateral shaking (Shown for H

_{tot}= 5 m, B = 5.7 m, V

_{s}= 200 m/s).

**Figure 20.**Effect of varying B (m) on lateral footing response to lateral shaking (Shown for H

_{tot}= 5 m, D

_{f}= 2 m, V

_{s}= 200 m/s).

**Figure 21.**The effect of the D/B ratio on altering the DSSI based on (

**a**) peak amplitude and (

**b**) peak frequency, with other parameters held constant.

**Figure 22.**Effect of varying $\overline{h}$ on altering DSSI effects relative to a fixed base based on the peak lateral amplitude ratio of the deck due to lateral shaking. Shown for V

_{s}=200 m/s, B = 5.7 m, and D

_{f}= 2 m.

**Figure 23.**Extent of $\overline{h}$ on structural softening due to DSSI effects relative to a fixed base. Shown for V

_{s}=200 m/s, B = 5.7 m, and D

_{f}= 2 m.

**Figure 24.**Extent of the $\overline{s}$ alteration of lateral peak amplitude caused by lateral shaking due to DSSI effects relative to a fixed base. Shown for B = 5.7 m, D

_{f}= 2m, and varying h

_{tot}and V

_{s}.

**Figure 25.**Effect of varying $\overline{s}$ on altering DSSI effects relative to a fixed base based on the peak lateral amplitude ratio of the deck due to lateral shaking. Shown for B = 5.7 m, D

_{f}= 2 m, and varying V

_{s}and h

_{tot}.

Mode | Resonant Frequency (Hz) | Determination Method | Damping ξ_{s} (%) | Determination Method | MCF (%) |
---|---|---|---|---|---|

1 | 2.75 | Peak Picking | 1.74 | Half-power | - |

2 | 3.39 | Peak Picking | 1.18 | Half-power | - |

3 | 4.39 | RFP Model | 3.71 | RFP Model | 35.909 |

4 | 4.44 | RFP Model | 3.73 | RFP Model | 22.375 |

5 | 4.69 | RFP Model | 3.67 | RFP Model | 20.249 |

6 | 8.27 | Peak Picking | 1.49 | Half-power | - |

7 | 8.81 | Peak Picking | 2.41 | Half-power | - |

**Table 2.**Eigenfrequencies obtained from numerical simulations of DSSI-incorporating and fixed-base models.

Mode | Resonant Frequency—Experimental (Hz) | Resonant Frequency *—DSSI (Hz) | Resonant Frequency **—Fixed-Base (Hz) |
---|---|---|---|

1 | 2.75 | 2.98 (+8%) | 3.13 (+13%) |

2 | 3.39 | 3.2700 + 5.07 × 10^{−5}i (−4%) | 3.28 (−3%) |

3 | 4.39 | 4.1779 + 3.20 × 10^{−5}i (−5%) | 4.20 (−4%) |

4 | 4.44 | 4.3734 + 2.07 × 10^{−3}i (−2%) | 4.39 (−1%) |

5 | 4.69 | 4.7502 + 3.74 × 10^{−3}i (+1%) | 4.81 (+2%) |

6 | 8.27 | 8.00 (−3%) | 8.00 (−3%) |

7 | 8.81 | 8.58 + 4.56 × 10^{−4}i (−3%) | 8.59 (−3%) |

Mode | MPF_{x} | MPF_{y} | MPF_{z} | MPF_{x-x} | MPF_{y-y} | MPF_{z-z} |
---|---|---|---|---|---|---|

1 | 4.79886 × 10^{−6} | 0.003778069 | 6.63835 × 10^{−8} | 0.678519839 | 8.89294 × 10^{−8} | 5.1748 × 10^{−5} |

2 | 1.23837 × 10^{−5} | 3.56323 × 10^{−6} | 2.20892 × 10^{−7} | 0.000898653 | 1.47175 × 10^{−6} | 0.037969573 |

3 | 0.438255866 | 1.27803 × 10^{−6} | 0.002418895 | 3.65114 × 10^{−6} | 0.184974325 | 4.93414 × 10^{−7} |

4 | 0.003515275 | 1.10738 × 10^{−9} | 0.645697958 | 2.32779 × 10^{−9} | 7.3578 × 10^{−7} | 4.26308 × 10^{−9} |

5 | 0.305994111 | 4.32857 × 10^{−7} | 0.000675773 | 9.06041 × 10^{−7} | 0.722411423 | 3.51911 × 10^{−9} |

6 | 7.00489 × 10^{−8} | 9.30383 × 10^{−7} | 1.34622 × 10^{−8} | 1.55898 × 10^{−6} | 2.41889 × 10^{−8} | 1.76563 × 10^{−5} |

7 | 2.74277 × 10^{−8} | 5.4032 × 10^{−8} | 0.000318148 | 1.57805 × 10^{−7} | 2.3009 × 10^{−5} | 8.37671 × 10^{−8} |

Σ | 0.747782532 | 0.003784328 | 0.649111074 | 0.679424768 | 0.907411078 | 0.038039562 |

**Table 4.**Lateral response due to lateral shaking at the temporal trace of frequency of select frequencies.

Frequency (Hz) | Response [cm/s] (% Error) | ||
---|---|---|---|

Fixed | DSSI | Test | |

4.16 | 2.41 (−8.1%) | 2.53 (−3.5%) | 2.62 (−) |

4.49 | 2.83 (32.4%) | 2.41 (12.9%) | 2.13 (−) |

4.69 | 0.79 (11%) | 0.74 (4.2%) | 0.71 (−) |

**Table 5.**Summary of factors affecting the extent of DSSI effects on altering rocking motion in typical overpass bridges.

Parameter | Effect on Peak Amplitude Ratio | Effect on Resonant Frequency |
---|---|---|

Embedment Ratio (D/B) | Increasing D/B leads to increasing the amplitude ratio, with a diminishing effect beyond 0.6, where the response becomes less dependent on DSSI effects. Increasing footing depth (D) decreases the peak amplitude of the footing rocking motion. | Increasing D/B causes the resonant frequency ratio to approach 1 or approach the fixed-base condition. This is due to the substantial increase in damping the embedment provides, and the system becomes stiffer overall through increased impedance. Increasing footing depth (D) increases the rocking resonant frequency of the footing rocking motion. |

$\mathrm{Slenderness}\text{}\mathrm{Ratio}\text{}\left(\overline{h}\right)$ | Increasing $\overline{h}$ leads to a decrease in the peak amplitude ratio. The ratio becomes 1 beyond a ratio of 1.2, confirming that DSSI diminishes beyond a certain threshold, and the response becomes a function of the structure. This threshold depends on the structure type and site conditions. | Increasing $\overline{h}$ leads to a decrease in the resonant frequency ratio since larger rotations are present when DSSI effects are partially controlling the rocking motion. However, the drop in frequency ratio starts to diminish, and structural aspects (i.e., structural height [h]) overtake and control the overall response. Therefore, DSSI effects are less prominent in slender bridges. |

$\mathrm{Rigidity}\text{}\mathrm{Ratio}\text{}\left(\overline{s}\right)$ |
For a set value of V_{s}, the effect of
$\overline{s}$ on the overall response follows the effect of
$\overline{h}$ with respect to rocking motion. Increasing the rigidity ratio leads to a decrease in the peak amplitude ratio, implying that DSSI effects start to diminish, generally beyond $\overline{s}$ = 0.2. However,
$\overline{s}$
can have different effects on other vibration modes since it is evaluated at the resonant frequency of the motion. Coupled modes may have a decrease or increase in the peak amplitude ratio.
| For different values of ($\overline{s}$), the extent of DSSI altering the response would depend on the mode(s) of vibration considered. For rocking motion, there was a decrease in resonant frequency ratio up to $\overline{s}$ = 0.2, beyond which the response starts to approach fixed-base response. A drop from a ratio of 1.9 to 1.65 was observed for the dimensionless frequency range covered in the current study at a steady decrease, implying that a fixed-base condition would be achieved at a certain $\overline{s}$ . |

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## Share and Cite

**MDPI and ACS Style**

Farrag, S.; Gucunski, N.
Evaluation of DSSI Effects on the Dynamic Response of Bridges to Traffic Loads. *Constr. Mater.* **2023**, *3*, 354-376.
https://doi.org/10.3390/constrmater3040023

**AMA Style**

Farrag S, Gucunski N.
Evaluation of DSSI Effects on the Dynamic Response of Bridges to Traffic Loads. *Construction Materials*. 2023; 3(4):354-376.
https://doi.org/10.3390/constrmater3040023

**Chicago/Turabian Style**

Farrag, Sharef, and Nenad Gucunski.
2023. "Evaluation of DSSI Effects on the Dynamic Response of Bridges to Traffic Loads" *Construction Materials* 3, no. 4: 354-376.
https://doi.org/10.3390/constrmater3040023