4.2. ADINA FSI Finite Element Model
4.2.5. Deterministic Response for a Case Study Bridge
This section illustrates the deterministic response of a case study bridge from
Houston/Galveston Bay area. Figure 4-9 depicts the case study bridge geometry.
The deck width is 11m and consists of a 0.2m thick slab with six AASHTO type
III girders. The bent beam is supported by three square columns of 0.8m dimension
and 5m height. The columns are reinforced with twelve 28mm diameter rebars in
the longitudinal direction and a spiral with a pitch of 0.25m in the transverse
direction. The columns are supported by a pile foundation consisting of a 2.6m
square pile cap and eight 0.4m diameter piles of 14m length. The bridge is assumed
to be located at a soft clay site which is the typical soil type in the
Houston/Galveston region. This means a cohesive soil condition with undrained
Figure 4-9. Case study bridge geometry.
The deterministic response of the developed FSI model is provided here in
order to increase insight on the response of simply supported span bridges under
extreme wave and surge condition. Key quantities monitored include the
hydrodynamic forces on superstructure and substructure, and horizontal and
vertical displacement of the superstructure. Since there is no connection between
the super- and substructure, the bridge is susceptible to deck movement and
unseating under extreme wave and surge loads. The hazard input parameters for
the considered scenarios are presented in Table 4-1. The second and third
combinations of wave and surge values are selected to produce vertical forces that
are larger than the uplift capacity (i.e., weight of the deck for the case study
bridge) according to AASHTO (2008). The fourth row, hazard scenario 3-L,
2.6m
AASHTO Type III Girder 0.2m
0.4m 5m
0.8m
contains the same hazard parameters as the third one; however, the turbulence is
neglected in this scenario (i.e., the fluid flow is considered to be laminar).
Table 4-1. Hazard parameters for the deterministic study.
No. Hmax (m) Tp (s) ds (m) Zc (m)
1 1.8 5.0 6.0 0
2 3.2 6.0 6.0 0
3 4.2 6.0 7.5 -1.5
3-L 4.2 6.0 7.5 -1.5 (laminar flow)
Figure 4-10 depicts the vertical and horizontal displacement time history of
the top waveward node of the deck for each scenario. The deck displacements for
scenario 1 are small and are in the order of magnitude of 10-4m. However, hazard
scenarios 2 and 3 (and 3-L) lead to deck shifting and unseating. Scenarios 3 and 3-
L lead to similar responses, as expected. Nevertheless, the vertical deck
displacement for the laminar flow (case 3-L) is slightly less than the turbulent flow
model (case 3). Therefore, it is recommended to use the turbulent flow model for
higher accuracy since it is believed to provide a better representation of the
problem than laminar flow model. More discussion on the flow assumption is
provided in the sensitivity study in Section 5.2.2. The results indicate that the
deck displacement is different for scenarios 2 and 3, underscoring the impact of
consequently vertical displacements are larger for the submerged deck (scenario 3).
On the other hand, lateral forces and displacements are larger for the deck located
at the waterline since the wave crest can directly impact the deck (scenario 2).
Additionally, the submerged bridge deck undergoes larger moments as well as
larger rotation. Due to these differences in responses, greater distortion occurs in
fluid elements for scenario 3, and the simulation stops before a full wave passage.
Figure 4-11 shows the displacement of the deck during wave and surge load for
hazard scenarios 2 and 3 at time 1.7s. Differences in the displacement response and
the rotation of the leeward part of the submerged deck are evident. After the
bridge deck is shifted, the contact area between super- and substructure decreases.
Thus the bridge deck has less resistance against the next wave cycle, as can be
seen in Figure 4-10 (b) for scenario 2.
Figure 4-10. (a) Vertical and (b) horizontal deck displacements under different wave and surge load scenarios.
Figure 4-11. Deck displacement under wave and surge action at time 1.7s: (a) scenario 2; and (b) scenario 3.
Figure 4-12. (a) Vertical and (b) horizontal forces per unit length on the superstructure. The bold line in (a) represents the weight of the bridge deck per unit
length.
In addition to displacements, the total forces per unit length imposed on the
bridge deck are plotted in Figure 4-12. The deck weight is shown in a bold line.
Vertical forces on the bridge deck for scenario 1 follow a sinusoid pattern, similar
to the trend observed in the experiment (Sheppard and Marin 2009). Nonetheless,
the force pattern is not sinusoidal for scenarios that lead to the rigid body
deck movement; therefore, this phenomenon has not been observed (A preliminary
experimental result of the unrestrained deck is presented by Cox et al. (2012) in
the ATC-SEI Advances in Hurricane Engineering Conference, which supports the
results of FSI the model). The vertical forces do not increase significantly after the
deck uplift. The initial vertical forces for scenarios 3 and 3-L are greater than zero
due to the buoyancy. By comparing Figure 4-10 and Figure 4-12, it can be
concluded that the waveward face of the deck can be uplifted before vertical forces
fully overcome the deck weight. However, significant transverse displacement
occurs after the full deck uplift. Also, Figure 4-12 (b) reveals an alteration in the
direction of horizontal forces on the submerged bridge deck (3 and 3-L) after deck
uplift. The vertical forces are almost identical for turbulent and laminar flow.
However, there is a slight difference in the horizontal forces. The maximum
difference in terms of horizontal forces between turbulent and laminar flow models
is 2.1%.
As mentioned in Chapter 3, wave and surge loads on the bridge
substructure can be approximated based on the Morison equation (1950). The fluid
domain should be solved to find the acceleration and velocity of water particles
spatially varying; i.e., they are not constant along the column height. Since the
water is relatively shallow, and large wave heights are expected during coastal
storms, linear wave theory is not applicable to solve the fluid domain. Therefore,
the stream function (Dean 1965) is employed to solve the fluid domain for the
velocity and acceleration. The stream function solves the Laplace equation with a
nonlinear free surface boundary condition by computing a series solution. A
computer code, originally developed by Chaplin (2012) is modified to calculate the
stream function and the velocity and acceleration of water particles at any given
point of time and space for the given wave profile. This computer code
automatically increases the stream function order to reach an accurate water
surface that matches the input profile. After solving the fluid domain, forces are
calculated by integrating Equation (3-5) over 50 points along the column height.
Figure 4-13 (a) compares the horizontal velocity of water particles at
elevation 1.2m from the channel bottom line obtained from the stream function
and the FSI model for the first hazard scenario. Also, Figure 4-13 (b) depicts the
water velocity profile for hazard scenario 1 under the wave crest. This figure shows
a good agreement between the theoretical (stream function) and numerical model.
Figure 4-13. Horizontal water velocity at 1.2m elevation; and (b) horizontal water velocity profile at wave crest for scenario 1. Theoretical values are calculated from the
stream function, where numerical values are the results from FSI model.
Horizontal forces on the waveward column for the first hazard scenario
resulting from the numerical model and the Morison equation are shown in Figure
4-14. The Cd and Cm coefficients in the Morison equation are taken equal to 1 and
2, respectively. Although the fluid domain is in good agreement, the forces from
the Morison equation are smaller than the numerical model. This is because the
Morison equation does not consider wave diffraction, which is not an accurate
assumption for the column width in the case study bridge. Therefore, the Morison
equation does not lead to conservative results for estimation of forces on
substructure. Nevertheless, the forces on the substructure are significantly smaller
than the forces on superstructure, and they do not govern the behavior of the
5.2.2. Therefore, no modification is proposed for the substructure forces that are
calculated from the Morison equation.
Figure 4-14. Horizontal forces on waveward column for scenario 1. Theoretical values are calculated from the Morison equation (1950), where numerical values are the
results from FSI model.
4.3. Summary
This chapter introduced two different numerical models that are developed to
assess the vulnerability of coastal bridges under storm wave and surge loads. The
first model only includes the structural domain, and thus, is more computationally
efficient. The second model includes a full fluid-structure interaction model which
is verified by comparison with experimental test data from the literature. This
model is computationally intense; however, it can be used for validation of the
and surge load models applied to the structure only simulations. The two
developed numerical models are used in the following chapters for probabilistic
76
Chapter 5
Hurricane Hazard Parameters and
Uncertainty Treatment
This chapter explores the hazard intensity measures that should be adopted to
condition the fragility models for coastal bridges. Also, this chapter defines the
random variables that should be considered in the probabilistic analysis.
Appropriate probability density functions are introduced for hazard and structural
random variables. The significance of different modeling parameters are identified