- Email: [email protected]
Modeling Tsunami Induced Debris Impacts on Bridge Structures using the Material Point Method
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 175 (2017) 175 – 181
1st International Conference on the Material Point Method, MPM 2017
Modeling tsunami induced debris impacts on bridge structures using the material point method Wen-Chia Yanga , Krishnendu Shekhara , Pedro Arduinob,∗, Peter Mackenzie-Helnweinb , Greg Millerb a Graduate
Student, Department of Civil and Environmental Engineering, University of Washington, Seattle, WA 98195-2700, United States of Civil and Environmental Engineering, University of Washington, Seattle, WA 98195-2700, United States
b Department
Abstract Bridges represent a key part of infrastructure, playing a critical role in emergency response and post-event reconstruction. In this context, it is important for bridges to be able to survive both ground shaking and the effects of tsunamis. This paper focuses on numerically modeling bridge loading due to tsunamis. Although several studies have addressed the effect tsunami loads on bridges, few have examined the influence of debris carried by the tsunami. These problems involve complex contact interactions between solids and fluids that are not easily accommodated using typical fluid-oriented or solids-oriented numerical frameworks. In this paper the material point method (MPM) is used to address fluid and solid (moving and stationary) interactions with emphasis in evaluating demands on bridge superstructures by tsunami-driven debris. Two parametric studies have been conducted in this study. The first is aimed at understanding the influence of the contact area and the eccentricity of impact on the impact forces. The results show that the effect of these two factors are reduced with an increase in longitudinal length of the solid object. The second part of the study tries to evaluate the difference between tsunami driven debris impact loads and in-air debris impact loads. The results indicate that tsunami driven debris impacts tend to be larger by upto 35% compared to only in-air impact forces calculated using empirical equations. © by Elsevier Ltd. This is an openB.V. access article under the CC BY-NC-ND license ©2017 2016Published The Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 1 st International Conference on the Material Point Method. Peer-review under responsibility of the organizing committee of the 1st International Conference on the Material Point Method Keywords: Material point method; debris; fluid; impact; tsunami; bridge; fluid-solid interaction.
1. Introduction Throughout history, strong earthquakes have struck countries all over the world and caused major damage. Most of this damage has been due to ground shaking, but in coastal areas tsunamis induced by earthquakes have resulted in greater loss of lives and infrastructure. As coastal populations continue to increase around the world, understanding and managing tsunami effects on infrastructure becomes increasingly important. The objective of this study is to give preliminary results for researchers and engineers trying to understand the demands on bridge superstructures by tsunami-driven debris. It is imperative that bridges are able to survive both ∗
Corresponding author. E-mail address: [email protected]
1877-7058 © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 1st International Conference on the Material Point Method
doi:10.1016/j.proeng.2017.01.050
176
Wen-Chia Yang et al. / Procedia Engineering 175 (2017) 175 – 181
(a)
(b)
Fig. 1: 3-D simulation of two pieces to debris impacting a column causing partial damming.
ground shaking and the effects of tsunamis caused by earthquakes. Many prior numerical and experimental studies considering tsunami induced loads on bridges have been done, especially after the Great East Japan Earthquake. However, few of these earlier studies have examined the influence of debris carried by the tsunami. The studies that have been completed have demonstrated that debris can cause strong impact forces on columns and walls [1–3]. Lessons from these studies should be straightforward to apply to bridge vertical structures (i.e. bridge piers), but need some adjustment when applied to bridge superstructures, since in this case flows carrying debris have different characteristics compared with flows considered in the previous studies. In addition, debris can also affect the fluid motion around the bridge and introduce additional forces with longer duration compared to the impact. This class of problems is not well studied yet in the literature and involve complex contact interactions between solids and fluids. These effects are not easily accommodated with typical fluid-oriented or solid-oriented numerical frameworks. In this research the material point method (MPM) is used to model these complex fluid/solid (moving and stationary) interactions. The ultimate goal of the work is to model tsunami induced debris impacts in three dimensions like shown in Figure 1 and to understand damming and more complex flows around the superstructure during a tsunami. The results presented in this study are obtained using a single-threaded MPM code. This limits the mesh and particle refinements used in the validation examples and the debris-induced load study is relatively coarse compared to general applications found in the literature. A program allowing for parallel computing and /or one that employs implicit algorithms is necessary for further study in the future. 2. Background With large peak values, impact forces due to debris (solid objects) can cause severe local damage on bridge components, even though they happen in a very short time. In the literature, impact forces are evaluated using simple equations based on fundamental physics. In ASCE/SEI 7-10 [4], for instance, maximum impact forces F I p are calculated using impulse-momentum based equations of the form FI p =
πmd vd 2ΔtI
(1)
where md and vd represent the mass and impact velocity of the debris, and ΔtI is the time interval over which the debris stopped from its original velocity vd . Recommended and measured values of ΔtI range from 10−3 seconds to 1.0 seconds. This diversity of ΔtI results in very different impact forces and hence demands on structural systems. To improve Equation (1) and avoid the need to select a ΔtI , the flexibility of the solid object representing the debris is considered. By simplifying the collision of a debris into a bridge deck by a 1D spring model with equivalent stiffness keq , Equation (1) can be rewritten as, F Ik = cI vd keq md (2)
177
Wen-Chia Yang et al. / Procedia Engineering 175 (2017) 175 – 181 Table 1: Variables considered in the study of impact forces due to an 2D solid block.
Variable height thickness length initial velocity mass density Young’s modulus Poisson’s ratio contact height [eccentricity]
Symbol hd td L vI ρd Ed νd hc [ec ]
Unit (m) (m) (m) (m/s) (kg/m3 ) (GPa) (m) [(m)]
Values 3.0 1.0 3, 6, 9, 12, 15 3, 6, 9, 12, 15 200, 400, 600, 800, 1000 10, 5, 3, 1, 0.5, 0.3, 0.1 0.1, 0.2, 0.3 3.0 [0.0] 1.5 [0.75, 0.00] 1.0 [1.00, 0.75, 0.00] 0.5 [1.25, 1.00, 0.75, 0.00]
Fig. 2: Illustration of model setup for studying the influences of contact areas on impact forces.
where cI is a constant. Piran Aghl et al.[3] suggested cI be taken as 1.0 for impact forces caused by a 20-ft shipping container colliding with a wall in air, and evaluated the equivalent stiffness keq with keq =
Ed Ac L
(3)
where Ed and L are the modulus of elasticity and length of debris (shipping container), respectively; and Ac is contact area. Consequently, with md = ρd Ad L Equation (2) becomes ⎛ ⎞ ⎜⎜ Ac ⎟⎟ ⎟⎟⎠ F Ic = vd Ed Ac ρd Ad = vI Ad Ed ρd ⎜⎜⎝ Ad
(4)
Ko et al.[2] verified Equation (2) in their study and used experimental data to determine the equivalent stiffness keq instead of Equation (3) for their 1/5-scale shipping container. Furthermore, they found that impact forces for in-water cases are no greater than 17% of the corresponding impact forces for in-air cases, and can be approximately predicted with Equation (2) using the same keq determined from in-air tests. 3. Contact area and eccentricity In this study, the influence of contact area and eccentricity on impact forces are evaluated with a parameter study using two-dimensional (plane strain) solid blocks. To simplify the problem, only in-air conditions are considered here (i.e. no water). Figure 2 illustrates the model setup. Model descriptions and symbols shown in this figure are listed in Table 1 along with the corresponding values considered in this study. This results in 26250 combinations (models). Debris is treated as solid objects and is initially adjacent to a rigid boundary simulating the moment right before contact. Analysis time tana of each model is taken as tana =
4L cd
(5)
178
Wen-Chia Yang et al. / Procedia Engineering 175 (2017) 175 – 181
(a)
(b)
(c)
Fig. 3: Comparison of simulated impact faces of models with (a) all lengths, (b) L = 3 m and (c) L = 12 m against theory values, F Ith . (hc = 3 m).
(a)
(b)
(c)
Fig. 4: Comparison of simulated impact faces of models with L = (a) 3 m, (b) 6 m and (c) 12 m against theory values, F Ith . (ec = 0 m).
(a)
(b)
(c)
Fig. 5: Comparison of simulated impact faces of models with L = (a) 3 m, (b) 6 m and (c) 12 m against theory values, F Ith . (hc = 0.5 m).
in which cd = El /ρd is the stress wave propagation speed and El is a longitudinal elastic modulus. For a plane strain problem, El can be calculated with Ed (6) El = 1 − νd2 Regular square grids with cell size 0.5 m by 0.5 m, and nine (3 by 3) particles per cell are used. Figure 3 compares the simulation results of all models with full contact surface (i.e. hc = 3 m) against theory values: F Ith = td vI hd ρd El (7) The simulation results have good agreement with the theory when the solid blocks have longitudinal lengths L greater or equal to 6 m, but have about 15% error when L = 3 m as shown in Figure 3(b). This could be caused by the coarse mesh size (which is controlled by the efficiency of self-developed single-threaded MPM program) and hence is identified as the accuracy limitation of this study. To study the influence of the contact area Ac = td hc and eccentricity ec on the impact forces, first the case of ec = 0 m is investigated. Figure 4(a) shows that a decrease in contact area
Wen-Chia Yang et al. / Procedia Engineering 175 (2017) 175 – 181
(a)
(b)
Fig. 6: Illustration of modified contact height hc in the case when debris length is (a) larger or (b) smaller than the transition zone.
(a)
(b) Fig. 7: Comparisons of simulated impact forces against (a) F Ith and (b) F Imod .
brings down the impact on the boundary, which is consistent with Equation (4) suggested by Piran Aghl et al.[3]. Comparing Figures 4(a) to (c) the influence of Ac is lessen with increases of the block lengths L. Next, with a fixed contact area Ac = 0.5 m2 (hc = 0.5 m), which most variants of ec , the eccentricity affect the impact forces slightly in this study and has less influence with longer longitudinal length L (Figure 5). According to the observation in Figures 4 and 5, a modified height (area) hc with tan θ = 5 (as illustrated in Figure 6) is suggested to be used in Equation (7), i.e. (8) F Imod = td vI hc ρd El = vI Ac ρd El Figure 7 shows this modified equation can capture the impact forces significantly better than Equation (7). Furthermore, because the influence of length L and eccentricity ec on impact force is considered, Equation (8) shall be a suitable equation for predicting impact forces introduced by solid objects. However, more study will be necessary to examine the capability of Equation (8) to predict impact forces introduced by objects containing spaces or gaps, e.g. shipping container, in which the stress waves have different propagation path.
Fig. 8: Illustration of model setup for studying the contribution of water flow to debris-induced impact forces.
179
180
Wen-Chia Yang et al. / Procedia Engineering 175 (2017) 175 – 181 Table 2: Variables considered in the study of impact forces due to tsunami-driven debris.
Variable height thickness length initial velocity mass density Young’s modulus Poisson’s ratio water depth
Symbol hd td L vI ρd Ed νd Dw
(a)
Unit (m) (m) (m) (m/s) (kg/m3 ) (GPa)
Values 3.0 1.0 3, 6, 9 3, 6, 9, 12, 15 200, 400, 600, 800 10, 5, 3, 1, 0.5, 0.3, 0.1 0.1, 0.2, 0.3 4, 6, 8
(m)
(b)
(c)
Fig. 9: Comparisons of tsunami-driven debris-induced impact forces against F Imod for in-water cases calculated from Equation (8).
Fig. 10: Influences of debris mass density on the impact forces for in-water cases (L = 9 m).
4. Contribution of water flow In the last part of the study, the influence of water flow on debris-induced impact force is discussed with a parameter study using two-dimensional (plane strain) solid blocks. Figure 8 illustrates the numerical model used in this study and Table 2 lists all variable values under consideration. This is a full scale model and has a 1 m-by-5 m bridge. The acceleration due to gravity is g = 9.81 m2/s; The water has Bulk modulus Kw = 2.2 GPa, mass density ρw = 1000 kg/m3 , and viscosity μ = 0.001 Pa · s. A cell-based anti-locking algorithm proposed by [5] is used here for fluid modeling; and a constant flux smoothing algorithm with linear and oscillation limiters presented by [6] is used here to stabilize the simulation. Regular square grids with cell size 0.5 m by 0.5 m, and nine (3 by 3) particles per cell are used. Figure 9 indicates the impact forces due to debris (solid objects) in the water flow are about 25 to 35 percent larger than the corresponding impact forces for in-air cases, evaluated with Equation (8). The variance shown in Figure 9(c) is related to the mass density ρd of debris. Figure 10 indicates the influence of debris mass density on the impact force when the length of debris is kept constant. Since, ρd in this case influences the contact area, eccentricities of impact
Wen-Chia Yang et al. / Procedia Engineering 175 (2017) 175 – 181
forces, and wet areas of the debris, more study will be necessary to thoroughly understand the contribution of water flow to the debris-induced impact forces. 5. Summary and conclusions In this study, we have discussed the influence of contact area and eccentricity of impact on the impact force. Moreover, the contribution of flow to the flow-driven-debris-induced impact is also studied. The results show the influence of contact area and eccentricity of reaction forces are lessened with an increase in the longitudinal length of the debris. Based on this observation, an improved equation using a modified (contact) height hc in the classic equation for impact force (Equation (7)) increases the capability of capturing the effect of contact area and eccentricity of impact. Tsunami-driven debris-induced impact forces are 25% to 35% larger than the corresponding in-air cases. Additional study is necessary to validate the proposed impact model (as illustrated in Figure 6) and understand more about the contribution of water flow to the debris-induced impact forces. References [1] T. Hiraishi, K. Haruo, E. Saitoh, Experimental study on impulsive force of drift body due to tsunami flow, Journal of Earthquake and Tsunami 04 (2010) 127–133. [2] H. Ko, D. Cox, H. Riggs, C. Naito, Hydraulic Experiments on Impact Forces from Tsunami-Driven Debris, Journal of Waterway, Port, Coastal, and Ocean Engineering (2014) 4014043. [3] P. Piran Aghl, C. Naito, H. Riggs, Full-Scale Experimental Study of Impact Demands Resulting from High Mass, Low Velocity Debris, Journal of Structural Engineering 140 (2014) 4014006. [4] Minimum Design Loads for Buildings and Other Structures, asce/sei 7-10 ed., American Society of Civil Engineers, Reston, VA, 2013. doi:10.1061/9780784412916. [5] C. M. Mast, P. Mackenzie-Helnwein, P. Arduino, G. R. Miller, W. Shin, Mitigating kinematic locking in the material point method, Journal of Computational Physics 231 (2012) 5351–5373. [6] W.-C. Yang, Study of Tsunami-Induced Fluid and Debris Load on Bridges using the Material Point Method, Ph.D. thesis, University of Washington, 2016.
181
ScienceDirect Procedia Engineering 175 (2017) 175 – 181
1st International Conference on the Material Point Method, MPM 2017
Modeling tsunami induced debris impacts on bridge structures using the material point method Wen-Chia Yanga , Krishnendu Shekhara , Pedro Arduinob,∗, Peter Mackenzie-Helnweinb , Greg Millerb a Graduate
Student, Department of Civil and Environmental Engineering, University of Washington, Seattle, WA 98195-2700, United States of Civil and Environmental Engineering, University of Washington, Seattle, WA 98195-2700, United States
b Department
Abstract Bridges represent a key part of infrastructure, playing a critical role in emergency response and post-event reconstruction. In this context, it is important for bridges to be able to survive both ground shaking and the effects of tsunamis. This paper focuses on numerically modeling bridge loading due to tsunamis. Although several studies have addressed the effect tsunami loads on bridges, few have examined the influence of debris carried by the tsunami. These problems involve complex contact interactions between solids and fluids that are not easily accommodated using typical fluid-oriented or solids-oriented numerical frameworks. In this paper the material point method (MPM) is used to address fluid and solid (moving and stationary) interactions with emphasis in evaluating demands on bridge superstructures by tsunami-driven debris. Two parametric studies have been conducted in this study. The first is aimed at understanding the influence of the contact area and the eccentricity of impact on the impact forces. The results show that the effect of these two factors are reduced with an increase in longitudinal length of the solid object. The second part of the study tries to evaluate the difference between tsunami driven debris impact loads and in-air debris impact loads. The results indicate that tsunami driven debris impacts tend to be larger by upto 35% compared to only in-air impact forces calculated using empirical equations. © by Elsevier Ltd. This is an openB.V. access article under the CC BY-NC-ND license ©2017 2016Published The Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 1 st International Conference on the Material Point Method. Peer-review under responsibility of the organizing committee of the 1st International Conference on the Material Point Method Keywords: Material point method; debris; fluid; impact; tsunami; bridge; fluid-solid interaction.
1. Introduction Throughout history, strong earthquakes have struck countries all over the world and caused major damage. Most of this damage has been due to ground shaking, but in coastal areas tsunamis induced by earthquakes have resulted in greater loss of lives and infrastructure. As coastal populations continue to increase around the world, understanding and managing tsunami effects on infrastructure becomes increasingly important. The objective of this study is to give preliminary results for researchers and engineers trying to understand the demands on bridge superstructures by tsunami-driven debris. It is imperative that bridges are able to survive both ∗
Corresponding author. E-mail address: [email protected]
1877-7058 © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 1st International Conference on the Material Point Method
doi:10.1016/j.proeng.2017.01.050
176
Wen-Chia Yang et al. / Procedia Engineering 175 (2017) 175 – 181
(a)
(b)
Fig. 1: 3-D simulation of two pieces to debris impacting a column causing partial damming.
ground shaking and the effects of tsunamis caused by earthquakes. Many prior numerical and experimental studies considering tsunami induced loads on bridges have been done, especially after the Great East Japan Earthquake. However, few of these earlier studies have examined the influence of debris carried by the tsunami. The studies that have been completed have demonstrated that debris can cause strong impact forces on columns and walls [1–3]. Lessons from these studies should be straightforward to apply to bridge vertical structures (i.e. bridge piers), but need some adjustment when applied to bridge superstructures, since in this case flows carrying debris have different characteristics compared with flows considered in the previous studies. In addition, debris can also affect the fluid motion around the bridge and introduce additional forces with longer duration compared to the impact. This class of problems is not well studied yet in the literature and involve complex contact interactions between solids and fluids. These effects are not easily accommodated with typical fluid-oriented or solid-oriented numerical frameworks. In this research the material point method (MPM) is used to model these complex fluid/solid (moving and stationary) interactions. The ultimate goal of the work is to model tsunami induced debris impacts in three dimensions like shown in Figure 1 and to understand damming and more complex flows around the superstructure during a tsunami. The results presented in this study are obtained using a single-threaded MPM code. This limits the mesh and particle refinements used in the validation examples and the debris-induced load study is relatively coarse compared to general applications found in the literature. A program allowing for parallel computing and /or one that employs implicit algorithms is necessary for further study in the future. 2. Background With large peak values, impact forces due to debris (solid objects) can cause severe local damage on bridge components, even though they happen in a very short time. In the literature, impact forces are evaluated using simple equations based on fundamental physics. In ASCE/SEI 7-10 [4], for instance, maximum impact forces F I p are calculated using impulse-momentum based equations of the form FI p =
πmd vd 2ΔtI
(1)
where md and vd represent the mass and impact velocity of the debris, and ΔtI is the time interval over which the debris stopped from its original velocity vd . Recommended and measured values of ΔtI range from 10−3 seconds to 1.0 seconds. This diversity of ΔtI results in very different impact forces and hence demands on structural systems. To improve Equation (1) and avoid the need to select a ΔtI , the flexibility of the solid object representing the debris is considered. By simplifying the collision of a debris into a bridge deck by a 1D spring model with equivalent stiffness keq , Equation (1) can be rewritten as, F Ik = cI vd keq md (2)
177
Wen-Chia Yang et al. / Procedia Engineering 175 (2017) 175 – 181 Table 1: Variables considered in the study of impact forces due to an 2D solid block.
Variable height thickness length initial velocity mass density Young’s modulus Poisson’s ratio contact height [eccentricity]
Symbol hd td L vI ρd Ed νd hc [ec ]
Unit (m) (m) (m) (m/s) (kg/m3 ) (GPa) (m) [(m)]
Values 3.0 1.0 3, 6, 9, 12, 15 3, 6, 9, 12, 15 200, 400, 600, 800, 1000 10, 5, 3, 1, 0.5, 0.3, 0.1 0.1, 0.2, 0.3 3.0 [0.0] 1.5 [0.75, 0.00] 1.0 [1.00, 0.75, 0.00] 0.5 [1.25, 1.00, 0.75, 0.00]
Fig. 2: Illustration of model setup for studying the influences of contact areas on impact forces.
where cI is a constant. Piran Aghl et al.[3] suggested cI be taken as 1.0 for impact forces caused by a 20-ft shipping container colliding with a wall in air, and evaluated the equivalent stiffness keq with keq =
Ed Ac L
(3)
where Ed and L are the modulus of elasticity and length of debris (shipping container), respectively; and Ac is contact area. Consequently, with md = ρd Ad L Equation (2) becomes ⎛ ⎞ ⎜⎜ Ac ⎟⎟ ⎟⎟⎠ F Ic = vd Ed Ac ρd Ad = vI Ad Ed ρd ⎜⎜⎝ Ad
(4)
Ko et al.[2] verified Equation (2) in their study and used experimental data to determine the equivalent stiffness keq instead of Equation (3) for their 1/5-scale shipping container. Furthermore, they found that impact forces for in-water cases are no greater than 17% of the corresponding impact forces for in-air cases, and can be approximately predicted with Equation (2) using the same keq determined from in-air tests. 3. Contact area and eccentricity In this study, the influence of contact area and eccentricity on impact forces are evaluated with a parameter study using two-dimensional (plane strain) solid blocks. To simplify the problem, only in-air conditions are considered here (i.e. no water). Figure 2 illustrates the model setup. Model descriptions and symbols shown in this figure are listed in Table 1 along with the corresponding values considered in this study. This results in 26250 combinations (models). Debris is treated as solid objects and is initially adjacent to a rigid boundary simulating the moment right before contact. Analysis time tana of each model is taken as tana =
4L cd
(5)
178
Wen-Chia Yang et al. / Procedia Engineering 175 (2017) 175 – 181
(a)
(b)
(c)
Fig. 3: Comparison of simulated impact faces of models with (a) all lengths, (b) L = 3 m and (c) L = 12 m against theory values, F Ith . (hc = 3 m).
(a)
(b)
(c)
Fig. 4: Comparison of simulated impact faces of models with L = (a) 3 m, (b) 6 m and (c) 12 m against theory values, F Ith . (ec = 0 m).
(a)
(b)
(c)
Fig. 5: Comparison of simulated impact faces of models with L = (a) 3 m, (b) 6 m and (c) 12 m against theory values, F Ith . (hc = 0.5 m).
in which cd = El /ρd is the stress wave propagation speed and El is a longitudinal elastic modulus. For a plane strain problem, El can be calculated with Ed (6) El = 1 − νd2 Regular square grids with cell size 0.5 m by 0.5 m, and nine (3 by 3) particles per cell are used. Figure 3 compares the simulation results of all models with full contact surface (i.e. hc = 3 m) against theory values: F Ith = td vI hd ρd El (7) The simulation results have good agreement with the theory when the solid blocks have longitudinal lengths L greater or equal to 6 m, but have about 15% error when L = 3 m as shown in Figure 3(b). This could be caused by the coarse mesh size (which is controlled by the efficiency of self-developed single-threaded MPM program) and hence is identified as the accuracy limitation of this study. To study the influence of the contact area Ac = td hc and eccentricity ec on the impact forces, first the case of ec = 0 m is investigated. Figure 4(a) shows that a decrease in contact area
Wen-Chia Yang et al. / Procedia Engineering 175 (2017) 175 – 181
(a)
(b)
Fig. 6: Illustration of modified contact height hc in the case when debris length is (a) larger or (b) smaller than the transition zone.
(a)
(b) Fig. 7: Comparisons of simulated impact forces against (a) F Ith and (b) F Imod .
brings down the impact on the boundary, which is consistent with Equation (4) suggested by Piran Aghl et al.[3]. Comparing Figures 4(a) to (c) the influence of Ac is lessen with increases of the block lengths L. Next, with a fixed contact area Ac = 0.5 m2 (hc = 0.5 m), which most variants of ec , the eccentricity affect the impact forces slightly in this study and has less influence with longer longitudinal length L (Figure 5). According to the observation in Figures 4 and 5, a modified height (area) hc with tan θ = 5 (as illustrated in Figure 6) is suggested to be used in Equation (7), i.e. (8) F Imod = td vI hc ρd El = vI Ac ρd El Figure 7 shows this modified equation can capture the impact forces significantly better than Equation (7). Furthermore, because the influence of length L and eccentricity ec on impact force is considered, Equation (8) shall be a suitable equation for predicting impact forces introduced by solid objects. However, more study will be necessary to examine the capability of Equation (8) to predict impact forces introduced by objects containing spaces or gaps, e.g. shipping container, in which the stress waves have different propagation path.
Fig. 8: Illustration of model setup for studying the contribution of water flow to debris-induced impact forces.
179
180
Wen-Chia Yang et al. / Procedia Engineering 175 (2017) 175 – 181 Table 2: Variables considered in the study of impact forces due to tsunami-driven debris.
Variable height thickness length initial velocity mass density Young’s modulus Poisson’s ratio water depth
Symbol hd td L vI ρd Ed νd Dw
(a)
Unit (m) (m) (m) (m/s) (kg/m3 ) (GPa)
Values 3.0 1.0 3, 6, 9 3, 6, 9, 12, 15 200, 400, 600, 800 10, 5, 3, 1, 0.5, 0.3, 0.1 0.1, 0.2, 0.3 4, 6, 8
(m)
(b)
(c)
Fig. 9: Comparisons of tsunami-driven debris-induced impact forces against F Imod for in-water cases calculated from Equation (8).
Fig. 10: Influences of debris mass density on the impact forces for in-water cases (L = 9 m).
4. Contribution of water flow In the last part of the study, the influence of water flow on debris-induced impact force is discussed with a parameter study using two-dimensional (plane strain) solid blocks. Figure 8 illustrates the numerical model used in this study and Table 2 lists all variable values under consideration. This is a full scale model and has a 1 m-by-5 m bridge. The acceleration due to gravity is g = 9.81 m2/s; The water has Bulk modulus Kw = 2.2 GPa, mass density ρw = 1000 kg/m3 , and viscosity μ = 0.001 Pa · s. A cell-based anti-locking algorithm proposed by [5] is used here for fluid modeling; and a constant flux smoothing algorithm with linear and oscillation limiters presented by [6] is used here to stabilize the simulation. Regular square grids with cell size 0.5 m by 0.5 m, and nine (3 by 3) particles per cell are used. Figure 9 indicates the impact forces due to debris (solid objects) in the water flow are about 25 to 35 percent larger than the corresponding impact forces for in-air cases, evaluated with Equation (8). The variance shown in Figure 9(c) is related to the mass density ρd of debris. Figure 10 indicates the influence of debris mass density on the impact force when the length of debris is kept constant. Since, ρd in this case influences the contact area, eccentricities of impact
Wen-Chia Yang et al. / Procedia Engineering 175 (2017) 175 – 181
forces, and wet areas of the debris, more study will be necessary to thoroughly understand the contribution of water flow to the debris-induced impact forces. 5. Summary and conclusions In this study, we have discussed the influence of contact area and eccentricity of impact on the impact force. Moreover, the contribution of flow to the flow-driven-debris-induced impact is also studied. The results show the influence of contact area and eccentricity of reaction forces are lessened with an increase in the longitudinal length of the debris. Based on this observation, an improved equation using a modified (contact) height hc in the classic equation for impact force (Equation (7)) increases the capability of capturing the effect of contact area and eccentricity of impact. Tsunami-driven debris-induced impact forces are 25% to 35% larger than the corresponding in-air cases. Additional study is necessary to validate the proposed impact model (as illustrated in Figure 6) and understand more about the contribution of water flow to the debris-induced impact forces. References [1] T. Hiraishi, K. Haruo, E. Saitoh, Experimental study on impulsive force of drift body due to tsunami flow, Journal of Earthquake and Tsunami 04 (2010) 127–133. [2] H. Ko, D. Cox, H. Riggs, C. Naito, Hydraulic Experiments on Impact Forces from Tsunami-Driven Debris, Journal of Waterway, Port, Coastal, and Ocean Engineering (2014) 4014043. [3] P. Piran Aghl, C. Naito, H. Riggs, Full-Scale Experimental Study of Impact Demands Resulting from High Mass, Low Velocity Debris, Journal of Structural Engineering 140 (2014) 4014006. [4] Minimum Design Loads for Buildings and Other Structures, asce/sei 7-10 ed., American Society of Civil Engineers, Reston, VA, 2013. doi:10.1061/9780784412916. [5] C. M. Mast, P. Mackenzie-Helnwein, P. Arduino, G. R. Miller, W. Shin, Mitigating kinematic locking in the material point method, Journal of Computational Physics 231 (2012) 5351–5373. [6] W.-C. Yang, Study of Tsunami-Induced Fluid and Debris Load on Bridges using the Material Point Method, Ph.D. thesis, University of Washington, 2016.
181