Dynamic Loading on Submerged Components

By John McLean, P.E., S.E., P.Eng. & Michael DePiero

Introduction

Assessing structures in air subjected to dynamic loading from earthquakes or other vibratory loading is typical for structural engineers. Being so light, air has little to no effect on a structure during expected oscillatory loading. When a structure is submerged in a heavier fluid, such as water or oil, the effects of the surrounding fluid cannot be ignored. Submerged dynamic loading is relevant to a host of applications, ranging from cooling pools to fuel tanks to aquariums. While this paper refers to the forces on tanks, this can apply to any fluid-containing structure. This paper focuses on the design of submerged structural components within tanks, not the design of the tanks or exterior tank supports, as those provisions are far more codified.

Before calculating the hydrodynamic loads, it is essential to understand the basic components of the loading. Other loads must be combined with the hydrodynamic loads to estimate the resultant forces on submerged components. These include typical loads such as dead, live, and inertial, but they must also include other submerged loads such as the fluid’s hydrostatic pressure and buoyant forces. Only a short description of these other loads will be discussed here. 

Oscillatory Submerged Loading Components

The standard components of submerged loading are dead, hydrostatic, inertial, and hydrodynamic (impulsive and convective), as illustrated in Figure 2 below from ACI 350.3.

For all intents and purposes, the dead loads are the same as when a structure is unsubmerged. The key difference is that when the object is submerged, it will be subjected to buoyant forces, which can be lumped in the dead load category. Not including buoyancy is often conservative, as the buoyancy loads are usually small and only decrease the gravity load. However, should the controlling element of loading be in uplift, it is unconservative for buoyancy not to be included. The inclusion of buoyancy in the dead load is at the discretion of the design engineer. 

The hydrostatic loads are due to the pressure of the fluid above. Most simply, it is calculated as the fluid’s density multiplied by the object’s depth in the fluid. If the object is entirely surrounded by fluid, it is unnecessary to include the hydrostatic pressures as the applied forces result in equilibrium. Hydrostatic forces should be considered if there is a concern that the object could internally collapse due to the surrounding pressures or if the fluid is not on all sides of the object (such as a window on a tank wall). 

Inertial forces are very similar to that of an unsubmerged structure. Using an equivalent lateral force method is usually sufficient for basic structures. Often, this is simply multiplying the masses by the appropriate accelerations, such as:

F₁ = mU

Where:

F₁ = Inertial force

m = Mass

U = Acceleration of the Object

The hydrodynamic forces are broken into two components: impulsive and convective. The impulsive forces are due to the fluid that moves with the object. This fluid becomes what is known as the hydrodynamic mass, which can be added to the total mass when calculating the inertial forces on the object. The convective component, also called sloshing, is due to fluid moving past the object and applying drag loads. Both components of hydrodynamic loading occur simultaneously, and if enough information is obtained, they can be summed together by using the Morison Equation, which can be simplified in one form as:

F_hydrodynamic= F_M+F_D

Where:

F_D = Convective Forces

F_M = Impulsive Forces

However, to properly use the Morision Equation the velocity and acceleration of the fluid and the object must be known at a specific point of time. Convective and impulsive forces are easiest to calculate at their maximum and occur at different frequencies. For this reason, the effects of the out-of-phase forces may be combined using a method such as the square-root-sum-of-the-squares (SRSS). 

Computational Modeling of Hydrodynamic Effects

Calculating the submerged loading with hydrodynamic effects is often complex, as the period of motion, the tank geometry, and the component geometry can significantly impact the applied loads. Creating a computational model is the most accurate way to calculate and combine submerged hydrodynamic forces, especially for the more complex geometries.

Creating a computation model that considers submerged hydrodynamic forces can be done by considering the various mode shapes of the waves and developing fluid velocity and acceleration fields. The impulsive, and convective forces can then be calculated with the Morison Equation. One method for doing this is described in “Seismically Induced Loads on Internal Components Submerged in Waste Storage Tanks” by M. A. Rezvani, J.L. Julyk, and E.O. Weiner. While creating a computational model is the most accurate way to solve these forces, it is also very time-consuming and complicated. Depending on the shape of the tank, it is often more efficient and conservative to use the other equations presented here. 

Hydrodynamic Loads on the Whole Tank

While not the focus of this paper, it is important to understand how the hydrodynamic forces affect the tank as a whole and how the hydrodynamic forces may affect the design of the tank shell and the tank’s anchorage. Thankfully, calculating these forces are well-codified. 

Within the tank, the mass of the fluid can be divided into a convective and impulsive mass. Once the masses are found, the resultant convective and impulsive forces and their centroids can be solved for. Numerous references help the engineer calculate these values. ASCE 7-16 has guidance in Section 15.7. Because the nuclear industry must always consider seismic loading and nuclear plants typically have cooling water tanks and spent fuel pools, guidance is provided on forces to consider in standards written for the nuclear industry such as Chapter 9 of ASCE 4 and Chapter 6 and Appendix F of TID-7024. Because many concrete structures are used to hold fluids, ACI has also published a standard on dynamic effects, ACI 350.3. This standard gives an excellent summary of the design method for tanks in Appendix A. 

Tanks that are not open at the top must have sufficient freeboard to prevent loading the top of the tank and changing the dynamics of the sloshing behavior. The maximum vertical displacement of the fluid within the tank can be estimated using Chapter 7 of ACI 350.3 or Appendix F of TID-7024. Although the equations below look different, they are very similar once you consider the length is based on the effective radius for the TID-7024 equation, which is L/2.

Assessment of the tanks with insufficient freeboard is further described in the commentary to Chapter 7 of ACI 350.3 and in a paper titled “Earthquake Induced Sloshing in Tanks with Insufficient Freeboard,” published in the Structural Engineering Journal from March of 2006. 

Calculating Loads on Submerged Components

Analyzing submerged structures or components within the tanks themselves is much less codified. ASCE 7-16 Section 15.7.6.1.4 covers internal components but only says that “major elements shall be designed for the lateral loads caused by the sloshing liquid in addition to the inertial forces by a substantiated analysis method. Thus, it is up to the engineer to decide what loads should be applied to internal submerged tank components. While the loading is directly specified, a methodology similar to that used for the whole tank can be used. Since the focus is on a localized region of the tank, the impulsive and convective forces may be calculated more specifically.   

As the fluid sloshes in the tank or other structure containing fluid, components towards the top of the tank see higher horizontal and vertical drag (convective) forces than those near the bottom of the tank. In addition to the higher hydrostatic forces, components near the bottom of the tank see higher impulsive forces than those at the top. The forces are also affected by the shape of the tank (rectangular vs. circular) and the ratio of the height of the fluid in the tank to the dimensions of the tank. For example, a tall and narrow tank may experience practically no sloshing forces toward the bottom, but a shallow, wide tank would undergo notable sloshing forces toward the bottom. 

Similarly, the vertical forces from sloshing are assumed to be zero in the center of the fluid-containing structure and are highest towards the outside of the structure, as illustrated by Figure 3 below. The motion of the fluid causes an overturning force on the base of the tank and dynamic fluid pressures on the base of the tank. These can be computed using the equations in ACI 350.3, which are shown in Figure 3 below.

The convective pressures acting on the tank’s walls can be calculated using several codified equations. ASCE 4-16 provides equation C9-16, which is only valid for cylindrical tanks. ACI 350.3 provides equations in Commentary Section R5.3 for the assumed linear distribution and the exact distribution for both circular and rectangular tanks. 

It is important to note that the equations for the rectangular tanks produce results with the units of force per unit length. The exact pressure can be found by dividing the result by the width of the tank. Since internal components are often mounted on or near a tank wall, these pressures can be easily used to calculate the convective forces (pressure times area) conservatively. For components mounted away from the tank walls, using the convective wall pressure to calculate the loading is still conservative as the convective forces are maximized at the fluid-wall interface. Alternatively, the drag forces from the convective motion can be calculated if there is a reliable estimate of the maximum fluid velocity at that location. The drag forces are computed by the equation:

Calculating the impulsive force on submerged components is done by finding the added mass. The treatment of the added mass from the surrounding fluid is best described in a paper by R. G. Dong titled “Effective Mass and Damping of Submerged Structures,” written in 1978 for the nuclear industry. In addition to added mass for single components, this gives equations for components grouped, such as fuel rods in a spent fuel pool. For many structures, this equates to the volume of liquid that fits within the component’s perimeter. Any fluid that may be trapped inside of the component should also be included. The added mass forces can be computed by the equation:

ACI 350.3 provides an impulsive tank wall pressure distribution similar to the convective tank wall pressure distribution. However, this distribution is for the mass of fluid that moves with the whole tank, not just a singular component. Using the component’s hydrodynamic mass to calculate the impulsive forces on each component is recommended, rather than using the pressure distribution for the whole tank on an individual component. Should the engineer also design the tank wall, then the impulsive pressure distribution would be a good equation.  

The combined expression for the in-line oscillatory forces is:

As noted above, the effects of horizontal impulsive loading, horizontal convective loading, and vertical hydrodynamic loading must be combined. ASCE 4-16 states that the seismically induced hydrodynamic pressures on the tank shell at any level shall be determined by the SRSS combination of the horizontal impulsive, horizontal sloshing, and vertical hydrodynamic pressures. The hydrodynamic pressure at any level shall be added to the hydrostatic pressure at that level to determine the hoop tension in the tank shell. ACI 350.3 also recommends a combination of horizontal and vertical accelerations using the SRSS method. Similarly, for most nuclear facilities or states requiring consideration of horizontal forces in two directions, the effect in each horizontal direction can be combined with the effects in the vertical direction using the SRSS method.

Conclusion

Despite not being wholly codified, there are reliable and conservative ways to estimate the hydrodynamic loading on internal submerged components, which can be combined with the other components of submerged loading. The convective component of the hydrodynamic loading can be calculated by using the wall pressure equations provided in ASCE 4-16 and ACI 350.3. The impulsive component can be calculated by multiplying the associated hydrodynamic mass by the corresponding acceleration. The convective and impulsive forces can be combined via SRSS along with the other submerged loads. Following this methodology, calculating the dynamic loading on submerged components can be performed without complex computational modeling.

Michael DePiero is a Structural Designer with EXP in Chicago, IL. He has experience in structural design and analysis for a wide range of project types, including new design, repair and rehabilitation, and nuclear. He can be reached at michael.depiero@exp.com.

John McLean, P.E., S.E., P.Eng., is a Technical Director with Simpson Gumpertz & Heger in Chicago, IL. He is an expert in structural design, analysis, and engineering project management for commercial, industrial, and nuclear facilities, specializing in blast analysis, security design, and nuclear containment. John also holds leadership roles in industry committees and professional groups. He can be reached at jbmclean@sgh.com.