This article focuses on threat of tank failure during an earthquake that threw public opinion against the idea of development in the hills of San Francisco. If the failure mechanism could be predicted, the real-estate developer was prepared to formulate a mitigation scheme to divert the water away from the development. Sloshing of the water was idealized as a system of radial springs attached to the inside surface of the tank at the height of the convective mass. To get a second set of data for comparison, the engineers also calculated the event including the effects of work hardening. The tank’s base steel was a quarter-inch thick. Modeling the effect of friction and relative movement between the tank base and the foundation slab could have added another dimension to the matrix and boosted its credibility.
In the hills south of San Francisco, over some of the Bay Area’s most desirable land for housing, sits a million-gallon steel tank, part of the local water supply. Beneath the hills lies the San Andreas Fault.
Although the real estate downhill is very valuable, the threat of tank failure during an earthquake threw public opinion against the idea of development. But just how grave the risk was, no one could say. So the developer called in a San Francisco engineering firm, URS/Dames & Moore. Led by Ahmed Nisar, an engineering team from the firm studied the problem.
From an analysis standpoint, it was difficult to solve because of a high degree of nonlinearity in tank response. Most simplified analysis techniques had already shown the tank was likely to fail. But how? That was the question.
“Seismic response of steel water tanks is very complex,” Nisar said, “and since this tank is not anchored to the foundation it can shift or lift off.”
If the failure mechanism could be predicted, the real estate developer was prepared to formulate a mitigation scheme to divert the water away from the development.
Paul Jacob, a senior engineer, and Paul B. Summers, engineering manager of the analysis team, used ANSYS/ LS-DYNA from ANSYS Inc. in Canonsburg, Pa., for their calculations and analysis. The underlying product, LS-DYNA from Livermore Software Technology Corp. of Livermore, Calif., was conceived by its author, John Hallquist, at the Lawrence Livermore National Laboratory to simulate high-energy phenomena.
The problem presented to URS/Dames & Moore was what could happen when 1.5 million gallons of water, weighing 7,100 tons, contained in 20 tons of 30-year- old steel, is violently shaken in an earthquake.
The final results predicted that the tank could withstand an earthquake of probable scale. The simulated earthquake took into account fault slip rates and historic earthquake data.
Checking the Tank
To make sure of actual conditions, Summers made a trip to San Francisco from the URS/Dames & Moore Houston office, where he and Jacob work, to meet Nisar and see the tank firsthand.
The tank is 75 feet in diameter and 53 feet high. “It sits unanchored on a 4-foot-thick ring of concrete,” Summers said, “but we couldn’t start until we knew if the slab was cracked in any way and under what conditions it might fail. The slab was sound and there was no erosion around it.” Nor were there any leaks in the shell, which could indicate a weakness in the steel, he added.
A material testing specialist approved the quality and workmanship of the welds.
According to Jacob, “If there were cracks in the slab or defects in the tank, we would have had a far bigger problem to solve. We would have had to model the slab, the ground beneath it, and the weldments. As it turned out, we only had to model the main steelwork of the tank and the water inside.”
The biggest modeling challenge was the water. “Backed by studies from the American Water Works Association and other industry standards organizations, we decided to idealize the water rather than actually model it.” Jacob said.
According to Summers, “The movement of water in a shaken tank is two-phased: convective or sloshing in the upper part of the tank and impulsive below.” The AWWA standards for evaluating the ratio of the two water components, and the heights at which they act, were used to approximate the water.
Sloshing of the water was idealized as a system of radial springs attached to the tank’s inside surface at the height of the convective mass.
Modeling the tank followed. The steel was late 1960s- vintage A36, widely used in tanks then and now. “Because of its age, its modulus of elasticity and post-yield data could only be estimated,” Jacob said. Jacob and Summers used a worst case, that the metal was perfectly plastic.
“That meant all potential energy absorption due to work hardening would not be taken into account,” Jacob explained.
Getting More Data
To get a second set of data for comparison, the engineers also calculated the event including the effects of work hardening.
The tank’s base steel was a quarter-inch thick. Shell courses were three-quarters of an inch at the base and 11/16-inch near the roof.
The resulting model, representing half of the tank, had 4,218 nodes, 4,255 elements, and 25,308 degrees of freedom.
Next to be accounted for were the loads on the structure. Hydrostatic pressures were applied as a linearly varying pressure distribution on the inside of the tank shells and a constant distribution on the tank base. Earthquake loads were then applied to nodes on the tank base as acceleration time histories in the horizontal and vertical directions.
“Not a huge model, but it added up to a respectable 500-megabyte solution database,” said Jacob. Running each analysis took about 15 hours on the workstation available, a six-year-old Power Indigo from Silicon Graphics Inc. of Mountain View, Calif. The computer contained SGI’s R8000 CPU with a clock speed of 75 MHz, not a powerful machine by today’s standards, but sufficient to do the job.
The solution evaluated the tank’s response to a 15-second earthquake with a stable model time step of 0.03 millisecond, Jacob said.
The calculations predicted that, at the height of the earthquake, the tank would bounce 1/2 to 3 inches.
An examination of analysis results indicated about 9 percent plastic strain in the base plate at the end of the earthquake.
Clearly, bottom plate bending is the critical factor in the tank’s earthquake safety and the predominant phenomenon in the analysis. The repeated uplift phenomenon led URS/Dames & Moore to perform a simplified low-cycle fatigue assessment on the bottom plate using the Coffin-Manson Law.
The response of the base plate was characterized by high-strain, low-cycle fatigue, where there is permanent deformation, Jacob said. “But 9 percent plastic strain is comfortably within the bounds of expected material plastic strain limits.”
Summers added, “In our experience, the most common failure mode of the tank is the buckling of the bottom part of the tank shell in the shape of an elephant’s foot.” The LS-DYNA model clearly showed the bulge at the lower portion of the tank shell from the tank’s bouncing up and crashing down. Sufficient forces were generated to plastically deform the steel shell. The elephant-foot outward budding was observed at the bottom of the tank shell, just above its joint with the base plate. There was also a compensating upward bend in the bottom plate adjacent to the elephant’s foot. In addition, there were convex and concave deformations at the top of the tank walls due to the convective sloshing.
Because the earthquake motion was synthetic, the team also chose to run a historical record for comparison. These two motions with the two material characteristics gave four runs for comparison purpose. “This gave us a matrix of four cases and thus the runs would yield enough data for making a sound judgment,” Jacob pointed out. “From this data, we were able to conclude that the model gave us a very good approximation of the real world.”
The project took six weeks to complete.
Complete Analysis is Elusive
The team members say there is never time or money enough for a perfectly complete analysis.
For instance, if they had more time, they would have modeled the water in the tank as a frictionless fluid, then meshed it, rather than representing it by a spring-mass system. “That could have been marginally more accurate if the true behavior of water under seismic shaking could be captured, but on this project, with its tight budget and time scales, it was not possible,” Summers said.
Failure in the welds and steel could have been modeled. However, “We could not determine the criteria for such a failure, nor verify its accuracy given the limited data we had,” Jacob said. “Therefore, the results from failure modeling would probably have been of only academic interest.” To do this properly would have required considerably more detailed information than they had on the material.
Modeling the effect of friction and relative movement between the tank base and the foundation slab could have “added another dimension to the matrix and boosted its credibility,” Jacob said. “But we didn’t do it, once again because of the project constraints. Furthermore, it would have been difficult to determine a coefficient of friction for steel sitting unanchored on concrete for 30 years.”