21st Century Dam Design - Advances and Adaptation

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21st Century Dam Design —

Advances and Adaptations

31st Annual USSD Conference

San Diego, California, April 11-15, 2011

On the CoverArtist's rendition of San Vicente Dam after completion of the dam raise project to increase local storage and provide

a more flexible conveyance system for use during emergencies such as earthquakes that could curtail the region’s

imported water supplies. The existing 220-foot-high dam, owned by the City of San Diego, will be raised by 117

feet to increase reservoir storage capacity by 152,000 acre-feet. The project will be the tallest dam raise in the

United States and tallest roller compacted concrete dam raise in the world.

The information contained in this publication regarding commercial projects or firms may not be used for

advertising or promotional purposes and may not be construed as an endorsement of any product or

from by the United States Society on Dams. USSD accepts no responsibility for the statements made

or the opinions expressed in this publication.

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To be the nation's leading organization of professionals dedicated to advancing the role of dams

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• Advancing the knowledge of dam engineering, construction, planning, operation,

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resources;

• Enhancing practices to meet current and future challenges on dams; and

• Representing the United States as an active member of the International Commission on

Large Dams (ICOLD).

Folsom Dam Auxiliary Spillway Piers 339

DESIGNING THE FOLSOM AUXILIARY SPILLWAY PIERS USING LS-DYNA® TIME-HISTORY RESULTS

Eric Kennedy, P.E.1

ABSTRACT

The Folsom Auxiliary Spillway is a joint project between the U.S. Army Corps of Engineers and the Bureau of Reclamation for flood damage reduction and dam safety purposes. The spillway will increase the discharge capacity from the Folsom Reservoir, enabling it to pass the probable maximum flood (PMF). The main feature is the concrete control structure, which consists of three non-flow-through monoliths and two flow-through monoliths, each of which contains three bulkhead and three tainter gates. An LS-DYNA model of the control structure was developed for evaluation, design, and verification of several components. Two water surfaces were modeled, one representing the PMF and the other the maximum normal pool; the latter was combined with two operational scenarios and 28 combinations of ground motions to envelope the expected demand on the structure.

This paper focuses on utilizing results from the LS-DYNA model to design the spillway piers. Because of site constraints and hydraulic requirements, the piers have a higher aspect ratio than would be expected, the obvious result of which is an increase in steel reinforcement. The model output included nodal forces at several locations in each pier; these were translated into axial, moment, and shear demands on tributary strips of the piers. The axial force on each strip was used to calculate both moment and shear capacities. At each output time step, instantaneous demand-to-capacity ratios were evaluated using Corps criteria for the allowable magnitude and number of exceedences. The resulting design concentrates the heaviest reinforcing only where necessary, resulting in a substantial cost savings.

INTRODUCTION

The Folsom Auxiliary Spillway control structure has two flow-through monoliths that house three bulkhead and three tainter gates each. The invert slab is cast directly against the foundation and serves as the bottom of the conduit and as the base for the piers. A mass concrete headwall, split in two by the bulkhead slots, sits atop the piers and helps carry cross-canyon loads from abutment-to-abutment. There are four piers per monolith: one is 5’-3”, the inner two are 5’-10 ½” each, and the pier at the joint between flow-through monoliths is 3’-9” wide. Figure 1 shows a cross-section with the reinforced pier areas hatched.

Since the focus of this paper is the analysis and design of the piers, the section discussing the development and features of the LS-DYNA model itself is brief, especially considering the enormity of the model. The basic model information is presented, followed by an overview of some of the verification and validation that was done. The 1 Civil Engineer, U.S. Army Corps of Engineers, Structural Design Section, 1325 J Street, Sacramento, CA 95814, [email protected]

340 21st Century Dam Design — Advances and Adaptations

discussion of the pier design follows, proceeding through demands, capacities, acceptance criteria, and output and results.

Figure 1. Flow-through Monolith Section (Pier Area Hatched)

THE LS-DYNA MODEL

The LS-DYNA finite element analysis (FEA) model, shown in Figure 2, includes solid, shell, and beam elements that represent the foundation, backfill, reservoir, and concrete control structure as well as steel components, such as gates and anchors. TrueGrid, an advanced meshing program, was used to generate the four different meshes that model the combinations of gate configurations (either bulkhead or tainter gates impounding the reservoir) at two water surface elevations.

Figure 2. Full View of the LS-DYNA Model

Model Features

The number of elements varies depending on the mesh, but each contains approximately 760,000 nodes and 685,000 elements and is 2,000 feet upstream-downstream x 2,000 feet cross-canyon x 750 feet vertical. The reservoir extends 443’ upstream of the


structure and is shaped to approximate the approach channel; a close-up showing the structure, reservoir, and backfill is shown in Figure 3. The solid mesh resolution is significantly greater in the control structure than in the foundation and reservoir; of the elements, 210,000 are in the two flow-through monoliths and forty percent of those make up the piers. All solid elements have a single integration point; the amount of time to run the dynamic model with fully integrated or selectively-reduced elements is impractical for design purposes, although both would provide greater accuracy. The bulkhead gates and tainter gate skin plate are represented by rigid shell elements with thicknesses appropriate to model the mass. Beam elements are used in several locations: the strut arms of the tainter gates, the trunnion anchorage, and pier struts, which run from pier-to-pier and keep them moving in phase during an earthquake. Prestressing forces are applied to the trunnion anchorage as an initial stress; this feature was important to include because it applies a large compressive force to the piers, affecting their behavior and capacity.

Figure 3. Close-up of the Control Structure, Reservoir, and Backfill

One benefit of using LS-DYNA is that it makes it relatively simple to define nonlinear contact surfaces between different parts of the model. The interfaces can be defined between parts, segments (element faces), or nodes; in this model, the majority is segment based. Additionally, most of the contacts are defined as “automatic surface-to-surface” (sliding) contacts; this definition allows compressive force transfer orthogonal to the contact surface with the option to transfer in-plane forces by defining static and dynamic friction coefficients. All sliding contacts that include the reservoir are frictionless; the dam-to-foundation and dam-to-backfill contacts use friction values obtained from geotechnical explorations and testing of the materials. Tied contacts constrain pairs of nodes on either side of the interface and produce the same results as if the elements were merged (defined using the same nodes); these were used to obtain nodal force output for design at the two lower locations outlined with red in Figure 4. Tied offset contacts, which approximate constraints for contacts involving one or more rigid bodies, are used to connect the bulkhead gates to the slots within the piers when the bulkheads impound the reservoir; they were also used at the two upper locations in Figure 4. Node-to-surface contacts are used at the ends of the trunnion anchorage and pier struts and to connect the bulkhead to the top of the headwall for the cases where the gates are raised and locked in place.


Figure 4. Locations of Output for Pier Design

Verification and Validation Procedures

Because the Sacramento District had never before used LS-DYNA, several procedures helped to verify the model was functioning correctly. First, the team compared static results from SAP2000 and LS-DYNA models of a single flow-through monolith, which was pinned at the base of the invert slab. The comparisons included distribution of self-weight through the piers and crest deflections. The results were essentially the same, so the verification progressed to a time-history analysis; crest deflection, velocity, and acceleration histories were also found to agree. In addition, Eigenvector modal analyses showed similar mode shapes and mass participation ratios.

At this point, the LS-DYNA model progressed beyond the capabilities of the SAP2000 model, so a different procedure was taken. All of the loads applied to the control structure (e.g., hydrostatic forces, the trunnion anchorage force, and backfill loads) were verified against hand calculations. With the basic function of the program already tested against SAP2000, this was deemed sufficient to gauge success and proceed to dynamic runs.

The LS-DYNA model is executed using the explicit integration solver; because of this, a massless foundation is not (and, arguably, should never be) an option. This presents a problem in that a foundation with mass affects ground motions as they propagate to the surface from the plane of application. The solution to this was to perform a frequency-based scaling of each direction of each motion.

To achieve this, a separate block model was meshed in TrueGrid, with the foundation of the full model as the starting point. The motion-scaling model has the same horizontal extents, bottom elevation, and element configuration as the full model, but the canyon was filled in and the block was cut at the elevation of the free-field motion. The steps listed below were followed to apply and scale the motions. Refer to Figure 5 for the plots referenced; note that units are not shown in the plots, but all plots of the same type use the same scale (so a visual comparison is possible).


1. Convert the free-field acceleration history (a) to a velocity history (b) using trapezoidal rule integration

2. Apply the free-field velocity history at depth within the foundation and record the acceleration history at a center node on the surface (c).

3. Divide the FFT of the free-field accelerations (d) by the FFT of the recorded accelerations (e) to obtain the correction factors (f).

4. Apply the correction factors by frequency to the free-field FFT to obtain the corrected FFT (g).

5. Use an inverse FFT to obtain the corrected acceleration history (h), then convert it to velocities (i).

The corrected velocities should be reapplied to the model as the input motion; LS-DYNA converts the velocities to traction forces internally. If the new recorded motion matches the free-field in the frequency domain, the process is complete; otherwise, it should be repeated until the results are acceptable. The results are easiest to compare with a response spectrum; the target was to limit the error to ±10% over the entire range, with a better match of ±5% around the natural period of the structure.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 5. Sequential Plots of the Ground Motion Scaling Process

PIER DESIGN

The results from LS-DYNA include dead loads, live loads (including soil and hydrostatic pressures), and earthquake loads for the seismic cases. Output is written to a database during the run at an interval of 0.01 seconds. This will be referred to as the time step in this section, but it should be noted that this output interval is different from the internal time step of the LS-DYNA calculations. This output is formatted into comma-separated


value (CSV) files using LS-PrePost, the companion pre-processor/post-processor from LSTC, the publisher of LS-DYNA.

Once the data is formatted, it is input into a Microsoft Excel worksheet that tabulates time-histories of the demands and capacities. Separate files are used to design the reinforcing for each section of the three pier geometries; this process is repeated for every load case, resulting in over 1,800 design files. The primary reference for the capacity calculations is the Corps Engineer Manual (EM) 1110-2-6053, “Earthquake Design and Evaluation of Concrete Hydraulic Structures.” Additional guidance was obtained from EM 1110-2-6051, “Time-History Dynamic Analysis of Concrete Hydraulic Structures,” and ACI 318-05.

Demand

Each pier is considered as a series of rectangular columns, defined by a tributary width for each set of nodes along the length ( henceforth referred to as a “strip”). Each strip has its own time-history of vertical axial load, biaxial shear (upstream/downstream and cross-canyon directions), and flexure about the vertical and upstream/downstream axes. By convention, the strips are numbered starting from one at the upstream end of the pier.

As mentioned previously, the output file contains the nodal forces for each of four contacts per pier. This information is sufficient to provide the axial and shear demands, which are simply the sums of the forces in each of the three principal directions (X, Y, Z), but the flexural demand must be calculated in Excel. A section of the pier with a uniform pressure will have an uneven nodal force distribution, as shown in the left half of Figure 6, but should result in zero moment. Note that the force distribution will not be symmetric, as it is shown, for an asymmetric nodal spacing.

Figure 6. Pier Force Distributions and Moments

The right half of the figure shows an uneven pressure distribution and its uneven force distribution, resulting in a moment equal to:


( 1 )

where x and F are the distance from pier centerline and nodal force, respectively, for each of the n nodes across the width of the pier (see Table 1). This results in a loading diagram similar to Figure 7, which shows a representative strip of a 5’-10 ½” pier; this particular strip is in compression with a negative Mx and positive Mz.

Table 1. Number of Nodes Across Width of Piers

Pier Width Typical Bulkhead

Slot 5'-3" 7 5

5'-10 1/2" 8 4 3'-9" 6 4

Figure 7. Pier Strip Loading Diagram

When calculating the demands, it is important to be aware of when the direction of forces is important. Shear and axial demands are simply sums of forces; for the shear demand, it is irrelevant whether the X force is directed upstream or downstream (or the Y force to the left or right abutment). However, the sign of the axial (Z) force indicates if the section is in compression or tension, which affects the moment and shear capacities. The sign of the moment demand is also irrelevant since the reinforcing is the same in each face. It is important to be consistent, though, and programming capacity equations in Excel is made easier by reporting the absolute values of demands for everything other than the axial load.


Capacity

Flexure: The moment capacity for flexure about the upstream-downstream axis is calculated by relating the axial demand, P, to a four-point interaction diagram generated for each reinforcing pattern. The four points and their associated equations are:

Point 1, pure compression, zero flexure:

( 2 )

Point 2, balanced failure: Eq. ( 3 ) and ( 4 ), plus the strain assignments ,

( 3 )

( 4 )

Point 3, pure flexure, zero axial load: Eq. ( 3 ) and ( 4 ), plus the strain assignments , , and the sum of forces requirement of

Point 4, pure tension, zero flexure:

( 5 )

Figure 8 shows interaction diagrams for four reinforcing patterns at the middle contact of Pier 1: three are with the full pier width and the fourth is for the reduced section at the bulkhead slot. There are actually five points shown for each diagram; this is because the pure compression capacity is limited to 80 percent of the calculated value.


Figure 8. Sample Interaction Diagram

The pure flexural capacity is calculated using Excel’s Goalseek function to iterate the value of c until the sum of forces on the section equals zero. For Goalseek to work properly, Excel automatically recalculates any cells affected by the changing value with each iteration. Because of this, the resulting capacity is not directly referenced by any cell in the sheet; the value is presented to another cell that is referenced by the interaction diagrams. This results in only two full-sheet recalculations instead of the tens or hundreds that would occur otherwise.

Shear and Shear Friction: The shear capacity of each column is determined using Eq. ( 6 ) (EM 6053, Eq. 5-2):

( 6 )

This is similar to ACI 2005 Eq. 11-4, the shear capacity of a concrete section under axial compression, with two key differences: k and (0.8 * Ag) replace 1.0 and bw*d, respectively. The value of k is based on the flexural displacement ductility demand on the section, μ, which is given by:

( 7 )

Assuming To = T, the value of μ equals the ratio of the moment demand to the nominal moment capacity, or the demand-capacity ratio (DCR) from the flexural design calculations. For μ < 1, meaning the flexural capacity has not been exceeded, k = 1.0, with a linear decrease to k = 0.5 corresponding to μ = 2. Thus, EM 6053 reduces the shear capacity when the section exceeds the elastic limit, which is permissible at limited magnitudes and for limited durations in seismic cases. The contribution of reinforcing


steel is calculated using Eq. ( 8 ) (EM 6053, Eq. 5-3), which gives the same result as ACI Eq. 11-31.

( 8 )

The shear friction capacity is of interest at the lift line interfaces and is calculated using a modified version of Eq. 5-5 from EM 6053:

( 9 )

A friction coefficient of 1.0 is used, which assumes the concrete is placed against hardened concrete with an intentionally roughened surface (ACI 11.7.4.3). Like its counterpart in the ACI code, Eq. ( 9 ) increases proportionally with a compressive load on the section, but it also includes 25 percent of the longitudinal (in this case, vertical flexural) steel area. In cases where the section is in tension, this prevents a zero shear friction capacity, but it prevents reliance on the full area of the flexural steel. Where this is insufficient, bars are provided at the centerline of the pier, with the full area contributing to the capacity (As-sf).

Separate DCRs are calculated for both shear and shear friction in the upstream/downstream and cross-canyon directions. A combined DCR accounts for interaction by taking the maximum of the following three cases: the square root of the sum of squares (SRSS) and the two combinations of the 100/30 method (DCRusds + 0.30DCRcc and 0.30DCRusds + DCRcc). This method presents one problem, however, because the LS-DYNA output does not separate nodal forces into their components (dead load, live loads, and earthquake loads). For the interaction DCR, the combination equations are only supposed to be applied to the earthquake component of the load; in the event that the 0.30DCRusds + DCRcc case controls, the large hydrostatic force is improperly reduced. However, EM 6053 suggests the use of the SRSS method as a more accurate approach; applying the combination equation to the entire force is conservative in this case.

The horizontal reinforcing bars run in the upstream/downstream direction and serve to resist shear in that direction as well as flexure about the vertical axis (Mz). Because they contribute to both capacities, the equation for the shear capacity is modified to reduce it relative to the flexural DCR at each time step:

( 10 )

Eq. ( 10 ) decreases the shear capacity linearly with an increasing flexural DCR; the shear capacity equals zero for a flexural DCR of 1, which is obviously unacceptable. Thus, DCR(Mz) can never reach 1 and is excluded from the cumulative inelastic duration calculations (discussed in the section on acceptance criteria).

Section 11.10 of the ACI code requires short walls with a height-to-length ratio less than 0.5 to have equal amounts of shear reinforcing in the horizontal and vertical directions.


This is because shear cracking in such walls will tend to occur at a shallower angle than in taller walls, rendering the horizontal steel much less effective than the vertical. Considering the minimum height from invert slab to the roof (~34 feet) of the conduit and the full upstream-downstream length (up to 144.75 feet), the piers have a ratio of approximately 0.23. This is very conservative, but it is deemed a prudent measure due to the nature of the structure; the effects of this requirement are discussed later. The amount of vertical shear steel required is calculated using Eq. ( 11 ):

( 11 )

A simpler explanation of this is that the area added to the vertical steel is equal to the horizontal area required minus the area of vertical steel not used for flexure. The first max() statement ensures the required area is positive and the second max() ensures that the utilized area of horizontal steel is positive. The min() statement is required because the DCR for the vertical steel is allowed to exceed one; this prevents the available area of vertical steel from being negative.

Acceptance Criteria

Allowable values for the demand-to-capacity ratios of reinforced concrete hydraulic structures are given in Table 2, which is reproduced from EM 6053-Table 6-1a. Both shear and shear friction are brittle failure modes and their allowable DCRs are limited to 1.0 for the Maximum Design Earthquake (MDE) load case. The flexural DCR is allowed to exceed 1.0 for the MDE, to a maximum of 2.0, and an additional evaluation is performed to determine the cumulative inelastic duration (CID) in accordance with EM 6051. The CID can be thought of as a way to unify considerations of the magnitude and quantity of stress excursions.

Table 2. Allowable Demand-to-Capacity Ratios Performance Objectives Action

(in terms of forces) Damage Control (MDE) Serviceability (OBE) Flexure 2.0 1.0 Shear 1.0 0.8 Shear Friction 1.0 0.8

In terms of the pier reinforcing, the CID is defined as the total length of time the DCR for any section exceeds a given value greater than 1.0. A specific length of time is allowed for a CID equal to 1.0 and the allowable time decreases linearly to zero at a DCR of 2.0. The maximum allowable CID for plain concrete gravity dams is 0.30 seconds, but this is overly conservative since the piers are heavily reinforced. Instead, the design uses a maximum of 0.70 seconds, which is still conservative since it is intended for lightly reinforced intake towers. Figure 9 shows the CID criteria for this design; allowable values fall into the shaded area.


Figure 9. Cumulative Inelastic Duration

The CID is calculated by counting the number of times the DCR of each strip exceeds a given value over the entire time history. In Excel, this is accomplished by setting up 101 “bins” for each strip, with each bin containing the number of DCR exceedences of 1.00, 1.01, 1.02…2.00. The value of each bin is compared to its maximum value; if any bin in a strip exceeds its allowable maximum, a value of “NG” is returned to a summary chart, indicating that additional reinforcement is required at that location.

Section 6-8 of EM 6053 contains minimum steel requirements for new reinforced concrete hydraulic structures to ensure ductile behavior. The suggested minimum reinforcement is that required to obtain a nominal moment capacity greater than or equal to 120 percent of the cracking moment. The section continues to say:

For concrete elements with deep cross-section dimension, i.e., spillway piers, the amount of reinforcing steel needed to satisfy minimum steel requirements may be excessive. When it is impractical to design for Mn/Mcr ≥ 1.2, sufficient quantities of reinforcing steel should still be provided to ensure nearly elastic performance under the MDE.

Because the nominal moment capacity is dependent on the amount of axial force the section experiences at a given time, the cracking moment is compared to the minimum moment capacity over the time history from halfway through the quiet time to the end of the record. Results show that the amount of steel needed to achieve this requirement would be impractical, on the order of two layers of #11 bars spaced at 6” on center, in each face, for the entire length of the pier, with increased local reinforcing in higher demand areas. Instead, to obtain ductile performance, the design includes only reinforcing patterns that yield the steel before the concrete reaches its ultimate strain.


Output and Results

Each Excel design spreadsheet is about 200 megabytes, has 23 tabs of calculations and output, and could produce an almost limitless supply of graphs, charts, and tables. This section presents and discusses selected plots and some interesting results that came from this analysis.

Summary Table: The summary table, a portion of which is shown in Table 3, is also the design spreadsheet equivalent of a command center. From here, the user can adjust the blue-shaded cells to change the vertical and horizontal reinforcing patterns and quantity of shear friction dowels for each strip of the pier. After the sheet recalculates, the peak DCRs for flexure, shear, shear friction, and compression are updated, as well as the “OK” or “NG” check for CID (if applicable). If a value exceeds criteria, the entire cell is shaded red; the DCR values under the “Concrete Only” column of the shear design do not consider the steel contribution and are only for informational purposes, hence red text instead of red fill.

Adjacent tables (not shown) offer the user the ability to change the actual size and spacing of reinforcement for each standard pattern. They also show the points of the interaction diagram and the strain condition in the extreme tensile steel, allowing virtually everything to be adjusted from a single tab.

Table 3. Sample Summary Table (9 of 60 Strips)

Force and DCR Envelopes: These plots, which show the peak forces (positive and negative) and peak DCRs across the length of the pier, provide a simpler representation of the demands than the summary table. When overlain, the envelope plots allow one to see how the demand changes from the lowest contact, at elevation 370 (blue), to the middle contact, at elevation 382 (red). Figure 10 and Figure 11 show the force (upstream/downstream and vertical) and moment envelopes, respectively, for Pier 1 during one of the earthquakes; the values are normalized by the tributary area of each strip to eliminate the effect of changes in node spacing. The flow direction in these plots is from left to right; the discontinuities at the first third-point are due to the decreased pier width at the bulkhead slot. The necessity of designing the steel at multiple elevations is apparent by inspection; the distribution and magnitude of the force and moment demands at the base of the pier are quite different than they are 12 feet above the invert.


Figure 10. Sample Output Plots: Fx Envelope (L) and Fz Envelope (R)

Figure 11. Sample Output Plots: Mx Envelope (L) and Mz Envelope (R)

Figure 12 shows the flexural and shear DCR plots for the middle contact of Pier 1 during the same earthquake. The portion of the tension curve with a DCR of zero corresponds to the location of zero tensile force on the Fz envelope. Also, note the discontinuities in both the Mz and US/DS shear DCRs around the second third point, which indicate an increase in the horizontal reinforcing. Again, this area experiences a higher demand, which is evident on the envelope plot.

Figure 12. Tension & Flexure (L) and Shear (R) DCR Plots

Hydrodynamic Loads: The added mass approach is a traditional method of accounting for dynamic loads on a structure and can be applied pseudostatically or dynamically. For


example, the Westergaard method defines the added mass due to hydrodynamic loads on the vertical face of a dam, with a resultant load of:

( 12 )

where H is the total depth of water, z is the height above the base of the dam, and A is the tributary area of the node (EM 6051). The upstream/downstream force history on the vertical face of a 5’-10 ½” pier is shown in Figure 13; the calculated reservoir forces and the results from LS-DYNA are presented in Table 4.

Figure 13. Force History on Upstream Face of Pier 2

Table 4. Reservoir Load Comparison Eq. ( 12 ) LS-DYNA % Diff.

Static 1763 kips 1863 kips 5.64% Hydrodynamic 473 kips 1180 kips 149.42%

While the static loads are comparable, there is a significant difference in the hydrodynamic loads; the piers experience a much higher peak force from the reservoir in the LS-DYNA model than they would in an added mass approach. The LS-DYNA hydrodynamic force exceeds the calculated value 90 times; even after adjusting for the slight shift at the end of the record, there are still 42 exceedences. This is only the upstream face of a single pier, too, which accounts for less than 4% of the exposed width of the flow-through monoliths. The force applied to the headwalls exhibits a similar increase, which applies even more shear to the piers at the same time that it reduces the compressive force, thereby decreasing moment capacities.

CONCLUSIONS

The Folsom Auxiliary Spillway is a critical structure upstream of a major metropolitan area. This, in conjunction with the complex geometry and loading, lead the Sacramento District to use LS-DYNA for analysis and design. Currently used mostly for mechanical


applications, LS-DYNA is versatile enough to handle seismic analyses for civil and structural engineering. It is not without its problems, and the learning curve is steep, but it helped perform the most sophisticated structural analysis and design ever completed by the Sacramento District of the Corps.

The pier design itself, taken to its most basic form, is about two hundred million column designs that consider triaxial forces and biaxial flexure. The Excel design spreadsheets were tailored to the LS-DYNA output format, but could easily be adapted to accept input from other finite element software. In the end, the procedures presented in this paper allowed the piers to be considered independently and the reinforcing to be optimized as much as practical.

There are still issues to be worked out – for example, how to efficiently compute the combined DCR without affecting the static contribution – but this approach proved a good, if sometimes cumbersome, starting point. Future analyses could use this framework to develop a self-contained program to input force histories and perform the repetitive demand and capacity calculations, with summary tables and plots as output. This would greatly increase the efficiency of the operation and could allow more design locations to be considered.

REFERENCES

American Concrete Institute. “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary.” 2005.

U.S. Army Corps of Engineers. “Earthquake Design and Evaluation of Concrete Hydraulic Structures,” Engineer Manual 1110-2-6053. 1 May 2007.

U.S. Army Corps of Engineers. “Time-History Dynamic Analysis of Concrete Hydraulic Structures,” Engineer Manual 1110-2-6051. 22 December 2003.

21st Century Dam Design - Advances and Adaptation

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