2008 Schotanus Maffei

Tailor Made Concrete Structures – Walraven & Stoelhorst (eds)© 2008 Taylor & Francis Group, London, ISBN 978-0-415-47535-8

Computer modeling and effective stiffness of concrete wall buildings

M.IJ. Schotanus & J.R. MaffeiRutherford & Chekene Consulting Engineers, San Francisco, California, USA

ABSTRACT: Shake-table tests of a full-scale seven-story wall structure at the University of California at SanDiego (UCSD) provide a crucial benchmark in evaluating methods that are currently being used to design mid-riseand high-rise concrete buildings in seismically active areas. The authors compare properties and characteristicsof the UCSD test structure with twelve tall concrete core-wall buildings that have recently been designed for thewestern United States, and find that the test results are applicable to this type of structure. Using assumptions,methods, and software that are typical in design practice, the authors constructed linear and non-linear computeranalysis models of the UCSD test structure. Iterations of assumptions for the linear models lead to recommendedconcrete stiffness properties, which are then compared to published recommendations that are often used indesign. Recommended stiffness properties are lower than those commonly used in practice. Comparison of thenon-linear models to test results shows a difficulty in matching building deformations while also matchingoverturning moments and shear forces. Both types of models show a significant influence of slabs engagingcolumns, and acting as outriggers, increasing overturning resistance and shear demand on the wall.

1 INTRODUCTION

In recent years a number of tall concrete wall buildingshave been designed for high seismic areas in Cali-fornia, Washington and Utah using “non-prescriptive”seismic design methods (Maffei & Yuen 2007). Theseismic design uses a capacity design process thatincludes both linear and non-linear dynamic analyses.A linear response-spectrum analysis is used at the codelevel of earthquake ground motion, and a non-linearresponse-history analysis is used at the MaximumConsidered Earthquake ground motion (SEAONC2007).

This paper compares the results from shake-tableexperiments on a seven-story test structure to analysismodels that are typically used in the structural engi-neering design practice for tall concrete wall buildings.The objective of the comparison is to find whichmodeling assumptions best predict response.

Buildings in the United States are typicallydesigned for earthquakes using a force-based proce-dure where the strength of elements is determinedby linear analysis according to building code require-ments.The analysis takes account of expected inelasticbehavior through application of the force reductionfactor R, which is applied to the spectrum in the analy-sis. For tall buildings, where wind forces or a minimumbase shear governs the lateral strength, an effectivereduction factor Reff can be calculated as the baseshear from an analysis at the unreduced code responsespectrum, divided by the governing design base shear.

The appropriate stiffness for the linear analysis isan effective initial stiffness consistent with the bilin-ear force-deformation assumptions that were usedto establish this traditional design approach. See forexample Blume et al. (1961).

Thus the appropriate stiffness for linear seismicanalysis is an effective initial stiffness of the structureas it reaches the limit of essentially linear behavior.Such a stiffness is illustrated by the EI effective line inFigure 1.

The effective initial stiffness should account forcracking of the concrete and other phenomena thatreduce stiffness, such as strain penetration, bond slip,

Figure 1. Modeling of structural response in design.

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Figure 2. Base moment vs. roof displacement experimentalbehavior for EQ3 and EQ4 loading.

shear deformation, and tension shift associated withdiagonal cracking.

Though non-linear response is not explicitly eval-uated in conventional design, safe non-linear seis-mic performance under the design ground-motion isexpected based on historic calibration of the forcereduction factors in the building code. When vali-dated R-factors are not available, as is the case for verytall concrete wall buildings, the need arises to verifynon-linear response as part of the design process.

The UCSD test structure was subjected to severalground motions (Panagiotou et al. 2006). Figure 2shows that the earthquake ground motion of mediumintensity, EQ3, demonstrates essentially linear forceversus deformation response, and EQ4 shows non-linear response. Comparing Figures 1 and 2, thebehavior for EQ3 response, which approaches globalyielding, can be described as representative of elasticdesign. By contrast, EQ4 shows significant excursionsinto the non-linear domain.

To evaluate the applicability of the test structureto actual tall concrete buildings, a number of char-acteristic parameters are compared in Table 1. Foreach parameter, both a range and a typical value havebeen determined from the review of twelve recentlydesigned tall buildings, which are compared to thosefor the test structure.

The quantity Vyield/W in Table 1 provides a com-parison of the design strength of the test structurecompared to the typical high rise. Typical tall build-ings on the West Coast are designed for a minimumbase shear of between 4% and 6% of the buildingweight. Considering over-strength, typical high-risebuildings have a yield base shear capacity correspond-ing to about 7% to 12% of the building weight. Theover-strength results from: (1) expected strength ofreinforcing steel which is about 15% higher than nomi-nal design strength, (2) the use of a strength-reductionfactor of about 0.9 for flexural design, and (3) con-tribution of elements not designated as part of the

Table 1. Comparison of key properties for UCSD teststructure and typical tall concrete core-wall buildings.

UCSD High-Rise High-RiseParameter Wall (range) (typical)

h/lw 5.2 9–13 10(weak way) (weak way)

ρvert hinge zone 0.7% 0.7%–2.0% 1%ρvert above hinge 0.8% 0.8%–2.2% 1.1%ρhoriz hinge zone 0.3% 0.3%–2.6% 1%ρhoriz above hinge 0.4% 0.3%–1.2% 0.7%Vu/(

√f′cAg) at hinge 3.0 3–8 6

Axial load ratio 0.05 0.06–0.13 0.09(P/Ag f′c)Floor span-to-depth 17 30–45 40ratioEffective elastic 1 sec 4–9 sec 6 secperiod (weak way) (weak way)Vyield /W 26% 6%–12% 8%

Seismic-Force-Resisting System. The upper range ofyield capacity (9% to 12% of building weight) isreached for buildings with IBC S1 factors greater than0.6 (ICC 2006), for which a larger minimum base shearis specified, and for the stronger direction of buildinglayouts that have a rectangular concrete core.

The Vyield/W value for the test structure was com-puted by performing a response spectrum analysison the building, using the code spectrum for soilclass C (very dense soil and soft rock). The response-spectrum results are then scaled to compute Vyieldequal to the base shear that corresponds to a wallbase moment equaling the wall’s expected momentcapacity: Vyield = Vu (Mn,exp/Mu).

The values in Table 1 show that typical high-risecore-walls have, for most parameters, similar proper-ties to the test structure. Dissimilar properties includethe floor span-to-depth ratio and Vyield/W . The effectsof these differences are considered when extrapolat-ing the results for the test structure to actual high-risebuildings.

In practice, different engineers use differentassumptions for the effective initial stiffness of awall, based on any of the conflicting recommendationsin various published guidelines. Table 2 lists refer-ence documents and approaches proposed or used inpractice.Three classes of recommendations can be dis-tinguished, the first with fixed stiffness modifiers, thesecond with modifiers depending on axial load level,and the third that includes the reinforcement ratio inaddition to the axial load level.

Table 2 shows the results of applying each publishedrecommendation to the test structure, both for the firstand the sixth story. Table 2 also shows the results ofapplying each published recommendation to the typi-cal tall concrete core-wall building. Wall dimensionsfor each case are shown in Figure 3. The test specimen

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Table 2. Effective elastic flexural stiffness properties forstrong way bending of structural walls: Comparison forUCSD test specimen and typical high-rise.

Effective Stiffness Considers:

6th Typ.Reference Base floor High-Rise P ρvert

ASCE 41, uncracked 0.80Ig 0.80Ig 0.80Ig NO NOASCE 41, cracked 0.50Ig 0.50Ig 0.50Ig NO NOMicroys et al. 0.64Ig 0.61Ig 0.68Ig YES NO(Eq. on p. 302)NZS:3101 (µ = 6) 0.29Ig 0.26Ig 0.33Ig YES NONZS:3101 (µ = 3) 0.54Ig 0.51Ig 0.58Ig YES NOPaulay & Priestley 0.25Ig 0.23Ig 0.29Ig YES NO(Eq. 5.7 on p. 376)Restrepo (Eq. 7) 0.22Ig 0.26Ig 0.42Ig NO YESACI 318 (Eq. 9–8) 0.32Ig 1.0Ig – YES YESFIB 27 (Eq. on p. 83) 0.21Ig 0.20Ig 0.28Ig YES YESMoment-Curvature 0.20Ig 0.21Ig 0.27Ig YES YES

Figure 3. UCSD test versus typical layout of high-riseconcrete core-wall.

could be considered to represent a scaled down versionof the hatched part of the core wall.

2 ELASTIC MODELING FOR WALLBUILDINGS

A linear-elastic model of the test structure is builtin the ETABS (2006) analysis software. For the walland floor slabs wall/slab elements are used, which areshell-type elements with a membrane and a bendingcomponent. The analysis model is shown in Figure 4.

Mass and forces are calculated by ETABS fromthe concrete self-weight. The mass contribution fromthe post-tensioned wall that provides out-of-planerestraint to the test is added to the central node of theweb wall, at each story.

Six models with different sets of assumptions onthe effective wall and slab stiffness are created, andsubjected to linear time-history analysis using an inte-gration time step of 1/240s. With each elastic modelthe medium level ground motions EQ3 from the tests(Panagiotou et al. 2006) is used, as discussed in theintroduction. All cases analyzed are summarized inTable 3. For simplicity the effective stiffness factorreported is applied to the material stiffness Ec, thusaffecting the stiffness in both shear and flexure for

Figure 4. UCSD Wall (left) and ETABS model (right).

Table 3. Analysis models and runs used to determineappropriate stiffness modeling assumptions.

Wall Wall1st 2nd–7th FloorFloor Floor slabs Record Results/comments

Factor on Ig0.8 0.8 0.5 EQ3 Typical practice

assumption;too stiff

0.2 0.2 0.2 EQ3 OK match ofdisplacement

0.2 0.2 0.1 EQ3 Better match ofdisplacementhistory

0.13 0.3 0.3 EQ3 Accounts for bondslip; limitedimprovementin match

0.13 0.3 0.1 EQ3 Best match ofdisplacementhistory and profile

0.1 0.3 0.15 EQ3 Improves match ofoutrigger factor

fiber fiber 0 EQ3&EQ4 Slightly over predictspeak displacement,under predictsmoment and shear

fiber fiber 0.1 EQ3&EQ4 Matches peakdisplacement andprofile, underpredicts momentand shear

fiber fiber 1 EQ3&EQ4 Under predictsdisplacement,matches systemmoment and shear

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Figure 5. Comparison of roof displacement response forEQ3, using 80% of gross wall stiffness and 50% of grossslab stiffness (top), and 20% of gross wall stiffness and 10%of gross slab stiffness (bottom).

all degrees of freedom. Comparisons are made to theUCSD test wall response for the same earthquakerecord.

The first three models apply the same effectivestiffness multiplier over the entire wall height. Theyrepresent the extremes of the published effective stiff-ness values from Table 2. The effective stiffness val-ues used for the slab bound a fully cracked and anun-cracked response. Key response quantities areshown in Figures 5 through 7. Figure 5 includesresponse time-histories for roof displacement, Figure 6shows the peak displacement envelope over the build-ing height, and Figure 7 shows peak moment and shearvalues at each story. All values are shown togetherwith the experimental results. The moment in Figure7 is the system moment, including overturning resis-tance from the slab and column system. Included inthe figure showing the system moments are the peakweb-wall moment at the base (from analysis) and theexpected wall yield moment under gravity loads. Thedifference between the system moment and these lattermoments are indicators of the slab-outrigger effect onthe structural response.

The results show that use of the lower stiffness val-ues predicts the response much better. In particular,the results for the 20% wall stiffness model using 10%

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Figure 7. Comparison of peak moment (left) and peak storyshear (right) for EQ3, using 80% of gross wall stiffness and50% of gross slab stiffness, and 20% of gross wall stiffnessand 10% of gross slab stiffness.

of gross slab stiffness show very good agreement inthe displacement response, see Figure 5.

Looking at the displaced shape in Figure 6 (right-hand side), it can be seen that the displacement at thelower floors is slightly under-predicted, while the gra-dient at higher floors is slightly over-predicted. Theadditional displacement in the first floor could resultfrom both bond-slip at the lap splice and strain penetra-tion into the foundation. To account for such behavior,the stiffness of the ground floor is reduced to add thisadditional flexibility, based on an estimate of this con-tribution from strain penetration calculations. To keepthe match of roof displacement, the effective stiffnessfor the upper floors is somewhat increased.Three addi-tional models include this effect, as summarized inTable 3. Results are shown in Figure 8.

3 NON-LINEAR MODELING FOR WALLBUILDINGS

In addition to the linear modeling, a non-linear modelof the structure is built, using the Perform 3D anal-ysis software (2006). The model uses vertical fiber

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Figure 8. Comparison of key response quantities for EQ3,using 13% and 30% of gross wall stiffness and 10% of grossslab stiffness.

elements to explicitly model the non-linear propertiesof the wall cross section. The base of the modelreflects the essentially fixed support at the shake tablethat was observed in the tests. The model restrainsout-of-plane translational degrees of freedom to repli-cate the one-dimensional input and two-dimensionalresponse.

As shown in Figure 9, two non-linear models areevaluated – one with, and one without, the slab andcolumns. The wall-only model corresponds to whatis often modeled in practice, where only those ele-ments designated as part of the seismic-force-resistingsystem are assumed to resist earthquake forces.

P-delta effects are explicitly included in theanalyses.

3.1 Wall modeling

The wall is modeled with the Perform 3D shear wallcompound element and includes inelastic sectionsover the height of the building. The compound ele-ment consists of three components: (1) an inelasticaxial/bending fiber section containing vertical fibersmodeling the vertical reinforcing steel and the grossconcrete area, (2) an elastic shear component, and(3) an elastic out-of-plane flexural component. Sim-ilar elements have been shown to model wall flexuralbehavior well (Orakcal and Wallace, 2006).

In modeling the wall, one element is used over thestory height, except at the first floor, where two ele-ments are used. This is to better model the plastichinge, which is expected to extend to about half theheight of the first floor. Two planar elements are usedover the wall length.

The inelastic fiber section is described by the wallthickness and reinforcement ratio, and a material lawfor both the steel and concrete. Since reinforcementis not distributed uniformly, the program’s option tolocate fibers manually is used to better replicate thereinforcement distribution.

Expected material properties are used for theparameters in the material models. Detailed

Figure 9. PERFORM 3D models of wall web, with andwithout slab.

information for concrete batches and steelreinforcement heats is available for the test structure(Restrepo 2006), and is used to determine an averagevalue for the entire building.

Effective stiffness values are used for the elasticcomponents of the wall and slab elements. For thewall in flexure, the fiber element properties determineeffective initial stiffness.

3.2 Analysis parameters

After an initial static load step applying gravity loads,the earthquake records from the test structure areapplied to the model. A Rayleigh damping model isused, with damping coefficients chosen to result in 2%equivalent viscous damping at the initial fundamentalperiod of the structure T1, and at 0.2T1.The integrationtime-step used is 1/240 second, equal to the time stepat which the ground motion records were sampled.

Table 3 shows the three analysis cases considered.

3.3 Results

From modal analysis, the initial fundamental periodT1 of each model is as follows: 0.490 seconds forthe wall-only model, 0.485 seconds for the model withthe slab at 10% gross stiffness, and 0.468 secondsfor the model with the slab at 100% gross stiffness.Theperiod found at the beginning of the testing cycle wasreported in Panagiotou et al. (2006) to be 0.502 sec-onds, while the period of the structure prior to runningEQ4 was approximately 1 second.

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Figure 10. Comparison of peak displacement envelopes forEQ4.

Key response quantities for the three models forEQ4 are shown in Figures 10 and 11.

Figure 10 shows that the wall-only model over-predicts the displacement response. Figure 11 showsthat this model under-predicts system forces. The wallshear values for the wall-only model for EQ4 under-estimate the experimental values by up to 40%. Theresults for the wall-only model indicate that the inter-action of the gravity system in the seismic responseshould not be neglected.

Results for the models including slab behaviorimprove the prediction of system forces. However,none of the models are able to capture both thedisplacement and force demands accurately.

A summary of the performance of all three modelsanalyzed is included in Table 3.

4 CONCLUSIONS

The properties and characteristics of the UCSD seven-story test structure are applicable to tall concretebuildings that use core walls as their seismic-force-resisting system. The shake-table tests of the structureprovide a valuable benchmark against which engineerscan study the assumptions typically used in the seismicanalysis and design of concrete wall structures.

Linear and non-linear analyses show that it is impor-tant to model the gravity framing system in order toinclude the influence it has on the structure’s seismicresponse. Floor slabs engaging columns, and acting asoutriggers, increase overturning resistance of the sys-tem, axial forces in columns, and shear demands onthe wall. This effect is somewhat more pronounced inthe test structure than it would be in structures wherefloor span-to-depth ratios are higher.

The results for the linear-elastic analyses confirmthat effective stiffness formulations that include theeffect of axial load and longitudinal reinforcement

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Figure 11. Comparison of peak moment (left) and peakstory shear (right) for EQ4, using 100% gross slab stiffness.

ratio better predict effective initial stiffness. Actualstiffness values can be significantly lower than thoseused in some current published guidelines.

REFERENCES

ACI 318-05 2005. Building Code Requirements for StructuralConcrete, Second Printing, American Concrete Institute,Farmington Mills.

ASCE 41 2006. Seismic Rehabilitation of Existing Build-ings, American Society of Civil Engineers, StructuralEngineering Institute, Reston, VA.

Blume, J.A., Newmark, N.A. & Corning, L.H. 1961, Designof Multistory Reinforced Concrete Buildings for Earth-quake Motions, Portland Cement Association, Skokie,Illinois.

ETABS Plus version 9.1.1 2006. Extended Analysis of 3DBuilding Systems, Computers & Structures, Inc. Berkeley,California.

FIB Bulletin 27 2003. Seismic Design of Precast ConcreteBuilding Structures, State-of-art report prepared by FIBTask Group 7.3, Lausanne, Switzerland.

ICC 2006. International Building Code (IBC), InternationalCode Council, Falls Church, Va.

Maffei, J. & Yuen, N. 2007. “Seismic Performance andDesign Requirements for High-Rise Buildings.” StructureMagazine, April: 28–32.

Microys, H.F., Michael, D. & Saiidi, M. 1992. Cast-in-placeconcrete tall building design and construction, Councilon Tall Buildings and Urban Habitat, Committee 21D.McGraw-Hill.

NZS:3101 1995. Concrete Structures Standard Part 2 – Com-mentary on the Design of Concrete Structures, Prepared byConcrete Design Committee P3101 for the New ZealandStandards Council, New Zealand.

Orakcal, K. &Wallace, J.W. 2006. Flexural Modeling of Rein-forced Concrete Walls – Experimental Verification. ACIStructural Journal, 3(2): 196–206.

Panagiotou, M., Restrepo, J.I., Conte, J.P. & Englekirk, R.E.2006. “Shake Table Response of a Full Scale ReinforcedConcrete Wall Building Slice.” In: Proceedings of the 75thAnnual SEAOC Conference, CA: 285–299.

Paulay, T. & Priestley, M.J.N. 1992. Seismic design of rein-forced concrete and masonry buildings, John Wiley &Sons.

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Perform 3D version 4.0.1 (2006) Nonlinear Analysis andPerformance Assessment for 3D Structures, Computers &Structures, Inc. Berkeley, California.

Restrepo, J.I. 2000. Issues Related to the Seismic Design ofReinforced Concrete Structural Systems. Sesoc Journal,13(1): 50–58.

Restrepo, J.I. 2006. Seven-story Building-slice EarthquakeBlind Prediction Contest. NEES@UCSD, <http://nees.ucsd.edu/7Story.html> (March, 2006).

SEAONC 2007. Recommended Administrative Bulletin onthe Seismic Design and Review of Tall BuildingsUsing Non-Prescriptive Procedures, Structural Engi-neers Association of Northern California, San Francisco,California.

APPENDIX I. NOTATION

The following symbols are used in this paper:Ag = area of gross concrete section;Cd = deflection amplification factor (ICC 2006);δx = design deflection (ICC 2006);

δxe = deflection determined by linear analysis;Ec = concrete modulus of elasticity;f′c = nominal concrete peak stress;h = wall height;

Ig = moment of inertia of gross concrete section;lw = wall length;µ = structure ductility capacity;

Mn,exp = expected wall moment strength;Mu = peak wall moment demand;

P = wall axial load;ρvert = longitudinal wall reinforcement ratio;

ρhoriz = transverse wall reinforcement ratio;R = response modification coefficient;

S1 = mapped MCE spectral response accelerationparameter at a period of 1s (ICC 2006);

Sa = design spectral response acceleration;tw = wall thickness;T1 = first period of vibration;Vu = peak wall shear force demand;

Vyield = shear force demand at flexural yielding;W = total building weight.

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