Design Seismic Response Evaluation of Wall Systems Including Foundation Flexibility

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DESIGN SEISMIC RESPONSE EVALUATION OF WALL SYSTEMS

INCLUDING FOUNDATION FLEXIBILITY

Zdravko Bonev ([email protected])University of Architecture, Civil Engineering and Geodesy,Hr. Smirnenski Blvd. 1, 1046 Sofia, Bulgaria

Golubka Necevska-Cvetanovska ([email protected])Institute of Earthquake Engineering and Engineering Seismology,IZIIS, University “Ss. Cyril and Methodius”Salvador Aljende St. 73, 1000, Skopje, F.Y.R. of Macedonia∗

Elena VasevaCentral Laboratory of Seismic Mechanics and Seismic Engineering,Bulgarian Academy of Sciences, Acad. Georgi Bonchev St., Bl. 3,1113 Sofia, Bulgaria

Roberta Apostolska ([email protected])Institute of Earthquake Engineering and Engineering Seismology,IZIIS, University “Ss. Cyril and Methodius”Salvador Aljende St. 73, 1000, Skopje, F.Y.R. of Macedonia∗

Dilyan BlagovUniversity of Architecture, Civil Engineering and Geodesy,1, Hristo Smirnenski Blvd., 1046 Sofia, Bulgaria

Abstract. The paper deals with application of capacity spectrum method which is extendedto provide estimate for seismic demands for specific type of structure and variable boundaryconditions. Flat slab—R/C wall systems are considered. Flexible shallow wall foundations areused because of soil deformations. Only elastic soil properties are considered. Foundation mo-tion is implemented assuming new additional degrees of freedom instead of standard fixed end.The only rocking motion of the footing is accounted for as the most essential. The influence offoundation flexibility on the seismic demands is evaluated using capacity spectrum method.Numerical results for a single wall and for overall structure are graphically represented. Itis concluded that soil conditions are very important for the design seismic response of wallsystems. Some important conclusions are made.

Keywords: capacity spectrum method, effects of foundation flexibility, design seismicperformance including soil effects

∗Turkey recognises the Republic of Macedonia with its constitutional name.

T. Schanz and R. Iankov (eds.), Coupled Site and Soil-Structure Interaction Effects 241with Application to Seismic Risk Mitigation, NATO Science for Peace and SecuritySeries C: Environmental Security, c© Springer Science+Business Media B.V. 2009

Author's personal copy242 Z. BONEV ET AL.

1. Introduction

The Capacity Spectrum Method (CSM) is a tool to predict the design seismicperformance of structures subjected to design seismic action. This method isrecommended by the new generation of seismic resistant design codes suchas EC8, 2004 for evaluation of seismic demands and for capacity assessmentof newly designed or existing building structures. The theoretical backgroundand application of CSM can be found in Chopra and Goel (2003) and Fajfar(1999). In Fajfar (1999) the seismic hazard data such as design ground ac-celeration and soil amplification effects are taken into account by the shapeand scaling coefficient of the demand spectrum. The idea of the present pa-per is to account for the foundation flexibility in the capacity curve sincesoil deformations themselves influence overall stiffness of the structure. Thusthe capacity curve is obtained for coupled foundation-R/C building system.Analyses in this direction are carried out in Bonev et al. (2005) and Schanzet al. (2007) for plane frame structures. In this paper the subject of interestis one 3D structure being designed as a wall system. It is shown in FEMA450 (2003) the wall systems are much sensitive to soil deformations becausethe most stiff elements—walls dictate internal force distribution. The imple-mentation of the soil conditions into numerical models of entire structureis discussed in FEMA 273 (1997), FEMA 274 (1998), FEMA 357 (2000),FEMA 450 (2003), Geotechnical Engineering (1997) and Kramer (1996).FEMA 357 (2000) provides foundation stiffness coefficients as dependenton the effective shear modulus, G, and Poisson’s ratio. The model can beused for static and dynamic analyses. FEMA 273 (1997), FEMA 274 (1998)and FEMA 450 (2003) propose the constitutive relationship moment-rotationwhich is related to the rocking motion of the single shallow footing. Soilplasticity and uplifting phenomena can be taken into account in the result-ing moment-rotation curve which can be used for static loads only. In thispaper only linear soil properties are taken into account being represented bythe unit foundation modulus (Winkler’s constant). The paper deals with thechanges in the capacity curves which are more or less steep depending on theelastic soil properties. Due to footing flexibility the target displacements areincreased and the global ductility of the structure is reduced. It is shown thatfor soft soils the design performance of the structure may remain completelyelastic. In contrary, for stiff soils the seismic demands of the structure arelarge enough to develop significant inelastic deformations. It is found thatthe influence of accidental eccentricity is much essential after collapse of themost loaded walls. The results are graphically illustrated and discussed.

Author's personal copyDESIGN SEISMIC RESPONSE EVALUATION OF WALL SYSTEMS 243

2. Model Description

The general view of the model can be seen in Fig. 1. The numerical modelused in calculations can be described starting with assumptions listed below:

1. Floor slab is treated as a rigid diaphragm in its own plane. The mem-brane stiffness of the floors is practically infinitely large and the slab maymove horizontally as absolutely rigid body. On the other hand the slabsdistribute the seismic loads between the walls.

2. The vertical loads are carried by shearwalls and columns. Lateral loadsare carried by the shearwalls only. In case when the slabs are flat (nobeams are used) the slab to column connection is not designed as momentresisting. It is assumed that columns are pinned at both ends and can resistto vertical loads only.

3. Shearwalls are modelled by vertical frame elements. The potential loca-tions of plastic hinges are considered at each floor level.

Figure 1. (a) General 3D view of the numerical model—walls, columns and slabs;(b) shearwall system only.


4. The structure is symmetric in plan with respect to X- and Y-axes. If thelocation of the centre of resistance (CR) is coincident with the centre ofmasses (CM) the structure is regular. The reference co-ordinate system islocated in CR-point (level 0). If the centre of the masses CM has positiondifferent from the position of CR the structure is irregular with accidentaleccentricity.

5. Bending stiffness of the slab is taken into account only to obtain the verti-cal loads distribution between the vertical elements—walls and columns.

6. Single footing under each wall is used. The foundation is supported bythe soil with its vertical resistance. The elastic soil properties are im-plemented by the unit foundation modulus (Winkler’s constant). Eachfoundation has 3 degrees of freedom in XZ-vertical plane and 3 degreesof freedom in YZ-vertical plane. The stiffness of the springs is determinedusing the unit foundation modulus. This approach is considered in FEMA273 (1997), FEMA 274 (1998) and FEMA 450 (2003).

7. Loading pattern used for pushover analysis in both X- and Y-directionshas the shape of inverted triangle and implies linear force distribution inelevation. Forces are applied in CM for each floor level.

8. The calculation of the spring stiffness implies that only rocking motionof the footing is considered.

9. Torsion effects due to different disposition of CM with respect to CR aretaken into account. The influence of accidental eccentricities is accountedfor as a source of torsion.

10. Axial forces are remaining constant during the lateral pushover analy-sis and plastic hinge properties once determined after application of thevertical loads are remaining the same.

11. The interface footing surface is unable to move horizontally because ofthe friction, passive pressure and embedment effect.

The general 3D view of the building model is shown in Fig. 1. The modelis consisting of reinforced concrete walls, columns and slabs. Pin-joint con-nection is used and columns are unable to resist lateral loads. Foundations aremodeled as single footings including the elastic support of the soil. Consid-ering a single wall in X–Z plane for example, see Fig. 2a FEMA 274 (1998)the soil resistance is represented by equivalent soil springs whose stiffnesscan be calculated on the basis of the unit foundation modulus. This is shownin Fig. 3. The potential location of the first plastic hinge is at the base of thewall (in the centre of the plastic zone). It is indicated by the black point on


Figure 2. (a) Single footing under the wall in the plane of rocking; (b) structural layout.

Figure 3. A single wall designed as dissipative wall—moment and curvature distribution.Plastic zone modelled by plastic hinge and spring elements including the soil resistance.

300

250

200

150

100

50

00 0.1 0.2 0.3

Top Displacement, [m]

Bas

e She

ar, [k

N]

0.4 0.5 0.6 0.7

rigidc=60000

c=50000

c=40000

c=30000

c=20000

0.8 0.9

Figure 3a. Capacity curves for a single wall using different values for Winkler’s modulus.


Figure 4. Typical constitutive relationships: (a) moment M—curvature φ for distributedplasticity model; (b) moment M—rotation θ for concentrated plasticity model.

the Z-axis.The plastic zone approach is based on distributed plasticity model.The typical constitutive relationship is moment-curvature. The distribution ofelastic/inelastic curvature for a simple wall element is shown in Fig. 3.

The plastic zone length LP is calculated following Paulay and Priestley(1992)

LP = 0.08L + 0.022db fy, (1)

where L is the total wall length in (m), db is the bar diameter in (m), andfy is the yield strength of steel in (MPa). For concentrated plasticity mod-els (zero-length plastic hinges) used herein the basic constitutive relation-ship is moment-rotation, see Fig. 4. The parameters of the plastic hinges arecalculated as follows:

θy =MyLP

EI,

θP = LP(ϕu − ϕy)(

1 − LP

2L

)

, (2)

θu = θy + θP,

whereφy and φu the yielding and ultimate curvatures correspondingly,My and Mu the yielding and ultimate moments,θy and θu the yielding and ultimate rotations, see Fig. 4,θP the plastic rotation component.For wall elements plastic hinges are implemented only in their own plane.

In order to simulate very large initial stiffness of rotation springs it is rec-ommended to implement the “computational yield rotation” being defined as

θ′y =MyL′P

EI. (3)


It is accepted in calculation of θ′y that L′P = 0.10 m. The moment-rotationcurve obtained by calculations is given in Fig. 4b as a solid line. Themoment-rotation curve implemented in calculations is shown in the samefigure with a dotted line.

The structural layout and wall/columns distribution is shown in Fig. 2b.The eccentricity is defined by the distances between CM center and CR cen-ters. The behaviour of a single shearwall subjected to in-plane loading canbe seen in Figs. 3 and 3a and is represented by the capacity curve. It isevident that the flexible foundations do not influence the base shear strength(capacity). An essential influence is found in target displacements and ductil-ity. Target displacement is drastically increased for soft soils because elasticdeformations are growing up. At the same time inelastic displacement de-mands are relatively constant. Ductility factor is significantly reduced andpure ductile behaviour of a single wall is practically not achieved.

The behaviour of single wall elements is in a large extent important forthe behaviour of overall system (see Fig. 1).

3. Analysis Method

Capacity spectrum method is used as evaluation tool (Fajfar, 1999) wherethe use of design demand spectra is recommended. A version of the methodbased on elastic demand spectrum is available in Annex B of EC8, 2004.The basic relationship considering the global behaviour is “Base shear forceV—Roof displacement u” derived numerically from pushover analysis. Thenormalized vector of lateral displacements distribution in elevation is denotedby {Φ}, see Fig. 4a. For a single wall system it is reasonable to accept invertedtriangle as a shape of the {Φ}-vector.

Figure 4a. Structural system subjected to two independent lateral loading patterns in mainorthogonal directions. The direction is depicted by x and y subscripts.


The governing system of equations for dynamic equilibrium assumingthat the effect of the damping takes part implicitly could be written as follows:

[m]{u} + { f } = −[m]{1}ug, (4)

where [m] is the diagonal mass matrix of floor masses, { f } is the vectorof internal (restoring) forces, {u} is the vector of level accelerations, ug isthe ground acceleration. It is assumed further that the distribution of lateraldisplacements {u} follows the shape of {Φ}-vector with proportionality mul-tiplier u(t). This quantity is the roof displacement. Thus after replacement of{u(t)} = {Φ}u(t) in Eq. (4) and after multiplication of both sides of the newequations by {Φ}T on the left the following scalar equation is obtained:

{1}T [m]{Φ}u + {1}T { f } = −{1}T [m]{1}ug. (5)

It is easy to implement the following quantities related to equivalent single de-gree of freedom system, namely: mass parameter m∗ = {1}T [m]{Φ} =

∑

imiΦi,

total mass of the original system {1}T [m]{1} =∑

imi = m and base shear force

V = {1}T { f }. Note that the mass of the original system is equal to the mass ofthe equivalent system. After taking into account the above notations Eq. (5)is transformed to give

u∗ +Vm

= −ug. (6)

The modified displacement u∗ of equivalent single degree of freedom systemis introduced by the following expression:

u∗ =uΓ, (7)

where Γ = m/m∗ is modification factor used to transform the displacements inoriginal system into displacement for equivalent system. Finally, the systemof Eq. (4) provides the result:

Vm

= −(ug + u∗). (8)

According to Eq. (8) it is implied in capacity spectrum method that capac-ity curve could be plotted following the rule: use the quantity V/m on theacceleration axis and use the quantity u∗ on the displacement axis.

Two simplified and independent analyses in both X- and Y-directionsare carried out. Two equivalent single degree of freedom systems are used(see Fig. 4a). After that the capacity spectrum method is applied in bothorthogonal directions as well.


4. Numerical Results

The numerical model described in Section 2 is subjected consequently tomonotonically increasing vertical and horizontal loads. Six values of unitfoundation modulus are used in calculations: fixed base (infinitely large mod-ulus), 60,000, 50,000, 40,000, 30,000 and 20,000 in kN/m3 metric units. Aftercompletion of vertical loading procedure the horizontal loading pattern isapplied. The effects of accidental torsion are studied considering eccentricityof 15% (large eccentricity and irregular structure) and 0% (regular structure).The modification factors Γ are dependent on the plastic mechanism reachedduring the loading. For structure with flexible foundation Γ is very close to2.0. In case of fixed base structure this factor is slightly higher than 2.0. Thetotal mass of the structure is 875 kN · s2/m (125 per each storey level). Thecapacity spectrum method (Fajfar, 1999) is applied using the design demandspectrum in acceleration-displacement format. The analyses results are rep-resented graphically in Figs. 5–14. Numerical calculations are based on theresults obtained in Blagov et al. (2008).

Figure 5. Capacity spectrum method applied to fixed-base structure: (a) in X-direction;(b) in Y-direction.

Figure 6. Capacity spectrum method applied to structure with flexible foundations (unitfoundation modulus 60,000 kN/m3). Left side plot is for X-direction, right side plot is forY-direction. Solid line implies 15% eccentricity; dotted line implies 0% eccentricity.


Figures 5–10 illustrate the results obtained through capacity spectrummethod. It is evident that the greatest inelastic seismic demands in the RC






700

600

500

400

300

200

100

00 0.1 0.2 0.3

Roof Displacement, [m]

Base S

hear,

[kN

]

0.4 0.5 0.6 0.7

rigid basec=60000c=50000c=40000

c=30000c=20000

0.8 0.9 1

Figure 11. Capacity curves in X-direction (eccentricity 0%) obtained for different Winkler’sconstants.

0 0.1 0.2 0.3


0.4 0.5 0.6 0.7 0.8 0.9 1

700

600

500

400

300

200

100

0

Base S

hear,

[kN

]

Figure 12. Capacity curves in X-direction (eccentricity 15%) obtained for different Winkler’sconstants.


0 0.1 0.2 0.3


0.4 0.5 0.6 0.7 0.8 0.9

700

600

500

400

300

200

100

0

Ba

se

Sh

ea

r, [

kN

]

Figure 13. Capacity curves in Y-direction (eccentricity 0%) obtained for different Winkler’sconstants.

0 0.1 0.2 0.3


0.4 0.5 0.6 0.7 0.8 0.9 1

700

600

500

400

300

200

100

0

Base S

hea

r, [

kN

]

Figure 14. Capacity curves in Y-direction (eccentricity 15%) obtained for different Winkler’sconstants.

members is achieved when the Winkler’s constant is infinitely large (fixedbase). When foundation flexibility is increased the inelastic seismic de-mands are reduced. Elastic component of roof displacement is growing up.Figures 11–14 illustrate that the global ductility demand is reduced withincreasing the footing flexibility. The same figures show that the initial(elastic) stiffness is also reduced due to flexibility. At the same time the baseshear strength is relatively slightly influenced by the footing flexibility.

Figures 8–10 illustrate that when the soil is soft and structure reachesthe target displacement the global behaviour of the structure may remaincompletely elastic. This mode of deformation implies that the soil failureprecedes yielding of the structure. Figure 5 shows that the largest values forthe behaviour factor could be achieved if the base is fixed and foundationflexibility does not exist. The smallest target displacements are observed inthe same case. If the foundations are flexible (see Figs. 6–10) target displace-ment is increased but the behaviour factor decreases. The behaviour factorreduction is not on the side of safety. It is seen that the wall system is sensitiveto soil deformations and the negative effect of flexibility can be avoided bymaking foundations less flexible.


5. Conclusions

Considering the analysis of the numerical results obtained in Section 4, thefollowing conclusions can be made:

1. The foundation flexibility caused by soil deformations influence essen-tially the capacity curves. The global ductility factor is seriously reduceddue to significant increase of elastic part of deformations. In general,the participation of soil deformation in overall structural deformationsis essential. It is expected this participation to become more essential incase of plastic soil deformations and foundation uplift in new shearwallsin existing buildings, where the vertical load is not sufficient to preventthe footing from uplifting.

2. As a rule, target displacements are increased if the soil is softer. The be-haviour factor however shows decreasing tendency. Safe design solutionscould be expected if soil deformations are taken into account in capacitycurves.

3. The influence of accidental torsion effects is small considering the elasticbehaviour of the structure. More essential influence is observed whensome plastic hinges yield and when some walls are collapsed. The capac-ity curves are sensitive to accidental torsion when wall elements yield orand plastic mechanism takes place.

4. The global structure strength is relatively independent of soil stiffnessand accidental torsion effects influence slightly the strength.

5. It is concluded that wall systems are sensitive to flexible foundations ina large extent. It is demonstrated that overstrength factor of a single wallis 1.0 and no plastic redistribution mechanism is observed. This is easyto explain this by the fact that each single wall is not redundant system.Structural assemblages composed of separate walls and flat slabs in largedegree follow this property (Figs. 11–14) and plastic force redistribu-tion mechanism does not contribute significantly towards seismic loadingreduction.

Acknowledgements

The project grant No. BM-6/2006 of the National Science Foundation at theBulgarian Ministry of Education and Sciences is greatly acknowledged bythe authors.

The project grant No. BN-84/2008 of the University of Architecture, CivilEngineering and Geodesy, Sofia, Bulgaria, is greatly acknowledged by theauthors.


References

Blagov, D., Georgiev, V., and Bonev, Z. (2008) Influence of flexible foundations on the designresponse of buildings with accidental eccentricity. In Proc. 5th European Workshop onIrregular and Complex Structures, 16–17 September 2008, Catania, Italy.

Bonev, Z., Ganchev, S., Blagov, D., and Zerzour, A. (2005) Behaviour factor evaluationaccounting for the elastic foundation, EE-21C, Topic 4: Structural Modelling, Analysis,Design and Seismic Safety, Technical report, Skopje, Republic of Macedonia.

Chopra, A. K. and Goel, R. K. (2003) Evaluation of the modal pushover analysis proce-dure using vertically “regular” and irregular generic frames, Technical Report No. EERC2003–03, University of California, Berkeley.

Fajfar, P. (1999) Capacity spectrum method based on inelastic demand spectra, EarthquakeEngineering and Structural Dynamics 28, 979–993.

FEMA 273 (1997) NEHRP Guidelines for the Seismic Rehabilitation of Buildings, Issued byFEMA in Furtherance of the Decade for National Disaster Reduction.

FEMA 274 (1998) Seismic Rehabilitation Commentary C4: Foundations and GeotechnicalHazards.

FEMA 357 (2000) Global Topics Report on the Prestandard and Commentary for the SeismicRehabilitation of Buildings.

FEMA 450 (2003) NEHRP Recommended Provisions for Seismic Regulations for NewBuildings and Other Structures, Commentary C7A.

Geotechnical Engineering, Circular No. 3 (1997) Design Guidance: Geotechnical EarthquakeEngineering for Highways, Vol. 1: Design Principles, US Department of Transportation,Federal Highway Administration.

Kramer, S. L. (1996) Geotechnical Earthquake Engineering, Prentice-Hall InternationalSeries in Civil Engineering and Engineering Mechanics, Upper Saddle River, New Jersey.

Paulay, T. and Priestley, M. J. N. (1992) Seismic Design of Reinforced Concrete and MasonryBuildings, John Wiley & Sons, New York.

Schanz, T., Bonev, Z., Georgiev, V., and Iankov, R. (2007) Application of capacity spectrummethod to soil-foundation-structure interaction problems. In Proc. of the Jubilee Scien-tific Conference, Devoted to 65 Years Anniversary of the University of Architecture, CivilEngineering and Geodesy, 16–17 May 2007, Sofia, Bulgaria.

Design Seismic Response Evaluation of Wall Systems Including Foundation Flexibility

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