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Hindawi Publishing CorporationJournal of StructuresVolume 2013, Article ID 679859, 14 pageshttp://dx.doi.org/10.1155/2013/679859

Research ArticleUltimate Seismic Resistance Capacity for Long Span LatticeStructures under Vertical Ground Motions

Yoshiya Taniguchi

Osaka City University, Sugimoto-cho 3-3-138, Sumiyoshi-ku, Osaka 5588585, Japan

Correspondence should be addressed to Yoshiya Taniguchi; [email protected]

Received 26 March 2013; Accepted 15 August 2013

Academic Editor: Aurelio Araujo

Copyright © 2013 Yoshiya Taniguchi. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Seismic resistance capacities of frame structures have been discussed with equilibrium of energies among many researchers. Theearly one is the limit design presented by Housner, 1956; that is, frame structures should possess the plastic deformation abilityequivalent to an earthquake input energy given by a velocity response spectrum. On such studies of response estimation by theenergy equilibrium, the potential energy has been generally abandoned, since the effect of self-weight or fixed loads on the potentialenergy is negligible, while ordinary buildings usually sway in the horizontal direction. However, it could be said that the effect ofgravity has to be considered for long span structures since the mass might be concerned with the vertical response. In this paper, asfor ultimate seismic resistance capacity of long span structures, an estimation method considering the potential energy is discussedas for plane lattice beams and double-layer cylindrical lattice roofs. The method presented can be done with the information ofstatic nonlinear behavior, natural periods, and velocity response spectrum of seismic motions; that is, any complicated nonlineartime history analysis is not required. The value estimated can be modified with the properties of strain energy absorption and thesafety static factor.

1. Introduction

Long span and spatial structures have been utilized as a roofstructure of buildings including large space. They are oftenused as a place of refuge or stronghold of rescue in a disasterarea. Then it is important for government or caretaker tograsp ultimate seismic resistance capacity of such buildingswithout regard to new or existing buildings in advance. Theymight wish to know concretely the seismic motion level atwhich structures reach a limit state if it would be subjected toover design loads. The information would be just an ultimateseismic resistance capacity of structures.

Seismic resistant capacities for long span structures havebeen studied by many researchers all over the world. Amongthem early on, Kato et al. [1] studied the static and dynamicbehaviors of long span beams against vertical loads to expressthe quantitative earthquake resistant capacity in terms ofthe first natural period and the slenderness ratio of upperchord members. The selected measure was peak groundacceleration (PGA) at dynamic collapse. Ishikawa and Kato[2] studied the resistance capacity of double-layer latticedomes under static loading and vertical earthquake motions

to present an estimation method for PGA at collapse. Themethod was based on the results of static load-deflectioncurves to reach the deflection level below the dead load.Murata [3] numerically studied the maximum accelerationsof input earthquake motions leading to collapse for single-layer lattice domes with varying the static safety factors.Taniguchi et al. [4] also carried out time history responseanalyses for double-layer cylindrical lattice roofs to estimatethe maximum acceleration of an input wave (PGA) at thecollapse recognized by a sudden increase of nodal displace-ments and presented a prediction method of initial yieldand dynamic collapse accelerations with the limit state loadand response spectrum. Kumagai et al. [5] investigated thestatic and dynamic buckling behavior of double-layer latticedomes with various mesh patterns to compare the predictionaccuracy with the modified Dunkerley formulation.

The seismic response of structures has been analyzedby many researchers in the past using methods of energyequilibrium instead of a time history analysis. Among them,the limit design presented by Housner [6] is found as anearly one. The method was to design the structure so thatit could plastically absorb energy equal to the earthquake

2 Journal of Structures

3,00

0

27,000

Figure 1: Plane lattice beam of X type (X).

2,59

8

24,000

Figure 2: Plane lattice beam of Warren type (W).

input energy estimated by a velocity response spectrum. Katoand Akiyama [7] defined the energy absorption associatedto plastic deformations as the energy that contributed to thedevelopment of structural damage. They carried out numeri-cal studies with a 5-mass model for many cases, to confirmvalidity of the limit design. As for the estimation methodwith such energy index with respect to spatial structures,Tada et al. [8] introduced gravity energy, defined by theproduct of the self-weight and vertical displacements, intothe input energy as a collapse index for double-layer grids.It was shown that the double-layer grid began to collapsewhen the earthquake energy input to the grid exceeded acertain amount. Qiao et al. [9] investigated the dynamiccollapse behavior of a single-layer shallow lattice dome tomake clear the relationships between themaximum absorbedenergies and the vibration modes and pointed out that themaximum absorbed energies would change corresponding tovibration modes. As a further study of estimation methodfor dynamic collapse level of seismic motions, Taniguchi[10] treated plane lattice arches and double-layer cylindricallattice roofs and defined a limit state load and a limitstate deformation representing an ultimate state, given bythe information of static nonlinear behavior under verticalloading. An estimationmethod of ultimate seismic resistancecapacity was presented with the static absorbed energy untilan initial yield state and ultimate state, which is a kind of anextrapolation method. The method includes a modificationto improve the accuracy, considering the properties of elasticand plastic strain energies of structures during a pushoveranalysis until an ultimate state. However, themethod involvesa retrogression equation which includes an unknown quan-tity. Then in this paper, the effect of static safety factors isinvestigated to make clear the meanings of the unknownquantity in themodification equation, for lattice beams of twotypes, plane lattice arches, and double-layer cylindrical roofsdescribed in [10], to establish a consistent estimation methodof the ultimate seismic resistance capacity.

2. Numerical Model

Numerical models are shown in Figures 1 and 2. They aresupported at the side ends, by roller and pinsupports. Themodels consist of two member types; all the members havethe same section properties denoted as small letter a, and the 3

𝛿𝛿LEO

P

PGY

PLE

PDL

𝛼 · 𝛿LE

PGY(= 𝛽 · PDL )

PGY= avg.(max, min)

Figure 3: Limit state load and limit state deformation.

00

800600400200 1000

Tota

l loa

ds (k

N)

Vertical displacements (mm)

XaXb

WaWb

100

200

300

400

Figure 4: Load-deformation curves of models.

center top chordmembers are larger than the others, denotedas small letter b. All nodes are assumed to be rigid jointedsince the joints may have sufficient strength and stiffness.Thestatic safety factor ], that represents the ratio of initial yieldload against the dead load including the self-weight, is treatedas a numerical parameter ] = 2, 3, 4. The section propertiesof models are shown in Table 1.

3. Analysis of the Nonlinear Behavior

Nonlinear static analyses were carried out to grasp thenonlinear behavior of models, under vertical distributedloads, which were nodal loads corresponding to the covered,area. In the static analysis the energy equilibrium is expressedas follows:

𝐸

𝑒

− 𝐸

𝐺

= 𝐸

𝐹

, (1)

where 𝐸𝑒 is the strain energy and 𝐸𝐺 is the potential energyperformed by the product of the self-weight and verticaldisplacements. 𝐸𝐹 is the energy done by the external loads.𝐸

𝑒 consists of elastic strain energy𝑠𝑊𝑒and the dissipation

Journal of Structures 3

100

200

300

400

500

Equi

vale

nt v

eloci

ty o

f ene

rgy

(cm

/s)

00

200 400 600 800 1000Vertical displacements (mm)

(a) Xa4

100

200

300

400

500

Equi

vale

nt v

eloci

ty o

f ene

rgy

(cm

/s)

00


(b) Xb4

100

200

300

400

500

Equi

vale

nt v

eloci

ty o

f ene

rgy

(cm

/s)

00


sVe

sVF

sVG

(c) Wa4

100

200

300

400

500

Equi

vale

nt v

eloci

ty o

f ene

rgy

(cm

/s)

00


sVe

sVF

sVG

(d) Wb4

Figure 5: Equivalent velocities of energy and center vertical deformations.

energy𝑠𝑊𝑝done by plastic deformations. Each energy is

expressed as an equivalent velocity as follows:

𝑠𝑉

𝑒

=√

2𝐸

𝑒

𝑀

,𝑠𝑉

𝐺

=√

2𝐸

𝐺

𝑀

,𝑠𝑉

𝐹

=√

2𝐸

𝐹

𝑀

,

(2)

where 𝑀 is the total mass of each model. The formersubscript 𝑠 denotes the static analysis. In this paper,

𝑠𝑉

𝐹 isdefined as static absorbed energy, and the maximum valueof𝑠𝑉

𝐹 is considered as the maximum energy input to thestructure. The equivalent velocity of strain energy 𝐸𝑒 at themaximum

𝑠𝑉

𝐹 is denoted as𝑠𝑉𝑓. Further the equivalent

velocities of strain energy at the elastic limit load 𝑃LE andthe limit state load 𝑃GY are denoted as

𝑠𝑉LE and

𝑠𝑉GY,


Table 1: Section properties of models.

Model Safety factor ] Dead load Section size Section area Moment inertia𝑃DL (kN) 𝜑 × 𝑡 (mm) 𝐴 (cm2) 𝐼 (cm4)

Xa2 155.22

89.1 × 4.5 11.96 1073 103.484 77.61

Xb2 155.17

89.1 × 4.5

114.3×4.5

11.9615.52

1072343 103.45

4 77.59

Wa2 135.91

89.1 × 4.5 11.96 1073 90.614 67.95

Wb2 135.38

89.1 × 4.5

114.3×4.5

11.9615.52

1072343 90.25

4 67.69Young’s modulus 𝐸 (N/mm2) 205,000Yield stress 𝜎

𝑦(N/mm2) 300

0.01 0.1 1 10Natural period (s)

0

20

40

60

Velo

city

resp

onse

(cm

/s)

Figure 6: Velocity response spectrum of BCJ-L2.

respectively. The limit state load, as shown in Figure 3, is theload bearing capacity at an ultimate state after peak.The limitstate deformation corresponding to the limit state load 𝑃GYis represented by the limit state deformation factor 𝛼 and theelastic limit deformation 𝛿LE. It should be noted that 𝑃LE maybe defined as another phenomenon, that is, elastic buckling.

The load-deformation curves of plane lattice beams areshown in Figure 4.The horizontal axis represents the verticaldisplacements of center bottom node. The results of Xb andWb do not show any reduction since they are yielded intensile axial loads. The results of Xa and Wa show somereduction because of compressive member failure. Xa modelshows relatively gentle reduction than Wa since it has bothtensile and compressive member failures. The relationshipsbetween three energies and vertical deformations of eachmodel are shown in Figure 5.ThemodelWa4 shows the peakof𝑠𝑉

𝐹, and the other models do not show any peak in thepresent work.

The equivalent velocities of strain energy are listed inTable 2. The values

𝑠𝑉𝑓of Xa, Xb, and Wb are given by the

condition of tensile strain 3%, since it corresponds to aboutthe value of 𝛼 = 5 in the present work and may be in thestrain hardening region for usual steel materials. The values𝑠𝑉GY are estimated at the two factors 𝛼 = 3 and 6.The values𝑠𝑊𝑒/𝑠𝑊𝑝represent the ratio of the elastic strain energy

𝑠𝑊𝑒

and the plastic strain energy𝑠𝑊𝑝at𝑠𝑉GY.

4. Analysis of the Dynamic Properties

The results of free vibration analyses are shown in Table 3.The top 3 of effective mass ratios are shown in each table.The natural periods are almost equal to each other since thestiffness of models is almost equal as shown in Figure 4.

5. Time History Analysis

The dynamic elastoplastic behaviors are estimated by thegeometrical andmaterial nonlinear analysis [10, 11].The inputseismic waves are artificial waves, The building center ofjapan (BCJ) level 2 and the two sin waves of the 1st naturalperiods and the 110% of 1st ones. They are denoted as BCJ-L2, SIN, and SIN10, respectively. The acceleration data from0 to 60 seconds of BCJ-L2 are adopted.The velocity responsespectrum at 2% damping ratio is shown in Figure 6. Thesinusoidal waves are 20 seconds including the period of 4second amplification. The sinusoidal wave SIN10 is adoptedto study the effect of lengthening natural periods by structuralplasticization. Consequently, the effect was not confirmed inthe present work.

The relationships between maximum input accelerationsand maximum vertical displacements are shown in Figure 7.The tensile yield model b shows larger values than thecompressive yield model a. The compressive yield models,


Table 2: Equivalent velocities of strain energy and𝑠𝑊𝑒/𝑠𝑊𝑝.

Model Safety factor ]𝑠𝑉LE (cm/sec)

𝑠𝑉𝑓(cm/sec) 𝑠

𝑉GY (cm/sec)𝑠𝑊𝑒/𝑠𝑊𝑝

𝛼 = 3.0 𝛼 = 6.0 𝛼 = 3.0 𝛼 = 6.0

Xa2 99.15 236.98 174.27 244.24

0.21 0.153 121.43 290.25 213.44 299.144 140.26 335.21 246.51 345.52

Xb2 95.99 291.38 201.89 297.00

0.30 0.123 117.37 356.72 247.06 363.334 135.46 411.81 285.05 419.34

Wa2 103.30 103.30 118.69 143.94

0.06 0.043 126.41 126.41 145.33 176.234 146.04 146.04 167.79 203.49

Wb2 100.01 310.82 212.54 312.88

0.29 0.123 122.27 380.71 260.12 382.954 141.10 439.55 300.28 442.03

Table 3: Natural vibration property.

(a) Xa

Mode no. Natural period (sec) Effective mass ratio (%) Order] = 2 ] = 3 ] = 4 𝑋 direction 𝑍 direction 𝑋 𝑍

1 0.367 0.300 0.260 6.06 77.74 — 12 0.133 0.108 0.094 71.82 2.10 1 —3 0.095 0.077 0.067 12.68 0.04 2 —4 0.054 0.044 0.038 0.03 4.46 — 35 0.045 0.037 0.032 7.47 7.75 3 2

(b) Xb


1 0.356 0.290 0.252 6.86 77.42 3 12 0.131 0.107 0.092 70.50 2.70 1 —3 0.093 0.076 0.066 14.09 0.13 2 —4 0.054 0.044 0.038 0.11 3.97 — 35 0.044 0.036 0.031 6.63 7.98 — 2

(c) Wa


1 0.361 0.295 0.256 4.55 88.91 — 12 0.128 0.104 0.090 72.98 1.60 1 —3 0.100 0.082 0.071 15.59 0.27 2 —4 0.057 0.047 0.041 0.41 5.73 — 25 0.045 0.036 0.032 5.05 2.13 3 3

(d) Wb


1 0.350 0.286 0.248 5.29 88.68 3 12 0.126 0.103 0.089 71.34 2.13 1 33 0.099 0.081 0.070 17.76 0.48 2 —4 0.057 0.047 0.040 0.21 5.24 — 2


400

800

1200

1600

2000

Max

. inp

ut ac

cele

ratio

n (g

al)

00

200 400 600 800 1000Max. vertical displacements (mm)

Xa2Xa3Xa4

Xb2Xb3Xb4

(a) Model X

400

800

1200

1600

2000

Max

. inp

ut ac

cele

ratio

n (g

al)

00

200 400 600 800 1000Max. vertical displacements (mm)

Wa2Wa3Wa4

Wb2Wb3Wb4

(b) Model W

Figure 7: Maximum input acceleration and maximum vertical displacement (BCJ-L2).

Table 4: Rise amount 𝑏 and𝑠𝑊𝑒/𝑠𝑊𝑝.

Model Safety Factor ]𝛼 = 3.0 𝛼 = 6.0

Rise amount b from 𝑦 = 𝑥𝑠𝑊𝑒/𝑠𝑊𝑝

Rise amount 𝑏 from 𝑦 = 𝑥𝑠𝑊𝑒/𝑠𝑊𝑝BCJ-L2 SIN SIN10 BCJ-L2 SIN SIN10

Xa2 0.05 0.02 0.01 0.21 0.08 0.03 0.02 0.153 0.07 0.04 0.06 0.21 0.11 0.07 0.06 0.154 0.57 0.28 0.25 0.21 0.46 0.41 0.47 0.15

Xb2 0.01 −0.02 −0.07 0.30 0.02 −0.01 −0.10 0.123 0.00 −0.01 −0.03 0.30 0.02 0.00 −0.03 0.124 0.02 0.03 −0.01 0.30 0.05 0.03 −0.01 0.12

Wa2 0.05 0.02 0.02 0.06 0.06 0.02 0.03 0.043 0.07 0.08 0.03 0.06 0.07 0.09 0.04 0.044 0.06 0.05 0.07 0.06 0.07 0.09 0.07 0.04

Wb2 0.05 0.04 0.01 0.29 −0.13 −0.14 −0.30 0.123 0.04 0.04 0.06 0.29 −0.04 −0.05 −0.14 0.124 0.04 0.05 0.02 0.29 0.05 0.08 −0.11 0.12

especially models Wa, show dynamic collapse phenomenonrepresenting a sudden increase of displacements.

The relationships of the strain energy and potentialenergy are shown in Figure 8. In the figure, the curves givenby the static pushover analyses are also drawn as gray colorlines.The black trianglemarks represent the initial yield pointin the static analyses. The curves by time history analysesalmost coincide with the static curves until reaching theinitial yield point. After the initial yield, the time historyresponses are above the static results as for models Xa and

Wa showing compressive failure. The two results are not sodifferent for Xb and Wb showing tensile failure.

6. Effect of Static Safety Factors on UltimateSeismic Resistance Capacity

The ratio 𝑉GY/𝑉LE given by the time history analyses is com-pared with the ratio

𝑠𝑉GY/ 𝑠𝑉LE given by the static analyses,

as shown in Figure 9. The relationships of both ratios might


00

100 200 300 400VG (cm/s)

150

300

450

600Ve

(cm

/s)

(a) Xa4

00

100 200 300 400VG (cm/s)

150

300

450

600

Ve

(cm

/s)

(b) Xb4

50

100

150

200

250

Ve

(cm

/s)

00

50 100 150 200 250VG (cm/s)

BCJ-L2SINSIN10

(c) Wa4

100

200

300

400

500Ve

(cm

/s)

00

50 100 150 200 250

VG (cm/s)

BCJ-L2SINSIN10

(d) Wb4

Figure 8: Relationships between 𝑉𝑒 and 𝑉𝐺.

be on the diagonal line 𝑦 = 𝑥, if the dynamic effect wouldbe negligible. However, the model Xa shows the rise from thediagonal line 𝑦 = 𝑥, and some dynamic effect is confirmed.The rise amount and the ratio

𝑠𝑊𝑒/𝑠𝑊𝑝are listed in Table 4.

Although the models Xa2 and Wa show clearly dynamiccollapse, the limit state deformation determined by the factor𝛼 was adopted in order to compare with each other. The riseamount b becomes larger as the safety factor ] is larger, forcompressive yield models Xa and Wa. It may be due to the

reason that the hysteresis dissipation energy becomes largeras the dead load is smaller. The rise amount 𝑏 is small as fortensile yield models Xb and Wb, regardless of any seismicwave and safety factor ].

The differences between the strain energy at dynamicbehavior and static behavior are shown in Figure 10, to studythe relationships of the rise amount 𝑏 and components ofstrain energy. The data treated is at the limit state defor-mations. In the vertical axis, Δ represents the difference


BCJ-L2SIN

SIN10

0

1

2

3

4

0 1 2 3 4

VG

Y/V

LE

y = x

sVGY/sVLE

Xa2 (𝛼 = 3)Xa3 (𝛼 = 3)Xa4 (𝛼 = 3)

Xa2 (𝛼 = 6)Xa3 (𝛼 = 6)Xa4 (𝛼 = 6)

(a) Xa

BCJ-L2SIN

SIN10

0

1

2

3

4

VG

Y/V

LE

y = x

0 1 2 3 4sVGY/sVLE

Xb2 (𝛼 = 3)Xb3 (𝛼 = 3)Xb4 (𝛼 = 3)

Xb2 (𝛼 = 6)Xb3 (𝛼 = 6)Xb4 (𝛼 = 6)

(b) Xb

BCJ-L2SIN

SIN10

0

1

2

3

4

VG

Y/V

LE

y = x

0 1 2 3 4sVGY/sVLE

Wa2 (𝛼 = 3)Wa3 (𝛼 = 3)Wa4 (𝛼 = 3)

Wa2 (𝛼 = 6)Wa3 (𝛼 = 6)Wa4 (𝛼 = 6)

(c) Wa

BCJ-L2SIN

SIN10

0

1

2

3

4

VG

Y/V

LE

y = x

0 1 2 3 4sVGY/sVLE

Wb2 (𝛼 = 3)Wb3 (𝛼 = 3)Wb4 (𝛼 = 3)

Wb2 (𝛼 = 6)Wb3 (𝛼 = 6)Wb4 (𝛼 = 6)

(d) Wb

Figure 9: Relationship between𝑠𝑉GY/ 𝑠𝑉LE and 𝑉GY/𝑉LE.


−0.4 −0.2 0 0.2 0.4 0.6 0.8−0.4

−0.2

0

0.2

0.4

0.6

0.8

b

ΔEe/ sEe

(a) 𝑏 − Δ𝐸𝑒/𝑠𝐸𝑒

−0.4 −0.2 0 0.2 0.4 0.6b

−1

−0.5

0

0.5

1

1.5

ΔW

e/ sW

e(b) 𝑏 − Δ𝑊

𝑒/𝑠𝑊𝑒

−0.4 −0.2 0 0.2 0.4 0.6 0.8b

−0.2

0

0.2

0.4

0.6

0.8

1

ΔW

p/ sW

p

(c) 𝑏 − Δ𝑊𝑝/𝑠𝑊𝑝

−0.4 −0.2 0 0.2 0.4 0.6 0.8ΔEe/sE

e

−0.2

0

0.2

0.4

0.6

0.8

1

ΔW

p/ sW

p

(d) Δ𝐸𝑒/𝑠𝐸𝑒

− Δ𝑊𝑝/𝑠𝑊𝑝

Figure 10: Rise amount 𝑏 and Δ𝐸𝑒/𝑠𝐸

𝑒, and Δ𝑊𝑒/𝑠𝑊𝑒, Δ𝑊𝑝/𝑠𝑊𝑝.

between the dynamic results and static ones. In the figures,the interrelation is confirmed for total strain energy ratioΔ𝐸

𝑒

/𝑠𝐸

𝑒 (Figure 10(a)) and plastic energy ratio Δ𝑊𝑝/𝑠𝑊𝑝

(Figure 10(c)), and any interrelation is not confirmed forelastic strain energy ratio Δ𝑊

𝑒/𝑠𝑊𝑒(Figure 10(b)), against

rise amount 𝑏. Since some interrelation is confirmed betweenΔ𝐸

𝑒

/𝑠𝐸

𝑒 and Δ𝑊𝑝/𝑠𝑊𝑝(Figure 10(d)), the increase of strain

energy at dynamic behavior is due to the dissipation energyby plastic deformations.

Then the relationships between Δ𝑊𝑝/𝑠𝑊𝑝and the rise

amount 𝑏 are illustrated for each model and safety factor, asshown in Figure 11.

In Figure 11, the tensile yield models Xb and Wb aredistributed in the small range of two axes. However, thecompressive yield models Xa and Wa are widely distributedin the positive range of horizontal axis.The factmay be due tothe plastic dissipation energy by yield hinges in compressivemembers. As the safety factors are larger, they are distributed


00

0.2 0.4 0.6 0.8b

0.25

0.5

0.75

1ΔW

p/ sW

p

(a) XaΔW

p/ sW

p

−0.12 −0.08 −0.04 0 0.04 0.08b

−0.04

−0.02

0

0.02

0.04

0.06

(b) Xb

ΔW

p/ sW

p

0 0.025 0.05 0.075 0.1

b

−0.05

0

0.05

0.1

0.15

n = 2

n = 3

n = 4

(c) Wa

ΔW

p/ sW

p

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2b

n = 2

n = 3

n = 4

(d) Wa

Figure 11: Rise amount 𝑏 − Δ𝑊𝑝/𝑠𝑊𝑝.

in the right and upper range of the figures. It should benoted that any interrelation was not confirmed between thedissipation energy of damping and rise amount 𝑏.

7. Estimation Method of UltimateSeismic Capacity

The previous results of [10] are combined with the presentwork to investigate the effect of safety factor ] on the rise

amount 𝑏. The previous results are listed in Table 5 for latticearch and double-layer cylindrical lattice roof as shown inFigure 12. The letter P denotes both pin supports, and PRdenotes pin supports and roller supports. The number 1represents all member sections being equal, and number2 represents members consisting of several section prop-erties. The same relationships are confirmed between therise amount 𝑏 and member yield type. As for model PR2,although the static result shows tensile yield, the dynamic


Table 5: Previous results of lattice arch and double layer cylindrical lattice roof [10].

Model Safetyfactor ] 𝑠

𝑉GY/𝑠𝑉LE𝑉GY/𝑉LE Rise amount 𝑏 from𝑦 = 𝑥

𝑠𝑊𝑒/𝑠𝑊𝑝𝛼

BCJ-L2 KOBE TAFT BCJ-L2 KOBE TAFTPlane lattice arch

P1 9.68 1.74 1.82 1.89 — 0.08 0.15 — 0.01 3.01.99 2.80 2.64 — 0.80 0.65 — 0.01 6.0

P2 10.51 1.29 1.48 1.55 — 0.19 0.27 — 0.07 3.01.58 2.01 1.78 — 0.44 0.21 — 0.02 6.0

PR1 5.17 1.28 1.33 1.47 — 0.06 0.19 — 0.03 3.01.58 1.61 1.73 — 0.04 0.15 — 0.01 6.0

PR2 3.56 2.08 2.07 2.71 — −0.01 0.63 — 0.29 3.03.05 2.93 3.85 — −0.12 0.81 — 0.14 6.0

Double-layer lattice roof

SSR1 6.34 1.51 2.61 2.46 2.49 1.10 0.95 0.98 0.13 3.01.97 3.08 2.65 3.36 1.11 0.68 1.39 0.05 6.0

SSR2 6.99 1.42 2.65 2.21 2.11 1.23 0.79 0.70 0.10 3.01.82 2.65 2.42 2.29 0.83 0.60 0.47 0.04 6.0

X

Y

Z

30,0

00

26,540

(a)

X

Z

Y

5,64

0

2,12

0

30∘

(b)

Figure 12: Double-layer cylindrical lattice roof [10].

behavior includes compressive member yield to increase therise amount 𝑏 than model PR1.

The total results of compressive yield models are plot-ted in Figure 13, according to (3) [10]. The horizontal axisrepresents the rise amount 𝑏 that means the increase ratioof dynamic results against static results. The vertical axis

1.5

00

0.3 0.6 0.9 1.2 1.5b

0.3

0.6

0.9

1.2

k = 10.97

k = 7.17

k = 1.84

k = 0.51

k = 0.33k = 0.32k = 0.11

�=2.00

�=3.00

�=4.00

� =5.1

7

� =6.34

� = 6.99

� = 9.67

𝛼· sW

e/ sW

p

Figure 13: Relationships between rise amount 𝑏 and 𝛼 ⋅𝑠𝑊𝑒/𝑠𝑊𝑝.

represents the strain absorption property of structures at alimit state deformation. Consider

𝛼 ⋅

𝑠𝑊𝑒

𝑠

𝑊𝑝

= 𝑘 ⋅ 𝑏. (3)


00

20 40 60 80 100

(Safety factor �)2

2

4

6

8

10

1/k

y = 0.086x

R2 = 0.94

Figure 14: Relationships between safety factor ] and 1/𝑘.

In Figure 13, the data of Figure 9 and Tables 4 and 5 is plotted,and the safety factor ] and the slope 𝑘 in (3) are shown. Thelarger the safety factor ] is, the smaller slope 𝑘 becomes. Itshows that the large safety factors enlarge the rise amounts,because more dissipation energy by cyclic deformations isoccurrs until a limit state, and consequently the rise amount𝑏 becomes large.

In order to study the value of slope 𝑘, the relationshipsbetween slope 𝑘 and safety factor ] are drawn in Figure 14.The slope 𝑘 can be estimated with the safety factor since thecorrelation coefficient is large.

Consequently the ultimate seismic capacity can be accu-rately estimated with the information of

𝑠𝑊𝑒/𝑠𝑊𝑝given by

a nonlinear static analysis and the limit state deformationfactor 𝛼 decided by a designer. The value estimated is finallymodified by the static safety factor ].

The flow chart of the estimation method presented isshown as follows, (Figure 15).

Step 1. The static elastoplastic behavior is estimated underthe vertical loads corresponding to the distribution of mass,until the static absorbed energy of (2) shows maximumvalue or the limit state deformations are reached. The elasticcomponent of strain energy

𝑠𝑊𝑒and the plastic dissipation

energy𝑠𝑊𝑝are calculated at the limit state.

Step 2. The seismic motion level at which structures becomein initial yield can be estimated with the equivalent velocity𝑠𝑉LE and the velocity response spectrum of seismic waves.The equivalent velocity

𝑠𝑉LE is determined at initial yield by

the nonlinear static analysis. If the natural mode of the largesteffective mass ratio would be adopted, the value estimatedmight be in the safety region [10].

Step 3. The seismic motion level at which structures reachthe limit state deformation can be estimated with the value𝑠𝑉GY/ 𝑠𝑉LE. The seismic motion level obtained at Step 2 maybe multiplied by this value to obtain the seismic motion levelcorresponding to the limit state deformation. If the value

𝑠𝑉𝑓

is adopted instead of𝑠𝑉GY, that of dynamic collapse could be

obtained.

Step 4. The value obtained at Step 3 could be modified by therise amount 𝑏 that could be given by (3) and Figure 13. Themodification with the rise amount 𝑏 is not necessary in thecase that structures would reach a limit state deformation bytensile member yield.

8. Conclusions

Themain conclusions in the presentwork are listed as follows.

(1) The equivalent velocities 𝑉𝑓, 𝑉GY of strain energy at

which structures reach dynamic collapse or a limitstate deformation could be accurately estimated withthe static safety factor being the ratio of initial yieldload against dead load.

(2) The increase of𝑉𝑓,𝑉GY, at the case that structures are

subjected to the seismic motion level correspondingto dynamic collapse or a limit state deformation, isdue to the plastic dissipation energy.The effect is smallat the conditions that the static safety factor V is smallor structures are in tensile yield.

(3) The ultimate seismic capacity can be estimated byFigure 15 without any time history analysis.


(Judgment of modification) In the static analysis, is the structure yielded

in tensile members?

Is the analysis needed until the max. of static absorbed energy?

End

Static pushover analysis

Yes No

Yes

Modification is not needed

No

Estimation of sVLE at the initial yield

Estimation of sVf

at the max. of static absorbed energy

Estimation of sVGY

at the limit state deformation 𝛼𝛿LE

Estimation of elastic strain energy sWe and plastic energy sWp

Estimation of seismic motion level at initial yieldwith sVLE and velocity response spectrum

Estimation of seismic motion levelat dynamic collapse or limit state deformation

by considering the proportion of sVf/sVLE or sVGY/sVLE and Vf/VLE

Modification with the following equation and

Safety factor �, limit state deformation factor 𝛼 , and sWe/sWp

I

II

III

IV

Figure 13: 𝛼 · (sWe/sWp) = k · b [1/k ∝ �2]

Figure 15: Flow chart of estimation method for ultimate seismic capacity.

Acknowledgments

The author gives special thanks to Ms. Risa Fukushima andMs. Yuki Kadotsuka for their numerical works. This work ispartially supported by project research 2010 of the GraduateSchool of Engineering, Osaka City University, and JSPSKAKENHI Grant 24656325, Grants-in-Aid for ExploratoryResearch, Japan.

References

[1] S. Kato, K. Ishikawa, and Y. Yokoo, “Earthquake resistantcapacity of long span trusses structures a study on trsussedbeam due to vertical earthquake motions,” Journal of Structuraland Construction Engineering, no. 360, pp. 64–74, 1986.

[2] K. Ishikawa and S. Kato, “Earthquake resistant capacity andcollapse mechanism of dynamic buckling on double layerlatticed domes under vertical motions,” in Proceedings of theSEIKEN-IASS Symposium on Nonlinear Analysis and Design forShell and Spatial Structures, pp. 569–576, 1993.

[3] M. Murata, “Dynamic characteristics of single layer reticu-lar domes subjected to vertical and horizontal earthquakemotions,” Journal of Structural and Construction Engineering,no. 571, pp. 103–110, 2003.

[4] Y. Taniguchi,M.Kurano, F. Zhang, andT. Saka, “Limit state loadand dynamic collapse estimation for double-layer cylindricallatticed roofs,” in Proceedings of the IASS International Sympo-sium New Olympics New Shell and Spatial Structures, p. DR13,2006, Extended Abstracts and CD-ROM of IASS-APCS.


[5] T. Kumagai, Y. Taniguchi, T. Ogawa, and M. Masuyama,“Static and dynamic buckling behavior of double-layer latticeddomes with various mesh patterns,” in Proceedings of the IASSInternational Symposium New Olympics New Shell and SpatialStructures, p. BK10, 2006, Extended Abstracts and CD-ROM ofIASS-APCS.

[6] G. W. Housner, “Limit design of structures to resist earth-quakes,” in Proceedings of the World Conference on EarthquakeEngineering, pp. 5-1–5-13, Berkley, Calif, USA, 1956.

[7] B. Kato and H. Akiyama, “Energy input and damages instructures subjected to sever earthquakes,” Transactions ofArchitectural Institute of Japan, no. 235, pp. 9–18, 1975.

[8] M. Tada, M. Hayashi, and T. Yoneyama, “An improvement ofseismic capacity of double-layer space trusses using force limit-ing devices,” in Proceedings of the IASS International Symposiumon Spatial Structures: Heritage, Present and Future, vol. 2, pp.1085–1092, Milan, Italy, 1995.

[9] F. Qiao, N. Hagiwara, and T. Matsui, “On the relation betweenabsorbed energy and dynamic collapse of a single-layer shallowlatticed domes,” Journal of Structural and Construction Engi-neering, no. 531, pp. 117–124, 2000.

[10] Y. Taniguchi, “Seismic motion level of dynamic collapse orlimit state deformation for lattice arch and cylindrical roof,” inProceedings of the Structural Engineers World Congress, p. 176,Como, Italy, 2011, Abstract Book and CD-ROM.

[11] M. Murata, SPACE, Meijo University, Nagoya, Japan, http://wwwra.meijo-u.ac.jp/labs/ra007/space/index.htm.

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