Seismic resistance capacities of frame structures have been discussed with equilibrium of energies among many researchers. The early one is the limit design presented by Housner, 1956; that is, frame structures should possess the plastic deformation ability equivalent to an earthquake input energy given by a velocity response spectrum. On such studies of response estimation by the energy equilibrium, the potential energy has been generally abandoned, since the effect of self-weight or fixed loads on the potential energy is negligible, while ordinary buildings usually sway in the horizontal direction. However, it could be said that the effect of gravity has to be considered for long span structures since the mass might be concerned with the vertical response. In this paper, as for ultimate seismic resistance capacity of long span structures, an estimation method considering the potential energy is discussed as for plane lattice beams and double-layer cylindrical lattice roofs. The method presented can be done with the information of static nonlinear behavior, natural periods, and velocity response spectrum of seismic motions; that is, any complicated nonlinear time history analysis is not required. The value estimated can be modified with the properties of strain energy absorption and the safety static factor.
1. Introduction
Long span and spatial structures have been utilized as a roof structure of buildings including large space. They are often used as a place of refuge or stronghold of rescue in a disaster area. Then it is important for government or caretaker to grasp ultimate seismic resistance capacity of such buildings without regard to new or existing buildings in advance. They might wish to know concretely the seismic motion level at which structures reach a limit state if it would be subjected to over design loads. The information would be just an ultimate seismic resistance capacity of structures.
Seismic resistant capacities for long span structures have been studied by many researchers all over the world. Among them early on, Kato et al. [1] studied the static and dynamic behaviors of long span beams against vertical loads to express the quantitative earthquake resistant capacity in terms of the first natural period and the slenderness ratio of upper chord members. The selected measure was peak ground acceleration (PGA) at dynamic collapse. Ishikawa and Kato [2] studied the resistance capacity of double-layer lattice domes under static loading and vertical earthquake motions to present an estimation method for PGA at collapse. The method was based on the results of static load-deflection curves to reach the deflection level below the dead load. Murata [3] numerically studied the maximum accelerations of 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 response analyses for double-layer cylindrical lattice roofs to estimate the maximum acceleration of an input wave (PGA) at the collapse recognized by a sudden increase of nodal displacements and presented a prediction method of initial yield and dynamic collapse accelerations with the limit state load and response spectrum. Kumagai et al. [5] investigated the static and dynamic buckling behavior of double-layer lattice domes with various mesh patterns to compare the prediction accuracy with the modified Dunkerley formulation.
The seismic response of structures has been analyzed by many researchers in the past using methods of energy equilibrium instead of a time history analysis. Among them, the limit design presented by Housner [6] is found as an early one. The method was to design the structure so that it could plastically absorb energy equal to the earthquake input energy estimated by a velocity response spectrum. Kato and Akiyama [7] defined the energy absorption associated to plastic deformations as the energy that contributed to the development of structural damage. They carried out numerical studies with a 5-mass model for many cases, to confirm validity of the limit design. As for the estimation method with such energy index with respect to spatial structures, Tada et al. [8] introduced gravity energy, defined by the product of the self-weight and vertical displacements, into the input energy as a collapse index for double-layer grids. It was shown that the double-layer grid began to collapse when the earthquake energy input to the grid exceeded a certain amount. Qiao et al. [9] investigated the dynamic collapse behavior of a single-layer shallow lattice dome to make clear the relationships between the maximum absorbed energies and the vibration modes and pointed out that the maximum absorbed energies would change corresponding to vibration modes. As a further study of estimation method for dynamic collapse level of seismic motions, Taniguchi [10] treated plane lattice arches and double-layer cylindrical lattice roofs and defined a limit state load and a limit state deformation representing an ultimate state, given by the information of static nonlinear behavior under vertical loading. An estimation method of ultimate seismic resistance capacity was presented with the static absorbed energy until an initial yield state and ultimate state, which is a kind of an extrapolation method. The method includes a modification to improve the accuracy, considering the properties of elastic and plastic strain energies of structures during a pushover analysis until an ultimate state. However, the method involves a retrogression equation which includes an unknown quantity. Then in this paper, the effect of static safety factors is investigated to make clear the meanings of the unknown quantity in the modification equation, for lattice beams of two types, plane lattice arches, and double-layer cylindrical roofs described in [10], to establish a consistent estimation method of the ultimate seismic resistance capacity.
2. Numerical Model
Numerical models are shown in Figures 1 and 2. They are supported at the side ends, by roller and pinsupports. The models consist of two member types; all the members have the same section properties denoted as small letter a, and the 3 center top chord members are larger than the others, denoted as small letter b. All nodes are assumed to be rigid jointed since the joints may have sufficient strength and stiffness. The static safety factor ν, that represents the ratio of initial yield load against the dead load including the self-weight, is treated as a numerical parameter ν=2,3,4. The section properties of models are shown in Table 1.
Section properties of models.
Model
Safety factor ν
Dead load
Section size
Section area
Moment inertia
PDL (kN)
φ×t (mm)
A (cm^{2})
I (cm^{4})
Xa
2
155.22
89.1×4.5
11.96
107
3
103.48
4
77.61
Xb
2
155.17
89.1×4.5114.3×4.5
11.96 15.52
107 234
3
103.45
4
77.59
Wa
2
135.91
89.1×4.5
11.96
107
3
90.61
4
67.95
Wb
2
135.38
89.1×4.5114.3×4.5
11.96 15.52
107 234
3
90.25
4
67.69
Young’s modulus E (N/mm^{2})
205,000
Yield stress σy (N/mm^{2})
300
Plane lattice beam of X type (X).
Plane lattice beam of Warren type (W).
3. Analysis of the Nonlinear Behavior
Nonlinear static analyses were carried out to grasp the nonlinear behavior of models, under vertical distributed loads, which were nodal loads corresponding to the covered, area. In the static analysis the energy equilibrium is expressed as follows:
(1)Ee-EG=EF,
where Ee is the strain energy and EG is the potential energy performed by the product of the self-weight and vertical displacements. EF is the energy done by the external loads. Ee consists of elastic strain energy Wse and the dissipation energy Wsp done by plastic deformations. Each energy is expressed as an equivalent velocity as follows:
(2)Ves=2EeM,VsG=2EGM,VsF=2EFM,
where M is the total mass of each model. The former subscript s denotes the static analysis. In this paper, VsF is defined as static absorbed energy, and the maximum value of VsF is considered as the maximum energy input to the structure. The equivalent velocity of strain energy Ee at the maximum VsF is denoted as Vsf. Further the equivalent velocities of strain energy at the elastic limit load PLE and the limit state load PGY are denoted as VsLE and VsGY, respectively. The limit state load, as shown in Figure 3, is the load bearing capacity at an ultimate state after peak. The limit state deformation corresponding to the limit state load PGY is represented by the limit state deformation factor α and the elastic limit deformation δLE. It should be noted that PLE may be defined as another phenomenon, that is, elastic buckling.
Limit state load and limit state deformation.
The load-deformation curves of plane lattice beams are shown in Figure 4. The horizontal axis represents the vertical displacements of center bottom node. The results of Xb and Wb do not show any reduction since they are yielded in tensile axial loads. The results of Xa and Wa show some reduction because of compressive member failure. Xa model shows relatively gentle reduction than Wa since it has both tensile and compressive member failures. The relationships between three energies and vertical deformations of each model are shown in Figure 5. The model Wa4 shows the peak of VsF, and the other models do not show any peak in the present work.
Load-deformation curves of models.
Equivalent velocities of energy and center vertical deformations.
Xa4
Xb4
Wa4
Wb4
The equivalent velocities of strain energy are listed in Table 2. The values Vsf of Xa, Xb, and Wb are given by the condition of tensile strain 3%, since it corresponds to about the value of α=5 in the present work and may be in the strain hardening region for usual steel materials. The values VsGY are estimated at the two factors α=3 and 6. The values Wse/Wsp represent the ratio of the elastic strain energy Wse and the plastic strain energy Wsp at VsGY.
Equivalent velocities of strain energy and sWe/sWp.
Model
Safety factor ν
sVLE (cm/sec)
sVf (cm/sec)
sVGY (cm/sec)
sWe/sWp
α=3.0
α=6.0
α=3.0
α=6.0
Xa
2
99.15
236.98
174.27
244.24
0.21
0.15
3
121.43
290.25
213.44
299.14
4
140.26
335.21
246.51
345.52
Xb
2
95.99
291.38
201.89
297.00
0.30
0.12
3
117.37
356.72
247.06
363.33
4
135.46
411.81
285.05
419.34
Wa
2
103.30
103.30
118.69
143.94
0.06
0.04
3
126.41
126.41
145.33
176.23
4
146.04
146.04
167.79
203.49
Wb
2
100.01
310.82
212.54
312.88
0.29
0.12
3
122.27
380.71
260.12
382.95
4
141.10
439.55
300.28
442.03
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 the stiffness of models is almost equal as shown in Figure 4.
Natural vibration property.
Xa
Mode no.
Natural period (sec)
Effective mass ratio (%)
Order
ν=2
ν=3
ν=4
X direction
Z direction
X
Z
1
0.367
0.300
0.260
6.06
77.74
—
1
2
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
—
3
5
0.045
0.037
0.032
7.47
7.75
3
2
Xb
Mode no.
Natural period (sec)
Effective mass ratio (%)
Order
ν=2
ν=3
ν=4
X direction
Z direction
X
Z
1
0.356
0.290
0.252
6.86
77.42
3
1
2
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
—
3
5
0.044
0.036
0.031
6.63
7.98
—
2
Wa
Mode no.
Natural period (sec)
Effective mass ratio (%)
Order
ν=2
ν=3
ν=4
X direction
Z direction
X
Z
1
0.361
0.295
0.256
4.55
88.91
—
1
2
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
—
2
5
0.045
0.036
0.032
5.05
2.13
3
3
Wb
Mode no.
Natural period (sec)
Effective mass ratio (%)
Order
ν=2
ν=3
ν=4
X direction
Z direction
X
Z
1
0.350
0.286
0.248
5.29
88.68
3
1
2
0.126
0.103
0.089
71.34
2.13
1
3
3
0.099
0.081
0.070
17.76
0.48
2
—
4
0.057
0.047
0.040
0.21
5.24
—
2
5. Time History Analysis
The dynamic elastoplastic behaviors are estimated by the geometrical and material nonlinear analysis [10, 11]. The input seismic waves are artificial waves, The building center of japan (BCJ) level 2 and the two sin waves of the 1st natural periods and the 110% of 1st ones. They are denoted as BCJ-L2, SIN, and SIN10, respectively. The acceleration data from 0 to 60 seconds of BCJ-L2 are adopted. The velocity response spectrum at 2% damping ratio is shown in Figure 6. The sinusoidal waves are 20 seconds including the period of 4 second amplification. The sinusoidal wave SIN10 is adopted to study the effect of lengthening natural periods by structural plasticization. Consequently, the effect was not confirmed in the present work.
Velocity response spectrum of BCJ-L2.
The relationships between maximum input accelerations and maximum vertical displacements are shown in Figure 7. The tensile yield model b shows larger values than the compressive yield model a. The compressive yield models, especially models Wa, show dynamic collapse phenomenon representing a sudden increase of displacements.
Maximum input acceleration and maximum vertical displacement (BCJ-L2).
Model X
Model W
The relationships of the strain energy and potential energy are shown in Figure 8. In the figure, the curves given by the static pushover analyses are also drawn as gray color lines. The black triangle marks represent the initial yield point in the static analyses. The curves by time history analyses almost coincide with the static curves until reaching the initial yield point. After the initial yield, the time history responses are above the static results as for models Xa and Wa showing compressive failure. The two results are not so different for Xb and Wb showing tensile failure.
Relationships between Ve and VG.
Xa4
Xb4
Wa4
Wb4
6. Effect of Static Safety Factors on Ultimate Seismic Resistance Capacity
The ratio VGY/VLE given by the time history analyses is compared with the ratio VsGY/VsLE given by the static analyses, as shown in Figure 9. The relationships of both ratios might be on the diagonal line y=x, if the dynamic effect would be negligible. However, the model Xa shows the rise from the diagonal line y=x, and some dynamic effect is confirmed. The rise amount and the ratio Wse/Wspare listed in Table 4. Although the models Xa2 and Wa show clearly dynamic collapse, the limit state deformation determined by the factor αwas adopted in order to compare with each other. The rise amount b becomes larger as the safety factor νis larger, for compressive yield models Xa and Wa. It may be due to the reason that the hysteresis dissipation energy becomes larger as the dead load is smaller. The rise amount b is small as for tensile yield models Xb and Wb, regardless of any seismic wave and safety factor ν.
Rise amount b andsWe/sWp.
Model
Safety Factor ν
α=3.0
α=6.0
Rise amount b from y=x
sWe/sWp
Rise amount b from y=x
sWe/sWp
BCJ-L2
SIN
SIN10
BCJ-L2
SIN
SIN10
Xa
2
0.05
0.02
0.01
0.21
0.08
0.03
0.02
0.15
3
0.07
0.04
0.06
0.21
0.11
0.07
0.06
0.15
4
0.57
0.28
0.25
0.21
0.46
0.41
0.47
0.15
Xb
2
0.01
−0.02
−0.07
0.30
0.02
−0.01
−0.10
0.12
3
0.00
−0.01
−0.03
0.30
0.02
0.00
−0.03
0.12
4
0.02
0.03
−0.01
0.30
0.05
0.03
−0.01
0.12
Wa
2
0.05
0.02
0.02
0.06
0.06
0.02
0.03
0.04
3
0.07
0.08
0.03
0.06
0.07
0.09
0.04
0.04
4
0.06
0.05
0.07
0.06
0.07
0.09
0.07
0.04
Wb
2
0.05
0.04
0.01
0.29
−0.13
−0.14
−0.30
0.12
3
0.04
0.04
0.06
0.29
−0.04
−0.05
−0.14
0.12
4
0.04
0.05
0.02
0.29
0.05
0.08
−0.11
0.12
Relationship between VsGY/VsLE and VGY/VLE.
Xa
Xb
Wa
Wb
The differences between the strain energy at dynamic behavior and static behavior are shown in Figure 10, to study the relationships of the rise amount b and components of strain energy. The data treated is at the limit state deformations. In the vertical axis, Δrepresents the difference between the dynamic results and static ones. In the figures, the interrelation is confirmed for total strain energy ratio ΔEe/Ees (Figure 10(a)) and plastic energy ratio ΔWp/Wsp (Figure 10(c)), and any interrelation is not confirmed for elastic strain energy ratio ΔWe/Wse (Figure 10(b)), against rise amount b. Since some interrelation is confirmed between ΔEe/Ees and ΔWp/Wsp (Figure 10(d)), the increase of strain energy at dynamic behavior is due to the dissipation energy by plastic deformations.
Rise amount band ΔEe/Ees, and ΔWe/Wes, ΔWp/Wsp.
b−ΔEe/Ees
b−ΔWe/Wes
b−ΔWp/Wsp
ΔEe/Ees− ΔWp/Wsp
Then the relationships between ΔWp/Wspand the rise amount b are illustrated for each model and safety factor, as shown in Figure 11.
Rise amount b−ΔWp/Wsp.
Xa
Xb
Wa
Wa
In Figure 11, the tensile yield models Xb and Wb are distributed in the small range of two axes. However, the compressive yield models Xa and Wa are widely distributed in the positive range of horizontal axis. The fact may be due to the plastic dissipation energy by yield hinges in compressive members. As the safety factors are larger, they are distributed in the right and upper range of the figures. It should be noted that any interrelation was not confirmed between the dissipation energy of damping and rise amount b.
7. Estimation Method of Ultimate Seismic Capacity
The previous results of [10] are combined with the present work to investigate the effect of safety factor ν on the rise amount b. The previous results are listed in Table 5 for lattice arch and double-layer cylindrical lattice roof as shown in Figure 12. The letter P denotes both pin supports, and PR denotes pin supports and roller supports. The number 1 represents all member sections being equal, and number 2 represents members consisting of several section properties. The same relationships are confirmed between the rise amount b and member yield type. As for model PR2, although the static result shows tensile yield, the dynamic behavior includes compressive member yield to increase the rise amount b than model PR1.
Previous results of lattice arch and double layer cylindrical lattice roof [10].
Model
Safety factor ν
sVGY/sVLE
VGY/VLE
Rise amount b from y=x
sWe/sWp
α
BCJ-L2
KOBE
TAFT
BCJ-L2
KOBE
TAFT
Plane lattice arch
P1
9.68
1.74
1.82
1.89
—
0.08
0.15
—
0.01
3.0
1.99
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
3.0
1.58
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
3.0
1.58
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
3.0
3.05
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
3.0
1.97
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
3.0
1.82
2.65
2.42
2.29
0.83
0.60
0.47
0.04
6.0
Double-layer cylindrical lattice roof [10].
The total results of compressive yield models are plotted in Figure 13, according to (3) [10]. The horizontal axis represents the rise amount b that means the increase ratio of dynamic results against static results. The vertical axis represents the strain absorption property of structures at a limit state deformation. Consider
(3)α·WesWps=k·b.
In Figure 13, the data of Figure 9 and Tables 4 and 5 is plotted, and the safety factor ν and the slope k in (3) are shown. The larger the safety factor ν is, the smaller slope k becomes. It shows that the large safety factors enlarge the rise amounts, because more dissipation energy by cyclic deformations is occurrs until a limit state, and consequently the rise amount b becomes large.
Relationships between rise amount b and α·Wse /Wsp.
In order to study the value of slope k, the relationships between slope k and safety factor ν are drawn in Figure 14. The slope k can be estimated with the safety factor since the correlation coefficient is large.
Relationships between safety factor ν and 1/k.
Consequently the ultimate seismic capacity can be accurately estimated with the information of Wse/Wspgiven by a nonlinear static analysis and the limit state deformation factor α decided by a designer. The value estimated is finally modified by the static safety factor ν.
The flow chart of the estimation method presented is shown as follows, (Figure 15).
Flow chart of estimation method for ultimate seismic capacity.
Step 1.
The static elastoplastic behavior is estimated under the vertical loads corresponding to the distribution of mass, until the static absorbed energy of (2) shows maximum value or the limit state deformations are reached. The elastic component of strain energy Wes and the plastic dissipation energy Wsp are calculated at the limit state.
Step 2.
The seismic motion level at which structures become in initial yield can be estimated with the equivalent velocity VsLE and the velocity response spectrum of seismic waves. The equivalent velocity VsLE is determined at initial yield by the nonlinear static analysis. If the natural mode of the largest effective mass ratio would be adopted, the value estimated might be in the safety region [10].
Step 3.
The seismic motion level at which structures reach the limit state deformation can be estimated with the value VsGY/VsLE. The seismic motion level obtained at Step 2 may be multiplied by this value to obtain the seismic motion level corresponding to the limit state deformation. If the value Vsf is adopted instead ofVsGY, that of dynamic collapse could be obtained.
Step 4.
The value obtained at Step 3 could be modified by the rise amount b that could be given by (3) and Figure 13. The modification with the rise amount b is not necessary in the case that structures would reach a limit state deformation by tensile member yield.
8. Conclusions
The main conclusions in the present work are listed as follows.
The equivalent velocities Vf, VGY of strain energy at which structures reach dynamic collapse or a limit state deformation could be accurately estimated with the static safety factor being the ratio of initial yield load against dead load.
The increase of Vf, VGY, at the case that structures are subjected to the seismic motion level corresponding to dynamic collapse or a limit state deformation, is due to the plastic dissipation energy. The effect is small at the conditions that the static safety factor v is small or structures are in tensile yield.
The ultimate seismic capacity can be estimated by Figure 15 without any time history analysis.
Acknowledgments
The author gives special thanks to Ms. Risa Fukushima and Ms. Yuki Kadotsuka for their numerical works. This work is partially supported by project research 2010 of the Graduate School of Engineering, Osaka City University, and JSPS KAKENHI Grant 24656325, Grants-in-Aid for Exploratory Research, Japan.
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