bam report

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ECOLE POLYTECHNIQUE FEDERALEDE LAUSANNE

EPFL

INFORMATIQUE ET MECANIQUEAPPLIQUEES A LA CONSTRUCTION -

IMAC

A THREE DIMENSIONAL SEISMIC ANALYSIS OF

A MODULAR BUBBLE SYSTEM FOR HOUSING

IN BAM CITY IN IRAN

Authors:

* Dr. Belmouden Y., * Dr. Lestuzzi P., ** Dr. Sellami S.

[email protected] , [email protected]

*Ecole Polytechnique Fdrale de Lausanne (EPFL)

ENAC-IS-IMAC, EPFL, CH-1015, Lausanne, Switzerland

[email protected] ** Zurich, Switzerland

Date: 31/03/2005

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A three dimensional seismic analysis of a bubble system of housing 1

Contents

1. Introduction.2

2. Description of the Bubble system ..2

2.1. Geometrical characteristics of the Bubble system2

2.2. Reinforced concrete shell structures .6

2.2.1. Design of reinforced concrete shells structures ..6

2.2.2. Non linear method of RC shell structures reinforcements design ..6

3. Finite element model....7

4. Strong motion values resulting from earthquake activity at the site of bam..8

5. Response spectrum analysis with ANSYS of the BUBBLE system..9

5.1. Modal analysis of the tubular unit.9

5.2. Modal analysis of the cupola unit....10

5.3. Modal spectrum analysis of the tubular unit..11

5.4. Modal spectrum analysis of the cupola unit...12

6. Modal spectrum results combination....12

7. Results..13

8. Conclusion...16

Appendix 1 SHELL FINITE ELEMENT MODEL

Appendix 2 ESTIMATION OF STRONG MOTION VALUES IN BAM (IRAN)

Appendix 3 RESPONSE SPECTRUM ANALYSIS OF THE BUBBLE SYSTEM WITH ANSYS

Appendix 4 TUBULAR AND DOME RESULTS

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1. Introduction

A powerful earthquake has occurred in the southeastern region of Iran on the 26 th December 2003.

The magnitude of this event was reported to be 6.6 by USGS. In Bam Township, more than 40,000 waskilled and more than 25,000 was injured. The buildings in these regions are highly vulnerable even to

moderate earthquakes and most of them completely collapse when subjected to this earthquake.

The population of Bam was estimated to be 100,000 at time of the earthquake in 2003. So a

project of a new concept of construction is developed and proposed for this area.

In this study, a seismic analysis of a modular bubble system of housing is presented. This type

structure is designed by the architect Mr. Justus Dahinden. It consists of a medium tubular main unit as

cylindrical structure, and number of cupola units or monolithic domes that are connected to the cylindrical

main unit. This system is based on the shell concept. This concept is known as differentiated shellconstruction.

It is well known that Shells structures gain their strength by virtue of the three dimensional

development of their surfaces, with a resulting ability to carry external loads primarily through in-plane

stresses rather bending. The internal force and stress distribution in shell structures, and especially for

domes, is in general, spatial. Then a careful study must be preformed to catch the real behavior of such

structures under lateral forces when bending will be a non-negligible effect. In general, relatively simple

and idealized cases could readily be solved analytically. However, for more complex structures under

combined loading paths, these are relegated to the domain of numerical analysis.The main difficulty in designing structures that are subjected to seismic action is to calculate the

forces generated by this phenomenon. Some codes do not contain specifications for uncommon structures,

such as domes. So, a three dimensional finite element model for seismic analysis is then required. A

modal spectral analysis is performed on the basis of the SIA Swiss codes using ANSYS finite element

package commercial software.

Regarding seismic actions in Bam region, a specific seismological investigation was performed in

order to update the seismic hazard according to the new data gained by the Bam earthquake.

2. Description of the Bubble system

2.1. Geometrical characteristics of the Bubble system

The concept is a one story concrete tubular and concrete cupolas modular system for housing. The

cupola units are jointed according to need around the tubular main unit. The connecting links between the

tubular and cupola forms act as flexible joints. Then, the both tubular and domes units are studied

separately.

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The cupola structure is pierced by a skylight with two large openings at the lateral sides. One of

these openings is used to have an access to the tubular unit. The tubular unit has also two large openings

for domes connection. Both, tubular and domes units have no bearing walls. This allows the subdivision of the units into a number of partitions.

The structures are assumed to be rigidly connected to the floor foundation. Then, dome and

tubular units are considered to have continuous support. However, the structural behavior of domes is

represented by the so-called arch action mechanism of the shell along the meridional direction. The

tubular and cupola unit dimensions for structural modeling are given in the following figures.

Fig. 1 Different views of the Bubble system for hosing

Z

X

m5.10

m20.3

m35.3 m20.0

m20.0

Fig. 2 View of the tubular plan

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m30.2

m906.0

m30.2

m206.3

m114.2

X

Y

Fig. 3 Layout and transverse dimensions of the tubular unit

= 075.27 m60.1

m90.3

X

Z

m50.3

m20.0

m45.0m10.3

Fig. 4 View of the dome plan

m60.3m50.3

m10.3

Y

X

m45.0

m20.3

Fig. 5 Layout and transverse dimensions of the dome unit

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Fig. 6 Tubular unit meshing views

Fig. 7 Dome unit meshing views

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2.2. Reinforced concrete shell structures

2.2.1. Design of reinforced concrete shells structures

In concrete shell structures, reinforcement is required to resist tensile stresses, as well as to control

shrinkage and thermal cracking. In shell structures, the reinforcement bars which resist the in-plane stress

resultants should be placed in two or more directions and should ideally be oriented in the general

directions of the principal tensile stresses, especially in regions of high tension. Reinforcement to resist

stress couples should be placed near both faces even though moment reversal is not anticipated, since the

bending may vary rapidly along the surface. Under seismic loading, the two layers are also include the

membrane reinforcement. The provision of adequate clearance and cover may necessitate increasing the

shell thickness. In the current structure, the thickness of the bubble system was taken to be equal to 20 cm.

A special attention is required for edges members and opening frames that must be proportioned to resist

the forces imparted by the shell.

In practice, we can consider two regions in shell structures where the stresses are primarily in-

plane or membrane and regions where there is significant bending action. In the first case, direct tensile

stresses should be resisted entirely by reinforcing steel in concrete shells. Regions with direct compressive

stresses are generally controlled by stability requirements. In the second case, the moments or stress

couples may resist by considering a concrete section with reinforcement near the surfaces to act as a wide

flexural member. So, a suitable depth is required for facilitate the provision of ample reinforcing steel.

2.2.2. Non linear method of RC shell structures reinforcements design

The values of internal stress resultants and distribution are necessary to perform the design of

reinforcement. Under lateral seismic loading with gravity loads, reinforcement design in RC shell is more

complex that the case with gravity loads only.

For most of the proposed methods, the shell resultants are computed from elastic analysis of the

structure, while the design of the bending and membrane reinforcing takes into account the inelastic

behavior of steel and concrete. Stephan J. Medwadowski and al. [Design of reinforcement in concrete

shells: a unified approach, Jour. of the International Association for Shell and Spatial Structures, IASS,

pp. 41-50, Vol. 45 (2004) April n. 144] have proposed a unified approach for design of reinforcement in

concrete shells. This method is applicable to shells subjected to both bending and membrane effects.

Chang-shik Min has developed an iterative numerical computational algorithm to design shell element

subjected to membrane and flexural forces is used ([Design and ultimate behavior of RC plates and

shells, Nuclear Eng. and Design 228, pp. 207-223, 2004]). This algorithm is based on equilibrium

considerations for the limited state of the reinforcement and cracked concrete. In this method, the

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reinforcement design is based on combined membrane forces and bending moments obtained from a finite

element analysis.

For these methods, the main ingredient to reinforcement designing are the principal in-planestresses, in-plane forces or membrane stress resultants per unit length, bending stress resultants or normal

shear forces per unit length and bending couples or moments per unit length. The direction of the principal

in-plane stresses (angle between the principal direction and the local element axis) is of interest to

determine the crack directions in the concrete. The reinforcement consists of two orthogonal layers placed

at the top surface and the bottom surface, with appropriate covers. The capacity of the reinforcements will

be designed according to the internal forces and principal stresses. The reinforcements are calculated per

unit width of the shell.

3. Finite element model

A three dimensional shell finite element with shear deformations was used to model the bubble

system. The analysis of the bubble system material is based on the assumption that the shell material is

linearly elastic, isotropic and homogeneous material (See Appendix 1).

According to SIA262 , the Young modulus of the concrete (concrete C30/37 type) is given by the

following relation:

x E = y E = z E = MPa f k E cm E cm 0.270003 = (art. 3.1.2.3.3 SIA262)

8000= E k is a factor for concrete Young modulus determination

MPa f cm 38= , mean concrete compression strength (art. 3.1.2.3.5 SIA262)

MPa f ctm 9,2= , mean concrete tension strength (art. 3.1.2.3.5 SIA262)

The shear modulus is: = xyG ( )MPa E

E GGG cm

cm xz yz 117404348.012

==+

===

The material Poissons ratio is: = xy = yz = xz =0,15

MPacd 1,1= , concrete shear strength (art. 3.1.2.3.5 SIA262)

In this study, we are interested to the following results:

1- displacements of the structures in Cartesian system of coordinate (nodal displacements):

lateral displacements X U , Z U , vertical displacement Y U

2- internal forces per unit length (element resultants):

Bending moments x M , y M and twisting moment xy M , Shear out-of-plane forces x N , y N and shear in-

plane force xyT , Membrane in-plane forces xT , yT

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3- principal stresses (nodal stresses): 1 , 2 and 3

4- reaction forces at the structures base (nodal forces): X F , Y F , Z F , X M , Y M , Z M

4. Strong motion values resulting from earthquake activity at the site of bam

According to the conclusions and recommendations of the seismological investigation, on the

basis of the new data of the Bam earthquake, in this study we have considered a peak ground acceleration

equal to 0,45g with amplification 3 to 4 for the type of soil (See Appendix 2). This amplification gives a

range of 1,2g to 1,8g for the maximal acceleration (PGA). The local soil corresponds to the class IIa at the

best (Table 1).

Iranian soils classification Swiss soils classification

Class

N

Soil

description

Predominant

Frequency

(Hz)

Vs

(m/s)

Geological

Condition

Class

SIA

Vs

(m/s)

S BT

(sec)

C T

(sec)

DT

(sec)

I Soft soil 750

Well cemented and

compacted soil, old

quaternary outcrop

A >800 1.0 0.15 0.4 2.0

Table 1 Equivalence between Iranian and SIA class of soils

It can be observed that the peak value of the elastic spectrum estimates the maximumamplifications of Bam spectra with a suitable accuracy. The SIA261 elastic spectrum eS for soil type

class D is chosen:

( ) ( )

+= B

gd e T T

SaT S15,2

1

, 0 T BT

( ) SaT S gd e 5,2= , BT T C T

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( )T T

SaT S C gd e 5,2= , C T T DT

( ) 25,2 T T T SaT S DC gd e = , DT T

S5,2 is the amplification factor, BT and C T are the transition periods, gd a the peak ground acceleration

(PGA) for the horizontal seismic component.

The fraction of critical damping is equal to 5%, such that:

55.0105.0

1 +

=

Where is a correction factor taking into account for the fraction of critical damping value .The vertical acceleration is taken to be equal 0.7 times of the horizontal acceleration gd a

according to the SIA261 considerations.

5. Response spectrum analysis with ANSYS of the BUBBLE system

In the following, an elastic modal spectral analysis is performed with an elastic spectrum. In the elastic

range, the behavior factor is the equal to unit (See Appendix 3). The stiffness reduction due to initial

cracks was not considered in this study.

5.1. Modal analysis of the tubular unit

The total mass of the tubular unit is equal to 43.93 tones. In the X-X direction, the most

significant mode is mode 1. The corresponding effective mass is equal to 34.47 tones. This represents

78.46% of the total tubular unit mass. In the Y-Y direction, the most significant mode is mode 3. The

corresponding effective mass is equal to 12.2 tones. This represents 27.77% of the total tubular unit mass.

In the Z-Z direction, the most significant mode is mode 8. The corresponding effective mass is equal to

16.35 tones. This represents 37.21% of the total tubular unit mass (See Appendix 3).

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Fig. 8 Mode shape for mode 1 (Front view, Oblique view)

0

1

2

3

4

0.01 0.1 1 10PriodeT [s]

S e

/

a g

d

classe de sol A

classe de sol B

classe de sol C

classe de sol D

classe de sol E

8 2 1

Period (s)

Soil Class A

Soil Class B

Soil Class C

Soil Class D

Soil Class E

3

Modes

Fig. 9 Elastic design spectrum and tubular unit periods corresponding to the soil class D

5.2. Modal analysis of the cupola unit

The total mass of the tubular unit is equal to 43.26 tones. In the X-X direction, the most

significant mode is mode 4. The corresponding effective mass is equal to 21.1 tones. This represents

48.77% of the total dome unit mass. In the Y-Y direction, the most significant mode is mode 2. The

corresponding effective mass is equal to 18 tones. This represents 41.6% of the total dome unit mass. In

the Z-Z direction, the most significant mode is mode 1. The corresponding effective mass is equal to 34.21

tones. This represents 79% of the total dome unit mass (See Appendix 3).

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Fig. 10 Mode shape for mode 1 (Isometric view, Left side view)

0

1

2

3

4

0.01 0.1 1 10PriodeT [s]

S e

/

a g

d

classe de sol A

classe de sol B

classe de sol C

classe de sol D

classe de sol E

Period (s)

Soil Class A

Soil Class B

Soil Class C

Soil Class D

Soil Class E

1234

Modes

Fig. 11 Elastic design spectrum and dome unit periods corresponding to the soil class D

5.3. Modal spectrum analysis of the tubular unit

The modal spectral accelerations for the first modes among the most significant modes in the three

principal directions are given in the following (See Appendix 3):

a/ The most significant mode in the X-X direction is the mode 1 having a spectral acceleration

equal to 9.2606 m/s 2.

b/ The most significant mode in the Y-Y direction is the mode 2 having a spectral acceleration


c/ The most significant mode in the Z-Z direction is the mode 2 having a spectral acceleration


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5.4. Modal spectrum analysis of the cupola unit

The modal spectral accelerations for the first modes among the most significant modes in the three

principal directions are given in the following (See Appendix 3):

a/ The most significant mode in the X-X direction is the mode 3 having a spectral acceleration equal

to 7.1886 m/s 2.

b/ The most significant mode in the Y-Y direction is the mode 2 having a spectral acceleration equal

to 5.0424 m/s 2.

c/ The most significant mode in the Z-Z direction is the mode 1 having a spectral acceleration equal to

7.4385 m/s 2.

6. Modal spectrum results combinationA well-designed structure should be capable of equally resisting earthquake motions from all

possible directions. For non-rectangular three dimensional structures, a simplification in the current code

is lack of definition of the principal horizontal directions which produces the maximum stresses. In

practice, we are allowed to select an arbitrary reference system. If a major principal direction for the

structure was chosen, the minor principal direction will be, by definition, ninety degrees from the major

axis (Fig. 12). So that, both tubular and dome unit are modeled such that x and z axes are the principal

directions.

( )0or X

( )90or Z

901S2S

X S

Z S

Y S

Fig. 12 Major and minor principal directions

In general, a structure must resist an earthquake motion of magnitude S 1 for all possible directions

with angle of , and at the same point in time, resist earthquake motions of magnitude S 2 at the

orthogonal direction to the angle (Fig. 12). Three consecutive modal spectrum analyses were performed

in the three principal directions x, y and z. In the both horizontal directions, we have used the same

spectrum excitation defined in the paragraph (same PGA). However, in the vertical direction, a reduction

of 70% in the PGA value is adopted according to the SIA seismic code. Finally, a static analysis under self

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weight load due to gravity load by a applying a linear vertical acceleration equal to g=10 m/s 2 was

performed.

The spectrum combination was performed trough 63 possible load cases. The load cases aredefined to represent the response produced by 100% of the lateral input spectrum in one direction and

respectively 30%, 70% and 100% in the other directions with negative and positive signs. The maximum

positives and negatives values, respectively, are obtained for the following load cases:

Case A: SW+Sy+Sx+Sz

Case B: SW-Sy-Sx-Sz

This result confirms the previous choice of the major and minor principal directions. But these

load cases correspond to a conservative hypothesis. Moreover, for complex three dimensional structures

the use of the 100/30, 100/40 or 100/70 percentage combination rules will produce member designs whichare not equally resistant to earthquake motions from all possible directions and can also underestimate the

design forces in certain regions which are relatively weak in a certain direction.

Note: SW represents the self weight response; Sx, Sy and Sz are respectively modal responses in the three principal

directions.

7. Results

The most significant results are given in the Appendix 4. The maximum absolute values are:

For tubular unit:

Maximum absolute displacements U

Horizontal displacement in the X-direction : Ux 1.40E-3 (m)

Vertical displacement in the Y-direction : Uy 0.657E-3 (m)

Horizontal displacement in the Z-direction : Uz 0.0687E-3 (m)

Table 2

The X-direction is more flexible than Z-direction.

Maximum absolute forces N/T/M

Membrane force in the x-direction xT 106.61 (kN/m)

Membrane force in the y-direction yT 1047.5 (kN/m)

In-plane shear force xyT 243.38 (kN/m)

Out-of-plane shear force x N 64.997 (kN/m)

Out-of-plane shear force y N 53.212 (kN/m)

Bending moment x M 10.618 (kNm/m)

Bending moment y M 25.568 (kNm/m)

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Twisting moment xy M 5.4443 (kNm/m)

Table 3

The out-of-plane shear forces are negligible in the comparison with membrane forces; the bending

and twisting moment are very low. However, in-plane shear force can not be negligible. This confirms the

membrane resisting mechanism in shell structures. The figures from 1 to 48 (Appendix 4) shows that the

highly stressed zones are always confined at the opening frames.

Maximum absolute reaction nodal forces F/M

Horizontal reaction nodal force in the X-direction FX 31.4 (kN)

Vertical reaction nodal force in the Y-direction FY 166.0 (kN)

Horizontal reaction nodal force in the Z-direction FZ 22.3 (kN)

Horizontal reaction nodal bending moment MX 0.68 (kNm)Vertical reaction nodal bending moment MY 0.725 (kNm)

Horizontal reaction nodal bending moment MZ 13.03 (kNm)

Table 4

The higher value of the nodal reaction force FY is due to the fact that the tubular unit behaves as

under gravity loading. The MZ value confirms the fact that X-X is the weakest direction.

According to the plasticity condition for reinforced concrete members, the tensile stresses must be

taken over by the reinforcement, while the compression ones by concrete. For concrete, we must check up

if the tension and compression strength condition are satisfied (table 5).

Maximum absolute principal stresses

1 (tension) 7323 (kN/m2) > 2900 (kN/m 2)

3 (compressive) 1550 (kN/m2)

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The dome unit is more deformable horizontally in Z-direction that represents the weakest one,

while the X-direction is the stronger. Both tubular and dome unit does not exhibit the same weakestdirection with the respect to the location of the openings.

Maximum absolute forces N/T/M

Membrane force in the x-direction xT 185.87 (kN/m)

Membrane force in the y-direction yT 220.72 (kN/m)

In-plane shear force xyT 96.316 (kN/m)

Out-of-plane shear force x N 44.713 (kN/m)

Out-of-plane shear force y N 69.924 (kN/m)

Bending moment x M 6.7816 (kNm/m)

Bending moment y M 10.98 (kNm/m)

Twisting moment xy M 3.5661 (kNm/m)

Table 7

We can extract the same conclusions as tubular unit concerning the in-plane shear force, the out-

of-plane shear forces, the membrane forces, the bending and twisting moment. The figures from 49 to 64

(Appendix 4) show that the highly stressed zones are exclusively confined inside and near the opening

frames. The dome unit exhibits a much better behavior in comparison with tubular unit. The dome unit

presents a uniform distribution of stresses in the cupola region.

Maximum absolute reaction nodal forces F/M

Horizontal reaction nodal force in the X-direction FX

cupola region 23.30 (kN)

opening frame region 7.33 (kN)

Vertical reaction nodal force in the Y-direction FY

cupola region: 28.11 (kN)opening frame region 36.20 (kN)

Horizontal reaction nodal force in the Z-direction FZ

cupola region 23.40 (kN)

opening frame region 12.22 (kN)

Horizontal reaction nodal bending moment MX

cupola region 3.79 (kNm)

opening frame region 2.034 (kNm)

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Vertical reaction nodal bending moment MY


opening frame region 0.1337 (kNm)Horizontal reaction nodal bending moment MZ


opening frame region 0.549E-4 (kNm)

Table 8

The higher value of the nodal reaction force FY is due to the fact that the dome unit behaves as

under gravity loading. The MZ value confirms the fact that X-X is the weakest direction (table 8).

The principal stresses developed in the dome unit are:

Maximum absolute principal stresses

Maximum absolute principal stress 1

cupola region 1584 (kN/m 2) < 2900 (kN/m 2)

opening frame region 1969 (kN/m 2) < 2900 (kN/m 2)


cupola region 604 (kNm 2)

opening frame region 682.6 (kN/m 2)


cupola region 699.4 (kN/m2

)

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1/ For the tubular unit, it exhibits a relatively uniform stresses distribution, however, a special attention is

required for opening frame region. In these regions, highly membrane stresses are confined essentially in

small zones.2/ For Dome unit, it exhibits a uniform stresses distribution. Stress concentrations are observed

exclusively in the opening frames. However, the dome unit exhibits a very good behavior with

compression resisting mechanism. The tension stresses are found very low.

In general, we have observed that the bubble system can carry the external seismic actions

exclusively by membrane mechanism. Globally, both tubular and unit can be considered as free of

bending actions. Membrane reinforcement is still sufficient. However, the bending and shear field were

developed in some regions to satisfy the global equilibrium (monolithic behavior) and deformationrequirements. The bending and shear field tend to be localized and confined is some regions to the vicinity

of loading and geometrical discontinuities and deformation incompatibilities as opening connections, units

base regions. However, the calculated internal forces are found to be not higher.

Finally, a very satisfactory behavior under seismic actions is observed for the BUBBLE system of

housing. We can conclude that the proposed system for housing can be recommended as a housing system

for regions with a high seismicity. This is due to the most profound and efficient structural performance of

shell concept.

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APPENDIX 1

SHELL FINITE ELEMENT MODEL

A shell element is defined as a three dimensional solid element in the form of a surface with a

thickness small compared to its other dimensions. In general, in a shell structure subjected to applied

external loads, internal stresses and forces may develop. A three dimensional shell element with shear

deformations is shown in the following figure.

Middle-surface

yT

xyT

xT

xyT xyT

y N

x N

y M xyT

x M

xy M

xy M

bot x,

top x,

top y,

bot y,

i

j

k

l nr

Top surface

Bottom surface

x

y z

Fig. 1 Shell finite element showing the components of internal force resultants and stress field

(quadrilateral element - four nodes ijkl)

For conveniency, we are working under Cartesian system. The so-called Mindlin finite element is

used for the structural analysis. It is well known that shells structures carry the applied loads mostly-called

membrane forces and shear mechanisms by comparison with so-called framed structures. In the shell

structures under lateral actions, the bending field and shear field are crucial in the design of the

reinforcements. The internal membrane (in-plane) forces consists of two membrane normal resultant

forces xT , yT and a membrane in-plane shear force xyT per unit length such that:

=2 /

2 /

t

t x x dzT , =

2 /

2 /

t

t y y dzT , =

2 /

2 /

t

t xy xy dzT

where : t is the shell thickness at midpoint of element, computed normal to center plane)

These forces are the resultant internal forces which lie inside the middle-surface of the shell. The

membrane forces cause the stretching of the shell without producing any bending and / or local curvature

changes. These forces are associated to the membrane and shear stresses which are assumed to be

uniformly distributed through the thickness of the shell. The bending and membrane mechanism are

decoupled. The internal bending and transverse shear forces are expressed as follows:

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=2 /

2 /

t

t

x x dz z M ,

=2 /

2 /

t

t

y y dz z M ,

=2 /

2 /

t

t

xy xy dz z M ,

=2 /

2 /

t

t

xy x dz N ,

=2 /

2 /

t

t

yz y dz N

Where: x , y and z are direct stresses, xy , yz et xz are shear stresses. The thru-thickness stress ( z )

is set equal to the negative of the applied pressure at the surfaces of the shell elements, and linearly

interpolated in between.

The bending forces field consists of two bending moments x M , y M per unit length, a twisting

moment xy M of the shell cross-sections per unit length, and two transverse shear forces x N , y N per unit

length. The symbols on the left hand sides of the previous defined equations can be used to represent the

stress resultants at the point in study obtained from elastic analysis of the shell elements.

For linear elastic shells, the internal forces can be calculated by the following equations without

integration procedure:

)6

4 ,,, bot xmid xtop x x

t T

++= ,

6

4 ,,, bot ymid ytop y y

t T

++= ,

6

4 ,,, bot xymid xytop xy xy

t T

++=

( )12

,,2

bot xtop x x

t M

= ,

( )12

,,2

bot ytop y y

t M

= ,

( )12

,,2

bot xytop xy xy

t M

=

6

4 ,,, bot xzmid xztop xz x

t N

++= ,

6

4 ,,, bot yzmid yztop yz y

t N

++=

The component of stress normal to the shell surface (out-of-plane stress), z , is neglected in the

classical shell finite element formulation.

It is well-known for RC structures, that the reinforcing bars will be placed along the general

directions of principal tensile stresses. In ANSYS, the principal stresses are calculated from the resolving

of a cubic equation such that:

0

0

0

0

=

z yz xz

yz y xy

xz xy x

The three principal stresses are called 1 , 2 and 3 . It is important to know that, in ANSYS, 1 is

the most positive stress (tensile), 3 the most negative (compressive).

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However, the element resultants do not represent the maximum values of stresses developed into

the shell elements. Thus, in some cases, the design of shell elements will be done using principal moments

and membrane forces such that:

2

2

3,1 22 xy y x y x M

M M M M M +

++

= for moments

2

2

3,1 22 xy y x y x T

T T T T T +

++

= for membrane forces

1 M , 1T , 3 M , 3T are the largest values of the moments and membrane forces in the two principal direction.

These values correspond to an isotropic homogeneous material.

The bending moments assume maximum values in shell element sections where twisting moment

is equal to zero. The angle defining the principal directions are:

( )

= y x

xy M

M M

M arctg

2

21

1

( ) ( ) 012 90+=

M M

The membrane (axial) forces assume maximum values in shell element sections where shear

forces are zero. The angle defining the principal directions are:

( )

= y x

xyT

T T

T arctg

2

21

1

( ) ( ) 012 90+=

T T

xyT

xyT

x

y

xT

yT

Diagonal crack

31

3T ( )3 1T ( )1

1

1

3

Fig. 2 Principal stresses and concrete crack in a planar shell element

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Estimation of strong motion values in Bam (IRAN)

Report: Estimation of Strong Motion Values Resulting fromEarthquake Activity at the Site of Bam (Iran)

Dr. Souad SellamiSeismologist (Zurich)

I. Introduction

The aim of this report is to give an estimate of the expected strong motion in thecity of Bam, to be taken into account for the reconstruction of houses destroyedduring the earthquake of December 26, 2003. This earthquake allowed newinformation to be developed. We will first present a summary of the earthquakecharacteristics done by an Iranian team for preliminary report after the earthquake.In the second part, we will give our estimation of the strong motion values with

justifications.

II. Earthquake of the 26.12.2003 in Bam (Iran) (extract from[1])

The Bam earthquake occurred on 26 th of December 2003, at 07:56:56 GMT (05:26:26 local time) near the city of Bam, which is located in the southeast of

Iran (Eshghi and Zare, 2003). The coordinates of the epicentre of this earthquakehave been determined by IIEES (IIEES, 2003) at 29.01N and 58.26E in 10km SW of the town of Bam , which is close to the coordinates reported by USGS (28.99N,58.29 E (USGS, 2003)), but based on the surface evidences reported by Zare(Eshghi and Zare, 2003), the epicentre is located under the city of Bam.

The Moment Magnitude of 6.5 for this earthquake (Mw) have been measured based on the preliminary evaluations and the focal depth is estimated to be 8kmbased on S-P evaluation on the records obtained from the main shock (Eshghi and

Zare, 2003). The macro-seismic intensity of the earthquake is estimated to be I 0=IX according to the EMS98 scale. The attenuation of the strong motion seemsto be considerable as the surface evidences and damages decrease sharply at thedirection perpendicular to the Bam fault (Figure 1).

Although the focal mechanism of Bam earthquake was reported to be strike slip having a small vertical component (USGS, 2003), the strong motionsrecorded at the Bam station show a considerable vertical component. Themaximum PGA for the horizontal components are 0.7 and 0.8g, and 1.01g for the vertical component (Corrected values; BHRC, 2003). The Bam earthquakehas been accompanied by some geotechnical phenomena such as landslides,liquefaction and land subsidence.

III. Estimation of strong motion values in Bam

Hazard values

The official hazard map of Iran elaborated by the BRHC (figure 1)considers Bam in the region with high hazard with a value of 0.3g. The highestvalue shown is 0.35g. The hazard map of Iran presented in the GSHAP study(figure2) shows a maximum acceleration (PGA) of the order 0.3g for a return

S.Sellami Leinen 1 [email protected]

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period of 500 years. In regard to the last event of December 2003 we couldextend the nearby zone with higher hazard and take for Bam a value of 0.45g.The hazard level depicted on the map, 500years return period (exactly 475 years)corresponds to a hazard value, which has 90% probability of non exceedanceduring 50 years. This is the standard level chosen for ordinary constructions

(private houses).

This hazard value is computed for hard rock. To evaluate the hazard at a sitewe have to take into account the amplification due to the local geology, the siteeffect.

Soil characteristics (amplification)

The soils at Bam area are unconsolidated. From the satellite picture (figure3) andsites pictures (figure 4) shown on the report of the Bam earthquake (geology andgeotectonics).

A soil study has been done in the region north of Bam (table 1). The localsoil corresponds to the class IIa at the best. An amplification of 3 to 4 times is

possible. Figure 5 shows peaks of acceleration in the low frequencies 2-6 Hz andat 10 Hz.

Response spectrum of the Bam earthquake

From the December 26th earthquake, some information has been gathered. Thisearthquake, or the damages have a very strong directivity in the north-south

direction for example as shown on the isoseismal map (figure 6).The Iranian strong motion network is well developed and many instruments

are in the vicinity of Bam (table 2). One station was located in the Governors building in Bam [8.4]. This house has been damaged but the data were safe. Thedata showed a strong PGA of 1.02 vertical, 0.7 and 0,8g respectively in thedirections NS and EO (figures 7 and 8). The Fourier spectrum and responsespectrum for this earthquake are also available (figures 9, 10, 11 and12)

Discussion

If we consider the hazard value up to 0.45g and the amplification (3 to 4) for thetype of soil, it gives a range of 1.2 to 1.8g for the maximal acceleration (PGA).We can compare this estimation to the last earthquake. This earthquake wasstrong with 6.5 Moment magnitude and rather shallow at 8km, which explains thehigh acceleration values. The accelerations recorded in Bam, 0.7 to 1.01g, areclose to the accelerations expected during a longer period or a stronger earthquake7 (figure 13). The directivity is typical of the type of faulting (strike slip) and thevicinity of the fault. The acceleration values at the epicentre are higher than themodel for a magnitude of 6.5 (figure 13). We also note the large verticalmovement respectively to the horizontal values at the epicentre, which is not thecase anymore further away.


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IV. References

[1] BamEReport1.pdf The Very urgent Preliminary report on Bam earthquake of Dec. 26-2003. Ministry of Housing and Urban Development Building and HousingResearch Center Iran Strong Motion Network (ISMN)

[2] Seismic macrozonation hazard map of Iran. Iranian code of practice for seismicresistant design of building. (Standard 2800)

[3] Seismic hazard assessment of Iran, B. Tavakoli & M. Ghafory- Ashtiany. Giardiniet al. Ed. GSHAP Summary Volume Annali di Geofisica, 1999

[4] Engineering Geology and Geotechnical Aspects of Bam Earthquake (PreliminaryReport) Kambod Amini Hosseini, Mohammad Reza Mahdavifar, MohammadKeshavarz Bakhshayesh, Masomeh Rakhshandeh, 10/01/2004. International Instituteof Earthquake Engineering and Seismology

[5] Geological Survey of Iran (GSI), Geological quadrangles of Bam, Sabzevaran,allah Abad, Jahan Abad , Scale:1:250000

[6] Site Effect Classification in East-Central of Iran A. Komak Panah1, N. Hafezi Moghaddas2, M.R. Ghayamghamian3, M. Motosaka4, M.K. Jafari5, and A.Uromieh6. JSEE: Spring 2002, Vol. 4, No. 1

[7] Characterization of Site Response: General Site Categories. Adrin Rodrguez-Marek. Jonathan D. Bray, and Norman Abrahamson.. PEER Report 1999/03. PacificEarthquake Engineering Research Center. College of Engineering. University of

California, Berkeley. February 1, 1999.[8] Building and Housing Research Center (BHRC). 2003. Website on: Bamearthquake, December,26,2003.

[8.1] http://www.bhrc.gov.ir/bhrc/reports/bam/bam_pdf.pdf ;

[8.2] http://www.bhrc.gov.ir/Bhrc/d-stgrmo/shabakeh/earthquake/bam/bam.htm

[8.3] http://www.bhrc.gov.ir/Bhrc/d-stgrmo/shabakeh/earthquake/bam/graph.pdf

[8.4]

http://www.bhrc.gov.ir/Bhrc/dstgrmo/SHABAKEH/earthquake/bam/Bam%20Accelerograph%20Station%20History.pdf

[9] Bam (SE Iran) earthquake of 26 December 2003, Mw6.5: A PreliminaryReconnaissance Report, Eshghi, S. and Zare, M. (2003), Website:http://www.iiees.ac.ir/English/Bam_report_english.html

V. Tables and figures

Figure 1. Seismic macrozonation of Iran to be use for hazard code zonation (BHRC1989).[2]

Figure 2. Seismic hazard map of Iran with a 475 return period (GSHAP 2001). [3]

http://www.bhrc.gov.ir/Bhrc/d-stgrmo/shabakeh/earthquake/bam/bam.htmhttp://www.bhrc.gov.ir/Bhrc/d-stgrmo/shabakeh/earthquake/bam/graph.pdfhttp://www.iiees.ac.ir/English/Bam_report_english.htmlhttp://www.iiees.ac.ir/English/Bam_report_english.htmlhttp://www.bhrc.gov.ir/Bhrc/d-stgrmo/shabakeh/earthquake/bam/graph.pdfhttp://www.bhrc.gov.ir/Bhrc/d-stgrmo/shabakeh/earthquake/bam/bam.htm

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Figure 3. Tectonic faults and strong earthquakes (bigger than M=4) in the Bam area.[91]

Figure 4. Macroseismic Intensity map of the Bam earthquake 26.12.2003 showingthe strong directivity and an epicentral intensity of IX to X. [4]

Table 1. Site classification and description of Iranian soils in the central eastern part.[6]

Figure 5. Right: Comparison of average transfer functions for the different classesof table 1. Link: The relation between dominant frequencies estimated by 2different methods with respect to Vs. [6]

Figure 6. Satellite view of the city of Bam and area. [1]

Figure 7. Picture showing an example of soil in the Bam area.[4]

Table 2. Strong motion data from the Bam earthquake.[1]

Figure 8. Acceleration of the main shock recorded in Bam 26.12. 2004.[1]

Figure 9. Acceleration data sheet (station specification and values recorded) for thestation Bam and earthquake of Bam. .[8.3]

Figure 10. Acceleration a) uncorrected b) corrected, c)Velocity and d) Displacementof the Bam Earthquake recorded at the station Bam. .[8.3]

Figure 11. Response spectrum (with different damping values) of the acceleration,velocity and displacement for the Bam earthquake registered at the stationBam. .[8.3]

Figure 12. Fourier amplitude of the acceleration for the three components depictedwith two different scales.[8.3]

Figure 13. Attenuation model of the peak acceleration for the eastern part of Iranfitted to the Bam earthquake data.[1]


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Figure 1. Seismic macrozonation of Iran to be use for hazard code zonation (BHRC1989).

Figure

2. Seismic hazard map of Iran with a 475 return period (GSHAP 2001).


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Figure 3. Tectonic faults and strong earthquakes (bigger than M=4) in the Bam area


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Figure 4. Macroseismic Intensity map of the Bam earthquake 26.12.2003 showingthe strong directivity and an epicentral intensity of IX to X.

Table 1. Site classification and description of Iranian soils in the central eastern part.

Figure 5. Right: Comparison of average transfer functions for the different classesof table 1. Link: The relation between dominant frequencies estimated by 2different methods with respect to Vs.


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Figure 6. Satellite view of the city of Bam and area

Figure 7. Picture showing an example of soil in the Bam area.


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Table 2. Strong motion data from the Bam earthquake.


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Figure 8. Acceleration of the main shock recorded in Bam 26.12. 2004.


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Figure 9. Acceleration data sheet (station specification and values recorded) for thestation Bam and earthquake of Bam.


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a)

b)

c)

d)

Figure 10. Acceleration a) uncorrected b) corrected, c)Velocity and d) Displacementof the Bam Earthquake recorded at the station Bam.


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Figure 11. Response spectrum (with different damping values) of the acceleration,velocity and displacement for the Bam earthquake registered at the stationBam.


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Figure 12. Fourier amplitude of the acceleration for the three components depictedwith two different scales.


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Figure 13. Attenuation model of the peak acceleration for the eastern part of Iranfitted to the Bam earthquake data.


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APPENDIX 3

RESPONSE SPECTRUM ANALYSIS OF THE BUBBLE SYSTEM WITH ANSYSThe type of the analysis performed is the response spectrum analysis. This method is used for the

prediction of displacements and element forces in structures. The method involves the calculation of only

the maximum values of the displacements and members in each mode using smooth design spectra. The

analysis consists of a three dimensional mode shapes and natural frequencies of vibration calculation.

These are the undamped free vibration response of the structure. The structure has constant stiffness and

mass effects. The mass is taken distributed along the structures by a density. Then, the structure is excited

by a spectrum of known directions and frequency components. A single-point response spectrum method

based on an acceleration spectrum introduced as a known function is used and implemented on ANSYS asa MACRO SCRIPT function.

The selected modal analysis method is FULL SUBSPACE. This method did not require a set of

master degrees of freedom, and it gives more accurate answers with the comparison with reduced method

for eigenvalues calculation. But, it takes somewhat longer to solve. In this method a number of modes is

considered for the structural response calculation. For each principal direction, the square-root-of-sum-of-

the-squares (SRSS) modes combination method is used for the purpose of structural design. This approach

assumes that all the maximum modal values are statistically independent.

The significant modes are determined, in ANSYS, such that mode coefficients (MCOEF) for allmodes are compared with the most significant mode having the maximum mode coefficient

(MCOEFmax). So that, a significant factor for combining modes (SIGNIF) is defined and taken equal to

0,5% in this study. For a given mode i, having a mode coefficient MCOEFi, if

SIGNIF>MCOEFi/MCOEFmax, then the current mode will be not considered as significant, and will be

ignored in the global structural response (in the SRSS Mode Combination). The significant modes for both

structures are given in tables and.

Another parameter is the modal participation factor, PFACT that is used for the modal

coefficient determination. This parameter is calculated by: PFACT = i

ijiam ,

With: ija is the N2 mode shape matrix, im the mass matrix.

So, the modal coefficient, MCOEF, is calculated by: MCOEF = PFACT x S e(T) / ( )22 f

f is the natural frequency, T the corresponding period and S e(T) the spectral displacement value at the

current mode

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0

50

100

150

200

250

1 3 5 7 9 11 13 15 1 7 19 2 1 23 25 27 29 31 3 3 35 37 39

Modes

F r e q u e n c y

( H z )

Dome

Tubular

Fig. 1 Variation of dome and tubular frequencies by means of modes

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

7.00E-02

8.00E-02

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Modes

P e r

i o d ( s )

Dome

Tubular

Fig. 2 Variation of dome and tubular periods with regards to modes

1. Modal analysis of the tubular unit

In the X-X direction:

MODE FREQUENCY(Hz)

PERIOD(s)

PARTICIPATIONFACTOR RATIO

EFFECTIVEMASS(tones)

CUMULATIVEMASS

FRACTION1 14.3267 6.98E-02 5.8711 1 34.4702 0.8841582 32.3132 3.09E-02 -3.13E-06 0.000001 9.80E-12 0.8841583 33.2473 3.01E-02 -7.86E-06 0.000001 6.18E-11 0.8841584 38.2482 2.61E-02 -2.31E-02 0.003937 5.34E-04 0.8841725 59.2625 1.69E-02 -1.2267 0.208939 1.50481 0.9227716 63.6301 1.57E-02 6.99E-05 0.000012 4.88E-09 0.9227717 64.9751 1.54E-02 -1.4037 0.239084 1.97035 0.973318 67.1482 1.49E-02 3.79E-05 0.000006 1.44E-09 0.973319 69.6891 1.43E-02 -5.26E-02 0.008964 2.77E-03 0.973381

10 70.8775 1.41E-02 -3.04E-05 0.000005 9.23E-10 0.973381

SUM OF EFFECTIVE MASSES= 38.9864 tones Table 1 Modal results for tubular unit in X-X direction

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In the Y-Y direction:

MODE FREQUENCY

(Hz)

PERIOD

(s)


EFFECTIVEMASS

(tones)

CUMULATIVEMASS

FRACTION1 14.3267 6.98E-02 2.52E-05 0.000007 6.35E-10 2.15E-112 32.3132 3.09E-02 0.36125 0.103397 0.130499 4.43E-03

3 33.2473 3.01E-02 3.4938 1 12.2066 0.4184424 38.2482 2.61E-02 1.64E-06 0 2.69E-12 0.4184425 59.2625 1.69E-02 -1.08E-05 0.000003 1.16E-10 0.418442

6 63.6301 1.57E-02 0.35873 0.102676 0.128688 0.4228067 64.9751 1.54E-02 -4.60E-05 0.000013 2.12E-09 0.4228068 67.1482 1.49E-02 -0.73366 0.209989 0.538256 0.4410639 69.6891 1.43E-02 -3.87E-05 0.000011 1.50E-09 0.441063

10 70.8775 1.41E-02 2.3722 0.678987 5.62755 0.631933

SUM OF EFFECTIVE MASSES= 29.4835 tonesTable 2 Modal results for tubular unit in Y-Y direction


In the Z-Z direction:

MODE FREQUENCY(Hz)

PERIOD(s)



CUMULATIVEMASS

FRACTION1 14.3267 6.98E-02 -1.26E-06 0 1.58E-12 4.34E-142 32.3132 3.09E-02 -1.9946 0.493187 3.97847 0.1095613 33.2473 3.01E-02 0.21082 0.052128 4.44E-02 0.1107854 38.2482 2.61E-02 1.23E-05 0.000003 1.51E-10 0.110785

5 59.2625 1.69E-02 4.27E-06 0.000001 1.83E-11 0.1107856 63.6301 1.57E-02 -2.094 0.517761 4.38481 0.2315367 64.9751 1.54E-02 -2.06E-04 0.000051 4.25E-08 0.231536

8 67.1482 1.49E-02 -4.0443 1 16.3566 0.6819729 69.6891 1.43E-02 7.47E-05 0.000018 5.57E-09 0.681972

10 70.8775 1.41E-02 -1.0233 0.253014 1.04708 0.710807

SUM OF EFFECTIVE MASSES= 36.3128 tones

Table 3 Modal results for tubular unit in Z-Z direction

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2. Modal analysis of the cupola unit


MODE FREQUENCY(Hz)

PERIOD(s)



CUMULATIVEMASS

FRACTION1 41.8199 2.39E-02 2.55E-08 0 6.50E-16 1.93E-172 59.698 1.68E-02 -3.01E-06 0.000001 9.04E-12 2.68E-133 61.0767 1.64E-02 3.4453 0.75 11.8702 0.351988

4 85.3363 1.17E-02 -4.5938 1 21.1026 0.9777445 97.333 1.03E-02 -2.18E-06 0 4.76E-12 0.9777446 101.045 9.90E-03 -2.58E-05 0.000006 6.65E-10 0.9777447 101.748 9.83E-03 3.65E-05 0.000008 1.33E-09 0.9777448 115.84 8.63E-03 0.26397 0.057463 6.97E-02 0.979819 121.604 8.22E-03 7.90E-07 0 6.24E-13 0.97981

10 125.799 7.95E-03 -1.53E-05 0.000003 2.34E-10 0.97981


Table 4 Modal results for dome unit in X-X direction

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In the Y-Y direction:

MODE FREQUENCY(Hz)

PERIOD(s)

PARTICIPATION

FACTOR RATIOEFFECTIVE

MASS(tones)

CUMULATIVEMASS

FRACTION1 41.8199 2.39E-02 -2.22E-07 0 4.91E-14 1.62E-15

2 59.698 1.68E-02 4.2412 1 17.9881 0.5931723 61.0767 1.64E-02 8.15E-06 0.000002 6.64E-11 0.5931724 85.3363 1.17E-02 -4.29E-06 0.000001 1.84E-11 0.5931725 97.333 1.03E-02 -8.63E-05 0.00002 7.44E-09 0.5931726 101.045 9.90E-03 -1.2904 0.30425 1.66512 0.6480817 101.748 9.83E-03 1.04E-04 0.000024 1.07E-08 0.6480818 115.84 8.63E-03 -1.51E-06 0 2.29E-12 0.6480819 121.604 8.22E-03 1.1099 0.261687 1.23182 0.688701

10 125.799 7.95E-03 -3.74E-05 0.000009 1.40E-09 0.688701

SUM OF EFFECTIVE MASSES=30.3252 tones

Table 5 Modal results for dome unit in Y-Y direction


In the Z-Z direction:

MODE FREQUENCY(Hz)

PERIOD(s)



CUMULATIVEMASS

FRACTION1 41.8199 2.39E-02 5.8488 1 34.208 0.9425522 59.698 1.68E-02 -6.93E-07 0 4.80E-13 0.9425523 61.0767 1.64E-02 -2.75E-06 0 7.58E-12 0.942552

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4 85.3363 1.17E-02 -3.36E-06 0.000001 1.13E-11 0.9425525 97.333 1.03E-02 4.28E-06 0.000001 1.83E-11 0.9425526 101.045 9.90E-03 -1.65E-05 0.000003 2.73E-10 0.9425527 101.748 9.83E-03 -0.50533 0.086399 0.255355 0.9495888 115.84 8.63E-03 1.89E-07 0 3.56E-14 0.949588

9 121.604 8.22E-03 5.55E-06 0.000001 3.08E-11 0.94958810 125.799 7.95E-03 -5.48E-07 0 3.00E-13 0.949588


Table 6 Modal results for dome unit in Z-Z direction


5.3. Modal spectrum analysis of the tubular unit

0

5

10

15

20

25

30

35

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Modes

E f f e c

t i v e m a s s e

( t o n e s

)

Meff X-X

Meff Y-Y

Meff Z-Z

Fig. 9 Effective masses with the respect of number of modes for the tubular unit

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-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Modes

M o

d e

C o e f

f i c i e n

t

MCOEF X-X

MCOEF Y-Y

MCOEF Z-Z

Fig. 10 Mode coefficients with the respect of number of modes for the tubular unit

-6

-4

-2

0

2

4

6

8

1 3 5 7 9 11 13 1 5 17 19 21 2 3 25 27 29 31 3 3 35 37 39

Modes

P a r

t i c i p a t

i o n

F a c

t o r

Pfact X-X

Pfact Y-Y

Pfact Z-Z

Fig. 11 Participation factors with the respect of number of modes for the tubular unit

The modal spectral accelerations for the three most significant modes in the three principal

directions are given in the following tables.


MODE FREQUENCY(Hz)

( )T Se (m/s2)

PARTICIPATIONFACTOR(PFACT)

MODECOEFFICIENT

(MCOEF)


CUMULATIVEMASS

FRACTION1 14.33 9.2606 5.871 6.71E-03 34.4702 0.8841582 32.31 7.6135 -3.13E-06 -5.78E-10 9.80E-12 0.8841583 33.25 7.594 -7.86E-06 -1.37E-09 6.18E-11 0.8841584 38.25 7.4986 -2.31E-02 -3.00E-06 5.34E-04 0.884172

5 59.26 7.2082 -1.227 -6.38E-05 1.50481 0.9227716 63.63 7.1621 6.99E-05 3.13E-09 4.88E-09 0.922771

7 64.98 7.1486 -1.404 -6.02E-05 1.97035 0.973318 67.15 7.1275 3.79E-05 1.52E-09 1.44E-09 0.973319 69.69 7.1036 -5.26E-02 -1.95E-06 2.77E-03 0.973381

10 70.88 7.0928 -3.04E-05 -1.09E-09 9.23E-10 0.973381

Table 7 Modal spectrum results for tubular unit in X-X direction

The significant modes, in the X-X direction, are modes 1, 5 and 7 such that:

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For mode 1: MCOEF = PFACT x S e(T) / ( )22 f =5.871x9.2606/(2 x14.33) 2=6.71E-3

For mode 5: MCOEF = PFACT x S e(T) / ( )22 f =-1.227x7.2082/(2 x59.26) 2=-6.386E-5


For the Y-Y direction:

MODE FREQUENCY(Hz)

( )T Se (m/s2)


MODECOEFFICIENT

(MCOEF)


CUMULATIVEMASS

FRACTION1 14.33 6.4824 2.52E-05 2.02E-08 6.35E-10 2.15E-11

2 32.31 5.3295 0.3612 4.67E-05 0.130499 4.43E-03

3 33.25 5.3158 3.494 4.26E-04 12.2066 0.418442

4 38.25 5.249 1.64E-06 1.49E-10 2.69E-12 0.418442

5 59.26 5.0458 -1.08E-05 -3.91E-10 1.16E-10 0.418442

6 63.63 5.0135 0.3587 1.13E-05 0.128688 0.422806

7 64.98 5.004 -4.60E-05 -1.38E-09 2.12E-09 0.422806

8 67.15 4.9892 -0.7337 -2.06E-05 0.538256 0.441063

9 69.69 4.9725 -3.87E-05 -1.01E-09 1.50E-09 0.441063

10 70.88 4.965 2.372 5.94E-05 5.62755 0.631933

Table 8 Modal spectrum results for tubular unit in Y-Y direction

There are 11 significant modes in the Y-Y direction (2, 3, 6, 8, 10, 13, 14, 21, 24, 27 and 33). Among the

ten first given modes are:






For the Z-Z direction:

MODE FREQUENCY(Hz)

( )T Se (m/s2)


MODECOEFFICIENT

(MCOEF)


CUMULATIVEMASS

FRACTION1 14.33 9.2606 -1.26E-06 -1.44E-09 1.58E-12 4.34E-14

2 32.31 7.6135 -1.995 -3.68E-04 3.97847 0.109561

3 33.25 7.594 0.2108 3.67E-05 4.44E-02 0.110785

4 38.25 7.4986 1.23E-05 1.60E-09 1.51E-10 0.110785

5 59.26 7.2082 4.28E-06 2.22E-10 1.83E-11 0.110785

6 63.63 7.1621 -2.094 -9.38E-05 4.38481 0.231536

7 64.98 7.1486 -2.06E-04 -8.84E-09 4.25E-08 0.231536

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8 67.15 7.1275 -4.044 -1.62E-04 16.3566 0.681972

9 69.69 7.1036 7.47E-05 2.77E-09 5.57E-09 0.681972

10 70.88 7.0928 -1.023 -3.66E-05 1.04708 0.710807

Table 9 Modal spectrum results for tubular unit in Z-Z direction

There are 13 significant modes in the Y-Y direction (2, 3, 6, 8, 10, 13, 14, 15, 17, 20, 21, 24 and 36).

Among the ten first given modes are:




For mode 8: MCOEF = PFACT x S e(T) / ( )2

2 f =-4.044x7.1275/(2 x67.15)2

=-1.619E-4


5.4. Modal spectrum analysis of the cupola unit

0

5

10

15

20

25

30

35

40

1 3 5 7 9 11 13 15 17 19 2 1 23 25 27 29 31 33 35 37 39

Modes

E f f e c t

i v e m a s s e

( t o n e s )

Meff X-X

Meff Y-Y

Meff Z-Z

Fig. 12 Effective masses with regard to the number of modes for the dome unit

-2.00E-04

-1.00E-04

0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

6.00E-04

7.00E-04

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Modes

M o

d e

C o e

f f i c i e n

t

MCOEF X-X

MCOEF Y-Y

MCOEF Z-Z

Fig. 13 Mode coefficients with the respect of number of modes for the dome unit

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-6

-4

-2

0

2

4

6

8

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Modes

P a r

t i c i p a t

i o n

f a c t o r

Pfact X-X

Pfact Y-Y

Pfact Z-Z

Fig. 14 Participation factors with the respect of number of modes for the dome unit

For the X-X direction:

MODE FREQUENCY(Hz)

( )T Se (m/s2)


MODECOEFFICIENT

(MCOEF)


CUMULATIVEMASS

FRACTION1 41.82 7.4385 2.55E-08 2.75E-12 6.50E-16 1.93E-17

2 59.7 7.2035 -3.01E-06 -1.54E-10 9.04E-12 2.68E-13

3 61.08 7.1886 3.445 1.68E-04 11.8702 0.351988

4 85.34 6.975 -4.594 -1.12E-04 21.1026 0.977744

5 97.33 6.8927 -2.18E-06 -4.02E-11 4.76E-12 0.977744

6 101 6.8695 -2.58E-05 -4.39E-10 6.65E-10 0.977744

7 101.7 6.8652 3.65E-05 6.12E-10 1.33E-09 0.977744

8 115.8 6.7854 0.264 3.38E-06 6.97E-02 0.97981

9 121.6 6.7557 7.90E-07 9.14E-12 6.24E-13 0.97981

10 125.8 6.7351 -1.53E-05 -1.65E-10 2.34E-10 0.97981

Table 10 Modal spectrum results for dome unit in X-X direction

There are 7 significant modes in the Y-Y direction (3, 4, 6, 16, 24, 29 and 39). Among the ten first given

modes are:



2 f =-4.594x6.975/(2 x85.34)2

=-1.115E-4


For the Y-Y direction:

MODE FREQUENCY(Hz)

( )T Se (m/s2)


MODECOEFFICIENT

(MCOEF)


CUMULATIVEMASS

FRACTION1 41.82 5.2069 -2.22E-07 -1.67E-11 4.91E-14 1.62E-15

2 59.7 5.0424 4.241 1.52E-04 17.9881 0.593172

3 61.08 5.032 8.15E-06 2.79E-10 6.64E-11 0.593172

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4 85.34 4.8825 -4.29E-06 -7.28E-11 1.84E-11 0.593172

5 97.33 4.8249 -8.63E-05 -1.11E-09 7.44E-09 0.593172

6 101 4.8087 -1.29 -1.54E-05 1.66512 0.648081

7 101.7 4.8057 1.04E-04 1.22E-09 1.07E-08 0.648081

8 115.8 4.7498 -1.51E-06 -1.36E-11 2.29E-12 0.648081

9 121.6 4.729 1.11 8.99E-06 1.23182 0.688701

10 125.8 4.7146 -3.74E-05 -2.82E-10 1.40E-09 0.688701

Table 11 Modal spectrum results for dome unit in Y-Y direction

There are 11 significant modes in the Y-Y direction (2, 6, 9, 11, 17, 19, 21, 23, 27, 35 and 38). Among the

ten first given modes are:


For mode 6: MCOEF = PFACT x S e(T) / ( )22 f =-1.29x4.8087/(2 x101) 2=-1.541E-5


For the Z-Z direction:

MODE FREQUENCY(Hz)

( )T Se (m/s2)


MODECOEFFICIENT

(MCOEF)


CUMULATIVEMASS

FRACTION1 41.82 7.4385 5.849 6.30E-04 34.208 0.942552

2 59.7 7.2035 -6.93E-07 -3.55E-11 4.80E-13 0.942552

3 61.08 7.1886 -2.75E-06 -1.34E-10 7.58E-12 0.942552

4 85.34 6.975 -3.36E-06 -8.16E-11 1.13E-11 0.942552

5 97.33 6.8927 4.28E-06 7.88E-11 1.83E-11 0.942552

6 101 6.8695 -1.65E-05 -2.82E-10 2.73E-10 0.942552

7 101.7 6.8652 -0.5053 -8.49E-06 0.255355 0.949588

8 115.8 6.7854 1.89E-07 2.42E-12 3.56E-14 0.949588

9 121.6 6.7557 5.55E-06 6.42E-11 3.08E-11 0.949588

10 125.8 6.7351 -5.48E-07 -5.91E-12 3.00E-13 0.949588

Table 12 Modal spectrum results for dome unit in Z-Z direction

There are 4 significant modes in the Y-Y direction (1, 7, 12, and 15). Among the ten first given modes are:


2 f =5.849x7.4385/(2 x41.82) 2=6.307E-4


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APPENDIX 4

TUBULAR AND DOME RESULTS

1. Tubular unit

1.1. Nodal displacements

Load casesParameter Maximum

values SW Sx Sy Sz Case A Case B

(+) .125E-3 .125E-2 .107E-3 .103E-3 .140E-2 0.0Ux

(m) (-) -.125E-3 0.0 0.0 0.0 0.0 -.140E-2

(+) 0.0 .340E-3 .169E-3 .163E-3 .376E-3 0.0Uy

(m) (-) .204E-3 0.0 0.0 0.0 -.148E-4 -.657E-3

(+) .950E-5 .156E-4 .119E-4 .474E-4 .637E-4 0.0Uz

(m) (-) -.989E-5 0.0 0.0 0.0 0.0 -.687E-4

Table 1 Nodal displacements envelop values for the tubular unit in global coordinate

1.2. Elements solicitations



(+) 26.759 64.252 16.415 30.082 69.241 -.79399 x

T

(kN/m)(-) -20.490 .2838E-1 .13210 .17642 -.32474 -106.61

(+) 36.923 696.29 112.76 61.893 684.7 -7.1685 yT

(kN/m)(-) -181.41 .11884 1.3387 1.6330 -8.9274 -1047.5

(+) 47.464 151.27 32.848 16.107 243.38 -1.1815 xyT

(kN/m)(-) -24.805 .33428 .21591 .30434 1.1810 -150.45

(+) 2.0265 5.8507 1.9365 .80991 10.618 -.1740E-1 x M

(kNm/m)(-) -2.0264 .1996E-1 .1013E-1 .21438E-1 .17371E-1 -10.618

(+) 4.5693 15.276 3.1945 3.2650 25.568 -.32075 y M

(kNm/m)(-) -2.9084 .2438E-1 .9017E-1 .75594E-1 .32065 -18.018

(+) .068910 3.3092 1.0310 .67311 5.4443 -.12433 xy M

(kNm/m)(-) -.68908 .1862E-1 .297E-1 .79694E-1 .12438 -5.4443

(+) 7.4902 38.797 12.134 7.0247 64.996 -.14317 x N

(kN/m)(-) -7.2112 .1028E-1 .4425E-1 .565E-1 0.18087 -64.997

y N (+) 12.736 34.548 10.217 15.19153.212 1.0597

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(-) -12.735 .57202 .36621 .38628 -1.0586 -53.212

Table 2 Element resultant forces envelope values for tubular unit

Load Case A:

Fig. 1 Membrane normal resultant force in x-element direction xT per unit length (Isometric view)

Fig. 2 Normal force xT per unit length (Top view) Fig. 3 Normal force per unit length xT (Left view)

Fig. 4 Membrane normal resultant force in y-element direction yT per unit length (Isometric view)

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Fig. 5 Normal force yT per unit length (Top view) Fig. 6 Normal force yT per unit length (Left view)

Fig. 7 Membrane in-plane shear force xyT per unit length (Isometric view)

Fig. 8 shear force xyT per unit length (Top view) Fig. 9 shear force xyT per unit length (Left view)

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Fig. 10 bending moment around y-element axis x M

per unit length (Isometric view)

Fig. 11 Bending moment x M per unit length (Top view) Fig. 12 Bending moment x M per unit length (Left view)

Fig. 13 bending moment around x-element axis y M per unit length (Isometric view)

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Fig. 14 bending moment y M per unit length (Top view) Fig. 15 bending moment y M per unit length (Left view)

Fig. 16 Twisting moment xy M per unit length (Isometric view)

Fig. 17 Twisting moment xy M per unit length (Top view) Fig. 18 Twisting moment xy M per unit length (Left view)

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Fig. 19 Transverse shear x N per unit length (Isometric view)

Fig. 20 Transverse shear x N per unit length (Top view) Fig. 21 Transverse shear x N per unit length (Left view)

Fig. 22 Transverse shear y N per unit length (Isometric view)

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Fig. 23 Transverse shear y N per unit length (Top view) Fig. 24 Transverse shear y N per unit length (Left view)

Load Case B:

Fig. 25 Membrane normal resultant force in x-element direction xT per unit length (Isometric view)

Fig. 26 Normal force xT per unit length (Top view) Fig. 27 Normal force per unit length xT (Left view)

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Fig. 29 Normal force yT per unit length (Top view) Fig. 30 Normal force yT per unit length (Left view)


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Fig. 32 shear force xyT per unit length (Top view) Fig. 33 shear force xyT per unit length (Left view)

Fig. 34 bending moment around y-element axis x M per unit length (Isometric view)

Fig. 35 Bending moment x M per unit length (Top view) Fig. 36 Bending moment x M per unit length (Left view)

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Fig. 38 bending moment y M per unit length (Top view) Fig. 39 bending moment y M per unit length (Left view)

Fig. 40 twisting moment xy M per unit length (Isometric view)

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Fig. 41 Twisting moment xy M per unit length (Top view) Fig. 42 Twisting moment xy M per unit length (Left view)


Fig. 44 Transverse shear x N per unit length (Top view) Fig. 45 Transverse shear x N per unit length (Left view)

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Fig. 47 Transverse shear y N per unit length (Top view) Fig. 48 Transverse shear y N per unit length (Left view)

1.3. Base shear and nodal reaction forces (Edge effects)


values SW Sx Sy Sz Case A Case B(+) .630E+01 .1926E+2 .385E+1 .218E+1 .314E+2 0.0

(-) -.63E+1 0.0 0.0 0.0 0.0 -.314E+2

FX

(kN)

Cumul 0.0 .320E+3 .878E+2 .620E+2 - -

(+) .297E+2 .109E+3 .187E+2 .964E+1 .166E+3 .2748E+1

(-) -.102E+2 0.0 0.0 0.0 0.0 -.1067E+3

FY

(kN)

Cumul .439E+3 .1326E+4 .252E+3 .236E+3 - -

FZ (+) .2436E+1 .138E+2 .271E+1 .419E+1 .223E+2 0.0

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(-) -.2257E+1 0.0 0.0 0.0 0.0 -.203E+2(kN)

Cumul 0.0 .1538E+3 .678E+2 .138E+3 - -

(+) .123E+0 .301E+0 .989E-1 .156E+0 .6806E+0 0.0(-) -.1163E+0 0.0 0.0 0.0 0.0 -.603E+0

MX(kNm)

Cumul .262E-1 .3845E+1 .1172E+1 .200E+1 - -

(+) .1758E+0 .393E+0 .972E-1 .692E-1 .725E+0 0.0

(-) -.1758E+0 0.0 0.0 0.0 0.0 -.725E+0

MY

(kNm)

Cumul 0.0 .689E+1 .176E+1 .266E+1 - -

(+) .270E+1 .7138E+1 .163E+1 .181E+1 .1303E+2 0.0

(-) -.270E+1 0.0 0.0 0.0 0.0 -.1303E+2

MZ

(kNm)

Cumul -.259E-3 .219E+3 .505E+2 .402E+2 - -

Table 3 Nodal reactions forces envelop values for tubular unit

1.4. Principal stresses



(+) .5951E+3 .5893E+4 .9707E+3 .5003E+3 .7323E+4 0.01

(kN/m 2)(-) -.5417E+2 0.0 0.0 0.0 0.0 -.7315E+4

(+) .1238E+3 .47585E+3 .1712E+3 .141E+3 .7937E+3 .59068E+12

(kN/m2)

(-) -.2693E+3 -.4377E+1 -.186E+1 -.171E+1 -.585E+2 -.8410E+3

(+) .39168E+2 .15831E+3 .1215E+3 .6311E+2 .1101E+3 .58144E+33

(kN/m 2)(-) -.1498E+4 -.6804E+3 -.118E+3 -.677E+2 -.155E+4 -.1491E+4

Table 4 Principal stress envelop values in the tubular unit

2. Dome unit

2.1. Nodal displacements


values SW Sx Sy Sz Case A Case BCupola region

(+) .288E-4 .356E-4 .303E-4 .248E-4 .9697E- 4 0.0

(-) -.288E-4 0.0 0.0 0.0 0.0 -0.970E-4

Openings frame region

(+) .289E-4 0.355E-4 .304E-4 .261E-4 .9691E-4 0.0

Ux

(m)

(-) -.289E-4 0.0 0.0 0.0 0.0 -.970E-4

Cupola region

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(+) .711E-6 .726E-4 .924E-4 .356E-4 .601E-4 0.0

(-) -.113E-3 0.0 0.0 0.0 -.197E-4 -.279E-3

Openings frame region(+) .137E-6 .945E-4 .101E-3 .293E-4 0.821E-4 0.0

Uy

(m)

(-) -.126E-3 0.0 0.0 0.0 0.0 -.330E-3

Cupola region

(+) .392E-4 .2048E-4 .297E-4 .137E-3 .196E-3 0.0

(-) -.392E-4 0.0 0.0 0.0 0.0 -.1965E-3


(+) .329E-4 .1779E-4 .259E-4 .130E-3 .184E-3 0.0

Uz

(m)

(-) -.329E-4 0.0 0.0 0.0 0.0 -.1819E-3

Table 5 Nodal displacements envelop values for the dome unit in global coordinate

2.2. Elements solicitations

Load casesParameter Envelope


Max 72.074 45.399 63.056 45.276 185.87 2.9269 xT

(kN/m)Min -19.267 .5078E-1 .28354E-1 .55245E-2 -1.0459 -76.858

Max 71.585 65.686 62.552 94.052 184.98 -.43048 yT

(kN/m)Min -83.186 .97533E-2 .18031 .93774E-2 -3.7871 -220.72

Max 31.755 18.415 19.988 51.293 96.304 -.39545 xyT

(kN/m)Min -19.227 .23287E-1 .81375E-1 .26868E-1 .38199 -96.316

Max 2.3472 1.3579 1.8845 2.290 6.7816 .156E-3 x M

(kNm/m)Min -1.6434 .64363E-3 .46502E-2 .39353E-3 -.2426E-1 -6.282

Max 3.0471 2.2594 2.6688 6.3298 10.398 .2658 y M

(kNm/m)Min -3.0611 .68979E-3 .40862E-1 .11424E-2 -.6706E-3 -10.98

Max .78156 .93745 1.0946 .60465 2.3068 .2486E-1 xy

M

(kNm/m)Min -1.3448 .15548E-2 .11416E-2 .85148E-3 -.2495E-1 -3.5661

Max 18.067 11.310 13.833 12.317 44.713 .7167E-1 x N

(kN/m)Min -13.929 .78075E-2 .32623E-2 .4277E-2 -.6741E-1 -42.990

Max 16.246 16.847 13.220 37.725 69.924 3.0978 y N

(kN/m)Min -16.237 .18871E-2 .11522 .49896E-2 -.2886E-2 -69.922

Table 6 Element resultant forces envelop values for dome unit

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Load Case A:

Fig. 49 Membrane normal resultant force in x-element direction xT per unit length(isometric view -top left side-, top view -top right side-, left view -bottom left side-, front view -bottom right side-)

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(isometric view -top left side-, top view -top right side-, left view -bottom left side-, front view -bottom right side-)



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Fig. 54 Twisting moment xy M per unit length (Isometric view)


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Fig. 62 twisting moment xy M per unit length (Isometric view)




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2.3. Base shear and nodal reaction forces (Edge effects)



Cupola region

(+) .45E+1 .484E+1 .310E+1 .126E+2 .2337E+2 0.0

(-) -.45E+1 0.0 0.0 0.0 0.0 -.233E+2


(+) .2345E+1 .167E+1 .131E+1 .226E+1 .7323E+1 0.0(-) -.2345E+1 0.0 0.0 0.0 0.0 -.733E+1

Cumul

Fx

(kN)

0.0 .1965E+3 .1158E+3 .409E+3 - -

Cupola region

(+) .875E+1 .2411E+1 .392E+1 .156E+2 .2811E+2 .240E+1

(-) -.394E+1 0.0 0.0 0.0 0.0 -.2215E+2

Fy Openings frame region

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(+) .1333E+2 .1026E+2 .677E+1 .178E+2 .362E+2 0.0

(-) -.307E+1 0.0 0.0 0.0 0.0 -.284E+2

Cumul

(kN)

.432E+3 .190E+3 .239E+3 .7726E+3 - -

Cupola region

(+) .395E+1 .325E+1 .354E+1 .127E+2 .234E+2 0.0

(-) -.395E+1 0.0 0.0 0.0 0.0 -.234E+2


(+) .2156E+1 .1533E+1 .186E+1 .666E+1 .1222E+2 0.0

(-) -.2156E+1 0.0 0.0 0.0 0.0 -.1222E+2

Cumul

Fz

(kN)

0.0 .1115E+3 .925E+2 .343E+3 - -

Cupola region

(+) .109E+1 .239E+0 .4995E+0 .218E+1 .379E+1 .405E-2

(-) -.109E+1 0.0 0.0 0.0 -.310E-2 -.379E+1


(+) .470E+0 .226E+0 .350E+0 .988E+0 .2034E+1 0.0

(-) -.4702E+0 0.0 0.0 0.0 0.0 -.2034E+1

Cumul

Mx

(kNm)

-0.621E-3 .115E+2 .2405E+2 .759E+2 - -

Cupola region

(+) .367E-1 .182E-1 .152E-1 .778E-1 .114E+0 .6345E-2

(-) -.367E-1 0.0 0.0 0.0 -.632E-2 -.114E+0


(+) .3365E-1 .1654E-1 .210E-1 .624E-1 .1336E+0 0.0

(-) -.3667E-1 0.0 0.0 0.0 0.0 -.1337E+0

Cumul

My

(kNm)

.8628E-4 .655E+0 .6155E+0 .231E+1 - -

Cupola region(+) .654E+0 .201E+0 .293E+0 .778E+0 .179E+1 0.0

(-) -.6544E+0 0.0 0.0 0.0 0.0 -.179E+1


(+) .170E-4 .1238E-4 .1422E-4 .114E-4 .549E-4 0.0

(-) -.1654E-4 0.0 0.0 0.0 0.0 -.5371E-4

Cumul

Mz

(kNm)

.1906E-2 .4964E+1 .8652E+1 .209E+2 - -

Table 7 Nodal reaction forces envelop values in dome unit

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2.4. Principal stresses



Cupola region

(+) .4426E+3 .3015E+3 0.4006E+3 .1209E+4 .1584E+4 0.0

(-) -.1374E+2 0.0 0.0 0.0 0.0 -.1573E+4


(+) .5424E+3 .5936E+3 .6741E+3 .1073E+4 .1969E+4 0.0

1

(kN/m 2)

(-) -.9163E+2 0.0 0.0 0.0 0.0 -.133E+4

Cupola region

(+) .1512E+3 .1717E+3 .20546E+3 .19051E+3 .4326E+3 .407E+2

(-) -.1385E+3 -.2743E+1 -.9305E+0 -.4209E+1 -.4654E+2 -.604E+3


(+) .2095E+3 .1802E+3 .1796E+3 .2256E+3 .6826E+3 .32795E+2

2

(kN/m 2)

(-) -.1993E+3 -.5180E+1 -.3548E+1 -.7335E+1 -.1189E+2 -.6415E+3

Cupola region

(+) .2506E+1 .6301E+2 .6403E+2 .65308E+2 .7668E+2 .2205E+3

(-) -.6059E+3 -.8272E+2 -.6181E+2 -.2328E+3 -.6994E+3 -.5671E+3


(+) .1138E+3 .8169E+2 .1061E+3 .14758E+3 .4086E+3 .8943E+2

3

(kN/m 2)

(-) -.683E+3 -.7228E+2 -.636E+2 -.7717E+2 -.6842E+3 -.682E+3

Table 8 Principal stresses envelop values in dome unit

bam report

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