Combined approximations method for reanalysis of natural

Transkrypt

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
Combined approximations method for reanalysis of natural frequencies
of reinforced buildings in mining areas
1,2
Krystyna Kuzniar1, Maciej Zajac2
Institute of Technology, Pedagogical University of Cracow, Podchorazych 2, 30-084 Krakow, Poland
email: [email protected], [email protected]
ABSTRACT: In the paper combined approximations hybrid method (CA) is proposed as an effective reanalysis approach for
computation of natural frequencies of actual reinforced 12 storey prefabricated buildings located in the most seismically active
mining region in Poland – Legnica-Glogow Copperfield (LGC). The reanalysis method is intended to analyse modified
structures using some information about the structures before modification and without solving the complete set of equations in
case of the new (modified) object. This idea leads to the computational effort reduction. Changes of building stiffness and mass
resulted from the inner or exterior structural reinforcements were taken into account. Buildings were modelled using finite
element method and the special code of CA algorithm was drawn up in Matlab software. CA procedure was applied in case of
the same number as well as in case of increasing number of degrees of freedom (DOF) in structure after modification. Computed
values of natural frequencies of building vibrations were compared with experimental ones obtained for actual structures. The
results show that in case of the small as well as the large modifications of building structure, the eigenvalue analysis using CA
method is much faster with no significant decrease of the accuracy.
KEY WORDS: Eigenvalue reanalysis; Combined approximations method; Paraseismic excitations; Prefabricated buildings.
1
INTRODUCTION
Precast reinforced concrete apartment and public utility
buildings constructed in 20th century represent a significant
part of all buildings, especially in Central and Easter Europe.
Using precast reinforced concrete panels technology made the
building production process much easier, faster and cheaper.
However some disadvantages of this technology solution
are observed nowadays. The limited building design
flexibility as well as the age of the structures seem to be the
biggest problem for current users. Taking into account for
instance new environmental regulations, safety requirement
and expectations of today’s lodgers, the modernization of
precast concrete buildings become necessary. Mainly,
modernization deals with heat insulation, additional storey,
new door openings or widening the existing ones. For public
utility buildings main works deal with adaptation to the needs
of the disabled.
A lot of prefabricated buildings are located in mining areas
and they are subjected to paraseismic excitations induced by
mining tremors. Although these tremors are strictly connected
with the human activity and can usually be observed only in
the mining regions, they differ considerably from other
paraseismic vibrations. They are not subject to human control
and they are random events with respect to the time, place and
magnitude likewise earthquakes. Most of the structures have
not been designed to carry this kind of load. So modernization
and reinforcement of the buildings becomes necessary to
assure safety in use of them [6, 17]. Such modernization
increases the stiffness of building.
For instance, various changes in structures of prefabricated
buildings are carried out in the most seismically active mining
region in Poland – Legnica-Glogow Copperfield (LGC) where
mine-induced rockbursts excite tremors comparable to low
intensity earthquakes, characterized by energies run up to 1E10
J, surface horizontal vibrations reaching even 0.2 acceleration
of gravity (g) and vertical components reaching 0.3g.
Every significant change in structure causes the changes in
its dynamic properties, among them – in natural frequencies
of vibrations which are used in response spectrum method [8]
in case of kinematic excitations. Therefore new computations
are necessary in every case of significant modifications of
building structure, especially for buildings subjected to
paraseismic or seismic excitations.
Computation of natural frequencies of vibrations can be
carried out using various methods. In case of simple models of
structures it is possible to use analytic formulas. The
eigenproblem for more complicated ones needs to be solved
using numerical methods. Nowadays finite element method
(FEM) [23, 24] is one of the most popular. In spite of fast
progress in hardware and software, computation of the natural
vibration frequencies of the modified actual buildings using
finite element method usually needs a very long
computational time because of a huge number of degrees of
freedom of the models, the large scale problem and various
modification variants discussed.
That is why application of reanalysis techniques
[4, 9, 10, 12] for computation of natural frequencies of
modified structures is proposed in this paper. Reanalysis
methods are intended to analyse modified structures using
some information about the structure before modification and
without solving the complete set of equations in case of the
new (modified) object. This idea leads to the computational
effort reduction.
In the paper combined approximations (CA) hybrid method
was applied and verified as an eigenvalue reanalysis approach
in case of modifications of actual typical apartment high (12
3861
storey) buildings located in LGC mining region in Poland.
Changes of building stiffness and mass resulted from the inner
or exterior structural reinforcements were taken into account.
Buildings were modelled using finite element method and the
special code of CA algorithm was drawn up in Matlab [25]
software. CA procedure was applied in case of the same
number as well as in case of increasing number of degrees of
freedom (DOF) in structure after modification. Computed
values of natural frequencies of building vibrations were
compared with experimental ones obtained for actual
structures.
2
COMBINED APPROXIMATIONS HYBRID METHOD
The main aim of combined approximations (CA) method
[10, 13] application is to quicken computing by reduction of
the number of eigenproblem equations.
In the CA algorithm, basis vectors (global approximations),
containing information about the initial structure and
modifications, are computed by the terms of the binomial
series (local approximation).
Equations (1) and (2) describe the eigenproblems for
structure before and after modification, respectively,
KΦ i = λi MΦ i , i = 1, K , p,
(1)
K M Φ Mi = λMi M M Φ Mi , i = 1,K, p,
(2)
where: K, KM - stiffness matrices; M, MM - mass matrices;
Φi, ΦMi - mode shapes; λi, λMi - eigenvalues before and after
structure modification, respectively.
Substitution matrices KM and MM in equation (2) for the
relations (3)
K M = K + ΔK
M M = M + ΔM
,
(3)
(where ΔK, ΔM - changes in stiffness and mass matrices
corresponding to the geometrical changes of the structure)
allows to write the eigenproblem for the modified structure
using equation (4) with information about the initial model
parameters and changes caused by modifications.
( K + Δ K )Φ M i = λ M i ( M + Δ M )Φ M i .
(4)
Computation of the basis vectors matrix rB according to
equation (5) is the next step of CA approach [14].
rB = [r1 , r2 , K, rs ],
(5)
where: r1,…,rs – the basis vectors, s – the number of basis
vectors; s is much smaller than the number of degrees of
freedom.
The first basis vector r1 is given by formula (6) [14]:
r1 = K −1 M M Φ i .
(6)
Each next vector is calculated using the previous one and
matrix B:
rk = −Brk-1 , k = 2,3, K , s,
3862
(7)
B = K −1 Δ K .
(8)
The following step is the evaluation of reduced stiffness KR
and mass MR matrices:
K R = rBT K M rB
M R = rBT M M rB
.
(9)
The first (lowest) eigenvalue λR1 of reduced eigenproblem
described by Equation (10), represents the value of natural
vibration frequency corresponding to the adequate value
which is obtained from full modified eigenproblem using
Equation (2) (y1 - vector of coefficients):
K R y1 = λR1M R y1.
(10)
In case of modification where the number of DOF is
increased [2, 18, 19, 22], the modification of CA algorithm is
necessary because of changing of the number of equations.
Sizes of the stiffness matrix and mass matrix are increased
accordingly in such case of structure modification. Having
a different number of DOF of the models of the initial and
modified structure, it is impossible to define ΔK, ΔM
matrices which represent real changes between initial and
modified structure and are necessary for basis vectors
computing, cf. Equations (3)-(8).
The new pseudo initial model with added degrees of
freedom needs to be created, because the sizes of
eigenproblem before and after structure modification have to
be equal.
Pseudo initial model consists of Kf (Eq. 11) and Mf (Eq. 12)
matrices that are created using stiffness and mass matrices of
initial model (K, M) and new sub-matrices: KMn, KMp, KMnp,
MMn, MMp, MMnp that represent the parts of KM, MM
corresponding to the new DOF.
αK Mp ⎤
,
αK Mnp ⎥⎦
(11)
α M Mp ⎤
⎡ M
Mf =⎢
⎥.
⎣αM Mn αM Mnp ⎦
(12)
⎡ K
Kf =⎢
⎣αK Mn
In case of the modification where the number of DOF is
increased it is necessary to choose the proper value of α
parameter that is a scalar multiplier to be selected from the
range (0, 1] [11, 15].
Now, equations (3) can be written as follow:
K M = K f + ΔK f
M M = M f + ΔM f
,
(13)
where ΔKf, ΔMf represent the changes between pseudo initial
and modified model.
Additionally it is important to define new pseudo initial
eigenvector Φfi for pseudo initial model [3], which is applied
for calculation of the first basis vector according to Equation
(5). Vector Φfi consists of two parts: the elements of initial
model eigenvector Φi and the vector ΔΦi containing elements
related to new degrees of freedom.
Further part of CA algorithm remains unchanged.
3
ANALYSED BUILDINGS
In the paper three actual 12 storey precast concrete apartment
buildings constructed according to one of Polish prefabricated
systems – system WWP [6, 7] were analysed.
The first one (building “A”) is typical for WWP system.
The second one (building “B”) was reinforced because of
mine-induced kinematic excitations with the inner and
exterior structural reinforcements [5, 28]. In the building “C”
(the third one) only the inner structural reinforcements were
introduced as the modifications. All the analysed buildings are
located in the seismically active mining region in Poland –
Legnica-Glogow Copperfield (LGC).
Transverse-longitudinal load-bearing concrete wall system
has been applied in WWP prefabricated system. The thickness
of the reinforced concrete walls is 14cm and the storey height
is 2.7m [6, 7]. Precast gable and curtain walls consist of three
layers: the thermal insulation layer and the inner and outer
reinforced concrete layers.
The buildings are founded directly on the ground using
concrete strip foundations and there are solid reinforced
concrete basements (wall thickness - 30cm) in the analysed
buildings [5, 28].
Figure 2. Floor plan of typical segment of WWP system [6].
a)
x
Detail “1”
y
Detail “2”
a)
b)
b)
Figure 1. Analysed structures: a) building “A” - typical for
system WWP; b) building “B” with inner and exterior
structural reinforcements.
c)
Figure 3. Reinforcement of building “B”: a) Floor plan with
reinforcements marked with bold line, where: (1) and (7) –
new monolithic concrete spans; (3) and (5) – new monolithic
load-bearing longitudinal walls; (2), (4), (6) – reinforcements
of load-bearing transverse walls; b) Detail “1”; c) Detail “2”.
3863
Each of the three buildings is divided into single and double
stairway segments by expansion gaps. The segments are
connected to each other using the lightweight “wood
concrete” called arbolit [26] in each floor plan.
Examples of the analysed buildings are presented in Figure
1. Figure 2 shows the floor plan of typical segment of such
buildings.
The range of strengthening in case of the building "B" is
presented in Figure 3. New monolithic concrete spans, built at
both gable walls are labelled as (1) and (7) in Figure 3a. The
height of extra spans is 7 and 10 storeys respectively [5, 28].
New monolithic load-bearing longitudinal walls described as
(3) and (5) in Figure 3a (thickness of 20cm) were also
constructed. Additionally, some of the existing transverse
load-bearing walls (described as 2, 4, 6 in Figure 3a) as well
as the basement walls were reinforced by wall thickening [5,
28]. FEM models of the building “B” parts with
reinforcements are shown in Figure 3b and Figure 3c.
4
NUMERICAL MODELS OF BUILDINGS
Numerical computations of natural frequencies of vibrations
of analysed buildings were carried out in known FEM system
– Ansys [27]. Building elements such as continuous concrete
strip foundations, monolithic walls of basement, precast
concrete load-bearing walls, curtain walls, stairways, lift
shafts, air-shafts, flat roof were taking into account and
modelled using 4-node elastic shell elements with 6 degrees of
freedom.
a)
Figure 5. Model of foundation and basement of reinforced
building “B”.
The connections of load-bearing prefabricated walls with
the floors were modelled as fixed and 40% of changing load
was attached at each floor level.
The number of degrees of freedom of models of buildings
before modification (model “A”) and after strengthening of
the load-bearing walls and reinforcing by the extra building
spans (model “B”) is 612732 and 672138 respectively. Finite
element models of analysed structures are presented in Figure
4.
In case of elements of structural monolithic concrete
members (foundation, basement, new walls and wall
strengthening) one value of Young modulus E equal 23.1GPa
was used. Figure 5 shows the model of foundation and
basement of reinforced building “B”.
The influence of panel joints on the wall stiffness was taken
into consideration in the models. Young modulus (E)
reduction of structural elements was proposed for this
purpose. Several attempts to select the proper value of the
reduced Young modulus were carried out and E = 20 GPa was
applied in the computations.
In the analysed models non-structural elements such as for
instance division walls, stairways, lift shafts and air-shafts
were also taken into consideration. From the results of
investigations presented in [16] it is visible that the influence
of non-structural elements on dynamic characteristics of
buildings can be significant.
The division walls in the analysed buildings are made of
cavity brick, thickness – 6.5cm. They were modelled as shells
fixed connected with load-bearing panels. Shell models of the
division walls (dark color) in one-stairway segment of
building are presented in Figure 6.
The applied material parameters of structural elements are
shown in Table 1.
Soil flexibility was taken into consideration using
translation and rotation spring elements according to Savinov
soil model [20].
b)
Figure 4. Finite element models of analysed structures:
a) building “A”; b) building “B”.
3864
Figure 6. Shell models of division walls (dark color) in onestairway segment of building.
Table 3. Comparison of computed values of the natural
vibration frequencies of the model of typical building “A”
and the model of modified building “C”.
Table 1. Material parameters for FEM elements.
FEM element
Foundation
Basement
Prefabricated
elements
Reinforced elements
Cavity brick wall
Arbolit [26]
3
E [GPa]
23.1
23.1
20
ν
0.2
0.2
0.2
ρ [kg/m ]
2500
2500
2500
23.1
0.72
0.9
0.2
0.25
0.25
2500
1400
700
To verify finite element models material and structure
assumptions, the computed results were compared with
experimental ones obtained in [21] for actual structures.
Values of natural frequencies of horizontal vibrations of
analysed buildings are presented in Table 2. It is visible that
the computational accuracy is sufficient in engineering
practice.
Table 2. Comparison of computed and experimental values of
natural frequencies of building horizontal vibrations before
and after modification.
Building
“A”
“B”
Natural frequencies of horizontal building
vibrations [Hz]
computed
experimental [21]
f1x
1.59
1.60 – 1.63
f1y
1.73
1.71 – 1.76
f1x
1.79
1.64 – 1.71
f1y
2.47
2.17 – 2.28
The another type of modifications (introduced to typical
WWP building – building “A”) does not change the number
of DOF of the FEM model. As was said, in the building “C”
only the inner structural reinforcements were introduced. The
modifications relate to strengthening by thickening of the
existed monolithic basement and precast load-bearing
concrete walls (thickening of shell elements in the model of
the building “C”). Thus the number of DOF of the model of
building “A” and the number of DOF of the model of building
“C” are the same and equal 612732.
Floor plan of the model “C” with reinforcements marked
with bold line is shown in Figure 7. Thickened gable walls are
labelled as (1) and (3) and the thickened transverse loadbearing wall is described as (2).
x
y
Natural frequencies of horizontal building vibrations [Hz]
Building “A”
Building “C”
f1x
1.59
1.62
f1y
1.73
1.74
Comparison of computed values of the natural horizontal
vibration frequencies of the model of typical building “A” and
the model of modified building “C” is presented in Table 3.
It is visible that the inner reinforcements (small
modifications of the structure) applied in building “C”
increase the values of natural frequencies of horizontal
building vibrations, especially in transverse (“x”) direction.
5
REANALYSIS RESULTS
The accuracy of combined approximations hybrid method
algorithm (CA) as the reanalysis approach was verified in
evaluation of the first natural frequency of horizontal
vibrations of modified buildings subjected to mine-induced
tremors.
Two cases of eigenvalue reanalysis corresponding to two
types of building modifications were analysed. The first one
concerns strengthening of existing walls without changing the
number of DOF of FEM model (building ”C”) in relation to
the number of DOF of model of the initial structure (building
“A”). In the second one the inner and exterior structural
reinforcements (building “B”) are taken into account –
increasing the number of DOF of FEM model in relation to
the number of DOF of model of the initial structure.
To illustrate accuracy of the results obtained using CA
method fractions r1 and relative errors Errf1 were computed:
r1 = f1CA f1FEM ,
Errf1 =
f1FEM − f1CA
(14)
⋅ 100% ,
f1FEM
(15)
where: f1CA, f1FEM – the first natural frequency of building
vibrations computed using CA method and FEM, respectively.
As the example, relative errors Errf1 of the first natural
frequency of horizontal vibrations in transverse direction of
building “B” obtained using CA method, in dependence of
the various number of basis vectors are presented in Table 4.
As one can see, using CA method with only 10 basis vectors
ensures relative errors not greater than 8% by the reduction of
number of eigenproblem equations from almost 700000 to 10.
Comparison of computed and experimental values of the
first natural vibration frequency of the modified building “B”
is presented in Table 5.
Table 4. Relative errors Errf1 of the first natural frequency of
transverse vibrations of the modified building “B” obtained
using CA method.
Figure 7. Floor plan of the model “C” with reinforcements
marked with bold line; where: (1) and (3) – the thickened
gable walls, (2) – the thickened transverse load-bearing wall.
No. of basis vectors
2
5
10
20
Errf1 [%]
55
27
7.8
6.7
3865
Table 8. Values of fraction r1 in case of model of modified
building “B” in the function of number of basis vectors
and value of α parameter.
Table 5. Comparison of computed and experimental values of
the first natural vibration frequency of the building “B”.
Experimental
[21]
f1 [Hz]
FEM
model
1.64-1.71
CA
(20 basis vectors)
α =10-6
1.91
1.79
α
-3
It is visible that a little worse accuracy of the computed
frequency was achieved using CA method instead of FEM.
But the result (Errf1 less than 7%) is acceptable in engineering
practice.
In Table 6 fraction r1 and Errf1 for model “C” is presented.
As one can see, using CA method with only ten basis vectors
ensures relative errors not greater than 3% (reduction of
number of equations from almost 613000 to 10).
Table 6. Fraction r1 and Errf1 of the first natural vibration
frequency computed in case of the model “C”.
r1
2
1.13
5
1.04
10
1.03
20
1.01
Errf1 [%]
12.5
3.9
2.5
0.8
Additionally, the number of numerical operations which are
necessary to obtain natural frequencies of building vibrations
using CA was estimated and compared with the number of
numerical operations in very popular subspace iteration
method [1].
Values of fraction NSIM / NCA [13] (NSIM, NCA – number of
numerical operations for subspace iteration method and
combined approximations method in case of single eigenvalue
computation) depending on number of basis vectors in case of
the model of building “B” are presented in Table 7.
It is visible that using CA method with 20 basis vectors
reduces the number of necessary numerical operations more
then 20 times in proportion to application of subspace
iteration method.
In case of modification that increase the number of degrees
of freedom, it is very important to choose proper value of α
parameter that assure convergence of CA method.
In case of models of actual buildings with significant
number of DOF and large modifications, choosing α = 0.001
that is suggested in some papers [4, 12] does not guarantee to
obtain satisfying results. Therefore the tests to find
a relationship between the value of α parameter and “size” of
model and its modifications were carried out.
In Table 8 values of fraction r1 obtained in case of building
“B” in the function of number of basis vectors and value of α
parameter are presented. It is visible that the choice of α equal
or smaller than 10-6 provides the results accurate enough.
Table 7. Values of fraction NSIM / NCA in the function
of number of basis vectors in case of the model of modified
building “B”.
2
3
4
5
10
20
NSIM / NCA
216
144
108
86
43
21
3866
10
10-6
10-8
6
1
185
7.02
7.02
r1 = f1CA / f1FEM
3
5
10
180
174
116
1.49
1.27
1.08
1.41
1.31
1.09
20
82
1.07
1.07
CONCLUSIONS
In the paper combined approximations hybrid method (CA)
was proposed as a tool for reduction of computational effort in
case of reanalysis of natural frequencies of reinforced wallbearing prefabricated buildings subjected to mine-induced
tremors. Large and small modifications of high apartment
buildings were analysed.
It is clear from the obtained results that application of CA
method leads to significant decreasing of number of necessary
algebraic operations what makes the numerical analysis much
faster.
CA method is efficient reanalysis procedure for computing
of natural vibration frequencies of structures with an accuracy
sufficient in engineering practice.
7
ACKNOWLEDGMENTS
Numerical analysis in the paper has been supported by ACK
CYFRONET AGH software. Grant no.MNiSW/Zeus_lokalnie
/UP/018/2012, MNiSW/SGI3700/UP/018/2012.
REFERENCES
[1]
K.J. Bathe and S. Ramaswary, An accelerated subspace iteration
method, Computer Methods in Applied Mechanics and Engineering,
23(2), 313-331, 1980.
[2] S.H Chen, C. Huang and Z.J. Cao, Structural modal reanalysis of
topological modifications, Shock and Vibration, 7, 15-21, 2000.
[3] S.H. Chen and F. Rong, A new method of structural modal reanalysis
for topological modifications, Finite elements in analysis and design,
38, 1015-1028, 2002.
[4] S.H. Chen, X.W. Yang and H.D. Lian, Comparison of several
eigenvalue reanalysis methods for modified structures, Structural and
Multidisciplinary Optimization, 20(4), 253-259, 2000.
[5] A. Cholewicki, A. Bociąga and W. Kotlicki, Static analysis of XI-storey
external reinforced prefabricated buildings located in Polkowice,
Warsaw, Poland, 1982 (in Polish).
[6] A. Cholewicki, and J. Szulc, Provisioning rules of buildings in
designing phase, Kwiatek J., Protection of buildings in mining areas,
642-698, GIG, Katowice, Poland, 1998 (in Polish).
[7] Z. Dzierżewicz and W. Starosolski, Precast concrete panel building
systems in Poland in the years 1970-1985, Wolters Kluwer Polska,
Warsaw, Poland, 2010 (in Polish).
[8] A.K. Gupta, Response Spectrum Method In Seismic Analysis and
Design of Structure, CRC PRESS, NY, USA, 1992.
[9] U. Kirsch, Approximate reanalysis methods, in: M.P. Kamat (ed).
Structural optimization, AIAA, Washington D.C., USA, 1993.
[10] U. Kirsch, Combined approximations – a general reanalysis approach
for structural optimization, Structural and Multidisciplinary
Optimization, 20, 97-106, 2000.
[11] U. Kirsch, A unified reanalysis approach for structural analysis. Design
and optimization, Structural and Multidisciplinary Optimization, 25, 6785, 2003.
[12] U. Kirsch, Reanalysis of structure – A Unified Approach for Linear,
Nonlinear, Static and Dynamic System. Solid Mechanics And Its
Applications 151, Springer, Waterloo, England, 2008.
[13] U. Kirsch, M. Bogomolni and I. Sheinman, Efficient procedures for
repeated calculations of the structural response using combined
approximations, Structural and Multidisciplinary Optimization, 32, 435446, 2006.
[14] U. Kirsch, M. Bogomolni, and I. Sheinman, Efficient Dynamic
Reanalysis of Structures, Journal of Structural Engineering, 133(3),
440-448, 2007.
[15] U. Kirsch and P.Y. Papalambros, Exact and accurate reanalysis of
structures for geometrical changes, Engineering With Computers, 17,
363-372, 2001.
[16] A. Kowalska, Analysis of influence of the non-structural elements on
dynamic characteristics of buildings, Cracow, Poland, 2007 (in Polish).
[17] K. Kuźniar, E. Maciąg, and T. Tatara, Prediction of Building
Foundation Response Spectra from Mining-Induced Vibrations using
Neural Networks, Mining & Environment, 4(4), 50-64, 2010 (in Polish).
[18] P. Level, Y. Gallo, T. Tison and Y. Ravalard, On an extension of
classical modal reanalysis algorithms: the improvement of initial
models, Journal of Sound and Vibration, 186(4), 551-560, 1995.
[19] Z.G. Li, C.W. Lim and B.S. Wu, A comparison of several reanalysis
methods for structural layout modifications with added degrees of
freedom, Structural and Multidisciplinary Optimization, 36, 403-401,
2008.
[20] J. Lipiński, Machine Foundation, Arkady, Warsaw, Poland, 1985 (in
Polish).
[21] E. Maciąg and W. Kowalski, Estimation of changes of dynamic
properties of buildings subjected to mining tremors, Folia Scientiarium
Universitatis Technicae Resoviensis, Mechanics, 60, 405-414, 2002 (in
Polish).
[22] B.S. Wu and Z.G. Li, Static reanalysis of structures with added degrees
of freedom, Communications in Numerical Methods in Engineering, 22,
269-281, 2006.
[23] O.C. Zienkiewicz, The Finite Element Method, McGraw-Hill, London,
England, 1997.
[24] O.C. Zienkiewicz and R.L. Taylor, Finite Element Method for Solid and
Structural Mechanics, Elsevier Butterworth Heinemann, Oxford,
England, 2005.
[25] Manual Matlab 7.6.0.324, 2008.
[26] Novye stroitel’nye materialy: Arbolit. Moscow, 1968.
[27] Release 11.0 Documentation for Ansys, 2007.
[28] Technical Building Design, Structural design of reinforced XI-storey
apartment building, Miedziana Street, Polkowice, Poland, 1989 (in
Polish).
3867

Combined approximations method for reanalysis of natural

Transkrypt

Podobne dokumenty

MARK S.C. - Komplexe Bauinvestitionen, Neu

BoZPE-2012-1_Spis tresci

National fund for environmental protection

this PDF file - Archives of Mining Sciences