- Email: [email protected]
About Verification of Discrete-continual Finite Element Method of Structural Analysis. Part 1: Two-dimensional Problems
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 91 (2014) 2 – 7
XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) (TFoCE 2014)
About Verification of Discrete-Continual Finite Element Method of Structural Analysis. Part 1: Two-Dimensional Problems Pavel A. Akimova,b*, Marina L. Mozgalevaa, Mojtaba Aslamia, Oleg A. Negrozova a
b
Moscow State University of Civil Engineering, 26, Yaroslavskoe Shosse, Moscow, 129337, Russia Samara State University Architecture and Civil Engineering, 194, Molodogvardeyskaya Street, Samara 443001, Russia
Abstract This paper is devoted to verification of so-called discrete-continual finite element method (DCFEM) of structural analysis. Twodimensional problems of analysis of deep beams are under consideration. Formulation of the problem for deep beam with piecewise constant physical and geometrical parameters along one direction (so-called basic direction), solutions obtained by DCFEM and finite element method (FEM) /with the use of ANSYS Mechanical/, their comparison are presented. DCFEM is more effective in the most critical, vital, potentially dangerous areas of structure in terms of fracture (areas of the so-called edge effects), where some components of solution are rapidly changing functions and their rate of change in many cases can’t be adequately considered in the standard FEM. © 2014 2014The TheAuthors. Authors. Published by Elsevier © Published by Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil (http://creativecommons.org/licenses/by-nc-nd/3.0/). Engineeringunder (23RSP). Peer-review responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) Keywords: discrete-continual finite element method; structural analysis; multipoint boundary problem; two-dimensional problems; verification;
1. Operational formulation of multipoint boundary problem of deep beam analysis Let x2 be “basic” direction” while physical and geometrical parameters of structure can be changed arbitrarily along x1 . Operational formulation of corresponding resultant multipoint boundary problem of two-dimensional theory of elasticity at extended domain, embordering considering structure, within DCFEM has the form: U kc L~kU k S k , x2 ( x2b, k , x2b, k 1 ), k = 1, ..., nk 1 ~ ° ~ U k 1 ( x2b, k 0) BkU k ( x2b, k 0) g~k g~k , k = 2, ..., nk ®B k ~ ~ b ° B1 U 1 ( x2 ,1 0) Bn U n 1 ( x2b, n 0) g~1 g~n , ¯ ª 0 º ~ ª 0 E º ; Sk « 1 ~ » ; U k Lk « 1 1 ~ » ( ) L L C L L k k ,vv k ,uv ¼ ¬ k ,vv k ,uu ¬ Lk ,vv Fk ¼ k
k
k
(1)
1
k
ªu k º ; v k «¬ vk »¼
w 2uk ; Ukc w2Uk ;
(2)
* Pavel A. Akimov. Tel.: +7-495-183-5994; fax: +7-495-183-5994. E-mail address: [email protected]
1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) doi:10.1016/j.proeng.2014.12.002
3
Pavel A. Akimov et al. / Procedia Engineering 91 (2014) 2 – 7
ªP k «0 ¬
Lk ,vv ~ Lk ,uv
0 º ; Lk ,uv Ok 2P k »¼
L*k ,uv ; Lk ,vu
Lk ,uv Lk ,vu ; Lk ,vu
~ Fk
ª 0 «w * P ¬ 1 k
w1*Ok º ; Lk ,uu 0 »¼
ªO 2 P k w 1* « k ¬ 0
0º
w 1 ; Ck
P k »¼
0º ªc (Tk G Ƚ ,k )« k ,1 »; 0 c k ,2 ¼ ¬
L*k ,uv ;
(4)
x :k T k Fk G * ,k f k ; T k ( x) ® 1, G ( x) ¯ 0, x : k ; * , k
: is the domain, occupied by structure; x
(3)
wG * , k ; wnk
wT k ; G *c , k ( x ) wn k
(5)
( x1 , x2 ) ; x1 , x2 are coordinates ( x2 corresponds to basic dimension);
b 2, k
x , k = 1, ..., nk are coordinates of boundary cross-sections of structure (in particular, coordinates of cross-sections with discontinuities of the first kind of physical and geometrical parameters of structure; l2 is the length of structure along basic dimension, x2 [0, l2 ] ; : k , k = 1, ..., nk 1 are corresponding fragments of domain : with boundaries
x2b,k and x2
*k , obtained by separation from domain : by cross-sections x2
tended domains, embordering fragments : k , k = 1, ..., nk 1 ; T k : k ; G * ,k
boundary *k
T k ( x1 , x2 ) is the characteristic function of domain
[ nk ,1 nk ,2 ]T is unit normal vector of domain
w: k ; nk
G * , k ( x1 , x2 ) is the delta-function of border *k
x2b,k 1 ; Zk , k = 1, ..., nk 1 are ex-
w: k ; u k , k = 1, ..., nk 1 is the unknown vector of displacements in domain : k ;
~ ~ ~ ~ Bk , Bk , k = 2, ..., nk 1 , B1 , Bn are matrices (operators) of boundary conditions of the fourth order ( x2 independent); g~k , g~k , k = 2, ..., nk 1 , g~1 , g~n are right-side vectors of boundary conditions of the fourth order k
k
~ ( x2 -independent); Fk is the right-side vector in domain : k ; Fk is the vector of body forces in domain : k ; f k is
the boundary traction vector in domain : k ; Ok , Pk are Lame coefficients of material in domain : k ; Ck is the matrix of elastic parameters of the supports (if any); c k ,i is the coefficient of resistance in the direction of the axis Ox i ; w k
w / wxk , w k
w / wxk , k
1, 2 ; vk
w 2uk
u kc ; vkc
w 2 vk .
2. Discrete-continual formulation of multipoint boundary problem of deep beam analysis DCFEM presupposes finite element approximation of extended domain along direction of structure perpendicular to the basic direction, while along basic direction problem remain continual (thus extended domain is divided into discrete-continual finite elements). Resulting multipoint boundary problem for the first-order system of ordinary differential equations with piecewise-constant coefficients within DCFEM [1, 2] has the form: y k(1) Ak y k f k , x2 ( x2b, k , x2b, k 1 ), k 1, 2, ..., nk 1 ° b b ® Bk yk ( x2 , k 0) Bk yk ( x2 , k 0) g k g k , k = 2, ..., nk 1 b b °¯ B1 yk ( x2 ,1 0) Bnk yk ( x2 , nk 0) g1 g nk ,
(6)
where Ak , k 1, 2, ..., nk 1 are matrices of constant coefficients of order n 4 N ; f k , k 1, 2, ..., nk 1 are vectors of size n 4 N ; N 1 is the number of elements along x1 ;
yk
yk ( x2 ) [ ukT ( x2 ) vkT ( x2 ) ]T ;
uk
( k ,1) T n
u
uk ( x2 ) [ (u
(k , p) n
ui( k , p )
u
(k , p) n
)
(u
(k , p) 1
( x2 ) [ u
( k , 2) T n
)
u
(k , p) 2
(7)
... (u T
(k ,N ) T n
T
) ] ; vk
] , p 1, 2, ..., N ; v
( k ,1) T n
vk ( x2 ) [ (v (k , p) n
v
(k , p) n
( k , 2) T n
)
(v (k , p) 1
( x2 ) [ v
)
v
(k , p) 2
... (v
(k ,N ) T n
T
) ] ;
T
] , p 1, 2, ..., N ;
(8) (9)
ui( k , p ) ( x2 ), p 1, 2, ..., N , k 1, 2, ..., nk 1 are functions, which define component of displacement u i in the
b b node with coordinate ( x1p , x2 ) in the interval x2 ( x2,k , x2,k 1 ) ; N 1 is the total number of elements.
4
Pavel A. Akimov et al. / Procedia Engineering 91 (2014) 2 – 7
Solution of problem (6) is accentuated by numerous factors. They include boundary effects (stiff systems) and considerable number of differential equations (several thousands). Moreover, matrices of coefficients of a system normally have eigenvalues of opposite signs and corresponding Jordan matrices are not diagonal. Special method of solution of multipoint boundary problems for systems of ordinary differential equations with piecewise constant coefficients in structural analysis has been developed [1]. Not only does it overcome all difficulties mentioned above but its major peculiarities also include universality, computer-oriented algorithm and computational stability, optimal conditionality of resultant systems and partial Jordan decomposition of matrix of coefficient, eliminating necessity of calculation of root (principal) vectors [1]. 3. Software Discrete-continual finite element for analysis of two-dimensional structures with piecewise constant physical and geometrical parameters in one direction, considering in the distinctive paper, has been realized in software DCFEM2Dpc. Programming environment is Microsoft Visual Studio 2012 Professional and Intel Parallel Studio XE 2013 (Intel Visual Fortran Composer XE 2013). Program is designed for Microsoft Windows 7/8/8.1. Test, model and practically important problems of structural analysis have been solved with the use of DCFEM2Dpc. 4. Numerical Sample Let’s consider rectangular deep beam hinge-supported on two sides ( x2 0 and x2 l with zero displacements u1 u2 0 ). Length of structure ( l ) is equal to 600 cm (Figure 1). Height of structure ( h ) is equal to 300 cm. Additional geometrical parameters (Fig. 1): l1 l2 300 cm; a1 a2 150 cm. Elastic modules of material for the first ( x2 (0, l1 ) ) and the second ( x2 (l1 , l2 ) ) parts of structure ( E1 , E2 ) are equal to 3000 kN/cm2 and to 3500 kN/cm2 respectively. Poisson's ratios of material for the first and the second parts of structure (Q 1 ,Q 2 ) are equal to 0.16 and to 0.14 respectively. Structure is loaded by concentrated forces P1 100 kN and P2 100 kN.
Fig. 1. Design model of structure.
ANSYS Mechanical (ANSYS 15.0) simulation software has been used for solution of problem in terms of FEM. Cartesian coordinate system ( x, y ) has been used. We have x x2 , y x1 . Uniform square mesh 60x30 in ANSYS 15.0 has been constructed from PLANE182 finite element. DCFEM2Dpc simulation software has been used for solution of problem in terms of DCFEM. Uniform approximating mesh in DCFEM2Dpc along x1 includes 30 discrete-continual finite elements. Distributions of displacements u x (along x ), u y (along y ) and stresses V x , V y , W xy in DCFEM2Dpc are presented at Fig. 2.
Pavel A. Akimov et al. / Procedia Engineering 91 (2014) 2 – 7
a)
b)
c)
5
6
Pavel A. Akimov et al. / Procedia Engineering 91 (2014) 2 – 7
d)
e) Fig. 2. (a) DCFEM distribution of displacements u1 (cm); (b) DCFEM distribution of displacements u 2 (cm); (ɫ) DCFEM distribution of stresses V 1,1 (kN/cm2); (d) DCFEM distribution of stresses V 2, 2 (kN/cm2); (e) DCFEM distribution of stresses V1, 2 (kN/cm2).
Comparison of stresses and displacements, obtained by ANSYS Mechanical and DCFEM2Dpc at several crosssections of deep beam are presented at Fig. 3. Thus, we can conclude that the results of analysis obtained by the ANSYS Mechanical (ANSYS 15.0) and DCFEM2Dpc simulation software generally agree well with each other. It was confirmed that DCFEM is more effective in the most critical, vital, potentially dangerous areas of structure in terms of fracture (areas of the so-called edge effects), where some components of solution are rapidly changing functions and their rate of change in many cases can’t be adequately taken into account by the standard finite element method [3, 4]. Acknowledgements This work was financially supported by the Grants of Russian Academy of Architecture and Construction Sciences (7.1.7, 7.1.8) and by the Ministry of education and science of Russia under grant number No 2014/107.
Pavel A. Akimov et al. / Procedia Engineering 91 (2014) 2 – 7
a)
b)
c)
d)
e) Fig. 3. Comparison of results, obtained by ANSYS Mechanical and DCFEM2Dpc: (a) distribution of displacements u1 along section x1 10 cm (cm); (b) distribution of displacements u1 along section x1 0 cm (cm); (c) distribution of displacements u1 along section x2 150 (cm); (d) distribution of stresses V 2,2 along section x2 150 cm (kN/cm2); (e) distribution of stresses V 1,1 along section x1 150 cm (kN/cm2).
References [1] Akimov P.A.: Correct Discrete-Continual Finite Element Method of Structural Analysis Based on Precise Analytical Solutions of Resulting Multipoint Boundary Problems for Systems of Ordinary Differential Equations. // Applied Mechanics and Materials Vols. 204-208 (2012), pp. 4502-4505. [2] Akimov P.A., Sidorov V.N.: Correct Method of Analytical Solution of Multipoint Boundary Problems of Structural Analysis for Systems of Ordinary Differential Equations with Piecewise Constant Coefficients. // Advanced Materials Research Vols. 250-253, 2011, pp. 3652-3655. [3] Barbero E.J.: Finite Element Analysis of Composite Materials Using ANSYS. CRC Press, 2013, 366 pages. [4] Lawrence K.: ANSYS Tutorial Release 14. SDC Publications, 2012, 178 pages.
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ScienceDirect Procedia Engineering 91 (2014) 2 – 7
XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) (TFoCE 2014)
About Verification of Discrete-Continual Finite Element Method of Structural Analysis. Part 1: Two-Dimensional Problems Pavel A. Akimova,b*, Marina L. Mozgalevaa, Mojtaba Aslamia, Oleg A. Negrozova a
b
Moscow State University of Civil Engineering, 26, Yaroslavskoe Shosse, Moscow, 129337, Russia Samara State University Architecture and Civil Engineering, 194, Molodogvardeyskaya Street, Samara 443001, Russia
Abstract This paper is devoted to verification of so-called discrete-continual finite element method (DCFEM) of structural analysis. Twodimensional problems of analysis of deep beams are under consideration. Formulation of the problem for deep beam with piecewise constant physical and geometrical parameters along one direction (so-called basic direction), solutions obtained by DCFEM and finite element method (FEM) /with the use of ANSYS Mechanical/, their comparison are presented. DCFEM is more effective in the most critical, vital, potentially dangerous areas of structure in terms of fracture (areas of the so-called edge effects), where some components of solution are rapidly changing functions and their rate of change in many cases can’t be adequately considered in the standard FEM. © 2014 2014The TheAuthors. Authors. Published by Elsevier © Published by Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil (http://creativecommons.org/licenses/by-nc-nd/3.0/). Engineeringunder (23RSP). Peer-review responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) Keywords: discrete-continual finite element method; structural analysis; multipoint boundary problem; two-dimensional problems; verification;
1. Operational formulation of multipoint boundary problem of deep beam analysis Let x2 be “basic” direction” while physical and geometrical parameters of structure can be changed arbitrarily along x1 . Operational formulation of corresponding resultant multipoint boundary problem of two-dimensional theory of elasticity at extended domain, embordering considering structure, within DCFEM has the form: U kc L~kU k S k , x2 ( x2b, k , x2b, k 1 ), k = 1, ..., nk 1 ~ ° ~ U k 1 ( x2b, k 0) BkU k ( x2b, k 0) g~k g~k , k = 2, ..., nk ®B k ~ ~ b ° B1 U 1 ( x2 ,1 0) Bn U n 1 ( x2b, n 0) g~1 g~n , ¯ ª 0 º ~ ª 0 E º ; Sk « 1 ~ » ; U k Lk « 1 1 ~ » ( ) L L C L L k k ,vv k ,uv ¼ ¬ k ,vv k ,uu ¬ Lk ,vv Fk ¼ k
k
k
(1)
1
k
ªu k º ; v k «¬ vk »¼
w 2uk ; Ukc w2Uk ;
(2)
* Pavel A. Akimov. Tel.: +7-495-183-5994; fax: +7-495-183-5994. E-mail address: [email protected]
1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) doi:10.1016/j.proeng.2014.12.002
3
Pavel A. Akimov et al. / Procedia Engineering 91 (2014) 2 – 7
ªP k «0 ¬
Lk ,vv ~ Lk ,uv
0 º ; Lk ,uv Ok 2P k »¼
L*k ,uv ; Lk ,vu
Lk ,uv Lk ,vu ; Lk ,vu
~ Fk
ª 0 «w * P ¬ 1 k
w1*Ok º ; Lk ,uu 0 »¼
ªO 2 P k w 1* « k ¬ 0
0º
w 1 ; Ck
P k »¼
0º ªc (Tk G Ƚ ,k )« k ,1 »; 0 c k ,2 ¼ ¬
L*k ,uv ;
(4)
x :k T k Fk G * ,k f k ; T k ( x) ® 1, G ( x) ¯ 0, x : k ; * , k
: is the domain, occupied by structure; x
(3)
wG * , k ; wnk
wT k ; G *c , k ( x ) wn k
(5)
( x1 , x2 ) ; x1 , x2 are coordinates ( x2 corresponds to basic dimension);
b 2, k
x , k = 1, ..., nk are coordinates of boundary cross-sections of structure (in particular, coordinates of cross-sections with discontinuities of the first kind of physical and geometrical parameters of structure; l2 is the length of structure along basic dimension, x2 [0, l2 ] ; : k , k = 1, ..., nk 1 are corresponding fragments of domain : with boundaries
x2b,k and x2
*k , obtained by separation from domain : by cross-sections x2
tended domains, embordering fragments : k , k = 1, ..., nk 1 ; T k : k ; G * ,k
boundary *k
T k ( x1 , x2 ) is the characteristic function of domain
[ nk ,1 nk ,2 ]T is unit normal vector of domain
w: k ; nk
G * , k ( x1 , x2 ) is the delta-function of border *k
x2b,k 1 ; Zk , k = 1, ..., nk 1 are ex-
w: k ; u k , k = 1, ..., nk 1 is the unknown vector of displacements in domain : k ;
~ ~ ~ ~ Bk , Bk , k = 2, ..., nk 1 , B1 , Bn are matrices (operators) of boundary conditions of the fourth order ( x2 independent); g~k , g~k , k = 2, ..., nk 1 , g~1 , g~n are right-side vectors of boundary conditions of the fourth order k
k
~ ( x2 -independent); Fk is the right-side vector in domain : k ; Fk is the vector of body forces in domain : k ; f k is
the boundary traction vector in domain : k ; Ok , Pk are Lame coefficients of material in domain : k ; Ck is the matrix of elastic parameters of the supports (if any); c k ,i is the coefficient of resistance in the direction of the axis Ox i ; w k
w / wxk , w k
w / wxk , k
1, 2 ; vk
w 2uk
u kc ; vkc
w 2 vk .
2. Discrete-continual formulation of multipoint boundary problem of deep beam analysis DCFEM presupposes finite element approximation of extended domain along direction of structure perpendicular to the basic direction, while along basic direction problem remain continual (thus extended domain is divided into discrete-continual finite elements). Resulting multipoint boundary problem for the first-order system of ordinary differential equations with piecewise-constant coefficients within DCFEM [1, 2] has the form: y k(1) Ak y k f k , x2 ( x2b, k , x2b, k 1 ), k 1, 2, ..., nk 1 ° b b ® Bk yk ( x2 , k 0) Bk yk ( x2 , k 0) g k g k , k = 2, ..., nk 1 b b °¯ B1 yk ( x2 ,1 0) Bnk yk ( x2 , nk 0) g1 g nk ,
(6)
where Ak , k 1, 2, ..., nk 1 are matrices of constant coefficients of order n 4 N ; f k , k 1, 2, ..., nk 1 are vectors of size n 4 N ; N 1 is the number of elements along x1 ;
yk
yk ( x2 ) [ ukT ( x2 ) vkT ( x2 ) ]T ;
uk
( k ,1) T n
u
uk ( x2 ) [ (u
(k , p) n
ui( k , p )
u
(k , p) n
)
(u
(k , p) 1
( x2 ) [ u
( k , 2) T n
)
u
(k , p) 2
(7)
... (u T
(k ,N ) T n
T
) ] ; vk
] , p 1, 2, ..., N ; v
( k ,1) T n
vk ( x2 ) [ (v (k , p) n
v
(k , p) n
( k , 2) T n
)
(v (k , p) 1
( x2 ) [ v
)
v
(k , p) 2
... (v
(k ,N ) T n
T
) ] ;
T
] , p 1, 2, ..., N ;
(8) (9)
ui( k , p ) ( x2 ), p 1, 2, ..., N , k 1, 2, ..., nk 1 are functions, which define component of displacement u i in the
b b node with coordinate ( x1p , x2 ) in the interval x2 ( x2,k , x2,k 1 ) ; N 1 is the total number of elements.
4
Pavel A. Akimov et al. / Procedia Engineering 91 (2014) 2 – 7
Solution of problem (6) is accentuated by numerous factors. They include boundary effects (stiff systems) and considerable number of differential equations (several thousands). Moreover, matrices of coefficients of a system normally have eigenvalues of opposite signs and corresponding Jordan matrices are not diagonal. Special method of solution of multipoint boundary problems for systems of ordinary differential equations with piecewise constant coefficients in structural analysis has been developed [1]. Not only does it overcome all difficulties mentioned above but its major peculiarities also include universality, computer-oriented algorithm and computational stability, optimal conditionality of resultant systems and partial Jordan decomposition of matrix of coefficient, eliminating necessity of calculation of root (principal) vectors [1]. 3. Software Discrete-continual finite element for analysis of two-dimensional structures with piecewise constant physical and geometrical parameters in one direction, considering in the distinctive paper, has been realized in software DCFEM2Dpc. Programming environment is Microsoft Visual Studio 2012 Professional and Intel Parallel Studio XE 2013 (Intel Visual Fortran Composer XE 2013). Program is designed for Microsoft Windows 7/8/8.1. Test, model and practically important problems of structural analysis have been solved with the use of DCFEM2Dpc. 4. Numerical Sample Let’s consider rectangular deep beam hinge-supported on two sides ( x2 0 and x2 l with zero displacements u1 u2 0 ). Length of structure ( l ) is equal to 600 cm (Figure 1). Height of structure ( h ) is equal to 300 cm. Additional geometrical parameters (Fig. 1): l1 l2 300 cm; a1 a2 150 cm. Elastic modules of material for the first ( x2 (0, l1 ) ) and the second ( x2 (l1 , l2 ) ) parts of structure ( E1 , E2 ) are equal to 3000 kN/cm2 and to 3500 kN/cm2 respectively. Poisson's ratios of material for the first and the second parts of structure (Q 1 ,Q 2 ) are equal to 0.16 and to 0.14 respectively. Structure is loaded by concentrated forces P1 100 kN and P2 100 kN.
Fig. 1. Design model of structure.
ANSYS Mechanical (ANSYS 15.0) simulation software has been used for solution of problem in terms of FEM. Cartesian coordinate system ( x, y ) has been used. We have x x2 , y x1 . Uniform square mesh 60x30 in ANSYS 15.0 has been constructed from PLANE182 finite element. DCFEM2Dpc simulation software has been used for solution of problem in terms of DCFEM. Uniform approximating mesh in DCFEM2Dpc along x1 includes 30 discrete-continual finite elements. Distributions of displacements u x (along x ), u y (along y ) and stresses V x , V y , W xy in DCFEM2Dpc are presented at Fig. 2.
Pavel A. Akimov et al. / Procedia Engineering 91 (2014) 2 – 7
a)
b)
c)
5
6
Pavel A. Akimov et al. / Procedia Engineering 91 (2014) 2 – 7
d)
e) Fig. 2. (a) DCFEM distribution of displacements u1 (cm); (b) DCFEM distribution of displacements u 2 (cm); (ɫ) DCFEM distribution of stresses V 1,1 (kN/cm2); (d) DCFEM distribution of stresses V 2, 2 (kN/cm2); (e) DCFEM distribution of stresses V1, 2 (kN/cm2).
Comparison of stresses and displacements, obtained by ANSYS Mechanical and DCFEM2Dpc at several crosssections of deep beam are presented at Fig. 3. Thus, we can conclude that the results of analysis obtained by the ANSYS Mechanical (ANSYS 15.0) and DCFEM2Dpc simulation software generally agree well with each other. It was confirmed that DCFEM is more effective in the most critical, vital, potentially dangerous areas of structure in terms of fracture (areas of the so-called edge effects), where some components of solution are rapidly changing functions and their rate of change in many cases can’t be adequately taken into account by the standard finite element method [3, 4]. Acknowledgements This work was financially supported by the Grants of Russian Academy of Architecture and Construction Sciences (7.1.7, 7.1.8) and by the Ministry of education and science of Russia under grant number No 2014/107.
Pavel A. Akimov et al. / Procedia Engineering 91 (2014) 2 – 7
a)
b)
c)
d)
e) Fig. 3. Comparison of results, obtained by ANSYS Mechanical and DCFEM2Dpc: (a) distribution of displacements u1 along section x1 10 cm (cm); (b) distribution of displacements u1 along section x1 0 cm (cm); (c) distribution of displacements u1 along section x2 150 (cm); (d) distribution of stresses V 2,2 along section x2 150 cm (kN/cm2); (e) distribution of stresses V 1,1 along section x1 150 cm (kN/cm2).
References [1] Akimov P.A.: Correct Discrete-Continual Finite Element Method of Structural Analysis Based on Precise Analytical Solutions of Resulting Multipoint Boundary Problems for Systems of Ordinary Differential Equations. // Applied Mechanics and Materials Vols. 204-208 (2012), pp. 4502-4505. [2] Akimov P.A., Sidorov V.N.: Correct Method of Analytical Solution of Multipoint Boundary Problems of Structural Analysis for Systems of Ordinary Differential Equations with Piecewise Constant Coefficients. // Advanced Materials Research Vols. 250-253, 2011, pp. 3652-3655. [3] Barbero E.J.: Finite Element Analysis of Composite Materials Using ANSYS. CRC Press, 2013, 366 pages. [4] Lawrence K.: ANSYS Tutorial Release 14. SDC Publications, 2012, 178 pages.
7