Derivation of stiffness and flexibility for rods and beams by using dual integral equations

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1 Derivation of stiffness and flexibility for rods and beams by using dual integral equations Advisor:Jeng-zong Chen Undergraduate Student:Cheng-Chan Hsieh Department of Harbor and River Engineering, National aiwan Ocean University Keywords: Dual boundary integral equations, Stiffness, Flexibility, Rigid body mode, Singular value decomposition, aplace problem Abstract In this paper, the dual boundary integral formulation is used to determine the stiffness and flexibility for rods and beams by using the direct and indirect methods. he stiffness and flexibility matrices derived by the dual boundary integral equations (DBIEs) are compared well with those derived by the direct stiffness and flexibility methods after considering the sign convention. Since any two boundary integral equations can be chosen for the beam problem, six options by choosing two from the four equations in dual formulation can be considered. It is found that only two options, either displacement-slope (single-layer and double-layer) or displacement-moment (single-layer and triple-layer) formulations in the direct (indirect) method can yield the stiffness matrix except the degenerate scale and a special fundamental solution. he rank deficiency is examined for the influence matrices. Not only rigid body mode in physics but also spurious mode in numerical implementation are found in the formulation by using SVD updating term and document, respectively. Introduction Concept of stiffness and flexibility in mechanics of material is well-known for undergraduate students []. For graduate students, they revisited the stiffness and flexibility matrices in the finite element course []. Rigid body modes occur for free-free bodies in physics as well as spurious modes appear for degenerate scales in numerical implementation []. Felippa et al. constructed the free-free flexibility matrices by using the generalized stiffness inverse through the concept of finite element method []. Besides, Dumont also studied the stiffness by using generalized inverse matrices through variational boundary element formulation [5]. A note to construct the relationship of the stiffness matrice between the finite element method (FEM) and boundary element method (BEM) was published by Pozritidis in 6 [6]. A unified formulation to derive the stiffness and flexibility is not trivial and is the main concern of the present paper. Applications of direct and indirect BEMs to solve rod and beam problems were reported in the textbook of Banerjee [7]. However, only conventional BEM instead of dual BEM was used in the direct method. Besides, only single-layerand double-layer potentials were adopted instead of higher order layer potentials in the indirect method. Here, we will complete the possible alternatives to solve rod and beam problems. DBIEs were developed by Hong and Chen [8] for -D and -D elasticity problems. his formulation can be employed to formulate the one-dimensional problem of rod and beam. Since DBIEs provide more equations than the conventional one, we may wonder the role of additional equations in mathematical aspects. Regarding to the role of dual formulation in computational mechanics, readers can consult with the review article [9]. In the recent years, SVD technique has been applied to solve problems of continumn mechanics [], fictitious-frequency problems [], and spurious eigenvalues []. Based on these

2 successful experiences, SVD updating technique will be employed to study the mathematical structure of the influence matrices derived by using dual formulation. In this report, rank deficiency for the influence matrices is also our concern. he rigid body mode and spurious mode in the dual formulation will be examined through SVD technique. he relation between zero singular values of updating matrices (updating terms and updating document) and nontrivial modes (rigid body mode and spurious mode) will be constructed. Both the rod and beam structures are considered as illustrative examples. Dual boundary integral formulation for rod problems et us consider the rod problem as shown in Fig.. he governing equation for a rod is u () u ( ) EAu () EAu ( ) Fig. Generalized displacement and force dux ( ), x D, () dx where ux ( ) is the axial displacement of the rod, D is the domain between of < x<. By introducing one auxiliary system of the fundamental solution, we have U( x, s) δ( x s) x, < x <, () where δ is Dirac-delta function, x is the field point, and s is the source point. For simplicity, the fundamental solution is selected as U( x, s) x s, () and can be expressed in terms of degenerate kernel as shown in able.by multiplying the auxiliary system in Eq. () with respect to the governing equation and integrating by parts, we have the boundary integral equation as [ ] us () uxu () (,) xs u() xu(,) xs. () By differentiating with respect to the source point s, with to Eq. (), the dual boundary integral equations as shown below us () xsux (,)() U(,)() xstx, (5) [ ] [ ] ts ( ) M( xsux, ) ( ) xstx (, ) ( ), (6) where ts () dus ()/ ds, and the kernels are defined as U( x, s) ( x, s), (7) n EAu ( ) p u p x U( x, s) xs (, ), (8) n U( x, s) M( x, s). (9) nx ns and degenerate kernels are shown in able. able Degenerate kernels for rod problem. Kernels Domain s U( x, s ) ( x, s ) ( xs, ) M ( xs, ) x > s ( ) x s x < s ( ) s x p u By approaching s to + and into Eqs. (5) and (6), we have u() t() [ A] [ B] u ( ) t ( ), () where [ A ] and [ B ] can be found as shown in able. Also, the ranks of influence matrices are calculated.. he stiffness matrix of rods We utilize the simple structure in Fig. to define the notations of generalized displacement and generalized force to connect the FEM notations. EAu () p Fig. Notations of generalized displacements and generalized forces in a simple structure. For the degree of freedom of generalized displacements (d. o. f.) and generalized forces, we have u() u u ru, u ( ) u u () t() t t rt. t ( ) EA t t ()

3 By substituting Eqs. () and () into Eq.(), and the relation between generalized displacement and generalized force is shown below u p [ A] [ B], () u p where A A, () [ ] ru [ ] [ B ] [ B ] rt. (5) he stiffness matrices are defined as t() u() [ KB ] t ( ) u ( ), (6) p u [ KF ]. (7) p u It is found that Eq. (6) fail in constructing the stiffness matrix, and the stiffness matrix can be expressed as the same form of that derived by FEM as shown in able.. he flexibility matrix of rods he flexibility matrix can not be obtained, because the [ A ] matrix is singular in able. We utilize the SVD technique to calculate [ A] and try to get the flexibility matrix of the rod. By employing SVD technique, we have [ A ] [ Φ][ ][ Ψ ], (8) where [ Φ ] and [ Ψ ] are the right and left unitary matrices, and [ ] is a diagonal matrix composed of singular value. It is found that [ Φ ], (9) [ ], () able Stiffness matrix for rod problems using dual BEM. Eq. A B Ψ (). [ ] he [ A ] matrix can be expressed as r [ A] [ ui][ σ i][ vi]. () i where [ σ i ] is the singular value, [ u i ] and [ v i ] are the left and right unitary vectors, respectively. he inverse of the [ A ] matrix is r A [ vi][ σ i] [ ui] i. () he flexibility matrices are defined as u() t() [ FB ] u ( ) t ( ), () u p [ FF ] (5) u p. It is found that Eq. (6) fail in constructing the flexibility matrix, and the flexibility matrix can be expressed as the same form of that derived by FEM as shown in able. Dual boundary integral formulation for beam problems Based on the successful experience of deriving the stiffness for a rod using BEM, we extend the one-dimensional aplace equation to KB K F (6) Rank( A) Rank( B) Rank( K B ) EA Rank( KF ) (7) ( ) Rank A Rank( B) NA NA

4 biharmonic equation for a beam. et us consider the Euler beam problems as shown Fig.. U( x, s) U( x, s) U( x, s) us ( ) ux ( ) u ( x) + u ( x) u ( xuxs ) (, ). x x x () m () m ( ) θ() θ( ) v () u () u ( ) v ( ) Fig. Generalized displacement and force d.o.f. he governing equation for the Euler beam is dux ( ). (6) dx where is the length of the beam, ux ( ) is the lateral displacement, D is the domain between of < x<. By introducing one auxiliary system of the fundamental solution U( x, s) δ( x s), < x <, (7) x where δ is Dirac-delta function, x is field point, and s is the source point. For simplicity, the fundamental solution is selected as U( x, s) x s, (8) and can be expressed in terms of degenerate kernel as ( ) x s, x> s U( x, s). (9) ( ) s x, x< s By multiplying the auxiliary system in Eq. (8) with respect to the governing equation and integrating by parts, we have the boundary integral equation as U( x, s) u( x) us () ux () Uxs (,) dx. () x x he boundary integral equation is derived as. Direct method By rewriting the displacement field, we have () us ( ) U( xsu, ) ( x) + Θ ( xsu, ) ( x) M( xsu, ) ( x) + V( xsux, ) ( ) By differentiating the displacement, the slope, moment and shear force fields can be obtained, θ θ θ θ () u ( s) U ( xsu, ) ( x) + Θ ( xsu, ) ( x) M ( xsu, ) ( x) + V( xsux, ) ( ), m m m m () u ( s) U ( xsu, ) ( x) + Θ ( xsu, ) ( x) M ( xsu, ) ( x) + V ( xsux, ) ( ). v v v v (5) u ( s) U( xsu, ) ( x) + Θ ( xsu, ) ( x) M ( xsu, ) ( x) + V( xsux, ) ( ) where us () is the deflection, θ () s is the slope, ms () is the moment and vs () is the shear force, respectively, and the relations of the sixteen kernels are shown in Fig.. Degenerate kernels of the sixteen kernels in a one-dimensional biharmonic problem are shown in able. Any two boundary integral equations can be chosen, six options can be considered. We utilize the degenerate kernel expansion and substitute them into the two boundary integral equations which are chosen. By approaching s to + and, we have the matrix form as follows u() u () u () u () [ A] [ B], (6) u ( ) u ( ) u ( ) u ( ) where [ A ] and [ B ] are obtained through six formulation ( u θ, u m, u v, θ m, θ v, m v ) as shown in able 5... he stiffness matrix of the Euler beam We utilize a simple structure in the sign convention to define the notations of generalized able Flexibility matrix for rod problems using the dual BEM. Eq. [ A ] [ B ] [ F ] [ F ] B F (6) Rank( A ) Rank( B ) Rank( F B ) EA Rank( F F ) (7) Rank( A ) Rank( B ) NA NA

5 x x x U( x, s) Θ( x, s) M( x, s) V( x, s) s Uθ( x, s) Θθ( x, s) Mθ( x, s) Vθ( x, s) s Um( x, s) Θm( x, s) Mm( x, s) Vm( x, s) s U ( x, s) Θ ( x, s) M ( x, s) V ( x, s) v v v v Fig. Differential operators for the sixteen kernels of the Euler beam. able Kernels Domain x > s x < s U( x, s ) Uθ ( x, s) Um( x, s ) Uv ( x, s) ( x s) ( x s) ( x s) ( x s) x s s x displacement and generalized force to connect the FEM notations. For the degree of freedom of generalized displacements and generalized forces, we have u() u u u () θ θ bu u ( ) u u (7) u ( ) θ θ, u () v v u () m m, bt u ( ) EI v v (8) u ( ) m m since ux ( ) is defined downward. By substituting Eqs. (7) and (8) into Eqs.(6), and the relation between generalized displacement and generalized force is shown below as u v m [ A] θ [ B], (9) u v θ m where [ A] bu A, () [ B ] bt B. () he stiffness matrices are defined as u () u() u () u () [ KB ], () u ( ) u( ) u ( ) u ( ) v u m θ [ KF ]. () v u m θ It is found that only two combinations of Eqs. (), () and Eqs. (), (5) can construct the stiffness matrix, and the stiffness matrix can be expressed as the same form of that derived by FEM as shown in able 5.. Indirect Method Instead of choosing two equations from the dual formulation in the direct BEM, we can also adopt two potentials from single, double, triple and guadrupole potentials as denoted by U-Θ, U M, U V, Θ M, Θ V, and M V formulations. ()Single and double layer approach (U -Θ ) us ( ) U(, s) + Us (, ) + Θ (, s) ψ + Θ ( s, ) ψ. () ()Single and triple layer approach (U M ) us ( ) U(, s) + Us (, ) + M(, s) ψ + M( s, ) ψ. (5) ()Single and guadrupole layer approach (U V ) us ( ) U(, s) + Us (, ) + V(, s) ψ + Vs (, ) ψ. (6) ()Double and triple layer approach ( Θ M ) us ( ) Θ (, s) + Θ( s, ) + M(, s) ψ + M( s, ) ψ. (7) (5)Double and guadrupole layer approach ( Θ V ) us ( ) Θ (, s) + Θ( s, ) + V(, s) ψ + Vs (, ) ψ. (8) (6) riple and guadrupole layer approach ( M V ) us () M(,) s + M( s,) + V(,) sψ + Vs (,) ψ. (9) By approaching s to and to +, and he unknown fictitious densities (, ψ ) can be obtained by u() u() θ() θ() [ A] [ A], ψ u ( ) ψ u ( ) ψ θ( ) ψ θ( ) (5) v() v() m() m() ψ v ( ) ψ v ( ) ψ m ( ) ψ m ( ) [ B] [ B][ A] where [ A ] and [ ]. (5) B are obtained through six formulation as shown in able 6. It is found that the stiffness matrix can be obtained by selecting U Θ and U M formulations. 5

6 Discussion of the rigid body mode and spurious mode If the rigid body term, c, and the linear term, ax, are superimposed in the fundamental solution, we have Ur ( x, s) U( x, s) + ax+ c. By substituting the auxiliary system U( x, s ) into Eq. (), and setting EA,, we have + a a c ac u() u (). u() u () + a a + c ac (5) he [ B ] matrix for a rod is singular when (+ a) c. his results in the degenerate scale problem. According to the Fredholm alternative theorem, the degenerate scale depends on the rigid body term. When a and c /, [ B ] matrix is not invertible and results in a degenerate scale. By employing the SVD technique with respect to the influence B matrices. he spurious matrix, for [ ] mode [ ] satisfies A and [ ] A B [ ] where the spurious mode [ ] mode [ ψ ] are shown in able 7., (5) and the rigid body If the rigid body term, c, and the linear, quadratic and cubic terms, ax, bx and dx are superimposed in the fundamental solution, we have U (, ) (, ) b x s U x s + ax + bx + dx + c. By substituting the auxiliary system Ub ( x, s ) into Eqs. () for the u θ formulation, and setting EI,, we have the + 6d b 6d + b+ 6d u() + 6d b 6d b+ 6d u () u() u () c a cab d + a+ b+ d c a c a b d a b d u () u (). u () u () (5) he [ B ] matrix for a beam is singular when (+ d) a 8c. his results in the degenerate scale problem. When a, b, c /8 and d, [ B ] matrix is not invertible and results in a degenerate scale. By employing the SVD technique with respect to the influence matrix in the u θ formulation, for B matrices. [ ] A, [ A ] and [ ] able 5 Stiffness matrix for the Euler beams by using the direct method. Eqs. A B KB KF u-θ Eqs. () and () Rank( A ) Rank( B ) Rank( K B ) EI Rank( K F ) u-m Eqs. () and () Rank( A ) Rank( B ) Rank( K B) EI Rank( K F ) 6

7 able 6 Stiffness matrix for the Euler beam by using the indirect method. Portential [ A ] [ B ] [ K ] [ B][ A] U-Θ single and double layer U-M single and triple layer Rank( A ) Rank( A ) Rank( B ) Rank( B ) Rank( K ) Rank( K ) According to Fredholm alternative theorem [], the spurious mode [ ] satisfies and the rigid body mode [ ] A [ ] B, (55) ψ satisfies A [ ψ ] A. (56) he spurious mode [ ] and the rigid body mode [ ψ ] are shown in able 7. he mathematical framework of [ A ] and [ B ] is shown in Fig Conclusions Dual boundary integral equations were employed to derive the stiffness and flexibility of the rod and beam which match well with those of FEM. Not only the direct method but also the indirect method was used. It is found that displacement-slope ( u θ ) and displacement-moment ( u m) formulations in the direct method can construct the stiffness matrix. Similarly, the single-double layer approach (U Θ) and single-triple layer approach (U M ) work for the constructing of stiffness matrix in the indirect method. For choosing a special fundamental solution, the stiffness matrix can not be obtained for the degenerate scale. Rigid body mode and spurious mode were studied by using the SVD updating term and document technique. It is found that rigid body mode and spurious mode are imbedded in the right and left unitary vectors of the influence matrices through SVD. 6 References [] Gere, J. M., Mechanics of Materials, homson earning, Inc., California (). [] 陳正宗, 洪宏基, 邊界元素法, 新世界出版社, 台北 (99). [] Chen, J.., W. C. Chen, S. R. in and I.. Chen, Rigid Body Mode and Spurious Mode in the Dual Boundary Element Formulation for he aplace Problems, Computers and Structures, Vol. 8, pp. 95- (). [] Felippa, C. A., K. C. Park and M. R. Justino Filho, he Construction of Free-Free Flexibility Matrices as Generalized Stiffness Inverses, Computers and Structures, Vol. 68, pp. -8 (998). [5] Dumont, Ney. A., Generalized Inverse Matrices and Structural Analysis, Numerical Methods in Continuum Mechanics (). [6] Pozrikidis, C., A Note on he Relation Between he Boundary- and Finite-Element Method with Application to aplace s Equation in wo Dimensions, Engineering Analysis with Boundary Elements, Vol., pp. -7 (6). [7] Banerjee, P. K. and R. Butterfield, Boundary Element Methods in Engineering Science, McGraw-Hill, ondon (98). [8] Hong, H.-K. and J.. Chen, Derivation of Integral Equations in Elasticity. Journal of Engineering Mechanics, ASCE, Vol., pp. 8- (988). 7

8 able 7 Spurious modes and the rigid body modes for a rod and a beam in BEM. Rod Beam.6.6 Spurious mode (generalized force) u '().58 u '().58 Rigid body mode (generalized displacement) u () u () u ().77 u () A.58 ψ ψ A A.577 ψ u '().577 u '().577 u ().577 u () A B spurious mode A A SVD updating term A -.77 A ψ ψ A A [ A, B ] SVD updating document rigid body mode -.6 B -.6 ψ Fig. 5 Mathematical SVD structures of the influence matrices using updating techniques [9] Chen, J.. and H.-K. Hong, Review of Dual Boundary Element Methods with Emphasis on Hypersingular Integrals and Divergent Series, Applied Mechanics Reviews, ASME, Vol.5, No., pp.7-. [] Chen, J.., C. F. ee and S. Y. in, A New Point of View For he Polar Decomposition Using Singular Value Decomposition, Int. J. Comp. Numer. Anal. Appl, Vol., No., pp (). [] Chen, J.., I.. Chen and K. H. Chen, A Unified Formulation For he Spurious and Fictitious Frequencies in Acoustics Using he Singular Value Decomposition and Fredholm Alternative heorem, J. Comp. Acoustics, (6).(Acepted) [] Chen, J..,. W. iu and H.-K. Hong, Spurious and rue Eigensolutions of Helmholtz BIEs and BEMs for a Multiply-Connected Problem, Proc. Royal Society ondon Series A, Vol. 59, pp (). 8

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