LEM for twisted re-entrant angle sections

Computers and Structures 133 (2014) 149–155

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LEM for twisted re-entrant angle sections Antonina Pirrotta ⇑ Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale, Dei Materiali (DICAM), Università degli Studi di Palermo, Viale delle Scienze, 90128 Palermo, Italy

a r t i c l e

i n f o

Article history: Received 25 January 2013 Accepted 27 November 2013 Available online 2 January 2014 Keywords: Torsion Complex potential function Re-entrant angles Stress field

a b s t r a c t In this paper an innovative numerical method named as line element-less method, LEM, for finding solution of torsion problem has been extended to all shaped sections, including sections possessing re-entrant angles at their boundary. The response solution in terms of shear stress field or Prandtl function or warping function in all domain and for any kind of domain with arbitrary contour, may be performed quickly, calculating line integrals only. The method takes full advantage of the theory of analytic complex function and is robust in the sense that returns exact solution if this exists. Numerical implementation of LEM has been developed using Mathematica software without resorting to any discretization neither in the domain nor in the boundary. The latter means that you can use the same program for all sections just by changing the first few lines of program where you declare the geometry of the section. Some numerical applications have been reported to demonstrate the efficiency and accuracy of the method. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Very recently a novel numerical method has been introduced for carrying out the response solution for torsion problems. In particular, considering a potential function in terms of shear stress [1,2], instead of the usual one, involving the warping function and its harmonic conjugate [3,4], the stress distribution may be represented as the double-ended Laurent series in terms of harmonic polynomials. The evaluation of coefficients of the Laurent expansion is pursued imposing the weak condition that the total square net flux across the border is minimum, together with the fulfilment of static equivalence condition. To aim at this, no discretization of the contour or the domain is required at all, and since only line integrals are performed, this method is called line element-less method (LEM). Such a method is robust, in the sense that for all the cases in which the analytical solution is already known (circle, equilateral triangle, ellipse) it returns the exact solutions, while for the other cases, by using very few terms of Laurent series, it provides very accurate results. It is valid for simply-connected contours, where only the regular part of Laurent series has to be considered, and for multiply-connected contours, where the principal part is necessary for a correct evaluation of the stress field in the given domain. With respect to boundary methods as the complex variable boundary element method, CVBEM, [5–12] it has the great advantage of no need of discretization of the contour and it may capture the exact solution if this exists. However, in the present form, LEM cannot be applied to beams with sections possessing re-entrant angles at their boundary; this ⇑ Tel.: +39 3204395957. E-mail address: [email protected] 0045-7949/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2013.11.015

highlights the aim of this paper at extending LEM for these cases, delivering a method available for the evaluation of the stress field and the warping function or of all features of the solution for torsion problems, involving any kind of shaped section, as well. To assess the accuracy of the results, the evaluation of the shear stress field is then compared with results obtained by means of numerical methods for structural analysis, as finite element method (FEM) [13–16]. To aim at this, different shape sections have been studied, observing that results are in good agreement, that allows to rely on LEM, but more, for studying different shape sections one can use the same code with LEM, it needs only to change the first few lines of program where you declare the geometry of the section, while the FEM code needs to be totally restructured changing the number and the shape of meshes. 2. Governing equations Consider an elastic and isotropic De Saint-Venant cylinder of length L and cross section A with contour C, referred to a counter-clockwise coordinate system with x and y axes coincident with the principal axes of inertia of the cross section as shown in Fig. 1. Assume the cylinder is forced by a torsion moment Mz(L) acting at the end of bar, then stress field is completely defined by shear stress szx(x, y), szy(x, y). Further, introduce the complex potential function Fð^zÞ in terms of shear stress as

Fð^zÞ ¼ vx ðx; yÞ þ ivy ðx; yÞ ^z ¼ x þ iy

ð1aÞ

vx ðx; yÞ ¼ szx ðx; yÞ þ Ghy

ð1bÞ

vy ðx; yÞ ¼ szy ðx; yÞ þ Ghx

ð1cÞ

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3. Line element-less method Recently the line element-less method, LEM, has been successfully proposed for torsion solution of the De Saint Venant cylinder [4] and for orthotropic beams [20]. This method, with respect to boundary methods, returns the exact solution if this exists without contour discretization. However, in the present form, this method cannot be applied for sections having re-entrant angles, then this paper aims at extending LEM for these cases. For better intelligibility, all formulations regarding LEM are summarized in this section. The starting point of LEM is the expansion of the analytic complex potential function Fð^zÞ through the double-ended Laurent series as

Fð^zÞ ¼

þ1 X

ak ð^z  ^z0 Þk ; ak ; ^z0 2 C; k–  1

ð9Þ

1

Fig. 1. De Saint Venant beam under torsion.

where G is the shear modulus of the material and h is an unknown coefficient, (whose value will return the twist rotation per unit length). From the complex potential theory [17–19], the function Fð^zÞ is analytic in the domain A, then the following relations hold

r2 vx ðx; yÞ ¼ 0;

r2 vy ðx; yÞ ¼ 0 in A

ð2a; bÞ

! @ 2 ½ @ 2 ½ r ½ ¼ 2 þ 2 ; @x @y

P ^ ^ k In Eq. (9) the series þ1 k¼0 ak ðz  z0 Þ is called regular part and it is capable to express any analytic function everywhere, while the P2 ^ ^ k summation k¼1 ak ðz  z0 Þ is called principal part and accounts for singularities in ^z0 . Notice that, in the Laurent expansion the term proportional to (1=ð^z  ^z0 Þ), related to k = 1, has to be excluded; since it leads to a logarithm term in the integrated expressions and consequently a multi-valued function for the Prandtl function defined successively. k Moreover, powers ð^zÞ will be denoted as Pk + iQk where i is the imaginary unit, Pk and Qk are harmonic polynomials defined as follows:

2

k

Pk ðx; yÞ ¼ Reðx þ iyÞ ;

Q k ðx; yÞ ¼ Imðx þ iyÞ

k

ð10a; bÞ

or recursively as

@ vx @ vy ¼ ; @x @y

@ vy @ vx ¼ @y @x

in A

ð3a; bÞ

It is worth noting that, introducing Eqs. (1) into conditions (2) the Beltrami equations are restored as:

r2 vx ðx; yÞ ¼ r2 szx ðx; yÞ ¼ 0 2

in A

2

r vy ðx; yÞ ¼ r szy ðx; yÞ ¼ 0

ð4a; bÞ

while, introducing Eqs. (1) into the Cauchy–Riemann conditions (3) it leads to the following equations @ szx ðx;yÞ @x @ szy ðx;yÞ @x

þ

@ szy ðx;yÞ @y

¼0

 @szx@yðx;yÞ ¼ 2Gh

in A

ð5a; bÞ

meaning that the shear stress field

szx ðx; yÞ ¼ vx ðx; yÞ  Ghy ¼ Re½Fð^zÞ  Ghy

Pk ðx;yÞ ¼ Pk1 x  Q k1 y; Q k ðx; yÞ ¼ Q k1 x þ Pk1 y Pk ðx;yÞ ¼

ð10c;dÞ

Pk ðx;yÞ ; 2 Pk ðx;yÞ þ Q 2k ðx;yÞ

Q k ðx; yÞ ¼ 

Q k ðx;yÞ P2k ðx; yÞ þ Q 2k ðx;yÞ

k>0

ð10e;fÞ

with P0 = 1, Q0 = 0, P1 = x, Q1 = y. The derivative of the harmonic polynomials are

@Pk ¼ kPk1 ; @x

@Pk ¼ kQ k1 ; @y

@Q k ¼ kQ k1 ; @x

@Q k @y

8k

¼ kPk1

r2 Pk ¼ 0; ð6aÞ

k>0

ð11aÞ

r 2 Q k ¼ 0 8k

ð11bÞ

Based on the above considerations, assuming in Eq. (9) ^z0 ¼ 0 and

ak ¼ ak þ ibk ðak ; bk 2 RÞ the complex potential function and the

szy ðx; yÞ ¼ vy ðx; yÞ þ Ghx ¼ Im½Fð^zÞ þ Ghx

ð6bÞ

shear stress field are rewritten in terms of harmonic polynomials as:

satisfies both equilibrium and compatibility conditions in A. It follows that, this shear stress field will be the solution of the torsion problem, provided that the traction-free boundary conditions is fulfilled:

Fð^zÞ ¼ vx ðx;yÞþivy ðx;yÞ ¼

szx nx þ szy ny ¼ sT n ¼ 0 on C

szx ðx; yÞ ¼

ð7Þ

(being s ¼ ½ szx szy  and n ¼ ½nx ; ny  the outward normal vector to the contour C), together with the static equivalence condition T

Z

T

T

s gdA ¼ Mz

ð8Þ

A

having considered gT ¼ ½ y

x .

1 X k¼1 k–1

r2 X

ðak Pk bk Q k Þþi

1 X

ð12Þ

ðak Pk ðx; yÞ  bk Q k ðx; yÞÞ  Ghy

k¼r1 k–1

szy ðx; yÞ ¼ 

ðak Q k þbk Pk Þ

k¼1 k–1

r2 X

ð13a; bÞ ðak Q k ðx; yÞ þ bk Pk ðx; yÞÞ þ Ghx

k¼1 k–r1

Notice that, the latter is written truncating the series, then the shear field completely fulfills equilibrium and compatibility equations in

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the domain, but it may not fulfill Eq. (7) in each point of the contour C. The unknown coefficients ak, bk and h are evaluated by minimizing the squared value net flux of the shear stress vector s through the boundary of the domain, that is, introduce the following functional:

Iðak ; bk ; hÞ ¼

I

2

ðsT nÞ dC

ð14Þ

C

and minimize I(ak, bk, h) with respect to ak, bk and h under the static equivalence condition given in Eq. (8):

8 H 2 > Iðak ; bk ; hÞ ¼ C ðsT nÞ dC ¼ min > < ak ;bk ;h subjected to > > :R T s gdA ¼ Mz A

Hence the need to deepen and extend the line element-less method to sections possessing re-entrant angles at their boundary, as detailed in the next section. 4. LEM for twisted re-entrant angle sections When dealing with re-entrant angles on the contour, first of all it needs to select n nodes on the boundary, say ^zj ¼ xj þ iyj , that do not serve to discretize the boundary but serve to particularize the following complex potential function

Fð^zÞ ¼ F1ð^zÞ þ F2ð^zÞ ð15a; bÞ

¼

r2 X

ak ð^z  ^z0 Þk þ

r 1 k–1

n X bj ð^z  ^zj Þlog^zj ð^z  ^zj Þ

ð18Þ

j¼1

Further, a strength of the method is the need of calculating only line integrals as detailed in [4], then the static equivalence condition expressed in Eq. (8) in terms of harmonic polynomials through line integrals assumes the form

Notice that the above functional is composed of two parts: Pr2 k z  ^z0 Þ equal to the previous one Eq. (9) and r1 ak ð^ k–1 P F2ð^zÞ ¼ nj¼1 bj ð^z  ^zj Þlog^zj ð^z  ^zj Þ, being bj ¼ dj þ iej ðdj ; ej 2 RÞ, re-

I I r2 r2 X X ak uTk ndC þ bk

ferred to the chosen n nodes ^zj ¼ xj þ iyj on the contour C. As well known, the complex logarithm, log^zj ð^z  ^zj Þ ¼ log j^z  ^zj jþ

k¼r1 k–1

C

k¼r 1 k–1

C

v

T k ndC

þ GhIp ¼ M z

ð16aÞ

where IP represents the polar inertia moment and uk and v k are given as        yPkþ1 ðx  x0 ; y  y0 Þ ðk þ 1Þ    ; v k ¼  yQ kþ1 ðx  x0 ; y  y0 Þ ðk þ 1Þ  uk ¼    xP kþ1 ðx  x0 ;y  y0 Þ=ðk þ 1Þ xQ kþ1 ðx  x0 ;y  y0 Þ=ðk þ 1Þ  ð16b;cÞ However this formulation is highly valid for general cross section simply or multiple connected but if the section presents re-entrant angles like section L, H, T, C or epitrochoidal type in this case results are not satisfactory like those reported in the Fig. 2 where the Prandtl function contours lines are reported for an L shaped section having selected r1 = 0 and r2 = 11 in the following expression of Prandtl function

wðx; yÞ ¼

 r2  X  Q Pkþ1 Gh  2  ak kþ1 þ bk x þ y2 2 kþ1 kþ1 k¼r 1 k–1

ð17Þ

F1ð^zÞ ¼

i argð^z  ^zj Þ is normally a multi-valued function, that may be considered single valued, if the argument is opportunely defined with respect to a chosen branch cut. Usually the principal branch cut is considered, that is the negative real axis x, so that argð^z  ^zj Þ assumes values in the interval: p < argð^z  ^zj Þ  p; but, in this context, the argument of the complex logarithm must be defined in a different interval with respect to a chosen branch cut P^zj , for each node ^zj , defining an arbitrary continuous non-self-intersecting path P^zj , joining ^zj to infinity without intersecting either the domain A or the contour C (except for the point ^zj itself) as

cj < argð^z  ^zj Þ  cj þ 2p

ð19Þ

where cj is the angle that the line P^zj forms with the positive real axis x, as shown in Fig. 3. The specified choice of branch cut plays a fundamental role, since: (i) It intersects the contour C only at the point ^zj . (ii) Allows to define a complex logarithm that is analytic in all the complex plane with the except of the branch cut P^zj  ^zj itself. (iii) Assures that the function F2ð^zÞ is analytical in the entire domain A and continuous in A [ C, including the point ^zj since F2ð^zÞ tends to zero as z approaches ^zj . Accordingly, the stress distribution given in Eqs. (6) is rewritten as

szx ðx; yÞ ¼ Re½F1ðx; yÞ þ Re½F2ðx; yÞ  Ghy

ð20aÞ

szy ðx; yÞ ¼ Im½F1ðx; yÞ  Im½F2ðx; yÞ þ Ghx

ð20bÞ

the unknown coefficients ak, bk, dj, ej and h are evaluated by searching the stationary values of the functional expressed in Eq. (15). Specifically, for evaluating the coefficients we can use the Lagrange multiplier method, considering the enlarged functional

Iðak ; bk ; dj ; ej ; h; kÞ ¼

I C

Fig. 2. Prandtl function w(x, y) contour lines in L shape section (LEM).



2

sT n dC þ k

Z

sT gdA  Mz

 ð21Þ

A

being k the Lagrange multiplier, then equating to zero the partial derivatives of the functional with respect to the coefficients we can build the following system of equations,

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     1 @p @q @p @q i dA Im þ  2 @x @y @y @x A  Z  Z 1 @q @p 1 T ¼ dA ¼   ðcurlP2ð^zÞÞ iz dA 2 A @x @y 2 A

Z

ð26Þ

having considered a formal vector P2T ¼ ½ p q , whose components are the real and the imaginary part of the function P2ð^zÞ, and iz a unit vector normal to the cross-section. Finally, applying the Stokes’s theorem

γj

Z

Pz j

I

T

ðcurlP2ð^zÞÞ iz dA ¼

A

P2T ð^zÞsdC

ð27Þ

C

T where s ¼ ½ sx sy  is the unit vector tangent to the contour C of the cross-section, we obtain this fundamental relation



Z

Imð^zF2ð^zÞÞdA ¼  A

Fig. 3. Definition of angle cj: principal branch cut (dashed line) and arbitrary branch cut (solid line).

Z

sT gdA ¼ Mz ¼ 

A

Z

Imð^zFð^zÞÞdA þ Gh

Z

A

ðx2 þ y2 ÞdA

ð23Þ

A

and introducing Eq. (18) we get

Z

sT gdA ¼ Mz ¼ 

A

Z

Imð^zF1ð^zÞ þ ^zF2ð^zÞÞdA þ GhIp

ð24Þ

A

R H where, IP ¼ A ðx2 þ y2 ÞdA ¼ C ½ x3 =3 y3 =3 ndC [19] is the polar inertia moment. Since the first term in the integral of Eq. (24) is equal to the previous case already transformed into line integral in Eq. (16), let focus on the second term rewritten as:



Z A

Imð^zF2ð^zÞÞdA ¼ 

k¼r1 k–1

1 2

C

I

Z A

Im

  dP2ð^zÞ dA d^z

ð25Þ

where P2ð^zÞ ¼ p þ iq is the primitive function of (^zF2ð^zÞ). Substituting the derivative of the complex function [18] P2ð^zÞ ¼ p þ iq into Eq. (24), it reverts to

C

k¼r 1 k–1

vTk ndC

P2T ð^zÞsdC þ GhIp

ð29Þ

C

taking into account the above expression (29) the enlarged functional (21) is rewritten as 0 I I r2 2 BX Iðak ;bk ;dj ;ej ;h; kÞ ¼ ðsT nÞ dC þ k@ ak uTk ndC r2 X k¼r 1 k–1

Considering the static equivalence condition Eq. (8) expressed in terms of Fð^zÞ as

ð28Þ

I I r2 r2 X X ak uTk ndC þ bk

þ

4.1. Static equivalence condition through line integrals

P2T ð^zÞsdC

C

C

whose solution returns the coefficients ak, bk, dj, ej, h and the Lagrange multiplier k.In Appendix A there is reported an alternative procedure to evaluate the coefficients. Furthermore, to underscore the strength of the method inherent in the fact that only need to calculate line integrals, let transform the domain integral in (21) into line integrals as detailed in the next sub-section.

Z

sT gdA ¼ Mz ¼



ð22Þ

I

useful for evaluating the static equivalence condition only by line integrals as:

A

@Iðak ; bk ; dj ; ej ; h; kÞ ¼0 @ak @Iðak ; bk ; dj ; ej ; h; kÞ ¼0 @bk @Iðak ; bk ; dj ; ej ; h; kÞ ¼0 @dj @Iðak ; bk ; dj ; ej ; h; kÞ ¼0 @ej @Iðak ; bk ; dj ; ej ; h; kÞ ¼0 @h @Iðak ; bk ; dj ; ej ; h; kÞ ¼0 @k

1 2

k¼r 1 k–1

I

1 bk v Tk ndC  2 C

I C

C

1 C P2T ð^zÞsdC þ GhIp  M z A ð30Þ

The latter will be considered for writing the system (22) to calculate the values of coefficients ak, bk, h, dj, ej, and the Lagrange multiplier k, allowing the evaluation of the solution of the twisted section in the whole domain. 5. Numerical applications To assess the validity and the simple use of the aforementioned formulation, let start from the epitrochoidal section, having three re-entrant angles, and built through two circles of radius equal to 0.2 and 0.6 respectively. Introducing the stress field Eqs. (20) into the functional (30), the coefficients ak, bk, h, dj, ej, and the Lagrange multiplier k are determined by solving the system of equations (22) obtained equating to zero the partial derivatives of the functional with respect to coefficients. Choosing Mz = 1, and G = 1 the value of twist rotation per unit length h, (returned by the value of h coefficient), is reported in Table 1. In particular, in this Table 1 it has been reported: exact value of h calculated by means of conformal mapping [3], error measures eð%Þ depending on the number of terms used for the series expansion Eq. (18), error measures depending on the number of elements selected for discretization of the contour for the CVBEM. By observing data reported in Table 1 it is apparent that the proposed LEM is very competitive, in achieving, with few terms in the expanded series, the same value response reached by using CVBEM discretizing the contour through 100 elements. Notice that in Table 1 n is referred to the number of chosen points located starting from the re-entrant angles and very close one each other, furthermore, in the picture of the section it has been depicted the Prandtl function contour lines obtained by LEM.

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A. Pirrotta / Computers and Structures 133 (2014) 149–155

Moreover, reconsidering the L shaped section, selecting ^z0 ¼ 0; k from 0 to 11 and n = 14 in the expression (18) and determining the coefficients ak, bk, h, cj, dj, ej, and the Lagrange multiplier k by solving the system of equations (A4) obtained equating to zero the partial derivatives of the functional (A3) with respect to coefficients. Then, the Prandtl function contours lines are depicted in Fig. 4. Both results demonstrate how highly efficient the proposed method is and the extension part to capture the stress response for sections with re-entrant angles. The algorithm is very simple and implemented in ‘‘Mathematica’’ environment. Results have been obtained for any shaped section as C, T, H, + contour, but for saving space here beside the previous ones, it has been proposed a hollow section, being this case quite interesting. 5.1. Hollow section Considering any hollow section with re-entrant angles, at first sight, it seems that we cannot apply the method as above, since, any chosen branch cut crosses the section; but introducing imaginary cuts we divide the original section into simply-connected sections having re-entrant angles, then for each simply-connected section we can use the aforementioned method. For instance, consider a squared hollow section having the external side le = 2 mm, the internal li = 0.8 mm, G = 1 MPa and subjected to a torsion moment of Mz = 11 mm. Further, introduce an imaginary horizontal cut IN (Fig. 5) passing through the centroid, two C-shaped sections (superior and inferior) have been generated, for which we can apply the proposed procedure. These C-shaped sections have own domain, say Asup and Ainf, own boundary, Csup, summation of the external (IBCN) and internal (LFGM) contours and Cinf, summation of the external (NDAI) and internal (MHEL) contours, and common sides labelled Ccut (summation of IL and MN on the imaginary cut IN). To study these C shaped sections it has been considered the functional Fð^zÞ in Eq. (18) twice (for superior and inferior sections), with k from 3 to 3, (r1 = r2 = 3) and n = 20 for each section. The latter n = 20 indicates that 20 nodes have been chosen for each C-shaped sections, with 9 nodes close to the re-entrant angles and one node in each external corner as shown in Fig. 5. Expressing the tangential stress vector ssup and sinf in terms of the unknown coefficients ak,sup, bk,sup, dj,sup, ej,sup, hsup and ak,inf, bk,inf, dj,inf, ej,inf, hinf according to Eqs. (20) such coefficients have been evaluated solving the system of equations (22) obtained by differentiating the following functional with respect to themselves

Table 1 Error measures eð%Þ in evaluating the twist rotation per unit length, h by LEM and by CVBEM. Epitrochoidal section exact h = 1.22805

LEM r1 = 0 r2 = 3

h

e ð%Þ

n = 3 (1 per angle) n = 15 (5 per angle) n = 27 (9 per angle) n = 33 (11 per angle) LEM r1 = 0 r2 = 7 n = 3 (1 per angle) n = 15 (5 per angle) n = 27 (9 per angle) n = 33 (11 per angle) CVBEM 12 Elements 50 Elements 100 Elements 200 Elements

0.944 1.167 1.171 1.185

23 4.9 4.6 3.5

h 0.825 1.199 1.211 1.226

e ð%Þ

h 1.211 1.22 1.224 1.226

e ð%Þ

Fig. 4. Prandtl function w(x, y) contour lines in L shape section (updated LEM).

Z

2

C sup

ðsTsup nÞ dC þ

 sinf ÞdC þ k

2

C inf

Z Asup

ðsTinf nÞ dC þ T sup gdA

s

þ

Z

Z Ainf

C cut

ðsTsup  sTinf Þðssup !

T inf gdA

s

 Mz

ð32Þ

Notice that, the latter condition provides: (i) Through the first two elements, the fulfillment of the vectors ssup and sinf tangent in all boundary (external and internal contour). (ii) Through the third element, the continuity of the stress function along the imaginary cut (ssup = sinf). (iii) Through the fourth element the fulfillment of the static equivalence condition in whole domain of squared hollow section. Same considerations may be simply extended considering the Prandtl functions wsup(x, y) and winf(x, y) kernel of the integral (A3), however here not reported for saving space. Moreover, to assess the accuracy of results two kinds of investigation have been developed: one is relative to verify the precision of the solution performed introducing an imaginary cut, and the

B

C

C sup

I

32.9 2.36 1.4 0.1

1.4 0.6 0.3 0.1

Z

F

G

L

M

N

H

E

C inf

li A

D

le Fig. 5. Squared hollow section.

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A. Pirrotta / Computers and Structures 133 (2014) 149–155

evaluated by using the well known Bredt formula szx = 10.15 MPa, while using FEM and changing the dimension of the thickness only, results are very far from this value because it needs to be restructured increasing the number of meshes. 6. Conclusions

(a) FEM

t zy max = 6.7563 MPa

In this paper the numerical method named as line element-less method, LEM, has been extended to all shaped sections, with particular attention to section possessing re-entrant angles at their boundary. In such a way, the response solution in terms of shear stress field or Prandtl function or warping function in whole domain and for any kind of domain with arbitrary contour, may be performed quickly, calculating line integrals only. The method takes full advantage of the theory of analytic complex function and is robust in the sense that returns exact solution if this exists. Numerical implementation of the LEM has been developed using Mathematica software without resorting to any discretization neither in the domain nor in the boundary. Some numerical applications have been reported to demonstrate the efficiency and accuracy of the method. Interesting is the case of hollow section for which it has been considered an imaginary cut, results are compared with FEM and it has been observed that LEM is more versatile with respect to FEM, and with respect to CVBEM, as well. In fact, one can implement LEM and consider the same code for an arbitrary geometry of the section. It means that it needs to change only first lines where the geometry is declared, while using FEM or CVBEM one has to develop a program for each geometry, even for the same geometry but different sides, because the code has to be restructured changing the number and the shape of meshes in FEM, the elements in CVBEM to discretize the contour. It is worth noting that such a method can be applied for solving shear problems, provided the proper static equivalence conditions and stress representation as in [19]. Acknowledgements The author is very grateful to Emilio Greco for the results developed by FEM published in his Master thesis, under my guide.

(b) LEM

t zy max = 6.72635 MPa

Fig. 6. Squared hollow section (a) FEM solution; (b) updated LEM solution.

other one is relative to check the accuracy of results pertinent the chosen hollow section. Precisely, to rely on the results obtained introducing an imaginary cut, all sections having an analytic solution have been tested introducing a cut passing through the centroid and studying two sections labelled superior and inferior, respectively. In each case, LEM returned solution coincident with the exact solution, even for triangle where the two superior and inferior parts are not equal. Further, to assess the accuracy of results, the same hollow section has been studied through finite element method FEM using a three-dimensional simulation in Straus7 environment considering 33,600 cubic mesh element. As apparent in Figs. 6 results obtained by FEM (Fig. 6a) and by updated LEM (Fig. 6b) are in good agreement, but it is amazing how versatile LEM is with respect to FEM. In fact, for instance to consider a very thin thickness section, using LEM, one can use the same program implemented for the previous section, just it needs to change only one parameter: the thickness = 0.16 mm. Proceeding in this way, the tangential stress value obtained by LEM szx = 10.87 MPa is very close to that

Appendix A. Alternatively, considering the Prandtl function w(x, y) we can write the following functional I(ak, bk, h, cj) under the static equivalence condition:

I 8 > > Iða ; b ; h; c Þ ¼ ðwðx; yÞ  cj Þ2 dC ¼ min j k k > < C

subjected to > > > :R T s gdA ¼ Mz A

ak ;bk ;h;cj

ðA1Þ

restoring that the Prandtl function w(x, y) is constant on each jth contour, (for simply connected regions the contour C is only the external one (j = 1), while for s-hollow-section (j = s + 1) the various contour integrals are simply the summation of the contour integrals extended to external and internal contours).For this case consider valid the following expression of the Prandtl function [19] reported together with the warping function for completeness

Z  Gh  2 Fð^zÞd^z  wðx; yÞ ¼ Im x þ y2 2  Z 1 Fð^zÞd^z xðx; yÞ ¼ Re Gh

ðA2Þ

Therefore, for evaluating the coefficients we can use the Lagrange multiplier method, considering the enlarged functional

A. Pirrotta / Computers and Structures 133 (2014) 149–155

Iðak ; bk ; h; cj ; dj ; ej ; kÞ ¼

I

ðwðx; yÞ  cj Þ2 dC 0 I I r2 r2 X BX ak uTk ndC þ bk þ k@ C



1 2

k¼r1 k–1

I

C

k¼r1 k–1

P2T ð^zÞsdC þ GhIp  M z

C

v Tk ndC

 ðA3Þ

C

then equating to zero the partial derivatives of the functional with respect to the coefficients we can build the following system of equations,

@Iðak ; bk ; dj ; ej ; h; kÞ ¼0 @ak @Iðak ; bk ; dj ; ej ; h; kÞ ¼0 @bk @Iðak ; bk ; dj ; ej ; h; kÞ ¼0 @cj @Iðak ; bk ; dj ; ej ; h; kÞ ¼0 @dj @Iðak ; bk ; dj ; ej ; h; kÞ ¼0 @ej @Iðak ; bk ; dj ; ej ; h; kÞ ¼0 @h @Iðak ; bk ; dj ; ej ; h; kÞ ¼0 @k

ðA4Þ

whose solution returns the coefficients ak, bk, cj, dj, ej, h and the Lagrange multiplier k. References [1] Di Paola M, Pirrotta A, Santoro R. Line element-less method (LEM) for beam torsion solution (truly no-mesh method). Acta Mech 2008;195:349–63.

155

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