Torsion in multi-cell thin-walled girders

Acta Mech 206, 163–171 (2009) DOI 10.1007/s00707-008-0102-y Achille Paolone · Giuseppe Ruta · Stefano Vidoli Torsion in multi-cell thin-walled girders Received: 5 May 2008 / Revised: 17 July 2008 / Published online: 8 October 2008 © Springer-Verlag 2008 Abstract A technique for finding the stiffnesses and shear flows in multi-cell thin-walled girders subjected to linear elastic torsion is proposed. The girder is thought as the superposition of elementary closed tracks, just like open girders are the juxtaposition of thin strips. For each track there is a uniform value of the stress flow function, found by means of a linear system of compatibility equations resembling those for the redundant reactions in statically indeterminate frames. The connection between the proposed approach and fundamental properties of graphs is discussed; the advantages with respect to the standard procedures are also enlightened referring to two sample cross-sections. 1 Introduction It is well known that the torsion in a Saint-Venant cylinder can be dealt with by introducing either a warping or a stress flow function (e.g. [8,12,18]). Both approaches yield a boundary-value problem of the Neumann or of the Dirichlet type, respectively, defined on the cross-section S of the cylinder (e.g. [11]). Let us denote by S o the interior of S and P the plane containing it; in general, S encloses the cavities Ch , h = 1, 2, . . . , m. If we pose S̄ = S ∪m h=1 Ch , ∂ S is composed by m + 1 simply connected closed lines ∂ Sk , k = 0, 1, 2, . . . , m, where ∂ S0 = ∂ S̄ , while the ∂ Sh , h = 1, 2, . . . , m are superposed with the ∂ Ch . An orientation for P induces an orientation for the ∂ Sk and for the field of the unit outward normal n attached to any point of the boundaries. The orientation of the ∂ Sh is opposite to that of the ∂ Ch , h = 1, 2, . . . , m. The boundary value problem for the stress flow function (named after Prandtl) ( p), p ∈ P , is ( p) = −2 in S o , ( p) = k ∀∂ Sk , k = 0, 1, . . . , m,
[∇ ( p) · n( p)] dl = −2 ACh ∀Ch , h = 1, 2, . . . , m, (1.1) (1.2) (1.3) ∂ Ch where ∇ and  are the gradient and Laplace differential operators in P , respectively, · is the scalar product between vectors in P , and ACh is the area of the hth cavity. A. Paolone · G. Ruta (B) · S. Vidoli Dipartimento di Ingegneria Strutturale e Geotecnica, “Sapienza” University, Rome, Italy E-mail: giuseppe.ruta@uniroma1.it A. Paolone E-mail: achille.paolone@uniroma1.it S. Vidoli E-mail: stefano.vidoli@uniroma1.it 164 A. Paolone et al. The problem (1.1) solution modulo an inessential constant (e.g. [1]); hence,  may be assumed  admits  vanishing in ∂ S0 ∪ P \ S̄ and attaining nonzero uniform values h in Ch , h = 1, 2, . . . , m. The torsion factor J , the twisting torque T and the relevant shear stress field s are given by (see e.g. [13])  J = 2  d A, T = G J κ, s( p) = Gκ∇ (( p)) × a, ∀ p ∈ S, (2.1–3) S̄ where G is the tangential elastic modulus, κ is the torsion curvature (twist), a is the unit normal to P and × is the cross product between vectors in P . Remark that the torsion factor equals twice the volume enclosed by the surface described by . The solution of the problem (1.1) is available in closed form only for elliptical and circular cross-sections. Hence, it is important to find analogies or approximated methods to provide qualitative results. The most known analogy is due to [16]: Eqs. (1.1) describe the deflection of a pre-tensed inextensible membrane, rigid on Ck and subjected to uniform pressure. The most useful approximated methods consider the so-called thin-walled cross-sections, which are a Cartesian product of generally curvilinear pieces of line (middle line L) and of a straight segment orthogonal to L everywhere. The length of such segments is called thickness, supposed smoothly varying, yet always much smaller than the length of any piece of L: thus, the dominant geometry of the multi-cell is described by L. When the middle line is a circuit, the cross-section is called closed, when L is the join of pieces the crosssection is called open. When both circuits and pieces of line form L, the cross-section is called composed; when there are multiple circuits the cross-sections are also called multi-cells. The origin of the technical formulæ for thin-walled sections are in the hydrokinetical analogue established by [19], perfected by [2] for tubular sections and by [15] for open sections. Recently, some new approaches have been proposed, for instance by [10], who suggests a local method for solving partial differential equations of interest in solid mechanics, with some examples of applications in determining Prandtl stress flow function for bars in torsion. Another one is due to [7], who propose a numerical method taking advance of the theory of analytic complex functions and of a novel element-free weak form procedure. These two interesting procedures hold for bars with arbitrary shape of the cross-section; due to their generality, their implementation may require some computational effort. On the other hand, the aim of this paper is to propose a unifying procedure for the technical solution of torsion of thin-walled girders, often met in engineering applications. The obtained equations can be easily implemented in a few lines of standard algebraic manipulators such as Mathematica© . 2 Standard approximated techniques for thin-walled girders When dealing with the torsion of thin-walled girders, one admits that the shear stress s has a dominant component s parallel to the middle line (for a proof by means of differential geometry techniques, see e.g. [5,17]). Let us fix the abscissas ξ along the middle line (the choice of the origin was proven to be immaterial) and η along the thickness; pose t (ξ ) the thickness and l(ξ ) the field of the unit tangent vectors to the middle line. The shear flow associated with s is q(ξ ) = s(ξ ) · t (ξ )l(ξ ). (3) The harmonic problem (1.1) with the position (2.3) lets q(ξ ) vanish in each piece of open sections and be uniform in the regular branches of each circuit. Thus, s is linear in η for open sections and uniform along the thickness in circuits. These results are presented in all textbooks on strength of materials, see e.g. [4,20]. The quantities of interest are provided by Kelvin’s formulæ for open sections; when L reduces to a piece of length l and uniform thickness t, it is 1 2T J o = lt 3 , s(η) = o η, (4) 3 J where J o is the torsion factor for the piece. When L is the join of many pieces Li , i = 1, . . . , n one admits that each piece contributes to the total stiffness, while rotating by the same twist (the pieces are structural elements in parallel): thus, denoting J o,t the total torsion factor for the open section, J o,t = n  li t 3 i i=1 3 , Ti = Jio 2 Ti T, si (η) = o η. J o,t Ji (5) Torsion in multi-cell thin-walled girders 165 These formulæ were generalized by [17] in the case of variable thickness. When dealing with a thin tube with middle line L, the Prandtl stress flow function has a uniform nonzero value in the cavity C , say, 1 , vanishes outside the tube and is linear along the thickness. Thus, because of Eqs. (2.3) and (3), the dominant component of the shear stress and the shear flow are given by s(ξ ) = Gκ1 , q = Gκ1 . t (ξ ) Bredt’s formulae hold:
J c = 2  1 , T = 2  q, L 1 dξ = 2, t (ξ ) (6) (7.1–3) where is the torsion factor for the circuit and  is the area enclosed by L. Since the tube is thin, the difference between L and ∂ C is negligible and the formulæ can be written with respect to C instead of L. Equations (2.1) and (7.1) suggest that the torsion factor for thin tubes is obtained by neglecting, in the calculation of the volume under the surface described by Prandtl’s stress flow function, the contribution of the steep decrease of  from the value 1 in the cavity to the value 0 outside S̄ . For multi-cells, the hydrokinetical analogy by Kelvin suggests that one shall write: (a) as many local balance equations as the number of independent nodes (the points where the regular branches constituting circuits meet)—at each node the sum of the flows vanishes; (b) a global balance equation—the torque due to the shear flows must equal the outer torque; (c) as many compatibility conditions as the number of independent circuits—these conditions derive from Eq. (1.3). One finds this approach in [3,9], for instance; the unknowns of this technique are the stress flows in any regular branch of the multi-cell and its twist. Den Hartog [6] proposed a different approach, based on balance equations for the transverse force acting on each virtual membrane considered in the membrane analogy. Composed cross-sections are dealt with by superposing the solutions for multi-cells and open sections. The authors wish to propose a technique intended to be more general and more easily subjected to numerical implementation than both the traditional and Den Hartog’s ones. Moreover, the proposed technique is intended to represent a unified approach for all kinds of thin-walled girders. Jc 3 A solution technique for multi-cells The standard technique looks at the multi-cell as a circuit where a stationary shear stress flows, while Den Hartog considers the multi-cell as the juxtaposition of tubes, like an open section is the juxtaposition of thin strips. We remark, however, that L can be seen as a closed graph G , i.e., a set of oriented branches B p , p = 1, 2, . . . , r connecting nodes, through which closed paths may be tracked. We propose, with a wider generality with respect to both former techniques, to consider the multi-cell as a statically indeterminate superposition of closed paths in G . This equals to considering redundant frames as the superposition of statically determinate ones in the mechanics of structures. Then, one solves the problem by means of compatibility equations, assuming a unique approach for all kinds of thin-walled girders. rj Let us consider the multi-cell as the superposition of tracks (closed paths) T j = ∪ p=1 B p . This is equivalent to choosing a basis for the cycles of G ; each track can be seen as the middle line of a virtual thin tube, for which Bredt’s formulae (7) hold. The total plot of  is then the superposition of the plots of the Prandtl functions pertaining to each track, see Fig. 1. Once the T j are oriented, the connectivity matrix C j p among branches and tracks is defined as follows: ⎧ ⎨ +1, when the branch p is run in the same orientation as the track T j , 0, when the branch p does not belong to the track T j , (8) C jp = ⎩ −1, when the branch p is run in the opposite orientation as the track T . j One can interpret the rows of the connectivity matrix as elementary unit shear flows in the tracks T j . These elementary flows automatically satisfy the local balance equations at the nodes of the multi-cell, by construction. If  j is a multiplier, by linearity the actual flow in the pth branch of G is qp = m  j=1 C j p Gκ j (9) 166 A. Paolone et al. Fig. 1 The superposition of elementary circuits and Prandtl’s flow function and satisfies local nodal balance equations. To find the  j , m compatibility equations (as many as the considered tracks) can be used, expressing the equivalence of the twist for all the T j . Since it is always possible to write a global balance equation expressing one of the  j in terms of the remaining m − 1, there are actually only m − 1 statically indeterminate multipliers. The purpose of this technique is to reduce the solution of the problem to a set of equations of the same kind, in order to automatize the procedure. The standard linear elastic constitutive relations and Eq. (9) provide the dominant shear strain γ p in the pth branch of G : qp = γp = Gt p (ξ ) The equation of virtual work for each track reads
Ti∗ κ = m j=1 C j p κ j t p (ξ ) . q ∗p γ p dξ, (10) (11) Ti where T ∗ and q ∗p are a balanced virtual system of actions expressed by Bredt’s formula (7.2), q ∗p = Ti∗ Ci p . 2i (12) Then, if we let
ηi j = Ci p C j p Ti ri   dξ dξ Ci p C j p = , t (ξ ) t (ξ ) i, j = 1, . . . , m, (13) p=1B p Eqs. (10)–(13) provide: m  ηi j  j = 2i , i, j = 1, . . . , m, (14) j=1 where i is the area enclosed by Ti . The ηi j fill a symmetric table of coefficients depending only on the geometry of the tracks. The system (14) is linear in the unknowns  j and its equations are independent since Torsion in multi-cell thin-walled girders 167 each row represents one compatibility condition independent from the others. The result (14) resembles the so-called equations of Müller-Breslau [14] used in the solution of redundant frames. A suitable choice of the indexes i, j lets the matrix ηi j contain non-vanishing terms only in a narrow band surrounding the diagonal, thus simplifying automatic calculations. We remark that when we choose a basis for the tracks the elements of which are disjoint (Fig. 1b, c), all the  j will be strictly positive. On the other hand, when the basis for the tracks has elements which intersect in some track (Fig. 1d, e), some of the  j may be zero or negative. Indeed, with a view to Fig. 1, since the  j in (a) are both positive, the choice of a basis for the tracks such as in (d) and (e) obviously implies that the value of  in track (e) equals 1 − 2 < 0. Each Ti contributes to the total stiffness according to Eq. (7.1); since the twist is common to all tracks, one gets J c,t = m  Jic = i=1 m   Jic Ti T, q p = Ci p . c,t J 2i m 2i i , Ti = i=1 (15.1–3) i=1 Equations (15) represent a perfect parallel to Eqs. (5) for open girders, so that the proposed technique unifies the treatment of thin-walled beams. Remark that a necessary and sufficient condition for the Jic to be positive is to choose the tracks coinciding with the circuits of the middle line surrounding the boundaries of the cavities. From the mechanical point of view this is tantamount to see the multi-cell as juxtaposition of thin tubes. Then, as seen above, since all the i > 0, so are the torsion factors Jic . On the other hand, if the basis for the Ti are not disjoint, some i , hence some Jic , may be zero or negative. 3.1 Extension to composed cross-sections The extension of the proposed technique from multi-cells to composed cross-sections is immediate. Indeed, the join of a multi-cell and thin strips always represents a set of structural elements in parallel, since the different shear flow in closed and open thin sections renders the two essentially independent from each other. With the help of Eqs. (15), it is then possible to write J t = J c,t + J o,t = m  i=1 2i i + n l t3  j j j=1 3 , Tic = J jo Jic o T, T = T, j Jt Jt (16) where J t is the torsion factor for the composed cross-section and Tic , T jo are the contributions of the ith track and jth strip to the total torque, respectively. Once the i are obtained from Eqs. (13)–(14), the calculations in Eqs. (16) are straightforward. The mechanical quantities of interest are then obtained by using Eqs. (5.3) and (15.3). 4 Examples The advantages of the proposed technique and the aspects related to its numerical implementation are discussed referring to two multi-cells; for the sake of simplicity and readability the thickness is supposed uniform and the girders are supposed subjected to a unit twisting torque. The considered cross-sections are shown in Fig. 2; the first, Fig. 2a, finds widespread application in both civil and aeronautical engineering; the second, Fig. 2b, resembles a bee-nest, has less applicative relevance but lets us underline the main peculiarities of the proposed approach. For instance, referring to cross-section 2 (having 11 cycles and 30 branches), Fig. 3 compares the two linear systems to be solved in the standard and in the proposed approach, respectively. To this aim a greyscale representation of the matrices of the coefficients of the linear systems providing the solutions is adopted. The standard approach suggests to write the balance equation of the torque (one equation), all the flow balance equations in the nodes of the graph (nineteen equations) and the compatibility equations in the tubes (eleven equations), since the associated unknowns are the shear flows in each of the 30 branches and the twist. The resulting system provides a neither symmetric nor banded 31 × 31 matrix to be inverted, whilst the proposed approach yields a 11 × 11 symmetric banded matrix. Indeed, the proposed approach focuses only on the solution of compatibility equations in terms of the uniform values of Prandtl’s stress flow function on each tube, seen as indeterminate multipliers, automatically 168 A. Paolone et al. (a) (b) Fig. 2 The considered multi-cells 1 10 20 31 1 1 2 4 6 8 10 11 1 2 10 4 6 20 8 10 11 31 (a) (b) Fig. 3 Linear systems associated to the standard (a) and present (b) approaches satisfying balance equations. Hence the number of equations is reduced to the number of tracks, or independent cycles in the graph G , and the resulting linear system (14) is symmetric and banded. In particular the bandwidth of the matrix ηi j depends on the chosen basis of independent tracks; to reduce the bandwidth one is lead to choose the tracks with minimal mutual intersections. For instance, solving the first cross-section with the tracks chosen as in Fig. 4b leads to a tridiagonal matrix, since each track has a non-vanishing intersection only with two adjacent tracks. However, in order to set up an approximated solution of multi-cells, trying to reduce the ηi j bandwidth should not be a priority for a good choice of the basis cycles. Indeed, the choice in Fig. 4b produces a tridiagonal matrix but the percentage contributions of all the tracks to the torsion factor are comparable; hence, one cannot neglect the contribution of some tracks in favour of others. A smart alternative is to choose the Ti in such a way that their percentage contribution to the torsion factor decreases monotonically with their counter, i.e. |Ji+1 | < |Ji |. When the thickness is uniform, a rule of thumb to pursue this aim is to choose and order the tracks according to their areas. For the first cross-section, Fig. 4a gives a suggestion of a good choice of the Ti (the stress distribution is sketched using grey levels). In Fig. 4c such a good choice for the basis of tracks is reported; for instance limiting the computation to the first 3 cycles will produce errors in the stress below 1.5%. Figure 5 presents a smart choice of tracks for the second cross-section. Note that one can solve the problem with a good approximation using only the first two tracks (one obtains 96.8% of the twisting torque); this choice leads to a 2 × 2 diagonal η matrix, meaning a vanishing computational cost. Let qe (k) and qa (k) be the exact and the approximated values of the shear flow in the kth branch; the maximum percentage error in evaluating qa (k) using only the first two tracks, calculated according to e (k)| max |qa (k)−q , |qe (k)| k is 7.2%, showing how reasonable the error also in the shear flow is. Torsion in multi-cell thin-walled girders 73 169 93 73 98 100 100 100 98 93 20 73 73 20 93 98 100 100 100 98 73 93 73 (a) 12.1 12. 11.9 12. 11.9 11.2 11.2 8.9 8.9 (b) 50.8 19. 4.4 19. 4.4 1. 1. 0.2 0.2 (c) Fig. 4 a Stress solution for the first cross-section (greyscale according to the percentage stress intensity with respect to maximum). b A possible choice for the basis of tracks; the tracks are ordered according to their percentage contribution to the torsion factor. c A smarter choice for the basis of tracks 78.8 18. 2.3 1.6 1. 0.7 0.1 0 0 2.2 0.1 Fig. 5 A good choice for the basis of tracks; tracks are ordered according to their percentage contribution to the torsion factor 4.1 A ten-line implementation with Mathematica© The described procedure is suitable for implementation in standard Computer Algebra Systems. Here we present a sample implementation including functions for the visualization of input and output. The core part of the code stays within 10 lines of Mathematica© language. Inputs are assigned as follows: • • • • nodes are assigned as a list of coordinate pairs; edges are assigned as a list of ordered pairs of node numbers; thicknesses are assigned as a list of same of length of edges; each cycle is assigned as a list of oriented edges; the minus sign is used to mean the reverse orientation of an edge. 170 A. Paolone et al. Below there is a sample input representing a square cross-section with one diagonal; the thickness of the sides is 0.01 while the thickness of the diagonal is 0.002. The tracks chosen are two adjacent triangles with counterclockwise orientations. (*** SAMPLE INPUT ***) nodes = {{0., 0.}, {1., 0.}, {1., 1.}, {0., 1.}}; edges = {{1, 2}, {2, 3}, {3, 4}, {4, 1}, {1, 3}}; thicknesses = {0.01, 0.01, 0.01, 0.01, 0.02}; cycles = {{1, 2, -5}, {3, 4, 5}}; The following lines allow to display the cross-section in input as well as the orientations of its edges: (*** DISPLAY THE INPUT ***) Graphics[{Point[nodes], MapThread[{Thickness[0.005 #2], Arrow@Part[nodes, #1]} &, {edges, Normalize[thicknesses, Norm[#, Infinity] &]}], MapThread[Text[Style[#1, Red, Bold, 12], #2, {1, 1}] &, {Range[Length[nodes]], nodes}], MapThread[Text[Style[#1, Blue, Bold, 12], #2] &, {Range[Length[edges]], Map[Mean@Part[nodes, #] &, edges]}]}] The following ten lines of the Mathematica© code solve the problem for a multi-cell subjected to a unit torque with an arbitrary number of nodes, branches and tracks: (*** ACTUAL CODE ***) NC = Length[cycles]; Connectivity = Map[SparseArray[Abs[#]->Sign[#],{Length@edges}]&,cycles]; ELengths = Map[Norm[Subtract@@nodes[[#]]]&,edges]; EAreas = Map[Cross[nodes[[First[#]]]].nodes[[Last[#]]]&,edges]; Omega = Map[Total[Sign[#] Part[EAreas,Abs[#]]]&,cycles]; eta = Table[Total[ELengths Connectivity[[i]]* Connectivity[[j]]/thicknesses], {i,NC}, {j,NC}]; psi = LinearSolve[eta, 2 Omega]; J = Total[2 psi Omega]; s = (1/thicknesses) Transpose[Connectivity].psi/J; In these lines all the quantities of interest, respectively the areas  of the tracks, the matrix η, Prandtl’s stress flow function , the torsion factor J and the shear stresses s, are computed. Finally, the following lines allow to visualize the output: (*** DISPLAY THE OUTPUT ***) With[{sn = Round[100 Abs[s]/Max[Abs[s]]], rG = Mean[nodes]}, Graphics[{ MapThread[{Thick, GrayLevel[.9 - #1], Line@Part[nodes, #2]} &, {Rescale[sn], edges}], MapThread[Text[Style[#1, Purple, Bold], 1.1 #2 - 0.1 rG] &, {sn, Map[Mean@Part[nodes, #] &, edges]}]}]] 5 Final remarks We have presented a technique suitable for finding the quantities of interest in multi-cells subjected to torsion. The proposed technique generalizes both the standard and Den Hartog’s ones in that it uses an approach the resolutive equations of which constitute a linear system which naturally leads to numerical implementation. Indeed, the proposed technique focuses on the oriented graph associated to the middle line of the cross-section. A suitable choice of the basis for the cycles of the associated graph allows to determine the bandwidth of the resulting symmetric linear system. Two examples have been brought, showing how the proposed technique may be useful in shape optimization in that it enlightens the most meaningful tracks supporting the outer torque. We have also shown how the proposed procedure can be very easily implemented by means of standard computer algebra codes. References 1. Arnold, V.I.: Mathematical methods of classical mechanics. (Moscow, MIR; Italian translation: Metodi matematici della meccanica classica. Roma, Editori Riuniti, 1992) (1979) 2. Bredt, R.: Kritische Bemerkungen zur Drehungselastizität. Zeitschrift des Vereins deutscher Ingenieure 40, 785–790, 813–817 (1896) Torsion in multi-cell thin-walled girders 171 3. Bruhn, E.F.: Analysis and Design of Flight Vehicle Structures. S.R. Jacobs, Indianapolis (1973) 4. Case, J., Chilver, A.H.: Strength of Materials and Structures. Edward Arnold, London (1971) 5. Dell’Isola, F., Ruta, G.: Outlooks in Saint Venant theory I: Formal expansions for torsion of Bredt-like sections. Arch. 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