A displacement method for the analysis of flexural shear stresses in thin walled isotropic composite beams W. Wagner a, F.

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1 A diplacement method for the analyi of flexural hear tree in thin walled iotropic compoite beam W. Wagner a, F. Gruttmann b a Intitut für Bautatik, Univerität Karlruhe (TH), Kaiertraße 12, D Karlruhe, Germany b Intitut für Statik, Techniche Univerität Darmtadt, Alexandertraße 7, D Darmtadt, Germany Abtract Shear tree in cro ection of primatic beam can be evaluated for a given normal tre ditribution by integration of the equilibrium equation. The conidered thin walled cro ection have a contant thickne for each element and may be otherwie completely arbitrary. Introduction of a warping function yield a econd order differential equation with contant coefficient. The olution of the boundary value problem lead to element tiffne relation for two node element within a diplacement method. The computed hear tree are exact with repect to the underlying beam theory. It hould be emphaized that the preent formulation i epecially uited for programming. Keyword: Primatic compoite beam; Thin walled cro ection; Flexural hear tree; Two node element; Suitable for programming 1 Introduction Shear tree in primatic beam ubjected to bending without torion can be evaluated uing the equilibrium equation. The differential equation contain the tranvere hear tree and the normal tree, ee e.g. [1 3]. For arbitrary cro ection one obtain a partial differential equation which can be olved approximately uing the finite element method, e.g. [4,5]. In thin walled cro ection the flexural hear tree are aumed to be contant through the thickne. For multiple connected cro ection uually force method are applied. The unknown circular hear flux for each cell i determined by continuity condition. However the procedure i not convenient for programming. Thi hold epecially for open profile with interection and for cloed cro-ection. In thi paper we decribe a diplacement method which i well uited for programming. We conider traight beam ubjected to torionle bending. The hear tree are obtained from a derivative of a warping function. Hence, the equilibrium lead to a econd order ordinary differential equation with contant coefficient. The olution lead to tiffne matrice and load vector for two node element within a diplacement method. The computed olution are Correponding author. Preent Addre: Intitut für Bautatik, Univerität Karlruhe (TH), Kaiertraße 12, D Karlruhe, Germany. Tel: ; Fax: addre: ww@b.uni-karlruhe.de. 1

2 exact within the underlying beam theory and atify the tre boundary condition along free edge and the continuity condition for multiple connected domain. 2 Torionle bending of primatic beam We conider primatic beam where the aumption of the technical beam theory are given. Let x decribe the length direction of the rod and y and z be ection coordinate, which are not retricted to principal direction and lie arbitrarily with repect to the center of gravity S. The parallel ytem ȳ = y y und z = z z interect at S. The cro ection conit of m thin walled element and may be imply or multiple connected. Each element ha a contant thickne h and α denote the angle between the element and y-axi. Furthermore, we define a local coordinate and pecify the partial derivative, denoted by comma = (ȳ ȳ 1 ) 2 +( z z 1 ) 2,, y =coα,, z =inα. (1) Here ȳ 1, z 1 are the boundary coordinate of the conidered element. The beam i ubjected to bending moment M y and M z and hear force Q y and Q z. Further tre reultant are not preent. w z z z h z 1 z 1 z S S y 1 y y v y S y 1 y Fig. 1. Coordinate ytem Let β y (x) andβ z (x) denote rotation about the axe y and z, andv(x) andw(x) the tranvere diplacement. Hence, the tandard beam kinematic i extended by a warping function ϕ(y, z) 2

3 due to hear deformation, ee alo [4]: u x = β y (x) z β z (x)ȳ + ϕ(y, z) u y = v(x) u z = w(x). (2) The train are obtained by partial derivative a ε x = u x, x = β y z β z ȳ γ xy = u x, y +u y, x = β z + v + ϕ, y (3) γ xz = u x, z +u z, x = β y + w + ϕ, z, where the prime () decribe the derivative with repect to the beam coordinate x. The tree follow from the material law for linear iotropic elaticity σ x = E i ε x = E i (β y z β zȳ) τ xy = G i γ xy = G i ( β z + v + ϕ, y ) (4) τ xz = G i γ xz = G i ( β y + w + ϕ, z ), where E i and G i denote Young modulu and hear modulu of element i, repectively. Further tre component σ y, σ z and τ yz are et to zero within the beam theory. Introducing the warping function ϕ with the ubtitution ϕ, y = β z + v + ϕ, y ϕ, z = β y + w + ϕ, z, (5) applying the chain rule and uing (1) yield τ xy = G i ϕ, y = G i ϕ,, y = τ co α τ xz = G i ϕ, z = G i ϕ,, z = τ in α. (6) Here, τ() i the hear tre, which i contant through the thickne, and t() ithehearflux τ() =G i ϕ, t() =τh. (7) The equilibrium equation in term of the beam tree can be taken from the literature, e.g. [1]. Along free edge = a the hear tree mut vanih. Thu, the boundary value problem 3

4 i defined a follow: τ, +σ x = G i ϕ, +σ x =0 τ( a )=0 (8) with σ x = E i (β y z β z ȳ) :=n i (a y ȳ + a z z). (9) where n i = E i /E and E i a reference modulu. The contant a y and a z are derived uing the condition Q y = M z = Q z = M y = σ xȳ da σ x z da (10) or with eq. (9) Q y = Q z = [n i (a y ȳ 2 + a z ȳ z)da] (A i ) [n i (a z z 2 + a y ȳ z)da], (A i ) (11) where A i denote the area of the element ection. Introducing the moment of inertia Iȳ n = [n i (A i ) z 2 da], I n z = [n i (A i ) ȳ 2 da], Iȳ z n = [n i (A i ) ȳ z da], (12) the ytem of equation (11) can be olved for the unknown parameter a y und a z a y = Q yi n ȳ Q z I n ȳ z I n ȳ I n z I n ȳ zi n ȳ z a z = Q zi n z Q y I n ȳ z I n ȳ I n z I n ȳ zi n ȳ z. (13) The integral of the hear tree (6) yield the hear force, thu it hold Q y = τ xy da and Q z = τ xz da. The proof i given in the appendix. 4

5 i 3 The diplacement method The cro ection i dicretized with m two node element, ee Fig. 2. The node mut be poitioned only at the element interection, at free edge and at thickne jump. The element coordinate are r 1 = {y 1,z 1 }, r 2 = {y 2,z 2 }, the length i denoted by l and the thickne h i contant for each element. Furthermore, we introduce the local normalized coordinate 0 ξ = /l 1. z 2 z 2 t 2 z 1 z t 1 1 = l l y r n y 1 y 2 y Fig. 2. Two node element The linear inhomogeneou econd order differential equation (8) 1 can be olved exactly a ϕ = ϕ h + ϕ p ϕ h = c 1 + c 2 ξ ϕ p = c 3 ξ 2 + c 4 ξ 3. (14) The contant c 3 and c 4 of the particular olution are obtained with ȳ =ȳ 1 + yξ ȳ 1 = y 1 y y = y 2 y 1 z = z 1 + zξ z 1 = z 1 z z = z 2 z 1 (15) a c 3 = l2 2 G i n i (a y ȳ 1 + a z z 1 ) c 4 = l2 6 G i n i (a y y + a z z). (16) The contant c 1 and c 2 are replaced uing the element degree of freedom ϕ 1 ϕ 2 = ϕ(1) = ϕ(0) and c 1 = ϕ 1 c 2 = ϕ 2 ϕ 1 c 3 c 4. (17) 5

6 Hence, the hear flux t(ξ) i derived with (7) and (14) t(ξ) =G i hϕ, = G ih (c 2 +2c 3 ξ +3c 4 ξ 2 ). (18) l Evaluation of t(ξ) atξ =0andξ = 1 yield with (17) the nodal value t 1 = t(0) and t 2 = t(1) t 1 = G ih t 2 l 1 1 ϕ 1 G ih c 3 c ϕ 2 l c 3 2 c 4 (19) t i = k i v i f i. Introducing the element node matrix a i which relate the element array to the global array one obtain v i = a i V T i = a T i t i. (20) The number of component of V correpond to the number of node. Furthermore, T i denote the global hear flux vector of element i. The equilibrium in length direction require that the um of the hear fluxe mut vanih at the node. Conidering (19) and (20) one obtain T i = KV F = 0. (21) The global tiffne matrix K and the right hand ide vector F are obtained by tandard aembly procedure K = a T i k i a i F = a T i f i. (22) The ytem of equation (21) can be olved with V I =0whereI i an arbitrary node. A different node yield another olution for V which ditinguihe only by a contant. Thi decribe a rigid body motion, which ha no influence on the hear tree. The back ubtitution yield with (18) the hear flux and with τ(ξ) = t(ξ)/h the hear tree for every element. Furthermore, we introduce the unit warping function with ϕ da =0by ϕ = ϕ 1 A ϕ da, (23) 6

7 where A = A i. The integral can be computed by a ummation over the element a follow [ ϕ da = hl (c c c ] 4 c 4) i. (24) Thu, the arbitrary nodal point I with V I = 0 doe not influence the reult for ϕ. For ymmetric cro ection ymmetry condition for ϕ can be conidered to reduce the total number of unknown. Finally the function ϕ i pecified with (5) ϕ(y, z) =( β z + v )ȳ +(β y + w ) z + ϕ(y, z). (25) Note, that ϕ decribe the nonlinear part of the warping function and the derivative of ϕ the nonlinear part of the hear tree. 4 Center of hear ForthecenterofhearM with coordinate {y M,z M } the condition hold Q z y M Q y z M = (τ xz y τ xy z)da = τ (y in α z co α) da, (26) with the flexural hear tree according to (6). The integration yield [ τyin α da = G i h in α { y 1 (c 2 + c 3 + c 4 )+ y( 1 2 c c ] 4 c 4) } i (27) and correponding expreion for τzco α da.todeterminey M a computation with Q y =0 and Q z = 1 ha to be carried out and properly for z M. Uing Betty Maxwell reciprocal relation Weber [6] howed, that the coordinate of the center of hear and of the center of twit are identical, the latter being defined a point of ret in a cro ection of a twited beam. Baed on thi reult explicit formulae for y M and z M can be derived y M = Rn ȳ I n z R n z I n ȳ z I n ȳ I n z I n ȳ zi n ȳ z z M = Rn z I n ȳ R n ȳ I n ȳ z I n ȳ I n z I n ȳ zi n ȳ z. (28) 7

8 The proof i given in the appendix. In the following the o called warping moment uing the warping function ω(y, z) of the Saint Venant torion theory are pecified: Rȳ n := [n i (A i ) ω z da], R n z := [n i (A i ) ωȳ da]. (29) For thi purpoe the Saint-Venant torion tree are expreed with the derivative of ω(y, z) and the twit θ, e.g.[1] τ xy = G i θ (ω, y z) = τ co α τ xz = G i θ (ω, z +y) = τ in α. (30) We ue the tilde to ditinguih the torion quantitie from the bending quantitie. The hear tree τ() and the flux t() follow from τ() =G i θ(ω, r n ), t() = τh. (31) The element coordinate y, z can be written a y = r n in α and z = r n co α where the orthogonal ditance r n i expreed with the element coordinate, ee Fig. 2 r n =ign(z n y y n z) r n with r n = {y n,z n } r n = r 1 + ξ n n, ξ n = r 1 n, n =(r 2 r 1 )/l. (32) The equilibrium in length direction with σ x 0 and tre boundary condition of the Saint Venant torion theory read τ, = G i θω, =0 τ( a )=0. (33) The olution of the differential equation read ω(ξ) = c 1 + c 2 ξ. The contant are expreed through the element degree of freedom ω 1 = ω(0) and ω 2 = ω(1) a c 1 = ω 1 und c 2 = ω 2 ω 1. From (31) 2 and with a linear hape of ω we oberve, that t() i contant in each element. With boundary condition (33) 2, t 0 hold for open part of the cro ection and thu only cloed part of the cro ection contribute to the torion moment. Evaluation of (31) 2 yield the hear flux at the node t 1 = t(0) and t 2 = t(1) t 1 G i h = θ t 2 l 1 1 ω 1 G i h r n 1 1 ω 2 r n (34) t i = k i ṽ i f i. 8

9 For the aembly θ = 1 can be et. The aembly and the olution of the ytem of equation idecribedin(20)-(22). The unit warping function i defined a ω = ω ω da/a. Havingω() we are able to expre the warping moment a [ R n z = n i hl { ȳ 1 ( c c 1 2)+ y ( 2 c ] 3 c 2) } i (35) and correponding expreion for R n ȳ. Thu evaluating eq. (28), both coordinate can be calculated in one tep. Remark: An alternative direct method for the evaluation of warping propertie due to torion (y M,z M,F ωω = ω 2 da) of thin walled open and cloed profile ha been given in [8]. Their theory for open profile require two modification before it can be applied to cloed profile. For each cell of the profile the evaluation of a contant Ψ i neceary. Secondly one ha to omit one element from each cell for the element equation wherea thi i not the cae for the property equation, ee [8]. Furthermore the global ytem of equation i non ymmetric. 9

10 5 Example The firt three example deal with contant Young modulu and hear modulu. Within the fourth example we conider a compoite cro ection. 5.1 Open cro ection The firt example i taken from the textbook of Peteren [2]. The welded profile coniting of a U300 (DIN 1026) and L (DIN 1029) i replaced by a thin walled cro ection, ee Fig. 3. The dicretization i performed with 5 element and 6 node. The local coordinate follow from the element node relation. For thi example a ytem of equation with 5 unknown ha to be olved. We calculate the hear tree due to Q y = 120 kn and Q z = 200 kn, ee Fig. 4. The reult are exact within the underlying beam theory and correpond to the olution in [2], which i computed by hand. The coordinate of the center of hear are evaluated uing eq. (26) or (28). The warping function ϕ with G =1kN/cm 2 i depicted in Fig. 5. One can ee, that the nonlinear part of ϕ in comparion to the linear part i mall, ee alo eq. (25) h=16 h=12 z h=10 h=16 z S y h=12 y mm A = cm 2 Iȳ = cm 4 I z = cm 4 Iȳ z = cm 4 ȳ M = z M = cm cm Fig. 3. Open cro ection 10

11 Fig. 4. Shear tree τ() in kn/cm Fig. 5. Warping function ϕ in cm 11

12 5.2 Symmetric mixed open cloed cro ection The econd cro ection conit of open and cloed part, ee Ref. [2]. In [2] overlapping of the element with each other i avoided introducing gap in the idealized cro ection, ee Fig. 6. Without conidering ymmetry the ytem i dicretized with 10 element and 15 node. The node acro the gap are aumed to have the ame warping ordinate which i enforced when olving the global ytem of equation. The Figure 7 and 8 how the ditribution of the hear flux for given hear force Q y = 100 kn and Q z = 1000 kn, repectively. The coordinate z M i evaluated with eq. (26) letting Q y =1andQ z = 0 or uing eq. (28). Conidering ymmetry condition ȳ M = 0 hold. Due to mitake in [2] when calculating the tatically undetermined hear flux t 1 in the cloed part of the cro ection there are different reult. With the correct value t 1 (Q y )=0.561 kn and t 1 (Q z )=3.349 kn one obtain our number. A comparative analyi without the gap in the dicretization lead to negligible difference. Thu the effort to avoid overlapping of the different element i not jutified for thi example h=30 h=12 15 S z y h= h= z y [mm] A = cm 2 Iȳ = cm 4 I z = cm 4 z M = cm Fig. 6. Symmetric, mixed open cloed cro ection 12

13 Fig. 7. Shear flux due to Q y in kn/cm 13

14 Fig. 8. Shear flux due to Q z in kn/cm 14

15 5.3 Cro ection with two cell The third example according to Fig. 9 i alo taken from [2]. The conidered profile i unymmetrical and ha open and cloed part with two cell. We ue 9 node and 10 element for a dicretization. The coordinate of the center of hear are evaluated with eq. (26) or (28), repectively. Fig. 10 how the computed hear tree due to Q z = 1000 kn. The value at z = 0 in the web are given. Again, the olution i exact and correpond to the reult in [2], which are calculated by hand. The warping function ϕ for G =1kN/cm 2 i depicted in Fig. 11. A the plot how there i continuity of the diplacement field within the whole domain. z h=1.2 h=2.0 h=1.6 h=1.2 z y h=1.0 h=1.0 h=2.0 S y h=1.6 h= [cm] 200 A = cm 2 Iȳ = cm 4 ȳ M = cm I z = cm 4 z M = cm Iȳ z = cm 4 Fig. 9. Cro ection with two cell 15

16 Fig. 10. Shear tree τ() in kn/cm Fig. 11. Warping function ϕ in cm 16

17 5.4 Compoite cro ection The compoite cro ection according to Fig. 12 conit of a concrete part and a teel part, ee [3]. The ratio of Young module and hear module i E /E c =5.7 andg /G c =5.4. A reference modulu we chooe E = E. To avoid overlapping of the web with the flange a gap with l =12.5 cm and vanihing thickne h ha been introduced in [3], ee Fig. 12. Without conidering ymmetry the cro ection i dicretized with 8 node and 6 element. The node acro the gap are aumed to have the ame warping ordinate which i enforced when olving the global ytem of equation. The hear force i given a Q y = kn,thuwe obtain a contant a y = 1. The coordinate of the center of hear are evaluated with eq. (26) or (28), repectively. Due to ymmetry y M = 0 hold. The computed hear flux correpond to the olution in [3], ee Fig. 13. The warping function i depicted in Fig E = kn/cm h=25 h=1.5 z h=2.5 y [cm] G = kn/cm 2 I n z = cm 4 z S =66.37 cm z M =71.78 cm Fig. 12. Compoite cro ection 17

18 Fig. 13. Shear flux t() inkn/cm Fig. 14. Warping function ϕ in cm 18

19 6 Concluion For the numerical analyi of the hear tree in cro ection of primatic beam a diplacement method ha been developed. The approach hold for arbitrary open and cloed thin walled profile. The dicretization are performed uing two node element. The computed hear tree are exact within the underlying beam theory. The tre boundary condition along free edge and the continuity condition for multiple connected domain are automatically fulfilled. Furthermore the procedure yield the coordinate of the center of hear. The eential equation can eaily be implemented in a finite element program. Several example how the correctne of the implementation. A Appendix A.1 Integral of the hear tree The integral of τ xy accordingto(6) 1 i reformulated adding the equilibrium equation (8) 1 τ xy da = [τ co α +ȳ (τ, +σ x)] h d. (A.1) () With ȳ =ȳ 1 + co α and thu ȳ, =coα it hold τ xy da = [(τȳ), +σ xȳ]h d () (A.2) and with integration by part τ xy da = (τȳ h, )d +[(τȳ)h] b a + () () σ xȳ da. (A.3) The firt integral of the right hand ide vanihe with contant thickne h. Thi, alo hold for the boundary term conidering (8) 2. Therefore eq. (A.3) yield with (10) 1 the hear force Q y. In an analogou way one can how that integration of τ xz according to (6) 2 yield Q z. A.2 Coordinate of the center of hear In order to derive explicit formulae for y M,z M we introduce the tre function Φ and the warping function ω of the Saint Venant torion theory [1] by Φ, z = ω, y z Φ, y = ω, z +y (A.4) 19

20 where ω i olution of the boundary value problem (33). Now, eq. (26) can be reformulated inerting y and z from (A.4) Q z y M Q y z M = τ [( Φ, y ω, z )inα ( Φ, z +ω, y )coα] da (A.5) and with application of the chain rule and eq. (1) Q z y M Q y z M = τ [Φ, (in α co α in α co α) ω, (in 2 α +co 2 α)] da = τω, da. (A.6) Integration by part lead to Q z y M Q y z M = τ, ω da [τωh] b a. (A.7) The boundary term vanihe conidering (8) 2. Inerting (8) 1 and (9) it hold Q z y M Q y z M = [ (A i ) n i (a y ȳ + a z z) ω da]. (A.8) Introducing the warping moment according to (29) we get Q z y M Q y z M = a y R n z a z R n ȳ. (A.9) Inerting (13), with Q y = 0 one obtain y M and with Q z = 0 the coordinate z M according to (28) i given. For a homogeneou beam cro ection and auming principal axe the correponding formula have been derived by Trefftz [7] uing an energy criterion. Reference [1] Timohenko SP, Goodier JN. Theory of Elaticity. London: McGraw Hill; [2] Peteren C. Stahlbau. Braunchweig, Wiebaden: Vieweg Verlag; [3] Friemann H. Schub und Torion in geraden Stäben. 2. Auflage. Düeldorf: Werner Verlag; [4] Gruttmann F, Wagner W, Sauer R. Zur Berechnung der Schubpannungen au Querkräften in Querchnitten primaticher Stäbe mit der Methode der finiten Elemente. Bauingenieur 1998;73(11):

21 [5] Gruttmann F, Wagner W. Shear correction factor in Timohenko beam theory for arbitrary cro-ection. Computational Mechanic 2001;27: [6] Weber C. Übertragung de Drehmoment in Balken mit doppelflanchigem Querchnitt. ZAMM 1926;6: [7] Trefftz E. Über den Schubmittelpunkt in einem durch eine Einzellat gebogenen Balken. ZAMM 1935;15: [8] Murray, N.W., Attard M.M. A Direct Method of Evaluating the Warping Propertie of Thin Walled Open and Cloed Profile Thin Walled Structure 1987;