Euler-Bernoulli Beams

Transcription

1 Euler-Bernoulli Beam The Euler-Bernoulli beam theory wa etablihed around 750 with contribution from Leonard Euler and Daniel Bernoulli. Bernoulli provided an expreion for the train energy in beam bending, from which Euler derived and olved the differential equation. That work built on earlier development by Jacob Bernoulli. However, the beam problem had been addreed even earlier. Galileo attempted one formulation that aimed at determining the capacity of beam in bending, but miplaced the neutral axi. Earlier, Leonardo da Vinci alo eem to have addreed the problem of beam bending. The two key aumption in the Euler-Bernoulli beam theory are: The material i linear elatic according to Hooke law Plane ection remain plane and perpendicular to the neutral axi The latter i referred to a Navier hypothei. In contrat, Timohenko beam theory, which i covered in another document, relaxe the aumption that the ection remain perpendicular to the neutral axi, thu including hear deformation. In the following, the governing equation are etablihed, followed by the formulation and olution of the differential equation. Thereafter, a ection cro-ection analyi decribe the computation of tree and cro-ection contant. The tarting point i D beam bending with the following ign convention: The z-axi i increae upward Diplacement w i poitive in the direction of the z-axi Ditributed load q z i poitive in the direction of the z-axi Bending moment that impoe tenion at the bottom i poitive Clockwie hear force i poitive Counter-clockwie rotation θ i poitive, thu it can be interpreted a the lope of the deformed beam element Tenile tree and train are poitive, compreion i negative Equilibrium The equilibrium equation are obtained by conidering equilibrium in the x-direction for the infiniteimal beam element in Figure. Notice that the ditributed load, q, act in the downward direction, while the z-axi i in the upward direction. The notation q z i employed in other document to identify the cae where poitive load act in the poitive z-direction. Vertical equilibrium yield: q z = dv Moment equilibrium about the rightmot edge yield: V = dm () () Euler-Bernoulli Beam Updated January, 06 Page

2 In Eq. () the econd-order term that contain are neglected. q z M V V+dV M+dM Figure : Equilibrium for infiniteimally mall beam element. Section Integration Integration of axial tree over the cro-ection: M = σ z d (3) where the minu ign appear becaue it i compreive (negative) tree in the poitive z-axi domain that give a poitive bending moment, i.e., bending moment with tenion at the bottom. Figure i intended to explain thi further. z Negative compreion tre Poitive z-value M M x Minu ign in cro-ection integral i neceary to get poitive bending moment Poitive tenion tre Negative z-value Figure : The reaon for the minu ign in Eq. (3). Material Law The material law throughout linear elatic theory i Hooke law: Euler-Bernoulli Beam Updated January, 06 Page

3 σ = E ε (4) In the context of two-dimenional theory of elaticity, the ue of Eq. (4) implie a plane tre material law. It implie that there i zero tre, i.e., air on the ide of the beam. The alternative plane train verion of the two-dimenional Hooke law i more appropriate in cae where the beam i only a trip of a long rectangular plate that i upported along the two long edge. In that cae the train i retrained in the y-direction: ε yy = σ yy E ν σ xx E = 0 σ yy = ν σ xx (5) Which, ubtituted into the material law in the x-direction yield: ( ) ε xx = σ xx E ν σ yy E = σ xx E ν ν σ xx E = σ xx E ( ν E ) σ xx = ν ε xx (6) ll the derivation and reult in the following are baed on the material law σ xx =E. ε xx from Eq. (4). However, the plain train verion i eaily introduced by replacing the Young modulu, E, in any equation by E/(-ν ). Kinematic The relationhip between the axial train and the tranveral diplacement of a beam element i ought. It i firt recognized that bending deformation eentially implie hortening and lengthening of fibre in the cro-ection. Fibre on the tenion ide elongate, while fibre on the compreion ide horten. The tarting point for the conideration i to link the axial train to the change of length of the imaginative fibre that the cro-ection i made up of. The ame conideration a in kinematic of tru member, namely that train i elongation divided by original length yield: z, w ε = du (7) d! z x, u Figure 3: Navier hypothei for beam bending. Next, the axial diplacement u i related to the rotation of the cro-ection. In particular, conider the infiniteimal counter-clockwie rotation dθ of the infiniteimally hort beam Euler-Bernoulli Beam Updated January, 06 Page 3

4 element in Figure 3. In paing, it i noted that dθ i equal to the curvature, κ. Under the aumption that plane ection remain plane and perpendicular to the neutral axi during deformation, each fibre in the cro-ection change length proportional to it ditance from the neutral axi. The amount of hortening or elongation depend upon the rotation of the cro-ection. geometrical conideration of to Figure 3 how that the hortening and lengthening, i.e., axial diplacement, of each infiniteimally hort fibre i du = dθ z (8) Finally, the rotation θ i related to the tranveral diplacement. For thi purpoe, conider two point on a beam that i apart, a hown in Figure 4. The relative diplacement i dw, which i meaure poitive upward. Conequently, a geometrical conideration of Figure 4 how that: tan(θ) = dw θ (9) where the equation i implified by auming that the deformation are ufficiently mall o that tan(θ) θ.! dw! Figure 4: Rotation of the cro-ection of a beam element. Combination of the previou equation yield the following kinematic equation for beam member: ε = du = dθ z = d w z (0) Thi expreion implie an approximation of the exact curvature of the beam. Mathematically, curvature i defined a κ R () where R i the radiu of curvature of the beam. In the Euler- Bernoulli beam theory that i preented here, the curvature i approximated by κ dθ d w () Euler-Bernoulli Beam Updated January, 06 Page 4

5 Notice that there are two approximation ign. The firt allude to the fact that differentiation i carried out with repect to the x- axi. Unle the deformation are negligible thi i inaccurate; differentiation hould be carried out with repect to the - axi that follow the curving beam axi. The econd approximation i due to Eq. (9). From that equation it i oberved that the accurate expreion for θ i: θ = tan dw (3) If thi expreion wa utilized in the derivation above then the differentiation of the invere tan- function yield κ dθ = d w + dw which reduce to the expreion in Eq. () when the lope dw/ i mall. However, the curvature expreion in Eq. (4) i till approximate becaue the differentiation i carried out with repect to the x-axi and not the beam axi. From mathematic, the exact curvature expreion i: d w κ = + dw 3 (4) (5) Differential Equation The governing equation for beam bending, namely equilibrium, ection integration, material law, and kinematic are ummarized in Figure 5. The differential equation i obtained by combining them a follow: q z = dv = d M = d = d E ε z d = d = EI y d 4 w 4 σ z d E d w z where the modulu of elaticity i aumed contant over the cro-ection and the moment of inertia i defined: d (6) I y = z d (7) Euler-Bernoulli Beam Updated January, 06 Page 5

6 In Eq. (6) it i aumed that the cro-ection i homogeneou o that E i contant. For compoite cro-ection thi aumption i invalid, and the revied verion of Eq. (6) i q z = d 4 w E z d (8) 4 One approach to retaining the original definition of I i to firt elect a reference-value of the Young modulu and aume that all part of the cro-ection ha that E-value. Next, the width of each part of the cro-ection i modified if it Young modulu i different from the reference value. The change in width i proportional to the difference in E-value. E.g., if E i twice the reference value then the width hould be doubled. If thi procedure i followed then Eq. (7) remain valid for the determination of I. q z q z = dv V = dm M q z = EI y d 4 w 4 M = EI y d w w ε = d w z M = σ z d σ σ = E ε ε Figure 5: Governing equation in Euler-Bernoulli beam theory. lthough olving the differential equation for beam bending i rarely done in everyday engineering practice, it i intructive to tudy it olution for imple reference cae. In particular, the olution of the differential equation i the tarting point for the election of hape function in the finite element method. Thoe hape function are often approximate, while the olution of the differential equation reveal the exact hape when the member deform. The general olution of the differential equation reveal whether the finite element hape function are exact or not. For beam member, the general olution of the differential equation when the load q z i uniformly ditributed along the beam i obtained by integrating four time: w(x) = 4 q z EI y x 4 + C x 3 + C x + C 3 x + C 4 (9) It i oberved that under uniform load the diplaced hape of a beam i a fourth-order polynomial. Without ditributed load the diplaced hape i a third-order polynomial. To Euler-Bernoulli Beam Updated January, 06 Page 6

7 obtain the olution for a pecific beam problem it i neceary to pecify boundary condition. To precribe a rotation, hear force, or bending moment, the following equation are ueful, obtained by combining the governing equation etablihed earlier: θ = dw (0) M = EI y d w () V = EI y d 3 w 3 () Neutral xi The neutral axi i a concept in beam bending, but it ha two interpretation. ccording to one interpretation it i the axi along the beam with zero axial tre during bending; ee the left-mot beam in Figure 6. nother interpretation i the axi in the cro-ection with zero axial tre during bending; ee the right-mot beam in Figure 6 and imagine that the beam i ubjected to bi-axial bending that yield thi neutral axi in the croection. Common for both interpretation, i.e., common for both beam in Figure 6 i that the ought axi i the axi where the axial tre i zero. Neutral axi Neutral axi Figure 6: Two interpretation of the neutral axi. Neutral xi along the Beam For homogeneou cro-ection the neutral axi along the beam coincide with the geometrical centroid of the cro-ection. For compoite cro-ection, i.e., croection compoed of different material, it i poible to cale the area of each croection part about it local centroid. The caling i proportional to the E-value relative to a reference value. Subequently, the centroid of the caled cro-ection i determined, which i the ought neutral axi. The centroid i determined by requiring that the firt area-moment y d and z d i zero, where y and z are axi with origin at the centroid. In practice, it i ueful to firt elect a reference point in the cro-ection. Let y o and z o denote the ditance from the reference point to the centroid. Next, divide the cro-ection into convenient egment with area i and let y i and z i denote the ditance from the reference point to the centroid of each egment. When denoting the total cro-ection area by, the following expreion of the firt area-moment mut be equal to zero: Euler-Bernoulli Beam Updated January, 06 Page 7

8 y o = i y i = 0 z o = i z i = 0 where the um are taken over all part of the cro-ection. Solving for y o and z o yield: y o = z o = i y i i z i In word, the location of the neutral axi i obtained by umming area multiplied by ditance for all part of the cro-ection, and then dividing the um by the total area. Neutral xi in the Cro-ection While the neutral axi determined above i a point in the cro-ection, the line in the cro-ection along which the axial tre i zero i now ought. In uni-axial bending thi tak i trivial; the anwer i the axi around which the bending occur. Thu, thi ection addree the problem of bi-axial bending. The location of the ought axi depend on the loading, and conider firt the cae of a moment, M, applied at a clockwie poitive angle, θ, relative to the horizontal y-axi. The decompoition of M yield the following moment about each principal axi: ( ) ( ) M y = M co θ M z = M in θ Becaue y and z are the principal axe, the axial tre in the cro-ection i found by the following formula from fundamental beam theory: σ = M y I y (3) (4) (5) z + M z I z y (6) In paing, it i noted that the value of θ that would yield the larget axial tre at a particular location (y, z) can here be determined. The ought neutral axi i characterized by σ=0, which according to Eq. (5) and (6) yield the following condition: Solved for z, Eq. (7) yield: ( ) M co θ I y ( ) z + M in θ y = 0 (7) I z z = I y tan( θ ) y (8) I z which i the ought reult. From Eq. (8), the angle, ψ, between the y-axi and the neutral axi i olved from the equation tan( ψ ) = z y = I y tan( θ ) (9) I z Euler-Bernoulli Beam Updated January, 06 Page 8

9 Moment of Inertia The cro-ection of any beam member ha two moment of inertia, I y and I z. If y and z are the principal axe of the cro-ection, then thee two moment of inertia are the large and mallet among all poible orientation of the orthogonal y and z axe. The ubcript on I y and I z indicate the axi about which the cro-ection rotate under bending. a reult, their definition are: I y = z d I z = y d For general cro-ection it i often convenient to divide the cro-ection into part, each with local moment of inertia denoted by I i about it local centroid. The contribution to the global moment of inertia from each part, about the global centroid of the croection, are ummed in accordance with the parallel axi theorem, ometime referred to a Steiner at: I y = ( I y,i + z i i ) (3) I z = I z,i + y i i ( ) where y i and z i are ditance from the centroid of the entire cro-ection, i.e., the neutral axi along the beam, to the centroid of the part. In cae where I y and I z about the principal axe are ought, but y and z are not the principal axe, I y,principal and I z,principal can either be calculated from cratch once the orientation, φ, of the principal axe are determined, or they can be determined by tranformation of the original moment of inertia: I y,principal = I y + I z I z,principal = I y + I z + (I y I z ) co(φ) + (I z I y ) co(φ) I yz in(φ) I yz in(φ) Product of Inertia The product of inertia, I yz =I zy, i a quantity that i non-zero only for double-unymmetric cro-ection, i.e., cro-ection without any axi of ymmetry. If a ymmetry axi exit then it i a principal axi, and thu both principal axe are immediately known. The product of inertia hould not be confued with the polar moment of inertia, which i related to torion. The product of inertia appear in the derivation of the governing equation for bi-axial bending, where it i defined a: (30) (3) I yz = y z d (33) Euler-Bernoulli Beam Updated January, 06 Page 9

10 The coordinate y and z in Eq. (33) repreent ditance from the neutral axi, which i a point in the cro-ection, to the area element. For cro-ection with a ymmetry axi, I yz =0. The parallel axi theorem, i.e., Steiner formula, for product of inertia i: ( ) I yz = I yz,i + y i z i i (34) where y i and z i are ditance from the global centroid to the local centroid. For example, the value of I yz for a triangular cro-ection with ide-length b and h i I yz = b h 7 (35) Principal xe The identification of the principal axe of a cro-ection ha everal benefit; it implifie the equation for tre and train and it implie that I y and I z are the greatet and mallet moment of inertia of the cro-ection. The product of inertia i employed to determine the principal axe; once I yz i determined the following concluion can be made in regard to the principal axe: If I yz =0 then y and z are the principal axe If I yz 0 then y and z are NOT the principal axe The orientation of the principal axe relative to the original axe i φ = arctan I yz I y I z (36) xial Stre in Bending For uniaxial bending, a convenient approach for obtaining the axial tre in term of the bending moment i to combine material law and kinematic equation, which yield: σ = E d w z (37) Then ubtitute the differential equation without equilibrium equation to obtain: σ = M I z (38) The minu ign mean that a poitive bending moment, i.e., tenion at the bottom, correctly yield negative tree at the top, i.e., compreion, where z i poitive. Thi i the ame reaon that wa given for the minu ign in Eq. (38), which alo correctly give poitive tenion tree at the bottom when a poitive moment act on the cro-ection. For biaxial bending, Eq. (38) i valid for bending about both the y and z axe, a long a they are the principal axe. If they are not the principal axe but their orientation, φ, relative to the original axe i known, then it i poible to develop formula for tre in term of φ and the original axe. Thee formula will be more complicated than the Euler-Bernoulli Beam Updated January, 06 Page 0

11 fundamental one becaue they include the product of inertia, I yz. The tarting point i the expreion for train at an arbitrary location in the cro-ection: ε = ε o d v y d w z (39) The material law yield the preliminary expreion for tre: σ = E ε o E d v y E d w z (40) Integration of axial tre yield the bending moment My and Mz about the y and z axe, repectively. Firt conider the bending moment about the y- axi: M y = σ z d = E ε o z d + E d v y z d + E d w z d (4) The firt integral vanihe becaue z originate at the neutral axi, while the lat term i the ordinary bending moment. a reult, Eq. (4) and it counterpart for M z can be written a: M y = EI yz d v + EI d w y M z = EI yz d w + EI z d v where the product of inertia, I yz, ha been defined a: (4) I yz = y z d (43) Similarly, I yz appear in the expreion for M z. Eq. (4) repreent kinematic, material law, and ection integration. In contrat, Eq. (40) repreent only kinematic and material law. Thu, combination of Eq. (40) and Eq. (4) facilitate the iolation of the ection integration relationhip, namely the relationhip between axial tre and bending moment. Thi i done by olving for the curvature in Eq. (4) and ubtituting them into Eq. (40). Becaue the axial train i N/E the reult i σ = N + M y M ( ) z ( I y I z I yz ) ( I y y I yz z) (44) ( I y I z I yz ) I z z I yz y Shear Stre in Bending When approaching hear tree in bending, an anomaly in Euler-Bernoulli beam theory i oberved. The theory i baed on the aumption that plane ection remain plane and perpendicular to the neutral axi. In other word, the only train that take place i the axial hortening or elongation of the fibre in the cro-ection. Effectively, thi prevent hear train. With no hear train there i no hear tre, which add up to zero hear force. In other word, hear train and hear force i not part of the Euler-Bernoulli beam Euler-Bernoulli Beam Updated January, 06 Page

12 theory. Thi i an anomaly, becaue hear force will develop even in imple beam that are ubjected to tranveral load. The anomaly i addreed by recovering the hear force by equilibrium, once the bending moment i computed. Thi i een in the imple beam theory, where the hear force i equal to the derivative of the bending moment; thi i the equilibrium equation that recover the hear force. Equilibrium conideration will alo be employed in thi document, to determine the hear tre and hear flow at any point in the cro-ection. nother document on Timohenko beam theory decribe an approach to extend Euler-Bernoulli beam theory to include hear deformation in the beam deflection. Cro-ection with Edge Parallel to Shear Direction To obtain the mot popular expreion for the hear tre, τ, in term of the hear force, V, conider the infiniteimally hort beam element in Figure 7. Furthermore, conider a cut in the cro-ection and let q denote the hear flow at that location. σ σ + dσ M V q V+dV M+dM Figure 7: Shear flow by equilibrium of infiniteimal beam element. The hear flow, q, i the force per unit length of the beam that enure equilibrium with the axial tree, which are greater on one ide than the other due to dm: q = dσ d = dm z d (45) I where i the cro-ectional area outide the cut. Given that V=dM/, thi yield q = V I Q (46) where the firt moment of area, Q, ha been defined a Q = zd (47) To eae the evaluation of Q in practice, the cro-ection i often dicretized into everal part with area i and ditance z i from the neutral axi to the centroid of the part. Then N Q = z i i (48) i= Euler-Bernoulli Beam Updated January, 06 Page

13 where N i the number of cro-ection part. Once Q i determined at a particular location, the hear tre i calculated by ditributing the hear flow over the thickne, t, of the cro-ection at the particular location: τ = V Q I t General Cro-ection The expreion in Eq. (49) i omnipreent in tructural engineering, but it i not alway valid. Firt, the beam mut be primatic, i.e., traight with contant cro-ection. Second, the cro-ection mut have a certain hape. Specifically, the edge of the cro-ection mut be parallel with the z-axi, o that the hear tree align with that axi. In other word, the cro-ection for which Eq. (49) i mot accurate i rectangular, high, and thin. In contrat, it i more difficult to determine the hear tre ditribution within a triangular cro-ection. That being aid, it i often poible to apply Eq. (49) a an approximation. For example, for a olid circular cro-ection the edge are locally parallel to the z-axi at the neutral axi. Eq. (49) can therefore be ued a an approximation to determine the hear tre at that particular location in the cro-ection, i.e., at the neutral axi. nother type of cro-ection where Eq. (49) i utilized i thinwalled cro-ection. For uch problem, a new -axi i defined, which follow the contour of the cro-ection. It i aumed that the hear flow and hear tre follow that direction, thu atifying the aforementioned requirement. With that aumption, Eq. (49) can be directly applied. For thin-walled cro-ection that are cloed, i.e., coniting of one or more cell, the hear tre cannot be olved by equilibrium alone, i.e., they are tatically indeterminate, a dicued next. Cloed Thin-walled Cro-ection The determination of hear tre and hear centre for cloed cro-ection, i.e., croection with cell, can be carried out in two way. In the approach addreed firt, the hear flow i determined before the hear centre. Thi approach explicitly recognize that the calculation of hear flow in a cloed cro-ection i a tatically indeterminate problem. In other word, equilibrium equation alone are inufficient to determine the ought force. Each cell i aociated with one redundant. Similar to the flexibility method in fundamental tructural analyi, the olution approach involve removing the capacity of the tructure to carry the ought force, i.e., to introduce cut, and then to enforce compatibility equation that are olved for the unknown force. In the following, a thin-walled cro-ection with one cell i conidered, and one cut i introduced to make it an open cro-ection. t the cut there will develop a dicrepancy in u-diplacement at each ide of the cut. The compatibility equation require thi diplacement to be zero: u = (49) du = 0 (50) where du are the infiniteimal contribution that are integrated along the -axi, which run through the middle thickne along the cro-ection. Figure 8 how an infiniteimal part of the cro-ection of a beam element, een from the ide of the beam. The figure illutrate the kinematic relationhip between the hear train, γ, and the ought quantity du: Euler-Bernoulli Beam Updated January, 06 Page 3

14 γ = du d (5) x, u q d γ q du Figure 8: Kinematic relationhip between du and hear train γ. It i noted that generally there would be another contribution to the hear train, namely (dφ/)(h), where φ i the rotation of the cro-ection and h i the ditance from the neutral axi to the tangent of the cro-ection part. However, it i undertood that the rotation of the cro-ection mut be zero if the hear force act through the hear centre. Thu dφ/=0. In hort, the change in axial diplacement at two point located a ditance d apart i, from kinematic: du = γ d (5) Material law provide the following expreion for the hear train in term of the hear tre and thu the hear flow: γ = τ G = q G t where G=E/((+ν)) i the hear modulu and t i the thickne of the cro-ection-wall, which may vary around the circumference. Subtitution of Eq. (53) into Eq. (5) yield (53) du = q d (54) G t Integration around the cell yield the total gap opening at the cut, which i enforced to zero by compatibility: Euler-Bernoulli Beam Updated January, 06 Page 4

15 u = q G t d = 0 (55) The unknown hear flow, i.e., the redundant, at the cut i denoted q o. Becaue the cut cro-ection i open, the hear flow at other location i: q () = q o + V I Q det () (56) where Q det i the firt moment of area with zero value at the cut, evaluated according to Subtitution of Eq. (56) into Eq. (55) yield Q = z d = z t d (57) 0 0 Solving for q o yield: q u =! o G t d + V Q det! I G t d = 0 (58) q o = V I!! Q det G t d G t d By comparing Eq. (59) and Eq. (46) it become clear that the integral fraction in Eq. (59) repreent the value of the firt moment of area at the cut: Q o =!! Q det G t d = G t d Q det d t t d where contant hear modulu G i aumed in the lat equality. ccording to Eq. (60), the final Q-diagram for the cro-ection can be obtained a follow:. Draw the Q det -diagram for the tatically determinate (cut) cro-ection by evaluating Eq. (57). Obtain the numerator in Eq. (60) by integration of the tatically determinate Q- diagram divided by repective thicknee; thi integration can ometime be cumberome but i eentially the integral of Eq. (57) 3. Obtain the denominator in in Eq. (60), which i traightforward becaue, for example for a rectangular cloed cro-ection with width b, height h, and thickne t, it i imply b/t+b/t+h/t+h/t 4. Obtain the final Q-diagram by adding the contant Q o to the tatically determinate one from the firt tep, rememberingt the minu-ign in Eq. (60) nother way of evaluating Eq. (60) i to firt apply integration by part to the numerator, which eentially move the integration from Q to /Gt:!! (59) (60) Euler-Bernoulli Beam Updated January, 06 Page 5

16 ! ( Q) Gt d = z t d"! Gt d 0 = z t d" 0 Gt d" 0 o =! ( z t) 0 Gt d" d! ( z t) 0 Gt d" d where the boundary term vanihe becaue the firt moment of area for the entire croection i zero when z originate at the neutral axi. Subtitution into Eq. (60) yield: Q o = " 0 Gt d! z t d " G t d If the cro-ection ha ome open part, i.e., flange that tick out from the part that encloe the cell, then the following mut be noted: The part of Eq. (60) that relate to the cloed integral around the cell do not pick up contribution from the protruding flange, but Q doe. In other word, the integration of z. t along in the numerator in Eq. (6) pick up contribution from the protruding flange. Thi fact i reiterated hortly. For now, notice that when the material i homogeneou, o that G i contant throughout the cro-ection, the expreion in Eq. (6) implifie to: Q o = " 0 t d! z t d " t d To reorganize thi expreion for practical computation it i ueful to define the function t d 0 g() = t d where the denominator i a contant and the numerator varie with. Subtitution of g() into Eq. (63) yield the following final expreion for the firt moment of area at the cut: Q o = (6) (6) (63) (64)! g() z t d (65) It i reiterated that g() i unaffected by protruding open part of the cro-ection, while the integration of z. t. d mut include contribution from thoe part: ( ) Q o =! g() z t d + Q flange#i g i (66) Euler-Bernoulli Beam Updated January, 06 Page 6

17 where Q flange i the firt moment of area of the flange and g i i the value of the g-function where the flange attache to the cell. Once Q o i computed, the hear flow at other location i determined a for open cro-ection, relative to the location of the cut: q () = V I ( Q + Q() ) (67) o For multi-cell cro-ection, multiple cut are introduced to make the cro-ection tatically determinate. Specifically, one cut and one compatibility equation i introduced for each cell. t each cut, the value of the firt moment of area, Q o, i determined either by Eq. (60), i.e., direct integration of the tatically determinate Q-diagram, or by Eq. (65), e.g., integration of the auxiliary g-function. Both approache are viable for multi-cell cro-ection, although the ue of Eq. (65) can now be omewhat more error prone. The reaon i the additional complication ariing from the hear flow in the eparation-wall that have cell on both ide. The ue of Eq. (60) i firt addreed. The tep-wie procedure that wa decribed above for ingle-cell cro-ection i adopted here, with an important modification; the integral of Q mut include both the earlier tatically determinate diagram and now alo the tatically indeterminate hear flow in the eparation wall: Q! t d = Q det () d t (68)! + h wall #i Q o,cell # j t wall #i where Q det i the firt moment of area for the tatically determinate (cut) cro-ection, h wall and t wall are the length and thickne of a wall that eparate two cell, repectively, and Q o i the redundant in the cell on the other ide of the wall compared to the cell around which the integral i conducted. In other word, the total Q-value that i integrated conit of the tatically determinate value, Q det, plu contant Q o around each cell. Becaue the hared wall couple the redundant in neighbouring cell, a ytem of equation i formed, which i olved for the unknown Q o value at the cut. The main challenge in the ue of Eq. (60) i to integrate a Q-diagram. For ingle-cell cro-ection, that challenge wa addreed by the introduction of the g-function and evaluation of Eq. (65). Thi approach i alo poible for multi-cell cro-ection, but it i now eaier to commit ign error. In the application of Eq. (65) it i once again noted that g() i unaffected by protruding flange, while z. t. d mut include uch part. In fact, thi i how the neighbouring cell enter the continuity integral around a cell. before, the g-function i etablihed around each cell, varying from zero to unity according to Eq. (64). Eq. (65) i then evaluated, wall-by-wall around the cell, adding contribution from the protruding flange from neighbouring cell: Q o = g() z t d ( ) g j (69) ( )! + Q flange# j ± Q o, j where Qflange i the tatically determinate firt moment of area, Qo i the contribution from the redundant in that cell, and g i the value of the g-function where the flange from the neighbouring cell attache to the cell around with the compatibility integration i taking place. In hort, the integration around each cell include the part around the other cell() a if they were protruding flange. Euler-Bernoulli Beam Updated January, 06 Page 7

18 Known Shear Centre In thi approach, the hear centre location i determined firt, either from ymmetry or uing the omega diagram from warping torion. uming the cro-ection ha one cell, the hear flow i determined a follow. lo in thi approach, a cut i introduced, which yield an open cro-ection. The coordinate originate at the cut and trace the cro-ection around the cell. The unknown hear flow at the cut i denoted q o, and the hear flow at all other location i determined relative to q o in accordance with Eq. (46), o that q () = q o + V I Q() (70) Once q i determined at all location of the opened cro-ection, the moment of the hear flow about the known hear centre i computed a T = q hd = q + V I Q hd = q o hd + V Q hd I where the integral are made around the cell, tarting at =0, and h() i the ditance from the hear centre to the tangent line of the contour of the cro-ection at. By definition the moment, T, about the hear centre mut be zero, and olving for q o yield q o = V I Q hd = hd (7) V m I Q hd (7) where the lat equality i obtained by recognizing that the integral of h around the croection i twice the cell area, m. Having the value of q o, the hear flow i determined at other location with Eq. (70). Bi-axial Bending The formula above are derived for uni-axial bending, i.e., bending about one of the principal axe of the cro-ection. The problem of bi-axial bending can be decompoed into two cae of uni-axial bending by determining the principal axe and conider bending about each axi eparately. However, an alternative i to leave the principal axe unknown and rather develop tre formula that include the product of inertia, I yz, which i non-zero unle y and z are the principal axe. In another document it i hown that the reulting axial force expreion i σ = N + M y M ( ) z ( I y I z I yz ) ( I y I z y yz ) (73) ( I y I z I yz ) I z I y z yz Thi expreion for the axial tre yield the following expreion for hear tre, again derived by equilibrium, a done earlier in thi document for open cro-ection: Euler-Bernoulli Beam Updated January, 06 Page 8

19 τ = V y t Q I Q I z y y yz V z I y I z I yz t Q I Q I y z z yz (74) I y I z I yz For reference, thi expreion revert to the ummation of the ordinary hear tre expreion when the y and z axe are the principal axe: τ = V y Q z I z t V z Q y I y t (75) Shear Centre The hear centre of a cro-ection, ometime called the centre of twit, i the point where the reultant of the hear force mut act to avoid rotation of the cro-ection. The coordinate of the hear centre are denoted y c and z c, and there are everal technique to determine them. The implet cae i double-ymmetric cro-ection; for thee croection the hear centre coincide with the centroid. In fact, if a cro-ection ha an axi of ymmetry then the hear centre i located on thi axi. For general cro-ection, one approach to determine y c and z c i decribed in the document on warping torion, where the omega diagram i utilized. However, a omewhat impler approach, when warping torion i not conidered, i offered here. The underlying principle i that the moment of the hear flow about the hear centre mut be zero. Thi lead to the following procedure to determine the coordinate of the hear centre, provided y and z are the principal axe through the centroid of the cro-ection:. Select an arbitrary point a trial hear centre, and let y c and z c denote the coordinate of the hear centre relative to the centroid; in other word, let y c and z c denote the ditance from the centroid to the trial hear centre. Determine the hear flow in the cro-ection due to a hear force in the z- direction, uing the formula preented above 3. Write the equation that expree the moment of the hear flow about the trial hear centre; in general, both y c and z c will appear in thi expreion 4. Determine the hear flow in the cro-ection due to a hear force in the y- direction, again uing the formula preented above 5. Write the equation that expree the moment of the hear flow determined in the previou item about the trial hear centre; in general, both y c and z c will appear in thi expreion 6. Set the equation from Item 3 and 5 equal to zero and olve them for the two unknown y c and z c Only one moment equation i needed for ingle-ymmetric cro-ection; in that cae the procedure implifie to:. Select an arbitrary point along the ymmetry axi a trial hear centre, and let e denote the ditance from the centroid to that point. Determine the hear flow in the cro-ection due to a hear force in the direction perpendicular to the axi of ymmetry 3. Write the equation that expree the moment of the hear flow in Item about the trial hear centre; e will appear in thi expreion Euler-Bernoulli Beam Updated January, 06 Page 9

20 4. Set the equation from Item 3 equal to zero and olve for e Euler-Bernoulli Beam Updated January, 06 Page 0