7.1. General rudiments for the stability analysis

Transcription

1 7. STRUCTUR STBIITY 7.. General rudiments fr the stabilit analsis In stabilit analses traditinall, the equilibrium equatins f the structure cnsidered will be written in its defrmed cnfiguratin. This means that the influence f the defrmatin has t be taken int accunt in the equilibrium cnditins. We call it gemetrical nn-linearit. et s cnsider at first the Green-agrange s general nnlinear epressins f strain cmpnents in the Cartesian c-rdinate sstem,, with unit ectrs i,, jk in accrdance with the definitins gien in equatins (.3, (.5, (.8, (. u u u u u ( ( w ( ε = i+ = u u u u ( ( w ( ε = j+ = u u u w u ( ( w ( ε = k+ = γ γ γ u u u u u u u w w = j+ i+ = u u u u w u u w w = k+ j+ = u u u u u w u u w w = i+ k+ = (7. If we intrduce fr the linear parts f strains the ntatins u w u e =, e =, e =, e = +, e w u w = +, e = + (7. and appl still the rtatin cmpnents, defined b (.7

2 u u w ω = ω = k j = u u u w ω = ω = i k = u u u ω3 = ω = j i = (7.3 all the strain cmpnents, ε fr eample, can be epressed b using the linear strains and rtatins in the frm ε u u u u w u w u = + ( + ( ( ( ω ( ω = e + e + e + + e (7.4 nd the shear strain γ crrespndingl γ u u u u u u = + + ( ( + + w u w u w w + ( + + ( = e + e ( e ω + e ( e + ω + ( e ω ( e + ω (7.5 Thus, we see that the nn-linear parts f the strains can be epressed b the linear strain and rtatin cmpnents nl. Crrespndingl, the rest f strain cmpnents can be deried similarl, resulting finall in the frm ( ( ( ( ( ( = e + e + e + + e ε ω ω = e + e + e + e + ε ω ω = e + e + e + + e ε ω ω γ = e + e ( e ω + e ( e + ω + ( e ω ( e + ω γ = e + e ( e ω + e ( e + ω + ( e ω ( e + ω γ = e + e ( e ω + e ( e + ω + ( e ω ( e + ω (7.6 In stabilit analses, the strain cmpnents e, e, e, e, e, e are assumed t be small as cmpared t rtatin cmpnents ω, ω, ω. Thus, in nn-linear terms the quadratic terms f rtatins nl will be included in the analsis, ielding fr strains the epressins

3 = e + ( + = e + ( + = e + ( + ε ω ω ε ω ω ε ω ω γ γ γ = e ωω = e ωω = e ωω ( Fleural Buckling f a straight plane beam We cnsider nw a clumn, in which the aial c-rdinate cincides with the clumn ais, i.e. ges thrugh the centrid f each crss-sectin plane. C-rdinates and are the principal aes f the crss-sectin. Central aial frces lad the clumn at the initial state, nl neither bending, nr trsin eists. In buckling analses, these lads are cmpressie. When defining the kinematics fr the analsis, an additinal degree f freedm t the initial state has t be adpted. This will lead t the hmgeneus sstem f equatins with respect t this additinal degree f freedm, f which the critical lad, as a slutin f an eigenalue prblem will be determined. The kinematics adpted fr a straight plane beam including in additin t cmpressin (stretching als bending, in accrdance with the Euler-Bernulli beam ther is d u= ( u i+ j (7.8 d in which u= u ( and = ( are the displacement cmpnents in - and -directin, and the rtatin f each nrmal fllws the slpe f the beam ais. Calculating nw the strains, using (7. and (7.3, gies just tw nn-er cmpnents, while e = e = e = e = e = ω = ω = du d e = = u d d d ω = = d In the cntinuatin t simplif the ntatins, a prime ( will be used fr differentiatin with respect t the aial c-rdinate. The nn-er strain cmpnents are (7.9 ε = u + ( ε = ( (7. t the initial lading state, cnsisting nl f aial cmpressin, the nrmal stress is σ = N /, while all the ther stress cmpnents are er.

4 σ σ * σ * ε σ ο σ ο ε ε U= σ ο ε + σ * ε ε Figure. Strain energ b the Euler methd. The prcedure we appl fllws the linearised ther, called als Euler methd (see Figure, presented fr eample b Nhil s and Washiu s famus tet-bks accrding t which the incremental strain energ f the beam is * * * σε σ ε σ ε U = U + U = ( + + d This can further be split t gie * * = ( σ + σ ( ( + ( d u + σ 4 (7. * * * σ σ σ σ 4 σ U = ( u d + ( u d + ( d + ( + ( d (7. The last term can be neglected because incrprating the linear elastic material mdel int it gies higher (third rder terms f displacement functins. The equatin can be written in the frm U = U + U + U (7.3 N N in which the first term, strains and the tw fllwing nes, U N U N U, is the traditinal strain energ epressin due t linearised and, take int accunt bth the nnlinear terms f strain cmpnents and the initial stresses.

5 When we take int accunt the linearl elastic material law in (7. the strain energ takes the frm * σ = Eε = E( u + ( E( u (7.4 U = E( u d + ( u d + ( d (7.5 σ σ in which N = σ( d = ( σ d N = σ = ( U = E( u d ( E( u EI ( d = + U u u ( σ U ( d ( d (7.6 Here, we hae split the lume integral t be ealuated separatel er the crsssectin, and alng the ais f the clumn, and taken int accunt that d =, d= S =, d= I = I (7.7 The ptential due t the eternal lads at the initial stage, the distributed aial lad p ( and the cncentrated lads at the ends f the clumn P, is ( d (7.8 = p u Pu Nw, the ttal ptential energ is cmpsed f terms If we cnsider at first the term Π = U + = U + U + U + (7.9 N N N ( d (d (7. U + = σ u p u Pu and find fr the statinar alue f it. B taking the first ariatin it is btained

6 N ( (7. δ( U + = σ δu d p ( δud P δu While N ( = σ, integrating the first term b parts gies N + = ( + + = ( ( ( δ d + + ( δ δ( U ( σ p ( δud ( σ P δu N p u N P u (7. which results in the equilibrium equatin f the initial state with bundar cnditins dn d + p ( =, N = P r δu = at = and = (7.3 This part f equatin cnsiders the initial state f the clumn, and it disappears when the initial state is in equilibrium. S, we hae still the epressin N ( ( Π = U + U = ( E( u + EI( d + N ( d Taking here the first ariatin gies and integrating b parts finall (7.4 ( δπ = δ( U + U N = ( Eu δu + EI δ + N δ d (7.5 δπ = Eu δu EI N δ d ( ( ( (( + Eu δu + EI δ EI N δ = = (7.6 Since δu and δ are arbitrar, we get as a result the sstem f hmgeneus differential equatins

7 with the bundar cnditins ( Eu = = ( EI ( N (7.7 Eu= N = r δu = EI = M = r δ = ( EI N = Q = r δ = (7.8 The first ne f the equatins (7.7 likewise f bundar cnditins (7.8 cncerns nl the initial state f the beam and is thus meaningless. If the crss sectin f the beam is cnstant, and we hae at the ends as a lad a cmpressie frce nl, i.e. N = P, the equatin simplifies t the well-knwn frm EI + P = (7.9 which is the hmgeneus rdinar differential equatin t define the critical cmpressie lad f a beam. Its general slutin is ( = C sin k + C cs k + C + C ( where k = P / EI. hmgeneus sstem f equatins will be btained b appling the releant bundar cnditins at the ends f the clumn. n alternatie frmulatin fr the differential equatin (7.7 with bundar cnditins (7.8 can be btained directl b appling the energ integral frmulatin (7.4, which simplifies amng the kinematicall admissible functins ( t the minimiatin prblem f the functinal ( Π = EI( + N ( d (7.3 When cnsidering a clumn in three-dimensinal space where the buckling can take place in an ne f the directins f the principal aes, the differential equatin sstem (7.7 will be prided with an additinal equatin ( Eu = ( EI ( N ( EIw ( N w = = (7.3 with d = I, and crrespnding additinal bundar cnditins

8 Eu= N = r δu = EI = M = r δ = EI w = M = r δw = ( EI N = Q = r δ= ( EI w N w = Q = r δw= (7.33 The minimiatin prblem takes the frm ( Π = EI ( + EI ( w + N (( + ( w d (7.34 Here, bth ( and w ( hae t fulfil the requirements due t kinematics. 7.3.Trsinal buckling f a straight beam Trsinal buckling is characteristic fr the behaiur f beams with thin-walled crsssectin. In the trsinal buckling, a beam laded at the initial stage b an aial lad nl, buckles thrugh mechanisms f trsin and bending, Figure. We cnsider a thin-walled beam, in which the aial c-rdinate cincides with the beam ais, i.e. ges thrugh the centrid f each crss-sectin plane. The lading at the initial state cnsists f centric cmpressin (stretching. C-rdinates and are the principal aes Trsinal buckling Figure. Trsinal buckling f a clumn f the crss-sectin. s an additinal degree f freedm, the tw deflectins (- and - directins and trsin, including the effects f bth Saint-enant s trsin and f las s warping (sectrial trsin, is adpted. The kinematics will be defined fr the

9 displacement ectr f the centre-line f the wall in the crss-sectin, dented b u. Fr a straight beam in stretching, bending and trsin the kinematics is defined b ( ( u = ( u w ωφ i+ ( φ j+ w+ ( φ k (7.35 in which u = u(, = (, w= w( and φ = φ( are the three translatin cmpnents in the directins f the c-rdinate aes, and the angle f twist, and ω is the sectrial c-rdinate. C-rdinates (, define the lcatin f the shear centre f the crss-sectin. Calculating nw the strain and rtatin cmpnents b appling (7. and (7.3 gies, e = e = e = and n α s Figure 3. Thin-walled crss-sectin. e = u w ωφ ω e = ( φ = ω e = + ( φ = ω = φ ω ω = ( w w ( φ = φ ω ω = ( φ + = ( φ (7.36 Substituting these int the epressins f strains gies

10 [ ( ] [ w ( ] [ w ( φ ] φ [ w ( ][ ( ] [ ( φ ] φ [ ( ] [ ( ] = u w + w + ε ωφ φ φ ε = φ + φ ε = φ φ + γ = + γ = + φ φ γ = + (7.37 The linear strain cmpnents e = e =, accrding t the assumptins f the anishing shear strain cmpnents in the warping trsin ther b las. It can be seen directl f the definitin (3.68 f the sectrial c-rdinate ω = ( ω = ( Hweer, these shear strain cmpnents d nt anish utside the mid surface f each wall f the crss-sectin. Thus the displacement ectr defined fr the mid surface u (7.35 has t be prided with a cmpnent cering the material pints utside it. The additinal cmpnent in the aial directin applies the Euler-Bernulli beam ther (r Kirchhff s plate ther, while n the crss-sectin plane, the rtatin f the whle crss-sectin defines the tangential displacement t the mid surface. In the directin f the thickness f the wall, the wall is assumed t be incmpressible. The displacement ectr is thus u= u n i nφ e n s where n is the nrmal c-rdinate t the mid surface f the wall (see Figure, and the displacement in this directin, i.e. = ( ( φsin α + ( w+ ( φcsα n Inserting this int the definitins f strains gies additinal cmpnents u ε = i = nn u u n γ s = es + i = n( φ s s The first ne is cnnected t the fleure f each plate lcall, and will be drpped ut, while the secnd ne is meaningful, leading t Saint-enant s trsin rigidit f the crss-sectin. Substituting int it gies n n

11 γ s = nφ n φ sinα + φ csα s s = nφ n( φ sin α + φ cs α = nφ T cmbine the shear strain terms, we transfrm the glbal shear cmpnents γ and γ in (7.37 nt the cmpnents γ and γ. The transfrmatin is deried in When taking int accunt the transfrmatin between the c-rdinate sstems, and s,n, deried in eample (Chapter 4, page 9 we get the final epressins f strains s n γ = γ csα + γ sinα s γ = γ sinα + γ csα n [ ( ] [ w ( ] [ ( ] [ ( ] = u w + w + ε ωφ φ φ ε = φ + φ ε = φ φ + γ = φ + + ( φ φcs α ( φ φsinα n [ w ] [ ] [ w ( ][ ( ] [ w ( ] sin [ ( ] s γ = + φ φ γ = + φ φ α n φ φ cs α (7.38 t the initial lading state, cnsisting nl f centric aial cmpressin, the nrmal / stress is σ = N, while all the ther stress cmpnents are er. Fllwing the same prcedure as befre, the incremental strain energ f the beam takes the frm * * * * * * s s n n U = ( σε + σε + σε + σε + τ γ + τ γ + τ γ d ( [ ] [ ] * * τ ( d s nφ ( σ + σ u w ωφ + w + ( φ + ( φ d + When we take int accunt the linearl elastic material law (7.39 [ ( ] [ ( ] σ ε ωφ φ φ E( u w ωφ * = E = E u w + w + + * s τ = Gγ = ngφ s (7.4 this will be

12 U = E( u w ωφ d G( nφ d + + σ ( u w ωφ d ( [ ] [ ] + σ w + ( φ + ( φ d (7.4 This is cmbined f three parts as U = U + U + U (7.4 N N / with taking int accunt that σ = N, when U = E( u w ωφ d G( nφ d + N ( ω φ t φ = E( u + EI ( + EI ( w + EI ( + GI ( d ( ωφ ([ ] [ ] U = σ ( u w d = N u d N = + + = N ( w + ( U σ w ( φ ( φ d wφ + φ + r ( φ d (7.43 Here in additin t (7.7, the ntatins d=, d = I, d=, d = I, 4n d= It (7.44 ω ω ω are used, and ( / r = I + I + +. The ptential due t the eternal lads in the aial directin is ( d (7.45 = p u Pu Nw, the ttal ptential energ is cmpsed f terms Π = U + = U + U + U + (7.46 N N

13 If we cnsider at first the part N ( d ( d (7.47 U + = Nu p u Pu It is eactl the same as in the case f pure fleural buckling, and disappears when the initial structure is in equilibrium fulfilling the equatin f equilibrium with bundar cnditins dn d + p ( =, N = P r δu = at = and = This part f equatin disappears when the beam in the initial state is in equilibrium. S, we hae still the equatin (7.48 N ω φ t φ + N ( w + ( w φ + φ + r ( φ d Π = U + U = E( u + EI ( + EI ( w + EI ( + GI ( d (7.49 Taking at first the first ariatin f the first term U results in δu = ( Eu δu EI δ EI w δ w EI φ δφ GI φ δφ d (7.5 ω t and integrating this b parts gies ] ( δu = ( Eu δ u ( EI δ ( EI w δ w ( EI φ ( GI φ δφ d ω t + Eu δu + EI δ ( EI δ + EI w δw ( EI w δw + + EIωφ δφ ( EIωφ δφ + GItφ δφ = = (7.5 The first ariatin f the secnd term U N in (7.49 is N = ( δu N wδ w δ ( wδφ φ δ w ( δφ φ δ r φ δφ d which takes after integrating b parts the frm (7.5

14 (( ( (( ( δu = N w N φ δw+ N + N φ δ N ( φ ( N w ( N ( N r δφ d + N ( w φ δw+ ( + φ δ ( w r φ δφ = = (7.53 Cmbining finall (7.5 and (7.53 results in a hmgeneus sstem f rdinar differential equatins with bundar cnditins, which are btained since δ u, δ, δ w and δφ are arbitrar. We get ( Eu = ( EI ( N ( N φ = ( EI w ( N w + ( N φ = ( + ( ( = ( EIωφ GItφ N w N r ( N φ with crrespnding bundar cnditins (7.54 Eu = N = r δu = EI = M = r δ = EI w = M = r δw = EI φ = B = r δφ = ω φ ( EI + N ( + = Q = r δ= φ ( EI w + N ( w = Q = r δw= t ω t ( EI φ + GIφ N ( w r φ = M + M = r δφ = ω (7.55 at each end f the beam. The first equatin f (7.54 and (7.55 describe the initial state and are nt f interest in this cntet. If the crss sectin f the beam is cnstant, and we hae at the ends as a lad a cmpressie frce nl, i.e. N = P, the sstem f equatins simplifies t the well-knwn frm EI + P + Pφ = EIw + Pw Pφ = EI φ GI φ Pw + P + r Pφ = ω t (7.56 which is the hmgeneus sstem f equatins t define the critical cmpressie lad f the beam. n alternatie frmulatin fr the differential equatin (7.54 with bundar cnditins (7.55 can be btained directl f the energ integral frmulatin (7.49, which simplifies t the minimiatin prblem f the functinal

15 Π = EI( + EI ( w + EIω ( GIt( d φ + φ + N ( w + ( w φ + φ + r ( φ d ( Cmbined fleural and trsinal buckling The beam cnsidered in this cntet is laded at the initial state b an eccentric cmpressie lad ielding tw mutuall equal bending mments at each end f the beam. Thus, the bending mment distributins er the beam length are cnstant. The lading at the initial state is a cmbinatin f aial frce and bending. Fr stabilit analsis, the additinal degree f freedm is then trsin. The initial nrmal stress distributin takes the frm N M M P Pe Pe I I I I σ = + + = + + (7.58 ll the ther initial stress cmpnents disappear. The latter part f the presentatin (7.58 cncerns the eccentric aial lad, in which c-rdinates e, e define its lcatin n the crss-sectin plane. The cnsideratin deiates frm the ne f the preius sectin nl in the terms U and U in (7.43 thrugh the initial nrmal stress N N distributin σ, and in in (7.45, in which the end mments hae t be included. The terms in (7.43 will get supplements U and U ( N N M M UN = ( d ( ( d σ u w ωφ = + u w ωφ I I = M M w d U = w + [ ] [ ] N ( ( d σ φ φ M = [ φ ] [ φ ] + + I I ( φ β φ ( φ β φ ( M w ( ( d = M w ( M ( + d In these equatins, the ntatins β and β i.e. Wagner s cefficients, (7.59

16 β = ( + d I β = ( + d I are used. The ptential f the eternal lad in (7.45 takes a supplement = Pe + Pew (7.6 (7.6 Here, we hae utilised the definitins ωd=, ωd= (7.6 Ealuating the sum U + U + and taking the first ariatin f it gies N N N + N + = ( P δu+ eδ + eδw δ( U U Nδu Mδ Mδw d p( δu d and integrating b parts ( ( ( δ( U + U + = ( N + p ( δu+ M δ+ M δw d N N + ( N P δu+ ( M Pe δ ( M δ+ ( M Pe δw ( M δw (7.63 (7.64 results in the equilibrium equatins f the beam at the initial state with crrespnding bundar cnditins M d M dn d + p( =, =, =, d d d N = P r δu = M = Pe r δ = M = Pe r δw = at = and = ( M = Q = r δ= ( M = Q = r δw= (7.65 Taking the ariatin f the term U N gies

17 N ( δ ( U = M φ δ w M φ δ + M w M + ( β M + β M φ δφ d This will be refrmulated after integrating b parts as (7.66 N δ( U = ( M φ δw ( M φ δ+ ( + ( M w ( M + β ( M φ + β ( M φ δφ d + Mφδ w Mφδ + βmφδφ + βmφδφ = = (7.67 Cmbining this finall with (7.5 and (7.53 gies the hmgeneus sstem f rdinar differential equatins with bundar cnditins describing the fleuraltrsinal buckling ( Eu = φ ( EI ( N ( N + ( M φ = φ ( EI w ( N w + ( N ( M φ = ( EIωφ ( GItφ + ( N w ( N r ( N φ ( M w + ( M β ( M φ β ( M φ = (7.68 and bundar cnditins Eu= N = r δu = EI = M = r δ = EI w = M = r δw = EI φ = B= r δφ = ω ( EI + N ( + φ M φ = Q = r δ= ( EI w + N ( w φ + M φ = Q = r δw= t M Mω Mt ( EI φ + GIφ N ( w r φ + ω + β M φ + β φ = + = r δφ = (7.69 When keeping in the mind, that initial bending mment distributins M and M are cnstant with respect t the aial c-rdinate, the sstem (7.68 simplifies t the frm

18 ( Eu = φ ( EI ( N ( N + M φ = φ ( EI w ( N w + ( N M φ = ( EIωφ ( GItφ + ( N w ( N r ( N φ M w + M β M φ β M φ = (7.7 The energ methd frmulatin will be btained again b cmbining the equatins (7.57 and (7.59. This results in the minimising prblem: Find the minimum fr the functinal Π = EI( + EI ( w + EIω ( φ + GIt( φ d + N ( w + ( w φ + φ + r ( φ d ( φ β φ ( φ β φ + M w ( M ( + d (7.7 amng kinematicall admissible functins (, w(, φ (. This frmulatin will ield an apprimate slutin fr the prblem f fleural-trsinal buckling ateral buckling ateral buckling is a phenmenn, in which a beam, laded b an arbitrar bending including transerse distributed r cncentrated lads alng the beam ais, buckles b trsin and transerse deflectin, Figure 4. In a mre general case, als aial lads can be present. The prblem f nn-distrtinal lateral buckling f a straight beam with a thin-walled crss-sectin is inestigated. The same principle as abe is applied in deriing the equilibrium equatins. t the initial state, the eternal lading ields in the beam the nrmal stress distributin σ and shear stress distributinτ s. The cnsideratin deiates frm that f the fleural-trsinal buckling prblem in shear stresses and in crrespnding shear frce resultants Q, Q. The additinal shear stresses bring t the analsis additinal terms int the epressins f frm U N, and als in. The initial shear stress distributin is f the τ Q S ( Q S ( s = It It (7.7

19 ateral buckling τ s φ φ Figure 4. ateral buckling f a beam Here, S ( and S ( are the first mments f the crss-sectin, and are defined b t first, takes the frm S ( = ( sd, S ( = ( sd, (7.73 ( ( ( ( ( d (7.74 = p u+ p + p w P u+ P + P w+ M + M w Here t aid cnfusin, the ntatins mments at the ends f the beam. This as cmbined with takes the frm M and M are used fr eternal bending U N in (7.43 and (7.59 N + = ( + + P δu+ Pδ+ Pδw+ Mδ + Mδw δ( U N δu M δ M δw d p ( δu p ( δ p ( δw d Perfrming the integratin b parts results in (7.75

20 ( ( ( δ( U + = ( N p ( δu ( M p ( δ ( M p ( δw d N + ( N P δu+ ( M M δ (( M P δ + ( M M δw (( M P δw (7.76 Disappearance f this leads t the equilibrium equatins and bundar cnditins f the initial state d M d M dn + p ( =, + p ( =, + p ( =, d d d N = P r δu = M = M r δ = M = M r δw = at = and = ( M = Q = P r δ= ( M = Q = P r δw= (7.77 The stabilit analsis is based again n the term U + UN f which U has n change as cmpared t the crrespnding ne deried in the case f trsinal buckling in (7.5. Instead, the latter term U will take an additinal cmpnent due t the shear stresses at the initial cnfiguratin N {[ ] [ ] } N N s s s U = τ γ d = τ w + ( φ φcs α ( φ φsinα d QS ( QS ( = + { [ w + ( φ ] φcs α [ ( φ ] φsinα} d It It (7.78 N in which γ s is the nn-linear part f strain cmpnent defined in (7.38. Inserting (7.7 int (7.78 requires certain integrals t be calculated er the crss-sectinal area f the beam. These integrals are f the tpe d /ds ts d S ( sin { α d { = S ( t d = I (7.79 and further,

21 S ( csαd= I S ( sinαd = S ( csαd= 3 S ( sinαd= d S ( csαd= d S ( csαd= d S ( sinαd= d 3 (7.8 Thus (7.78 takes the frm ( β φφ φ d ( β φφ φ N U = Q + w + Q d (7.8 with β and β defined in (7.6. Taking the ariatin f (7.8 gies ( N δ ( U = Q β φ δφ + β φδφ + wδφ + φδ w d ( β φ δφ β φδφ δφ φδ + Q + d (7.8 and integratin b parts ields further ( δ ( U = β p φδφ + Q wδφ ( Q φ δ w d N ( β φδφ δφ φ δ + p Q + ( Q d + ( β Q + β Q φδφ + Qφδ w+ Qφδ = = (7.83 Cmbining this nw with (7.5, (7.53 and (7.67 results in a hmgeneus rdinar differential equatin sstem ( Eu= φ ( EI ( N ( N + ( M φ = φ ( EI w ( N w + ( N ( M φ = t ( EI φ ( GIφ + ( N w ( N r ( N φ ω M w + M β ( M φ β ( M φ + p p + β φ+ β φ = (7.84

22 with the bundar cnditins Eu= N = r δu = EI = M = r δ = EI w = M = r δw = EI φ = B = r δφ = ω ( EI + N ( + φ ( M φ = Q = r δ= ( EI w + N ( w φ + ( M φ = Q = r δw= t w βφ M ( EI φ + GIφ N ( w r φ + ω + M ( + ( βφ + ω t + Q β φ + Q β φ = M + M = r δφ = (7.85 the first nes f equatins (7.84 and (7.85 cncern the initial state nl, and are cnsequentl meaningless in the stabilit analsis. When deriing sstem (7.84, the facts ( M = Q ( M = Q = = ( Q p ( Q p hae been utilised. The energ principle frmulatin can be built up b cmbining the integrals (7.57, (7.59 and (7.8 t gie Π = EI( + EI ( w + EIω( GIt( d φ + φ φ φ φ M ( wφ β( φ M ( φ β( φ d Q ( β φφ w φ d Q ( β φφ φ d + N ( w + ( w + + r ( d (7.86 The tw last lines can still be cmbined, when the final epressin fr the ttal ptential energ t be minimised is

23 Π = EI( + EI ( w + EIω( GIt( d φ + φ + N ( w + ( w φ + φ + r ( φ d ( φ β φ φ ( φ β φ φ + ( M w ( M ( M ( M + d (7.87 In pure lateral buckling withut the cmpressie aial lad, these equatins will biusl be simpler. In final equatins (7.84 and (7.87, the psitin f the eternal lad in - and -crdinate directins has n rle. It is hweer bius and eas t understand that this psitin plas an imprtant rle in the lateral buckling phenmenn. Taking this int accunt means actuall a step utside the traditinal ne-dimensinal beam ther, twards tw- r three-dimensinal analsis. It can be dne b impring the kinematics due t the rigid bd rtatin, t include als the secnd rder terms t crrespnd t the secnd rder ther used elsewhere. B cnsidering Figure 5, we can deduce mre generall the c-rdinates f a pint when underging rigid bd rtatin. simple calculatin gies and crrespndingl ( cs( θ φ csθ = r = r(csθ csφ + sinθsinφ cs θ 4 3 = r(cs θ( φ + O( φ + sin θ( φ + O( φ rcs θ( φ + rsin θ( φ = ( φ + ( φ ( sin( θ φ sinθ = r = ( φ ( φ (, θ φ r ' = r cs θ = r sin θ (7.88 (7.89 Figure 5. Rtatin f a fiber

24 dpting these in the definitin f the kinematics applied ields the displacement ectr ( ( w ( φ ( φ u = ( u w ωφ i+ ( φ ( φ j + + k (7.9 The additinal underlined terms in (7.9 will prduce terms f same rder in the shear * strains nl. The are dented b a superscript (, and are γ γ * = e = ( φφ * = e = ( φφ (7.9 These can be cmbined int the shear in the mid plane f each wall f the crss-sectin [( cs ( sin ] * s γ = α + α φφ (7.9 This shear strain cmpnent results in an additinal term in eceptinall terms f secnd rder in φ. U N which includes {[ ] } * * N s s s U = τ γ d = τ ( cs α + ( sinα φφ d QS( QS( = + { [( cs α + ( sinα ] φφ } d I t It (7.93 Cmparing this with the epressin f U N in (7.78 shws that (7.93 is a part f the frmer prided with an ppsite sign and cancels it. This can be erified further b ealuating (7.93 t gie * N ( U = Q β + Q β φφ d (7.94 Taking the first ariatin an integrating b parts ields

25 * N ( δ ( U = Q β + Q β ( φδφ + φ δφd ( ( ( = β ( Qφ Qφ + β ( Qφ Qφ δφd ( β Q + βq φδφ ( = β ( Q + β( Q φδφd ( β Q + βq φδφ ( β p βp φδφd ( β Q β Q φδφ (7.95 = + + Nw, it is eas t see the cancelling terms bth in the integral and bundar terms as cmpared t the underlined terms in (7.83. The ptential f the eternal lads will include als additinal terms due t the impred kinematics taking the epressin ( φ ( φ = p ( p ( cs + p w ( p ( cs d (7.96 = p ( + p w ( d ( φ ( φ p p The part cmplementing this, dented again b star, is * = + φ ( pe pe d (7.97 where e = and e = are the distances f the lading pints frm the p p shear center. This will directl be included in the final differential equatin sstem φ ( EI ( N ( N + ( M φ = φ ( EI w ( N w + ( N ( M φ = t ( EI φ ( GIφ + ( N w ( N r ( N φ ω M w+ M β ( Mφ β ( Mφ + peφ + peφ = (7.98 priding the sstem with the psitin factr f the eternal transerse lad. The bundar cnditins crrespnding t these are

26 Eu= N = r δu = EI = M = r δ = EI w = M = r δw = EI φ = B = r δφ = ω ( EI + N ( + φ ( M φ = Q = r δ= ( EI w + N ( w φ + ( M φ = Q = r δw= t w M Mω Mt ( EI φ + GI φ N ( w r φ + ω + M ( + β φ ( β φ = + = r δφ = (7.99 The crrespnding mdificatins are als present in the epressin f the ttal strain energ Π = EI( + EI ( w + EIω( GIt( d φ + φ + N ( w + ( w φ + φ + r ( φ d ( φ β φ ( φ β φ + ( M w M ( ( M M ( + d ( φ φ + p e + p e d (7. The final epressins deried, bth the differential equatin sstem and the energ principle, frm the basis t ealuate the critical lad intensit fr rather general stabilit prblem f an ne-dimensinal beam. The are cmpatible with the equilibrium equatins deried b using the fairl cmplicated tl f differential gemetr in the general literature f structural stabilit. The prcedure presented here is er sstematic sering as an idelgicall simple wa t handle these prblems Buckling f plates Buckling f plates is as a prblem er similar t the fleural buckling f beams, but etended t tw dimensins nl. The prcedure applied here fllws rather eactl the ne used abe in beam analses. The structure cnsidered is a tw-dimensinal rectangular plate lcated in three-dimensinal Cartesian space s, that - and -crdinate aes cincide with the mid surface f the plate, and is in the directin f the nrmal f plate. inearl elastic material mdel is adpted, with the parameters E, ν, the Yung s mdulus and Pissn s rati, respectiel. The thickness f the plate is h. The kinematics f the plate is gien b appling the e-kirchhff plate mdel, in which each nrmal is assumed t remain nrmal t the defrmed mid surface f the defrmed gemetr. The displacement ectr is thus

27 u w w = ( u + ( + w i j k (7. Here, the displacement cmpnents are u= u (,, = (, and w= w (,, and the unit ectrs in the directin f c-rdinate aes i,, jk. The linear strain and rtatin cmpnents are calculated again directl b appling the definitins (7. and (7.3 with e = e = e = e e e u w = w = u w = + w ω = w ω = u ω = ( resulting in the nner strains ( γ = γ = accrding t (7.7 (7. u w w ε = + ( w w ε = + ( w ( w ( ε = + γ u w w w = + + (7.3 The simplificatin perfrmed is based n the assumptin that the rtatin in the plane f the plate is small as cmpared t the rtatins ut f the plate in buckling. The epressin f the strain energ U = U + U = ( ( ( d (7.4 * * * * * σ σ ε σ σ ε τ τ γ σε can further be split t gie

28 * u w * w * u w U = ( ( ( d σ + σ + τ + u w w u w + σ ( + σ ( ( d + τ + w w w w σ ( + ( + d σ τ * w w + + σ ( +( d This is cmpsed f three parts fllwing the practice abe as in which tw fllwing nes, N N (7.5 U = U + U + U (7.6 U is the traditinal strain energ epressin due t linearised strains and the and, take int accunt bth the nn-linear terms f U N U N strain cmpnents and the initial stresses, i.e. u w w u w U ( ( ( d * * * = σ + σ + τ + u w w u w U ( ( ( d U N = σ + σ + τ + w w w w d N = ( + ( + σ σ τ (7.7 When deriing the epressin fr U, the tw dimensinal linear elastic plane stress state as a material mdel is adpted. Then * E σ = ( ε + νε ν * E σ = ( ε + νε ν E τ γ γ ( + ν * = = G (7.8 Inserting this int the epressin f the linearised strain energ gies U * * * ( σ ε σ ε τ γ = + + d ( ε ε νεε νγ E = ( d ( ν (7.9

29 Incrprating still the linearised strains frm (7.3 r (7., and denting the 3 bending stiffness f the plate b D = Eh /( ν we get h/ E u u u U= d ( ( ( ( dd + + ν + ν + ν h/ 3 = h h/ ν ν h/ E w w w w w + d ( + ( + + ( ( dd ν = D (7. h/ Here, it is assumed that the integral d =. The nn-linear parts f the ptential h/ energ epressin are simplifing t the frm U u u = + + ( + d N σ σ τ u u = N + N + N ( + dd (7. where σ σ τ N = h, N = h and N = h, and cnsequentl U w w w w ( ( d N = σ σ τ + + w w w w = N ( N ( N dd + + (7. The ptential f the eternal lad acting n the plane f the mid surface f plate and including the lume frces and the lads n the edges f plate is = p u p t u t + + ds d S = Pu P Tu T + + ds d d s (7.3 Here, S cers the area f bundar surfaces, and s is the c-rdinate fllwing the bundar mid-line, and the ntatins,,, and P = php = ph T = th T = th fr the lading cmpnents are adpted. When cnsidering the term u u UN + = N ( dd + N + N + Pu + P (7.4 T u+ T ds s

30 taking the first ariatin δu δ δu δ δ( UN + = N ( dd + N + N + Pδu+ Pδ (7.5 T δu+ T δ ds s and integrating b parts gies N N N N δ( UN + = ( + + P ( dd δu+ + + P δ s + [( N T δu+ ( N T δ]d + [( N T δ+ ( N T δu]d s (7.6 In the equilibrium this must anish prducing the equilibrium cnditins f the initial state f the plate as N N N + + P = N + + P = (7.7 with initial bundar cnditins N = T r δu= n the bundaries N r = parallell t -ais = T δ N = T r δu= n the bundaries N r = parallell t -ais = T δ (7.8 The bundar cnditins can be epressed n an bundar generall b + = N n N n T + = N n N n T (7.9 with n and n the directin csines f the nrmal f the bundar surface. We hae still the terms

31 Eh u u u U + UN = ( ( ν ( ν( dd ν D w w w w w + ( ( ( ( + + ν + ν (7. N w N w N w w + ( + ( + dd D D D The first term in this equatin cncerns the initial state f the plate and will be drpped. The rest f (7. takes after ariatin the frm w δw w δw δw w w δw δ( U + UN = D ( + + ν + w δw + ( ν N w δw N w δw N δw w w δw ( + dd D D D Integrating twice b parts simplifies the epressin t ( w w w N w N w N w δ( U + UN = D w dd + + δ 4 4 D D D (7. withut bundar cnditins. The disappearance f the surface integral gies the final hmgeneus partial differential equatin, which is = w w w w w w D( N N N (7.3 The bundar cnditins n all edges f the plate appearing in integratin b parts fllw the plate ther f e-kirchhff and can be gien in the frm w w M δw + ν = =, r = n the D bundaries 3 3 w w M N N w w + ( ν = ( Q, r parallell 3 + = + δw= D D D t -ais (7.4

32 w w M δw + ν = =, r = n the D bundaries 3 3 w w M N w N w + ( ν = ( Q, r parallell 3 + = + δw= D D D t -ais with and the Kirchhff shear frces. These can be cmbined in bundar cnditins in general frm δw δw Mn + Mn = r n+ n= w w n + n = ( N n + Nn + ( Nn + Nn r δw= (7.5 Sling equatin (7.3 with bundar restrictins (7.4 determines finall the critical intensit fr the lading f a plate due t lateral buckling. The ttal ptential energ frmulatin is actuall in equatin (7. D w w w w w Π = ( ( ( ( ν ν N w N w N w w + ( + ( + dd D D D (7.6 which has t be minimised in kinematicall admissible deflectin functins accrding t the energ principle. Eample: The prblem is t find ut the critical cmpressie lad T fr a plate shwn with dimensins ( a b and bending stiffness D. The plate is simpl supprted alng the bundaries. T ~D~ T b a

33 The differential equatin fr the prblem takes the frm and the bundar cnditins T + + = 4 4 w w w w D w(, = wa (, = w (, = wb (, = w w w w (, = ( a, = (, = ( b, = The Naier s slutin fr a plate with all edges simpl supprted fulfils the bundar cnditins under cnsideratin. It is nπ nπ w(, = mn sin( sin( a b n= m= Inserting this int the differential equatin ields a cnditin n π n π m π m π T n π nπ nπ mn + + sin( sin( 4 4 = a a b b D a a b n= m= This cnditin will be fulfilled when the epressin in brackets disappears. Frm this we can sle the alue fr the cmpressie lad, which is a n m n ma = π π + = + T D D n a b a nb We hae t find ut the minimum alue fr the lad with respect t m and n. The minimum alue fr T is btained with respect t m, when m=, but fr n, it must be calculated b differentiating. Thus it is btained π π 3 4 d T = D a n + a = D n a = n= a dn a nb a nb a nb b Because n is an integer ariable, the minimum alue is btained, when it will be taken as clse t the alue a/b as pssible. If a/b is an integer, the minimum alue is 4π D T = b

34 The same prblem can be handled b the ptential energ frmulatin. Thus fr eample, a kinematicall admissible basic set f functins t be used is w(, = w ( a ( b This fulfils the bundar cnditins w(, = w( a, = w(, = w(, b = fr the deflectin f the plate. B incrprating this int the energ integral gies at first, the deriaties w(, = w( a b (, w (, = w ( b, w (, = w ( a ( b, w (, = w ( a, w (, = w ( a ( b, and then D w w w w w N w Π = ( ( ( ( ( dd + + ν + ν + D ab wd = 4 ( b + 4 ( a + 8 ν( a ( b + ( ν( a ( b T a b ( ( dd D Taking the first ariatin and presuming it t disappear, gies 3 5 Dw 4 4 Tab Π = 4 ab(3a + 5a b + 3 b D 3 5 δ Π 4 4 Tab w Dw 4 ab(3a 5a b 3 b w w δ Π = = + + D δ = This results in the critical alue f the lad parameter If a = b, the result is T T 44 / 4 4 4(3a + 5a b + 3 b D = ab 4 = D b, which is.45% higher than abe.