1 Theoy of Elsticity Exm Poblems nd Answes Lectue CT5 (Peviously B6) Delft Univesity of Technology Fculty of Civil Engineeing nd Geosciences Stuctul Mechnics Section D.i. P.C.J. Hoogenboom CT5 August
2 Pefce This lectue book contins the poblems nd nswes of the exms elsticity theoy fom June 997 until Jnuy 00. It hs been ssembled with ce. If nevetheless mistke is found it would be ppecited if this is epoted to the instucto.
3 Contents 9 June Tube, she, complementy enegy. Theoy questions. Hole nd plug, xisymmetic, displcement method o foce method Jnuy 998. Box-gide, tosion. Theoy question, plte flexue. Hole, xisymmetic plte. System, complementy enegy 8 June 998. Bem, tosion, complementy enegy. Theoy question. Axisymmetic plte, flexue, displcement method 5 Octobe Theoy question. Disk nd ing, xisymmetic plte, foce method. Steel bem, potentil enegy 9 Octobe Bem, potentil enegy o complementy enegy. Box-gide, tosion Jnuy High-ise, tube, tosion. Ach system, complementy enegy 7 Octobe Box-gide, tosion. Theoy question, xisymmetic plte, flexue. Bem, complementy enegy 9 Jnuy Axisymmetic plte, flexue, displcement method. Thick wll tube, tosion, complementy enegy. Tuss, complementy enegy 6 Octobe High-ise, sping-connected bems, displcement method. Hollow coe slb, tosion. Sque plte, flexue, potentil enegy 8 Jnuy High-ise, sping-connected bems, foce method. Ach system, complementy enegy. Tunnel, Sping-suppoted ing, potentil enegy Jnuy Cicul hole, foce method. Axisymmetic plte, flexue, tempetue loding, displcement method. Tingul plte, flexue, potentil enegy
5 Technische Univesiteit Delft Fculteit de Civiele Techniek Mechnic & Constucties Vemeld echtsboven op uw wek: Nm en Studienumme Tentmen Elsticiteitstheoie b6 9 juni 997 vn uu Poblem ( points) A hollow tube is loded by she foce. The wll thickness t is smll comped to the dius of the tube. In the coss-section she foce n pe unit of cicumfeence occus due to the esulting she foce Q. We ppoximte the foce n with the following function. n Q z ϕ n n n cosϕ The tube mteil cn be consideed s line elstic with she modulus G. Clculte the esulting she foce Q due to the she foce n. (The positive diection of Q is ϕ π.) b Give the expession of the complementy enegy of slice of the tube. The slice hs length z. c Show tht the expession of the complementy enegy cn be ewoked to the following esult. Ecompl n π. z Gt d Deive the fomul fo the she stiffness GA d. A e Wht is the quntity of the shpe fcto η in Ad η. Fomule π 0 π 0 π 0 cosϕdϕ 0 cos cos ϕdϕ π ϕdϕ 0 5
6 Tentmen Elsticiteitstheoie b6 9 juni 997 Poblem ( points) A stight tck is loded in compession becuse the ils hve expnded on hot dy. The tck might buckle s shown in the figue. To nlyse this sitution n ssumption is mde on the displcement field nd the potentil enegy of the buckled tck is detemined. The esult is N u p N EI l uei un Epot + π π u l p l l N l p EI u noml foce in the tck buckling length fiction foce between the tck nd the bllst bed bending stiffness of the tck lgest deflection of the tck In eltion to wht pmete of pmetes should the potentil enegy of the tck be miniml? Explin you nswe. (You do not need to clculte something.) A stticlly indetemined system cn be nlysed by the foce method o the displcement method. In the foce method we choose numbe of edundnts s fundmentl unknowns. In the displcement method the displcements e the fundmentl unknowns. The foce method is convenient if the poblem is stticlly indetemined to smll degee. Howeve, moden compute pogms use the displcement method without exception. b Why is the displcement method used in compute pogms fo stuctul nlysis? 6
7 Tentmen Elsticiteitstheoie b6 9 juni 997 Poblem (5 points) A sque plte of oom tempetue hs ound hole in it. The hole is filled with plug of the sme mteil nd the sme thickness s the sque plte. The plug exctly fits the hole if it is cooled down T degees fom oom tempetue. Afte while the plug ssumes the oom tempetue gin nd is stuck in the hole. Dt E ν α Rdius of the plug is smll comped to the dimensions of the plte. The thickness of both the plte s the plug is t. The elsticity modulus of the mteil is E. The Poison s coefficient of the mteil is ν. The line expnsion coefficient of the mteil is α. Questions Deive n expession fo the plte which eltes the stess on the edge of the hole to the displcement of this edge. (So, the plug is eplced by stess on to the edge of the hole.) b Deive n expession fo the plug which eltes the stess on the edge to the displcement of the edge. (So, the plte is eplced by stess on to the edge of the plug.) c Fomulte the tnsition conditions between the plug nd the plte. d Clculte the stess between the plte nd the plug. e Clculte the stess distibution in the plug nd the plte. Suggestions The genel solution of the dil displcement in n xil symmeticl thin plte is u () A + B The constitutive eltion of this plte is σ σ θθ E ν ν ν ε ε θθ. 7
8 Tentmen b6, 9 juni 997 Answes to Poblem She Foce The esulting she foce Q is the integl of the veticl component of the she foce nds ove the cicumfeence s of the tube. Q π s 0 n ds cosϕ ϕ ϕ nds Evlution gives π Q nˆ cosϕ dϕcosϕ ϕ 0 π Q n ˆ cos ϕ dϕ Q ϕ 0 n π b Complementy Enegy The complementy enegy is the she foce nd s times the displcement γ z ove, integted ove the cicumfeence s of the tube. E compl π s 0 nds γ z c Evlution of the complementy enegy gives π n z z Ecompl nd z n d Gt Gt Gt n ϕ ϕ cos E compl ϕ 0 z Gt n π π 0 π 0 ϕ d ϕ. d She Stiffness The complementy enegy due to the she foce Q is equl to the complementy enegy due to the she flow n. Fom this we deive the she stiffness GA d. (See lectue book Enegy Pinciples, pge.) E E compl, Q compl, n Q GAd z z Gt n π O, 8
9 GA GA d d Gt Q n π Gt π e Shpe Fcto The section e of thin tube is A πt so tht the shpe fcto η becomes GA Gπ t η GA Gt π d. η Answes to Poblem Potentil enegy should be minimised s to the pmetes tht descibe the displcement field. The buckling shpe of the tck is descibed by u nd l. b In the fist computes little memoy ws vilble, theefoe the system of equtions tht hd to be solved needed to be s smll s possible. The foce method often yields few unknown nd equtions so tht this method ws used in old compute pogms. Howeve, it poved complicted to utomticlly select the edundnts. Mny studies hve been devoted to this subject but soon computes with moe memoy wee developed so tht the displcement method could be used. The displcement method often needs moe memoy but is esie to pogm thn the foce method. Answes to Poblem Becuse the dius is smll comped to the plte dimensions the poblem cn be teted s n xil symmetic plte of which the oute edge is infinitely f fom the hole. Anlysis of the Plte Fo the displcement method holds tht u ( ) A + B so tht ε ε θθ du A + B d u A. + B 9
10 Fo vey lge the stesses nd stins e zeo, theefoe B 0. When the plug is t oom tempetue it is compessed nd exets foce p pe unit of edge length into the diection of the dius. p σ t whee t is the plte thickness. u Becuse σ E ε + νε θθ ν ( ) p we find fo p Et A( ν) Et A ν + ν We define u() u. Theefoe u A. So the eltion between p nd u is. o p u +ν Et u ( + ν) Et p. () b Anlysis of the Plug The plug is compessed in ll diections by distibuted foce p tht is diected inwds. This gives homogeneous stess distibution. So σ p σ θθ. t ν ε ( σ νσθθ E ) E p. t p The dil displcement of edge of the plug is (diected inwds) ε ( ν) Et p The plug becomes T degees wme. If expnding feely this would give n edge displcement of T αt so tht the totl outwd diected displcement becomes 0
11 u αt ( ν) Et p. () c Tnsition Conditions The displcements of the edge of the plug nd the edge of the hole need be equl. u u u () In ddition thee needs to be equilibium p p p. () d Clcultion of the Tempetue Poblem (Foce Method) When we substitute () nd () into () we find. α T ( ν) Et p ( + ν) Et p. Fom this nd () it follows tht the foce pe unit of edge length is p EtαT. Now u cn be clculted with () o (). u ( + ν) αt e Stess Distibution We found ledy fo the stins in the plte ε ε θθ A A. Substitution in the constitutive eltion gives the stesses σ σ θθ E A + ν E A + ν. The constnt A is A u ( + ν) α T
12 so tht σ σ θθ () () EαT EαT. On the edge of the hole the stess is σ EαT so tht the homogeneous nd isotopic stess in the plug is Eα T. Remks The solution is independent of the thickness t nd the Poison s tio ν. The stess in the plug is ppoximtely hlve the vlue tht would occu in completely estined plug. Altentive Answe to Poblem d (displcement method) When we substitute () nd () in () we find u u αt ( + ν) ( ν) EI EI. Fom this nd () it follows tht the displcement of the edge is u ( + ν) αt. Now p cn be clculted with () o (). p EtαT
13 Technische Univesiteit Delft Fculteit de Civiele Techniek Mechnic & Constucties Vemeld echtsboven op uw wek: Nm en Studienumme Tentmen Elsticiteitstheoie b6 jnui 998 vn uu Poblem ( points) A box-gide bem is loded t tosion. The thickness of ll wlls is h s shown in the figue. We clculte the box-gide with the membne nlogy. The weightless pltes in the cones of the box-gide will hve the sme displcement becuse of ottionl symmety You cn ssume tht the wll thickness h is smll comped to the width of the box-gide. Clculte the displcements w nd w of the weightless pltes. b Clculte the tosion stiffness GI w of the coss-section. c Clculte the she stesses in the coss-section nd dw them in the coect diection. d Suppose tht wping (Dutch: welving) of the box-gide is loclly estined by clmped boundy condition. Will this cuse the tosion stiffness to be lge, smlle o will it emin unchnged? h h h h h h h h h
14 Tentmen Elsticiteitstheoie b6 jnui 998 Poblem ( points) A stuctul enginee clcultes the stesses in einfoced concete floo using the finite element method. He uses line elstic elements. Undeneth concentted lod vey lge moments ppe. Wht is you dvise fo this stuctul enginee? Choose fom the following options nd explin you nswe. A Clculte the moments gin with fine element mesh ound the concentted lod. B Replce the concentted lod by distibuted lod ove smll e. C Use the moments t some distnce of the concentted lod to clculte the einfocement. D Use the esultnt of the moment ove some width ound the concentted lod to clculte the einfocement. Poblem ( points) An oil compny dills hole in deep ock lye. Due to the geologicl oigintion pessue p is pesent in ll hoizontl diections of the mteil. The hole chnges this stess distibution. We conside the ock to be of line elstic mteil. The sitution is xil symmeticl with coodinte in the dil diection nd the ngle ϑ in the hoizontl plne. The genel solution of the stess distibution is σ σ σ ϑϑ ϑ C + C + C ( + ln) C C + C + 0 ( ln ) Clculte the tngentil stess σ ϑϑ in the edge of the hole nd dw the stess distibution σ ϑϑ nd σ. ϑ
15 Tentmen Elsticiteitstheoie b6 jnui 998 Poblem ( points) A steel bem is loded by foce F (see figue). The bem hs flnges of thickness t nd width t the uppe nd lowe edge. The stiffenes t the middle nd t the ends of the bem hve dimensions x x t. The width of the bem is, the height is nd the spn is. stiffene stiffene F A flnge stiffene web web stiffene 6 flnge A 6 section A-A We wnt to clculte the deflection of the bem using complementy enegy. The ppoximted stess distibution in the bem is dwn below. The pnels hve homogeneous she stess. The foces in flnges nd stiffenes vy line ove the length F F F F t F t F F F F Show tht the complimenty enegy of flnge is Ec 6 N l Et whee N is the foce in the end of flnge o stiffene nd l is the length. b Clculte the totl complementy enegy of the bem. Choose pescibed displcement u whee the foce F is ttched. Neglect the Poison s effect so tht the she modulus is G E/. c Expess the deflection u in the foce F. 5
16 Tentmen b6, jnui 998 Answes to Poblem The box-gide of this poblem is known in The Nethelnds s nbl bem pplied in the Deltweken in the dm of the Hingvliet estuy. Weightless pltes We choose w of the middle cell lge thn w of the cone cell. Equilibium of the weight less pltes of the cone cell gives p s w s w s w w + h h h Equilibium of the weight less pltes of the middle cell gives p s w w s w w s w w + + h h h This we cn simplify to p s h w w s p h w ( ) ( w ) A fom which w nd w cn be solved. w 6 p s h w p s h A w w w Section A-A b Tosion Stiffness Fom the membne we go to the φ-bubble with the following substitutions. So w φ p ϑ s G φ φ ϑg h ϑg h The tosion moment equls two times the volume of the φ-bubble. 6
17 ( φ φ φ φ ) Mw ( φ φ ) + Substitution of the pevious eltions in the ltte gives ( ϑ ϑ ) M G h + G h w ( + ) ϑg h G 9 h ϑ Fo wie fme model of the bem the tosion moment is Mw GIw ϑ. Theefoe, the tosion stiffness is 9 GIw G h c She stess The she stess is the slope of the φ-bubble. We fist ewite the eltion of the tosion moment ϑg h 9 Mw τ τ nd expess φ nd φ in the tosion moment φ M w 9 τ τ τ τ τ φ 9 Mw In the outside wlls of the box-gide is the she stess τ τ φ h w M τ 7 h In the inteio wlls is the she stess φ φ h 9 M w h 7 M w M w τ 7 h 7
18 d Wping (Dutch: welving) When wping is loclly estined the box-gide will loclly be stiffe thn clculted in this poblem (see lectue book Diect Methods, pge 97). Answe to Poblem Undeneth concentted lod the bending moment goes to infinity (see lectue book Diect Methods, Figue.0). Pobbly the element mesh tht ws selected by the stuctul enginee ws vey fine becuse othewise the locl pek would not hve shown up. It is not useful to select n even fine mesh (nswe A) becuse this will esult in even lge moments. The explntion fo the vey lge moments is tht ound the concentted lod the plte theoy is not ccute ove distnce of ppoximtely the plte thickness. Redistibuting the concentted lod ove n e (nswe B) will indeed educe the moments, howeve this tkes much effot. The esulting moment ove distnce of two times the plte thickness (nswe D) is vey suitble to dimension the einfocement becuse this does not compomise equilibium. This cn be seen s speding the pek moment. Howeve, this lso tkes much effot to clculte. The moment is ccute t distnce of ppoximtely the plte thickness (nswe C). So, only nswe A is elly wong. Answe B nd D e impcticl but possible. Answe C is the best. Answe to Poblem We cn solve the constnts using the boundy conditions. The boundy conditions fo this cse e. - On the edge of the hole e no noml stesses σ. - F fom the hole the stesses σ nd σ ϑϑ equl -p. When becomes vey lge ln( ) does not ppoch specific vlue. Insted it continues to gow. The stess should become equl to -p fo lge. This cn only be if C equls zeo. C 0 When becomes vey lge thn σ C p C becomes vey smll. So p The sme esult would hve been found if we hd consideed σ ϑϑ. If thn So σ C C p C C p 8
19 σ C + C p + p p( ) σ ϑϑ C C p p p + ( ) On the edge of the hole σ ϑϑ p( + ) σ ϑϑ p The stess distibution becomes -p σ ϑϑ σ -p Answes to Poblem Complementy Enegy The foce in b is line fom zeo to some vlue N N x l N The complimenty enegy in b is x l l N E EA dx l N l N N N c dx x dx l x l Et l Et l Et 0 6 Et The complementy enegy in pnel is F E V V 6 t t ) ( τ c τγ G G b The totl complementy enegy is the enegy in the bs plus the enegy in the pnels minus the enegy of position. 8 F Gt 9
20 E E E c c c 8 ( F) 6 ( F) F Et 6 Et 6 Et F F F F Fu Et 6 Et Et Gt 9 F F + Fu Et Gt F Gt Fu Using G this becomes E E c F Fu Et c In ou minds u is pescibed displcement nd F is the esulting suppot ection. The complete stess distibution in the bem is expessed in F. The complementy enegy must be miniml s to the pmete F 0 dec F u df Et Theefoe, the eltion between foce nd displcement is u F Et 0
21 Technische Univesiteit Delft Fculteit de Civiele Techniek Mechnic & Constucties Vemeld echtsboven op uw wek: Nm en Studienumme Tentmen Elsticiteitstheoie b6 8 juni 998 vn uu Poblem ( points) A pismtic bem is loded by tosion. The coss-section of the bem is sque with dimension h (see figue). We wnt to clculte the tosion stiffness of the bem. Theefoe, we conside slice of length x. The tosion moment cuses she stesses in the coss-section, which cn be deived fom function φ. We choose the following function s n ppoximtion. y z φ A ( )( h h ) h whee A is yet unknown constnt. The she stesses in the coss-section e clculted by. σxy σxz φ z y φ Show tht φ fulfils the boundy conditions. h z y b Give fomul with which the esulting tosion moment cn be clculted (You do not need to evlute the fomul). The fomul is evluted fo you with the following esult M w 8 9 Ah c Clculte the lgest stess in the coss-section nd expess this in the tosion moment. d Give the fomul fo clculting the complimenty enegy of the slice (You do not need to evlute the fomul). The fomul hs been evluted fo you with the following esult E c 8 5 A G x whee G is the she modulus of the mteil. e Clculte the tosion stiffness GI w of the bem. Suggestion: Mke the complementy enegy of the slice equl to the complimenty enegy of pt of the wie fme model.
22 Poblem ( points) Elsticity theoy in thee dimensions hs numbe of vibles fo descibing displcements, stins nd so foth. Wite these quntities in the fmewok below nd nd wite the nmes of the eltions. (You do not need to give fomul). Poblem ( points) A stoge tnk fo liquid needs to be jcked up fo mintennce (see the figue t the next pge). The steel bottom plte is elevted fom the foundtion t which it is nomlly esting. We wnt to check the stesses in the bottom plte. The bottom plte is xil symmeticl nd the edges cn be ssumed clmped. The liquid is emoved fom the tnk so tht only the self-weight p of the plte is elevnt. The dius of the tnk is, the plte thickness is h, the elsticity modulus is E nd the Poison s tio is ν. The diffeentil eqution of the deflection w of the plte is d d d d d d dw d p K whee K is the plte stiffness. The genel solution of this diffeentil eqution is w C + C + C ln + C ln + p 6 K Give the boundy conditions of the bottom plte (You do not need to solve the constnts.).
23 The boundy conditions hve been pocessed fo you nd we find the deflection w p K 6 ( ) b Clculte the exteme moment in the plte. jcks tnk bottom plte foundtion c Clculte the steel stess bsed on the exteme moment in question b. Use the following quntities: 5 m p 900 N/m h 0.0 m E N/m ν 0. Clculte lso the lgest deflection of the plte. side elevtion top view Pobbly you will notice tht the clculted stess in question c is much lge thn cn be cied by noml stuctul steel. An obvious conclusion is tht jcking up cnnot tke plce in this wy. Howeve, ecently this poject hs been executed in the descibed wy to the complete stisfction of the owne (Dutch: opdchtgeve). d Explin why the bottom plte does not fil in the jck up pocess.
24 Tentmen b6, 8 juni 998 Answes to Poblem Boundy Conditions The stesses e found by diffeentiting φ. σ xy σ xz y z A ( )( 8 ) h h y z A( 8 )( h h ) The stess σ xy t the edges y h/ nd y -h/ is σ xy h z A ( )( 8 h h )0 The stess σ xz t the edges z h/ nd z -h/ is σ xz h y A( 8 )( h h )0 Theefoe, the she stess pependicul to the edge of the bem is zeo. This is indeed necessy becuse t the sufce of the bem, e no she stesses nd she stesses t pependicul plnes e equl. b Tosion Moment The tosion moment is the esultnt of the she stesses ove the section e h h σ σ M y dy dz z dy dz w xz xy h h M w z y σ xz σ xy y dz As poved in the lectue book this is equl to two times the volume of the φ-bubble (Diect Methods, pge 69). z dy c Lgest Stess
25 The stess σ xy is lgest if y 0 nd z -h/. h 0 A σ xy A ( )( 8 ) h h h The stess σ xz hs the sme exteme vlue. We cn ewite the expession fo the moment s A 9 8 M h w Using this the lgest stess becomes σ M h 9 w This esult is 6% smlle thn n ccute computtion by the finite element method. d Complementy Enegy The complimenty enegy is the integl of the enegy in ll pticles in the slice. The foce t pticle is σdy dz nd the displcement of the foce is γ x. h h c xy xy xz h h σ γ σ γ E dy dz x + dy dz x e Tosion Stiffness The moment in the wie fme model is xz M w GI ϑ w The enegy in length x of this model is 8 ( Ah ) M E M x w A h c w x 9 ϑ x x GI w GI 8 w GIw The enegy in the slice must be equl to the enegy in the wie fme pt O 8 5 A G x A h 8 x GIw GI G h w 5 6 A computtion with the finite element method is just % stiffe (Lectue book, Diect Methods, Fig. 6.8). 5
26 Answe to Poblem u u u x y z ε ε ε γ γ γ xx yy zz xy xz yz σ σ σ σ σ σ xx yy zz xy xz yz p p p x y z kinemtic eltion equilibium eltion constitutive eltion u u u x y z ε ε ε γ γ γ xx yy zz xy xz yz σ σ σ σ σ σ xx yy zz xy xz yz p p p x y z kinemtic eltion equilibium eltion constitutive eltion Answes to Poblem Boundy Conditions On the edge ( ) the plte is clmped, theefoe, the locl displcement nd slope both equl zeo. w0 dw dx 0 In the oigin of the efeence fme ( 0) the slope is zeo becuse of symmety. In the oigin the she foce q is lso zeo becuse smll cylinde in the oigin must be in equilibium. dw dx 0 q0 b Exteme Moment The deflection line is w p K 6 ( ) Diffeentition gives d d ( )(- ) - ( ) w p K p K 6 6 d d - ( ) 6 w p K Substitution in the kinemtic eltions gives 6
27 κ κ ϑϑ d w p K ( ) d 6 dw p d K ( 6 ) Substitution in the constitutive eltions gives m K K p K p ( + ) + ( ( )) ( ( + ) ϑϑ ( κ νκ ν ν ν m K K p p ( + ) ( ( ) + ( )) ( ( + ) ϑϑ ϑϑ ( + 6K 6 The moments in the middle of the plte ( 0) e νκ κ ν ν ν p m ( + 6 ν) m ϑϑ ν) p ( 6 + The moments t the edges of the plte ( ) e p p m + ( ( ν) ( 6 + ν)) ( 6 ) p p m ϑϑ ( ( + ν) ( + ν)) ( ν) 6 6 Theefoe, the exteme moment is )) )) m p ( 6 ) m - p 8 c Steel Stesses Stess is moment ove section modulus σ M W We conside plte pt with width of m. W m 6 bh 6 6 Theefoe m p σ 8 W W 6 0 7
28 6 σ055 0 N / m The lges deflection of the plte is w p 6K whee the plte stiffness is K 9 Eh ( ) ( 0.0) ν 055 Nm The deflection becomes w w m! d Explntion Membne stesses (Dutch: zeilweking) in the bottom plte cuse the deflection nd stesses to be much smlle thn follows fom the plte theoy. Howeve, even if these geometicl nonline effects e tken into ccount the clculted edge stesses will be lge thn the yield stess. Theefoe, t the edges plstic defomtion occus duing the jck up pocess of the tnk. Consequently we need to conclude tht the pplied line elstic model is not suitble to nlyse this poblem due to the lge deflections. 8
29 Technische Univesiteit Delft Fculteit de Civiele Techniek Mechnic & Constucties Vemeld echtsboven op uw wek: Nm en Studienumme Tentmen Elsticiteitstheoie Ctme5 5 oktobe 998 vn uu Poblem ( point) In the foce method one o moe comptibility equtions cn be deived by dding the kinemtic equtions eliminting the stins fom the kinemtic eltions eliminting the displcements fom the kinemtic eltions Poblem (5 points) We conside two xil symmeticl elements, ing nd disk loded in thei plne. These elements will be nlysed septely nd subsequently connected. Ring The dius of the ing is (Figue ). The coss-section e is A. It is loded by pessue p pe unit of cicumfeence. The ing mteil hs n elsticity modulus E. Disk The dius of the disk is nd its thickness is t (Figue ). The disk hs hole in the middle of dius. The disk is loded by pessue p pe unit of cicumfeence. The mteil of the disk hs n elsticity modulus E nd Poison s tio ν 0. p p A A Figue. Coss-section of the ing Kinemtic equtions of the disk ε ε θθ du d u Equilibium eqution of the disk p d ( σ ) σ θθ 0 d Using the foce method the solution is t Figue. Disk 9
30 φ dφ θθ d C + C σ σ φ Deive the kinemtic eqution ofε θθ fo the disk. (Suggestion, conside how the shpe chnges due to displcement u). b Deive the equilibium eqution fo the disk. (Suggestion, dw n elementy pt of the disk). c Clculte the stesses in the disk due to the lod p. d Deive the following eqution fo the disk. u is the displcement of the oute edge. u 0 p Et e Deive the following eqution fo the ing. p u EA The ing is heted T degees. It now fits exctly ound the disk. Subsequently the ing is cooled down to its noml tempetue. The line expnsion coefficient of both mteils is α. E 0E nd A t. 5 f Clculte the stesses in the connected disk nd the ing. Poblem ( points) Conside the stuctul system of Figue. It consists of two bems, which e connected by hinge. The left bem hs n infinite bending stiffness. The ight-hnd bem hs bending stiffness EI. The left suppot is fee hinge nd the ight-hnd suppot is clmp. The left bem is suppoted by sping. The sping stiffness is K. The system is loded by foce F t the middle hinge nd moment T t the left bem. The system will be clculted using the pinciple of minimum potentil enegy. The following displcement is ssumed fo the F T EI K / / Figue. Stuctul system EI x w(x) 0
31 deflection of the ight-hnd bem. (Note tht the x-xis stts t the middle hinge.) w x x + C The following expession cn be deived fo the potentil enegy of the system. + EI T Epot K C F 9 C Wht is the unit of the constnt C? How cn it be intepeted? b Deive the expession of the potentil enegy of the system. c In the following we ssume the vlues. 7 EI K T F Clculte the constnt C. d Mke dwing of the moment line nd give its exteme vlues. e Ae the clculted esults ppoximtions o exct solutions of the system? Explin you nswe.
32 Tentmen b6, 5 oktobe 998 Answe to Poblem The coect nswe is nswe. The comptibility equtions e deived by eliminting the displcements fom the kinemtic equtions. Answes to Poblem Kinemtic Reltion Conside cicle of dius. The cicle inceses due to the loding to dius + u. The cicle length befoe loding is π. The cicle length fte loding is π( + u). Theefoe, the stin is π ( + u) π u θθ π ε b Equilibium Equtions The esulting foce in the left section is σ td θ. The esulting foce in the ight-hnd d section is σ tdθ + ( σ tdθ) d. The esulting foces in the top nd bottom section d e σθθ td. The ltte poduce foce σθθ tdd θ due to the ngled θ. Equilibium in the diection gives d ( σ tdθ) d σθθ tddθ 0. d This cn be simplified by division bytdd θ c Stesses We know tht d ( σ ) σ θθ 0. d dθ σθθ td d φ C + C σ tdθ d σ tdθ + ( σ tdθ) d d Substitution gives σ σ φ C + C dφ θθ C + C d σθθ td Applying the boundy conditions we obtin two equtions with two unknown.
33 σ 0 C + C 0 C + C p p σ + C C C + C t ( ) t The solution is p C t p C t Theefoe, the stesses e σ σ p p t t θθ p p t t d Displcement of the Disk Edge We know thtε θθ u. Theefoe, σ σ p t θθ p + t σ ε θθ θθ p u ( ) + E t E p + p u ( + ) t ( ) E t E 0 p u te Note tht u is diected inwds. e Displcement of the Ring We ssume tht the ing is thin in the dil diection. Theefoe, moments cn be neglected. The noml foce N in coss-section of the ing cn be found fom equilibium of segmentd θ. The esultnt of the lod p is p d θ. The esultnt of the noml foces N t both sections is Nd θ. Equilibium gives N p. The ing hs the sme kinemtic eqution s the disk. u u ε θθ
34 The constitutive eqution is N E A ε θθ. Fom the pevious thee equtions we cn deive N p p u ε θθ. EA EA EA f System of Ring nd Disk The tempetue ise of the ing is included in the constitutive eqution. N EA( εθθ αt ) Fom the equilibium eqution, kinemtic eqution nd constitutive eqution we deive N p u εθθ + αt + T EA EA α Afte the ing is fitted its tempetue dops T degees. Theefoe, p u αt EA The displcements of disk nd ing will be equl. u u 0 p p αt te E A Using, E 0E nd A t we deive 5 0 p p α p p T αt αt te 0E t Et Et 5 p α + 0 p p T Et t E Et α T p Et p α TEt. Subsequently, the displcement nd stesses cn be clculted. u p te T
35 Stesses in the disk σ σ α p TEt t t α θθ p TEt + + t t Stesses in the ing N αtet σ θθ A t 5 N p αtet αtet σθθ 5αTE σ σ αte θθ αte + Remks The intenl dius of the ing hs been used in ll clcultions. We could lso hve used the dius to the centeline of the ing coss-section. Howeve, this hs vey little effect on the esults when the ing is thin in the dil diection. Dmged tin wheels e often tuned off on lthe (Dutch: fdien op een dibnk) nd fitted with steel tyes. The method of this poblem cn be used to model this pocess σ EαT σ θθ EαT Answes to Poblem Constnt C C hs the unit of length. Fo exmple mete [m]. When x 0 then w(0) C. Theefoe, C is the displcement of the middle hinge. b Potentil Enegy Epot Es + E p T ϕ F E s consists of two pts, due to the ighthnd bem nd due to the sping. E p consists of two pts, due to the foce F nd due to the moment T. Togethe this gives E + pot Kv EIκ dx Fw T ϕ x 0 v w v / / 5
36 whee, v C is the shotening of the sping, w C is the displcement of the foce F, ϕ C d w x κ C dx is the ottion of the left bem, is the cuvtue of the ight-hnd bem. Substitution in the eqution of the potentil enegy gives. + 9 C Epot K C EI x C dx FC+ T 9 6 x EIC C Epot K C x dx FC+ T 9 6 x EIC C pot E K C FC T + EIC C Epot K C FC+ T 9 + EI T Epot K C F 9 C c Constnt C Using 7 EI K nd T F the potentil enegy becomes 7EI + EI F E pot C F C 9 EI E pot C FC d EI Epot 6 C F 0 dc F C 8 EI d Deflections nd Moments x Lgest deflection F w(0) C 8 EI F 8 EI Moment in the ight-hnd bem 6
37 x x κ F M EI EI C EI Fx 8 EI 8 The foce in the sping is EI F F Kv K C EI F Moment equilibium of the left bem gives the she foce D left of the middle hinge F F 6 D F D 9 F 5 8 F fom which the moment t the sping cn be clculted. F 9 6 F 5 F 8 F M F F e Appoximtion o Exct The esults e exct if the ssumed displcement function w(x) gives moment line which is in equilibium with the lod nd fulfills the dynmic boundy conditions. d M d Equilibium q Fx 0, which is coect. 8 dx dx 8 Dynmic boundy condition M Fx x 0 M 0, which is coect. Theefoe, the esults e exct. 7
39 Technische Univesiteit Delft Fculteit de Civiele Techniek Mechnic & Constucties Vemeld echtsboven op uw wek: Nm en Studienumme Tentmen Elsticiteitstheoie Ctme5 9 oktobe 999 vn uu Poblem (5 points) A non-pismtic bem hs length L nd is simply suppoted t both ends (Figue ). The bending stiffness t the left suppot is EI o nd t the ight-hnd suppot EI o. Between the suppots the bending stiffness vies line. The bem is loded by moment M t the left suppot EI o M x ϕ L Figue. Non-pismtic bem EI o A stuctul designe clcultes the ottion ϕ of the bem t the left suppot. He o she uses the stndd fomul (Dutch: vegeet-mij-nietje) fo pismtic bems nd the vege bending stiffness of the bem. Wht is the pecentge eo tht the stuctul designe mkes? Apply one of Cstiglino s theoems to clculte the ottion ϕ. In this poblem you cn use eithe the displcement method o the foce method. Explin you selection of the intepoltion function. b If the displcement method wee used: Would the foce method give lge o smlle vlue fo the ottion? If the foce method wee used: Would the displcement method give lge o smlle vlue fo the ottion? Suggestion If you need to clculte n integl fo which you do not know the solution, use Simpson s ule fo numeicl integtion (Figue.). f L f( x) dx L( f+ f + f 6 ) 0 f x L f Figue. Simpson s ule 9
40 Poblem (5 points) Conside the composite bidge shown in Figue. The continuous lines e einfoced concete pltes nd the dotted lines e steel tusses. Figue shows pt of one of the tusses. The thickness of ll concete wlls is t nd the she modulus of the concete is G. 7 Figue. Coss-section of composite bidge We need to clculte the tosion stiffness nd the tosion stesses in the coss-section. Theefoe, we eplce the tusses by homogenous isotopic pltes with the sme she modulus G s the concete pltes nd fictitious thickness t f. The extension stiffness of the tuss bs is EA s. The fictitious thickness of the wll is EA t s f. Use enegy to deive this G fictitious thickness (Figue 5). F F F F F F F F F F F F b Assume tht t t. Clculte the f tosion stiffness of the coss-section of the bidge. c Clculte the she stesses due to tosion moment M. t Figue. Elevtion of one of the tusses n xy F F F n xy γ γ n xy Figue 5. Tuss nd isotopic plte n xy 0
41 Tentmen Ctme5, 9 oktobe 999 Answes to Poblem Method of the Stuctul Designe Fomul of the pismtic bem ϕ ML EI EI EI EI EI Avege bending stiffness ( o o ) o Substitution ϕ ML EI M L * 0.875EI o ϕ 0. 8 Foce Method (Minimum complementy enegy o Cstiglino ) Moment line ( x M M ) L The line moment line is in equilibium with the lod. Theefoe, it cn be used in complementy enegy. The bem is stticlly detemined. Theefoe, we cn esily see tht the line moment line cn descibe the exct solution too. Consequently, ppoximtions e not intoduced hee. x Bending stiffness EI EIo ( ) L Complementy enegy E c L 0 M dx EI Substitution x L M ( ) L M Ec dx x EI 0 EI o o ( ) L x ( ) f ( x) L x ( ) L Appoximtion of the integl using Simpson s ule ( 0) f (0) ( 0) L ) ( ) f ( L) ( ) 8 ( ) f ( L) 0 ( ) E c M EI o L ( + * 6 Complementy enegy E E c M ϕ ) E f ( x o dx M L EI o M L M L 0.79 EI EI Intemezzo The integl cn lso be clculted exctly M M L Ec (7ln 6ln 0) L 0.78 EI EI ϕ o E compl c compl 0 ϕ M M E c ML 0.79 * M ϕ EIo M L EI o o o
42 Eo of the stuctul designe * 00% 6% 0.58 too lge Displcement Method (Minimum potentil enegy o Cstiglino ) Altentive nswe x x Deflection line w ( x ) C( ) L L This is the simplest function tht still fulfils the kinemtic boundy conditions w(0) 0 The ottion of the bem Cuvtue Stin enegy Substitution Potentil enegy Eo of the stuctul designe w( L) 0 dw x x ϕ C + C ( ) dx L L L L ϕ (0) C ϕ L w C ϕ κ C + C dx L L L L L L d L E s EIκ dx 0 L 0 x ϕ E ( )( s EIo ) dx L L ϕ EIo ( ) ( L E pot E s M ϕ L 0 x L E ϕ M s 7 EI ϕ o L ϕ ) dx EIo ( ) L * 00% % L 8 Epot E 0 s M ϕ ϕ ϕ M L EI o too lge 7 EI o ϕ L b Displcement vesus Foce Method The displcement method gives too stiff solution, so too smll ϕ. The foce method gives the exct solution except fo the ppoximtion of the in the integtion ule. Answes to Poblem Fictitious Thickness The enegy in the plte pts should be equl to the enegy in the tuss wlls. Fom this we clculte the fictitious thickness t. f Enegy in plte pt
43 E c nxy nxy tf nxy ( ) t f G Gt F F F Ec Gtf Gtf Enegy in two tuss bs N + N E c EA s EAs N F N F ( F ) ( F ) + F Ec EA s EAs EAs To equte F F Gtf EAs EA t s f G b Tosion Stiffness We use the membne nlogy. w w + w w S S S p t f t t w w + w + w S S S p t t t f Simplified tp w w S tp w+ w S Fom which we cn solve 5 tp w S 9 tp w S w w w Substituting S nd p θwe obtin the G φ -bubble.
44 φ 5 tθg φ 9 tθg The tosion moment Mt is two times the volume of the φ -bubble. Mt ( φ + φ φ + φ 5 tθ G+ 9 ) ( ) ( tθg) Mt Theefoe, 6 tθg M I t t G θ It 6 t c She Stesses The she stess is the slope of the φ -bubble. At the pevious pge we found φ 5 tθg φ 9 tθg. To eliminte θ we substitute Mt 6 t G θ, which gives M φ 5 t 6. M φ 9 t 6 φ M The she stesses in the top nd bottom pltes e τ 9 t t 6 t 9 Mt M 5 t φ φ 6 6 M The she stesses in the veticl pltes e τ t t t 6 t 5 Mt φ 6 M The she stesses in the fictitious tusses e t M τ 0 5 t t t 6 6 t t M x wt 6 t
45 Technische Univesiteit Delft Fculteit de Civiele Techniek en Geowetenschppen Mechnic & Constucties Vemeld echtsboven op uw wek: Nm en Studienumme Tentmen Elsticiteitstheoie Ctme5 Dinsdg jnui 000, 9:00 :00 uu Poblem A high-ise building hs tube fme stuctue. The coss-section is modelled s tube loded in tosion (Figue ). The wlls e pltes of homogenous isotopic mteil of thickness h nd she modulus G. We clculte the tube using the membne nlogy. You cn ssume tht the wll thickness is smll comped to the width of the tube. h h h h h M w h h Figue. Coss-section of the tube Figue. Altentive coss-section Clculte the position of the weightless pltes. b Clculte the tosion stiffness GI t of the coss-section. c Clculte the she stesses in the coss-section due to tosion moment M t nd dw the stesses in the coect diection. As n ltentive it is suggested to leve out the inteio wll. This coss-section hs been dwn in Figue. d Does the inteio wll contibute much to the tosion stiffness? Explin you nswe. 5
46 Poblem An ch nd tension b e idelised ccoding to the figue below. The mteils e line elstic. The bending stiffness of the ch is EI nd the tension stiffness of the ch is infinitely lge. The tension stiffness of the tension b is EA. The stuctue is clculted by complementy enegy. We choose the foce in the tension bs s edundnt φ (Dutch: sttisch onbeplde). F EI w EA φ γ F/ F/ Expess the ch moment s function of φ,, F nd γ. (Due to symmety only hlve the ch needs to be consideed.) b Give the fomul fo the complimenty enegy of the stuctue. c Evlution of the complimenty enegy gives the following esult. Deive this esult. d Clculte φ. Ecompl [ φ π φf + F ( 6 π )] EI e Clculte the deflection w of the ch top. φ + EA f Assume tht the clcultion would hve been mde by the diect method insted of complementy enegy. Would we hve found diffeent nswes to question d nd e? Explin you nswe. π / 0 π / 0 π / 0 π / 0 π / 0 sin γ d γ cos γ d γ sin cos π γ d γ π γ d γ sin γ cos γ d γ 6
47 Tentmen Ctme5, jnui 000 Answe to Poblem Weightless pltes w s s w h p w w h w w Equilibium plte Equilibium plte w w w w w s + s + s + s s h h h h w w w w w s + s + s + s h h h h w h p p This cn be simplified to p h 6w w s p h 6w w s Fom which w nd w cn be solved. p w h 5 s p w h 5 s b Tosion Stiffness Fom the bubble we go the φ-bubble using the following substitutions. w φ p ϑ s G Theefoe φ φ hϑg 5 5 hϑg The tosion moment is two times the volume of the φ-bubble. ( φ + φ ) ( φ + φ ) ( + hϑg M w ) 5 5 Fo wie fme model of the bem we hve 5 hϑg 7
48 M w GIwϑ Theefoe the tosion stiffness is GI w G h 5 c She Stesses The she stess is the slope of the φ-bubble. Fist we ewite the eqution fo the tosion moment hϑ G 5 M w nd expess φ nd φ in the tosion moment. Mw φ φ hϑg 5 8 This gives fo the she stess φ Mw τ. 8 h h d Altentive The inteio wll does not contibute to the tosion stiffness. Afte ll φ nd φ e equl nd the she stess in the wll is zeo. τ τ 0 τ τ τ τ τ Answes to Poblem Moment line M φ sinγ F / ( cos γ) b Complementy enegy 0 < γ < π EI EA F w γ M φ sin γ E compl π / γ 0 M ds + EI x 0 φ dx EA F/ F/ c Evlution - cos γ E compl EI π / γ 0 π / M EI γ 0 π / φ M ds + EA φ x 0 dx d γ + EA F d EI )] φ [ φ sinγ ( cos γ γ + EA γ 0 8
49 π / [ φsinγ EI γ 0 π / [ φ sin EI γ 0 π / [ φ EI γ 0 sin F ( cos γ)] γ φsinγ φ d γ + EA F ( cos γ) + γ φf sin γ + φf sin γ cos γ + π π φ φ +φ + + π φ [ F F F F F ] + EI EA φ φ π φ + π [ F F ( )] + 6 EI EA F φ ( cos γ) ] d γ + EA F F cos γ + F cos φ γ] d γ + EA Theefoe, the nswe povided in question c is wong. We continue with the coect nswe. d Redundnt E 0 φ compl 0 φ [ φπ F] + EI EA F φ EI + π EA e Deflection Fo this we ssume tht the deflection w of the ch top is imposed. The complementy enegy becomes φ E φ π φ F + F π compl [ ( )] + Fw EI 6 EA We cn clculte the suppot ection F by minimising the complimenty enegy. E compl 0 [ φ+ F( π )] w F EI 6 We now know the eltion between w nd F. It does not mtte ny longe which hs been imposed. F EA w π EI EI +π EA f We would hve found the sme nswes becuse the moment line of question is not n ppoximtion. 9
51 Technische Univesiteit Delft Fculteit de Civiele Techniek en Geowetenschppen Mechnic & Constucties Vemeld echtsboven op uw wek: Nm en Studienumme Tentmen Elsticiteitstheoie Ctme5 Vijdg 7 oktobe 000, :00 7:00 uu Poblem A fou-cell box-gide is loded in tosion (Figue ). All wll pts hve the sme thickness t. The cente to cente (Dutch: ht-op-ht) distnces e ll. The thickness t cn be consideed vey smll comped to. Which pts of the fou cell boxgide cn be neglected in clculting the tosion stiffness? b How mny weightless pltes do we need to conside in clculting the tosion stiffness by the membne nlogy? t t t t c Clculte the tosion stiffness. d Detemine the she stesses in ll wll pts due to tosion moment M t. Expess the stesses in M t /( t). t Figue. Coss-section of box-gide Poblem Conside n xil symmeticl plte which is loded pependicul to its plne (Figue ). The d w cuvtue in the dil diection is κ. Wht is the cuvtue κ in the tngentil θθ d diection? A κ θθ 0 dw B κ θθ d w C κ θθ θ Figue. Axil symmetic plte 5
52 Poblem A non-pismtic bem is loded by moment M (Figue ). At the left end the bem is simply suppoted nd t the ight-hnd side it is clmped. The bending stiffness EI vies ccoding to x x EI( x) EI ( ) + EI. L L M ϕ x L Figue. Non-pismtic bem The following function is poposed fo the moment line. x M ( x) M ( ) + L x A L whee A is constnt tht will be detemined lte. Dw the moment line. To do so mke n estimte of constnt A. b Is the poposed moment line suitble o ppliction in the pinciple of minimum complimenty enegy? Is this moment line n ppoximtion? Explin you nswe. c Give the fomul of the complimenty enegy of the bem. d Show tht the complimenty enegy cn be evluted to the following esult. Use Simpson s ule (Figue ). L Ecompl (0.9 M + 0. MA A ) EI e Clculte constnt A. f (x) f Clculte the ottion ϕ of the left end of the bem. f (0) f ( L) f (L) L x 0 f ( x ) dx ( f (0) + f ( L ) + f ( ) ) L 6 L Figue. Simpson s ule fo ppoximtion of the integl of f (x) 5
53 Tentmen CTme5, 7 oktobe 000 Answes to Poblem ( points) The cells t the left top nd ight bottom e not closed. The contibution of these wll pts is clculted s tht of stip ( l t xt ). This is much smlle thn the contibution of the closed cells, theefoe the open cells cn be neglected. b Fo the clcultion is just one weightless plte equied. Afte ll, when we otte the coss-section ove π d we obtin the sme shpe. Would the weightless pltes of the closed cells hve diffeent displcements thn the dwn coss-section nd the otted section would hve diffeent solutions. This is not possible, so the weightless pltes will hve the sme displcements. c We conside the weightless plte s dw below. The totl cicumfeence O of the plte is O () + 6( ) 0 The sufce A of the plte is A ( ) Equilibium of the plte gives w S 0 p t Theefoe pt w 5 S Tnsition fom w to φ. p θ S G φ Gt θ 5 The tosion moment is two times the volume of the volume of the φ-bubble. M t ) 5 5 Aφ ( G t θ G t θ We lso know tht M t Theefoe I t 5 t GIt θ d The she stess in the wll pts is equl to the slope of the φ-bubble. In ll wll pts is the size of the she stess the sme. φ σ t σ 8 M t 5 t Gt θ t 5 G θ 5 M G t G 5 t 5
54 Answe to Poblem ( point) The coect nswe is B. (Lectue book, Diect Methods, pge ) Answes to Poblem (5 points) The constnt A is the moment t the ight-hnd side of the bem. Fom the expected cuvtue of the bem we cn conclude tht A will be negtive. -A M b Yes, the poposed moment line is suitble fo complementy enegy becuse it is in d M equilibium with the loding M M(0) nd q 0. The moment line is not n dx ppoximtion becuse it cn descibe the el moment line. c In genel the complimenty enegy of bem consists of n intenl pt nd n extenl pt. E compl L x o M dx F u EI o F u o M ϕ o whee u o, u o nd θ o e imposed displcements (fee suppot nd clmp) nd F, F nd M e the coesponding suppot ections. The complimenty enegy of the bem in this poblem is E compl L x o M dx EI Afte ll, the imposed displcements u o, u o nd θ o equl zeo. d Evlution of the complimenty enegy E E E compl compl compl L x x M( ) + A L L dx x x EI( ) EI L L x o + EI L x L x x M( ) + A L L x o L f ( x) dx EI x o dx x x M( ) + A L L f ( x) f ( 0) M L ) x L ( M ) f ( + A f ( L) A 5
55 ( (0) + f ( L) f ( L) ) E compl L f + EI 6 L M + ( M + A) A EI 6 L( 7 M MA A ) EI 6 ( )) E compl + E compl + L E compl (0.9 M + 0. MA A EI Intemezzo The integl cn lso be clculted exctly. The esult is L E compl [(M A) A + ( M A) (ln() )] EI L E compl (0.9 M MA A ) EI e Complementy enegy needs to be miniml with espect to the pmetes of the moment line. d E compl 0 d A Fom this A is solved. d E compl L ( 0. M A) 0 d A EI 0. M A 0. 99M f The ottion ϕ is not pt of the expession of the complementy enegy. Theefoe we use tick. We ssume tht ϕ is imposed, so ϕ ϕo. The complementy enegy becomes L Ecompl ( 0.9 M + 0. MA A ) Mϕ o. EI Two pmetes detemine the moment line. These e A nd the suppot moment M. The complimenty enegy gin needs to be miniml with espect to these pmetes. d E compl L ( 0. M A) 0 d A EI d Ecompl L ( 0.88 M + 0. A) ϕo 0 d M EI In pinciple, fom these two equtions we cn solve the unknown A nd M. Howeve, since we know the eltion between ϕ ο nd M it does not mtte which ws imposed nd which ws clculted. Theefoe we cn lso wite 0. M A 0. 99M L M L ϕ (0.88 M + 0. A) EI EI ) 55
57 Technische Univesiteit Delft Fculteit de Civiele Techniek en Geowetenschppen Mechnic & Constucties Vemeld echtsboven op uw wek: Nm en Studienumme Tentmen Elsticiteitstheoie Ctme5 Dinsdg 9 jnui 00, 9:00 :00 uu Poblem ( points) A new softwe tool hs become vilble fo clcultion of cicul pltes *. We wnt to check the pogm with mnul clcultion. Theefoe, the computtion of Figue hs been pefomed. Fomule C L 00 psi (pound / inch ) d d d dw p D ( ( ( ))) d d d d ν m D( w + w ) mϑϑ D( w + νw ) q D( w + w ) w p C w + C + + ( + ln( )) C 6D p C w + C + ( + ln( )) C 6 D p C C w + + 8D Eh D ( ν ) E psi v 0. thickness 0. in in 6 in (inch) Figue. Computtion esults of the softwe tool w m m ϑϑ q Wht e the units of w, m, m ϑϑ nd q in Figue. b Clely n xil symmeticl plte cn be modelled by diffeentil eqution. The totl solution of this diffeentil eqution is p w ( ) + C + C + C ln( ) + C 6D ln( ) Give the boundy conditions tht cn be used to clculte the constnts C, C, C nd C. Ae the gphs of Figue in geement with these boundy conditions? Explin you nswe. c Deive the fou equtions fom which the constnts cn be solved. (You cn leve D, ν nd p in the equtions. You do not need to solve the constnts.) * WinPlte, Achon Engineeing, Columbi, Missoui, U.S.A. Shewe, 57
58 d The constnts hve been solved fo you. The esult is C 0.8 C C C Use this to check the lgest deflection in Figue. Explin possible diffeences. t Poblem ( points) A pismtic tube with thick wlls is loded by tosion. The coss-section of the tube is sque with dimension h nd wll thickness t (Figue ). We wnt to clculte the tosion stiffness nd the lgest she stess. Theefoe, we considee slice of length x. t t t h The tosion moment cuses she stesses in the coss-section, which cn be deived fom function φ. We select the following function s n ppoximtion. h Figue. Coss-section of the tube y A( ) h z φ A( ) h t A( ( ) ) h y > z en z y en y, z < h t h t h t y z h h whee A is jet unknown constnt. Function φ hs been dwn in Figue. The she stesses in the coss-section e clculted by σ σ xy xz φ z φ y z y Figue. The φ bubble Use Figue to show tht φ fulfils the boundy conditions. b Give the fomul fo clculting the esulting tosion moment (You do not need to evlute the fomul). The fomul hs been evluted fo you with the following esult. 58
59 M w t Ah ( h ) c Clculte the lgest stess in the coss-section nd expess this in the tosion moment. d Give the fomul fo clculting the complementy enegy of the slice of length x (You do not need to evlute the fomul). The fomul hs been evluted fo you with the following esult E c A t ( ) x G h whee G is the she modulus of the mteil e Clculte the tosion stiffness GI w of the bem. Suggestion: Mke the complementy enegy of the slice equl to the complementy enegy of pt of wie fme model. Poblem ( points) A stticlly indetemined tuss is loded by concentted lod F (Figue ). All bs hve coss-section e A nd n elsticity modulus E. The bs e connected with hinges to the nodes. The digonl bs e not connected in the middle. The complementy enegy of the tuss is ( + ) F + ( 5 + ) φ ( + φ) l E compl ) F EA Explin the pmete φ in the eqution of the complementy enegy. How cn φ be clculted? b Clculte the deflection of the concentted lod. l l F l Figue. Stticlly indetemined tuss 59
60 Tentmen Ctme5, 9 jnui 00 Answes to Poblem Units w m, m θθ q inch; in inch-pound / inch; pound; lb pound / inch; lb / in b Boundy Conditions.... w(7) 0 dw (7) 0 d m () 0 q () 0 In Figue we see tht the deflection line w t the clmped suppot ( 7) does not hve slope. This gees with boundy conditions nd. We lso see tht m nd q equl zeo t the fee edge ( ). This gees with boundy conditions nd. Theefoe the gphs fulfil the boundy conditions. c System of Equtions ν m D[ w + w ] p ν p ν C ν C ν D[ + + C + C + + C ln + C + C + C ln 6D 6D p D( ν ) ( ν + ) D( ν + ) C C D[ln( ν + ) + ν + ] C 6 q D[ w + w w ] p C C p C D[ C + C ln + C 8D 6D p C C C ln C ] 6D p D C ν + C ] p7 w (7) + C + C7 + C ln7 + C7 ln7 0 6D p7 C w (7) + C C 7ln7 + C7 0 6D 7 p m () ( ν + ) D ( ν + ) C D( ν ) C D[ln( ν + ) + ν + ] C 6 p q () DC
61 d Numbes 6 5 * 0 * 0. D 85 in-lb (Ponounce: inch-pound) ( 0. ) 00 * w() + C+ C + Cln+ C ln 0.8 in 6D The gph shows deflection of 0.9 in. Appently the pogm uses diffeent positive sign convention thn the lectue book. The smll diffeence in the numbe is definitely cused by ound off eos. Encoe (not n exm question) The othe vlues in the figue e checked below. Fom the figue we estimte the mximum moment t. in. 00 *. 85 * 0.7 m (,). * 85 *. C C 85 [ln. *. +.] C + 69 C C 05 Coect C 00* 7 85* 0,7 m (7). * 85 *. C C 85 [ln7 *. +.] C C 50 C 607 Coect C mθθ D[ w + νw ] p C p C D[ ( + C + + ( + ln( )) C) + ν ( + C + (+ ln( )) C )] 6 6 D D p C ν p νc D[ + C + + (+ ln( )) C + + νc + ν(+ ln( )) C ] 6D 6D p ν D[( + ν) + ( + ν) C + C + [ + ln( ) + ν( + ln( ))] C] 6D 00 mθθ () 85 [ * 85 C + 0.7C +.9 C] 65 Coect 00 * mθθ (7) 85 [,9 +.6 C + C + [+ ln(7) + 0. * ( + ln(7))] 6 * 85 7 C ] C 6.6 C 76 C 8 Klopt 00 * 7 * 85 q ( 7) C 7 Coect too 6
62 Answes to Poblem Boundy Conditions At the intenl nd extenl edges the she stesses tht e pependicul to the edge should be zeo. σ xy 0 if y h nd z y, y h t nd z y σ 0 if nd, xz z h y z z h t nd y z This follows fom moment equilibium of infinitesiml cubes in the edges. The fomule of the stess shows tht the she stess is zeo when the φ bubble does not hve slope in the diection pependicul to the stess. The thick lines in the ight-hnd figue show tht the φ bubble coectly does not hve slope t the edges. Theefoe, the boundy conditions e fulfilled. b Tosion Moment The tosion moment is the esultnt of the she stess ove the coss-section e. h h M yσ dydz zσ dydz t xz xy h h As shown in the lectue book, this equls two times the volume of the φ bubble (Diect Methods, pge 69). Equilibium of infinitesiml cubes in the edges of the tube z The φ bubble does indeed not hve slope t the edges of the tube. c Lgest Stess The stess is lgest whee the slope of the φ bubble is lgest. Fo exmple, σ xz is lgest when y h/. y A8 y > z en h t y h φ h σxz 0 z y en h t z h y 0 y, z < h t A σ xz( y h, z z) h M w y y The expession fo the tosion moment cn be σ ewoked s xz z σ xy dz y A h M t t ( ) h z dy 6