Stress Analysis of Ramberg-Osgood and Hollomon 1-D Axial Rods

Transcription

1 Uiversity of New Orleas Uiversity of New Orleas Theses ad Dissertatios Dissertatios ad Theses Sprig Stress Aalysis of Ramberg-Osgood ad Hollomo -D Axial Rods Roald J. Giardia Jr Uiversity of New Orleas, Follow this ad additioal works at: Part of the Mechaics of Materials Commos, Numerical Aalysis ad Computatio Commos, ad the Structural Materials Commos Recommeded Citatio Giardia, Roald J. Jr, "Stress Aalysis of Ramberg-Osgood ad Hollomo -D Axial Rods" 3. Uiversity of New Orleas Theses ad Dissertatios This Thesis is brought to you for free ad ope access by the Dissertatios ad Theses at It has bee accepted for iclusio i Uiversity of New Orleas Theses ad Dissertatios by a authorized admiistrator of ScholarWorks@UNO. The author is solely resposible for esurig compliace with copyright. For more iformatio, please cotact scholarworks@uo.e.

2 Stress Aalysis of Ramberg-Osgood ad Hollomo -D Axial Rods A Thesis Submitted to the Graate Faculty of the Uiversity of New Orleas i partial fulfillmet of the requiremets for the degree of Masters of Sciece i Applied Mathematics by Roald Joseph Giardia, Jr. B.S. Uiversity of New Orleas, May, 3

3 I am grateful for every ecouragig word ad helpful had. ii

4 Cotets ist of Figures Abstract v viii Itroctio. Static Equilibrium Equatio Ramberg-Osgood Hollomo oads Costat oad Poit oad Stepped oad Aalytic Solutios 3. Ramberg-Osgood Costat oad Poit oad Stepped oad Hollomo Costat oad Poit oad Stepped oad Fiite Elemet Solutios 8 3. Ramberg-Osgood Costat oad Poit oad Stepped oad Hollomo Costat oad Poit oad Stepped oad Poit-wise Covergece ad Error Estimatio 3 4. Ramberg-Osgood Poit-wise Covergece Poit-wise Error Hollomo Poit-wise Covergece Poit-wise Error iii

5 5 Figures 8 5. iear Model Compariso Ramberg-Osgood Hollomo Coclusios 9 Bibliography 3 Vita 3 iv

6 ist of Figures 5. EFT: The liear Taget moli model as compared to the Ramberg-Osgood relatioship. RIGHT: The liear secat moli model as compared to the Ramberg-Osgood relatioship EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg-Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod of iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a costat load distributed over the legth of the rod o kip. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg-Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod of iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a costat load distributed over the legth of the rod of 33 kip. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg-Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod of iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a costat load distributed over the legth of the rod of 44 kip. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg-Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a poit load o kip placed at the ceter of the rod. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg-Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a poit load of 38 kip placed at the ceter of the rod. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg-Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a poit load of 4 kip placed at the ceter of the rod. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg-Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a stepped load o kip with the begiig of the step iterval placed at /3 ad the ed of the iterval placed at /3. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models v

7 5.9 EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg-Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a stepped load of 38 kip with the begiig of the step iterval placed at /3 ad the ed of the iterval placed at /3. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg-Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a stepped load of 4 kip with the begiig of the step iterval placed at /3 ad the ed of the iterval placed at /3. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models EFT: The liear Taget moli model as compared to the Hollomo relatioship. RIGHT: The liear secat moli model as compared to the Hollomo relatioship EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a costat load distributed over the legth of the rod of 45 kip. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a costat load distributed over the legth of the rod of 6 kip. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a costat load distributed over the legth of the rod of 8 kip. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a poit load of 3 kip placed at the ceter of the rod. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a poit load of 45 kip placed at the ceter of the rod. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a poit load of 8 kip placed at the ceter of the rod. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a stepped load of 3 kip with the begiig of the step iterval placed at /3 ad the ed of the iterval placed at /3. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models vi

8 5.9 EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a stepped load of 45 kip with the begiig of the step iterval placed at /3 ad the ed of the iterval placed at /3. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a stepped load of 8 kip with the begiig of the step iterval placed at /3 ad the ed of the iterval placed at /3. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models vii

9 Abstract I this paper we preset ovel aalytic ad fiite elemet solutios to -D straight rods made of Ramberg-Osgood ad Hollomo type materials. These material models are studied because they are a more accurate represetatio of the material properties of certai metals used ofte i maufacturig tha the simpler composite liear types of stress/strai models. Here, various types of loads are cosidered ad solutios are compared agaist some liear models. It is show that the oliear models do have maageable solutios, which proce importat differeces i the results - attributes which suggest that these models should take a more promiet place i egieerig aalysis. Ramberg-Osgood, Hollomo, Stress Aalysis, Fiite Elemets, Static Equilibrium, Poitwise Covergece viii

10 Chapter Itroctio It is well kow that may moder high stregth alloys ad steels ca be modeled by simple power-law fuctios betwee stress, σ, ad strai, ε. The two most widely used such power-laws are the Ramberg-Osgood power-law, ε = σ E + k σ E m, where m ad k are material costats ad E is Youg s molus, first oted by W. Ramberg ad W. R. Osgood i 943, ad the Hollomo power-law, σ = Kε, where K ad are material costats with K i Hollomo ot to be cofused with k i Ramberg-Osgood, oted by J. H. Hollomo i 945. Some of the more commo of these Ramberg-Osgood type materials are Titaium ad various Alumium alloys, both frequetly used where stregth coupled with light weight are desig requiremets, whereas a Hollomo power-law ca be used to model may heat treated, or aealed, metals. Curretly, these materials are modeled i may commercial software packages usig a compositio of liear models. This is sufficiet for very small amouts of strai where Hooke s aw is still relevat, but whe it is ecessary to model the behavior of these materials outside the comfort of the elastic rage, these models rely upo a extesio of the liear Hooke s-type relatioship to approximate the plastic behavior of the material. For properly chose material costats the Ramberg-Osgood ad Hollomo relatioships ca provide much more accurate models of the material uder cosideratio. We ote that i the Ramberg-Osgood relatioship we will use A to represet the costat coefficiet of the liear σ term ad B to represet the costat coefficiet of the oliear σ term ad keepig m the same i order to simplify our expressios. I chapter two we provide aalytic solutios to straight rods subject to costat, poit, ad stepped load axial forces. I chapter three we provide fiite elemet solutios for these same loads. I chapter four we discuss poitwise covergece of the fiite elemet models ad some error-estimatio. I chapter five we will compare these results to some liear models ad i chapter six we briefly discuss the implicatio of the results.. Static Equilibrium Equatio At equilibrium the steady state mometum equatio for a straight rod subject to axial force is give by d σx + f x = dx If we presume the rod to be fixed at oe ed ad free at the other the the rod would experiece zero displacemet at the poit where it is fixed ad zero strai at the poit where it is free. These will geerate our boudary coditios. We ote ε = /dx... Ramberg-Osgood We employ the absolute value i the oliear term of the Ramberg-Osgood equatio to the m power i order to preserve the directio of force. We defie φσ = ε = Aσ + B σ m σ ad subsequetly express σ i the static equilibrium equatio with the iverse of φ. d dx φ dx + f x = u =. dx =

11 .. Hollomo Similarly to Ramberg-Osgood, we shall take the absolute value of the ε term to the power times the ε term to preserve the directio of force. d dx K dx dx + f x = u =. dx =. oads There are three types of loads which we will cosider for f x i our steady state equatio to be applied over our rod: a costat load, a poit load, ad the combiatio of these two, a stepped load. We will make use of the step fuctio cetered at some poit, Hx x, ad it will be importat to ote that the itegral of the step fuctio is x x Hx x, which is referred to as the ramp fuctio. We will take this fuctio to be udefied at x which will allow us to make the followig observatios: Hx x a = Hx x ad Hx x a = Hx x, where a is some expoet... Costat oad A costat load is a load which is applied evely at every poit alog a rod. From this we might expect the stress iced upo a rod to be greater closer to the fixed poit of the rod because those poits experiece the load placed directly upo them ad the loads of every poit behid them. We express a costat load as follows... Poit oad f x = A poit load is a load applied at a sigle poit somewhere alog the rod. Every poit of the rod betwee the poit load ad the fixed poit will experiece equal stress, as each poit is beig pulled upo with the same load geerated by the poit at which the load is applied. Every poit behid the poit load will experiece o stress at all. We express a poit load as follows. f x = δx x, x..3 Stepped oad A stepped load is a load which is applied over some iterval alog the rod. We could expect all poits i betwee the iterval ad the fixed poit to behave like a poit load experiecig the sum total of the force applied over the step. Withi the iterval the load would behave like a costat load varyig over the legth of the iterval. Behid the iterval, there will be o stress at all, agai like the poit load. We express the stepped load as follows. f x = Hx a Hx b, b a where a ad b defie the eds of our iterval o the rod. a < b

12 Chapter Aalytic Solutios. Ramberg-Osgood.. Costat oad We begi by itegratig.. φ + dx x + c = With our boudary coditios we ca easily solve for the itegratio costat, where we shall fid that c =. With a bit of rearragig ad subsequetly takig φ of both sides we fid the followig. dx = φ f x To solve for u we itegrate oce more. = A x + B ux = A x m + B x m m x xm+ + c Agai, we ca easily fid our itegratio costat here usig our boudary coditios. c = A + m + B This gives the full solutio for the displacemet of our rod, u... Poit oad ε. ux = A x + m + B m m+ m φ + Hx x + c = dx m+ x m+ We fid the costat of itegratio to be the same as with the costat load, ad rewrite our expressio to solve for which ca be represeted as follows. dx = φ Hx x = A Hx x + B Hx x m Hx x 3

13 We itegrate oce more. { dx = A + B m, x < x, x > x ux = A x x x Hx x + B m x x x Hx x + c Whe we solve for the itegratio costat here we fid that it is zero. So, we have for our full solutio for u the followig. which gives the followig relatioship. ux = A x x x Hx x + B m x x x Hx x ux = { A x + B m x, x < x A x + B m x, x > x..3 Stepped oad φ + dx b a x ahx a x bhx b + c = The costat of itegratio is oce agai ad we set our expressio to solve for /dx. dx = φ x ahx a x bhx b b a = A x ahx a x bhx b b a +B m x ahx a x bhx b b a x ahx a x bhx b b a which ca be represeted as follows. A + B m, x < a dx = A f b x b a + B f m f b x m b a, a < x < b, x > b To fid u we will itegrate each expressio for /dx over their respective itervals rather tha attempt to itegrate the geeral expressio for /dx. Takig the itegral over the first iterval,,a, we fid the followig. ux = A x + B m x + c, x < a where, usig our boudary coditios, we fid that our itegratio costat here is zero. Whe solvig for u over the ext iterval we ca take ua from the expressio over the first iterval to be a boudary coditio for the secod iterval, a, b, to solve for the subsequet itegratio costat. We ca employ the same for the subsequet iterval, b,. We fid, the, the followig for u. ux = A b a bx x B m b a m b xm+ + c 3, a < x < b m + Solvig for our costat of itegratio usig ua as the boudary coditio o this iterval we fid the followig. a c 3 = A + B m b + am b a m + 4

14 This gives the solutio for u over the iterval a,b as follows. ux = A bx a x b a + B m + m b + am b x b x m, a < x < b b a Itegratig over our fial iterval we get a costat, which will be the value of ub from the previous iterval. We have the followig relatioship for the full expressio for u.. Hollomo.. Costat oad ux = A f x + B m x, A b a bx a x + m+ B m b + am b x b x b a A f b a b a We begi by itegratig.. x < a m, a < x < b + B m+ m b + am, b < x K dx dx + x + c = From our boudary coditio we fid the costat of itegratio is which gives the solutio for /dx as follows. We itegrate oce more to fid u. dx = K K x ux = + + K x + c K We agai solve for our itegratio costat, which gives the followig. which gives the full expressio for u as follows... Poit oad c = + K K + ux = f + K + x + K K dx dx + Hx x + c = The costat of itegratio is oce agai ad we rewrite to solve for /dx. which ca be represeted as follows. dx = K K Hx x 5

15 We itegrate oce more. dx = f K K, x < x, x > x ux = K K x x x Hx x + c We will fid that our costat of itegratio here is zero, so we have the full expressio for u as follows. which gives the followig relatioship...3 Stepped oad K dx ux = K K x x x Hx x f K K x, x < x ux = f K K x, x > x dx + b a x ahx a x bhx b + c = With the costat of itegratio agai, we solve for /dx. which ca be represeted as follows. dx = K x ahx a x bhx b K b a f K K, x < a dx = f b x K K, a < x < b b a, x > b Just as with Ramberg-Osgood before, we will itegrate over each iterval usig for boudary coditios, u, ua, ad ub where appropriate. Over the first iterval,,a, we fid the followig. ux = K K x + c, < x < a where we will fid the costat of itegratio to be zero. For the subsequet iterval, a,b, we fid the followig. ux = + K where we use ua to solve for the costat of itegratio. + b a + c 3, a < x < b K c 3 = f + K b + a K which gives the solutio for u over the iterval a.b as follows. 6

16 ux = + K b + a b x K b x, a < x < b b a Our fial iterval yields a costat after itegratio which will be the value of ub from the previous iterval. We have the the followig relatioship for the full expressio for u. f K K x, x < a ux = + f K K b + a b x b x b a, a < x < b + f K K b + a, b < x 7

17 Chapter 3 Fiite Elemet Solutios We will employ the well kow Galerki method to. ad. to costruct a fiite elemet model of these two systems. We ote that ψ e ad ψ e are the shape fuctios over some give iterval e. We will make use of the followig., x x i,x i χ i x =, x x i,xi, otherwise ad 3. Ramberg-Osgood Γ a,b x = We apply Galerki first to. for our three loads. 3.. Costat oad e x = x e x e x e d dx φ d dx φ dx {, x a,b, x a,b + dx ψ e ψ e dx + ψ e ψ e dx x e x e ψ e ψ e dx = We ext itegrate the left-most term by parts ad the take the remaiig itegrals ad simplify our expressio. e x x e d φ dx dx ψ e ψ e = φ u e ue x e xe dx + φ dx ψ e ψ e x e = xe From this result, we costruct the global fiite elemet system as follows. x e e x + x e + Q e Q e ψ e ψ e dx = 8

18 φ φ u φ u u x u x φ u u x. φ u N u N u N x N u N x N φ u N u N x N = x x + x x N. + x N x N + Q. Q N 3. where give the boudary coditios i., u ad Q N would both be zero makig the system easily solvable. 3.. Poit oad e x = x e x e x e d dx φ d dx φ dx We ow itegrate ad simplify our expressio. e x x e d φ dx dx φ ψ e ψ e + δx x dx ψ e ψ e dx + dx + φ dx = φ u e ue x e xe x e x e ψ e ψ e ψ e dx ψ e δx x x e x e = χ ix + ψ e ψ e dx = e x + δx x x e Q e Q e ψ e ψ e dx = where i is the idex associated with the poit load. We ote that the poit load lies upo a existig ode, so the force is couted twice betwee the two elemets as defied by χ i x. We split the force, half over each elemet, to have the sum total of the force we are applyig be oe o the ode rather tha coutig it twice. From this we costruct the global fiite elemet system. φ u u x φ u u φ u x u x. Q where agai u ad Q N are both zero Stepped oad e x = x e x e x e d dx φ. φ u N u N u N x N u N x N φ u N u N x N d dx φ dx dx ψ e ψ e + b a e x dx + x e = +.. Q N ψ e Hx a Hx b dx ψ e Hx a Hx b b a 9 ψ e ψ e dx = 3.

19 We oce more employ itegratio by parts ad simplify the expressio. φ u e ue x e = xe b a xe Γ a,b x + From which we oce agai costruct the global fiite elemet system. We defie the iterval immediately followig the poit a, which defies the begiig of our iterval ad sits upo a ode, as the iterval α, ad the iterval immediately precedig the poit b, which defies the ed of our iterval ad is also o a ode, as the iterval β. φ φ u φ u u x with u ad Q N both zero. 3. Hollomo u x φ u u x. φ u N u N u N x N u N x N φ u N u N x N We ow apply Galerki to. uder our three loads. 3.. Costat oad e x = x e d K dx dx K dx dx x e d dx x e Q e Q e. xα = f xα + xα+ b a. + x β + xβ xβ. dx ψ e ψ e + e x dx + x e ψ e ψ e dx ψ e ψ e dx = Q. Q N We itegrate the left-most term by parts ad take the remaiig itegrals ad simplify our expressio. x e x e = K K dx dx d dx e x + x e u e ue u e x e xe ψ e ψ e ue x e xe From this we costruct the global fiite elemet system. dx + K dx ψ e ψ e dx = = xe ψ e dx ψ e + Q e Q e x e x e 3.3

20 K x u u un u N x N = u u x u u x. x N x N u N u N un u N x x + x x N. + x N x N u u x u u Our boudary coditios imply u ad Q N are both zero. x un u N x N u N u N x N + Q. Q N u u x u N u N x N Poit oad e x = x e d K dx dx K dx dx x e d dx x e We itegrate by parts ad simplify our expressio. x e x e K K dx dx d dx ψ e + δx x dx ψ e dx ψ e e x ψ e ψ e dx + δx x x e ψ e ψ e ψ e e x + δx x x e = u e ue u e x e xe ue x e xe dx + K dx ψ e ψ e This allows us to costruct the global fiite elemet system. K x u u un u N x N u u x u u x. x N x N u N u N un u N dx = = χ ix + u u x u u x un u N x N u N u N x N ψ e dx ψ e Q e Q e u u x dx = x e x e u N u N x N

21 . = +. Q. Q N where agai u ad Q N are both zero ad x lies upo a ode Stepped oad e x d K x e dx dx dx e x d = K ψ e x e dx dx dx ψ e + Hx a Hx b b a e x dx + x e ψ e ψ e dx Hx a Hx b b a ψ e ψ e dx = Agai, we itegrate by parts ad will the take the remaiig itegrals ad simplify the expressio. K u e ue u e x e xe ue x e xe = b a xe Γ a,b x We use this result to costruct the global fiite elemet system as follows. K x u u un u N x N u u x u u x. x N x N u N u N un u N u u x u u x un u N x N u N u N x N. x α = f x α + xα+ b a. + xβ + xβ xβ. Q. Q N u u x + Q e Q e u N u N x N Both u ad Q N are zero ad, as i the Ramberg-Osgood case, the iterval immediately followig the poit a is the iterval α ad the iterval immediately precedig the poit b is the iterval β. 3.6

22 Chapter 4 Poit-wise Covergece ad Error Estimatio For the fiite elemet system of a costat load above i both the Ramberg-Osgood ad Hollomo models with a very small umber of odes, aroud four, the values calculated for each ode poit are grossly iacurate, but oce the umber of odes is icreased to aroud te we see the solutios proced for each ode are very close to the aalytic solutio for those same poits withi several sigificat digits. So, do our fiite elemet model solutios actually approach the aalytic solutio if we were to icrease the umber of odes to ifiity? The average legth of elemets i our fiite elemet system is the legth of the rod divided by the total umber of elemets. We defie a fuctio, ci, to be the percetage of the average value of elemets which defies a give elemet, i. We ote that this would satisfy the property x i = ci N N ci = N i= We ca choose ay value iitially for each ci so log as it meets the above coditios. So, we ca defie a system with elemets of ay size ad distributio i this maer. We wat to esure that as the umber of odes i our system icreases to ifiity that each elemet goes to zero at the same rate. Each time we add a ew ode we create a ew elemet, ad this ew elemet must defie some iterval. This iterval which the ew elemet defies must be proced from the existig domai, so it must come from poits withi other curretly existig elemets. That is, the elemets ad odes must be shifted aroud i some maer so as to accomadate the additio of a ew elemet. So, we further defie ci such that as a ode is added to our system the size of the elemets shifts such that bigger elemets would shrik ad smaller elemets would grow distributig the weight more equitably across the whole fiite elmet system. To that ed, we defie ci such that it satisfies the followig limit. lim ci =, i N We defie qn N such that for some poit x where there exists a ode we may defie x as follows. qn x = i= ci N If we take the limit of the right had side as the umber of odes approaches ifiity so as to ecompass all possible poits x withi the domai, the we may defie q as follows. q := x 3

23 We further defie the supremum ad ifimum of ci. 4. Ramberg-Osgood 4.. Poit-wise Covergece sup i ci = d ad ifci = p i We see that the fiite elemet represeatio of the Ramberg-Osgood equatio 3. ca be solved for each ode recursively. We will employ the covetio u = u, u = u = u,..., u N = u N = u N, u N = u N. We ca the express u i as follows. N u i = ci N φ ci N + c j + u i N This expressio has a base term added to which is the value of the previous ode. We ca see that the value of ay particular ode is just the sum of all of these base terms up to ad icludig said ode. Whereas qn is the elemet idex to which we are summig we defie τ = qn + to be the idex of the ode immediately followig this elemet. qn N u τ = cτ N φ ci i= N + c j j=i+ N We ca see the that u τ is bouded above ad below by replacig each ci with, respectively, the supremum, d, ad the ifimum, p. p qn N φ p N i= N + p j=i+ N cτ N d N qn φ i= qn φ i= j=i+ N ci N + c j j=i+ N N d N + d N j=i+ We begi by expadig the latter expressio with the supremum, d. qn N d N A d i= N + d j=i+ N +d B m N N d N + d N qn i= = d A q + d m+ N Bm+ j=i+ m m qn i m i= N We will focus our attetio ow solely o the right-most portio of the previous expressio ad the come back to the full expressio i a momet. First, we otice that we may expad the expoetiated term withi the sum ito a sum itself usig the biomial theorem. d m+ N Bm+ m qn i= j= m i j j j N 4

24 We ote that the absolute value of the fractioal term withi the sum is less tha oe so the ifite sum is absolutely coverget ad the precedig sum is fiite so we may swap their order. d m+ N Bm+ m j= m j qn j N We apply the biomial theorem oce more to the right-most term i the expressio. d m+ N Bm+ m j= m j qn j N i= i= j k= This ew sum is fiite, so we may swap its order with the sum that precedes it. d m+ N Bm+ m j= m j j k= i j j j+k qn k k N j We ow use Beroulli s formula to express the right-most sum o the ed. d m+ N Bm+ j k + j k h= m j k + h j= m j j k= j k i j k k i= j j+k k k N j B h qn j k h+ i j k We may ow factor out from this expressio qn j+ which results i the followig. d m+ B m+ m j= m j j k= j j+k q j+ k k j k + j k h= j k + h B h qn k h Fially, we are ready to take the limit of our expressio. We ote that the expoet i the right most term is k h where k ad h are both idices always greater tha or equal to zero. Whe both k ad h are zero this term is oe. Whe either of k or h are ot zero whe the limit is take this term will go to zero. So, after takig the limit our etire expressio is zero except for whe the idices k ad h are both zero. We also ote that whe the umber of odes icreases to ifiity the supremum will go towards oe ad we will also have q = x /. We have, the, the followig. B m+ m = m + Bm+ j= m m j j + j x j+ j= We shift the idex j up oe ad adjust our sum accordigly. m + Bm+ m j= m + j + m + j j x j x If the idex were zero i this ew expressio we would have a value of mius oe. We add ad subtract oe icorporatig the mius oe ito the sum ad factor out a mius oe. m + Bm+ m m + j x j j The sum is ow the biomial theorem expressio of x / m+. We rewrite it as such ad distribute the m+ term. j= j+ j 5

25 m + B m m+ x m+ We add to this the term which we had left behid previously otig that we have take the limit, so d will have goe to oe ad q will have goe to x /. A x + m + B m m+ x m+ We have the the full expressio for the upper boud is exactly the previously foud aalytic solutio for the Ramberg-Osgood relatioship. We ote that the lower boud differed from the upper boud oly i the usage of the ifimum, p. Whe the limit is take here p will go to oe as did the supremum, d. So, the lower boud coverges to the same thig as the upper boud. I the poit load case the fiite elemet system 3. provides the exact aalytic solutio to the problem for as may odes as ecessary for there to be a ode at the poit where the load is applied. So, for as few as two elemets the exact aalytic solutio is provided ad the system coverges poitwise to the aalytic solutio. The proof of this is obvious. The stepped load fiite elemet system 3.3 ca be proved by hadlig all elemets to the left of the poit a, the left most boud of the stepped load, as a poit load ad all elemets betwe a ad b o the step as a costat load. This provides covergece for the exact aalytic solutio as well. 4.. Poit-wise Error To fid the amout of error i 3. at some ode for a system of a particular size we ca take the differece betwee the kow aalytic solutio ad the recursively defied fiite elemet solutio at some ode. We will substitute x i the aalytic solutio for its comparable represetatio q to correspod with the poit of the ode i the fiite elemet solutio. The we will take advatage of the relatioship, τ = qn + to express the error as depedet upo as few terms as possible. 4. Hollomo E N τ = A m Poit-wise Covergece N τ cτ N N τ i= N τ m+ cτ N N N ci N + τ i= j=i+ N ci N + c j + B N m c j N We will approach the costat load case of the Hollomo fiite elemet system 3.4 just as we did above with Ramberg-Osgood. We ca solve this sytem recursively reultig i the followig expressio. u i = ci N K K j=i+ m N ci N + c j + u i N j=i+ We ca the express the value at ay particular ode as the sum of the base terms as follows. u τ = cτ N K K qn i= N ci N + c j N As with Ramberg-Osgood, we boud u τ above ad below replacig each ci with the supremum, d, ad the ifimum, p, respectively. j=i+ m 6

26 p N K cτ N K d N K K K K qn i= qn i= qn i= p N N + p N j=i+ N ci N + c j N j=i+ N d N + d N j=i+ Focusig o the upper boud, we may rewrite this expressio as follows. d + + N K qn K i= j= / j j i N We ca see that the sum expressios are the same as we had whe we bega with Ramberg-Osgood above, the two differig oly i the expoet, here istead of m previously. So, we will arrive at the same aswer as we did before but exchagig m + with + +. After distributig the term we will the fid the exact aalytic solutio. f + K + x + K The poit load 3.5 ad stepped load 3.6 will be hadled just as it was previously. 4.. Poit-wise Error To fid the amout of error i 3.4 at some ode for a system of a particular size we take the differece betwee the kow aalytic solutio ad the fiite elemet solutio at some ode i the same maer as which we hadled Ramberg-Osgood above. + N τ N E N τ = + + cτ N K τ i= K N ci N + j=i+ j c j N 7

27 Chapter 5 Figures 5. iear Model Compariso We will perform some examples below to demostrate visually where ad why error would appear whe usig a liear model as opposed to oe of the more precise power-law models. Geerally, i moder fiite elemet software it would be stadard to iput betwee five ad te liear segmets. This of course ca provide some fairly accurate approximatios. The liear models ca eve be cocetrated i a more complex area such as the kee bed i the Ramberg-Osgood relatioship to provide better results. Depedig o the specific eeds of a project this approach ca be sufficiet. However, usig the power laws which model much more precisely the behavior of the material leads to almost o error, at least i the case of the model itself. This ca provide much more exactig results where such results are ecessary as well as a higher degree of cofidece i the project desig. There are may ways i which liear models are used to costruct results i fiite elemet software. Below we will focus o two simple ad commo methods. We will use a series of three taget moli, these are three lies taget to poits alog the curve, ad we will use a series of three secat moli, which are three lies betwee a set of poits o the curve. These models, usig so few liear segmets will ted to exagerate the appearace of error especially i the Hollomo case because of the ature of the relatioship. It is almost assured that i moder egieerig o egieer would use such a model to model material behavior. However, these simple models besides beig easy to uderstad do serve to highlight the ature of the error betwee the differet approaches as well as the possibility for mitigatig said error. 5.. Ramberg-Osgood For the triliear taget molus model we take lie segmets taget to the poits alog the curve at the origi, where we assume Youg s molus as the slope, at the poit.475,36, ad at ultimate. The poits of trasitio where the liear taget moli itersect are first at 3.57 kip ad at kip. I the secat molus model we take four poits betwee which we put secat liear segmets. We start at the origi the.353,3,.534,38, ad to ultimate. The poits of trasitio i the secat molus model are at their respective stress values. We use material costats derived from material tests of a Titaium alloy doe i the Naval Architecture ad Marie Egieerig Departmet at the Uiversity of New Orleas with A =. 3, B = , ad m = Uits are i kip. 8

28 Figure 5.: EFT: The liear Taget moli model as compared to the Ramberg-Osgood relatioship. RIGHT: The liear secat moli model as compared to the Ramberg-Osgood relatioship. Costat oad Figure 5.: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg- Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a costat load distributed over the legth of the rod o kip. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models. 9

29 Figure 5.3: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg- Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a costat load distributed over the legth of the rod of 33 kip. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models. Figure 5.4: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg- Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a costat load distributed over the legth of the rod of 44 kip. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models.

30 Poit oad Figure 5.5: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg- Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a poit load o kip placed at the ceter of the rod. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models. Figure 5.6: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg- Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a poit load of 38 kip placed at the ceter of the rod. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models.

31 Figure 5.7: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg- Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a poit load of 4 kip placed at the ceter of the rod. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models. Stepped oad Figure 5.8: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg- Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a stepped load o kip with the begiig of the step iterval placed at /3 ad the ed of the iterval placed at /3. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models.

32 Figure 5.9: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg- Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a stepped load of 38 kip with the begiig of the step iterval placed at /3 ad the ed of the iterval placed at /3. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models. Figure 5.: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Ramberg- Osgood relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a stepped load of 4 kip with the begiig of the step iterval placed at /3 ad the ed of the iterval placed at /3. RIGHT: The error from the Ramberg-Osgood model as the baselie of each of the secat ad taget molus triliear models. 5.. Hollomo For the triliear taget molus model we take lie segmets taget to the poits alog the curve at the origi, where we assume Youg s molus as the slope, at the poit.7,4, ad at ultimate. The poits of trasitio where the liear taget moli itersect are first at 4.4 kip ad at 66.8 kip. I the secat molus model we take four poits betwee which we put secat liear segmets. We start at the origi the.99,7,.567,58, ad to ultimate. The poits of trasitio i the secat molus model are at their respective stress values. We use material costats for 34 Stailess Steel Aealed with K = 5 ad = Uits are i kip. 3

33 Figure 5.: EFT: The liear Taget moli model as compared to the Hollomo relatioship. RIGHT: The liear secat moli model as compared to the Hollomo relatioship. Costat oad Figure 5.: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a costat load distributed over the legth of the rod of 45 kip. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models. 4

34 Figure 5.3: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a costat load distributed over the legth of the rod of 6 kip. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models. Figure 5.4: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a costat load distributed over the legth of the rod of 8 kip. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models. 5

35 Poit oad Figure 5.5: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a poit load of 3 kip placed at the ceter of the rod. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models. Figure 5.6: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a poit load of 45 kip placed at the ceter of the rod. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models. 6

36 Figure 5.7: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a poit load of 8 kip placed at the ceter of the rod. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models. Stepped oad Figure 5.8: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a stepped load of 3 kip with the begiig of the step iterval placed at /3 ad the ed of the iterval placed at /3. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models. 7

37 Figure 5.9: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a stepped load of 45 kip with the begiig of the step iterval placed at /3 ad the ed of the iterval placed at /3. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models. Figure 5.: EFT: A compariso betwee the triliear secat molus ad taget molus models ad the Hollomo relatioship used i a Fiite Elemet model to fid displacemet with a rod o iches i legth with oe ed fixed ad the other free usig 5 odes ad elemets of equal size with a stepped load of 8 kip with the begiig of the step iterval placed at /3 ad the ed of the iterval placed at /3. RIGHT: The error from the Hollomo model as the baselie of each of the secat ad taget molus triliear models. 8

38 Chapter 6 Coclusios I this paper we have preseted what we believe to be ew results for the oe dimesioal Ramberg-Osgood axial rod uder various loads as well as show the viability of fiite elemet models for both Ramberg-Osgood ad Hollomo rods. We have touched upo some shortcomigs as to the depedece o the umber of odes i the system to the accuracy of the solutios as well as to the limitatios of some liear based models compared to this approach. It is the case that egieerig, by it s ature, is geerally a physical results drive disciplie. For liear based models that approximate oliear material behavior, such as those used i most moder fiite elemet software packages45, the output calculatios are good eough for may applicatios; especially i disciplies ad o projects that ever exit the very low elastic rage, such as most civil projects. Oce we leave the comfort of the Hooke s relatioship, however, ad move ito the higher regios of elasticity ad the plastic rage of these oliear materials errors begi to appear depedet upo the choice of segmetatio. Eve for a large umber of liear segmets to rece the error, some small amout of model depedet error is still preset. As we begi to veture ito the desig of icreasigly smaller ad more precise machiatios ad use more exotic specifically crafted alloys ad materials a more accurate modellig approach will be ecessary to perform a proper desig aalysis of the limitatios ad abilities of the give project. Modelig of physical materials is imprecise by ature. It ca be depedet upo the purity of the raw material or the eviromet it operates withi. It should ot be the case that whatever error exists i our modellig of the behavior of our desigs be the proct of iexact models of already well uderstood material behaviors. 9

39 Bibliography Ramberg W ad Osgood W R 943, Descriptio of stress-strai curves by three parameters. Techical Note No. 9, Natioal Advisory Committee For Aeroautics, Washigto DC Hollomo J H 945, Tras. AIME, vol. 6 pp Callister W ad Rethwisch D, Fudametals of Materials Sciece ad Egieerig: A Itegrated Approach. Wiley 4th Editio, pp 4 ANSYS, ANSYS Mechaical ADP Material Referece. 3.4., Release 4. 5 ANSYS, ANSYS Mechaical ADP Material Referece. 3.4., Release 4. 3

40 Vita The author was bor i New Orleas, ouisiaa. He obtaied his Bachelors of Sciece degree i Mathematics from the Uiversity of New Orleas i. He joied the Uiversity of New Orleas graate program to pursue a PhD i Egieerig ad Applied Scieces with a cocetratio i Mathematics, ad became ivolved i research with Dr. Dogmig Wei i. 3