At the end of this lesson, the students should have the knowledge of:

Transcription

1 Instuctional Objectives: At the end of this lesson, the students should have the knowledge of: Stesses in thick cylindes. Lame s equation fo adial and cicumfeential stesses. Distibution of adial and cicumfeential stesses fo diffeent bounday conditions. Methods of inceasing elastic stength of thick cylindes by pestessing Stesses in thick cylindes Fo thick cylindes such as guns, pipes to hydaulic pesses, high pessue hydaulic pipes the wall thickness is elatively lage and the stess vaiation acoss the thickness is also significant. In this situation the appoach made in the pevious section is not suitable. The poblem may be solved by consideing an axisymmety about z-axis and solving the diffeential equations of stess equilibium in pola co-odinates. In geneal the stess equations of equilibium without body foces can be given as σ 1 τθ τz σ σθ = 0 θ z τθ 1 τθ τθz τθ = 0 θ z τz 1 τzθ σz τz = 0 θ z Fo axisymmety about z-axis = 0 θ σ τz σ σθ + + = 0 z τθ τθz τθ + + = 0 z τz σz τz + + = 0 z and this gives In a plane stess situation if the cylinde ends ae fee to expand σ z = 0 and due to unifom adial defomation and symmety τ z = τ θz = τ θ = 0. The equation of equilibium educes to (1) () Vesion ME, IIT Khaagpu

2 σ σ σ + θ = 0 This can be witten in the following fom: σ +σ =σθ (3) If we conside a geneal case with body foces such as centifugal foces in the case of a otating cylinde o disc then the equations educe to σ σ σ + θ +ρω 0 σ σ +σ +ρω θ = = which may be witten as 0 (4) It is convenient to solve the geneal equation so that a vaiety of poblems may be solved. Now as shown in figue , the stains ε and ε θ may be given by u 1 E ε = = σ νσθ ( ) [ + u Δθ Δθ u 1 Δθ E θ ] since σ z = 0 (5) [ ] ε = = = σ νσ θ (6) u u u + δ A A ' B B ' A A ' Δθ B B ' F- Repesentation of adial and cicumfeential stain. Vesion ME, IIT Khaagpu

3 Combining equation (5) and (6) we have σ σ ν + ( +ν)( σθ σ ) = 0 (7) θ 1 Now fom equation (4) we may wite σθ σ σ = + + ρω and combining this with equation (7) we may aive at σ σ 3 ( 3 ) + + +ν ρω = 0 (8) Fo a non-otating thick cylinde with intenal and extenal pessues p i and p o we substitute ω = 0 in equation (8) and this gives σ σ 3 + = 0 (9) A typical case is shown in figue A standad solution fo equation (9) is σ = c n whee c and n ae constants. Substituting this in equation (9) and also combining with equation (3) we have c σ = c + c σ θ = c 1 1 whee c 1 and c ae constants. (10) i o p i p o 9..1.F- A thick cylinde with both extenal and intenal pessue. Vesion ME, IIT Khaagpu

4 Bounday conditions fo a thick cylinde with intenal and extenal pessues p i and p o espectively ae: at = i σ = -p i and at = o σ = -p o The negative signs appea due to the compessive natue of the pessues. This gives c p ii poo i o po pi 1 = c = o i o i ( ) The adial stess σ and cicumfeential stess σ θ ae now given by p ii poo i o ( po pi) 1 σ = + o i o i ii i oo o ( po pi) p p 1 σ θ = o i o i (11) It is impotant to emembe that if σ θ woks out to be positive, it is tensile and if it is negative, it is compessive wheeas σ is always compessive iespective of its sign. Stess distibutions fo diffeent conditions may be obtained by simply substituting the elevant values in equation (11). Fo example, if p o = 0 i.e. thee is no extenal pessue the adial and cicumfeential stess educe to p ii o σ = + 1 o i p σ θ = + 1 ii o o i (1) The stess distibution within the cylinde wall is shown in figue Vesion ME, IIT Khaagpu

5 i o σ θ σ p i F- Radial and cicumfeential stess distibution within the cylinde wall when only intenal pessue acts. It may be noted that σ + σ θ = constant and hence the defomation in z-diection is unifom. This means that the coss-section pependicula to the cylinde axis emains plane. Hence the defomation in an element cut out by two adjacent coss-sections does not intefee with the adjacent element. Theefoe it is justified to assume a condition of plane stess fo an element in section If p i = 0 i.e. thee is no intenal pessue the stesses σ and σ θ educe to σ = p 1 oo i o i p σ θ = + 1 oo i o i (13) The stess distibutions ae shown in figue Vesion ME, IIT Khaagpu

6 p o i o σ (negative) σ θ (negative) F- Distibution of adial and cicumfeential stesses within the cylinde wall when only extenal pessue acts. 9.. Methods of inceasing the elastic stength of a thick cylinde by pe-stessing In thick walled cylindes subjected to intenal pessue only it can be seen fom equation (1) that the maximum stesses occu at the inside adius and this can be given by σ ( ) = max pi = i σ = p o i θ(max) i = i o i + This means that as p i inceases σ θ may exceed yield stess even when p i < σ yield. Futhemoe, it can be shown that fo lage intenal pessues in thick walled cylindes the wall thickness is equied to be vey lage. This is shown schematically in figue This means that the mateial nea the oute edge is not effectively used since the stesses nea the oute edge gadually educe (Refe to figue ). Vesion ME, IIT Khaagpu

7 p i 9...1F- A schematic vaiation of wall thickness with the intenal pessue in a thick walled cylinde. t In ode to make thick-walled cylindes that esist elastically lage intenal pessue and make effective use of mateial at the oute potion of the cylinde the following methods of pe-stessing ae used: 1. Shinking a hollow cylinde ove the main cylinde.. Multilayeed o laminated cylindes. 3. Autofettage o self hooping. 1. Composite cylindes An oute cylinde (jacket) with the intenal diamete slightly smalle than the oute diamete of the main cylinde is heated and fitted onto the main cylinde. When the assembly cools down to oom tempeatue a composite cylinde is obtained. In this pocess the main cylinde is subjected to an extenal pessue leading to a compessive adial stess at the inteface. The oute cylinde o the jacket is subjected to an intenal pessue leading to a tensile cicumfeential stess at the inne wall. Unde this condition as the intenal pessue inceases the compession in the inne cylinde is fist eleased and then only the cylinde begins to act in tension. Gun baels Vesion ME, IIT Khaagpu

8 ae nomally pe-stessed by hooping since vey lage intenal pessues ae geneated. Hee the main poblem is to detemine the contact pessue p s. At the contact suface the oute adius si of the inne cylinde is slightly lage than the inside diamete so of the oute cylinde. Howeve fo stess calculations we assume that so si = s (say). The inne and oute cylindes ae shown in figue o so si i p s p s Jacket o oute cylinde Inne cylinde 9...F- Dimensions and the pessues at the contact suface of the intenal and oute cylindes. Fo the oute cylinde the adial and cicumfeential stesses at the contact suface may be given by p ss o σ = 1 = p = s o s s p ss o σ θ = 1+ = s o s s s Vesion ME, IIT Khaagpu

9 In ode to find the adial displacements of the cylinde walls at the u 1 contact we conside that ε ( θ = = σ ) E θ νσ. This gives the adial displacement of the inne wall of the oute cylinde as u p + = + ν E o s ss o s 1 Similaly fo the inne cylinde the adial and cicumfeential stesses at the oute wall can be given by σ = p = s σ = p s s i θ = s s s i + And following the above pocedue the adial displacement of the contact suface of the inne cylinde is given by u p + = ν E s i ss s i The total intefeence δ at the contact is theefoe given by p + + δ= + E o s s i ss o s s i. This gives the contact pessue in tems of the known vaiables as follows: p Eδ = o + s s + i s + o s s i s The combined stess distibution in a shink fit composite cylinde is made up of stess distibution in the inne and oute cylindes and this is shown in figue Vesion ME, IIT Khaagpu

10 o s σ θ o s s i p s + = σ σ θ σ σ i σ θ 9...3F- Combined stess distibution in a composite cylinde. Residual cicumfeential stess is maximum at = i fo the inne cylinde and is given by σ = p ss θ (max) = i s i Residual cicumfeential stess is maximum at = s fo the oute cylinde and is given by σ = p o s θ(max) s = s o s + Stesses due to fluid pessue must be supeimposed on this to find the complete stess distibution.. Multilayeed o Laminated cylinde The laminated cylindes ae made by stetching the shells in tension and then welding along a longitudinal seam. This is shown in figue Vesion ME, IIT Khaagpu

11 Welded junctions weld weld 9...4F- Method of constuction of multilayeed cylinde 3. Autofettage In some applications of thick cylindes such as gun baels no inelastic defomation is pemitted. But fo some pessue vessel design satisfactoy function can be maintained until the inelastic defomation that stats at inne boe speads completely ove the wall thickness. With the incease in fluid pessue yielding would stat at the inne boe and then with futhe incease in fluid pessue yielding would spead outwad. If now the pessue is eleased the oute elastic laye would egain its oiginal size and exet a adial compession on the inne shell and tension on the oute egion. This gives the same effect as that obtained by shinking a hoop ove an inne cylinde. This is known as Self- hooping o Autofettage. This allows the cylinde to opeate at highe fluid pessue. Fo a given autofettage fluid pessue a given amount of inelastic defomation is poduced and theefoe in sevice the same fluid pessue may be used without causing any additional inelastic defomation. Vesion ME, IIT Khaagpu

12 9..3 Summay of this Lesson Stesses and stains in thick cylindes ae fist discussed and Lame s equations ae deived. Radial and cicumfeential stess distibution acoss the wall thickness in thick cylindes have been illustated. Methods of inceasing elastic stength of a thick cylinde by pestessing ae then discussed. Inteface pessue and displacement duing shinking a hollow cylinde ove the main cylinde have been expessed in tems of known vaiables. Finally multilayeed o laminated cylindes and autofettage ae discussed. Vesion ME, IIT Khaagpu