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1 LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of Mterils Deprtment of Civil nd Environmentl Engineering University of Mrylnd, College Prk LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 1 Stresses in Thin-Wlled Vessels The thin-wlled pressure vessels provide n importnt ppliction of plne-stress nlysis. This their wlls offer little resistnce to bending, it my be ssumed tht the internl forces exerted on given portion of the wll re tngent to the surfce of the vessel, s shown in Fig. 32.

2 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 2 Stresses in Thin-Wlled Vessels Figure 32. Internl Forces re Tngent LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 3 Stresses in Thin-Wlled Vessels The resulting stresses on n element of the wll will thus be contined in plne tngent to the surfce of the vessel. Two types of thin-wlled vessels re investigted: Sphericl Pressure Vessels Cylindricl Pressure Vessels

3 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 4 Stresses in Thin-Wlled Vessels The stress in thin-wlled vessel vries from mximum vlue t the inside surfce to minimum vlue t the outside surfce of the vessel. It cn be shown tht if the rtio of the wll thickness to inner rdius of the vessel is less thn 0.1, the mximum norml stress is less thn 5% greter thn the verge stress. LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 5 Definition A pressure vessel is defined s thinwlled when the rtio of the wll thickness to the rdius of the vessel is so smll tht the distribution of norml stress on plne perpendiculr to the surfce of the vessel is essentilly uniform throughout the thickness of the vessel.

4 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 6 Generl Types of Vessels The following types of vessels cn be nlyzed s thin-wlled elements: Boilers Gs Storge Tnks Pipelines Metl Tires Hoops LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 7 Generl Types of Vessels The following types of vessels cn be treted s thick-wlled elements: Gun Brrels Certin High-pressure Vessels in Chemicl Processing Industry Cylinders nd Piping for Hevy Hydrulic Pressure

5 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 8 Sphericl Pressure Vessels A typicl thin-wlled sphericl vessel used for gs storge is shown in Fig. 33. If the weights of the gs nd vessel re negligible (in most cses), symmetry of loding nd geometry requires tht stresses on sections tht pss through the center of the sphere be equl. LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 9 Sphericl Pressure Vessels Figure 33

6 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 10 Sphericl Pressure Vessels Consider the element shown in Fig. 34. The stresses σ x, σ y, nd σ n re relted by the following eqution: σ σ = σ x = (40) y n LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 11 Sphericl Pressure Vessels r Figure 34 t σ σ n σ x y σ y () z x (b)

7 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 12 Sphericl Pressure Vessels Shering stresses on ny of these plnes re not present becuse there re no lods to induce them. The norml stress component in sphere is known s meridionl or xil stress nd is commonly denoted s σ m or σ LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 13 Sphericl Pressure Vessels Derivtion of Axil or Meridionl Stress in Sphericl Vessel Consider the thin-wlled sphericl pressure vessel with rdius r nd thickness t, shown in Fig. 34b. The free-body digrm of tht figure cn be used to compute the stresses σ = σ = σ = σ (41) x y n

8 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 14 Sphericl Pressure Vessels Derivtion of Axil or Meridionl Stress in Sphericl Vessel in terms of the pressure p, nd the inside rdius r nd thickness t of the sphericl vessel. The force R is the resultnt of the internl forces tht ct on the cross-sectionl re of the sphere tht exposed by pssing plne through the center of the sphere. LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 15 Sphericl Pressure Vessels Derivtion of Axil or Meridionl Stress in Sphericl Vessel The force P is the resultnt of the fluid forces cting on the fluid remining within the hemisphere. Since the vessel is under sttic equilibrium, it must stisfy Newton's first lw of motion. In other words, the stress round the wll must hve net resultnt to blnce the internl pressure cross the cross-section

9 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 16 t Sphericl Pressure Vessels Derivtion of Axil or Meridionl Stress in Sphericl Vessel σ F = 0; R P = 0 R = P σ 2 ()( t 2πr ) = p( πr ) pr σ = 2t (42) LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 17 Stress on Sphericl Pressure Vessels σ = pr 2t p = pressure of gs or fluid r = inside rdius of sphere t = thickness of thin-wlled sphere (42)

10 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 18 Exmple 12 A steel pressure vessel of sphericl shpe hs the following specifictions: inside rdius r of 36 inches thickness t of 3/16" llowble yield stress σ y of 50 ksi modulus of elsticity E of 29,000 ksi Poisson s rtio ν of 0.25 ) Wht is the mximum pressure p crried by the tnk before yielding occurs? b) If p = 100 psi, wht is the new outer rdius of the tnk? LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 19 Exmple 12 (cont d) () Norml in-plne stresses re given by Eq. 42. Rewrite the eqution to solve for the mximum p pr 2tσ σ = ρ = 2t r p = = ksi = 521 psi 36

11 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 20 Exmple 12 (cont d) (b) First find the norml in-plne stress in the shell: pr 100( 36) σ = = = 9,600 psi 2 t Now pply Hooke s lw for plne stress: 1 1 ν 1 ν pr ε x = ( σ x νσ y ) = σ = E E E 2t ε x = ( 9,600) = The circumference, nd therefore the rdius, of the sphere will increse by 1 + e, so 3 New r outer = 36 ( ) = in. 16 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 21 Cylindricl Pressure Vessels A typicl thin-wlled sphericl vessel used for liquefied gs storge is shown in Fig. 35. Norml stresses, s shown in Fig. 36, re esy to evlute by using pproprite freebody digrm. Agin, The norml stress component on trnsverse plne is known s meridionl or xil stress nd is commonly denoted s σ m or σ.,

12 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 22 Cylindricl Pressure Vessels Figure 35 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 23 Cylindricl Pressure Vessels Figure 36 t r σ σ σ y () z x (b)

13 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 24 Cylindricl Pressure Vessels The norml stress component on longitudinl plne is known s hoop, tngentil, or circumferentil stress, nd commonly denoted s σ h, σ t, or σ c. Agin, there re no shering stresses on trnsverse or longitudinl plnes. Stress determintion in this cse will be the sme s in the cse of sphericl shpe. LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 25 Cylindricl Pressure Vessels Derivtion of Norml Stress σ To determine the longitudinl stress σ, we mke cut cross the cylinder similr to nlyzing the sphericl pressure vessel. The free body, illustrted on the left (Fig. 36), is in sttic equilibrium. This implies tht the stress round the wll must hve resultnt to blnce the internl pressure cross the crosssection.

14 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 26 Cylindricl Pressure Vessels Derivtion of Norml Stress σ Applying sttics (Newton's first lw of motion, we hve Or Or σ F = 0, R P = 0 R = P x 2 ()( t 2πr ) = p( πr ) pr σ = 2t (43) LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 27 Cylindricl Pressure Vessels Derivtion of hoop or tngentil stress σ h To determine the hoop stress σ h, we mke cut long the longitudinl xis nd construct smll slice s illustrted Fig. 37. The free body is in sttic equilibrium. According to Sttics (Newton's first lw of motion), the hoop stress yields, 2( σ )( t)( dx) = p( 2r)( dx) h Therefore, σ = h pr t (44)

15 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 28 Cylindricl Pressure Vessels Derivtion of hoop or tngentil stress σ h 2 ( σ )( t)( dx) = p( 2r)( dx) h Therefore, σ = h pr t Figure 37 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 29 Stress on Cylindricl Pressure Vessels pr σ = 2t (45) σ = pr h t (46) p = pressure of gs or fluid r = inside rdius of sphere t = thickness of thin-wlled sphere

16 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 30 Exmple 1 A steel pipe with inside dimeter of 12 in. will be used to trnsmit stem under pressure of 1000 psi. If the hoop stress in the pipe must be limited to 10 ksi becuse of longitudinl weld in the pipe, determine the mximum stisfctory thickness for the pipe. LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 31 Exmple 1 (cont d) For the cylinder, the hoop stress is given given by Eq. 46 s Therefore, σ h = pr t pr 2 t = = σ 10,000 h = 0.6 in.

17 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 32 Remrks of Thin-Wlled Pressure Vessels 1. The bove formuls re good for thinwlled pressure vessels. Generlly, pressure vessel is considered to be "thinwlled" if its rdius r is lrger thn 5 times its wll thickness t (r > 5 t). LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 33 Remrks of Thin-Wlled Pressure Vessels 2. When pressure vessel is subjected to externl pressure, the bove formuls re still vlid. However, the stresses re now negtive since the wll is now in compression insted of tension.

18 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 34 Remrks of Thin-Wlled Pressure Vessels 3. The hoop stress is twice s much s the longitudinl stress for the cylindricl pressure vessel. This is why n overcooked hotdog usully crcks long the longitudinl direction first (i.e. its skin fils from hoop stress, generted by internl stem pressure).

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