† Text Eq. refers to Mechanical Engineering Design, 7 th edition text by Joseph Edward Shigley, Charles R. Mischke and Richard G. Budynas; equations and figures with the prefix T refer to the present tutorial. M ECHANICAL E NGINEERING D ESIGN T UTORIAL 4 –15: P RESSURE V ESSEL D ESIGN P RESSURE V ESSEL D ESIGN M ODELS FOR C YLINDERS : 1. Thick-walled Cylinders 2. Thin-walled Cylinders T HICK - WALL T HEORY • Thick-wall theory is developed from the Theory of Elasticity which yields the state of stress as a continuous function of radius over the pressure vessel wall. The state of stress is defined relative to a convenient cylindrical coordinate system: 1. t σ — Tangential Stress 2. r σ — Radial Stress 3. l σ — Longitudinal Stress • Stresses in a cylindrical pressure vessel depend upon the ratio of the inner radius to the outer radius ( / oi rr) rather than the size of the cylinder. • Principal Stresses ( 123 ,, σσσ ) 1. Determined without computation of Mohr’s Circle; 2. Equivalent to cylindrical stresses ( , , trl σσσ ) • Applicable for any wall thickness-to-radius ratio. Cylinder under Pressure Consider a cylinder, with capped ends, subjected to an internal pressure, p i , and an external pressure, p o , FIGURE T4-15-1 o p o r i r i p l σ r σ t σ r σ l σ t σ Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 2/10 The cylinder geometry is defined by the inside radius, , i r the outside radius, , o r and the cylinder length, l. In general, the stresses in the cylindrical pressure vessel ( , , trl σσσ ) can be computed at any radial coordinate value, r, within the wall thickness bounded by i r and , o r and will be characterized by the ratio of radii, / . oi rr ζ = These cylindrical stresses represent the principal stresses and can be computed directly using Eq. 4-50 and 4-52. Thus we do not need to use Mohr’s circle to assess the principal stresses. Tangential Stress: 22 22222 /)( io iooiooii t rr rpprrrprp − −−− = σ for io rrr≤≤ (Text Eq. 4-50) Radial Stress: 22 22222 /)( io iooiooii r rr rpprrrprp − −+− = σ for io rrr≤≤ (Text Eq. 4-50) Longitudinal Stress: • Applicable to cases where the cylinder carries the longitudinal load, such as capped ends. • Only valid far away from end caps where bending, nonlinearities and stress concentrations are not significant. 22 22 io ooii l rr rprp − − = σ for io rrr≤≤ (Modified Text Eq. 4-52) Two Mechanical Design Cases 1. Internal Pressure Only ( 0= o p ) 2. External Pressure Only ( 0= i p ) Design Case 1: Internal Pressure Only • Only one case to consider — the critical section which exists at i rr = . • Substituting 0= o p into Eqs. (4-50) and incorporating / , oi rr ζ = the largest value of each stress component is found at the inner surface: 22 2 ,max 22 2 1 () 1 oi tit i i iti oi rr rr p p pC rr ζ σσ ζ + + == = = = −− (T-1) Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 3/10 where 22 2 222 1 1 oi ti oi rr C rr ζ ζ + + == −− is a function of cylinder geometry only. irir prr −=== max, )( σσ Natural Boundary Condition (T-2) • Longitudinal stress depends upon end conditions: ili p C Capped Ends (T-3a) l σ = 0 Uncapped Ends (T-3b) where 2 1 1 li C ζ = − . Design Case 2: External Pressure Only • The critical section is identified by considering the state of stress at two points on the cylinder: r = r i and r = r o . Substituting p i = 0 into Text Eqs. (4-50) for each case: r = r i 0)( == ir rr σ Natural Boundary Condition (T-4a) () 2 2 ,max 22 2 2 2 1 o tit o o oto oi r rr p p pC rr ζ σσ ζ = = =− =− =− −− (T-4b) where, 2 2 222 2 2 1 o to oi r C rr ζ ζ == −− . r = r o oror prr −=== max, )( σσ Natural Boundary Condition (T-5a) 22 2 22 2 1 () 1 oi to o o oti oi rr rr p p pC rr ζ σ ζ + + ==− =− =− −− (T-5b) • Longitudinal stress for a closed cylinder now depends upon external pressure and radius while that of an open-ended cylinder remains zero: olo p C− Capped Ends (T-6a) l σ = 0 Uncapped Ends (T-6b) Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 4/10 where 2 2 1 lo C ζ ζ = − . Example T4.15.1: Thick-wall Cylinder Analysis Problem Statement: Consider a cylinder subjected to an external pressure of 150 MPa and an internal pressure of zero. The cylinder has a 25 mm ID and a 50 mm OD, respectively. Assume the cylinder is capped. Find: 1. the state of stress ( r σ , t σ , l σ ) at the inner and outer cylinder surfaces; 2. the Mohr’s Circle plot for the inside and outside cylinder surfaces; 3. the critical section based upon the estimate of max τ . Solution Methodology: Since we have an external pressure case, we need to compute the state of stress ( , r σ , t σ l σ ) at both the inside and outside radius in order to determine the critical section. 1. As the cylinder is closed and exposed to external pressure only, Eq. (T-6a) may be applied to calculate the longitudinal stress developed. This result represents the average stress across the wall of the pressure vessel and thus may be used for both the inner and outer radii analyses. 2. Assess the radial and tangential stresses using Eqs. (T-4) and (T-5) for the inner and outer radii, respectively. 3. Assess the principal stresses for the inner and outer radii based upon the magnitudes of ( , r σ , t σ l σ ) at each radius. 4. Use the principal stresses to calculate the maximum shear stress at each radius. 5. Draw Mohr’s Circle for both states of stress and determine which provides the critical section. Solution: 1. Longitudinal Stress Calculation: OD 50mm ID 25mm 25mm; 12.5mm 22 22 oi rr== = == = Compute the radius ratio, ζ 25 mm 2.0 12.5 mm o i r r ζ == = Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 5/10 Then, 22 22 2 2 2 (2) 1(2) 1 ( ) ( ) ( 150MPa)(1.3333 mm ) 1 lo lilo o olo C rr rr p pC ζ ζ ζ σσ ζ == = −− == = =− =− =− − 2 1.3333 mm MPa200 − −− −= == = l σ σσ σ 2. Radial & Tangential Stress Calculations: Inner Radius (r = r i ) 22 22 2 ,max 22 22(2) 1(2)1 2 ( ) ( 150MPa)(2.6667) to o tit o oto oi C r rr p pC rr ζ ζ σσ == = −− == =− =− =− − 2.6667 400 MPa ti σ (r r ) Compressive==− 0pforConditionBoundaryNatural i === 0)r(rσ ir Outer Radius (r = r o ) 2 2 22 22 ,min 22 1(2)1 1(2)1 ( ) ( 150MPa)(1.6667) ti oi tot o oti oi C rr rr p pC rr ζ ζ σσ ++ == = −− + = = =− =− = − − 1.6667 eCompressiv MPa250 − −− −= == == == = )r(rσ ot ConditionBoundaryNatural MPa150 − −− −= == =− −− −= == == == = oir p)r(rσ 3. Define Principal Stresses: Inner Radius (r = r i ) Outer Radius (r = r o ) MPa400 MPa200 MPa0 3 2 1 −== −== == t l r σσ σσ σσ MPa250 MPa200 MPa150 3 2 1 −== −== −== t l r σσ σσ σσ 4. Maximum Shear Stress Calculations: Inner Radius (r = r i ) 13 max 0 ( 400) () 22 i rr σσ τ − −− == = = 200 MPa Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 6/10 Outer Radius (r = r o ) 13 max ( 150) ( 250) () 22 o rr σσ τ − −−− == = = 50 MPa 5. Mohr’s Circles: Inner Radius (r = r i ) Outer Radius (r = r o ) Critical Section !RadiusInsideatisSectionCriticalrr i ⇐ ⇐⇐ ⇐= == == == = MPa200)( max τ ττ τ 1 150 MPa σ =− 3 σ = -250 MPa τ σ 2 σ = -200 MPa τ max = 50 MPa σ 2 σ = -200 MPa 3 σ = -400 MPa τ 1 0 MPa σ = max τ = 200 MPa FIGURE T4-15-2 FIGURE T4-15-3 • Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 7/10 T HIN - WALL T HEORY • Thin-wall theory is developed from a Strength of Materials solution which yields the state of stress as an average over the pressure vessel wall. • Use restricted by wall thickness-to-radius ratio: ¾ 1 According to theory, Thin-wall Theory is justified for 20 t r ≤ ¾ 1 In practice, typically use a less conservative rule, 10 t r ≤ • State of Stress Definition: 1. Hoop Stress, t σ , assumed to be uniform across wall thickness. 2. Radial Stress is insignificant compared to tangential stress, thus, 0. r σ  3. Longitudinal Stress, l σ S Exists for cylinders with capped ends; S Assumed to be uniformly distributed across wall thickness; S This approximation for the longitudinal stress is only valid far away from the end-caps. 4. These cylindrical stresses ( , , ) trl σσσ are principal stresses ( , , ) trl σσσ which can be determined without computation of Mohr’s circle plot. • Analysis of Cylinder Section 1 t F V F Hoop F Hoop Pressure Acting over Projected Vertical Area d i FIGURE T4-15-4 Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 8/10 The internal pressure exerts a vertical force, F V , on the cylinder wall which is balanced by the tangential hoop stress, F Hoop . tpdFFF ttAF pddppAF tiHoopVy ttstressedtHoop iiprojV σ σσσ ¦ −=−== === === 220 )}1)({( )}1)({( Solving for the tangential stress, Hoop Stress 2 i t pd t σ = (Text Eq. 4-53) • Comparison of state of stress for cylinder under internal pressure verses external pressure: Internal Pressure Only 2 0 (Text Eq.4-55) 42 i t r it l pd Hoop Stress t By Definition pd Capped Case t σ σ σ σ = = == External Pressure Only 2 0 42 o t r ot l pd Hoop Stress t By Definition pd Capped Case t σ σ σ σ = = == Example T4.15.2: Thin-wall Theory Applied to Cylinder Analysis Problem Statement: Repeat Example T1.1 using the Thin-wall Theory (p o = 150 MPa, p i = 0, ID = 25 mm, OD = 50 mm). Find: The percent difference of the maximum shear stress estimates found using the Thick-wall and Thin-wall Theories. Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 9/10 Solution Methodology: 1. Check t/r ratio to determine if Thin-wall Theory is applicable. 2. Use the Thin-wall Theory to compute the state of stress 3. Identify the principal stresses based upon the stress magnitudes. 4. Use the principal stresses to assess the maximum shear stress. 5. Calculate the percent difference between the maximum shear stresses derived using the Thick-wall and Thin-wall Theories. Solution: 1. Check t/r Ratio: 10 1 20 1 2 1 mm25 mm5.12 or r t ² == The application of Thin-wall Theory to estimate the stress state of this cylinder is thus not justified . 2. Compute stresses using the Thin-wall Theory to compare with Thick- wall theory estimates. definitionbyStress Radialb. mm)2(12.5 )mm50)(MPa150( 2 wall)acrossuniformstress,(averageStressHoopa. r 0 MPa300 = −= − = − = σ σ t dp oo t c. Longitudinal Stress (average stress, uniform across wall) 42 oo t l pd t σ σ − ===− 150 MPa 3. Identify Principal Stresses in terms of “Average” Stresses: MPa300 MPa150 MPa0 3 2 1 −== −== == t l r σσ σσ σσ 4. Maximum Shear Stress Calculation: MPa150 2 )MPa300(0 2 31 max += −− = − = σσ τ 5. Percent Difference between Thin- and Thick-wall Estimates for the Critical Section: Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 10/10 max,Thin max,Thick max,Thick % 100% ( 150) ( 200) (100%) ( 200) Difference ττ τ − =∗ +−+ =∗=− + 25%    Thin -wall estimate is 25% low! ⇐ ⇐⇐ ⇐ . 4-52) Two Mechanical Design Cases 1. Internal Pressure Only ( 0= o p ) 2. External Pressure Only ( 0= i p ) Design Case 1: Internal Pressure Only • Only. Hoop Pressure Acting over Projected Vertical Area d i FIGURE T4-15-4 Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design