ME 354, MECHANICS OF MATERIALS LABORATORY PRESSURE VESSELS AND COMPOUND STRESSES/STRAINS

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1 PURPOS M 354, MCHANICS O MATRIAS ABORATORY PRSSUR VSSS AND COMPOUND STRSSS/STRAINS 7 november 00 / mgj The purposes of this exercise are to evaluate the compound stress/strain states in a thin walled closed end clinder and to relate analticall and experimentall determined stresses and strains. The thin walled closed end clinder is loaded b both an axiallapplied load and an internal pressure. Strain gages arranged longitudinall and tangentiall are used to experimentall determine axial and hoop strains. Thin wall pressure vessel stress equations and uniaxial loading stress equations along with constitutive relations are used to analticall determine the strains. QUIPMNT Strain-gaged, closed-end thin-wall pressure vessel Bridge completion and strain gage conditioning equipment Dead-weight, lever arm axial-loading device Various dead-weight masses of 5.0 and 0.0 kg. Device for producing air pressure. PROCDUR xamine the test setup and take note of the following a) Strain gage orientation on the pressure vessel b) Strain gage connections and circuits c) Dead weight lever arm axial loading device and the mechanical advantage nsure that the loading sstem is disconnected from the freel-hanging pressure vessel Zero all the strain gage circuits ongitudinal oading Onl Connect the loading sstem to the freel hanging pressure vessel Appl total dead weight masses of m A 5, 0, 5, and 0 kg to the pan of the dead weight lever arm. Record both the applied mass, m A, the mass, m P, of the pan and the mass of the lever arm in kg. Record the applied mass and the resulting strains of all the strain gages for each applied mass. Determine the elastic modulus,, and the Poisson s ratio, ν, from the recorded strains and stresses calculated from the applied longitudinal forces and the dimensions of the pressure vessel Pressure oading Onl Disconnect the loading sstems from the freel-hanging pressure vessel nsure that the strain gage circuits are eroed Introduce a pressure of 500 to 700 kpa to the pressure vessel and record pressure and the resulting strains Pressure oading and ongitudinal oading While the pressure vessel is still pressuried, reconnect the loading sstem and appl a total dead weight mass of 5 kg to the pan of the deadweight lever arm and record the applied mass, the pressure and the resulting strains Remove the mass, disconnect the deadweight loading sstem and release the pressure

2 BACKGROUND AND ANAYSIS O RSUTS Clindrical or spherical pressure vessels are commonl used in engineering to serve as boilers or tanks. When under pressure, the material from which these vessels are made is subjected to stresses from all directions. Although this is the case, the vessel can be analed in a simple manner, provided that it has a thin wall. Thin wall refers to a vessel having a ratio of inner radius to wall thickness of 0 or more (i.e., r/t 0). Specificall, when r/t0, the equations resulting from the thin-wall pressure vessel analsis will predict a stress that is ~4% less than the actual maximum stress in the vessel. As r/t increases, this difference decreases. When the vessel wall is thin, the stress distribution throughout the thickness will not var significantl and is therefore assumed constant and uniform. In analing stresses in thin wall pressure vessels, the pressure is assumed to be gauge since it measures the pressure greater than atmpospheric pressure which is assumed to exist both inside and outside the vessel s wall. An element in the wall of a clindrical thin wall pressure vessel is subjected to three normal stresses when the vessel is internall pressuried: axial, hoop and radial (see igure ): Axial stress: σ a at rri and rro () t Hoop stress: σ h at rri and rro () t Radial stress: σ r p at rri and σ r 0 at rro (3) where r is the variable for radius,ri is the inner radius, Ro is the outer radius, p is the internal gauge pressure, t is the wall thickness, The three stresses are mutuall orthogonal and represent the principal normal stresses in the wall of a thinwall clindrical pressure vessel as shown in ig.. σ h σ r x σ a igure Illustration of three normal stresses acting on the wall of a thinwall pressure vessel Compound stress states exist in man engineering components. Because stress is a tensor, having both tpe (normal or shear), direction (e,g, x,, or,, 3) and magnitude, stresses calculated from engineering formulae (e.g., σp/a or qs -3) cannot be added or subtracted indiscriminatel. Instead onl stresses of the same tpe and in the same direction can be superposed to find the resulting stress.

3 or example if a longitudinal force,, is applied to an internall pressuried clindrical thin wall pressure vessel, the resulting longitudinal normal stress, σ π( R o R i ) can onl be added to the axial stress, σ a, resulting from the closed end pressure vessel. t The total stress state for a longitudinall loaded, internall-pressuried thin-wall pressure vessel is the radial stress, the hoop stress, and superposed longitudinal and axial stresses. If a Cartesian coordinate sstem is applied as shown in ig, the resulting coordinate stresses at the outer surface ( rro) of the pressure vessel can be expressed as: x- stress: σx σa σl (4) t π( Ro Ri ) pr - stress: σ σ i h (5) t - stress: σ σ 0 (6) r Note that if the constitutive relations are applied to the coordinate stresses of quations 4-6, the resulting strains in the x, and directions can be calculated and compared to measured strains such that: [ ( )] [ ( )] ( ) x- strain from coordinate stresses: εx σx ν σ σ - strain from coordinate stresses: ε σ ν σx σ - strain from coordinate stresses: ε σ ν σx σ [ ] (7) (8) (9) Alternativel, the constitutive relations can be applied to measured coordinate strains to calculate resulting stresses in the x, and directions such that: ( ) x- stress from coordinate strains: σ ν ε ν ν ν ε ε ε x x x ( ) ( )( ) - stress from coordinate strains: σ ν ε ν ν ν ε ε ε ( x ) ( ) ( )( ) - stress from coordinate strains: σ ν ε ν ν ν ε ε ε ( x ) ( ) ( )( ) (7) (8) (9)

4 M 354, MCHANICS O MATRIAS ABORATORY PRSSUR VSSS AND COMPOUND STRSSS/STRAINS DATA SHT NAM DAT 7 november 00 / mgj ABORATORY PARTNR NAMS QUIPMNT IDNTIICATION Note: Be sure to note units of each quantit. Mass of Pan, m P kg Mass of Bar, m B kg Half length of lever arm, mm Half length of lever arm, mm ongitudinal oading Onl Added Mass, ma (kg) Strain Gage, SG Strain Gage, SG Strain Gage 3, SG3 Strain Gage 4, SG4 Strain Gage 5, SG5 Strain Gage 6, SG6 Pressure oading Onl Applied Pressure, p (kpa) Strain Gage, SG Strain Gage, SG Strain Gage 3, SG3 Strain Gage 4, SG4 Strain Gage 5, SG5 Strain Gage 6, SG6 Pressure oading and ongitudinal oading Applied Pressure, p (kpa) Added Mass, ma (kg) Strain Gage, SG Strain Gage, SG Strain Gage 3, SG3 Strain Gage 4, SG4 Strain Gage 5, SG5 Strain Gage 6, SG6

5 M 354, MCHANICS O MATRIAS ABORATORY PRSSUR VSSS AND COMPOUND STRSSS/STRAINS WORKSHT 7 november 00/ mgj NAM DAT D o 30.0 mm t0.53 mm ) ongitudinal oading Onl The inner diameter of the pressure vessel is D o -td i mm. ) The cross sectional area of the pressure vessel is π D D o i 4 ( ) A mm. m A m B R m P 3) The longitudinal force,, imposed on the pressure vessel is calculated from the mechanical advantage of the lever arm sstem b summing moments about the fixed ( reaction point such that the force in Newtons is g ) mb ma m P where g9.86 m/s. The uniaxial normal stress applied longitudinall to the pressure vessel can be calculated as σ in MPa when is in Newtons and A is in mm. A ε ε The average longitudinal and transverse strains are calculated as ε and εt εt εt, respectivel, the and superscripts refer to rosette locations and.

6 Results are tabulated as follows: Added Mass, ma (kg) Calculated ongitudinal Stress, σ A (MPa) Average onditudinal Strain, ε ε ε Average Transverse Strain, εt εt εt Two elastic constants can be extracted from these test results... The slope of the plot of longitudinal stress vs longitudinal strain is the elastic modulus, such that σ ε where σ (, 000, 000) MPa for σ in MPa and ε in µm/m. ε The slope of the plot of transverse strain vs longitudinal strain is Poisson s ratio, ν such ε that ε νε where ν T. ε T 0 0 Transverse Strain, ε T (µm/m) ongitudinal Stress, σ (MPa) ongitudinal Strain, ε (µm/m) ongitudinal Strain, ε (µm/m) Measured lastic Modulus, (MPa) xpected lastic Modulus, (MPa) Measured Poisson s Ratio, ν xpected Poisson s Ratio, ν

7 4) Pressure oading Onl: or a thinwall pressure vessel, the stress in the axial direction due to internal pressure is: σ a t or a thinwall pressure vessel, the stress in the hoop direction due to the internal pressure is: σ h t Using constitutive relations for plane stress state at the free surface of the pressure vessel, the strain in the axial direction due to internal pressure onl is: εa σa ν σh [ ] ( ) Using constitutive relations for plane stress state at the free surface of the pressure vessel, the strain in the hoop direction due to internal pressure onl is: εh σh ν σa [ ] ( ) Applied Presssure, p (kpa) Calculated Axial Stress, σa (MPa) Calculated Hoop Stress, σh (MPa) Calculated Axial Strain, εa Measured Axial Strain ε ε Calculated Hoop Strain, εh Measured Hoop Strain εt ε T 5) Pressure oading ongitudinal oading: or a thinwall pressure vessel, the stress in the axial direction due to internal pressure is: σ a. The uniaxial normal t stress applied longitudinall to the pressure vessel can be calculated as σ. A or at thinwall pressure vessel, the stress in the hoop direction due to the internal pressure is: σ h t Using constitutive relations for plane stress state at the free surface of the pressure vessel, the total strain in the axial direction due to internal pressure onl is: εa [ ( σa σ ) ν( σh) ]

8 Using constitutive relations for plane stress state at the free surface of the pressure vessel, the strain in the hoop direction due to internal pressure onl is: εh [ σh ν( σa σ) ] Applied Presssure, p (kpa) Applied Mass, ma (kg) Calculated Axial Stess, σa σ (MPa) Calculated Hoop Stress, σh (MPa) Calculated Axial Strain, εa Measured Axial Strain ε ε Calculated Hoop Strain, εh Measured Hoop Strain εt ε T 6) Discuss the agreement or lack of agreement between the various measured and calculated results. Were the results what ou expected? Do ou think it is possible to superpose stress solutions and still predict actual stress or strain states?