Static Analysis of Piping Systems

Static Analysis of Piping SystemsA Brief IntroductionBasic Stress ConceptsStresses can be categorised as one of two types:Normal & ShearNormal StressesAct Normal (perpendicular) to face of the crystal structureCompressive or tensile (tend to be more in tension due to internal pressure)Shear StressesParallel to face of crystal structureCan be caused by more than one type of loadNormal StressesNormal Stresses Longitudinal/Axial

FaxSLLongitudinal (or Axial) StressActs along the axis of the pipeCaused by an internal force acting axially in the pipe

Internal PressureSLNormal Stresses - Internal PressureSpecific Instance of Longitudinal Stress is that due to internal pressureCaused by Internal Pressure force Acting Axially along pipeCauses Longitudinal Stress in the pipe wall acting axiallyNormal Stresses Bending StressBending Stress caused by the bending moment on the pipeZero at neutral axisVaries linearly across cross section from max compressive to max tensile

Longitudinal StressAxial

Pressure

BendingHoop StressAnother normal stress present in the pipeCaused by internal pressure. Acts in a direction parallel to the pipe circumference.

Magnitude varies through the pipe wall Can be calculated by Lames equation as:Can be approximated conservatively for thin-wall cylinders by assuming that the pressure force applied over an arbitrary length of pipe, l is resisted uniformly by the pipe wall over that same arbitrary length.Radial Stress

Radial stress is zero at the outer radius of the pipe, where the bending stresses are maximised. For this reason, this stress component has traditionally been ignored during the stress calculations.Third normal stress present in the pipeCaused by internal pressure. Varies stress equal to internal pressure at pipes inner surfaceStress equal to atmospheric pressure at external surfaceShear stressesShear Stresses - Force

Applied in a direction parallel to the face of the plane of the crystal structure of the material and tend to cause adjacent planes of the crystal to slip against each other. May be caused by more than one type of applied load. For example, shear stress may be caused by shear forces acting on the cross section.Maximum at the neutral axis of the pipe and zero at the maximum distance from the neutral axisUsually small, shear stresses due to forces are traditionally neglected during pipe stress analysis.

Shear Stress - TorsionShear Stresses may also be caused by torsional loads.

Worked examplePart 1Worked Example Pipe Properties

Consider a 6 Nominal size, standard schedule pipe:Worked Example Piping Loads

Worked Example Stress CalculationsLongitudinal Stress:Worked Example Stress CalculationsShear Stress (due to Torsion)Hoop Stress3d state of stress3D State of Stress in the Pipe WallDuring operation, pipes are subject to all these types of stresses.Examining a small cube of metal from the outer wall (the highest stressed part) would show the stresses distributed like so:

This imaginary cube could be oriented in any wayEach orientation has a different combination of normal and shear stresses on the facesIn one orientation (whatever that may be) the normal stress is at its maximumAnother orientation is where the normal stress is at its minimum

In both of these orientations all shear stress components are zeroIn these cases (where shear stress is zero), the normal stresses are termed the Principal Stresses3D State of Stress in the Pipe WallFor 3-dimensional analyses there are three principal stresses

maximumminimum regardless of the orientation of the stress axes, the sum of the orthogonal stress components is always equalThe opposite of these orientations are the two orientations where the shear stress is at its maximum and minimum (minimum is typically ignored)This is termed orientation of maximum shear stressThis is equal to difference between largest and smallest principal stresses (which are by definition S1 and S3

3D State of Stress in the Pipe WallMohrs CircleAll these values can be determined from Mohrs circleMohrs circle can be simplified by neglecting radial stress component allowing a simpler (i.e. 2D) state of stressPlot Normal vs. Shear for the two know orientationsLongitudinal vs. shearHoop vs. shearThe infinite number of points on the Mohrs circle represent the infinite orientations of the local stress axes our imaginary cube

Mohrs CircleS1 and S2 are equal to the centre of the circle, plus or minus the radius Worked examplePart 2Worked Example Continued

Mohrs Circle of StressWorked Example ContinuedMaximum Principal Stress S1Maximum Principal Stresses S2 and S3S3 = S2Failure theories

Failure TheoriesSo we have calculated stresses but these are not much use on their own. Is 128MPa High or Low?We need allowable stressesMaterial allowable stresses are related to strengths as determined by material uniaxial testsSo our calculated stresses must also be related to the uniaxial tensile testWe can relate the two using a failure theoryFailure TheoriesThere are 3 generally accepted failure theories used to predict onset of yieldingOctahedral Shear or Von Mises theoryMaximum Shear or Tresca TheoryMaximum Stress or Rankine TheoryThese theories relate failure in an arbitrary 3D stress state in a material to failure in the stress state found in a uniaxial tensile test specimenFailure here = when plastic deformation occurs

Von MisesFailure occurs when the octahedral shear stress in a body is equal to the octahedral shear stress at yield in a uniaxial tension testTrescaFailure occurs when the maximum shear stress in a body is equal to the maximum shear stress at yield in a uniaxial tension test.RankineFailure occurs when the maximum tensile stress in a body is equal to the maximum tensile stress at yield in a uniaxial tension testThe maximum tensile stress is the largest, positive principal stress, S1 (by definition, S1 is always the largest of the principal stresses.)Max Stress Intensity CriterionMax Stress Intensity CriterionWorked examplePart 3Worked Example - CompletionPipe 1

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