PURPOSE OF STRESS ANALYSIS
To prevent failure of piping or supports from over stress or fatigue.
To prevent leakage at joints due to excessive bending moment.
To prevent detrimental stress or distortion in piping or in connected equipment’s, resulting from excessive thrusts and movements in piping.
To determine the magnitude and direction of forces on anchors, supports and guides.
To determine requirements for piping loops, expansion joints or re-routing of piping.
To determine location, size and type of spring supports or rod hangers.
LOADS CLASSIFICATION
LOADS ON PIPING WEIGHT
Weight causes the pipe to sag, which puts stress into the piping material and forces onto equipment nozzles.
Proper spacing and design of supports, and careful attention to concentrated loads can take care of most weight problems.
THERMAL
Temperature cause growth of pipe which pushes against the nozzles and restraining supports and can cause wearing out of pump bearing, leakage vessel nozzles, rupture of pipe or vessels itself etc.
Restraint of growth causes unnatural deflection of pipe and hence additional stresses.
WIND LOAD
Piping guides and anchors resist the wind loading on the piping system.
EARTHQUAKE LOAD
Pipe support engineers design pipe anchors and guides to resist these forces.
ALLOWABLE DISPLACEMENT STRESS RANGE SA = 1.25 (Sc + Sh) 1 Sh reserved for the longitudinal stresses developed due to sustained loading.
f … stress range reduction factor N… Equivalent number of full displacement cycles during expected service life of piping system SC … Basic allowable stress at minimum metal temp, MPa Sh … Basic allowable stress at maximum metal temp, MPa PIPE ALLOWABLE SPANS
For uniformly distributed loads 1. Both ends pinned (free to rotate): Max. bending moment ,Mmax = wl^2 / 8 (at center) 2. Both ends fixed: Mmax = wl^2 / 12 (at end)
But the true condition is somewhere in-between Mmax = wl^2 / 10 Or,
Lall
= Ö (10ZSall / w)
For concentrated loads
1. Both ends pinned (free to rotate): Mmax = Pab / l (at point of loading)
2. Both ends fixed: Mmax = wl^2 / 12 (nearer to load)
SUPPORTING PIPING FOR WEIGHT 1. In the plan view, take pipe riser as a point load equal to the kg /m of pipe times the total length of the riser. 2. First support all concentrated loads in the system as closely as possible. 3. Break piping into one of three different span types. 4. Apply correct table for allowable span. 5. Determine location of support points. 6. Make sure that there is a structure available for the support to use. 7. Allowable weight stresses are not exceeded if the piping system is supported such that no straight span exceeds allowable span. 8. Provide a layout to reduce load on a nozzle. 9. On pumps, adjustable support within a few feet of the nozzle required, hence provide some room on the pipe free of fittings, drains and instrumentation. PIPE ALLOWABLE SPANS Beam bearing freely on 2 supports with uniformly distributed load.
DEAD LOAD CALCULATION
SELECTION OF PIPING SYSTEMS FOR ANALYSIS: The Piping Engineer has following choices to establish that the required flexibility has been provided in the piping layout
No formal analysis of adequate flexibility is required for a piping system which:
Duplicates, or replaces without significant changes, a system operating with a successful service record;
Can readily be judged adequate by comparison with previously analyzed systems;
is uniform size, has no more than two points of fixation, no intermediate restraints, and falls within the limitations of empirical equation:
Outside Dia for 6” Pipe, D = 168.4 mmLinear Expansion in all three directions: ΔX = 4 x 2.25 / 1000 = 9 mm ΔY = 5 x 2.25 / 1000 = 11.25 mm ΔZ = 1 x 2.25 / 1000 = 2.25 mmResultant expansion, yy =
Developed Length Between Anchors, LL = X + Y + Z= 14 m
Straight Line Dist between Anchors, U
= 14.6 mm
U = = 6.48 m
Conclusion: Detail Flexibility Analysis is Not Required.
FLEXIBILITY ANALYSIS – METHODS 1. Approximate methods – Guided Cantilever method – Nomographs 2. Exact Analytical Method – Using Finite Element Technique Approximate methods: The method is used for the following purposes:
For approximate assessment of the flexibility of average piping
On critical piping, for layout assistance in arriving at a suitable system for detailed analysis
On non-critical piping, to establish the location of restraints without unduly impairing the flexibility of the system.
FREE THERMAL EXPANSION
Flexibility of a piping system is a measure of the amount of thermal expansion it can safely absorb.
To find the “Free thermal expansion” pretend only one end is anchored and find the movement at the other end, assuming there is no friction and there are no guides.
Free thermal expansion depends only on the relative locations of anchor points.
Thermal expansion is calculated by:
Refer table C-1 , ASME B31.3 for coefficient of thermal expansion
FREE THERMAL EXPANSION Case 1:
In N-S direction expansion absorbed is: =eL = 0.046*30’ =1.38” In E-Wdirection expansion absorbed is: =eL = 0.046*20’ =0.92”
N-S expansion can be reduced by just shifting the anchor end of drum as shown in second case. Case 2:
In N-S direction expansion absorbed is: =eL = 0.046*10’ =0.46” In E-W direction expansion absorbed is: =eL = 0.046*20’ =0.92” Second arrangement requires less flexibility (less expansion to be absorbed) & has the potential of saving pipe & fittings.
MAGNITUDE OF THERMAL A pipe line held between anchors, when heated up tries to expand against its restraints resulting in forces, moments and stresses. (fig.2.1 )
Free expansion DL will take place when one of the anchors is released. (fig. 2.2)
If pipe is to be maintained in the original position then there will be an axial force P to compress the increase in pipe length of DL . (fig.2.3 )
The strain developed in the pipe, e = DL / L= a Internal stress developed due to this strain, f
= Ee (Hooke’s Law) = Ea
The force required to compress back is P = Af = AEa where, A = Metal Area (mm2) E = Modulus of elasticity of material (Kpa) P = Compressive force on pipe (N) f = Stress developed (Kpa) DL = Axial compression of pipe (mm) L = length of pipe (mm) EXAMPLE To evaluate the magnitude of such a force, consider carbon steel pipe of 600 mm outside diameter with 10mm thickness, operating at a temperature of 300 deg. c Referring to ASME B31.3, Table C6, E = 26.85 MSI (1.888 x 104 kg/mm2 ) Referring to Table C1, a = 3.625 x 10-3 mm/mm Area of the pipe , A = Pi [(600) 2 – (580) 2 ] / 4 = 18535.4 mm 2 Compressive Force on pipe, P = AEa = 18535.4 x 1.888 x 104 x 3.625 x 10-3 = 12,68,563 kg = 1269 tons
Introduction of Piping Stress Analysis-for Beginner by varun chandel · Published September 29, 2015 · Updated February 24, 2016
What is piping Stress analysis
1. Analytical procedure to evaluate the stress state at various points in a piping system. 2. Also known as flexibility analysis since it also helps ascertain the required flexibility in the piping system 3. Helps determine displacements and forces / moments on the hangers, supports, restraints, guides, stops and anchors in the piping system
Why do we Perform Pipe Stress Analysis?
1. In order to keep stresses in the pipe and fitting within allowable levels. 2. In order to keep nozzle loading on the attached equipment within allowables of manufacturers or recognized standrads (NEMA SM23,API 610,API 617, etc) 3. In order to keep vessel stresses at piping connections within ASME Section VIII allowable levels. 4. In order to calcu;ate design load for sizing supports and restrains. 5. In order to determine piping displacements for interference checks. 6. In order to solve dynamics problems in piping. Such as those due to mechanical vibration, acoustic vibration, fliuid hammer,pulsation,transient flow and relief valve discharge.
7. In oerder to help optimize the pipe design.
Identification of Stress Critical lines
The main factors which decide stress critical lines are as follows: 1. Line design/operating/upset temperature 2. Equipment connection 3. Pipe and Equipment material 4. Pipe condition 5. Pipe thickness 6. Design/Upset pressure Mostly the critical lines for which stress analysis is to be performed by formal computer analysis consists of the following lines: 1. All Pump (Centrifugal-API/ANSI, gear pump, Screw pump) suction and discharge piping (4 inch and larger). 2. Centrifugal Compressor inlet and outlet piping. 3. Lines to and from steam generators. 4. Reciprocating pump and compressor suction and discharge piping. 5. Piping requiring expansion joints or other proprietary expansion devices. 6. Steam and Gas Turbine inlet and outlet piping. 7. Air Cooler inlet and outlet piping (3 inch and larger). 8. Process Heater inlet and outlet piping 9. Lines classified as category M as per ASME B31.3. 10. Piping subjected to high cyclic temperature conditions. 11. All jacketed lines. 12. Lines that require nozzle load compliance as stipulated per applicable codes or equipment Vendor allowable (Heat exchanger, Pressure Vessel Connected systems).
13. Lines subject to dynamic loading (relief lines, line with large pressure drop at control valves, surge pressure, slug flow, churn, two phase flow, water hammer, flashing, etc.) 14. All Fiberglass, aluminium alloy, refractory or elastomer lined piping. 15. All piping systems connected to FRP, plastic, glass lined steel or brittle equipment 16. Lines subjected to non-thermal movements (Expected differential settlement between structures, structure-equipment, etc., process equipment growth, header growth, tower growth or other significant displacements, etc.) 17. All lines 8” and larger operating above 150 deg. C (300 deg. F) and greater. 18. All lines 20” and larger operating above 80 deg. C (200 deg. F) and greater. 19. All lines 36” and larger. 20. All lines operating below -45 deg. C (-50 deg. F) which require special “cold” supports. 21. All plastic lined piping systems. Special attention shall be given to add enough additional supports to limit the external forces and moments in the flange connections to avoid an extra risk of flange leaks 22. Lines with special design requirements 23. All Safety pressure relieving systems 4 inch and larger (not including thermal reliefs) 24. Lines judged by the lead piping engineer/stress engineer as not having sufficient inherent flexibility 25. In addition, the piping effects of other conditions such as temperature gradients that could cause thermal bowing or where piping is connected to equipment with significant thermal growth may warrant detailed computer analysis. 26. For thin wall piping, if the D/T ratio exceeds 100, following requirements are applicable: 1. Design and support of piping systems using this specification should be reviewed by a stress engineer. Support and spans of thin wall piping systems are not covered by current Project practices and therefore must be designed for each application. 2. Stub-in connections per 304.3.2 thru 304.3.4 of ASME B31.3, are not allowed for run pipe with D/T greater than or equal to 100 and the branch diameter is greater than one half of the header diameter. 27. Lines connected to non-ferrous equipments.
28. Underground process lines with more than 30 degree difference in between design and ambient temperature. 29. All vertical lines connected to vertical vessels that require pipe supports or guides from that vessel. 30. All lines 4 inch and larger subject to external pressure or vacuum conditions. 31. All lines subject to vibration, as specified by Process, due to high velocity flow, high pressure drop, water hammer or mixed phase flow. 32. All lines that are connected to equipment constructed of thermoset or thermoplastic materials or that is glass, refractory, or elastomer lined. 33. All pressure containing non-metallic lines. 34. All flare line headers 35. Lines for which an Alternative Leak Test has been specified.
Critical Line List
Many organisations have the practice of dividing these critical lines into three groups based on their criticality: 1. Highly critical lines or group C1 lines: Must be reviewed thoroughly 2. Moderately Critical lines or group C2 lines and 3. Lower critical lines or group C3 lines
Information Required for Stress Analysis
Here are needed information. 1. Outside diameter of piping, wall thickness (or nominal diameter, schedule number) 2. Temperature, internal pressure 3. Material of piping. (Expansion coeffcient, Young’s modulus, and material density will be selected for this material.) 4. Insulation thickness and insulation material. (If not given, standard thickness for calcium silicate will be selected.) 5. Specifc gravity of contents
6. Any wind load to be considered? If yes, the direction of application is important. 7. Any anchor initial translation. (For towers, exchangers, and so on, nozzle initial ranslation is important.) 8. Corrosion allowance for piping 9. Flange rating, (ANSI B16.5) 10. Standard valve weight and flange weight will be used. (For special valves mark the weight on pipe stress isometric.) 11. Long radius elbows will be used. (If short radius or any other bend radius, mark on the isometric.) For short-radius elbow, radius= diameter 12. Any allowable loading from manufacturers on pumps, turbines, compressors? (From the vendor drawing for equipment.) 13. Any preference to use expansion loops, expansion joints, and so on, if needed? 14. Mark type of intersection (reinforced fabricated tee, etc.) 15. Mark support locations (available steel crossing, and so on) on the isometric 16. Is hydraulic testing load condition to be considered to get structural support loads?
Pipe stress isometrics (x-, y-, z-axis) piping plans, and sections are necessary.
Piping Loads – Primary, Secondary, Sustained Loads, Occasional Loads, Static
Primary Load These are typically steady or sustained types of loads such as internal fluid pressure, external pressure, gravitational forces acting on the pipe such as weight of pipe and fluid, forces due to relief or blow down, pressure waves generated due to water/steam hammer effects. Sustained Loads: 1. Internal/External Pressure : A pipe used for transporting fluid would be under internal pressure load. A pipe such as a jacketed pipe core or tubes in a Shell & Tube ex-changer etc. may be under net external pressure. Internal or external pressure induces stresses in the axial as well as circumferential (Hoop Stress) directions. The pressure also induces stresses in the radial direction, but these are often neglected. The internal pressure exerts an axial force equal to pressure times the internal cross section of the
pipe. F =P[πd^2/4]. If outer diameter is used for calculating approximate metalcrosssection as Pressure well as pipe cross-section, the axial stress can often be approximated as follows : S =Pd /(4t) 2. Dead Weight : It is the self weight of pipe including fluid, weight of fittings & other inline components (say valve, insulation etc.). This type of loads act throughout the life cycle of pipe. These Loads cause bending and the bending moment is related to normal and shear stresses. Pipe bending is caused mainly due to two reasons : distributed weight load (e.g. fluid weight) and concentrated weight load (e.g. valve weight). Occasional Loads: 1. Wind Load : Piping which are located outdoors and thus exposed to wind will be designed to withstand the maximum wind velocity expected during the plant operating life. Wind force is modeled as a uniform load acting upon the projected length of the pipe perpendicular to the direction of the wind. Wind pressure for various elevations will be used to calculate wind force using the following formula. Fw = Pw x S x A, where Fw = The total wind force, Pw = The equivalent wind pressure, S = Wind shape factor, A = Pipe exposed area. 2. Seismic Load : Seismic load is one of the basic concepts of earthquake engineering which means application of an earthquake-generated agitation to a structure. It happens at contact surfaces of a structure either with the ground, or with adjacent structures, or with gravity waves from tsunami. 3. Water Hammer : Water hammer (or more generally, fluid hammer) is a pressure surge or wave caused when a fluid (usually a liquid but sometimes also a gas) in motion is forced to stop or change direction suddenly (momentum change). Water hammer commonly occurs when a valve closes suddenly at an end of a pipeline system, and a pressure wave propagates in the pipe. It’s also called hydraulic shock. 4. Steam hammer : Steam hammer, the pressure surge generated by transient flow of superheated or saturated steam in a steam-line due to sudden stop valve closures is considered as an occasional load. Through the flow is transient, for the purpose of piping stress analysis, only the unbalanced force along the pipe segment tending to induce piping vibration is calculated and applied on the piping model as static equivalent force. 5. Safety Valve Discharge : Reaction forces from relief valve discharge is considered as an occasional load. The reaction force due to steady state flow following the opening of safety relief valve in an open discharge installation can be calculated in accordance with ASME B31.1 Appendix II and applied on the piping model as static equivalent force. Secondary Load Just as the primary loads have their origin in some force, secondary loads are caused by displacement of some kind. For example, the pipe connected to a storage tank may be under load
if the tank nozzle to which it is connected moves down due to tank settlement. Similarly, pipe connected to a vessel is pulled upwards because the vessel nozzle moves up due to vessel expansion. Also, a pipe may vibrate due to vibrations in the rotating equipment it is attached to. Displacement Loads: 1. Load due to Thermal Expansion of pipe 2. Load due to Thermal movement of Equipment A pipe may experience expansion or contraction once it is subjected to temperatures higher or lower respectively as compared to temperature at which it was assembled. The secondary loads are often cyclic but not always.For example load due to tank settlement is not cyclic. The load due to vessel nozzle movement during operation is cyclic because the displacement is withdrawn during shut-down and resurfaces again after fresh start-up. A pipe subjected to a cycle of hot and cold fluid similarly undergoes cyclic loads and deformation.
Piping Stresses- Primary, Secondary
1. Primary stress (membrane and bending) – This is the stress due to external loading of the pipe like weight,point load, wind, earthquake – If this exceeds the allowable stress it will cause failure of the pipe through continuous yielding 2. Secondary stress – This stress is not caused by any external loading but by such physical tendencies as thermal expansion – This stress is self-limiting in nature. It relieves itself upon yielding. It is due to this fundamental difference in behavior between primary and secondary stress that these two stress categories are treated very differently. These stresses are never added up and have different allowable values 3. Peak stress – Peak stresses are cyclical stresses which cause fatigue failure in pipes
Piping Component Stress Intensification Factor
Stress intensification factors – SIF 1. Elbows, branch connections and reducers will have a higher level of stress when compared to a straight pipe for the same amount of bending moment.
2. The factor by which the stress in the pipe component exceeds that of the straight pipe is called SIF (stress intensification factor). 3. SIF of a component depends upon its geometry and is calculated using empirical formulae available in piping codes. 4. For special components like Y-piece where no empirical relations are available, SIF will have to be determined through a analytical procedure like FEM. Relation between Elbow geometry and SIF 1. Elbow / bend radius – Has inverse relation to SIF 2. Elbow diameter – Has direct relation to SIF 3. Elbow thickness – Has inverse relation to SIF Relation between Branch geometry and SIF 1. Header diameter – Has direct relation to header & branch SIFs 2. Header thickness – Has inverse relation to header & branch SIFs 3. Branch diameter – Has direct relation to branch SIF. Has no bearing on header SIF 4. Branch thickness – Has direct relation to branch SIF. Has no bearing on header SIFStress in piping components Relation between Branch type and SIF
1. The various branch types are listed with their SIF in the increasing order 2. Welding Tee 3. Integrally reinforced fitting as per MSS SP 97 4. Reinforced fabricated Tee 5. Unreinforced fabricated Tee
Theories of Failure
The analysis of piping under pressure, weight and thermal expansion is complex. This complexity can be understood by knovledge of Principal Axis System. Stress is considered as the ratio of Force to Area. To find the stress in the small element, say cube of a piece of pipe, construct a three-dimensional, mutually perpendicular principal axis system with each axis perpendicular to the face of the cube it intersects. Each force, acting on the cube can be resolved into force components, acting along each of the axis. Each force, acting on the face of the cube divided by area of the cube face is called the principal stress. The principal stress acting along the centerline of the pipe is called Longitudinal principal stress. This stress is caused by longitudinal bending, axial force loading or pressure. Radial principal stress acts on a line from a radial line from center of pipe through the pipe wall. This stress is compressive stress acting on pipe inside diameter caused by internal pressure or a tensile stress caused by vacuum pressure. Circumferential principal stress, some times called Hoop or tangential stress, acts along the circumference of the pipe. This stress tends to open-up the pipe wall and is caused by internal pressure. When
two
or
more
at a point on a pipe, a shear stress will be generated. Longitudinal Principal stress, LPS = PD/4T Circumferential Principal stress, CPS (Hoop) = PD/2T Radial Principal stress, RPS = P
Failure Theories
principal
stresses
act
1. The Code presents equations for determining the stress levels in a piping system & provides stress limits for comparison. These theories are maximum principal stress failure theory & maximum shear stress failure theory. 2. The maximum principal stress failure theory states that when anyone of the three mutually perpendicular principal stresses exceed the yield strength of the material at temperature, failure will occur. 3. The maximum shear failure theory states that when the maximum shear stress (arithmetic average of largest minus smallest principal stresses) exceeds one-half the yield strength of the material at temperature, failure will occur. Stress Types The B31.3 Code provides design guidance for primary & secondary stresses. The basic characteristic of a primary stress is that it is not self-limiting as long as the load is applied, the stress will be present & will not diminish with time or as deformation takes place. The failure mode of a primary stress is gross deformation progressing to rupture. Examples of a primary stress are circumferential stresses due to internal pressure & longitudinal bending stresses due to gravity. The basic characteristic of a secondary stress is that it is self- limiting. The stress will diminish with time and strain. The failure mode of a secondary stress is small crack leading to leakage. Secondary stresses are due to cyclic thermal expansion and contraction. Pipe Span Calculations
Span limitations based on Stress, Deflection
Pipe Supporting The objective during the pipe supports design phase is to prevent the following: 1. overstress of piping 2. leakage at joints 3. overstress of supports 4. excessive forces on equipment 5. excessive interference with thermal expansion 6. excessive pipe sag (especially for piping requiring drain) 7. excessive heat flow, exposing support to temperature outside their limits, Etc.
1. The purpose of pipe supports is to control the weight effects of the piping system, as well as loads caused by pressure thrust, vibration, wind, earthquake, shock, and thermal displacement. The weight effects to be considered shall be the greater of operating or hydro-test loads. 2. The B3 1.3 guidance for pipe support types and materials of construction is presented in the B31 .3 TABLE 326.1 LISTED STANDARD, MSS SP-58. 3. The material selection for clamps and bolts, for example, is of particular importance in elevated temperature service. SP-58 assistance would be in the selection of a clamp material for example 4. in 750F (400C) service. 5. A review of the tables in SP-58 reveal that Carbon Steel clamp material would not be suitable, nor would the common type bolting, ASTM A307 used in clamps. 6. The designer would be guided to use an alloy steel for the clamp such as ASTM A240 and ASTM A193-Grade B7 bolts. 1. Pipe Support Span, based on deflection Pipe support span is a decision that faces the designer in most pipe supporting jobs. As a guide to the selection of support spacing, the following equation based on permissible mid span deflection is offered. The permissible mid-span deflection, y, concept is one technique commonly selected for support spacing. This technique is based on a specified mid-span, y deflection of the supported pipe considering the pipe, contents, and insulation weights. The equation is: L= [y.E.I / 22.5.W] ¼ where: L = pipe support spacing, feet, y = permissible mid-span deflection, inches E = modulus of elasticity at design temperature, lb/in (TABLE C-6) I = moment of inertia of pipe. W = weight of supported pipe, including pipe, contents, insulation, lb/ft 2. Pipe Support Span, based on stress
As a guide to the selection of support spacing, the following equation based on permissible stress is offered. The permissible mid-span deflection, y, concept is one technique commonly selected for support spacing. This technique is based on stress of supported pipe material considering the pipe, contents, and insulation weights. The equation is: L= [0.33.Z.Sh / W]1/2 where: L = pipe support spacing, feet, Z = section modulus, in³ Sh = Allowable tensile stress for pipe materialat design temp., psi W = weight of supported pipe, including pipe, contents, insulation, lb/ft. 3. Suggested Pipe Support Spacing
Flexibility Analysis – Expansion Loops & Expansion Joints
Concept of Thermal Expansion
Thermal expansion is the tendency of matter to change in volume in response to a change in temperature,through heat transfer. Temperature is a monotonic function of the average molecular kinetic energy of a substance. When a substance is heated, the kinetic energy of its molecules increases. Thus, the molecules begin moving more and usually maintain a greater average separation. Materials which contract with increasing temperature are unusual; this effect is limited in size, and only occurs within limited temperature ranges (see examples below). The degree of expansion divided by the change in temperature is called the material’s coefficient of thermal expansion and generally varies with temperature.
Providing Flexibility in Piping
As per B31.1-“ Power piping systems shall be designed to have suffi-cient flexibility to prevent piping displacements from causing failure from overstress of the piping compo-nents,
overloading of anchors and other supports, leak-age at joints, or detrimental distortion of connected equipment. Flexibility shall be provided by changes in direction in the piping through the use of fittings, bends,loops, and offsets. When piping bends, loops, and offsets are not able to provide adequate flexibility, provisions may be made to absorb piping displacements by utilizing expansion, swivel, or ball joints, or flexible metal hose assemblies”.
Minimum Leg required to absorb Thermal Expansion
Guided cantilever Method-The method can be explained using the L-bend given in below figure as an example. When the system is not constrained the point B and C will move to B’ and C’ respectively due
to thermal expansion. The end point C moves dx and dy respectively in X- and Y- directions, but no internal force or stress will be generated. However, in the actualcase the ends of the piping are always constrained as shown in fig(b). This is equivalent in moving the free expanded end C’ back to the original point C forcing the point b to move B”. the dx is the expansion from leg AB and dy from lehCB. The deformation of each leg can be assumed to follow the guided cantilever shape. This is conservative because the end rotation is ignored. The force and stress of each leg can now be estimated by guided cantilever formula. The leg AB is a guided cantilever subject to dy displacement and leg CB a guided cantilever subject to dx displacement respectively. From the basic beam theory, the moment and displacement relation of a guided cantilever is
(1) For thin wall pipes, the above equation can be further reduced. By using I= r3 t and S=M/( r2 t), the above equation (1) becomes
(2) Where,
S= thermal expansion stress , psi
E= modulus of elasticity, psi r= mean radius of the pipe, in Δ= total expansion to be absorbed, in L= length of the leg perpendicular to, in = length in feet unit, ft D= outside diameter of the pipe, in The above equation(2) is a convinent formula for the quick estimation of the expansion stress. By pre-setting E=29.0×10 psi and S=20000 psi, the above equation (3) used in finding the leg length required for steel pipes. The other can be used for the quick check is the one given in ANSI B31 Piping Codes. The code uses Equation (3) a measure of adequate flexibility, subjects to other requirements of the Code
Where,
D= outside diameter of the pipe, in
Y= resultant of total displacement to be absorbed, in L= developed length of piping between anchors, ft U= straight line distance between anchors, ft Equation (3) has to be used with great care, because the same extra length of pipe can have very different effects depending on the ways the pipe is laid out. Normally more flexibility will be achieved if the pipe is placed farther away from the elastical or geometrical center.
Types of Expansion Loops 1. Full loop This is simply one complete turn of the pipe and, on steam pipework, should preferably be fitted in a horizontal rather than a vertical position to prevent condensate accumulating on the upstream side. The downstream side passes below the upstream side and great care must be taken that it is not fitted the wrong way round, as condensate can accumulate in the bottom. When full loops are to be fitted in a confined space, care must be taken to specify that wrong-handed loops are not supplied. The full loop does not produce a force in opposition to the expanding pipework as in some other types, but with steam pressure inside the loop, there is a slight tendency to unwind, which puts an additional stress on the flanges.
Fig. 1 Full loop This design is used rarely today due to the space taken up by the pipework, and proprietary expansion bellows are now more readily available. However large steam users such as power stations or establishments with large outside distribution systems still tend to use full loop type expansion devices, as space is usually available and the cost is relatively low. 2. Horseshoe or lyre loop When space is available this type is sometimes used. It is best fitted horizontally so that the loop and the main are on the same plane. Pressure does not tend to blow the ends of the loop apart, but
there is a very slight straightening out effect. This is due to the design but causes no misalignment of the flanges.If any of these arrangements are fitted with the loop vertically above the pipe then a drain point must be provided on the upstream side as depicted in Figure 2.
Fig. 2 Horseshoe or lyre loop 3. Expansion loops
Fig. 3 Expansion loop The expansion loop can be fabricated from lengths of straight pipes and elbows welded at the joints (Figure 3). An indication of the expansion of pipe that can be accommodated by these assemblies is shown in Figure 4. It can be seen from Figure 3 that the depth of the loop should be twice the width, and the width is determined from Figure 4, knowing the total amount of expansion expected from the pipes either side of the loop. 4. Expansion Loop Sizing for Hot Piping
Figure 4
Nozzle Thermal Growth Calculations – Pumps, Vessels, Heat Exchangers.
Heat expansion doesn’t depend on the shape of the item. Shape comes into picture if the expansion is restricted. Bend the wire as shown and heat it up in free air. It expands between end points same as if you had piece of pipe from one point to the other. We can provide thermal growth by two way at Nozzle Model a rigid element
By calculating displacement values Thermal Growth at Nozzle= Coefficient of thermal X Temperature Difference X Distance from reference point With above formula thermal growth calculated along each axis.
Cold Spring
Power piping is pften installed with cold spring to control the initial hot reaction and to protect the connected equipment. However , cold springing of a restrained or a branched system is a very sophisticated procedure which can lead to an unpredictable result if it is not done properly. Cold spring , prespring and cold pull are all referring to the process which stresses the piping at the installed or cold condition in order to reduce the stress at the operating or hot condition. The
process involves laying out the piping somewhat shorter than the installing space. This creats a gap at final weld location when system is erected. The system pulled or pushed according to a predetermined procedure to close the gap and to finish the final weld. The gap is sized depending on the cold spring factor desired. A 100% cold sprung system will have the gap size equal to the amount of the system expansion minus the differential anchor movements. A 100% cold sprung system, if installed properly, will have the expansion stress reduced to zero when system reaches the operating temperature. It will be free of any thermal expansion stress under the hot operating condition. Cold spring is often applied to a piping system to 1) reduce the hot stress to mitigate the creep damage, 2) reduce the initial hot reaction force on the connecting equipment, and 3) control the movement space. Cold Springing like Expansion Joints should be a last resort. Cold Springing is difficult to accurately achieve in the field. If used the Cold Spring should be carefully inspected during installation to insure the design has been accurately implemented otherwise it is usless Flexibility Analysis using ASME B 31.1/31.3 Code Equations
Thermal Expansion Stress (Se), Code Allowable Thermal Displacement Stress Range (Sa)
B31 Approach – Design by Rules In the earlier days when knowledge was not sufficient to precisely look into many stress details, piping systems were designed with rough calculations on basic items and a lot of rules and experiments. The rules include details on junction shapes, design specifications, standard support details, limit on support spacings, operating procedures, etc. Experiments from previous operations on the finished plants are eventually all put into design specifications and/or codes. Local stress behaviors on piping components are tested with real components. Stress range concept on secondary stress and definite thermal expansion stress evaluation were eventually adopted in 1955 edition of B31 code. Allowable stresses are set lower to accommodate uncertainties. While the Design by Analysis approach was first adopted by nuclear vessels in 1963 and nuclear piping in 1969, the non-nuclear industries would like to get away as far as possible from it. The main reason is cost and availability of manpower. The cost would be unheard of in non-nuclear industries to calculate all those additional stresses in local notch, weld detail, thermal discontinuity and thermal gradient during transient and steady state operations, dynamic earthquake, fluid transient load, etc. in precise and reliable manner. The added calculations, documentations, checking, and independent review
would increase the required man-hours by 20-fold and would extend the project schedule by three times. Therefore, the old B31 approach, although not a hundred percent theoretically defendable, is the mainstay of traditional industries. B31 code does not exactly follow the stress criteria given in Table 1. Nevertheless, it does the same types of protections as given in Table 1. That is, the membrane protection and fatigue protection are properly addressed. Unless otherwise noted, the following discussions follow B31.1[6] for simplicity. Also to compare with the Design by Analysis, the items applicable to Design by Analysis are termed as the “Stress Criteria.” B31 Membrane Protection After the pressure design, which is compatible to all codes, the membrane protection is checked with the following equation:
Where, P = Design pressure h = Flexibility characteristic ( = tR/r2 for bend ) ( = 0.203 for example elbow ) i = Stress intensification factor ( = 0.9/h2/3 for bend ) ( = 2.61 for example elbow ) MA = Resultant moment due to weight and mechanical load Sh = Allowable stress at hot (design) condition = 2/3 Sy,h or smaller (use 2/3 Sy,h for simplifying comparison) For the example elbow component, Equation (a) can be written as follows:
The above requirement is not directly comparable with the Stress Criteria requirement given in Equation (AA) without some manipulation. Since both requirements has PD/4t longitudinal pressure stress term (hoop stress does not need to be included ),which can be set as 0.5 Sh = 0.333Sy,h. After substituting PD/4t = 0.333Sy,h, the requirements become
These can be rearranged to obtain the allowable moment for the example 12” Std L.R. elbow as
From the above, it is clear that for membrane protection requirement, B31 and Stress Criteria are almost identical, except that the moment Mi,L in Stress Criteria includes also operational earthquake inertia load. B31 considers the earthquake (and/or wind) as occasional load and has an increased allowable of 1.2 Sh = 0.8 S y,h. By doing the same deduction as above, the comparable allowable occasional load moment ( MA + MB ) would be equal to 0.24 S y,h Z as compared to 0.178 Sy,h Z of the Stress Criteria. MB is the resultant moment of occasional loads, such as operational earthquake or/and wind, relief valve discharge force, turbine trip loads, etc. For earthquake and wind loads, several design levels may be provided depending damage level to be tolerated. At operational level, B31 is somewhat shy of the Stress Criteria requirement. It is important to note that the sustained stress calculated in B31 for moment loads is only about one-half of the theoretical or actual stress implied by the Stress Criteria. B31 Fatigue Protection B31 fatigue protection deviates considerably from the Stress Criteria. Only thermal expansion and anchor/support displacement are included. Gross thermal discontinuity and thermal gradient are not considered. Local notches are covered by adjusting stress basis and by testing actual components. The current evaluation approaches were developed mostly by Markl and George and Markl and were officially adopted by B31 code in 1955 edition. Very little modifications have been made since then. Since thermal expansion stress is self-limiting. Its mode of failure is fatigue due to repeated operating cycles. A self-limiting stress does not cause gross structural deformation when the yield strength is exceeded. By allowing higher than yield, the thermal stresses can yield or relax at hot operating condition, resulting in stress reversal throughout the operating cycle. Therefore, the stress range throughout the operating cycle should be used for the design evaluation. Cold spring that affects only the initial stress level is not credited for improving fatigue strength. As all local stresses affect the fatigue damage, quantitative evaluation of local expansion stresses was introduced through stress intensification factors, which are derived and obtained mainly through tests. However, since there are considerable reliable theoretical stress relationships
available on bend components, the theoretical bend formulas are used as guides for establishing test data correlations and code formulas. Through strain controlled fatigue tests of piping components, Markl and his coworkers found that a pipe with an as-weld girth weld had a stress intensification of about two as compared with polished rods. In order to save the effort of identifying all the welds, and also other minor notches and clamping locations, they chose to use the pipe with girth weld as basis to establish the stress intensification factors of all components. This, in essence, cut the calculated stress in half when stress intensification factor is involved. It is important to note that the thermal expansion and anchor displacement stress calculated by B31 is actually only one-half of the real and theoretical stress. For fatigue protection, B31 requires that the following evaluation be met:
Where, SE = Thermal expansion stress h = Flexibility characteristic ( = tR/r2 for bends ) ( =0.203 for example elbow ) i = Stress intensification factor ( = 0.9/h2/3 for bends ) ( = 2.61 for example elbow ) MC = Resultant moment due to thermal expansion and anchor displacement MC = Mi,E* of Equation (BB) f = Fatigue strength reduction factor, =1 for 7000 cycles or less Sc = Basic allowable stress at ambient (cold) condition (2/3 Sy,c or less. Use 2/3 Sy,c ) Sh = Basic allowable stress at operation (hot) condition (2/3 Sy.h or less. Use 2/3 Sy,h ) SL = Sustained stress from Equation (a) For the example elbow, Equation (b) can be rewritten for f = 1.0 as
To compare with the Stress Criteria, we assume that SL take up a stress of Sh. With SL =
Sh = 2/3 Sy,h, we have
This MC can be compared with Mi,E* of the Stress Criteria. For the example elbow, the protection for plastic hinge by the Stress Criteria is
The original Stress Criteria and Section III nuclear code applies only for 700°F and below for ferrite steel and 800°F and below for austenitic steel. With this temperature limitation the difference between Sy,c and Sy,h is not very great, so Sy,c is not used in the nuclear piping code. To be comparable to B31, the original 2S y is separated into Sy,c + Sy,h. By rearranging equations (bb1) and (BB-1), we have the maximum allowable moment loading for the example elbow as
From the above allowable moment comparison, current B31 thermal expansion allowable is comparable to the Stress Criteria for plastic hinge protection at lower temperature range, but somewhat short of the Stress Criteria requirement at high temperature range.This shortcoming is somewhat compensated by applying the highest stress intensification factor to all in-plane, outplane, and torsion moments in B31.1. ( At bends, B31.3 uses smaller SIF for out-of-plane bending and no SIF for torsion ) As for actual fatigue damage evaluation, B31 relies on fatigue tests of actual components.Although gross thermal discontinuity and thermal gradients are not included, the hoop pressure stress is considered in the allowable via SL, which has a maximum of Sh the same as for hoop pressure stress. With the allowable stress limit as established in B31, Markl found that based on tests, the implied safety factors are in the order of 2 in terms of stress, and in the order of 30 ( ~ 25) in terms of cyclic life. The very least factor available,considering the 25% spread encountered between individual test data, might be estimated as 1.25 in terms of stress and 3 in terms of cyclic life. However, these safety factors were based on the allowable using the basic allowable stress as 5/8 of the yield strength. The basic allowable stress has later been increased to 2/3 of the yield strength; based on new basic allowable stresses, the adjusted safety factor in terms of stress has to be reduced by a factor of 0.94, and in terms of life reduced by a factor of 0.75. Therefore, an accurate calculation of the stress and a conservative estimate of the number of operation cycles are important. B31.3 Appendix P
In the 2004 edition of ASME B31.3, an Appendix P was added to provide “Alternative Rules for Evaluating Stress Range.” This Appendix has later been substantially revised in the 2010 edition. It uses Edwards’ [11] paper as background to layout some rules and equations to evaluate the stress range in a way quite different from the existing code. Edwards’ main theoretical basis is the Stress Criteria (shown in Table 1) developed for alternative rules for pressure vessel code and code for nuclear components. However,it appears that the stress bases of B31.3 and the Stress Criteria have been confused,leading to a set of incorrect stress calculations and unsafe stress limits. Besides the main concern of stress range evaluation, the Appendix P also concerns the stress due to axial forces. Though axial forces are routinely considered in sustained stress calculation and are important thermally for situations such as a straight run in between two restraints and at jacketed piping system, they are generally ignored in thermal expansion stress calculations. The shear forces are also generally ignored. They can be included, of course, if needed, and will not be discussed here. In fact, most piping stress computer programs do calculate Tresca stress that includes not only the axial force but also shear force and pressure hoop stress. It is just not used by the code stress evaluation. The main concerns here are the Equations (P1a) and (P1b) given in Appendix P. Equation (P1b), which is more closely related to the Stress Criteria, will be discussed first. (a) Checking for Plastic hinge or Gross Ratcheting Appendix P stipulates that the allowable thermal expansion stress range shall be
From Table 1 Stress Criteria, the thermal expansion stress range is used to check the potential plastic hinge or gross ratcheting. The allowable is given as 2Sy as shown in Equation (B) and in Equation (BB) for the example elbow component. In order to compare with Stress Criteria, we have to know how the expansion range is calculated. Disregarding the axial and shear forces, the expansion stress is calculated in B31.3 as
Where, SE = Expansion stress range, including anchor displacement, but no sustained load ii = SIF for in-plane bending moment ( = 0.9/h2/3 for bends) ( = 2.61 for example elbow ) io = SIF for out-of-plane bending moment ( = 0.75/h2/3 for bends) ( = 2.17 for example elbow )
it = SIF for torsion moment ( = 1.0 for all components ) Mi = In-plane bending moment Mo = Out-of-plane bending moment Mt = Torsion moment As discussed previously, this SE is not a real stress. It is just a reference stress about onehalf of the theoretical real stress. In order to compare with the Stress Criteria, we will use i i value also for io and it. This will soften the difference between B31.3 and the Stress Criteria and greatly simplify the comparison. By doing so, Equation (1b) becomes
This is the same form as Equation (b) used by B31.1, except the resultant moment range M R is called MC in B31.1. For the example elbow, Equation (1b-1) becomes
The allowable for SE is SEA given by Equation (P1b). Since the protection is against plastic hinge or gross ratcheting, the number of allowable cycle is theoretically zero. That is f = 1.0. Also for simplifying the comparison, Sc and Sh are assumed to be governed by the yield strength. Therefore, Equation (P1b) can be written as
Making SE < SEA, we have Appendix P requirement as
We will then compare Equations (1b-4) against Equation (BB) to see if Appendix P meets the Stress Criteria. The two equations cannot be compared directly. One way to compare is to look at the magnitude of the moments allowed in each case. By converting Equations (1b-4) and (BB) to allowable moments for the example elbow, we have
MR and Mi,E* are exactly the same and cover only the thermal expansion and anchor and/or support displacement ranges. From the above, we know Appendix P is roughly 80% over the Stress Criteria. In other words, by using Appendix P, an 80% overstressed component would still be considered acceptable.
(b) Primary Plus Secondary Stress Intensity Range As Equation (P1b) already shows that Appendix P is a poorly conceived, unsafe alternative rules to the main code, there is no need to investigate Appendix P further.However to be complete, we will also take a look at the so-called operating stress limit as given by the following Equation (P1a).
From Table 1 Stress Criteria and Equation (C), the allowable for primary plus secondary ( PL+Pb+Pe+Q ) stress is two times the yield strength. This matches the allowable of Equation (P1a). The problem is the way and the number SO is calculated. First, we already know (explained many times previously in this paper) SO calculated by B31.3 is just one-half of the actual real stress. From this point only, the allowable stress in (P1a) should be cut in half to something like 0.75 (Sc + Sh ). Second, the mode of failure of secondary stress is fatigue. All fatigue evaluations have to use stress range rather than a one-shot operation stress. This is clearly given in Table 1 as dotted line flow process. Third, only cyclic loadings or stresses cause a fatigue failure. Although Stress Criteria calls for primary plus secondary stress intensity, the dead weight is not included. On the other hand, the pressure cycling form zero to full pressure is included. For the pressure, the participation stress is the hoop stress rather than the longitudinal pressure stress as in most of so-called operation stress calculations. Fourth, there are gross thermal discontinuity and thermal gradient stresses that need to be included, but is not done in B31.3. Equation (C) also includes operation base earthquake inertia together with anchor displacement and other fluid transient loads, which are not generally included in operational stress calculations. Fifth, when so many different load types are involved, a reliable conservative combination or superimposing approach is required. For instance, the yielding and relaxation nature of expansion stress, cold spring effect, dual directional effects of earthquake, etc. have to be considered. From the above, it should be clear that Equation (P1a) itself does not have a problem. The problem is that the SO stress is not properly calculated in B31.3. It is also roughly 100% deficient, assuming the stress calculation method is correct.
WHAT IS STRESS ANALYSIS–Part 2
by varun chandel · Published September 23, 2015 · Updated February 24, 2016
FLEXIBILITY ANALYSIS – METHODS
Guided Cantilever Method: The assumptions of the method are
Each piping leg is assumed as a guided cantilever
The thermal expansion of a given leg is absorbed by the legs in perpendicular direction, which act as guided cantilever; i.e they are subjected to bending under end displacements, but no end rotation is permitted
Flexibility of elbow is not accounted
This method is applicable for
A piping system having two anchor points without any intermediate support.
This method is applicable for system with uniform pipe size.
All pipe legs should be parallel to the given coordinate system.
FORCES AND STRESS Let pipe is to be run from nozzle of vessel T1 to nozzle of vessel T2 at the same elevation.
Most economical way is to join them by a straight pipe. When temperature of vessel T1 is raised to 200 deg C and valve is opened, there will be an expansion in the connecting pipeline as calculated below. Expansion of carbon steel from 21 deg C to 200 deg C
= 2.2mm/m Total Expansion = 20 x 2.2 = 44mm To absorb this expansion one of the following things can happen. 1.As the pipe expands it will dent the sides of the vessel.
2.The pipe will buckle if the vessels are of large diameter and thickness and the pipe is small.
Another approach is to run the pipe in two different sections at right angles to each other as shown in fig.
With this configuration of piping , as the point B moves out to B’ , it is able to bend the leg BC to position B’C. Minimum bend length (BC) to absorb expansion of length AB (d) can be calculated by considering B’C as guided cantilever while restricting the stresses to a given value. MINIMUM PERPENDICULAR As per Elastic Theory, d = Pl³ / (12 EI) Where d = Movement, inches P = Force required to bend BC, lbs l = Length of BC, inches E = Young’s Modulus, lbs/in² I = Moment of inertia about bending axis, in 4 If L is length of BC in ft. (l = 12L) δ = 144PL³ / (EI) Hence,
P = EI δ/(144L³ )
Maximum bending moment at B or C, M = ±PL/2 = ft-lb Maximum bending Stress, f = (MY x 12) / I lbs/in² Y = OD of pipe/2 = D / 2 f = 12 MY / I = (12/I) x (PL/2) x (D/2) substituting P = EI δ /(144L³) f = (12/I) x (EI δ /(144L³)) x (L/2) x (D/2) i.e. f = (DE δ ) / 48L² L = √(DE δ ) / (48 f)
e.g. :- In the previous layout if we restrict the stress at 16,000 psi and consider modulus of elasticity of carbon steel as 29.5 x 106 psi and assume the pipe size as 6″ NB (6.625″ OD) and Temperature = 200 deg C Expansion of piping between T1 and T2,
This indicates that the length BC should not be less than 4.6 m. We can also calculate the stress developed in such a system of known dimensions of leg BC by the same method. δ= PL³ / (12 EI) Hence , P = 12 EI δ / L³ M = PL/ 2 =(12 EI δ / L³) X ( L/2) = 6 EI δ / L² Stress, f = M / Z = 6 EI δ / L²Z Replacing I/ Z by R f (SE) = 6 ER δ / L² Where; R = Outer radius of pipe, inches I = Moment of inertia of cross section, inΛ4 E = Modulus of elasticity,lbs/in² L = Length, inches EXAMPLE
In Fig1 if the vessels are arranged in such a way that AB and BC are equal and 10 M each, then the stress developed can be calculated as; (assume the pipe size as 6″ NB (6.625″ OD) and Temperature = 200 deg C ) l = AB = BC = 10 m = 394 inches E = 29.5 x 10Λ6 lbs/ in² R = 6.625/2 inches δ = 1.73/2 inches
= 3267 psi
NOMOGRAPH
These nomographs are developed based on guided cantilever equations. They are generally used in layout development stage as a quick reference to calculate the followings:
Perpendicular leg required to absorb expansion
Forces & Stresses at the restraint location
Forces & Stresses at the restraint location
EXAMPLE STEPS: 1.Align straight edge with nominal pipe size (pt. 1), pipe length in bending (pt. 2)& mark point on pivot line (pt. 3). 2.Align point on pivot line (pt. 3) with total thermal expansion (3.62 in) (pt. 4) & read off thermal force(1500#) (pt. 5). 3.The force against the anchor is equal to 1500# but is pushing in the opposite direction.
STEPS: 1.Align straight edge with nominal pipe size (pt. 1), pipe length in bending (pt. 2)& mark point on pivot line (pt. 3). 2.Align point on pivot line (pt. 3) with total thermal expansion (3.62 in) (pt. 4) & read off thermal force(1500#) (pt. 5). 3.The force against the anchor is equal to 1500# but is pushing in the opposite direction.
Perpendicular leg required to absorb expansion NOMOGRAPHS
Two separate nomographs for pumps and other equipments are used. To enter the nomograph, the data required is: 1. Total thermal expansion of the concerned span. 2. Allowable loading on the nozzle. 3. Type of equipment. (Pump or other to decide nomograph used). 4. NPD of pipe.
Step 1: Calculate total thermal expansion of piping span
Determine the temperature to use.
Determine the material of pipe.
Determine the thermal growth between the anchors for each global direction (north-south, east-west and up-down) using growth chart.
Step 2 : Calculate allowable loading on Nozzle.
Use equipment manufacturer’s published allowable loads.
If not available at the planning stage of the layout, make assumptions
as: Material : Carbon Steel (CS) Rotating Equipment Nozzle o Max allowed load: 200lb X Nominal nozzle size. (Up to maximum of 2000 lb). o Example: 18” 150# RF, CS nozzle: 200 X 18 = 3600 lb (Exceeds 2000 lb hence use 2000 lb).
EXPANSION LOOP : PURPOSE
To provide flexibility to the piping for expansion or contraction
Case 1:Horizontal
Case 2:Combined
Case 3:Vertical ( Not preferable)
FLEXIBILITY IN EXPANSION
POINTS FOR PIPEWAY LAYOUT
1.Provide limit stops at staggered locations for different lines. Calculate line spacing at the corners. 2.If the line spacing is wasting berthing room at the turns, determine which lines are giving the most trouble. (movement restricted to 4”) 3.Move the anchors of these lines closer to the corners. Place one or more loops between these two anchors & size the loops to fit the available space.
4.Lines requiring biggest loop should be grouped near the outside, with the lines requiring smaller loops progressing toward the center. 5.Check the anchor forces. LOCATING LOOP ANCHORS
Ideally loops shall be centered between anchors with equal legs on either side of anchor. If it isn’t practical , make legs on either side of anchor as equal as possible.
Loop width should always be based on utilizing existing supports.
Thermal expansion must be allowed for when spacing adjacent loops.
