Thin Walled Pressure Vessels

Spherical Pressure Vessel

Thin-walled pressure vessels are one of the most typical applications of plane stress.

Consider a spherical pressure vessel with radius r and wall

thickness t subjected to an internal gage pressure p.

For reasons of symmetry, all four normal stresses on a small

stress element in the wall must be identical. Furthermore, there

can be no shear stress.

The normal stresses  can be related to the pressure p by

inspecting a free body diagram of the pressure vessel. To simplify the analysis, we cut the

vessel in half as illustrated.

Since the vessel is under static equilibrium, it must satisfy

Newton's first law of motion. In other words, the stress around the

wall must have a net resultant to balance the

internal pressure across the cross-section.

Cylindrical Pressure Vessel

Consider a cylindrical pressure vessel with radius r and wall thickness t subjected to an

internal gage pressure p.

The coordinates used to describe the cylindrical

vessel can take advantage of its axial symmetry. It

is natural to align one coordinate along the axis of

the vessel (i.e. in the longitudinal direction). To

analyze the stress state in the vessel wall, a second

coordinate is then aligned along the hoop direction.

With this choice of axisymmetric coordinates, there is no shear stress. The hoop

stress h and the longitudinal stress l are the principal stresses.

To determine the longitudinal stress l, we make a cut

across the cylinder similar to analyzing the spherical

pressure vessel. The free body, illustrated on the left,

is in static equilibrium. This implies that

the stress around the wall must have a resultant to

balance the internal pressure across the cross-section.

Applying Newton's first law of motion, we have,

To determine the hoop stress h, we

make a cut along the longitudinal axis

and construct a small slice as illustrated

on the right.

The free body is in static equilibrium.

According to Newton's first law of

motion, the hoop stress yields,

Remarks

  • The above formulas are good for thin-walled pressure vessels. Generally, a pressure

vessel is considered to be "thin-walled" if its radius r is larger than 5 times its wall

thickness t (r > 5 · t).

  • When a pressure vessel is subjected to external pressure, the above formulas are still

valid. However, the stresses are now negative since the wall is now in compression

instead of tension.

  • The hoop stress is twice as much as the longitudinal stress for the cylindrical pressure

vessel. This is why an overcooked hotdog usually cracks along the longitudinal

direction first (i.e. its skin fails from hoop stress, generated by internal steam

pressure).

Pressure vesselFrom Wikipedia, the free encyclopedia

Horizontal pressure vessel in steel

A pressure vessel is a closed container designed to hold gases or liquids at a pressure substantially different from the ambient pressure.

The pressure differential is dangerous, and fatal accidents have occurred in the history of pressure vessel development and operation. Consequently, pressure vessel design, manufacture, and operation are regulated by engineering authorities backed by legislation. For these reasons, the definition of a pressure vessel varies from country to country, but involves parameters such as maximum safe operating pressure and temperature.

Contents

  [hide] 

1 History of pressure vessels 2 Pressure vessel features

o 2.1 Shape of a pressure vesselo 2.2 Construction materialso 2.3 Safety features

2.3.1 Leak before burst 2.3.2 Safety valves

o 2.4 Maintenance features 2.4.1 Pressure vessel closures

3 Uses 4 Alternatives to pressure vessels 5 Design

o 5.1 Scaling 5.1.1 Scaling of stress in walls of vessel 5.1.2 Spherical vessel 5.1.3 Cylindrical vessel with hemispherical ends 5.1.4 Cylindrical vessel with semi-elliptical ends 5.1.5 Gas storage

o 5.2 Stress in thin-walled pressure vesselso 5.3 Winding angle of carbon fibre vesselso 5.4 Operation standards

5.4.1 List of standards 6 See also 7 Notes 8 References 9 Further reading 10 External links

History of pressure vessels[edit]

A 10,000 psi (69 MPa) pressure vessel from 1919, wrapped with high tensile steel banding and steel rods to

secure the end caps.

Large pressure vessels were invented during the industrial revolution, particularly in Great Britain, to be used as boilers for making steam to drive steam engines.

Design and testing standards and a system of certification came about as the result of fatal boiler explosions.

In an early effort to design a tank capable of withstanding pressures up to 10,000 psi (69 MPa), a 6-inch (150 mm) diameter tank was developed in 1919 that was spirally-wound with two layers of high tensile strength steel wire to prevent sidewall rupture, and the end caps longitudinally reinforced with lengthwise high-tensile rods.[1]

Pressure vessel features[edit]

Shape of a pressure vessel[edit]

Pressure vessels can theoretically be almost any shape, but shapes made of sections of spheres, cylinders, and cones are usually employed. A common design is a cylinder with end caps called heads. Head shapes are frequently either hemispherical or dished (torispherical). More complicated shapes have historically been much harder to analyze for safe operation and are usually far more difficult to construct.

Theoretically, a spherical pressure vessel has approximately twice the strength of a cylindrical pressure vessel with the same wall thickness.[2] However, a spherical shape is difficult to manufacture, and therefore more expensive, so most pressure vessels are cylindrical with 2:1 semi-elliptical heads or end caps on each end. Smaller pressure vessels are assembled from a pipe and two covers. For cylindrical vessels with a diameter up to 600 mm, it is possible to use seamless pipe for the shell, thus avoiding many inspection and testing issues. A disadvantage of these vessels is

that greater breadths are more expensive, so that for example the most economic shape of a 1,000 litres (35 cu ft), 250bars (3,600 psi) pressure vessel might be a breadth of 91.44 centimetres (36 in) and a width of 1.7018 metres (67 in) including the 2:1 semi-elliptical domed end caps.

Spherical gas container.

 

Cylindrical pressure vessel.

 

Picture of the bottom of an aerosol spray can.

Construction materials[edit]

Composite overwrapped pressure vessel with titanium liner.

Many pressure vessels are made of steel. To manufacture a cylindrical or spherical pressure vessel, rolled and possibly forged parts would have to be welded together. Some mechanical properties of steel, achieved by rolling or forging, could be adversely affected by welding, unless special precautions are taken. In addition to adequate mechanical strength, current standards dictate the use of steel with a high impact resistance, especially for vessels used in low temperatures. In applications where carbon steel would suffer corrosion, special corrosion resistant material should also be used.

Some pressure vessels are made of composite materials, such as filament wound composite using carbon fibre held in place with a polymer. Due to the very high tensile strength of carbon fibre these vessels can be very light, but are much more difficult to manufacture. The composite material may be wound around a metal liner, forming a composite overwrapped pressure vessel.

Other very common materials include polymers such as PET in carbonated beverage containers and copper in plumbing.

Pressure vessels may be lined with various metals, ceramics, or polymers to prevent leaking and protect the structure of the vessel from the contained medium. This liner may also carry a significant portion of the pressure load.[3][4]

Pressure Vessels may also be constructed from concrete (PCV) or other materials which are weak in tension. Cabling, wrapped around the vessel or within the wall or the vessel itself, provides the necessary tension to resist the internal pressure. A "leakproof steel thin membrane" lines the internal wall of the vessel. Such vessels can be assembled from modular pieces and so have "no inherent size limitations".[5] There is also a high order of redundancy thanks to the large number of individual cables resisting the internal pressure.

Safety features[edit]

Leak before burst[edit]

Leak before burst describes a pressure vessel designed such that a crack in the vessel will grow through the wall, allowing the contained fluid to escape and reducing the pressure, prior to growing so large as to cause fracture at the operating pressure.

Many pressure vessel standards, including the ASME Boiler and Pressure Vessel Code and the AIAA metallic pressure vessel standard, either require pressure vessel designs to be leak before burst, or require pressure vessels to meet more stringent requirements for fatigue and fracture if they are not shown to be leak before burst.[6]

Safety valves[edit]

Example of a valve used for gas cylinders.

As the pressure vessel is designed to a pressure, there is typically a safety valve or relief valve to ensure that this pressure is not exceeded in operation.

Maintenance features[edit]

Pressure vessel closures[edit]

Pressure vessel closures are pressure retaining structures designed to provide quick access to pipelines, pressure vessels, pig traps, filters and filtration systems. Typically pressure vessel closures allow maintenance personnel.

Uses[edit]

Pressure vessels are used in a variety of applications in both industry and the private sector. They appear in these sectors as industrial compressed air receivers and domestic hot water storage tanks. Other examples of pressure vessels are diving cylinders, recompression chambers, distillation towers, pressure reactors, autoclaves, and many other vessels in mining operations, oil refineries and petrochemical plants, nuclear reactor vessels, submarine and space ship habitats, pneumatic reservoirs, hydraulic reservoirs under pressure, rail vehicle airbrake

reservoirs, road vehicle airbrake reservoirs, and storage vessels for liquified gases such as ammonia, chlorine, propane, butane, and LPG.

A unique application of a pressure vessel is the passenger cabin of an airliner; The outer skin carries both the aircraft maneuvering loads and the cabin pressurization loads.

A pressure tank connected to a water well and domestic hot water system.

 

A few pressure tanks, here used to holdpropane.

 

An expansion vessel for heating systems.

 

A pressure vessel used as a kier.

 

A pressure vessel used for The Boeing Company’s CST-100 spacecraft.

Alternatives to pressure vessels[edit]

Natural gas storage Gas holder

Depending on the application and local circumstances, alternatives to pressure vessels exist. Examples can be seen in domestic water collection systems, where the following may be used:

Gravity controlled systems[7] which typically consist of an unpressurized water tank at an elevation higher than the point of use. Pressure at the point of use is the result of the hydrostatic pressure caused by the elevation difference. Gravity systems produce 0.43 pounds per square inch (3.0 kPa) per foot of water head (elevation difference). A municipal water supply or pumped water is typically around 90 pounds per square inch (620 kPa).

Inline pump controllers  or pressure-sensitive pumps.[8]

Design[edit]

Scaling[edit]

No matter what shape it takes, the minimum mass of a pressure vessel scales with the pressure and volume it contains and is inversely proportional to the strength to weight ratioof the construction material (minimum mass decreases as strength increases[9]).

Scaling of stress in walls of vessel[edit]

Pressure vessels are held together against the gas pressure due to tensile forces within the walls of the container. The normal (tensile) stress in the walls of the container is proportional to the pressure and radius of the vessel and inversely proportional to the thickness of the walls.[10] Therefore pressure vessels are designed to have a thickness proportional to the radius of tank and the pressure of the tank and inversely proportional to the maximum allowed normal stress of the particular material used in the walls of the container.

Because (for a given pressure) the thickness of the walls scales with the radius of the tank, the mass of a tank (which scales as the length times radius times thickness of the wall for a cylindrical tank) scales with the volume of the gas held (which scales as length times radius squared). The exact formula varies with the tank shape but depends on the density, ρ, and maximum allowable stress σ of the material in addition to the pressure P and volume V of the vessel. (See below for the exact equations for the stress in the walls.)

Spherical vessel[edit]

For a sphere, the mass of a pressure vessel is

,

where:

 is mass,  is the pressure difference from ambient (the gauge pressure),  is volume,  is the density of the pressure vessel material,  is the maximum working stress that material can tolerate.[11]

Other shapes besides a sphere have constants larger than 3/2 (infinite cylinders take 2), although some tanks, such as non-spherical wound composite tanks can approach this.

Cylindrical vessel with hemispherical ends[edit]

This is sometimes called a "bullet"[citation needed] for its shape, although in geometric terms it is a capsule.

For a cylinder with hemispherical ends,

,

where

R is the radius W is the middle cylinder width only, and the overall width is W + 2R

Cylindrical vessel with semi-elliptical ends[edit]

In a vessel with an aspect ratio of middle cylinder width to radius of 2:1,

.

Gas storage[edit]

In looking at the first equation, the factor PV, in SI units, is in units of (pressurization) energy. For a stored gas, PV is proportional to the mass of gas at a given temperature, thus

. (see gas law)

The other factors are constant for a given vessel shape and material. So we can see that there is no theoretical "efficiency of scale", in terms of the ratio of pressure vessel mass to pressurization energy, or of pressure vessel mass to stored gas mass. For storing gases, "tankage efficiency" is independent of pressure, at least for the same temperature.

So, for example, a typical design for a minimum mass tank to hold helium (as a pressurant gas) on a rocket would use a spherical chamber for a minimum shape

constant, carbon fiber for best possible  , and very cold helium for best

possible  .

Stress in thin-walled pressure vessels[edit]

Stress in a shallow-walled pressure vessel in the shape of a sphere is

,

where   is hoop stress, or stress in the circumferential direction,   is stress in the longitudinal direction, p is internal gauge pressure, r is the inner radius of the sphere, andt is thickness of the cylinder wall. A vessel can be considered "shallow-walled" if the diameter is at least 10 times (sometimes cited as 20 times) greater than the wall depth.[12]

Stress in the cylinder body of a pressure vessel.

Stress in a shallow-walled pressure vessel in the shape of a cylinder is

,

,

where:

 is hoop stress, or stress in the circumferential direction

 is stress in the longitudinal direction p is internal gauge pressure r is the inner radius of the cylinder t is thickness of the cylinder wall.

Almost all pressure vessel design standards contain variations of these two formulas with additional empirical terms to account for wall thickness tolerances, quality control of welds and in-service corrosion allowances.

For example, the ASME Boiler and Pressure Vessel Code (BPVC) (UG-27) formulas are:[13]

Spherical shells:

Cylindrical shells:

where E is the joint efficient, and all others variables as stated above.

The factor of safety is often included in these formulas as well, in the case of the ASME BPVC this term is included in the material stress value when solving for pressure or thickness.

Winding angle of carbon fibre vessels[edit]

Wound infinite cylindrical shapes optimally take a winding angle of 54.7 degrees, as this gives the necessary twice the strength in the circumferential direction to the longitudinal.[14]

Operation standards[edit]

Pressure vessels are designed to operate safely at a specific pressure and temperature, technically referred to as the "Design Pressure" and "Design Temperature". A vessel that is inadequately designed to handle a high pressure constitutes a very significant safety hazard. Because of that, the design and certification of pressure vessels is governed by design codes such as the ASME Boiler and Pressure Vessel Code in North America, the Pressure Equipment Directive of the EU (PED), Japanese Industrial Standard (JIS), CSAB51 in Canada, Australian Standards in Australia and other international standards like Lloyd's, Germanischer Lloyd, Det Norske Veritas, Société Générale de Surveillance (SGS S.A.), Stoomwezen etc.

Note that where the pressure-volume product is part of a safety standard, any incompressible liquid in the vessel can be excluded as it does not contribute to the potential energy stored in the vessel, so only the volume of the compressible part such as gas is used.

List of standards[edit]

EN 13445 : The current European Standard, harmonized with the Pressure Equipment Directive (97/23/EC). Extensively used in Europe.

ASME Boiler and Pressure Vessel Code  Section VIII: Rules for Construction of Pressure Vessels.

BS 5500 : Former British Standard, replaced in the UK by BS EN 13445 but retained under the name PD 5500 for the design and construction of export equipment.

AD Merkblätter: German standard, harmonized with the Pressure Equipment Directive.

EN 286 (Parts 1 to 4): European standard for simple pressure vessels (air tanks), harmonized with Council Directive 87/404/EEC.

BS 4994 : Specification for design and construction of vessels and tanks in reinforced plastics.

ASME PVHO: US standard for Pressure Vessels for Human Occupancy.

CODAP: French Code for Construction of Unfired Pressure Vessel.

AS/NZS 1200 : Pressure equipment.[15]

AS 3788 Pressure equipment - In-service inspection API 510.[16]

ISO 11439: Compressed natural gas (CNG) cylinders[17]

IS 2825-1969 (RE1977)_code_unfired_Pressure_vessels.

FRP tanks and vessels . AIAA S-080-1998: AIAA Standard for Space Systems -

Metallic Pressure Vessels, Pressurized Structures, and Pressure Components.

AIAA S-081A-2006: AIAA Standard for Space Systems - Composite Overwrapped Pressure Vessels (COPVs).

B51-09 Canadian Boiler, pressure vessel, and pressure piping code.

HSE guidelines for pressure systems. Stoomwezen: Former pressure vessels code in the

Netherlands, also known as RToD: Regels voor Toestellen onder Druk (Dutch Rules for Pressure Vessels).

See also[edit]

American Society of Mechanical Engineers  (ASME) Bottled gas Composite overwrapped pressure vessel Compressed natural gas Demister Fire-tube boiler Gas cylinder Gasket Head (vessel) Minimum Design Metal Temperature  (MDMT) Pressure bomb  - a device for measuring leaf water

potentials Rainwater harvesting Relief valve Safety valve Shell and tube heat exchanger Vapor-Liquid Separator  or Knock-Out Drum Vortex breaker

Water well Water-tube boiler

Cylinder stressFrom Wikipedia, the free encyclopedia

Components of cylinder orcircumferential stress.

In mechanics, a cylinder stress is a stress distribution with rotational symmetry; that is, which remains unchanged if the stressed object is rotated about some fixed axis.

Cylinder stress patterns include:

Circumferential stress or hoop stress, a normal stress in the tangential (azimuth) direction; Axial stress, a normal stress parallel to the axis of cylindrical symmetry; Radial stress, a stress in directions coplanar with but perpendicular to the symmetry axis.

The classical example (and namesake) of hoop stress is the tension applied to the iron bands, or hoops, of a wooden barrel. In a straight, closed pipe, any force applied to the cylindrical pipe wall by a pressure differential will ultimately give rise to hoop stresses. Similarly, if this pipe has flat end caps, any force applied to them by static pressure will induce a perpendicular axial stress on the same pipe wall. Thin sections often have negligibly small radial stress, but accurate models of thicker-walled cylindrical shells require such stresses to be taken into account.

Contents

  [hide] 

1 Definitionso 1.1 Hoop stress

2 Relation to internal pressureo 2.1 Thin-walled assumptiono 2.2 Thick-walled vessels

3 Practical effectso 3.1 Engineeringo 3.2 Medicine

4 Historical development of the theory 5 See also

6 References

Definitions[edit]

Hoop stress[edit]

The hoop stress is the force exerted circumferentially (perpendicular both to the axis and to the radius of the object) in both directions on every particle in the cylinder wall. It can be described as:

where:

F is the force exerted circumferentially on an area of the cylinder wall that has the following two lengths as sides:

t is the radial thickness of the cylinder l is the axial length of the cylinder

An alternative to hoop stress in describing circumferential stress is wall stress or wall tension (T), which usually is defined as the total circumferential force exerted along the entire radial thickness:[1]

Cylindrical coordinates

Along with axial stress and radial stress, circumferential stress is a component of the stress tensor in cylindrical coordinates.

It is usually useful to decompose any force applied to an object with rotational symmetry into components parallel to the cylindrical coordinates r, z, and θ. These components of force induce corresponding stresses: radial stress, axial stress and hoop stress, respectively.

Relation to internal pressure[edit]

Thin-walled assumption[edit]

For the thin-walled assumption to be valid the vessel must have a wall thickness of no more than about one-tenth (often cited as one twentieth) of its radius. This allows for treating the wall as a surface, and subsequently using the Young–Laplace equation for estimating the hoop stress created by an internal pressure on a thin-walled cylindrical pressure vessel:

 (for a cylinder)

 (for a sphere)

where

P is the internal pressure t is the wall thickness r is the mean radius of the cylinder.  is the hoop stress.

The hoop stress equation for thin shells is also approximately valid for spherical vessels, including plant cells and bacteria in which the internal turgor pressure may reach several atmospheres.

Inch-pound-second system (IPS) units for P are pounds-force per square inch (psi). Units for t, and d are inches (in). SI units for P are pascals (Pa), while t and d=2r are in meters (m).

When the vessel has closed ends the internal pressure acts on them to develop a force along the axis of the cylinder. This is known as the axial stress and is usually less than the hoop stress.

Though this may be approximated to

Also in this situation a radial stress   is developed and may be estimated in thin walled cylinders as:

Thick-walled vessels[edit]

When the cylinder to be studied has a r/t ratio of less than 10 (often cited as 20) the thin-walled cylinder equations no longer hold since stresses vary significantly between inside and outside surfaces and shear stress through the cross section can no longer be neglected.

In order to calculate the stresses and strains here a set of equations known as the Lamé equations must be used.

where

A and B are constants of integration, which may be discovered from the boundary conditions

r is the radius at the point of interest (e.g., at the inside or outside walls)

A and B may be found by inspection of the boundary conditions. For example, the simplest case is a solid cylinder:

if   then   and a solid cylinder cannot have an

internal pressure so 

Practical effects[edit]

Engineering[edit]

Fracture is governed by the hoop stress in the absence of other external loads since it is the largest principal stress. Note that a hoop experiences the greatest stress at its inside (the outside and inside experience the same total strain which however is distributed over different circumferences), hence cracks in pipes should theoretically start from insidethe pipe. This is why pipe inspections after earthquakes usually involve sending a camera inside a pipe to inspect for cracks. Yielding is governed by an equivalent stress that includes hoop stress and the longitudinal or radial stress when present.

Medicine[edit]

In the pathology of vascular or gastrointestinal walls, the wall tension represents the muscular tension on the wall of the vessel. As a result of the Law of Laplace, if an aneurysmforms in a blood vessel wall, the radius of the vessel has increased. This means that the inward force on the vessel decreases, and therefore the aneurysm will continue to expand until it ruptures. A similar logic applies to the formation of diverticuli in the gut.[2]

Historical development of the theory[edit]

Cast iron pillar ofChepstow Railway Bridge, 1852. Pin-

jointed wrought ironhoops (stronger in tension than cast iron)

resist the hoop stresses.[3]

The first theoretical analysis of the stress in cylinders was developed by the mid-19th century engineer William Fairbairn, assisted by his mathematical analyst Eaton Hodgkinson. Their first interest was in studying the design and failures of steam boilers.[4] Early on Fairbairn realised that the hoop stress was twice the longitudinal stress, an important factor in the assembly of boiler shells from rolled sheets joined by riveting. Later work was applied to bridge building, and the invention of the box girder. In the Chepstow Railway Bridge, the cast iron pillars are strengthened by obvious bands of wrought iron. The vertical, longitudinal force is a compressive force, which cast iron is well able to resist. The hoop stress though is tensile, and so wrought iron, a material with better tensile strength is added.

STRESSES IN THIN-WALLED PRESSURE VESSELS

A pressure vessel is a type of container which is used to store liquids or gases under a pressure

different from the ambient pressure. Examples of pressure vessels can be diving cylinder, autoclave,

nitrogen tanks, submarine and storage vessels for liquefied gases such as LPG. Different shapes of

pressure vessels exist but most generally cylindrical and spherical shapes are used. Spherical

vessels are theoretically 2 times stronger than cylindrical ones but due to the manufacturing

difficulties, cylindrical ones are generally preferred in the industry.

A pressure vessel is assumed to be a thin wall pressure vessel when the thickness of the vessel is

less than 1/20 of its radius. [Ref-2] The walls of thin-walled pressure vessels have little resistance to

bending so it may be assumed that the internal forces exerted on a given portion of the wall are

tangent to the surface of the vessel. The resulting stress state on vessel is plane stress situation

since all stresses are tangent to surface of vessel.

The calculation tool was developed to analyze two types of vessels, cylindrical and spherical type.

According to geometric properties and pressure, principal stresses and maximum shear stress on

the surface of the vessel can be calculated. The formulas used for the calculations are given in the

List of Equations section.

Vessel type Cylindrical Spherical

INPUT PARAMETERS

Parameter Symbol Value Unit

Gage pressure of fluid pg

Vessel wall thickness t

Vessel inside radius r

Note: Use dot "." as decimal separator.

RESULTSParameter Symbol Value Unit

Hoop stress (Principal stress-1) σ1 ---

MPaLongitudinal stress (Principal stress-2) σ2 ---

Maximum shear stress (in plane) τmax(in plane) ---

Maximum shear stress (out plane) τmax(out plane) ---

Thickness to inner radius ratio t/r --- * ---

Note: * Red color :t/r > 1/20 , Green color : t/r < 1/20

10

10

100

Calculate

Definitions:

Gauge(Gage) Pressure: The pressure relative to atmospheric pressure. Eq: pg=pa-patm : pa is the

absolute pressure of the system and patm is atmospheric pressure.

Hoop Stress: Stress acts in tangential direction.  It's the 1st principal stress.

Longitudinal stress: Stress acts in longitudinal direction. It's the  2nd principal stress.

Principal Stress: Maximum and minimum normal stress possible for a specific point on a structural

element. Shear stress is 0 at the orientation where principal stresses occur.

Shear stress: A form of a stress acts parallel to the surface (cross section) which has a cutting

nature.

Supplements:

Link Usage

Yield Criteria for Ductile Material

After calculation of principal stresses on pressure vessel, yield criteria can be checked for ductile material.

Plane Stress Transformations

After calculation of principal stresses  on pressure vessel, plane stresses in different orientation can be checked .

List Of Equations:

Parameter Symbol Formula

Cylindrical pressure vessel

Hoop stress σ1 pgr/t

Longitudinal stress σ2 (pgr)/(2t)

Maximum in-plane shear stress τmax(in plane) (pgr)/(4t)

Maximum out-plane shear stress τmax(out plane) (pgr)/(2t)

Spherical pressure vessel

Hoop stress σ1 (pgr)/(2t)

Longitudinal stress σ2 (pgr)/(2t)

Maximum in-plane shear stress τmax(in plane) 0

Maximum out-plane shear stress τmax(out plane) (pgr)/(4t)

Examples:

Link Usage

Pressure Vessel

An example about the calculation of stresses on a pressure vessel, evaluation of yield criteria of material and stress transformation to find shear and perpendicular stresses on welding of the cylindrical body of the pressure vessel.

Pressure Vessel , Thin Wall Hoop and Longitudinal Stresses

Mechanics of Materials

For the thin-walled assumption to be valid the vessel must have a wall thickness of no more than about one-tenth (often cited as one twentieth) of its radius. The classic equation for hoop stress created by an internal pressure on a thin wall cylindrical pressure vessel is:

 = PDm /2t for the Hoop StressThin Wall Pressure Vessel Hoop Stress Calculator

where:

P = is the internal pressure t = is the wall thickness r = is the inside radius of the cylinder. Dm = Mean Diameter (Outside diameter - t). Mean diameter of OD and ID...  = is the hoop stress.

The hoop stress equation for thin shells is also approximately valid for spherical vessels, including plant cells and bacteria in which the internal turgor pressure may reach several atmospheres.

Inch-pound-second system (IPS) units for P are pounds-force per square inch (psi). Units for t, and d are inches (in). SI units for P are pascals (Pa), while t and d=2r are in meters (m).

Longitudinal Stress Thin Walled Pressure Vessel:

When the vessel has closed ends the internal pressure acts on them to develop a force along the axis of the cylinder. This is known as the axial or longitudinal stress and is usually less than the hoop stress.

Though this may be approximated to

Thin Wall Pressure Vessel Longitudinal Stress Calculator

Where:

P = Pressure

d = Mean Diameter (Outside diameter - t). Mean diameter of OD and ID... t = Wall Thickness

= Logitudinal Stress

Thin Wall Pressure Vessels

A thin wall pressure vessel is a container that is under pressure that is considered to have a thin wall. To classify as a thin wall pressure vessel, the inner radius r is more than or 10 times greater than the thickness t; r/t > 10. If r/t = 10 then the predicted stress can be up to 4 % less than the actual stress. This error is due to the fact that as the thickness grows in relationship to the inner radius, there will be a greater variation of stress across the thickness. However, if it is said to be a thin wall pressure vessel then the stress difference isn't significant and it can be assumed as uniform.

When a thin wall pressure vessel is under stress, there can be multiple stresses that need to be considered. The first stress is called the circumferential or hoop stress. Refer to the figure and equation 1 below.

(1)

r = radius

t = thickness

The second stress that needs to be considered is the longitudinal stress. Refer to the figure and equation 2 below.

(2)

Types of Pressure Vessels

There are different types of pressure vessels, but the two that will be discussed here are cylinders and spheres. A pressure vessel that is cylindrical in shape has both a hoop stress and a longitudinal stress. The hoop stress however is normally always two time greater than the longitudinal stress. Due to this, hoses and other cylindrical type vessels will split on the wall instead of being pulled apart like it would under an axial load. Refer to the picture below.

On the other hand pressure vessels that are spheres do not have a hoop stress. Instead they only have longitudinal stress. Due to this spherical pressure vessels can withstand larger pressures then pressure vessels that are cylinders.

LECTURE 15

Members Subjected to Axisymmetric Loads

Pressurized thin walled cylinder:

Preamble : Pressure vessels are exceedingly important in industry. Normally two types of pressure vessel are used in common practice such as cylindrical pressure vessel and spherical pressure vessel.

In the analysis of this walled cylinders subjected to internal pressures it is assumed that the radial plans remains radial and the wall thickness dose not change due to internal pressure. Although the internal pressure acting on the wall causes a local compressive stresses (equal to pressure) but its value is neglibly small as compared to other stresses & hence the sate of stress of an element of a thin walled pressure is considered a biaxial one.

Further in the analysis of them walled cylinders, the weight of the fluid is considered neglible.

Let us consider a long cylinder of circular cross - section with an internal radius of R 2 and a constant wall thickness‘t' as showing fig.

This cylinder is subjected to a difference of hydrostatic pressure of ‘p' between its inner and outer surfaces. In many cases, ‘p' between gage pressure within the cylinder, taking outside pressure to be ambient.

By thin walled cylinder we mean that the thickness‘t' is very much smaller than the radius R i and we may quantify this by stating than the ratio t / Ri of thickness of radius should be less than 0.1.

An appropriate co-ordinate system to be used to describe such a system is the cylindrical polar one r,  , z shown, where z axis lies along the axis of the cylinder, r is radial to it and is the angular co-ordinate about the axis.

The small piece of the cylinder wall is shown in isolation, and stresses in respective direction have also been shown.  

Type of failure:

Such a component fails in since when subjected to an excessively high internal pressure. While it might fail by bursting along a path following the circumference of the cylinder. Under normal circumstance it fails by circumstances it fails by bursting along a path parallel to the axis. This suggests that the hoop stress is significantly higher than the axial stress.

In order to derive the expressions for various stresses we make following  

Applications :

Liquid storage tanks and containers, water pipes, boilers, submarine hulls, and certain air plane components are common examples of thin walled cylinders and spheres, roof domes.

ANALYSIS : In order to analyse the thin walled cylinders, let us make the following assumptions :

•  There are no shear stresses acting in the wall.

•  The longitudinal and hoop stresses do not vary through the wall.

•  Radial stresses r which acts normal to the curved plane of the isolated element are neglibly small as compared to

other two stresses especially when 

The state of tress for an element of a thin walled pressure vessel is considered to be biaxial, although the internal pressure acting normal to the wall causes a local compressive stress equal to the internal pressure, Actually a state of tri-axial stress exists on the inside of the vessel. However, for then walled pressure vessel the third stress is much smaller than the other two stresses and for this reason in can be neglected.

Thin Cylinders Subjected to Internal Pressure:

When a thin – walled cylinder is subjected to internal pressure, three mutually perpendicular principal stresses will be set up in the cylinder materials, namely

•  Circumferential or hoop stress

•  The radial stress

•  Longitudinal stress

now let us define these stresses and determine the expressions for them

Hoop or circumferential stress:

This is the stress which is set up in resisting the bursting effect of the applied pressure and can be most conveniently treated by considering the equilibrium of the cylinder.

In the figure we have shown a one half of the cylinder. This cylinder is subjected to an internal pressure p.

i.e.         p = internal pressure

d = inside diametre

L = Length of the cylinder

t  = thickness of the wall

Total force on one half of the cylinder owing to the internal pressure 'p'

= p x Projected Area

= p x d x L

= p .d. L                       -------  (1)

The total resisting force owing to hoop stresses H set up in the cylinder walls

= 2 .H .L.t                 ---------(2)

Because H.L.t. is the force in the one wall of the half cylinder.

the equations (1) & (2) we get

   2 . H . L . t = p . d . L

                  H = (p . d) / 2t

Circumferential or hoop Stress (H) = (p .d)/ 2t

Longitudinal Stress:

Consider now again the same figure and the vessel could be considered to have closed ends and contains a fluid under a gage pressure p.Then the walls of the cylinder will have a longitudinal stress as well as a ciccumferential stress.

Total force on the end of the cylinder owing to internal pressure

= pressure x area

= p x d2 /4

Area of metal resisting this force = d.t. (approximately)

because d is the circumference and this is multiplied by the wall thickness

Hoop, Axial and Radial Stresses in Thick-Walled Pressure VesselsCreated using ANYS 14.0

Problem SpecificationConsider the following pressurized thick-walled hydraulic cylinder. The following figure shows a section through the mid-plane. 

Stress directions in cylindrical coordinates:

  

σhoop is in the circumferential direction (out of the plane here)

a = inner radius = 1.5 in

b = outer radius = 2 in

Assume the cylinders are 18 inches long and the vessel is pressurized to 1000 psi. Here, we will be interested in finding the hoop, axial and radial stresses at the mid-length of the cylinders (@ 9 inches), to neglect the local effects of the end caps.

Compare the finite element results obtained from axisymmetric analysis to those calculated with the theoretical formulae for both thin-wall and thick-wall approximations.

Note: For this problem, the material choice will not affect the stresses; it will only affect the displacements and strains. 

Learning GoalsThe purpose of this tutorial is to showcase, in a relatively simple situation, where thin-wall pressure vessel theory is no longer as valid as it is in the limit of large radius-to-thickness ratios. The point is that inadequate theory should not be used for validation purposes in the limit that the physical assumptions on which the theory is based break down. In this problem, this happens gradually as the vessel walls become thicker. This tutorial is meant to highlight where it is relatively straightforward to apply axisymmetric FEA and resolve a solution correctly that disprove analytical treatment with simple formulae derived for thin-walled vessels.

Pre-Analysis & Start-Up

Pre-AnalysisThe equations for stresses in thin- and thick-wall cylinders can be found in many mechanics of materials references, and are summarized here, with a = inner radius, b = outer radius, r = radial position where stress is to be found, and t = wall thickness.

Notice that in thick-wall theory, the hoop stress varies with the radial position, while the stress is assumed to be constant in thin-wall theory. Comparing the substitution of a and b for r in the hoop stress thick-wall equation will convince you that stress is greater on the inner surface. The hoop stress variation in thick-walled vessels can be depicted as follows (the view shown corresponds to looking from above the pressure vessel):

By using the parameters given in the problem statement and the above formulae for hoop stress, we find that the maximum hoop stresses using the thin-wall and thick-wall approximations yield 3000 psi and 3571 psi, respectively. This corresponds to a 16% difference which tells us that the thin wall theory might not be adequate for this geometry. Thin-wall theory actually gives good results when b/a ratio is less than 1.10, and that is not the case here.

Notice that the axial stresses are constant for both theories since they do not depend on radius. For this example, the thin-wall and thick-wall approximations yield 1500 psi and 1285 psi, respectively. 

The radial stresses at the inner and outer surfaces can be deduced from the boundary conditions:

The radial stress at the outer surface is 0 psi since the traction is zero at a free surface. The radial stress at the inner surface is -1000 psi since it has to equal the applied

normal traction (radial direction is also the normal direction here). The negative sign indicates that the applied traction is compressive. 

The following tables display the results of these approximations: 

We will fill in the missing information by performing an axisymmetric analysis using ANSYS. We will then compare the numerical results from ANSYS to the analytical results.  

GeometryIcon

For users of ANSYS 15.0, please check this link for procedures for turning on the Auto Constraint feature before creating sketches in DesignModeler.

For axisymmetric models, we use cylindrical polar coordinates (r, θ, z) with no variation in the θ direction. So we can just model a slice in the (r, z) plane as shown below.

 

In ANSYS, the radial direction is x (rather than r) and the axial direction is y (rather than z). Confusing! We recommend that in an axisymmetric analysis, you think of the directions in ANSYS as radial & axial rather than “x” and “y”.

Note that for axisymmetric models in ANSYS, the y-axis is always the axis of symmetry. The corresponding 3D geometry can be generated by revolving the 2D section 360° about the y-axis (we’ll do this later in the Numerical Results step).

 

Below is the 2D geometry we need to model:

 We use symmetry to model only half the total length of the cylinder.

ANSYS Learning ModulesSkip to end of metadata

Created by Steve Weidner, last modified by Sebastien Lachance-Barrett on Dec 31, 2014

Go to start of metadata

List of Learning Modules

Each learning module below contains a step-by-step tutorial that shows details of how to solve a selected problem using ANSYS, a popular tool for finite-element analysis (FEA). The tutorial topics are drawn from Cornell University courses, the Prantil et al  textbook, student/research projects etc. If a tutorial is from a course, the relevant course number is indicated below. All tutorials have a common structure and use the same high-level steps starting with Pre-Analysis and ending with Verification and Validation . Pre-Analysis includes hand calculations to predict expected results while Verification and Validation can be thought of as a formal process for checking computer results. Both these steps are extremely important in practice though often

overlooked. The pedagogical philosophy behind these modules is discussed in this article from the ANSYS Advantage magazine.

Finite Element Analysis Using ANSYS Mechanical: Results-Interpretation

The following ANSYS tutorials focus on the interpretation and verification of FEA results (rather than on obtaining an FEA solution from scratch). The ANSYS solution files are provided as a download. We read the solution into ANSYS Mechanical and then move directly to reviewing the results critically. We are particularly interested in the comparison of FEA results with hand calculations.

Tensile Bar MAE 3250 Static Structural

Plate With a Hole MAE 3250 Static Structural

Bending of a Curved Beam         

MAE 3250             Static Structural   

 

Finite Element Analysis Using ANSYS Mechanical

The following ANSYS tutorials show you how to obtain an FEA solution from scratch using ANSYS Mechanical. 

Introductory Tutorials

Plate With a Hole MAE 3250/ MAE 4700- 5700

Static Structural

Bike Crank MAE 3250/ MAE 3272

Static Structural

Bike Crank: Part 2 MAE 3272 Static Structural

Cantilever Beam MAE 4700- 5700 Static Structural

Plane Frame MAE 4700- 5700 Static Structural

A stepped shaft in axial tension

Prantil et al textbook

Static Structural

A non-slender cantilever beam under point tip loading

Prantil et al textbook

Static Structural

Hoop and axial stresses in thick-walled pressure vessels

Prantil et al textbook

Static Structural

A four-point bend test on a T-beam

Prantil et al textbook

Static Structural   

Planar approximations for a two-dimensional beam analysis

Prantil et al textbook

Static Structural

Three-dimensional analysis of combined loading in a signpost

Prantil et al textbook

Static Structural

Plate With a Hole: Optimization

MAE 3250/ MAE 4700- 5700

Optimization

Heat Conduction in a Cylinder

MAE 4700- 5700 Heat Transfer

2D Steady Conduction in a Rectangular Domain

MAE 3240/ MAE 6510

Heat Transfer

Cantilever Beam Modal Analysis

MAE 4700- 5700 Dynamics

Modal Analysis of a Wing

  Dynamics

Advanced Tutorials

High Resolution FE Model of Bone

MAE 6640 Static Structural

Hertz Contact Mechanics

Undergrad Project

Static Structural

Wind Turbine Blade M.Eng Project Static Structural

Stress due to Gravity   Static Structural

Advanced FEA for Large Telescope Truss

CCAT Telescope Project

Static Structural 

Crack Between Neo-Hookean Material and Rigid Body

MAE 5700 Static Structural 

Wind Turbine Blade FSI (Part 2)

MAE 4020- 5020

Static Structural, FSI

Linear Column Buckling

  Structural

Thermal Stresses in a Bar

  Coupled Static  Structural and Heat Transfer

Transient 2D Conduction

  Heat Transfer

3D Conduction   Heat Transfer

Radiation Between Surfaces

  Heat Transfer

Modal Analysis of a Satellite

Cornell CubeSat Team

Dynamics

Modal Analysis of a Composite   Monocoque

Cornell Formula SAE team

 

 

 

Tips and tricks

Tips and Tricks                           

 

Finite Element Analysis Using ANSYS APDL (These tutorials are no longer being updated)

Two-Dimensional Static Truss

ANSYS 11.0 12.0 APDL

Basic

Plate with a hole ANSYS 11.0 12.0 APDL

Basic

Three-dimensional bicycle crank

ANSYS 12.0 APDL Intermediate

Three-dimensional curved beam

ANSYS 11.0 APDL Intermediate

Vibration analysis of a frame ANSYS 7.0 Intermediate

Semi-monocoque shell ANSYS 10.0 APDL Intermediate

Semi-monocoque shell, Part 2: Parametric study

ANSYS 10.0 APDL Intermediate

Orthotropic plate with a hole ANSYS 11.0 12.0 APDL

Intermediate

Disks in point contact ANSYS 7.1 Classic Intermediate

Frequently Asked Questions

About the ANSYS learning modules

This ANSYS short course consists of a set of learning modules on using ANSYS to solve problems in solid mechanics. The learning modules lead the user through the steps involved in solving a selected set of problems using ANSYS. We not only provide the solution steps but also the rationale behind them. It is worthwhile for the user to understand the underlying concepts as she goes through the learning modules in order to be able to correctly apply ANSYS to other problems. The user would be ill-served by clicking through the learning modules in zombie-mode. Each learning module is followed by problems which are geared towards strengthening and reinforcing the knowledge and understanding gained in the learning modules. Working through the problem sets is an intrinsic part of the learning process and shouldn't be skipped.

These learning modules have been developed by the Swanson Engineering Simulation Program in the Sibley School of Mechanical and Aerospace Engineering at Cornell University. The Swanson Engineering Simulation Program has been established with the goal of integrating computer-based simulations into the mechanical engineering curriculum. This program has been endowed by Dr. John Swanson, the founder of ANSYS Inc. and an alumnus of the Sibley School. The development of these learning modules is being supported by a Faculty Innovation in Teaching award from Cornell University.

What is ANSYS?

ANSYS is a finite-element analysis package used widely in industry to simulate the response of a physical system to structural loading, and thermal and electromagnetic effects. ANSYS uses the finite-element method to solve the underlying governing equations and the associated problem-specific boundary conditions.

Stability of thin-walled high-pressure vessels subjected to uniform corrosion

AbstractThe stressed state in real metal constructions changes in the process of operation even

under permanent external loading. It takes place due to changes in the cross-sections of the

loaded elements, resulting from the surface corrosion. This paper proposes a method for

determining the critical time of stability loss in thin-walled high-pressure vessels subjected

to uniform corrosion from the inside. The method is based upon the model of a thin elastic

cylindrical shell. It is shown that this critical time can be established if the solution of the

respective problem of the static stability loss for the vessel, not subjected to corrosion, and

the law of corrosion rate change are known. Several special cases of the law of corrosion

rate are examined.

Thin-walled pressure vessels and method of manufactureUS 3184092 AABSTRACT  available in

IMAGES(1)

 

CLAIMS  available in

DESCRIPTION  (OCR text may contain errors)

May 18, 1965 H. J. c. GEORGE THIN-WALLED PRESSURE VESSELS AND METHOD OF

MANUFACTURE Filed Sept. 9. 1960 INVENTOR. HENRI d. C. GEORGE W mm mm B W W n m

ATTORNEYS United States Patent 3,184,092 THIN-WALLED PRESSURE VESSELS AND METHDD 9F

MANUFACTURE Henri J. C. George, Paris, France, assignor to Quartz &

Silice S.A., Paris, France, a corporation of France Filed Sept. 9, 1960, Ser. No. 55,080

Claims priority, application France, Sept. 10, 1959,

804,785, Patent 1,243,920 1 Claim. (Cl. 220-3) This invention relates to thin-walled pressure vessels and

the manufacture thereof, and more particularly to improved means for reinforcing a pressure vessel

comprising a thin wall of metal or other material which is to be used under high temperature conditions.

Many metals used in making hollow pressure vessels, even those having high melting points, exhibit a

rapid decrease in mechanical properties with rising tempera ture, particularly with regard to elasticity and

yield point. The material of an unrcinforced metallic pressure vessel must therefore be of sufiicient

thickness to retain the design pressure on the basis of considerably lowered elasticity and yield point

when the vessel is intended to be subjected to greatly elevated temperatures in use.

It is the primary object of the present invention to provide an improved reinforced thin-walled pressure

vessel, which will maintain relatively high mechanical strength under high temperature operating

conditions, in spite of reduction in the mechanical strength of the material of the vessel.

It is a further object of this invention to provide an improved method for manufacturing a reinforced

thinwalled pressure vessel.

Further objects and advantages of the invention will become apparent as the following description

proceeds.

According to a preferred embodiment of the present invention, a thin-walled tube, receptacle, or other

pressure vessel is formed of thin metal, and is subsequently wound circumferentially with one or more

windings of fine quartz filaments which are subjected to a high tension, which is preferably the maximum

tension compatible with the yield point of the filaments. Because quartz has a considerably lower

coetficient of expansion than metals, the metal will expand more rapidly than the quartz filaments with a

rise in temperature, and the compression caused by the winding will increase. A very substantial increase

in the mechanical strength of the pressure vessel is provided even at these high temperatures. Quartz

maintains its elastic properties even at temperatures of 1000 C., at which most metals become relatively

soft. The resulting pressure vessel is thus capable of withstanding high pressures at relatively higher

temperatures than those which can be contained by comparable metal pressure vessels of equal or even

greater thickness, particularly at elevated temperatures.

The quartz filaments may have widely varying diameters, within a range which is compatible with good

winding characteristics. The most convenient range of diameters falls between one and about thirty

microns; thinner filaments are difficult to manufacture and to wind, while thicker filaments have higher

fragility and are also quite difiicult to wind.

According to an additional feature of this invention, the winding of the filament about the pressure vessel

can be accomplished concurrently with the drawing of the filament from a heated quartz rod, by

employing the pressure vessel as a drawing drum. The vessel is rotated to draw the filament under

tension from a crucible in which the heated quartz rod is maintained at a drawing temperature.

Alternatively, the filament may be drawn onto a drawing drum and then rewound under tension onto the

pressure vessel.

3,184,92 Patented May 18, 1955 "ice While the specification concludes with claims particularly pointing

out and distinctly claiming the subject matter which I regard as my invention, it is believed that the

invention will be better understood from the following detailed description taken in connection with the

accompanying drawing, in which:

FIG. 1 is a view showing a method of reinforcing a pressure vessel according to the invention;

FIG. 2 is a sectional view taken along line 2-2 in FIG. 1, looking in the direction of the arrows;

FIG. 3 is a schematic view showing a modified method of reinforcing a pressure vessel; and

FIG. 4 is a schematic view showing a further modification of the improved method.

Referring to the drawing, a metallic pressure vessel 1 is shown by way of illustration, and is formed with

hemispherical end walls 2 joining a cylindrical wall 3 to enclose a pressure chamber 4- which is intended

to contain gases under high pressures at elevated temperature conditions. Such a vessel has good

strength characteristics at low temperatures, but these characteristics diminish rapidly with increasing

temperature, particularly with regard to the elasticity and yield point of the metal. A metal pressure vessel

to be used at elevated temperatures on the order of hundreds of degrees centigrade must therefore be

made with relatively thick and heavy walls.

According to the method of this invention, I wind the vessel 1 with a filament 5 of quartz to form a winding

6 about at least the cylindrical portion of the vessel, and carry the winding on under a substantial tension.

The winding may be subjected to the maximum tension compatible with the yield point of the filament, and

I prefer to carry on the winding under a filament tensile stress at least within the range of to 300 kilograms

per square millimeter. A winding 2 millimeters thick produces a tangential compression force on the

vessel in the order of to 400 kilograms per millimeter of length of the vessel. While the winding may be

formed of a single layer, I prefer to wind the filament at least several layers thick, depending upon the

strength required in the vessel.

The win-ding is carried on by rotatably mounting the vessel 1 on the axis shown by any suitable shaft and

bearing means (not shown), and rotating the vessel to wind the filament thereon in a helical conformation.

The filament 5 is supplied by means of a reel 7 rotatably supported upon an axle 8 and constrained by

suitable tension braking means of any well-known type (not shown) to provide the desired winding stress.

The spool 7 is prepared in a conventional manner by drawing the filament from a suitably heated quartz

rod, using the spool as a drawing drum. In an alternative method which is a feature of this invention,

however, the filament may be drawn directly onto the vessel 1 from a heated quartz rod, the latter being

suitably heated within a crucible according to conventional practice. The method of directly winding and

drawing the filament onto the vessel is illustrated in alternative forms in FIGS. 3 and 4. In FIG. 3, a vessel

1 is shown mounted for rotation in the direction shown by the arrow upon a shaft 10, and arranged to

draw a quartz filament 12 from a suitably sup orted quartz rod 14, which is heated by a torch 16 to a

fusing temperature. In FIG. 4, a vessel 1" is supported upon a shaft 18 for rotation in the direction shown

by the arrow to draw a filament 20 from a molten charge of quartz 22, contained within a suitably heated

crucible 24, the bottom of which has an orifice 26 to form the filament in a manner well known in the art. I

have found that the direct drawing of the filament onto the vessel may afford a tensile strength as much

as two or three times greater than that of filaments which have been wound on a spool and subsequently

rewound on the vessel. Upon completion 3 of the winding operation, an end or" the filament is fused to

the winding by local heating means, such as a torch.

The diameter of the quartz filament may vary within a Wide range, but should be compatible with good

winding characteristics. A particularly convenient range of diameters is from one to thirty microns.

Filaments of lesser diameter are difficult to prepare and wind, while filaments of greater diameter are

relatively fragile and also present difficulty in winding.

I have found that my improved pressure vessel provides greater resistance to bursting than a metal

pressure vessel having a wall thickness equal to the total thickness of the metal wall and the quartz

filament winding of the pressure vessel of this invention. Furthermore, my improved pressure vessel

preserves its strength at high temperatures, because the quartz filament maintains its elasticity and

strength up to temperatures which may reach l000 Q, at which most metals become relatively soft.

Under high temperature conditions, the metal or other material of the pressure vessel 1, having a higher

edeflicient of expansion than the quartz, expands more rapidly and increases the compression of the

quartz winding on the vessel. The compressional strength of my improved pressure vessel is thus

increased with rising temperature, in a range of temperatures lower than the melting point of the metal or

other material of the vessel.

It will be apparent to those skilled in the art that the vessel may assume any desired shape, and is not

limited to a cylindrical vessel. Furthermore, the ends as well as the side wall of the vessel may be wound

with the quartz filament.

Various other changes and modifications will occur to those skilled in the art, and I intendto cover all such

changes and modifications in the appended claim.

What I claim and desire to secure by Letters Patent of the United States is:

A pressure vessel capable of maintaining high bursting strength at temperatures in the order of about

1000 (3., comprising a hollow metallic vessel which would normally be weakened at temperatures in the

order of about 1000 C. to a point where said metallic vessel would rupturc under its designed pressure

limit, said metallic vessel having a coefiicient of thermal expansion substantially greater than that of

quartz, said pressure vessel having a plurality of windings of a quartz filament wound under a tension less

than the yield point of said filament, said windings maintaining their tensile strength at a tem- 1 perature in

the order of about 1000 C. and restraining thermal expansion of said metallic vessel.

References Cited by the Examiner UNITED STATES PATENTS THERON E. CONDON, Primary

Examiner.

EARLE I. DRUMMOND, Examiner.

PATENT CITATIONS

Cited Patent

Filing datePublication

dateApplicant Title

US2405036 * Oct 1, 1941 Jul 30, 1946Linde Air Prod Co

Method of and apparatus for making glass products, such as fibers and rods

US2569612 * Oct 27, 1945 Oct 2, 1951Pont A Mousson Fond

Manufacture of reinforced concrete pipes

US2579183 * Jun 8, 1945 Dec 18, 1951Eugene Freysainet

Method for tensioning reinforcements

US2652943 * Jan 9, 1947 Sep 22, 1953Williams Sylvester Vet

High-pressure container having laminated walls

US2744043 * Jan 23, 1950 May 1, 1956 Fels & Company

Method of producing pressure

Cited Patent

Filing datePublication

dateApplicant Title

containers for fluids

US2827195 * Jul 7, 1954 Mar 18, 1958Thomas F Kearns

Container for high pressure fluids

US2848133 * Oct 28, 1954 Aug 19, 1958Einar M Ramberg

Pressure vessels and methods of making such vessels

US2984868 * Mar 20, 1958 May 23, 1961Engelhard Ind Inc

Method of making fused quartz fibers

US3045278 * Apr 3, 1959 Jul 24, 1962Engelhard Ind Inc

Fiber forming torch

Thin-walled Pressure Vessels  Key Concepts:  A pressure vessel is generally a "thin˗walled" structure  (the ratio of radius to thickness is large)  subject to internal pressure, p.  The pressure produces normal stressesin the plane of the structure which are determined from equilibrium of an element. 

In a Nut Shell:  Thin-walled pressure vessels store and transport gases or liquids underpressure such as pipelines, water towers, silos, and tanks.  For example, compressors store air pressure in tanks used at gas stations for tire inflation.  Pressure vessels include: 

                               Spherical Pressure Vessels                             Cylindrical Pressure VesselsCapped Pressure Vessels – Cylinder capped at each end by a hemisphere

If   r  is the inner radius of the sphere or cylinder and  t  is the wall thickness, then thesphere or cylinder is considered to be “thin” provided   r/t  ≥  10.First consider the case of a thin-walled spherical pressure vessel with wall thickness, t, internal radius, r, and internal pressure, p, as shown below.

                                                 σ1  =  σ2  =  pr/2t 

where                 σ1,  σ2  are the normal stresses on the outer surface in (lb/in2), (lb/ft2), (N/mm2), (N/m2)                  p  is the internal pressure  in (lb/in2), (lb/ft2), (N/mm2), (N/m2)                r   is the internal radius of the spherical pressure vessel in (in), (mm), etc                t   is the wall thickness of the spherical pressure vessel in (in), (mm), etcNote:  The normal stresses, σ1 and σ2 are the same in any direction tangent to the outer surface and constant throughout the thickness.  They are the principal stresses at every point. Click here to continue with discussion of a thin-walled, cylindrical pressure vessel.

Thin-walled Pressure Vessels  (continued)

Next consider the case of a thin-walled cylindrical pressure vessel with wall thickness, t,  inner radius, r, and internal pressure, p,  as shown below.

                        

       σ1  =  pr/t    (hoop stress)      σ2  =  pr/2t   (longitudinal stress) 

where       

                  σ1  is the hoop stress on the outer surface in (psi), (N/mm2), etc                σ2  is the longitudinal stress on the outer surface in (psi), (N/mm2), etc                   p  is the internal pressure  in (lb/in2), (lb/ft2), (N/mm2), (N/m2)                   r   is the internal radius of the cylindrical pressure vessel in (in), (mm),etc                   t   is the wall thickness of the cylindrical pressure vessel in (in), (mm),etc Note:  The normal stress, σ1 and σ2 are constant throughout the thickness. Note further:  In the case of a cylindrical pressure vessel the hoop and longitudinal stresses on outer surface of each face of are the principal stresses.  However an element rotated by 45o will carry the maximum shear stress as well as a normal stress as seen on a Mohr’s Circle.

 Click here for discussion of the use of Mohr’s Circle for pressure vessels.

Thin-walled Pressure Vessels  (continued)

Recall Mohr’s Circle provides a convenient way to determine the normal and shear stresseson any face on the outer surface of a pressure vessel.Mohr’s Circle for a spherical pressure vessel for in-plane stresses on the outer surface ofthe cylinder is as follows:

                    where   σ1  =   σ2  =   pr/2t          Note:  σ1  and  σ2  are the principal stresses and are the same in any direction for anyelement on the outer surface of the spherical pressure vessel.  There are no in-plane shearing stresses on any element on the outer surface of the sphere.

 Click here for Mohr’s Circle involving cylindrical pressure vessels. 

Thin-walled Pressure Vessels  (continued)

Recall Mohr’s Circle provides a convenient way to determine the normal and shear stresseson any face of the element.  Consider the following cylindrical pressure vessel of inner radius, r, thickness, t, and with internal pressure, p.

                              Mohr’s Circle for a cylindrical pressure vessel for in-plane stresses on the outer surfaceof the cylinder is as follows:

                                          where   σ1  =  pr/t (circumferential)  and   σ2  =   pr/2t    (longitudinal)  Note:  σ1  and  σ2  are the in-plane principal stresses.   The maximum in-plane shearingstress is rotated 45o from the faces of the element shown above.  The value of the maximumin-plane shearing stress for an element on the outer surface of the cylinder is just    

                               τ max   =    ( σ2  –  σ1 ) / 2  =  pr/4t    Click here for examples.

Pressure Vessels 

Example: The cylindrical, steel tank shown below is under a gage pressure of 1.5 MPa.  Itsinner diameter is 750 mm with a wall thickness of 9 mm.  The seams forming the tank usingbutt welds are at an angle of  θ with the longitudinal axis of the tank.  Determine the normalstress perpendicular to the weld and the shearing stress parallel to the weld for  θ = 60o. 

                                                            Strategy:  Use the expression  σC = pr/t and σL = pr/2t to determine the circumferential andlongitudinal, normal components of stress on an element oriented along the cylinder as shownbelow.  Then use Mohr’s circle to find the components of stress for an element oriented alongand perpendicular to the weld seam. 

                            σC =  pr/t   =  (1.5)(375)/9  =  62.5 MPa σL =  pr/2t  =  (1.5)(375)/(2)(9) = 31.25 MPa Next construct Mohr’s Circle for this element in plane stress. Use it to determine the normal and shearing stresses on the face of an element rotatedso that a face of the element aligns with the weld.  Remember that rotation of an elementthrough an angle  θ  is equivalent to rotation through an angle  2θ on Mohr’s Circle withboth rotations in the same direction.   Click here to continue with this example.

Pressure Vessels Example:  (continued) Identify “face A” and “face B” on the element oriented along and perpendicular to the axis ofthe cylinder.  Then show the circumferential and longitudinal stresses on each face. 

 Plot Mohr’s Circle.  Rotate the element so it aligns with the weld line as shown below. Note “face A” becomes “face A’ “ along the weld line with a rotation of 30o on the elementor 60o on Mohr’s Circle. 

                    The radius of the circle is   r  =  (62.5 – 31.25)/2  =  15.625 MPa The center of the circle is at   (31.25 + 15.625), 0)  =  (46.875,0) The shearing stress on face A’ along the weld line is then  15.625 sin(60)  =  13.53 MPa  The normal stress on face A’ perpendicular to the weld line is then   46.875 – 15.625 cos(60) = 39.1 MPa Suppose the allowable shearing stress in the cylindrical pressure vessel without failure is 15 MPaand instead of 60o the weld line is at 45o to the longitudinal axis of the vessel.  For the sameinternal pressure and wall thickness will the tank rupture?    Yes, since the shearing stress on aface along this new weld line is 15.625 MPa which exceeds the allowable shearing stress.                          Click here for another example.        

Pressure Vessels *Example: A spherical pressure vessel has an I.D. of 220 mm, a wall thickness of

5 mm, and an internal pressure of 4.0 MPa.  Find the maximum in-plane shearingstress and the maximum absolute shearing stress in the pressure vessel.  Strategy:  Use the expression  σ1  =  σ2  =  pr/2t to determine the  normal components of stresson an element of the sphere as shown below.  Then use Mohr’s circle to find the componentsof stress at any point on the sphere.                  σ1  =  σ2  =  pr/2t   =  (4.0)(110)/(2)(5)  =  44.0 MPa Mohr’s Circle for this element is shown below. 

              For “in-plane” stress Mohr’s Circle is just the common point with concident normal stressesσ1 and σ2 .  So the “in-plane” shearing stress is   (σ1 – σ2 )/ 2  =  0 MPa. For an element with internal pressure acting on a face of the element, the Mohr’s Circle has adiameter of  σ1 – ( – p)  =  σ1 +   p .   So Mohr’s Circle for “out-of-plane”  surfaces is a circlewith radius  r  =  (σ1 +   p )/ 2 .   Thus the maximum absolute shearing stress                         τmax  =  ( 44 +  4 ) / 2  =  24 MPa.   (result) Note:  If the “in-plane” surface is the xy-plane, then the “out-of-plane” surfaces are theplanes in xz and yz surfaces.

PRESSURE VESSELS

Pressure vessels have two stresses to check - hoop stress and axial stress. Hoop stress is always twice the axial stress.

STRESS IN A THIN-WALLED PRESSURE VESSEL

Hoop Stress H (MPa)

H = PD/2t 

Axial Stress A (MPa)

A = PD/4t 

Where: P = pressure of the fluid (MPa)D = diameter of the tank (mm). This is the inside diameter, but with a "thin-wall" it doesn't matter. t = thickness of tank wall. (mm) Should be pretty small compared to D. 

 

 

Here is a typical pressure vessel made of welded steel. The cylindrical portion is made from flat plate rolled into a circle and then welded (Joints A). These joints must withstand the hoop stress.

The vertical joints (Joints B) handle only axial stresses, which is half the hoop stress.

The ends are pressed to form a partially spherical shape (a complete hemisphere would have 1/2 the stress of the cylinder - so they make an ellipsoid, which also happens to be much easier to press into shape).

PipesPipes, hoses and tubes under pressure are another type of pressure vessel. Being cylinders, they will fail in hoop stress. Reinforced flexible hoses have cross-ply fibres that absorb both hoop and axial forces.

Hydraulic hose assembly undergoing a pressure test to failure. Hoses typically have a safety factor of 4. This 2 inch (50mm) hose is rated at 420 bar (42 MPa) and reached 1650 bar (165MPa) before it failed...like this:

Bird's nest failure of hydraulic hose. This is typical for an overpressure (burst) failure, the ruptured fibres release the pressurized oil with considerable energy (explosively). At the same pressures, compressed air would be even more dangerous than pressurized oil because air expands, creating an even more explosive effect. In hydraulics, the level of danger is in proportion to how much fluid is constrained elastically - in flexible hoses and accumulators. A large diameter, long hose will explode more violently than a small, short one, which is one reason to use solid steel tube wherever possible.

Whiteboard

SUMMARY

 

MECHANICS - THEORY

 Thin-walled Pressure Vessels

Cylindrical Pressure Vessel withInternal Pressure

 

Both cylinderical and spherical pressure vessels are common structures that are used ranging from large gas storage structures to small compressed air tanks in industrial equipment. In this section, only thin-walled pressure vessels will be analyzed.

A pressure vessel is assumed to be thin-walled if the wall thickness is less than 10% of the radius (r/t > 10). This condition assumes that the pressure load will be transfered into the shell as pure tension (or compression) without any bending. Thin-walled pressure vessels are also known as shell structures and are efficient storage structures.

If the outside pressure is greater than the inside pressure, the shell could also fail due to buckling. This is an advanced topic and is not considered in this section.

    

   Cylindrical Pressure Vessels

Cylindrical Vessels will Expierence Both Hoop and Axial Stress in

the Mid-section

 

Only the middle cylindrical section of a cylinder pressure vessel is examined in this section. The joint between the end caps and the mid-section will have complex stresses that are beyond the discussion in this chapter.

In the mid-section, the pressure will cause the vessel to expand or strain in only the axial (or longitudinal) and the hoop (or circumferential) directions. There will be no twisting or shear strains. Thus, there will only be the hoop stress, σh and the axial stress, σa. as shown in the diagram at the left.

    

Cross Section Cut ofCylindrical Vessel

 Pressure vessels can be analyzed by cutting them into two sections, and then equating the pressure load at the cut with the stress load in the thin walls. In the axial direction, the axial pressure from the discarded sections will produce a total axial force of p(πr2) which is simply the cross section area times the internal pressure. It is generally assumed that r is the inside radius.

The axial force is resisted by the axial stress in the vessel walls which have a thickness of t. The total axial load in the walls will be σa(2πrt). Since the cross section is in equilbrium, the two axial forces must be equal, giving

     p(πr2) = σa(2πrt)

This can be simplified to

   

where r is the inside radius and t is the wall thickness.    

Hoop Section Cut from Cylindrical Vessel

 

In addition to the axial stress, there will be a hoop stress around the circumference. The hoop stress, σh, can be determined by taking a vertical hoop section that has a width of dx. The total horizontal pressure load pushing against the section will be p(2r dx) as shown in the diagram.

The top and bottom edge section will resist the pressure and exert a load of σh(t dx) (each edge). The edge loads have to equal the pressure load, or

     p(2r dx) =σh(2t dx)

This can be simplified to

   

where r is the inside radius and t is the wall thickness.    

   Spherical Pressure Vessel 

Spherical Pressure Vessel Cut in Half

 

A spherical pressure vessel is really just a special case of a cylinderical vessel. No matter how the a sphere is cut in half, the pressure load perpendicular to the cut must equal the shell stress load. This is the same situation with the axial direction in a cylindrical vessel. Equating the to loads give,

     p(πr2) = σh(2πrt)

This can be simplified to

   

Notice, the hoop and axial stress are the same due to symmetry.