April 1st, 2024
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Failure criterion

Failure criterion refers to the principle according to which point of view in the design of vessel elements, to judge that the vessel can not bear its normal operating load.

There are two kinds of failure criteria for pressure vessels: strength failure criterion and stiffness failure criterion.

Strength failure generally refers to the failure caused by tensile (including bending, shear) stress of container elements.

The strength failure criteria include the following nine types.

When using elastic failure criterion to design the pressure vessel, the maximum stress (or maximum equivalent stress) shall be smaller than the yield strength of the material or or allowable stress of the material [σ] ( when the safety factor is considered ).

The stress of container element may occur in the area that far away from structural discontinuity area including the connection area, nozzle area, support area or other local load action area far away from the cylinder and head.

Once the stress higher than the yield strength of the material or or allowable stress of the material [σ] , it will lead to excessive deformation of the component leading to leakage at the sealed joint , or even an excessive deformation at the welded joint or unwelded integrity until it explodes.

The container is considered to be failed if leakage or bursting occurs.

Plastic failure criterion considers that when the stress on the wall surface reaches the yield strength of the material:

For the element with uniform stress distribution along the section, the yield of one point means the simultaneous yield of the whole loaded section of the element .

But for the element subjected to bending stress, the yield of the wall surface is far from exerting the bearing potential of the whole section of the material. And it can continue to carry the load. Along with the increasing of the applied bending moment, the yield layer of the element extends from the surface to the neutral plane. Failure occurs only when the yield layer extends to the neutral plane, i.e. the entire section yields.

From this point of view, only when the maximum stress of the plate that is subjected to bending moment (because the surface has already yielded, the elastic formula is no longer applicable, so the stress calculated by elastic formula is called virtual stress) calculated by elastic formula is 1.5σ, the whole section will yield. After introducing the safety factor ns=1.5, the strength check condition can reach 1.5[σ].

The design criterion of limiting the maximum stress calculated according to elastic formula to 1.5[σ] is derived according to plastic failure criterion.

Elastoplastic failure applies to repeated loads.

The elastoplastic failure criterion considers:

For the secondary stress, because it has self-limiting, for example, when the cylinder stress is much lower than the yield strength of the material, the total stress is greatly increased due to the existence of edge stress in the overall structural discontinuous area where the cylinder is connected with the head or the nozzle, and the plastic deformation may occur when it reaches the yield strength of the material.

But the adjacent area of this high stress area is still elastic. Plastic deformation under repeated loading does not necessarily lead to vessel failure.

If incremental deformation does not occur, it is called "stability". The structure is stable. The response of the structure after this is elastic or elastoplastic.

The maximum virtual stress threshold value for judging whether the structure is stable or not is 2σ, i.e. 3[σ]. The vessel shall be deemed to have failed only if stability has been lost.

Because this failure criterion allows local plastic deformation, and because of the inhomogeneity of stress distribution, the local plastic zone is surrounded by a large elastic zone, so it is called elastoplastic failure criterion.

If the range of secondary stress intensity is limited below 3[σ], it is derived according to elastoplastic failure criterion.

This failure criterion is first applied to stress analysis design criteria.

This failure mode is actually caused by elastoplastic failure.

For example, in the edge area where the molded head and cylinder are connected, there is additional edge stress in a certain range of the head and cylinder due to the overall structural discontinuity.

When the total stress exceeds 2σs, plastic instability-progressive collapse will occur during loading and unloading, that is, plastic deformation will occur at the connecting edge and the meridian of the head or cylinder will be obviously twisted.

The failure mode of elastic-plastic collapse and instability collapse is included in the design of forming head in JB4732《Steel Pressure Vessel-Analysis Design Standard》and ASME VIII-2.

Due to the existence of strain hardening phenomenon in material either large or small , so even if the container wall reaches full yield, it still will not blast. The blasting failure criterion is designed with the wall blasting as the limit condition.

In fact, the blasting failure criterion is only used for the design of thick wall vessels such as high pressure and ultra-high pressure.

Because for thick wall vessels, the uneven distribution of stress along the wall thickness is more obvious, and it cannot be considered by uniform distribution of stress along the wall thickness as in thin wall vessels.

The thicker of the wall, the more uneven the stress distribution along the wall thickness, so that when the inner wall yields, the outer wall is far away from yielding and is in an elastic state.

When the pressure is very high, in order to bear the pressure, increase the wall thickness can not avoid the inner wall yield. Once the wall yield, it can not be calculated according to the elastic formula. So, it can only be designed according to plastic failure or explosion failure criteria.

That is, caculate the full yield pressure or explosion pressure of the cylinder (or spherical shell) from the thick wall stress formula and yield conditions, then introduce the matching safety factor to find out the cylinder (or spherical shell) design pressure.

No walls yield during is allowed during normal operation of the vessel. Therefore, the safety factor introduced is to ensure that the maximum stress of the inner wall is still in the elastic state.

Although the pressure vessel or all kinds of chemical equipment in the entire life cycle load will not reach a very high number of alternating cycles, due to various reasons, such as, opening nozzle area, weld penetration, wrong edge, undercut, etc., the local stress concentration greatly increased. So, even the number of alternating cycles is only (103~105) times, it will also cause low cycle high stress (strain) fatigue failure.

Fatigue failure criteria limit the maximum alternating stress amplitude (or number of load cycles required) that may occur on the vessel to the allowable stress amplitude (or number of cycles required) derived from the fatigue design curve.

The vessel standard (such as GB150 -1998 Steel Pressure Vessel) designed according to rules is not suitable for vessels requiring fatigue analysis.

When fatigue analysis is required due to the high number of alternating cycles of load during the whole service life, the design shall be done as per stress analysis of vessel standard (such as JB4732 Steel Pressure Vessel-Analytical Design Standard, ASME V VIII-2).

The various failure criteria mentioned above are based on the calculation of traditional mechanics, that is, there is no defect in the material, calculated by the mechanics of materials, plate and shell theory or elastic mechanics method, when the maximum stress of the component reaches the yield strength of the material, yield occurs, and fracture occurs when the strength limit of the material is reached.

A large number of vessel explosion tests show that for vessels made of medium and low strength steel, even if there are tiny undetected defects, the results obtained by traditional strength calculation method are basically in line with reality.

However, with the using of medium and high strength steels and the increasing of vessel wall thickness, brittle fracture and low stress failure may occur when the working stress is lower than the yield strength of the material or even lower than the allowable stress of the material during operation or pressure test due to the reduction of toughness and possible missing defects.

This phenomenon has occurred many times in practical vessels, which shows that the traditional strength design method may not be applicable sometimes. Therefore, the calculation method of fracture mechanics is proposed as the criterion for evaluating failure or not.

According to fracture failure criterion, the basic point of designing pressure vessel is to limit the fracture parameters of vessel wall, including crack size and stress level, within the corresponding fracture toughness index of material. ASME V VIII-3 Another rule for the construction of high pressure vessels has applied this design method to the design of certain high pressure components.

Under the long-term action of high temperature and internal pressure, pressure vessels undergo plastic deformation slowly and cumulatively, resulting in continuous thinning of vessel wall thickness and finally rupture.

Under the long-term action of certain temperature and stress, plastic deformation accumulates continuously with the passage of time, and the bearing capacity decreases continuously, which finally leads to failure phenomenon called creep failure.

The component (such as the fastening bolt in the seal connector) in the stressed state of its total strain remains unchanged, in the long-term high temperature elastic strain is constantly converted into plastic strain, so that the elastic stress in the fastener decreases and leads to seal failure phenomenon, known as stress relaxation failure.

Stress relaxation and creep are different manifestations of the same problem, both of which are the process of elastic strain transforming into plastic strain with the passage of time under long-term high temperature, and the bearing capacity of loaded components decreases continuously with the passage of time.

According to creep failure criterion, the creep value (or equivalent stress calculated according to creep equation) of wall is limited within allowable range. The conventional design method only selects suitable high temperature steel at a certain high temperature, or limits the service temperature of general steel, that is, it does not adopt obvious design like the above failure modes, but adopts implicit design that is not calculated but only limits certain conditions.

The corrosion failure of pressure vessels includes the destruction of the wall in contact with the vessel medium by corrosion of corrosive media, and intergranular corrosion or stress corrosion caused by the structural characteristics of the material. It can be uniform corrosion or localized corrosion. The prevention of corrosion failure is generally through the selection of appropriate materials, or various types of heat treatment to materials, or protective measures including anodic protection or various corrosion protection linings, generally not limited by strength calculations, that is, by limiting certain conditions or taking certain protective measures implicit design methods.

Stiffness failure refers to the loss of normal operation function of the compressed component due to excessive deformation, and the leakage of the sealed connection caused by excessive deformation can also be said to be stiffness failure. However, stiffness failure usually refers to the instability of components under compressive stress.

Instability can occur when the compressive stress is below the proportional limit of the material, or when it exceeds the proportional limit and is below the yield strength (reaching yield means that compressive strength failure has occurred). The former is linear elastic instability, while the latter is non-linear elastic instability. Because the elastic modulus E of a material is constant in the linear elastic range, it is not constant in the non-linear elastic range and varies with its strain value. Therefore, the design methods of elastic and inelastic buckling ranges are different.

All kinds of pressure vessel standards basically adopt the same method to consider stiffness failure (instability), that is, linear elastic instability can be calculated, and non-linear elastic instability can be calculated by graph method.

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