FAQs of Finite Element Analysis

Unique Characteristics of Plastics Stress Analysis

- You cannot simply follow  metals analysis principles

Metals have enjoyed centuries of successful service history in large, heavy duty equipments of sorts.  Much of the fundamental understanding behind such applications has been buried deep under the easy-looking handbook formulas and thumb rules.   Their applicability depends on the metals behavior, and other assumptions in mechanics that hold well for metals, but not necessarily for plastics. 

Plastics is a different animal, with different material behavior.  Don't be fooled by the outward similarity of basic formulas of the handbooks since there are more failure mechanisms for plastics.  Or, the formulas may not apply for the particular product.   In short, the numbers may be derived in a similar manner to that of metals, but the interpretation is different.  

- Handbook Formulae Have Limited Applicability For Plastics

In the metals, Steel, and Aluminum are the two major structural materials.  It may be noted that the heat treatment conditions and alloying do not greatly change the material stiffness (Tensile Modulus, E).  Almost all plastics have a E that are one to two orders of magnitude lower than these metals.

Consequently the plastics structures must be designed with a variety of ribs, stiffeners, bosses etc.  The familiar formulas of the handbooks are no longer applicable for such configurations.

- Molding Technology and Part Integration Make Product Shapes Complex

Metals structures have been designed with plates, bars, extrusions in mind as the starting point, and with machining, forging, pressing and welding as the fabrication processes.  Plastics and composites side step these "fabrication" process almost entirely, and the tools take advantage of complex mold shapes using EDM, and multi-axes NC milling process.  As a result, it is possible to obtain, new, unconventional and complex shapes of parts. If the part design is a brand new one, there is no easy way of guessing the initial sizing the thickness.

- Viscoelasticity Effects 

Creep and relaxation in metals become significant at operating temperatures close to 50% of its melting point, which is close to 400^oC (750^oF).  Where they become significant, the design employs thermal shields and creep-resistant materials.   Contrarily with the plastics, these behavior prevail at almost all temperatures.  Thus, while in the metals the objective is to avoid creep, with plastics we seek to cope with it. 

In general therefore, it is never enough with plastics to compare stress with short term strength.

- Effect of Material Non-Linearity

The stress-strain relationship for plastics is non-linear, even within the elastic (recoverable) range.   With metals, it is linear all the way up to yield point.  This perhaps leads to the most important, and little realized philosophical difference between metals and plastics.  

The strength of any material is really the allowable strain, and not stress.  Because strain is real, and has a meaning all the way down to the molecular scale, and can be expressed as relative stretch in the distance between molecules.  But, the traditional meaning of stress as load per unit area breaks down at small scales.  Traditionally with metals, the strength of metals is reported as a stress value, and this has been followed in plastics also.  This works well for metals, because stress and strain simply translate to each other easily, and also because stress - and not strain - is a large enough quantity to measure.  However, stress is not enough information for plastics.  For example, at a given stress, strain can grow with time, and result in failure.

The basic formulation needed for plastics therefore is the allowable strain, and not stress which is hardly freely available.

- Effect of Low Modulus On Load Transfer By Contact

In the case of metals, at small scale of load transfer, the small area of contact is almost never a great structural problem that calls for analysis.  Rather, the local yielding is assumed to enlarge the area of contact enough to redistribute the load, 

............and then all is well.  

Not so with plastics.  In plastics, load transfer by contact produces considerable deformation in the structure in addition to (possibly) local yield. Therefore the mechanism of survival  of the structure is different and calls for a detailed analysis of multiple non-linearity.

Examples: (i) Plastic Gears, (ii) Filled 55 gallon drums falling off a stack  (iii) An occupied wheel chair climbing over a bump

What is meant by Non-Linearity?

First of all, let us examine the term 'linear' finite element stress analysis. In the linear analysis, two assumptions are involved. Firstly the stress in the material is exactly proportional to strain. Secondly, the structure deforms exactly proportional to the loads, as long as the supports are the same.  The second assumption may be mistaken to derive from the first - it does not.  (Fishing rod is an example of a non-linear structure made of linear material).

A stress analysis problem is linear only if both conditions hold.  If any one of them is violated, then we have a Non-Linear problem.

Most real life structures, especially plastics, are non-linear, perhaps both in structure and in material. Most plastic materials have a non-linear (see figure) stress strain relationship.  The non-linearity arising from the nature of material is called 'Material Non-linearity'.  Furthermore, due to the generally thin walls, plastic structures exhibit a non-linear load-deflection relationship, which could arise even if the material were linear (fishing rod).  This kind is called the 'Geometric Non-Linearity'

Do I need Non-Linear FEA for my product?

There is no cut-and-dried answer for this question.  It depends on the nature of the material, the product, and the kind of service load to be simulated.  For the same product, one service load may need linear FEA, and another a non-linear analysis.

How is Non-linearity handled in the FEA?

Both material and geometric non-linearity are handled in the same way in FEA.  The idea is to go by small increments of loads. For a small increment of load, the material is assumed to behave linearly, corresponding to the local tangent modulus (see figure). The change in the shape is calculated.   If so needed, the new shape is taken as a different structure, and the additive stresses due to another small increment in load is calculated,.. and so on, until the full applied load is reached.

Due to the iterative nature of calculations, the non-linear FEA is time-consuming, and uses up computer resources, but reflects the real life conditions more accurately than a linear analysis.

Is it not possible to do Product Design by linear analysis alone?

In grossly over-designed (over-sized) products, it is difficult to see the utility or relevance of non-linear FEA.  Linear analysis is justified when the stress in the part is within the linear stress-strain range, and stress concentrations are small and can be treated as trivial.   When a minimum weight design is involved, or a strain-based fatigue situation is involved, we need to push the material to higher stress levels, at which linear analysis will lead to unconservative conclusions. Furthermore, when one needs to detect buckling and post-buckling behavior, non-linear analysis is the only way.

What are 'Falling Impact' problems?

Imagine a glass-filled nylon wheel of a wheel-chair which can be strolled over bumps and pot holes. Each time it experiences an impact and the material as well as design must be suitable to withstand such impacts several thousand times. Less important-looking products like a bleach bottle may also be required to withstand falling impact without losing the cap, lest the spill could be a menace or even a hazard.  Plastic lawn chairs are another example.  For almost all sports goods, the falling impact (falling does not have to be free fall) is the very basis of design.

Several plastics product manufacturers have come to recognize the importance of 'falling impact capability' and consequently, currently there are appropriate ASTM standards specifying the relevant requirements.

How can FEA be used for 'Falling Impact' or 'Collision' Problems?

Falling impact or Collision problems are important for many plastics utilitarian products, and applies to falling objects, as well as those subjected to impact loads.   Examples: Large 55-gallon drums, bleach bottles, milk bottles, lawn chairs, tennis rackets, Wheel Chair Wheels, automobile hoods and trunks, bumpers, helmets etc.

This class of problems in FEA present a combination of at least three kinds of non-linearities - material non-linearity, geometric and boundary non-linearity, occurring together.  The last-mentioned kind refers to the small increase in the area of contact, as the colliding bodies squeeze into each other, and thereby spread the stress over larger area, which in turn reduces the stress.  This mechanism enables many plastics structures to withstand impact without breaking, in a large number of cases.   Without considering this non-linearity, it would be hard to distinguish a rigid material from a pliable plastic.  

Iterative procedure is still the technique, except that additional checks for contact being established, contact reactions, increase in area due to deformation caused at the point of contact etc. are involved.   Therefore, these problems entail much smaller load increments.  Consequently, many more cycles of calculation are to be expected.  Every problem is unique.

Use of explicit finite elements is also to be explored for this family of problem.

When are vibration problems important to consider?

Vibration, when known to exist, cannot be neglected at all, and must be designed against.  The reason for this is that vibration is directly related to fatigue, and fatigue failure can occur at stress levels much lower than applicable for static cases.

As an example, a 33% glass-filled nylon 6/6, in a static environment can be pushed to stress levels of 7000 psi, whereas under fatigue conditions, can be loaded only up to 4500 psi, including stress concentration effects.

Since vibration problems are closely tied to determining the natural frequency of the structure, the structure needs to be designed to be stiff enough to avoid resonance by a reasonable factor of separation.

What do the words Resonance and Factor of Separation mean?

These two words are significant only in the context of regular, cyclic loads, as in the case of a hand held electric tool.  In this context, when the natural frequency of the part is close enough to the rotational frequency of the motor, then very high amplitudes are encountered.  If the structure is designed with added stiffness so that the natural frequency is much higher than the rotational frequency, then the ratio of the natural to the rotational frequency is called the factor of separation.  A desirable value of factor of separation is between 2.5 to 3.0.

It appears that the factor of separation is higher than the normal factor of safety used for tensile strength in static designs.  Why is it so?

From the adjoining figure of the so-called 'transmissibility factor' diagram, it is seen that at 2.5 or above, the transmissibility factor is approximately 1.0, which means that the forces experienced by the part are very close to the static case.  No magnification is implied.  Furthermore as a consequence, stress calculations based on static analysis are valid.  The entailing material economy is also evident.

What is Random Vibration ?

What you experience when driving over rubbles in a country road can be called 'random vibration'. The important characteristic is that at any speed there is some vibration.  Smaller, lighter cars experience it more, while heavier, bigger cars experience is less.  Instead of a single cyclic frequency, there a continuous bandwidth of frequencies.  Depending on the road conditions, and the structure of the car, a certain range of frequency would cause the most hardship on the structure.  

More importantly, from the design viewpoint, it would make no sense to attempt to provide a 'factor of separation' on the highest encountered frequency.  The established practice is to perform a "Three-Sigma" statistical estimate of maximum stress in the component.  The input for such problems is typically a "Power Spectral Density" (PSD) curve, which is obtained from tests.  For example, an instrumented automobile can provide the frequencies encountered in good to rough road conditions, or over rubbles and rocks.  This data is usually enveloped by a smooth curve for conservatism, and specified as the Design PSD Curve.

The range of frequency encountered is quite wide that it is impractical to apply the "Factor of Separation" to the design.  Rather, the estimated Three Sigma stress should be below the fatigue limit.

Is Random Vibration analysis relevant to plastics products at all?

More and more plastic structural components are employed in the automobiles, aircrafts as dashboards and instrument and control housings. Future cars will use a real time computer on which most of the car's functions will depend.  It is absolutely important that such computerized components be designed to sustain vibrations, or be isolated from it.  

Ponder the question - How come the headlights and its bulbs in the car survive the slam of the hood? 

What is non-linear buckling?

Buckling mode of failure may be lurking where the stresses tend to be compressive.  Like in the linear stress analysis, linear buckling is highly idealized analysis, giving the entire benefit of all doubts to the structure.  Some of the assumptions are that the structure is perfectly centered for the applied load, and that all the material in the part 'polarize' their strengths, so to say, exactly against the applied load etc. etc. Knowing well that real life structures are far from this ideal case, we should expect that several effects interplay to reduce the 'ideal' strength.

Among those causes are the material non-linearity, real life non-uniformity of loads, large displacements, loads trying to follow the deformed shape of the part, local yielding at supports, process-induced orientations, variations in properties, residual stresses, etc.

The real life buckling strength is therefore to be obtained by including such effects - generically called non-linear buckling analysis.  A reduction of the ideal buckling strength (obtained from handbooks) by a factor of almost 20 times can be expected in very thin walled plastics, should a combination of the above factors exist. 

Thus,  the handbook formulas represent the ideal maximum strength, and not the real. 

So, how does one design against non-linear buckling?

By analysis using more than one approach, assumption, and topping it with conservatism.

An interesting variation might arise in the case of automotive hoods which in the case of front end collision is expected to crumple (buckle) in order to absorb the energy of collision, as well as to save the passenger compartment.  In such cases, we are not designing against, but for buckling.

What is buckling?

Buckling is a critical state of stress and deformation, at which a slight disturbance causes a gross additional deformation, or perhaps a total structural failure of the part.  Buckling is a critical state.  Structural behavior of the part near or beyond 'buckling' is not self-evident from the normal arguments of statics.  For example, a thin stick cannot stand on its feet on a flat floor, but a stout stump can; statics alone cannot explain this difference. Also, the lateral deflection of an axially compressed column cannot be explained from statics.

In the lingo of structural mechanics, buckling occurs when 

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When tensile stress prevails predominantly, buckling cannot happen.  It can happen only in compressive stress environments.  Since small disturbances are present all the time in the support, in the load, in the environment, the structure would jump to the configuration dictated by the disturbance, rather than that predicted by statics calculation. 

What factors affect Buckling Strength of a Structure?

We want to be aware that any structure is most efficient when subjected to evenly distributed tensile or compressive stress, such as occurring in cables, strings etc.  Evidently, such modes of loading makes the best use of the material, and its strength. On the other hand bending or flexing, is the least efficient mode of loading a structure.  In the figure (a) below, the internal force in the material exactly opposes the applied load, and therefore is the best mode.  In the figure (b), the flexural stress is oriented perpendicular to the applied load, and equilibrium equations demand very high flexural stresses in order to support a small load.

If Flexural stiffness is so important to prevent buckling, why can we not use deeper ribs?

Call it the way nature balances out your odds.  The Deeper-Ribs idea is hit by two problems sooner or later -- Stress, and Buckling, in particular 'Torsional Buckling'.  The first one is easily understood. Stress increases at the tip of the ribs as the square of the rib height, whereas the stiffness increases as the cube of the rib height.  Thus undue increase in rib height can result in excessive stress.  

The second problem relates to another form of buckling called "torsional buckling".  In this mode of failure, the rib twists and folds over flat on the panel which it is supposed to stiffen.  

What is a good design procedure for plastic components to protect against buckling?

Pioneer Technologies has been involved in a number of projects involving buckling.  Based on our experience, we recommend as follows.  

An important point to remember is (1) NOT to mix concepts of stress and static strength with Buckling, since they do not indicate anything about Buckling.  Designing  only for strength tends to be an under-design against buckling.  It is a necessary but not sufficient recommendation to apply a factor of safety on yield strength of about 5.0.   This is not sufficient because a detailed buckling analysis - non-linear one if necessary - is required to understand the product-specific modes of buckling and protect against it.  Depending on the first iteration design, the design may need further beefing up.  Most importantly, it is to recognized that any handbook formula on elastic instability will under-predict the part size for plastics.

What are composites, and advanced composites?

Composites are a synergetic combination of weak plastics and stronger fibers. When used for moderate enhancement in strength and stiffness of the plastics, they may be called 'filled plastics'.  When the combination occurs as a means of deploying the strength of the fiber, then they are called 'composites.'  In other words, if the resulting material, and the product is viewed as a plastic, and is processed similar to other plastics, then it is "filled plastic".  When the resulting product and process characteristics are dominated by the fiber properties predominantly,  then it is a "composite".  Advanced composites use very long - or even continuous fibers - to give the resulting product the maximum possible orientation of properties along and across the fibers.  Wood is a natural composite with distinct properties along and across fibers.  

Such materials, highly favored for the aircrafts, and sports goods, are very light, strong, stiff and pose special problems in material behavior in the macro and micro scale, in mechanical testing,  fabrication, joining, in product design, finite element analysis, etc.   Furthermore, the conceivable variety of combinations is very large. Only a handful of combinations are practiced in the industry.