Stress-strain Diagram & safety factors

Stress-strain Diagram

• elastic limit
• proportinal limit
• rupture strength
• stress-strain diagram
• ultimate strength
• yield point

Suppose that a metal specimen be placed in tension-compression-testing machine. As the axial load is gradually increased in increments, the total elongation over the gauge length is measured at each increment of the load and this is continued until failure of the specimen takes place. Knowing the original cross-sectional area and length of the specimen, the normal stress σ and the strain ε can be obtained. The graph of these quantities with the stress σ along the y-axis and the strain ε along the x-axis is called the stress-strain diagram. The stress-strain diagram differs in form for various materials. The diagram shown below is that for a medium-carbon structural steel.

Metallic engineering materials are classified as either ductile or brittle materials. A ductile material is one having relatively large tensile strains up to the point of rupture like structural steel and aluminum, whereas brittle materials has a relatively small strain up to the point of rupture like cast iron and concrete. An arbitrary strain of 0.05 mm/mm is frequently taken as the dividing line between these two classes.


Stress-strain diagram of a medium-carbon structural steel

Proportional Limit (Hooke's Law)
From the origin O to the point called proportional limit, the stress-strain curve is a straight line. This linear relation between elongation and the axial force causing was first noticed by Sir Robert Hooke in 1678 and is called Hooke's Law that within the proportional limit, the stress is directly proportional to strain or

The constant of proportionality k is called the Modulus of Elasticity E or Young's Modulus and is equal to the slope of the stress-strain diagram from O to P. Then

Elastic Limit
The elastic limit is the limit beyond which the material will no longer go back to its original shape when the load is removed, or it is the maximum stress that may e developed such that there is no permanent or residual deformation when the load is entirely removed.

Elastic and Plastic Ranges
The region in stress-strain diagram from O to P is called the elastic range. The region from P to R is called the plastic range.

Yield Point
Yield point is the point at which the material will have an appreciable elongation or yielding without any increase in load.

Ultimate Strength
The maximum ordinate in the stress-strain diagram is the ultimate strength or tensile strength.

Rapture Strength
Rapture strength is the strength of the material at rupture. This is also known as the breaking strength.

Modulus of Resilience
Modulus of resilience is the work done on a unit volume of material as the force is gradually increased from O to P, in N·m/m³. This may be calculated as the area under the stress-strain curve from the origin O to up to the elastic limit E (the shaded area in the figure). The resilience of the material is its ability to absorb energy without creating a permanent distortion.

Modulus of Toughness
Modulus of toughness is the work done on a unit volume of material as the force is gradually increased from O to R, in N·m/m³. This may be calculated as the area under the entire stress-strain curve (from O to R). The toughness of a material is its ability to absorb energy without causing it to break.

Working Stress, Allowable Stress, and Factor of Safety
Working stress is defined as the actual stress of a material under a given loading. The maximum safe stress that a material can carry is termed as the allowable stress. The allowable stress should be limited to values not exceeding the proportional limit. However, since proportional limit is difficult to determine accurately, the allowable tress is taken as either the yield point or ultimate strength divided by a factor of safety. The ratio of this strength (ultimate or yield strength) to allowable strength is called the factor of safety.

factor of safety

It is the ratio of the breaking stress of a material or structure to the calculated maximum stress when in use Also called safety factor

Factor of safety (FoS), also known as (and used interchangeably with) safety factor (SF), is a term describing the structural capacity of a system beyond the expected loads or actual loads. Essentially, how much stronger the system is than it usually needs to be for an intended load. Safety factors are often calculated using detailed analysis because comprehensive testing is impractical on many projects, such as bridges and buildings, but the structure's ability to carry load must be determined to a reasonable accuracy.
Many systems are purposefully built much stronger than needed for normal usage to allow for emergency situations, unexpected loads, misuse, or degradation.

Contents 1 Definition 2 Calculation 2.1 Design factor and safety factor 2.2 Margin of safety 2.3 Reserve factor 3 Yield & Ultimate Calculations 4 Choosing design factors 5 See also 6 Notes 7 Further reading

Definition

There are two distinct uses of the factor of safety: One as a ratio of absolute strength (structural capacity) to actual applied load. This is a measure of the reliability of a particular design. The other use of FoS is a constant value imposed by law, standard, specification, contract or custom to which a structure must conform or exceed.
The first sense (a calculated value) is generally referred to as a factor of safety or, to be explicit, a realized factor of safety, and the second sense (a required value) as a design factor, design factor of safety or required factor of safety, but usage is inconsistent and confusing. It is important to keep track of which of the two definitions is being used. The cause of much confusion is that various reference books and standards agencies use the factor of safety definitions and terms differently. Design codes and structural and mechanical engineering textbooks often use "Factor of Safety" to mean the fraction of total structural capability over that needed^[1][2][3] (first sense). Many undergraduate Strength of Materials books use "Factor of Safety" as a constant value intended as a minimum target for design^[4][5][6] (second sense).
This may sound similar, but consider this: Say a beam in a structure is required to have a safety factor of 3. The engineer chose a beam that will be able to withstand 10 times the load. The required safety factor is still 3, because it is the requirement that must be met, the beam just happens to exceed the requirement and its safety factor is 10. The realized safety factor should always meet or exceed the required safety factor or the design is not adequate. Meeting the required safety factor exactly implies that the design meets the minimum allowable strength. A high safety factor well over the required design factor sometimes implies "overegineering" which can result in excessive weight and/or cost. In colloquial use the term, "required safety factor" is functionally equivalent to the design factor.

Calculation

There are several ways to compare the factor of safety for structures. All the different calculations fundamentally measure the same thing: how much extra load beyond what is intended a structure will actually take (or be required to withstand). The difference between the methods is the way in which the values are calculated and compared. Safety factor values can be thought of as a standardized way for comparing strength and reliability between systems.
The use of a factor of safety does not imply that an item, structure, or design is "safe". Many quality assurance, engineering design, manufacturing, installation, and end-use factors may influence whether or not something is safe in any particular situation.

Design factor and safety factor

The difference between the safety factor and design factor (design safety factor) is as follows: The safety factor is how much the designed part actually will be able to withstand (first "sense" from above). The design factor is what the item is required to be able to withstand (second "sense"). The design factor is defined for an application (generally provided in advance and often set by regulatory code or policy) and is not an actual calculation, the safety factor is a ratio of maximum strength to intended load for the actual item that was designed.

Factor Of Safety =Material Strength/Design Load

• Design load being the maximum load the part should ever see in service.

Margin of safety

Many government agencies and industries (such as aerospace) require the use of a margin of safety (MoS or M.S.) to describe the ratio of the strength of the structure to the requirements. There are two separate definitions for the margin of safety so care is needed to determine which is being used for a given application. One usage of M.S. is as a measure of capacity like FoS. The other usage of M.S. is as a measure of satisfying design requirements (requirement verification). Margin of safety can be conceptualized (along with the reserve factor explained below) to represent how much of the structure's total capacity is held "in reserve" during loading.
M.S. as a measure of structural capacity: This definition of margin of safety commonly seen in textbooks basically says that if the part is loaded to the maximum load it should ever see in service, how many more loads of the same force can it withstand before failing. In effect, this is a measure of excess capacity. If the margin is 0, the part will not take any additional load before it fails, if it is negative the part will fail before reaching its design load in service. If the margin is 1, it can withstand one additional load of equal force to the maximum load it was designed to support (i.e. twice the design load).
Margin of Safety= [(Failure Load)/(Design load)] -1
Margin of Safety= Factor Of Safety -1

M.S. as a measure of requirement verification: Many agencies such as NASA and AIAAdefine the margin of safety including the design factor, in other words, the margin of safety is calculated after applying the design factor. In the case of a margin of 0, the part is at exactly the required strength (the safety factor would equal the design factor). If there is a part with a required design factor of 3 and a margin of 1, the part would have a safety factor of 6 (capable of supporting two loads equal to its design factor of 3, supporting six times the design load before failure). A margin of 0 would mean the part would pass with a safety factor of 3. If the margin is less than 0 in this definition, although the part will not necessarily fail, the design requirement has not been met. A convenience of this usage is that for all applications, a margin of 0 or higher is passing, one does not need to know application details or compare against requirements, just glancing at the margin calculation tells whether the design passes or not.

Design Safety Factor = [Provided as requirement]
Margin of Safety= [(Failure Load)/(Design load*Design Safety Factor)] -1
Margin of Safety= [(Realised Factor Of Safety)/( Design Safety Factor)] -1
For a successful design, the realized Safety Factor must always equal or exceed the required Safety Factor (Design Factor) so the Margin of Safety is greater than or equal to zero. The Margin of Safety is sometimes, but infrequently, used as a percentage, i.e., a 0.50 M.S is equivalent to a 50% M.S. When a design satisfies this test it is said to have a "positive margin," and, conversely, a “negative margin” when it does not.
In the field of Nuclear Safety (as implemented at U.S. government owned facilities) the Margin of Safety has been defined as a quantity that may not be reduced without review by the controlling government office. The U.S. Department of Energy publishes DOE G 424.1-1, "Implementation Guide for Use in Addressing Unreviewed Safety Question Requirements" as a guide for determining how to identify and determine whether a margin of safety will be reduced by a proposed change. The guide develops and applies the concept of a qualitative margin of safety that may not be explicit or quantifiable, yet can be evaluated conceptually to determine whether an increase or decrease will occur with a proposed change. This approach becomes important when examining designs with large or undefined (historical) margins and those that depend on 'soft' controls such as programmatic limits or requirements. The commercial U.S. nuclear industry utilized a similar concept in evaluating planned changes until 2001, when 10 CFR 50.59 was revised to capture and apply the information available in facility-specific risk analyses and other quantitative risk management tools.

Reserve factor

A measure of strength frequently used in Europe is the Reserve Factor (RF). With the strength and applied loads expressed in the same units, the Reserve Factor is defined as:
RF = Proof Strength / Proof Load
RF = Ultimate Strength / Ultimate Load
The applied loads have any factors, including factors of safety applied.

Yield & Ultimate Calculations

For ductile materials (e.g. most metals), it is often required that the factor of safety be checked against both yield and ultimate strengths. The yield calculation will determine the safety factor until the part starts to plastically deform. The ultimate calculation will determine the safety factor until failure. On brittle materials these values are often so close as to be indistinguishable, so is it usually acceptable to only calculate the ultimate safety factor.

Choosing design factors

Appropriate design factors are based on several considerations, such as the accuracy of predictions on the imposed loads, strength, wear estimates, and the environmental effects to which the product will be exposed in service; the consequences of engineering failure; and the cost of over-engineering the component to achieve that factor of safety. For example, components whose failure could result in substantial financial loss, serious injury, or death may use a safety factor of four or higher (often ten). Non-critical components generally might have a design factor of two. Risk analysis, failure mode and effects analysis, and other tools are commonly used. Design factors for specific applications are often mandated by law, policy, or industry standards.
Buildings commonly use a factor of safety of 2.0 for each structural member. The value for buildings is relatively low because the loads are well understood and most structures are redundant. Pressure vessels use 3.5 to 4.0, automobiles use 3.0, and aircraft and spacecraft use 1.2 to 3.0 depending on the application and materials. Ductile, metallic materials tend to use the lower value while brittle materials use the higher values. The field of aerospace engineering uses generally lower design factors because the costs associated with structural weight are high (i.e. an aircraft with an overall safety factor of 5 would probably be too heavy to get off the ground). This low design factor is why aerospace parts and materials are subject to very stringent quality control and strict preventative maintenance schedules to help ensure reliability. A usually applied Safety Factor is 1.5, but for pressurized fuselage it is 2.0, and for main landing gear structures it is often 1.25.
In some cases it is impractical or impossible for a part to meet the "standard" design factor. The penalties (mass or otherwise) for meeting the requirement would prevent the system from being viable (such as in the case of aircraft or spacecraft). In these cases, it is sometimes determined to allow a component to meet a lower than normal safety factor, often referred to as "waiving" the requirement. Doing this often brings with it extra detailed analysis or quality control verifications to assure the part will perform as desired, as it will be loaded closer to its limits.
For loading that is cyclical, repetitive, or fluctuating, it is important to consider the possibility of metal fatigue when choosing factor of safety. A cyclic load well below a material's yield strength can cause failure if it is repeated through enough cycles.