Chapter 2 - Stress & Strain: Axial Loading
Plotting Stress and Strain Diagram
A curve that shows the characteristic of the properties of the material but does not depend upon the dimensions of the specimen used.
Superposition Method. - Statically Indeterminate Problems
A structure is statically indeterminate whenever it is held by more supports than are required to maintain its equilibrium. This results in more unknown reactions than available equilibrium equations. It is often convenient to designate one of the reactions as redundant and to eliminate the corresponding support. Since the stated conditions of the problem cannot be changed, the redundant reaction must be maintained in the solution. It will be treated as an unknown load that, together with the other loads, must produce deformations compatible with the original constraints. The actual solution of the problem proceeds by considering separately the deformations caused by the given loads and those caused by the redundant reaction, and then adding—or superposing—the results obtained. The general conditions under which the combined effect of several loads can be obtained in this way are discussed in Sec. 2.5.
How to obtain the stress-strain diagram of a material
A tensile test is conducted on a specimen of the material.
Example of Brittle Material in Compression Test
An example of brittle material with different properties in tension and compression is provided by concrete, whose stress-strain diagram is shown in Fig. 2.9. On the tension side of the diagram, we first observe a linear elastic range in which the strain is proportional to the stress. After the yield point has been reached, the strain increases faster than the stress until rupture occurs. The behavior of the material in compression is different. First, the linear elastic range is significantly larger. Second, rupture does not occur as the stress reaches its maximum value. Instead, the stress decreases in magnitude while the strain keeps increasing until rupture occurs. Note that the modulus of elasticity, which is represented by the slope of the stress-strain curve in its linear portion, is the same in tension and compression. This is true of most brittle materials
Fiber-Reinforced Composites
An important class of anisotropic materials consists of fiber-reinforced composite materials. These are obtained by embedding fibers of a strong, stiff material into a weaker, softer material, called a matrix. Typical materials used as fibers are graphite, glass, and polymers, while various types of resins are used as a matrix
Phases of Stress-Strain Diagram for Ductile Materials
As the specimen is subjected to an increasing load, its length first increases linearly with the load and at a very slow rate. Thus the initial portion of the stress-strain diagram is a straight line with a steep slope (Fig. 2.6). However, after a critical value σY of the stress has been reached, the specimen undergoes a large deformation with a relatively small increase in the applied load. This deformation is caused by slippage along oblique surfaces and is due primarily to shearing stresses. After a maximum value of the load has been reached, the diameter of a portion of the specimen begins to decrease, due to local instability (Photo 2.4a). This phenomenon is known as necking. After necking has begun, lower loads are sufficient for the specimen to elongate further, until it finally ruptures (Photo 2.4b). Note that rupture occurs along a cone-shaped surface that forms an angle of approximately 45° with the original surface of the specimen. This indicates that shear is primarily responsible for the failure of ductile materials, confirming the fact that shearing stresses under an axial load are largest on surfaces forming an angle of 45° with the load (see Sec. 1.3). Note from Fig. 2.6 that the elongation of a ductile specimen after it has ruptured can be 200 times as large as its deformation at yield.
Temperature elongation between two fixed supports.
Assume the same rod AB of length L is placed between two fixed supports at a distance L from each other (Fig. 2.26a). Again, there is neither stress nor strain in this initial condition. If we raise the temperature by ΔT, the rod cannot elongate because of the restraints imposed on its ends; the elongation δT of the rod is zero. Since the rod is homogeneous and of uniform cross-section, the strain εT at any point is εT = δT∕L and thus is also zero. However, the supports will exert equal and opposite forces P and P′ on the rod after the temperature has been raised, to keep it from elongating (Fig. 2.26b). It follows that a state of stress (with no corresponding strain) is created in the rod.
Phases of Stress-Strain Diagram for Brittle Materials
Brittle materials, comprising of cast iron, glass, and stone rupture without any noticeable prior change in the rate of elongation (Fig. 2.7). Thus, for brittle materials, there is no difference between the ultimate strength and the breaking strength. Also, the strain at the time of rupture is much smaller for brittle than for ductile materials. Note the absence of any necking of the specimen in the brittle material of Photo 2.5 and observe that rupture occur long a surface perpendicular to the load. Thus, normal stresses are primarily responsible for the failure of brittle materials.
Normal Strain (ε)
Deformation of the member per unit length
Examples of examination of test specimens for fatigue
Examination of test specimens, shafts, springs, and other components that have failed in fatigue shows that the failure initiated at a microscopic crack or some similar imperfection. At each loading, the crack was very slightly enlarged. During successive loading cycles, the crack propagated through the material until the amount of undamaged material was insufficient to carry the maximum load, and an abrupt, brittle failure occurred. For example, Photo 2.6 shows a progressive fatigue crack in a highway bridge girder that initiated at the irregularity associated with the weld of a cover plate and then propagated through the flange and into the web. Because fatigue failure can be initiated at any crack or imperfection, the surface condition of a specimen has an important effect on the endurance limit obtained in testing. The endurance limit for machined and polished specimens is higher than for rolled or forged components or for components that are corroded. In applications in or near seawater or in other applications where corrosion is expected, a reduction of up to 50% in the endurance limit can be expected.
Fatigue
Fatigue must be considered in the design of all structural and machine components subjected to repeated or fluctuating loads. The number of loading cycles expected during the useful life of a component varies greatly. For example, a beam supporting an industrial crane can be loaded as many as 2 million times in 25 years (about 300 loadings per working day), an automobile crankshaft is loaded about 0.5 billion times if the automobile is driven 200,000 miles, and an individual turbine blade can be loaded several hundred billion times during its lifetime
Compression Test - Brittle Materials
For most brittle materials, the ultimate strength in compression is much larger than in tension. This is due to the presence of flaws, such as microscopic cracks or cavities that tend to weaken the material in tension, while not appreciably affecting its resistance to compressive failure
What can be taken from a Stress-Strain Diagram
From this diagram, some important properties of the material, such as its modulus of elasticity and whether the material is ductile or brittle, can be determined. rials depend upon the direction of the load. From the stress-strain diagram, you also can determine whether the strains in the specimen will disappear after the load has been removed—when the material is said to behave elastically—or whether a permanent set or plastic deformation will result
Compression Test - Ductile Materials
If a specimen made of a ductile material is loaded in compression instead of tension, the stress-strain curve is essentially the same through its initial straight-line portion and through the beginning of the portion corresponding to yield and strain-hardening. Particularly noteworthy is the fact that for a given steel, the yield strength is the same in both tension and compression. For larger values of the strain, the tension and compression stress-strain curves diverge, and necking does not occur in compression.
Statically Indeterminate Problems
In the problems considered in the preceding section, we could always use free-body diagrams and equilibrium equations to determine the internal forces produced in the various portions of a member under given loading conditions. There are many problems, however, where the internal forces cannot be determined from statics alone. Oftentimes, even the reactions themselves—the external forces—cannot be determined by simply drawing a free-body diagram of the member and writing the corresponding equilibrium equations, because the number of constraints involved exceeds the minimum number required to maintain static equilibrium. In such cases, the equilibrium equations must be complemented by relationships involving deformations obtained by considering the geometry of the problem. Because statics is not sufficient to determine either the reactions or the internal forces, problems of this type are called statically indeterminate. The following concept applications show how to handle this type of problem.
Elastic Behavior of Material
Material behaves elastically if the strains in a test specimen from a given load disappear when the load is removed. In other words, the specimen returns to its original undeformed shape upon removal of all load. The largest value of stress causing this elastic behavior is called the elastic limit of the material. If the material has a well-defined yield point as in Fig. 2.6a, the elastic limit, the proportional limit, and the yield point are essentially equal. In other words, the material behaves elastically and linearly as long as the stress is kept below the yield point. However, if the yield point is reached, yield takes place as described in Sec. 2.1B. When the load is removed, the stress and strain decrease in a linear fashion along a line CD parallel to the straight-line portion AB of the loading curve (Fig. 2.13
Stress-Strain Diagram
Plotting the stress σ versus the strain ε as the load applied to the member is increased produces a stress-strain diagram for the material used
Deformations of a Structural Member
Rods,Bars, or Plates, tend to deform when they are under axial loading.
Unit for Modulus of Elasticity
Since the strain ε is a dimensionless quantity, E is expressed in the same units as stress σ—in pascals or one of its multiples for SI units and in psi or ksi for U.S. customary units.
Temperature Changes
Solid bodies subjected to increases in temperature will expand, while those experiencing a reduction in temperature will contract
Loadings with Fluctuating Nature
Some loadings are of a fluctuating nature. For example, the passage of traffic over a bridge will cause stress levels that will fluctuate about the stress level due to the weight of the bridge. A more severe condition occurs when a complete reversal of the load occurs during the loading cycle. The stresses in the axle of a railroad car, for example, are completely reversed after each half-revolution of the wheel
Ductile and Brittle Materials in Stress and Strain Diagram
Stress-strain diagrams of materials vary widely, and different tensile tests conducted on the same material may yield different results, depending upon the temperature of the specimen and the speed of loading. However, some common characteristics can be distinguished from stress-strain diagrams to divide materials into two broad categories: ductile and brittle materials.
Modulus of rigidity
The components of shearing stress and shearing stress.
Common Tensile Test
The cross-sectional area of the cylindrical central portion of the specimen is accurately determined and two gage marks are inscribed on that portion at a distance L0 from each other. The distance L0 is known as the gage length of the specimen
endurance limit
The endurance limit is the stress for which failure does not occur, even for an indefinitely large number of loading cycles. For a low-carbon steel, such as structural steel, the endurance limit is about one-half of the ultimate strength of the steel.
Plastic Deformation or Permanent Set
The fact that ε does not return to zero after the load has been removed indicates that a permanent set or plastic deformation of the material has taken place. For most materials, the plastic deformation depends upon both the maximum value reached by the stress and the time elapsed before the load is removed. The stress-dependent part of the plastic deformation is called slip, and the time-dependent part—also influenced by the temperature— is creep
When to use Hooke's Law
The most significant value of stress for which Hooke's law can be used for a given material is the proportional limit of that material. For ductile materials possessing a well-defined yield point, as in Fig. 2.6a, the proportional limit almost coincides with the yield point. For other materials, the proportional limit cannot be determined as easily, since it is difficult to accurately determine the stress σ for which the relation between σ and ε ceases to be linear.
Modulus of Elasticity
The ratio of stress to strain when deformation is totally elastic; is also a measure of the stiffness of a material. The coefficient E of the material
Statically Indeterminate Problems
The reactions and the internal forces cannot be determined from statics alone. Here the equilibrium equations derived from the free-body diagram of the member must be complemented by relationships involvi
Thermal Strain
The strain εT is called a thermal strain, as it is caused by the change in temperature of the rod. However, there is no stress associated with the strain εT
Breaking Strength
The stress σB corresponding to rupture
Ultimate Strength
The stress σU corresponding to the maximum load applied
Yield Strength
The stress σY at which yield is initiated
δ = ∑ i PiLi/AiEi
Using the internal force Pi, length Li, cross-sectional area Ai, and modulus of elasticity Ei, corresponding to part i, the deformation of the entire rod is
fatigue
When structural or machine components to fail after a very large number of repeated loadings, even though the stresses remain in the elastic range.
Repeated Loadings and Fatigue
You might think that a given load may be repeated many times, provided that the stresses remain in the elastic range. Such a conclusion is correct for loadings repeated a few dozen or even a few hundred times. However, it is not correct when loadings are repeated thousands or millions of times. In such cases, rupture can occur at a stress much lower than the ordinary static breaking strength; this phenomenon is known as fatigue. A fatigue failure is of a brittle nature, even for materials that are normally ductile.
δ = PL /AE
can be used only if the rod is homogeneous (constant E), has a uniform cross section of area A, and is loaded at its ends. If the rod is loaded at other points, or consists of several portions of various cross sections and possibly of different materials, it must be divided into component parts that satisfy the required conditions for the application of Eq. (2.9).
Bulk Modulus
characterizing the change in volume of a material under hydrostatic pressure
Anisotropic
exhibiting different mechanical properties in response to loads from different directions
ductile materials
including structural steel and many alloys of other materials, are characterized by their ability to yield at normal temperatures.
Isotropic Materials
materials that have the same optical properties when observed from any direction
Poisson's Ratio
relating lateral and axial strain
Loading Cycles
revolution of the wheel. The number of loading cycles required to cause the failure of a specimen through repeated loadings and reverse loadings can be determined experimentally for any given maximum stress level. If a series of tests is conducted using different maximum stress levels, the resulting data are plotted as a σ-n curve. For each test, the maximum stress σ is plotted as an ordinate and the number of cycles n as an abscissa. Because of the large number of cycles required for rupture, the cycles n are plotted on a logarithmic scale.
Fatigue Limit
the stress corresponding to failure after a specified number of loading cycles.
Deformations of Members Under Axial Loading
δ = PL /AE
Temperature Change Equation
δT = α(ΔT)L For example, consider a homogeneous rod AB of uniform cross section that rests freely on a smooth horizontal surface (Fig. 2.25a). If the temperature of the rod is raised by ΔT, the rod elongates by an amount δT. This elongation is proportional to both the temperature change ΔT and the length L of the rod (Fig. 2.25b). where α is a constant characteristic of the material called the coefficient of thermal expansion. Since δT and L are both expressed in units of length, α represents a quantity per degree C or per degree F, depending whether the temperature change is expressed in degrees Celsius or Fahrenheit.
Hooke's Law Equation
σ = Eε
