STRESS AND STRAIN OF STRUCTURAL MATERIALS DEFINITION AND BASIC INFORMATION

Stress.

Stress is the intensity at a point in a body of the internal forces or components of force that act on a given plane through the point. Stress is expressed in force per unit of area (pounds per square inch, kilograms per square millimeter, etc.).

There are three kinds of stress: tensile, compressive, and shearing.

Flexure involves a combination of tensile and compressive stress. Torsion involves shearing stress. It is customary to compute stress on the basis of the original dimensions of the cross section of the body, though “true stress” in tension or compression is sometimes calculated from the area of the time a given stress exists rather than from the original area.

Strain.

Strain is a measure of the change, due to a force, in the size or shape of a body referred to its original size or shape. Strain is a nondimensional quantity but is frequently expressed in inches per inch, etc.

Under tensile or compressive stress, strain is measured along the dimension under consideration. Shear strain is defined as the tangent of the angular change between two lines originally perpendicular to each other.

Stress-Strain Diagram.

A stress-strain diagram is a diagram plotted with values of stress as ordinates and values of strain as abscissas. Diagrams plotted with values of applied load, moment, or torque as ordinates and with values of deformation, deflection, or angle of twist as abscissas are sometimes referred to as stress-strain diagrams but are more correctly called load-deformation diagrams.

The stress-strain diagram for some materials is affected by the rate of application of the load, by cycles of previous loading, and again by the time during which the load is held constant at specified values; for precise testing, these conditions should be stated definitely in order that the complete significance of any particular diagram may be clearly understood.

Modulus of Elasticity.

The modulus of elasticity is the ratio of stress to corresponding strain below the proportional limit. For many materials, the stress-strain diagram is approximately a straight line below a more or less well-defined stress known as the proportional limit.

Since there are three kinds of stress, there are three moduli of elasticity for a material, that is, the modulus in tension, the modulus in compression, and the modulus in shear.

The value in tension is practically the same, for most ductile metals, as the modulus in compression; the modulus in shear is only about 0.36 to 0.42 of the modulus in tension.

The modulus is expressed in pounds per square inch (or kilograms per square millimeter) and measures the elastic stiffness (the ability to resist elastic deformation under stress) of the material.

ALLOWABLE STRESSES FOR LUMBER BASIC INFORMATION AND TUTORIALS

The National Design Specification for Wood Construction (NDS) (AF&PA, 1997) makes comprehensive recommendations for engineered uses of stress-graded lumber. Stress values for all commercially available species groups and grades of lumber produced in the U.S. are tabulated in the NDS.

The moduli of elasticity for all species groups and grades are also included in these tables. These tabulated values of stresses and moduli of elasticity are called base design values. They are modified by applying adjustment factors to give allowable stresses for the graded lumber.

The adjustment factors reduce (or in some cases increase) the base design stress values to account for specific conditions of use that affect the behavior of the lumber. A list of these adjustment factors and a discussion of their use follows.

Load Duration — CD

The stress level that wood will safely sustain is inversely proportional to the duration that the stress is applied. That is, stress applied for a very short time (e.g., an impact load) can have a higher value than stress applied for a longer duration and still be safely carried by a wood member. This characteristic of wood is accounted for in determining allowable stresses by using a load duration factor, CD.

The load duration factor varies from 20 for an impact load (duration equal to one second) to 0.9 for a permanent load (duration longer than 10 years). ACI Committee 347 recommends that for concrete formwork, a load duration factor appropriate for a load of 7 days should be used. This corresponds to a value for CD of 1.25.

ACI Committee 347 says this load duration factor should only be applied to concrete forms intended for limited reuse. No precise definition of limited reuse is given by the ACI committee, but the no increase for duration of load should be used for concrete forms designed to be reused a high number of cycles.

Moisture — CM

Wood is affected by moisture content higher than about 19%. Higher moisture content significantly softens the wood fibers and makes it less stiff and less able to carry stresses. The reduction in allowable strength depends on the type of stress (e.g., shear stress is affected less than perpendicular to grain compressive stress) and the grade of the lumber.

Size — CF

Research on lumber allowable stresses has shown that as cross-sectional size increases, allowable stresses are reduced. A size factor, CF, is used to increase base design values for different sizes of lumber.

Repetitive Members — Cr

The NDS allows bending stresses to be increased for beams that share their loads with other beams. The increased allowable stress is referred to as a repetitive member stress. For a beam to qualify as a repetitive member, it must be one of at least three members spaced no further apart than two feet and joined by a load-distributing element such as plywood sheathing.

When these three requirements are met, the allowable bending stress can be increased by 15%. This corresponds to a value for Cr of 1.15. Repetitive member stresses may be appropriate for some formwork components. Because the intent of allowing increased stress for repetitive flexural members is to take advantage of the load sharing provided by continuity, gang panels assembled securely by bolting or nailing and intended for multiple reuse would seem to qualify for this increase.

ACI Committee 347 specifies that they should not be used where the bending stresses have already been increased by 25% for short duration loads.

Perpendicular to Grain Compression — Cb

Allowable perpendicular to grain bearing stress at the ends of a beam may be adjusted for length of bearing according to: lb is the length of bearing parallel to grain.

Horizontal Shear Constant — CH

Shear stress in lumber beams used as components of concrete forms is usually highest at the ends of the members. For beams having limited end defects (e.g., splits, checks, cracks), the values of allowable shear stress can be increased. This is done by using a shear constant CH that depends on the size of end defects and varies from 1 to 2.

Temperature — CT

Sustained high temperatures adversely affect some properties of wood. It is unusual for concrete forms to be exposed to temperatures high enough to require the use of a temperature adjustment factor. For temperatures in excess of 100°F, the stresses and moduli should be adjusted using CT.

Stablity — CP

Like all columns, wood shores will safely carry axial loads in inverse proportion to their effective slenderness. The more slender a wood shore is, the less load it will support because of the increased influence of buckling. Prior to the 1997 edition of the NDS, wood columns were divided into three categories (short, intermediate, and long) according to their slenderness.

Allowable stresses and loads were then found using three different formulas — one for each category. Beginning with the 1997 NDS, allowable loads for all wood columns are found using a stability adjustment factor, CP, that reduces the base stress to account for the buckling tendency of the column. It is no longer necessary to divide wood shores into three categories to find allowable loads.

POISSON’S RATIO BASIC INFORMATION AND TUTORIAL

POISSON'S RATIO TUTORIALS AND SAMPLE PROBLEM
What Is Poisson's Ratio? Sample Problem And Solution Using Poisson's Ratio


When a homogeneous slender bar is axially loaded, the resulting stress and strain satisfy Hooke’s law, as long as the elastic limit of the material is not exceeded.

In all engineering materials, the elongation produced by an axial tensile force P in the direction of the force is accompanied by a contraction in any transverse direction (Fig. 2.36).† In this section and the following sections (Secs. 2.12 through 2.15), all materials considered will be assumed to be both homogeneous and isotropic, i.e., their mechanical properties will be assumed independent of both position and direction.

It follows that the strain must have the same value for any transverse direction.\ Therefore, for the loading shown in Fig. 2.35 we must have Py 5 Pz. This common value is referred to as the lateral strain.

An important constant for a given material is its Poisson’s ratio, named after the French mathematician Siméon Denis Poisson (1781–1840) and denoted by the Greek letter n (nu). It is defined as

v = - lateral strain / lateral stress.

Sample Problem:


A 500-mm-long, 16-mm-diameter rod made of a homogenous, isotropic material is observed to increase in length by 300 mm, and to decrease in diameter by 2.4 mm when subjected to an axial 12-kN load. Determine the modulus of elasticity and Poisson’s ratio of the material.

Solution:

Click on the image to enlarge and see the solution.

HOOKE’S LAW; MODULUS OF ELASTICITY BASICS AND TUTORIALS

HOOKE’S LAW; MODULUS OF ELASTICITY BASIC INFORMATION
What Is Hooke's Law? How To Apply Hooke's Law?


Most engineering structures are designed to undergo relatively small deformations, involving only the straight-line portion of the corresponding stress-strain diagram. For that initial portion of the diagram, the stress s is directly proportional to the strain P, and we can write

σ = ΞE

This relation is known as Hooke’s law, after Robert Hooke (1635–1703), an English scientist and one of the early founders of applied mechanics. The coefficient E is called the modulus of elasticity of the material involved, or also Young’s modulus, after the English scientist Thomas Young (1773–1829). 

Since the strain P is a dimensionless quantity, the modulus E is expressed in the same units as the stress s, namely in pascals or one of its multiples if SI units are used, and in psi or ksi if U.S. customary units are used.

The largest value of the stress for which Hooke’s law can be used for a given material is known as the proportional limit of that material. In the case of ductile materials possessing a well-defined yield poin, the proportional limit almost coincides with the yield point. 

For other materials, the proportional limit cannot be defined as easily, since it is difficult to determine with accuracy the value of the 63 stress s for which the relation between s and P ceases to be linear. But from this very difficulty we can conclude for such materials that using Hooke’s law for values of the stress slightly larger than the actual proportional limit will not result in any significant error.

Some of the physical properties of structural metals, such as strength, ductility, and corrosion resistance, can be greatly affected by alloying, heat treatment, and the manufacturing process used. For example, we note from the stress-strain diagrams of pure iron and of three different grades of steel (Fig. 2.11) that large variations in the yield strength, ultimate strength, and final strain (ductility) exist among these four metals. 

All of them, however, possess the same modulus of elasticity; in other words, their “stiffness,” or ability to resist a deformation within the linear range, is the same. Therefore, if a high-strength steel is substituted for a lower-strength steel in a given structure, and if all dimensions are kept the same, the structure will have an increased load-carrying capacity, but its stiffness will remain unchanged.

For each of the materials considered so far, the relation between normal stress and normal strain, s 5 EP, is independent of the direction of loading. This is because the mechanical properties of each material, including its modulus of elasticity E, are independent of the direction considered. Such materials are said to be isotropic.

Materials whose properties depend upon the direction considered are said to be anisotropic. An important class of anisotropic materials consists of fiberreinforced composite materials. These composite materials are obtained by embedding fibers of a strong, stiff material into a weaker, softer material, referred to as a matrix. 

Typical materials used as fibers are graphite, glass, and polymers, while various types of resins are used as a matrix. Figure 2.12 shows a layer, or lamina, of a composite material consisting of a large number of parallel fibers embedded in a matrix.

An axial load applied to the lamina along the x axis, that is, in a direction parallel to the fibers, will create a normal stress sx in the lamina and a corresponding normal strain Px which will satisfy Hooke’s law as the load is increased and as long as the elastic limit of the lamina is not exceeded. 

Similarly, an axial load applied along the y axis, that is, in a direction perpendicular to the lamina, will create a normal stress sy and a normal strain Py satisfying Hooke’s law, and an axial load applied along the z axis will create a normal stress sz and a normal strain Pz which again satisfy Hooke’s law. 

However, the moduli of elasticity Ex, Ey, and Ez corresponding, respectively, to each of the above loadings will be different. Because the fibers are parallel to the x axis, the lamina will offer a much stronger resistance to a loading directed along the x axis than to a loading directed along the y or z axis, and Ex will be much 

larger than either Ey or Ez.

A flat laminate is obtained by superposing a number of layers or laminas. If the laminate is to be subjected only to an axial load causing tension, the fibers in all layers should have the same orientation as the load in order to obtain the greatest possible strength. But if the laminate may be in compression, the matrix material may not be sufficiently strong to prevent the fibers from kinking or buckling. 

The lateral stability of the laminate may then be increased by positioning some of the layers so that their fibers will be perpendicular to the load. Positioning some layers so that their fibers are oriented at 308,

45 deg, or 60 deg to the load may also be used to increase the resistance of the laminate to in-plane shear. 

STRESS - STRAIN CURVE FOR BRITTLE MATERIALS BASICS AND TUTORIALS

STRESS - STRAIN CURVE FOR BRITTLE MATERIALS BASIC INFORMATION
What Is The Stress-Strain Curve Of Brittle Materials?


Many of the characteristics of a material can be deduced from the tensile test. It is more convenient to compare materials in terms of stresses and strains, rather than loads and extensions of a particular specimen of a material.

The stress at any stage is the ratio of the load of the original cross-sectional area of the test specimen; the strain is the elongation of a unit length of the test specimen.

For stresses up to about 750 MN/m2 the stress-strain curve is linear, showing that the material obeys Hooke’s law in this range; the material is also elastic in this range, and no permanent extensions remain after removal of the stresses.

The ratio of stress to strain for this linear region is usually about 200 GN/m2 for steels; this ratio is known as Young’s modulus and is denoted by E. The strain at the limit of proportionality is of the order 0.003, and is small compared with strains of the order 0.100 at fracture.

FIG 1.5

We note that Young’s modulus has the units of a stress; the value of E defines the constant in the linear relation between stress and strain in the elastic range of the material. We have

for the linear-elastic range. If P is the total tensile load in a bar, A its cross-sectional area, and Lo its length, then

where e is the extension of the length Lo. Thus the expansion is given by

If the material is stressed beyond the linear-elastic range the limit of proportionality is exceeded, and the strains increase non-linearly with the stresses. Moreover, removal of the stress leaves the material with some permanent extension; h range is then bothnon-linear and inelastic.

The maximum stress attained may be of the order of 1500 MNlm’, and the total extension, or elongation, at this stage may be of the order of 10%. The curve of Figure 1.5 is typical of the behaviour of brittle materials as, for example, area characterized by small permanent elongation at the breaking point; in the case of metals this is usually lo%, or less.

When a material is stressed beyond the limit of proportionality and is then unloaded, permanent deformations of the material take place. Suppose the tensile test-specimen of Figure 1.5 is stressed beyond the limit of proportionality, to a point b on the stress-strain diagram. If the stress is now removed, the stress-strain relation follows the curve bc; when the stress is completely removed there is a residual strain given by the intercept Oc on the &-axis.

If the stress is applied again, the stress-strain relation follows the curve cd initially, and finally the curve df to the breaking point. Both the unloading curve bc and the reloading curve cd are approximately parallel to the elastic line Oa; they are curved slightly in opposite directions.

The process of unloading and reloading, bcd, had little or no effect on the stress at the breaking point, the stress-strain curve being interrupted by only a small amount bd, Figure 1.6. The stress-strain curves of brittle materials for tension and compression are usually similar in form, although the stresses at the limit of proportionality and at fracture may be very different for the two loading conditions.

Typical tensile and compressive stress-strain curves for concrete are shown in Figure 1.7; the maximum stress attainable in tension is only about one-tenth of that in compression, although the slopes of the stress strain curves in the region of zero stress are nearly equal.

STEEL STRUCTURES BRITTLE FRACTURES UNDER IMPACT LOAD BASICS AND TUTORIALSER

BRITTLE FRACTURES OF STEEL STRUCTURES UNDER IMPACT LOAD BASIC INFORMATION
What Are Brittle Fractures Of Steel Structures?


Structural steel does not always exhibit a ductile behaviour, and under some circumstances a sudden and catastrophic fracture may occur, even though the nominal tensile stresses are low. Brittle fracture is initiated by the existence or formation of a small crack in a region of high local stress.

Once initiated, the crack may propagate in a ductile (or stable) fashion for which the external forces must supply the energy required to tear the steel. More serious are cracks which propagate at high speed in a brittle (or unstable) fashion, for which some of the internal elastic strain energy stored in steel is released and used to fracture the steel.

Such a crack is self-propagating while there is sufficient internal strain energy, and will continue until arrested by ductile elements in its path which have sufficient deformation capacity to absorb the internal energy released.

The resistance of a structure to brittle fracture depends on the magnitude of local stress concentrations, on the ductility of the steel, and on the three-dimensional geometrical constraints. High local stresses facilitate crack initiation, and so stress concentrations due to poor geometry and loading arrangements (including impact loading) are dangerous.

Also of great importance are flaws and defects in the material, which not only increase the local stresses, but also provide potential sites for crack initiation.

The ductility of a structural steel depends on its composition, heat treatment, and thickness, and varies with temperature and strain rate. Figure 1.11 shows the increase with temperature of the capacity of the steel to absorb energy during impact.

At low temperatures the energy absorption is low and initiation and propagation of brittle fractures are comparatively easy, while at high temperatures the energy absorption is high because of ductile yielding, and the propagation of cracks can be arrested.

Between these two extremes is a transitional range in which crack initiation becomes increasingly difficult. The likelihood of brittle fracture is also increased by high strain rates due to dynamic loading, since the consequent increase in the yield stress reduces the possibility of energy absorption by ductile yielding.

The chemical composition of steel has a marked influence on its ductility: brittleness is increased by the presence of excessive amounts of most non-metallic elements, while ductility is increased by the presence of some metallic elements.


Steel with large grain size tends to be more brittle, and this is significantly influenced by heat treatment of the steel, and by its thickness (the grain size tends to be larger in thicker sections). EC3-1-10 [18] provides values of the maximum thickness t1 for different steel grades and minimum service temperatures, as well as advice on using a more advanced fracture mechanics [34] based approach and guidance on safeguarding against lamellar tearing.

Three-dimensional geometrical constraints, such as those occurring in thicker or more massive elements, also encourage brittleness, because of the higher local stresses, and because of the greater release of energy during cracking and the consequent increase in the ease of propagation of the crack.

The risk of brittle fracture can be reduced by selecting steel types which have ductilities appropriate to the service temperatures, and by designing joints with a view to minimising stress concentrations and geometrical constraints.

Fabrication techniques should be such that they will avoid introducing potentially dangerous flaws or defects. Critical details in important structures may be subjected to inspection procedures aimed at detecting significant flaws.

Of course the designer must give proper consideration to the extra cost of special steels, fabrication techniques, and inspection and correction procedures.

STRESS AND STRAIN BEHAVIOR OF STRUCTURAL STEEL BASICS AND TUTORIALS

STRUCTURAL STEEL STRESS AND STRAIN BEHAVIOR
Stress and Strain Behavior Of Structural Steel Tutorials


Structural steel is an important construction material. It possesses attributes such as strength, stiffness, toughness, and ductility that are very desirable in modern constructions.

Strength is the ability of a material to resist stresses. It is measured in terms of the material’s yield strength, Fy, and ultimate or tensile strength, Fu.

For steel, the ranges of Fy and Fu ordinarily used in constructions are 36 to 50 ksi (248 to 345 MPa) and 58 to 70 ksi (400 to 483 MPa), respectively, although higher strength steels are becoming more common. Stiffness is the ability of a material to resist deformation. It is measured as the slope of the material’s stress-strain curve.

With reference to Figure 3.1 in which uniaxial engineering stress-strain curves obtained from coupon tests for various grades of steels are shown, it is seen that the modulus of elasticity, E, does not vary appreciably for the different steel grades.

Therefore, a value of 29,000 ksi (200 GPa) is often used for design. Toughness is the ability of a material to absorb energy before failure. It is measured as the area under the material’s stress-strain curve.

As shown in Figure 3.1, most (especially the lower grade) steels possess high toughness which is suitable for both static and seismic applications. Ductility is the ability of a material to undergo large inelastic, or plastic, deformation before failure.

It is measured in terms of percent elongation or percent reduction in area of the specimen tested in uniaxial tension. For steel, percent elongation ranges from around 10 to 40 for a 2-in. (5-cm) gage length specimen.

Ductility generally decreases with increasing steel strength. Ductility is a very important attribute of steel.

The ability of structural steel to deform considerably before failure by fracture allows an indeterminate structure to undergo stress redistribution. Ductility also enhances the energy absorption characteristic of the structure, which is extremely important in seismic design.

HIGH STRENGTH BOLT AND NUTS INSTALLATION BASICS AND TUTORIALS

GUIDE IN INSTALLATION OF HIGH STRENGTH BOLTS AND NUTS
What Is the Proper Way Of Installing Nuts and Bolts?


Washer requirements for high-strength bolted assemblies depend on the method of installation and type of bolt holes in the connected elements. These requirements are summarized in Table 7.5.

Bolt Tightening. Specifications require that all high-strength bolts be tightened to 70% of their specified minimum tensile strength, which is nearly equal to the proof load (specified lower bound to the proportional limit) for A325 bolts, and within 10% of the proof load for A490 bolts.

Tightening above these minimum tensile values does not damage the bolts, but it is prudent to avoid excessive uncontrolled tightening. The required minimum tension, kips, for A325 and A490 bolts is given in Table 7.6.

There are three methods for tightening bolts to assure the prescribed tensioning:

Turn-of-Nut.
By means of a manual or powered wrench, the head or nut is turned from an initial snug-tight position. The amount of rotation, varying from one-third to a full turn, depends on the ratio of bolt length (underside of heat to end of point) to bolt diameter and on the disposition of the outer surfaces of bolted parts (normal or sloped not more than 1:20 with respect to the bolt axis). Required rotations aretabulated in the ‘‘Specification for Structural Steel Joints Using A325 of A490 Bolts.’’

Calibrated Wrench.
By means of a powered wrench with automatic cutoff and calibration on the job. Control and test are accomplished with a hydraulic device equipped with a gage that registers the tensile stress developed.

Direct Tension Indicator.
Special indicators are permitted on satisfactory demonstration of performance. One example is a hardened steel washer with protrusions on one face. The flattening that occurs on bolt tightening is measured and correlated with the induced tension.

STRESS - STRAIN RELATIONS CIVIL ENGINEERING BASICS AND TUTORIALS

STRESS - STRAIN RELATIONS BASIC INFORMATION
What is Stress - Strain Relations?

Materials deform in response to loads or forces. In 1678, Robert Hooke published the first findings that documented a linear relationship between the amount of force applied to a member and its deformation.

The amount of deformation is proportional to the properties of the material and its dimensions. The effect of the dimensions can be normalized. Dividing the force by the cross-sectional area of the specimen normalizes the effect of the loaded area.

The force per unit area is defined as the stress in the specimen (i.e., ). Dividing the deformation by the original length is defined as strain of the specimen (i.e., length/original length). Much useful information about the material can be determined by plotting the stress–strain diagram.

Figure 1.2 shows typical uniaxial tensile or compressive stress–strain curves for several engineering materials. Figure 1.2(a) shows a linear stress–strain relationship up to the point where the material fails. Glass and chalk are typical of materials exhibiting this tensile behavior.

Figure 1.2(b) shows the behavior of steel in tension. Here, a linear relationship is obtained up to a certain point (proportional limit), after which the material deforms without much increase in stress.

On the other hand, aluminum alloys in tension exhibit a linear stress–strain relation up to the proportional\ limit, after which a nonlinear relation follows, as illustrated in Figure 1.2(c).

Figure 1.2(d) shows a nonlinear relation throughout the whole range. Concrete and other materials exhibit this relationship, although the first portion of the curve for concrete is very close to being linear.

Soft rubber in tension differs from most materials in such a way that it shows an almost linear stress–strain relationship followed by a reverse curve, as shown in Figure 1.2(e).