In the preceding examplem the stress normal to the interface AOB (Fig.2.2) was everywhere compressive and equal to w/2b except at the ends of the beam .The stress would have been tensile if the load w had been applied to the lower member. Such loading is unlikely, except when traveling cranes are suspended from the steelwork of a composite floor above: but there are other situations in which stresses tending to cause uplift can occur at the interface. These arise from complex effects such as the torsional stiffness of reinforced concrete slabs forming flanges of composite beams, the triaxial stresses in the vicinity of shear connectors and, in box-girder bridges, the torsional stiffness of the steel box.
Tension across the interface can also occur in beams of non-uniform, Section or with partially completed flanges. Two members without shear connection, as shown in Fig. 2.5,provide a simple example. AB. Is sup-ported on CD and carries distributed loading. It can easily be shown by elastic theory that if the flexural rigidity of AB exceeds about one-tenth of that of CD, then the whole of the load on AB is transferred to CD at points A and B, with separation of the beams between these points. If AB was connected to CD, there would be uplift forces at midspan.
Almost all connectors used in practice are therefore so shaped that they provide resistance to uplift as well as to slip. Uplift forces are so much less than shear forces that it is not normally necessary to calculate or estimate them for design purposes, provided that connectors with some uplift resistance are used.
The behavior of structural steels subjected to short-time loadings at elevated temperatures is usually determined from short-time tension tests. In general, the stress-strain curve becomes more rounded and the yield strength and tensile strength are reduced as temperatures are increased. The ratios of the elevated-temperature value to room-temperature value of yield and tensile strengths of several structural steels are shown in Fig. 1.9a and b, respectively.
Modulus of elasticity decreases with increasing temperature, as shown in Fig. 1.9c. The relationship shown is nearly the same for all structural steels. The variation in shear modulus with temperature is similar to that shown for the modulus of elasticity. But Poisson’s ratio does not vary over this temperature range.
The following expressions for elevated-temperature property ratios, which were derived by fitting curves to short-time data, have proven useful in analytical modeling (R. L. Brockenbrough, ‘‘Theoretical Stresses and Strains from Heat Curving,’’ Journal of the Structural Division, American Society of Civil Engineers, Vol. 96, No. ST7, 1970):
In these equations Fy / F’y and E/E’ are the ratios of elevated-temperature to room-temperature Fy yield strength and modulus of elasticity, respectively, is the coefficient of thermal expansion per degree Fahrenheit, and T is the temperature in degrees Fahrenheit.
Ductility of structural steels, as indicated by elongation and reduction-of-area values, decreases with increasing temperature until a minimum value is reached. Thereafter, ductility increases to a value much greater than that at room temperature. The exact effect depends on the type and thickness of steel. The initial decrease in ductility is caused by strain aging and is most pronounced in the temperature range of 300 to 700F. Strain aging also accounts for the increase in tensile strength in this temperature range shown for two of the steels in Fig. 1.9b.
Under long-time loadings at elevated temperatures, the effects of creep must be considered.
When a load is applied to a specimen at an elevated temperature, the specimen deforms rapidly at first but then continues to deform, or creep, at a much slower rate. A schematic creep curve for a steel subjected to a constant tensile load and at a constant elevated temperature is shown in Fig. 1.10. The initial elongation occurs almost instantaneously and is followed by three stages. In stage 1 elongation increases at a decreasing rate. In stage 2, elongation increases at a nearly constant rate. And in stage 3, elongation increases at an increasing rate. The failure, or creep-rupture, load is less than the load that would cause failure at that temperature in a short-time loading test.
Table 1.9 indicates typical creep and rupture data for a carbon steel, an HSLA steel, and a constructional alloy steel. The table gives the stress that will cause a given amount of creep in a given time at a particular temperature.
For special elevated-temperature applications in which structural steels do not provide adequate properties, special alloy and stainless steels with excellent high-temperature properties are available.
Tensile properties of structural steels are usually determined at relatively slow strain rates to obtain information appropriate for designing structures subjected to static loads. In the design of structures subjected to high loading rates, such as those caused by impact loads, however, it may be necessary to consider the variation in tensile properties with strain rate.
Figure 1.8 shows the results of rapid tension tests conducted on a carbon steel, two HSLA steels, and a constructional alloy steel. The tests were conducted at three strain rates and at three temperatures to evaluate the interrelated effect of these variables on the strength of the steels. The values shown for the slowest and the intermediate strain rates on the roomtemperature curves reflect the usual room-temperature yield stress and tensile strength, respectively.
(In determination of yield stress, ASTM E8 allows a maximum strain rate of 1⁄16
in per in per mm, or 1.04 x 10^3 in per in per sec. In determination of tensile strength, E8 allows a maximum strain rate of 0.5 in per in per mm, or 8.33 x 10^3 in per in per sec.)
The curves in Fig. 1.8a and b show that the tensile strength and 0.2% offset yield strength of all the steels increase as the strain rate increases at 50F and at room temperature. The greater increase in tensile strength is about 15%, for A514 steel, whereas the greatest increase in yield strength is about 48%, for A515 carbon steel. However, Fig. 1.8c shows that at 600F, increasing the strain rate has a relatively small influence on the yield strength. But a faster strain rate causes a slight decrease in the tensile strength of most of the steels.
Ductility of structural steels, as measured by elongation or reduction of area, tends to decrease with strain rate. Other tests have shown that modulus of elasticity and Poisson’s ratio do not vary significantly with strain rate.
In the fabrication of structures, steel plates and shapes are often formed at room temperatures into desired shapes. These cold-forming operations cause inelastic deformation, since the steel retains its formed shape. To illustrate the general effects of such deformation on strength and ductility, the elemental behavior of a carbon-steel tension specimen subjected to plastic deformation and subsequent tensile reloadings will be discussed. However, the behavior of actual cold-formed structural members is more complex.
As illustrated in Fig. 1.6, if a steel specimen is unloaded after being stressed into either the plastic or strain-hardening range, the unloading curve follows a path parallel to the elastic portion of the stress-strain curve. Thus a residual strain, or permanent set, remains after the load is removed. If the specimen is promptly reloaded, it will follow the unloading curve to the stress-strain curve of the virgin (unstrained) material.
If the amount of plastic deformation is less than that required for the onset of strain hardening, the yield stress of the plastically deformed steel is about the same as that of the virgin material. However, if the amount of plastic deformation is sufficient to cause strain hardening, the yield stress of the steel is larger. In either instance, the tensile strength remains the same, but the ductility, measured from the point of reloading, is less. As indicated in Fig. 1.6, the decrease in ductility is nearly equal to the amount of inelastic prestrain.
A steel specimen that has been strained into the strain-hardening range, unloaded, and allowed to age for several days at room temperature (or for a much shorter time at a moderately elevated temperature) usually shows the behavior indicated in Fig. 1.7 during reloading.
This phenomenon, known as strain aging, has the effect of increasing yield and tensile strength while decreasing ductility.
Most of the effects of cold work on the strength and ductility of structural steels can be eliminated by thermal treatment, such as stress relieving, normalizing, or annealing. However, such treatment is not often necessary.
(G. E. Dieter, Jr., Mechanical Metallurgy, 3rd ed., McGraw-Hill, Inc., New York.)
The ratio of shear stress to shear strain during initial elastic behavior is the shear modulus G. According to the theory of elasticity, this quantity is related to the modulus of elasticity E and Poisson’s ratio V by
Thus a minimum value of G for structural steels is about 11 x 10^3 ksi. The yield stress in shear is about 0.57 times the yield stress in tension. The shear strength, or shear stress at failure in pure shear, varies from two-thirds to three-fourths the tensile strength for the various steels. Because of the generally consistent relationship of shear properties to tensile properties for the structural steels, and because of the difficulty of making accurate shear tests, shear tests are seldom performed.
The tensile properties of steel are generally determined from tension tests on small specimens or coupons in accordance with standard ASTM procedures. The behavior of steels in these tests is closely related to the behavior of structural-steel members under static loads. Because, for structural steels, the yield points and moduli of elasticity determined in tension and compression are nearly the same, compression tests are seldom necessary.
Typical tensile stress-strain curves for structural steels are shown in Fig. 1.1. The initial portion of these curves is shown at a magnified scale in Fig. 1.4. Both sets of curves may be referred to for the following discussion.
Strain Ranges. When a steel specimen is subjected to load, an initial elastic range is observed in which there is no permanent deformation. Thus, if the load is removed, the specimen returns to its original dimensions. The ratio of stress to strain within the elastic range is the modulus of elasticity, or Young’s modulus E. Since this modulus is consistently about 29 103 ksi for all the structural steels, its value is not usually determined in tension tests, except in special instances.
The strains beyond the elastic range in the tension test are termed the inelastic range.
For as-rolled and high-strength low-alloy (HSLA) steels, this range has two parts. First observed is a plastic range, in which strain increases with no appreciable increase in stress.
This is followed by a strain-hardening range, in which strain increase is accompanied by a significant increase in stress. The curves for heat-treated steels, however, do not generally exhibit a distinct plastic range or a large amount of strain hardening.
The strain at which strain hardening begins (st) and the rate at which stress increases with strain in the strain-hardening range (the strain-hardening modulus Est) have been determined for carbon and HSLA steels. The average value of Est is 600 ksi, and the length of the yield plateau is 5 to 15 times the yield strain. (T. V. Galambos, ‘‘Properties of Steel for Use in LRFD,’’ Journal of the Structural Division, American Society of Civil Engineers, Vol. 104, No. ST9, 1978.) Yield Point, Yield Strength, and Tensile Strength. As illustrated in Fig. 1.4, carbon and HSLA steels usually show an upper and lower yield point. The upper yield point is the value usually recorded in tension tests and thus is simply termed the yield point.
The heat-treated steels in Fig. 1.4, however, do not show a definite yield point in a tension test. For these steels it is necessary to define a yield strength, the stress corresponding to a specified deviation from perfectly elastic behavior. As illustrated in the figure, yield strength is usually specified in either of two ways: For steels with a specified value not exceeding 80 ksi, yield strength is considered as the stress at which the test specimen reaches a 0.5% extension under load (0.5% EUL) and may still be referred to as the yield point. For higherstrength steels, the yield strength is the stress at which the specimen reaches a strain 0.2% greater than that for perfectly elastic behavior.
Since the amount of inelastic strain that occurs before the yield strength is reached is quite small, yield strength has essentially the same significance in design as yield point.
These two terms are sometimes referred to collectively as yield stress.
The maximum stress reached in a tension test is the tensile strength of the steel. After this stress is reached, increasing strains are accompanied by decreasing stresses. Fracture eventually occurs.
Proportional Limit. The proportional limit is the stress corresponding to the first visible departure from linear-elastic behavior. This value is determined graphically from the stressstrain curve. Since the departure from elastic action is gradual, the proportional limit depends greatly on individual judgment and on the accuracy and sensitivity of the strain-measuring devices used. The proportional limit has little practical significance and is not usually recorded in a tension test. Ductility. This is an important property of structural steels. It allows redistribution of stresses in continuous members and at points of high local stresses, such as those at holes or other discontinuities.
In a tension test, ductility is measured by percent elongation over a given gage length or percent reduction of cross-sectional area. The percent elongation is determined by fitting the specimen together after fracture, noting the change in gage length and dividing the increase by the original gage length. Similarly, the percent reduction of area is determined from crosssectional measurements made on the specimen before and after testing.
Both types of ductility measurements are an index of the ability of a material to deform in the inelastic range. There is, however, no generally accepted criterion of minimum ductility for various structures. Poisson’s Ratio. The ratio of transverse to longitudinal strain under load is known as Poisson’s ratio . This ratio is about the same for all structural steels—0.30 in the elastic range and 0.50 in the plastic range.
True-Stress–True-Strain Curves. In the stress-strain curves shown previously, stress values were based on original cross-sectional area, and the strains were based on the original gauge length. Such curves are sometimes referred to as engineering-type stress-strain curves.
However, since the original dimensions change significantly after the initiation of yielding, curves based on instantaneous values of area and gage length are often thought to be of more fundamental significance. Such curves are known as true-stress–true-strain curves.
A typical curve of this type is shown in Fig. 1.5.
The curve shows that when the decreased area is considered, the true stress actually increases with increase in strain until fracture occurs instead of decreasing after the tensile strength is reached, as in the engineering stress-strain curve. Also, the value of true strain at fracture is much greater than the engineering strain at fracture (though until yielding begins true strain is less than engineering strain).