2.9 PROPERTIES IN TENSION
While concrete is best employed in a manner that uses its favorable compressive strength, its behavior in tension is also important. The conditions under which cracks form and propagate on the tension side of reinforced concrete flexural members depend strongly on both the tensile strength and the fracture properties of the concrete, the latter dealing with the ease with which a crack progresses once it has formed. Concrete tensile stresses also occur as a result of shear, torsion, and other actions, and in most cases member behavior changes upon cracking. Thus, it is important to be able to predict, with reasonable accuracy, the tensile strength of concrete and to understand the factors that control crack propagation.
a. Tensile Strength
There are considerable experimental difficulties in determining the true tensile strength of concrete. In direct tension tests, minor misalignments and stress concentrations in the gripping devices are apt to mar the results. For many years, tensile strength has been measured in terms of the modulus of rupture fr, the computed flexural tensile stress at which a test beam of plain concrete fractures. Because this nominal stress is computed on the assumption that concrete is an elastic material, and because this bending stress is localized at the outermost surface, it is apt to be larger than the strength of concrete in uniform axial tension. It is thus a measure of, but not identical with, the real axial tensile strength.
More recently the result of the so-called split-cylinder test has established itself as a measure of the tensile strength of concrete. A 6times;12 in. concrete cylinder , the same as is used for compressive tests, is inserted in a compression testing machine in the horizontal position, so that compression is applied uniformly along two opposite generators. Pads are inserted between the compression platens of the machine and the cylinder to equalize and distribute the pressure. It can be shown that in an elastic cylinder so loaded, a nearly uniform tensile stress of magnitude 2P/pi;dL exists at right angles to the plane of load application. Correspondingly, such cylinders, when tested, split into two halves along that plane, at a stress fct that can be computed from the above expression. P is the applied compressive load at failure, and d and L are the diameter and length of the cylinder respectively. Because of local stress conditions at the load lines and the presence of stresses at right angles to the aforementioned tension stresses, the results of the split -cylinder tests likewise are not identical with ( but are believed t o be a good measure of ) the true axial tensile strength. The results of all types of tensile tests show considerably more scatter than those of compression tests.
Tensile strength, however determined, does not correlate well with the compressive strength frsquo;c. It appears that for sand-and-gravel concrete, the tensile strength depends primarily on the strength of bond between hardened cement paste and aggregate, whereas for lightweight concret es it depends largely on the tensile strength of the porous aggregate. The compressive strength, on the other hand, is much less determined by these particular characteristics.
Better correlation is found between the various measures of tensile strength and the square root of the compressive strength. The direct tensile strength, for example, ranges from about 3 to 5 root f′c for normal-density concretes, and from about 2 to 3 root f′c for all-lightweight concrete. Typical ranges of values for direct tensile strength, splitcylinder strength, and modulus of rupture are summarized in Table 2.2. In these expressions, f′c is expressed in psi units, and the resulting tensile strengths are obtained in psi.
These approximate expressions show that tensile and compressive strengths are by no means proportional, and that any increase in compressive strength, such as that achieved by lowering the water-cement ratio, is accompanied by a much smaller percentage increase in tensile strength.
The ACI Code contains the recommendat ion that the modulus of rupture f′r be taken to equal 7.5 f′c for normal-weight concrete, and that this value be multiplied by 0.85 for “sand-lightweight” and 0.75 for “all-lightweight” concretes, giving values of 6.4 root f′c and 5.6 root f′c respectively for those materials.
b. Tensile Fracture
The failure of concrete in tension involves both the formation and propagation of cracks. The field of fracture mechanics deals with the latter. While reinforced concrete structures have been successfully designed and built for over 150 years without the use of fracture mechanics, the brittle response of high-strength concretes ( Section 2.12), in tension as well as compression, increases the importance of the fracture properties of the material as distinct from tensile strength. Research dealing with the shear strength of high-strength concrete beams and the bond between reinforcing steel and high-strength concrete indicates relatively low increases in these structural properties with increases in concrete compressive strength ( Refs. 2.28 and 2.29). While shear and bond strength are associated with the root f′c for normal-strength concrete, tests of high-strength concrete indicate that increases in shear and bond strengths are well below values predicted using root f′c, indicating that concrete tensile strength alone is not the governing factor. An explanation for this behavior is provided by research at the University of Kansas and elsewhere ( Refs. 2.30 and 2.31) that demonstrates that the energy required to fully open a crack ( i.e., after the crack has started to grow ) is largely independent of compressive strength, water-cement ratio, and age. Design expressions reflecting this research are not yet available. The behavior is, however, recognized in the ACI Code by limitations on the
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Alignment Design
The alignment of a road is shown on the plan view and is a series of straight lines called tangents connected by circular curves. In modern practice it is common to interpose transition or sprial curves between tangents tangents and circular curves.
Alignment must be consistent. Sudden changes from flat to sharp curves and long tangents followed by sharp curves are to be avoided; otherwise accident hazards are created. Likewise, placing circular curves of different radii end to end (compound curves) or having a short tangent between two curves is poor practice unless suitable transition between them are provided. Long, flat curves are preferable at all times, as they are pleasing in appearance and decrease the possibility of future obsolescence. However, alignment without tangents is undesirable on two-lane roads because some drivers hesitate to pass on curves. Long, flat curves should be used for small changes in direction, as short curves appear as “kinks”. Also, as indicated above, horizontal and vertical alignment must be considered together, not separately. For example, a sharp horizontal curver beginning near a crest can create a serious accident hazard.
A vehicle traveling in a curved path is subject to centrifugal force. This is balanced by an equal and opposite force developed through superelevation and side friction. From a highway design standpoint, neither superelevation nor side friction can exceed certain maximums, and these controls place limits on the sharpness of a given circular curve is indicated by its radius. However, for alignment design, sharpness is commonly expressed in terms of degree of curve. Degree of curve is inversely proportional to the radius.
Tangent sections of highway carry normal cross slope; curved sections are superelevated. Provision must be made for gradual change from one to other. A common method is to maintain the centerline of each individual roadway at profile grade while raising the outer edge and lowering the inner edge to produce the desired superelevation. This involves first raising the outside edge of the pavement with relation to the centerline until the outer half of the cross section is flat, next the outer edge is raised further until the cross section is straight, then the entire cross section is rotated as a unit until full superelevation is reached. For smoother riding, A Policy on Geometric Design recommends that short vertical curves having a length in feet equal to the design speed in miles per hour be introduced into the edge profile at their break points.
Where the alignment consists of tangents connected by circular curves, introduction of superelevation usually is begun on tangent before the curve is reached and full superelevation is attached some distance beyond the point of curve. It is recommended that 60 to 80% of the run off be on tangent.
Where the alignment includes easement curve (see below), superelevation is applied entirely on the easement curve, except for bringing the outer pavement edge to a level position. It follows that, at times, the superelevation application rate sets the minimum length of the easement curve.
If a vehicle at high speed on a carefully restricted path made up of tangents connected by sharp circular curves, riding is extremely uncomfortable. As the car approaches a curve, superelevation begins and the vehicle is tilted inward, but the passengers must remain vertical since there is no centrifugal force requiring compensation. When the vehicle the curve, full centrifugal force develops at once, and pulls the riders outward form their vertical positions. To achieve a position of equilibrium, the riders must force their bodies far inward. As the remaining superelevation takes effect, future adjustments in position are required. This process is repeated in reverse order as the vehicle leaves the curve. When easement curve are employed, the change in radius form infinity on the tangent to that of the circular curve is effected gradually so that centrifugal force also develops gradually. By introducing superelecation along the spiral, a smooth and gradual application of centrifugal force can be had and the roughness avoided.
Easement curves have been used by the railroads for many years, but their adoption by highway agencies came much later. Many agencies do not use them today. This is understandable. Railroad trains must follow the precise alignment of the tracks, and the discomfort describe here can be avoided only by adopting easement curves. On the other hand, motor-vehicle operators are free to alter their lateral positions on the road and can provide their own easement curves by steering into circular curves gradually. However, this weaving within a traffic lane (sometimes into their lanes) is dangerous. Properly designed easement curves make weaving unnecessary. Safety, then, is an argument favoring them. Another is that they give alignments a smother, more flowing appearance.
The points beginning of an ordinary circular curve is usually labeled the PC (point of curve) or BC (beginning of curve). Its end is marked the PT (point of tangency) of EC (end of curve). For curve that include easements, the common notation is , as stationing increase, TS (tangent to spiral), SC (spiral to circular curve), CS (circular curve to spiral), and ST (spiral to tangent).
The sharpness of the commonly used easement curve, measured in terms of degree of curvature, increases uniformly from their beginners. If, for instance, easement curves 400 ft (120m) long are selected to connect each end of 4° circular curve to its tangents, the sharpness of the easement curves will increase by i° each 100 ft (30m). At the TS or ST, wh
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