钢筋混凝土结构-受扭构件的强度及变形原文

Strength and Deformation of Members with Torsion8.1 INTRODUCTIONTorsion in reinforced concrete structures often arises from continuity between members. For this reason torsion received; relatively scant attention during the first half of this century, and the omission from design considerations apparently had no serious consequences. During ;the last 10 to 15 years, a great increase in research activity has advanced the understanding of the problem significantly. Numerous aspects of torsion in concrete have been,and currently are being, examined in various parts of the world. The first significant organized pooling of knowledge and research effort in this field was a symposium sponsored by the American Concrete Institute. The symposium volume also reviews much of the valuable pioneering work.Most code references to torsion to date have relied on ideas borrowed from the behavior of homogeneous isotropic elastic materials. The current ACI code8.2 incorporates for the first time detailed design recommendations for torsion. These recommendations are based on a considerable volume of experimental evidence, but they are likely to be further modified as additional information from current research efforts is consolidated.Torsion may arise as a result of primary or secondary actions. The case of primary torsion occurs when the external load has no alternative to being resisted but by torsion. In such situations the torsion, required to maintain static equilibrium, can be uniquely determined. This case may also be refer-red to as equilibrium torsion. It is primarily a strength problem because the structure, or its component, will collapse if the torsional resistance cannot be supplied. A simple beam, receiving eccentric line loadings along its span,cantilevers and eccentrically loaded box girders, as illustrated in Figs. 8.1and 8.8, are examples of primary or equilibrium torsion.In statically indeterminate structures, torsion cart also arise as a secondary action from the requirements of continuity. Disregard for such continuity in the design may lead to excessive crack widths but need not have more serious consequences. Often designers intuitively neglect such secondary torsional effects. The edge beams of frames, supporting slabs or secondary-beams, are typical of this situation (see Fig.8.2). In a rigid jointed space structure it is hardly possible to avoid torsion arising from the compatibility of deformations. Certain structures, such as shells elastically restrained by edge beams," are more sensitive to this type of torsion than are other.The present state of knowledge allows a realistic assessment. of the torsion that may arise in statically indeterminate reinforced concrete structures at various stages ofthe loading.Torsion in concrete structures rarely occurs. without other actions.Usually flexure, shear, and axial forces are also present. A great many of the more recent studies have attempted to establish the laws of interactions that may exist between torsion and other structural actions. Because of the large number of parameters involved, some effort is still required to assess reliably all aspects of this complex behavior.8.2PLAIN CONCRETE SUBJECT TO TORSIONThe behavior of reinforced concrete in torsion, before the onset of cracking,can be based ors the study of plain concrete because the contribution of rein-force ment at this stage is negligible.8.2.1 Elastic BehaviorFor the assessment of torsional effects in plain concrete, we can use the well-known approach presented inmost texts on structural mechanics. The classical solution of St.Venant can be applied to the common rectangular concrete section. Accordingly, the maximum torsional shearing stress v t is generated at the middle of the long side and can be obtained fromwhere T=torsional moment at the sectiony,x =overall dimensions of the rectangular section, x <yΨt =a stress factor being a function y/x, as given in Fig. 8.3It may be equally as important to know the load-displacement relationship for the member. This can be derived from the familiar relationship.where θt,= the angle of twistT = the applied torque, which may be a function of the distance along the spanG = the modulus in shear as defined in Eq. 7.37C = the torsional moment of inertia, sometimes referred to as torsion constant or equivalent polar moments of inertiaz = distance along memberFor rectangular sections, we havein which βt, a coefficient dependent on the aspect ratio y/x of the section (Fig.8.3), allows for the nonlinear distribution of shear strains across the section.These terms enable the torsional stiffness of a member of length section. l to be defined as the magnitude of the torque required to cause unit angle of twist over this length asIn the general elastic analysis of a statically indeterminate structure, both the torsional stiffness and the flexural stiffness of members may be required.Equation 8.4 for the torsional stiffness of a member may be compared with the equation for the flexural stiffness of a member with far end restrained,defined as the moment required to cause unit rotation, 4EI/1, where EI =flexural rigidity of a section.The behavior of compound sections, T and L shapes, is more complex.However, following Bach's suggestion, it is customary to assume that a suitable subdivision of the section into its constituent rectangles is an accept-able approximation for design purposes. Accordingly it is assumed that each ,rectangle resists a portion of the external torque in proportion to its torsional rigidity. As Fig. 8.4a shows, the overhanging parts of the flanges should be taken without overlapping. In slabs forming the flanges of beams, the effective length of the contributing rectangle should not be taken as more than three times the slab thickness. For the case of pure torsion, this is a conservative approximation.Using Bach's approximation,8.5 the portion of the total torque T resisted by element 2 in Fig. 8.4a isand the resulting maximum torsional shear stress is from Eq. 8.1The approximation is conservative because the "junction effect" has been neglected.Compound sections in which shear must be subdivided in a different way.The elastic torsional shear stress flow can occur, as in box sections,Figure 8.4c illustrates the procedure.distribution over compound cross sections may be best visualized by Prandtl's membrane analogy, the principles of which may be found in standard works concrete structures, we seldom encounter the on elasticity." In reinforced foregoing assumptions associated with linear conditions under which the elastic behavior are satisfied.8.2.2 Plastic BehaviorIn ductile materials it is possible to attain a state at which yield in shear occur over the whole area of a particular cross section. If yielding occurs over the whole section, the plastic torque can be computed with relative ease.Consider the square section appearing in Fig. 8.5, where yield in shear V ty has set in the quadrants. The total shear force V acting over one quadrant isThe same results may be obtained using Nadai's ‘sand heap analogy.’ According to this analogy the volume of sand placed over the given cross section is proportional to the plastic torque sustained by this section.the heap (or roof) over the rectangular section (see Fig. 8.6) has a height xv.where x = small dimension of the cross section.mid over the square section (Fig.8.5) isThe volume of the heap over the oblong section (Fig. 8.6) isIt is evident that Ψty=3 when x/y= I and O,y =2when x/y=0It may be seen that Eq. 8.7 is similar to the expression obtained for elastic behavior, Eq. 8.1.Concrete is not ductile enough, particularly in tension, to permit a perfect plastic distribution of shear stresses. Therefore the ultimate torsional strength of a plain concrete section will be between the values predicted by the membrane (fully elastic) and sand heap (fully plastic) analogies. Shear stresses cause diagonal (principal) tensile stresses, which initiate, the failure. In the light of the foregoing approximations and the variability of the tensile strength of concrete, the simplified design equation for the determination of the nominal ultimate sections, proposed by shear stress induced by torsion in plain concrete ACI 318-71, is acceptable:where x ≤y.The value of 3 for t is or ty,3, is a minimum for the elastic theory and a maxi-mum for the plastic theory (see Fig. 8.3 and Eq. 8.7a).The ultimate torsional resistance of compound sections can be mated by the summation of the contribution of the constituent sections such as those in Fig. 8.4, the approximation iswhere x ≤y for each rectangle.The principal stress (tensile strength) concept would suggest that failure cracks should develop at each face of the beam along a spiral running at 450 to the beam axis. However, this is not possible because the boundary of the failure surface must form a closed loop. Hsu has suggested that bending occurs about an axis parallel to the planes that is at approximately 450 to the beam axis and of the long faces of a rectangular beam. This bending causes compression beam. The latter tension cracking eventually and tensile stresses in the 450 plane across the initiates a surface crack. As soon as flexural occurs the flexural strength of the section is reduced, the crack rapidly propagates, and sudden failure follows. Hsu observed this sequence of failure with the aid of high-speed motion pictures. For most structures little use can be made of the torsional (tensile) strength of unreinforced concrete members.8.2.3 Tubular SectionsBecause of the advantageous efficient in resisting distribution of shear stresses, tubular sections are most resisting torsion. They are widely used in bridge construction .Figure8.7 illustrates the basic forms used for bridge girders. The torsional properties of the girders improve in progressing from Figs. 8.7a to 8.7g.When the wall thickness h is small relative to the overall dimensions of the section, uniform shear stress across the thickness can be assumed. By considering the moments exerted about a suitable point by the shear stresses,acting over infinitesimalelements of the tube section, as in Fig. 8.8a, the torque of resistance can be expressed.asThe product hv t = v o is termed the shear flow,.and this is constant; thuswhere Ao = the area enclosed by the center Jine of the tube wall (shaded area in Fig.8.8).The concept of shear flow around the thin wall tube is useful when the role of reinforcement in torsion is considered.The ACI code 8.2 suggests that the equation relevant to solid sections. 8.8, be used also for hollow sections, with the following modification when the wall thickness is not less than x/l0 (see Fig. 8.8c):where x ≤y.Equation 8.9b follows from first principles and has the advantage of being applicable to both the elastic and fully plastic state of stress.The torque-twist relationship for hollow sections may be readily derived from strain energy considerations. By equating the work done by the applied torque (external work) to that of the shear stresses (internal work), the torsion constant C O for tubular sections can be found thus:Hence by equating the two expressions and using Eq. 8.9b, the relationship between torque and angle of twist is found to beand the torsional stiffness of such member is thereforewhere C0is the equivalent polar moment of inertia of the tubular section and is given bywhere s is measured around the wall centerline. The same expression for the more common form of box section (Fig. 8.8b) becomesFor uniform wall thickness Eq. 8.11 reduces further towhere p is the perimeter measured along the tube centerline.It is emphasized that the preceding discussion on elastic and plastic behavior relates to plain concert .and the propositions are applicable only at low load intensities before cracking. They may be used for predicting the one of diagonal cracking.8.3 BEAMS WITHOUT WEB REINFORCEMENT SUBJECT TOFLEXURE AND TORSIONThe failure mechanism of beams subjected to torsion and bending depends on the predominance of one or the other. The ratio of ultimate torque to moment, T J/M U,is a suitable parameter to measure the relative magnitude of these actions. The flexural resistance depends primarily on the amount of flexural reinforcement. The -torsional behavior of a concrete beam without web reinforcement is more difficult to assess in the presence of flexure.Flexural stresses initiate diagonal cracks in the case of torsion, much as they do in the case of shear. In the presence of flexure these cracks are arrested in the compression zone. For this reason a diagonally cracked beam is capable of carrying a certain amount or torsion. The manner in which this torsion is resisted is, at present, a matter, of speculation. Clearly the compression zone of the beam is capable of resisting a limited amount of torsion,.and horizontal reinforcement can also contribute to torsional resistance by means of dowel action.It has been found (e.g., by Mattock") that the torsional resistance of a cracked section is approximately one-half the ultimate torsional strength of the uncracked section, provided a certain -amount of bending is present.Thus one half the torque causing cracking can be sustained after the formation of cracks. The torque thus carried is so small that its influence on flexure on can be ignored.The nominal torsional shear stress, corresponding to this limited torsion is conservatively assumed by ACI 318-71. to be 40 % of a cracking stress ofand the torque supplied by the .concrete section only, after the onset of cracking, is revealed by Eq. 8.8 to beSimilarly, for compound sections, Eq. 8.8a giveswith the limitations on overhanging parts as indicated in Fig. 8.4.When T./M > 0.5 (i.e., when torsion is significant), brittle failure has been observed. When the bending moment is more pronounced, (i.e., when T/Mu < 0.5), a more ductile failure can be expected. The torsional strength of abeam can be increased only with the addition of web reinforcement. The amount of flexural reinforcement appears to have no influence on the torsional capacity of the concrete section, T .In T or L beams the overhanging part of the flanges contribute to torsional. strength. This has been verified on isolated beams. The effective width of flanges, when theseare part of a floor slab, is difficult to assess.When a yield line can develop along an edge beam because of negative bending moment in the slab, as illustrated in Fig. 8.9 it is unlikely that much of the flange can contribute toward torsional strength.8.4 TORSION AND SHEAR IN BEAMS WITHOUT WEB REINFORCEMENTIt is evident that in superposition ; the shear stresses generated by torsion and shearing force are additive along one side and subtractive along the opposite side of a rectangular beam section .The critical diagonal tensile stresses that ensue are further affected by flexural tensile stresses in the concrete, because it is impossible to apply shearing forces without simultaneously inducing flexure. A fully rational theory for the interaction of shear and torsion in the presence of bending is not known to have yet been developed. For this reason reliance must be placed on empirical information derived from tests. By providing more than adequate flexural reinforcement, it is possible to experimentally study the failure criteria for combined shear and torsion. It is usual in such tests to keep the torsion to, shear ratio constant while the load is being increased to failure. However, in practice one action may occur first,imposing its own crack pattern before the other action becomes significant. For the time being, it is advisable to be conservative in the interpretation of test results.Figure 8.10 plots the scatter obtained in typical combined torsion-shear tests. It also indicates that a circular interaction relationship (normalized for this particular group of tests) can be useful for design purposes, provided sufficiently low stress values for diagonal cracking by shear and torsion are chosen. For these beams,which contained no web reinforcement, the shear and torsional stresses which formed an approximate lower bound for the plotted experimental points, as computed from Eqs. 7.5 and 8.8, were found to be, respectively,The circular interaction action relationship is the basis of the current ACI code provisions. 8.2 For convenience, the magnitude of the interaction shear and torsional forces carried by a cracked section at ultimate load can be expressed in terms-of nominal stress aswherevt = induced nominal torsional stress carried by the concrete at ultimate, given by Eq. 8.8vu = induced nominal shear stress carried by the concrete at ultimate given by Eq. 7.5。