钢筋混凝土结构受扭构件的强度及变形原文.docx

1、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 INTRODUCTION Torsion 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

2、 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 worl

3、d. 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 t

4、he 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 inf

5、ormation 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 equilibriu

6、m, 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,cantilev

7、ers 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 lea

8、d 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

9、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

10、 statically indeterminate reinforced concrete structures at various stages of the 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 i

11、nteractions 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 TORSION The behavior of reinforced concrete in torsion, befo

12、re 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 mecha

13、nics. The classical solution of St.Venant can be applied to the common rectangular concrete section. Accordingly, the maximum torsional shearing stress vt is generated at the middle of the long side and can be obtained fromwhere T=torsional moment at the section y,x =overall dimensions of the rectan

14、gular section, x y t =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 t,= the angle of twist T = the applied torque, which may be a function of

15、 the distance along the span G = the modulus in shear as defined in Eq. 7.37 C = the torsional moment of inertia, sometimes referred to as torsion constant or equivalent polar moments of inertia z = distance along memberFor rectangular sections, we havein which t, a coefficient dependent on the aspe

16、ct 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 as In the

17、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,d

18、efined 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 Bachs suggestion, it is customary to assume that a suitable subdivision of the section into its constituent rec

19、tangles 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

20、 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 Bachs approximation,8.5 the portion of the total torque T resisted by element 2 in Fig.

21、8.4a isand the resulting maximum torsional shear stress is from Eq. 8.1 The 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,Figur

22、e 8.4c illustrates the procedure.distribution over compound cross sections may be best visualized by Prandtls 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 linea

23、r conditions under which the elastic behavior are satisfied.8.2.2 Plastic Behavior In 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

24、 relative ease. Consider the square section appearing in Fig. 8.5, where yield in shear Vty has set in the quadrants. The total shear force V acting over one quadrant is The same results may be obtained using Nadais sand heap analogy. According to this analogy the volume of sand placed over the give

25、n 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.

26、 8.6) isIt is evident that ty=3 when x/y= I and O,y =2when x/y=0 It 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 ultimat

27、e 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

28、 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 th

29、eory 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 is where x y for each rectangle. The principal str

30、ess (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 a

31、xis 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 th

32、e 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 Sections Because 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 b