衢州学院本科毕业设计（论文）外文翻译(4)

where T=torsional moment at the section

y,x =overall dimensions of the rectangular section, x

It 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 twist

T = the applied torque, which may be a function of 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 member For rectangular sections, we have

in 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 as

In 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 is

and the resulting maximum torsional shear stress is from Eq. 8.1

The approximation is conservative because the \ 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 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 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 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) is

The volume of the heap over the oblong section (Fig. 8.6) is

It is

evident that Ψty=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 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 is

where x ≤ y for each rectangle.

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