Bending moments and torque

The assumed positive sense of the internal bending moments and torques acting on each element are as shown ... 

From Eq. (13.1.13), the earthquake-induced bending moments and torques at the base due to the n mode are ... 

Because moment of force (bending moment) and torque are equal to a force times a distance (moment arm or lever arm), their SI unit is N m. The Joule (J = N m), which is a special name for the SI unit of energy and work, should... 

Slabs are stmcturally continuous in both directions X and Y, and they resist applied loads by shear forces, bending moments and torques which are coupled with each other. For this reason it was previously mentioned that the transverse slab elements of the grillage models should not be used for the final design of the slab. 

When the stalling side load is applied to the shaft at the agitator, the bending moment My, at the driving end will be the product of force and length of shaft, i.e. M = FJ f,. Thus, at the driving end of the shaft, the forces, moments and torques are shown in Figure 13.4. 

Below, we first introduce the most general mechanical description of the surface moments (torques) exerted on the boundary between two fluid phases. Then, we consider the thermodynamics of a curved interface (membrane) in terms of the work of flexural deformation. Next, we specify the bending rheology by means of the model of Helfiich [202]. Finally, we review the available expressions for the contributions of the electrostatic, steric, and van der Waals interactions to the interfacial bending moment and curvature elastic moduli. These expressions relate the interfacial flexural properties to the properties of the adsorbed surfactant molecules. 

Torque transducer units are available that contain strain-gauged sections of shaft that can be fitted in line with the mixer shaft. They must be suitably isolated from any bending moments and axial loads (Figure 4-9). They are generally subject to the same calibration and care during use precautions as are sensitive strain gauges (Chappie et al., 2002). 

Since the bending moment and the torque act simultaneously, these loads must be combined and resolved into a combined shear stress and a combined tensile stress acting on the shaft. The minimum shaft diameter for the allowable shear stress can be calculated as follows ... 

The Joule is equal to a 1 N m, but is reserved for a unit of energy and can have more than one application, as discussed later. When we get into thermal stresses and heat transfer, it is confusing to nse Jonle as a bending moment of torque and as a thermal unit. We will spend more time later on the proper use of metric units. 

Vector Influence Some units, such as torque, involve vector quantities, but in common with other systems of measurement units, the sjrmbols and names do not indicate this fact. As a result, an apparent anomaly exists in the use of newton-metre for torque or bending moment, and the use of joule for work (1 J = 1 N-m). These are entirely different units, since the unit of work results from unit force moving through unit distance, while the unit of bending moment involves a force applied normal to a lever of unit length. This would be readily seen if vectors were incorporated in the unit symbols. For this reason it is important not to express moments in joules. 

Figure 4-260. Sketch of theoretical strain gage position in a sub to read WOB, torque, and bending moment.

Example 10.2 compares data of Table 10.4 with calculations based on Figures 10.6 and 10.7 for all-liquid mixing. Power and rpm requirements at a given superficial liquid velocity are seen to be very sensitive to impeller diameter. When alternate combinations of HP/rpm are shown in the table for a particular performance, the design of the agitator shaft may be a discriminant between them. The shaft must allow for the torque and bending moment caused by the hydraulic forces acting on the impeller and shaft. Also, the... 

Thus, a maximum bending moment of 8272 in lb (935 N m) occurs just below the agitator drive, at the top of the shaft extension. The hydraulic loads on the shaft result in a torque and bending moment that the shaft must be strong enough to handle. 

In equilibrium, the sum of all forces and torques are zero. For bent levers, rotational equilibrium should be considered, which is related to the so-called area moment of inertia. The area moment of inertia (7) measures the beam s ability to resist bending. The larger its value, the less the beam will bend. 

A cadaveric study comparing the range of motion in the L5-S1 motion segment in five human spines, before and after implantation with the PRODISC, has indicated that implantation of fhis device did not significantly affect the ROM (Lipman et al. 2003). Specimens were tested in an apparatus that applied pure bending moments. The specimens were cycled in torque to a maximum of 10 Nm in flexion-extension, lateral bending, and torsion with 600 N and 1200 N compressive loads. The ROM at 8 Nm after the fifth cycle of loading was documented and used for comparison between intact and implanted specimens. Summary data is shown in Table 10.3. 

The adequacy of the vertical shaft diameter. This depends not only on the torque, which is easily calculated, but also on the required bending moment, which is not. The bending moment is caused by eddying and unstable flow within the tank thus it can be promoted by insufficient size or number of baffles. Research articles have been published on the subject, but it is more practical to be guided by relevant experience. 

The maximum bending moment, M ax. for an overhung shaft is the sum of the products of the hydraulic forces and the distance from the individual impellers to the bottom bearing in the mixer drive (see Figure 21-32). The following expression computes an empirical hydraulic force related to the impeller torque acting as a load at a distance related to the impeller diameter. 

The method for determining the deflection is very similar to calculating the critical speed. However, the hydraulic forces on the shaft are now taken into account. Also, the frequency is a known value. Hydraulic forces are determined based on the impeller torque. Because hydraulic forces used by mixer manufacturers already include the effect of dynamics, speed (frequency) is already included in the magnitnde. Consequently, a forced response at the shaft speed would not be appropriate because the results would reflect the effect of frequency twice. Therefore, determining the forced response for the static condition is necessary (frequency = 0). From the bending moment for each position along with the torque from the impeller(s), the tensile and shear stresses can be calculated for each position. The static condition can only be calcnlated where the forcing freqnency is effectively zero compared with the natnral frequency, and such an analysis requires a 4 x 5 matrix. 

The analysis of spline couplings subjected to applied torque and misalignment or bending moment has been carried out by two methods classical complete slip theory and an elastostatic model with stick-slip Action based on the boundary element method. 

Figure 13.1 Internal bonded and unbonded composite repair for a pipeline subjected to internal pressure P, axial force N, torque T, bending moment M and thermal loading Ar(r).

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