Bending-moment

When a load is placed on a partially supported board, it sets up bending moment (M) along the board s length. This in turn causes a stress in the cross section of the board. 

The bending stress is zero at the neutral axis (see above), increases linearly toward the outer surface of the board, and reaches maximum at the outer surface. This causes the board to bend, to deflect. 

The bending stress at any point of the board is determined by the following formula  

For rectangular deck boards, solid or hollow, the distance c is typically equal to d/2, that is, half-a-depth of the board. Hence, 

The bending moment is determined from standard beam diagrams for different types of beam, snch as fixed end beam (cantilever), guided end, simply supported end, partial distribnted load, and their various combinations. We will consider only maximum bending moment, which results in a board failure, and only for four types of load application to the board resting on two supports with the distance (span) L between them  

The force acting on the transverse section S of the beam is given in the viscoelastic case by 

On the other hand, the bending moment M on a cross section of the beam can be written as 

By applying the differential operator to both sides of Eq. (17.5) and using (17.3b), we obtain 

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Let (9 C be a bounded domain with smooth boundary 7 and outward normal n = (ni,ri2). We introduce the following notation for the bending moment and transverse forces on 7 ... 

Now we are in a position to give a formulation of the boundary value problem. We recall formulae for the transverse force and the bending moment at Tc,... 

Here i —> i is the convex and continuous function describing a plasticity yield condition. The function w describes vertical displacements of the plate, rriij are bending moments, (5.139) is the equilibrium equation, and equations (5.140) give a decomposition of the curvatures —Wjj as a... 

Here i —> i is the convex and continuous function describing a plasticity yield condition, the dot denotes a derivative with respect to t, n = (ni,ri2) is the unit normal vector to the boundary F. The function v describes a vertical velocity of the plate, rriij are bending moments, (5.175) is the equilibrium equation, and equations (5.176) give a decomposition of the curvature velocities —Vij as a sum of elastic and plastic parts aijkiirikiy Vijy respectively. Let aijki x) = ajiki x) = akuj x), i,j,k,l = 1,2, and there exist two positive constants ci,C2 such that for all m = rriij ... 

The stress in this pipe resulting from the bending moment caused by the thermal expansion is... 

Moment of Force (Torque or Bending Moment). Moment of force is force times moment arm (lever arm). Its SI unit is N-m. 

Pu -on joints do not resist bending moments or axial forces tending to separate the joints but yield to them to an extent limited by the... 

Figure 9.3 Transverse stress-corrosion cracks in a condenser tube the presence of the cracks along just one side indicates that a bending moment provided the stress.

Belts should not be tightened more than necessary, otherwise the drive and the driven shafts will come under torsion and excessive bending moment. The bearings would also be subjected to excessive stresses. 

Bending moment at the motor shaft due to pulley and load... 

Figure 12.2 shows that the stress in the beam is zero along the neutral axis at its centre, and is a maximum at the surface, at the mid-point of the beam (because the bending moment is biggest there). The maximum surface stress is given by... 

Fig. 28.15. Rim design. The left-hand design is poor because the large bending moments will distort the rim by creep. The right-hand design is better the bending moments are less and the ribs stiffen the rim.

M, = bending moment to cause rupture b = width of beam d = depth of beam. Compressive strength... 

Some typieal load histories for meehanieal eomponents and systems are shown in Figure 4.21. The load (dead load, pressure, bending moment, ete.) is subjeet to variation in all eases, rather than a unique value, the likely shape of the final... 

M = bending moment y = distanee from x-x axis to extreme fibre /jx = seeond moment of area about the axis x-x F = load I = eouple length d = depth of seetion. 

Therefore, the bending moments resolved about the prineipal axes are ... 

There are three relevant bending moments caused by a gear coupling when transmitting torque with angular or parallel misalignment ... 

The contoured diaphragm coupling causes two bending moments ... 

In the previous question the use of the 2% limiting strain will produce a conservative estimate for the beam length because the actual strain in the beam will be less than 2%. If the T-section is 25 mm wide and 25 mm deep with a general wall thickness of S mm, what is the % error incurred by using the 2% modulus . Calculate the likely beam deflection after 1 week. The central bending moment on the beam is given by WL/24. 

If the stress in the composite beam in the previous question is not to exceed 7 MN/m estimate the maximum uniformly distributed load which it could carry over its whole length. Calculate also the central deflection after 1 week under this load. The bending moment at the centre of the beam is lVL/24. 

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