Bending Of Rectangular Plates

The theory of bending of rectangular plates is beyond the scope of this text and will not be introduced here. The standard formulae obtained from the theory however, may be presented in simple form and are relatively easy to apply. The results for the two most frequently used loading conditions are therefore summarised below. 7.15. Rectangular plates with simply supported edges carrying uniformly For a rectangular plate length d, shorter side b and thickness t, the maximum deflection is found...

Info

(7) and (10) are of similar form and will be identical provided that Three values of 9 may be obtained to satisfy (12), Then, from (9), three corresponding values of z are obtained, namely (6) then yields appropriate values of y and hence the required values of x via (2). 8.14. Stress invariants Eigen values and Eigen vectors Consider the special case of the stress at a point tetrahedron Fig. 8.3 where plane ABC is a principal plane subjected to a principal stress ap and, by definition, zero...

Introduction To The Finite Element Method

So far in this text we have studied the means by which components can be analysed using so-called Mechanics of Materials approaches whereby, subject to making simplifying assumptions, solutions can be obtained by hand calculation. In the analysis of complex situations such an approach may not yield appropriate or adequate results and calls for other methods. In addition to experimental methods, numerical techniques using digital computers now provide a powerful alternative. Numerical techniques...

J V

The only departure of eqn. (9.93) from the previous expressions is the replacement of the modulus of elasticity, E, by the elasticity matrix > , due to the change from a one- to a two-dimensional stress system. Recalling, for the present case that the displacement fields are linearly varying, then matrix 5 is independent of the x and y coordinates. The assumption of isotropic homogeneous material means that matrix D is also independent of coordinates. It follows, assuming a constant...

La

(For a flat plate R2 is infinite, for a cylinder in a cylindrical bearing Ri is negative). Stress conditions at the surface on the load axis are then ax 2vpo (along cylinder length) The maximum shear stress is occurring at a depth beneath the surface of 0.786 b and on planes at 45 to the load axis. In cases such as gears, bearings, cams, etc. which (as will be discussed later) can be likened to the contact of parallel cylinders, this shear stress will reduce gradually to zero as the rolling...

La aJcrj ff ii Vax ay J

A similar equation can be obtained for the plane stress case, namely If the body forces X and Y have constant values the same equation holds for both plane stress and plane strain, namely (ar2 + dy2 xx + yy) Vl xx + yy) This equation is known as the Laplace differential equation or the harmonic differential equation. The function (oxx + oyy) is referred to as a harmonic function. It is interesting to note that the Laplace equation, which of course incorporates all the previous equations, does...

O

Residual rodiol stresses re-plotted on horizontal base line Fig. 3.32. Determination of residual radial stresses by elastic unloading. Finally, if the stress distributions due to an elastic internal working pressure Pw are superimposed on the residual stress state then the final working stress state is produced as in Figs. 3.33 and 3.34. The elastic working stresses are given by eqns. (3.42) and (3.43) with PA replaced by Pw. Alternatively a Lam line solution can be adopted. The final stress...

P

Where P is the applied load, D is the disc diameter and t is the thickness. Thus, comparing with the photoelastic eqn. (6.1), The slope of the load versus fringe order graph is given by 6.18. Fractional fringe order determination - compensation techniques The accuracy of the photoelastic technique is limited, among other things, to the accuracy with which the fringe order at the point under investigation can be evaluated. It is not sufficiently accurate to count to the nearest whole number of...

R rdr rdp J Vdr r dr r

The stresses oyr, and r are related to the stress function 0 in a similar manner to ox and oVy. The resulting values are _ 1 3< > 1 d2< f> rJr + 7-W 1 3 4> 1 3 2< p P-w ' drde The derivation of these from the corresponding cartesian coordinate values is a worthwhile exercise for a winter evening. 8.27.4. Forms of stress function in polar coordinates In cylindrical polars the stress function is, in general, of the form 4> f (r) cos n9 or < p f(r)sinnd (8.104) where f(r) is a...

Tf

Negative Nominal Strains

To the substrate as a result of which they are usually more stable. Additionally, the grids of foil gauges can be made much smaller and there is almost unlimited freedom of grid configuration, solder tab arrangement, multiple grid configuration, etc. Figure 6.8 shows but a few of the many types and size of gauge which are available. So vast is the available range that it is difficult to foresee any situation for which there is no gauge suitable. Most manufacturers' catalogues'13' give full...

U

Typical residual stress distribution with depth for the shot-peening process. For lower-strength steels and alloys om can initially reach the yield stress or 0.1 proof stress but this will fade under cyclic loading. Cold rolling of threads, crankpins and axles relies on similar principles to those outlined above with, in this case, continuous pressure of the rollers producing controlled amounts of cold working. Further examples of cold working are the bending of pipes and conduits,...

Oul

(b) Hence, determine the nodal reactions. -0.387, -13.557 mm, -1.116,0.644, 1.233,0.712, -1.116,0.644 kN 9.2 Figure 9.51 shows a roof truss, all members of which are made from steel, and have the same cross-sectional area, such that AE 10 MN, throughout. For the purpose of analysis the truss can be treated as a pin-jointed plane frame. Using the displacement based finite element method, taking advantage of any symmetry and redundancies and treating each member as a rod element, determine the...

[B d[iVbc

In the present case of the two-node linear rod element, eqn. (9.32) shows iV - L x, x , and hence Note that in this case the derivative matrix contains only constants and does not involve functions of x and hence the strain, given by eqn. (9.33), will be constant along the length of the rod element. Assuming Hookean behaviour and utilising eqn. (9.33) It follows for the linear rod element that the stress will be constant and is given as

T Section Structural Material

A cantilever is to be constructed from a 40 mm x 60 mm T-section beam with a uniform thickness of 5 mm. The cantilever is to carry a u.d.l. over its complete length of 1 m. Determine the maximum u.d.l. that the cantilever can carry if yielding is permitted over the lower part of the web to a depth of 10 mm. ay 225 MN m. 3.7 B . A 305 mm x 127 mm symmetrical I-section has flanges 13 mm thick and a web 5.4 mm thick. Treating the web and flanges as rectangles, calculate the bending moment...

Experimental Stress Analysis

We live today in a complex world of manmade structures and machines. We work in buildings which may be many storeys high and travel in cars and ships, trains and planes we build huge bridges and concrete dams and send mammoth rockets into space. Such is our confidence in the modern engineer that we take these manmade structures for granted. We assume that the bridge will not collapse under the weight of the car and that the wings will not fall away from the aircraft. We are confident that the...

Unsymmetrical Bending

Similarly, repeating the process with OQ perpendicular to OQ gives the result sin 6 Iv sin 6 cos 9 - I - sin 26 Ixy Thus the construction shown in Fig. 1.8 can be used to determine the second moments of area and the product second moment of area about any set of perpendicular axis at a known orientation to the principal axes. Consider again the general plane surface of Fig. 1.7 having radii of gyration ku and kv about the U and V axes respectively. An ellipse can be constructed on the principal...

Disc With Hole Rotating Radial Displacement

1.1 Product second moment of area 3 1.2 Principal second moments of area 4 1.3 Mohr's circle of second moments of area 6 1.4 Land's circle of second moments of area 7 1.5 Rotation of axes determination of moments of area in terms of the principal values 8 1.6 The ellipse of second moments of area 9 1.8 Stress determination 11 1.9 Alternative procedure for stress determination 11 1.10 Alternative procedure using the momental ellipse 13 1.11 Deflections 15 Examples 16 Problems 24 2.2 Equivalent...

Euler And Rankine-gordon

Euler Curve

Having derived the result for the buckling load of a strut with pinned ends the Euler loads for other end conditions may all be written in the same form, where I is the equivalent length of the strut and can be related to the actual length of the strut depending on the end conditions. The equivalent length is found to be the length of a simple bow half sine-wave in each of the strut deflection curves shown in Fig. 2.6. The buckling load for each end condition shown is then readily obtained. The...