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...

E

In the fully plastic condition, therefore, when the stress is equal throughout the section, the above equation reduces to y) areas above N.A. areas below N.A. and in the special case shown in Fig. 3.5 the N.A. will have moved to a position coincident with the lower edge of the flange. Whilst this position is peculiar to the particular geometry chosen for this section it is true to say that for all unsymmetrical sections the N.A. will move from its normal position when the section is completely...

Cos ex pxnp I

Line of action of normal stress By definition the normal stress is that which acts normal to the plane, i.e. the line of action of the normal stress has the same direction cosines as the normal to plane viz I, m and n. 8.4.3. Line of action of shear stress As shown in 8.4 the resultant stress pn can be considered to have two components one normal to the plane (cr ) and one along the plane (the shear stress r ) - see Fig. 8.6. Let the direction cosines of the...

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

This text is a revised and extended third edition of the highly successful text initially published in 1977 intended to cover the material normally contained in degree and honours degree courses in mechanics of materials and in courses leading to exemption from the academic requirements of the Engineering Council. It should also serve as a valuable reference medium for industry and for post-graduate courses. Published in two volumes, the text should also prove valuable for students studying...

Bnb

The integral part of the expression is the second moment of area of the shaded portion of Fig. 3.24 about the vertical axis. Thus, determination of this quantity for a given > 'max value yields the corresponding value of the applied torque T. As for the case of inelastic bending, the form of the shear stress-strain curve, Fig. 3.24, is identical to the shear stress distribution across the shaft section with the y axis replaced by radius r. 3.16. Residual stresses - strain-hardening materials...

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...

Ox

(b) In polar coordinates with displacements ur, ug and uz along r, 0 and z respectively these equations become 8.25. The strain equations of transformation Using the experimental or theoretical procedures described in earlier sections it is possible to derive the values of the direct and shear stresses acting at a point on a body. These are normally obtained with reference to some convenient set of X, Y coordinates which, for example, may be parallel to the edges of the component considered....

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Reprinted with corrections 1980, 1981, 1982 All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing...

P

A three-dimensional complex stress system has principal stress values of 280 MN m2, 50 MN m2 and - 120 MN m2. Determine (a) analytically and (b) graphically (i) the limiting value of the maximum shear stress (ii) the values of the octahedral normal and shear stresses. (i) The limiting value of the maximum shear stress is the greatest value obtained in any plane of the three-dimensional system. In terms of the principal stresses this is given by (ii) The octahedral normal stress is given by...

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...

T y R d ERS

Now - and, for small deflections, tan0 9 (radians). R dxl dx Consider now the diagram Fig. 7.2 in which the radii of the concentric circles through C and D, on the unloaded plate increase to (x + dx) + (9 + d9)u and x + u9 , respectively, when the plate is loaded. ( circumferential strain) (7.4) Substituting eqns. (7.3) and (7.4) in eqns. (7.1) and (7.2) yields

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,...

W

Partial Plastic Stress

Other structures may require even more plastic hinges depending on their particular support conditions and degree of redundancy, but these need not be considered here. It should be evident, however, that there is now even more strength or load-carrying capacity available beyond that suggested by the shape factor, i.e. with a knowledge of the yield stress and hence the maximum elastic moment for any particular cross-section, the shape factor determines the increase in moment required to produce...

1

Mohr's circle of second moments of area. The procedure is therefore identical to that for determining the direct stress on some plane inclined at a. to the plane on which ax acts in Mohr's stress circle construction, i.e. angles are DOUBLED on Mohr's circle. 1.4. Land's circle of second moments of area An alternative graphical solution to the Mohr procedure has been developed by Land as follows (Fig. 1.6) Fig. 1.6. Land's circle of second moments of area. Fig. 1.6. Land's circle of...

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...

Z

The above equations have been derived by consideration of equilibrium of forces only, and this does not represent a complete check on the equilibrium of the system. This can only be achieved by an additional consideration of the moments of the forces which must also be in balance. Consider, therefore, the element shown in Fig. 8.17 which, again for simplicity, shows only the stresses which produce moments about the Y axis. For convenience the origin of the cartesian coordinates has in this case...

[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

J L L

For very short struts Pe is very large l Pe can therefore be neglected and PR Pc. For very long struts Pe is very small and 1 Pe is very large so that 1 P, can be neglected. Thus The Rankine formula is therefore valid for extreme values of L k. It is also found to be fairly accurate for the intermediate values in the range under consideration. Thus, re-writing the formula in terms of stresses,

T

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...

Membrane analogy

It has been stated earlier that the mathematical solution for the torsion of certain solid and thin-walled sections is complex and beyond the scope of this text. In such cases it is extremely fortunate that an analogy exists known as the membrane analogy, which provides a very convenient mental picture of the way in which stresses build up in such components and allows experimental determination of their values. It can be shown that the mathematical solution for elastic torsion problems...

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...