The Shear Analysis of Beams

The Shear Analysis of Beams The Shear Analysis of Beams A Statement of the Limitations of the three It is not convenient in practice to take end- Current Methods loads at stations an infinitely small distance apart, wing rib or fuselage frame bays being the usual distance chosen. This introduces an error, which in the preceding example is demonstrated By B. A. Noble below. Sections x, x+δh finite distance apart between the two stations is uniform. This rate of HE limitations of the various methods of Apparent change of end-load of any member is then equated Beam Analysis are often improperly under­ to the change of shear flux. At section x Tstood and it is felt that an explanation and criticism of them would be of some value. Apparent This article is not intended to cover the problem of shear lag, but if a lag analysis is to be carried But using equation (1) out it is obviously necessary that the analyst shall base his work on sound 'no lag' distributions, This necessarily gives results consistent with and shall be aware of the limitations of the end-load diffusion assumptions, and auto­ methods used. This, too often, is not the case. matically includes the depth taper effect. Hence apparent q is underestimated and web There are three current methods of analysis The difference in result by this method and by needs to be corrected by multiplying by the which are here propounded and their limitations the two previous methods is best illustrated by explained. A conclusion-is then drawn as to their the following example of a box beam tapering in factor usefulness and applicability. width, but not in depth, with constant thickness skins, all bending being taken by the top and At section x+δx bottom skins. Method I Both the bending moment and the section Apparent An applied shear 'S' is distributed per moment of inertia are directly proportional to the distance x. Hence the flanges are at constant This is derived in the standard textbooks from a stress; i.e. if the top skin were divided into uniform parallel beam, and is correctly applicable parallel elementary strips, the end-load in each only when those conditions are satisfied. Air­ But from equation (1) would be constant. This implies that there is no craft structures such as wings, or fuselages, un­ fortunately for the stressman, rarely do satisfy TrueS web these conditions, and when the method is ap­ plied to them serious errors result. The major sources of error are as follows: 1. When taper in depth is present, no account is taken of the vertical component of flange end- load. Thus a summation of all vertical com­ ponents of shear will give a value greater than the applied shear. 2. The assumptions made regarding the diffusion of end-load at any discontinuities used in the Hence apparent q is overestimated and web calculation of section constants are violated. change of shear across the top and bottom skins. needs a correcting factor the reciprocal of that at For example, consider a station at which a skin This distribution may be seen to be quite different Section x stringer finishes, since at this particular station from that obtained by Methods I and II, and well its area is omitted from the section constants, illustrates the absurdity of basing shear lag Conclusions calculations on a foundation of those two i.e., no load in the stringer; the method gives methods. With most types of aircraft 'beam' structure, A simple case is chosen to demonstrate the Method III yields far more reliable results. It no change of shear flux at this point, whereas the should be noted that the evaluation of correction assumptions regarding diffusion of load into the validity of the method; and to offer criticism factors involves extra labour, but the over-all stringer imply a change of shear, usually a heavy depth will give a sufficiently accurate measure. one. on, and correction to, its usual application. In any case, the error involved by its complete Example. Flanged beam tapering in depth, omission is not usually serious. Method II bending taken by end-loads in the flanges only. This is really a variation of Method I, and the Assumption. Constant rate of change of end- procedure is as follows: load in the flanges between x and x+δx . The components in the direction of the applied shear of the end-loads are estimated and sub­ tracted from the applied shear. The residue is then distributed as in Method I. Sskin =Sapplied — Σ(Ptanθ) P being end-load in any member of slope tan θ; shear flux, In the limit δx, δh→0, and the stated assumption tends to truth. The first source of error in Method I is then obviated, but the second criticism still holds true; that is, the distribution does not line up with the assumptions made in the calculation of the section constants. Method III or as one would expect web shear flux= The basis of this method is the estimation of element end-loads at two stations, and the as­ sumption that the rate of change of end-load March 1947 85 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Aircraft Engineering and Aerospace Technology Emerald Publishing

The Shear Analysis of Beams

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Publisher
Emerald Publishing
Copyright
Copyright © Emerald Group Publishing Limited
ISSN
0002-2667
DOI
10.1108/eb031483
Publisher site
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Abstract

The Shear Analysis of Beams A Statement of the Limitations of the three It is not convenient in practice to take end- Current Methods loads at stations an infinitely small distance apart, wing rib or fuselage frame bays being the usual distance chosen. This introduces an error, which in the preceding example is demonstrated By B. A. Noble below. Sections x, x+δh finite distance apart between the two stations is uniform. This rate of HE limitations of the various methods of Apparent change of end-load of any member is then equated Beam Analysis are often improperly under­ to the change of shear flux. At section x Tstood and it is felt that an explanation and criticism of them would be of some value. Apparent This article is not intended to cover the problem of shear lag, but if a lag analysis is to be carried But using equation (1) out it is obviously necessary that the analyst shall base his work on sound 'no lag' distributions, This necessarily gives results consistent with and shall be aware of the limitations of the end-load diffusion assumptions, and auto­ methods used. This, too often, is not the case. matically includes the depth taper effect. Hence apparent q is underestimated and web There are three current methods of analysis The difference in result by this method and by needs to be corrected by multiplying by the which are here propounded and their limitations the two previous methods is best illustrated by explained. A conclusion-is then drawn as to their the following example of a box beam tapering in factor usefulness and applicability. width, but not in depth, with constant thickness skins, all bending being taken by the top and At section x+δx bottom skins. Method I Both the bending moment and the section Apparent An applied shear 'S' is distributed per moment of inertia are directly proportional to the distance x. Hence the flanges are at constant This is derived in the standard textbooks from a stress; i.e. if the top skin were divided into uniform parallel beam, and is correctly applicable parallel elementary strips, the end-load in each only when those conditions are satisfied. Air­ But from equation (1) would be constant. This implies that there is no craft structures such as wings, or fuselages, un­ fortunately for the stressman, rarely do satisfy TrueS web these conditions, and when the method is ap­ plied to them serious errors result. The major sources of error are as follows: 1. When taper in depth is present, no account is taken of the vertical component of flange end- load. Thus a summation of all vertical com­ ponents of shear will give a value greater than the applied shear. 2. The assumptions made regarding the diffusion of end-load at any discontinuities used in the Hence apparent q is overestimated and web calculation of section constants are violated. change of shear across the top and bottom skins. needs a correcting factor the reciprocal of that at For example, consider a station at which a skin This distribution may be seen to be quite different Section x stringer finishes, since at this particular station from that obtained by Methods I and II, and well its area is omitted from the section constants, illustrates the absurdity of basing shear lag Conclusions calculations on a foundation of those two i.e., no load in the stringer; the method gives methods. With most types of aircraft 'beam' structure, A simple case is chosen to demonstrate the Method III yields far more reliable results. It no change of shear flux at this point, whereas the should be noted that the evaluation of correction assumptions regarding diffusion of load into the validity of the method; and to offer criticism factors involves extra labour, but the over-all stringer imply a change of shear, usually a heavy depth will give a sufficiently accurate measure. one. on, and correction to, its usual application. In any case, the error involved by its complete Example. Flanged beam tapering in depth, omission is not usually serious. Method II bending taken by end-loads in the flanges only. This is really a variation of Method I, and the Assumption. Constant rate of change of end- procedure is as follows: load in the flanges between x and x+δx . The components in the direction of the applied shear of the end-loads are estimated and sub­ tracted from the applied shear. The residue is then distributed as in Method I. Sskin =Sapplied — Σ(Ptanθ) P being end-load in any member of slope tan θ; shear flux, In the limit δx, δh→0, and the stated assumption tends to truth. The first source of error in Method I is then obviated, but the second criticism still holds true; that is, the distribution does not line up with the assumptions made in the calculation of the section constants. Method III or as one would expect web shear flux= The basis of this method is the estimation of element end-loads at two stations, and the as­ sumption that the rate of change of end-load March 1947 85

Journal

Aircraft Engineering and Aerospace TechnologyEmerald Publishing

Published: Mar 1, 1947

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