A method of shear-lag analysis of box beams for axial stresses, shear stresses, and shear center
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Fifteen boreholes have been drilled to the base of Ice Stream B in the vicinity of UpB Camp. The boreholes are spread over an area of about 500 x 1000 m. Several till cores were retrieved from the bottom of the 1000-m-deep holes. Laboratory tests using a simple shear box revealed a yield strength of basal till of 2 kPa. This agrees well with in-situ measurements using a shear vane. Since the average basal shear stress of Ice Stream B with a surface slope of 0.1 degree is about 20 kPa, the ice stream cannot be supported by till that weak. Additional support for this conclusion comes from the basal water pressure that has been measured in all boreholes as soon as the hot water drill reached bottom. In several boreholes, the water pressure has been continuously monitored; in two of them, over several years. The water pressure varies but stays within 1 bar of flotation where ice overburden pressure and water pressure are equal. The ratio of water and overburden pressure lies between 0.986 and 1.002. This is an extremely high value as compared to other fast-moving ice masses; e.g., Variegated Glacier in surge has a ratio of 0.8, and Columbia Glacier - a fast-moving tidewater glacier - has a ratio of 0.9. It implies that water flow under the glacier occurs in a thin film and not in conduits that would drain away water too rapidly. It also implies that basal sliding must be very effective. Water flow under the glacier was measured in a salt-injection experiment where a salt pulse was released at the bottom of a borehole while 60 m down-glacier, the electrical resistance was measured between two other boreholes. A flow velocity of 7 mm/s was obtained.
Soil from the scoop of Surveyor 3, returned to earth by Apollo 12 astronauts, has been tested in a miniature shear box at five bulk densities, from 0.99 to 1.87 g/cu cm. Cohesion increased with bulk density from .03 to .3 N/sq cm; internal friction angle increased from 13 to 56 deg. Shear stress vs normal stress data fit a logarithmic relationship better than a linear one, at normal stresses of .003 to 3 N/sq cm. Results of these tests, in air, show no systematic differences from those for tests made elsewhere in vacuum and nitrogen. Results agree with those obtained in remotely controlled lunar surface operations with Surveyor 3 and other spacecraft provided that the bulk density was slightly underestimated for the on-surface measurements.
The distribution of stress in the cover sheets of thin-wall box beams is analyzed, with regard to the effect of shear deformation in the cover sheets, by the method of least work. Explicit results are obtained for a number of representative cases that show the influence of the following factors on the stress patterns. (1) Variation of stress in spanwise direction as given by elementary beam theory. (2) Value of a parameter called shear-lag aspect ratio which designates the product of span-width ratio of the beam and of the square root of the ratio of effective shear modulus and tensing modulus of the cover sheets. (3) Value of ratio of cover-sheet stiffness to side-web stiffness. (4) Variation of beam height in span direction. (5) Variation of beam width in span direction. (6) Variation of cover-sheet thickness in span direction. General conclusions are drawn from the results obtained. Among them the most important one appears to be the fact that the shear-lag effect depends primarily on the following tow quantities: (1) the value of the shear-lag aspect ratio. (2) the shape of the curve representing the product of the stress of elementary beam theory and of the cover-sheet thickness.
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The bending stresses in the covers of box beams or wide-flange beams differ appreciably from the stresses predicted by the ordinary bending theory on account of shear deformation of the flanges. The problem of predicting these differences has become known as the shear-lag problem. The first part of this paper deals with methods of shear-lag analysis suitable for practical use. The second part of the paper describes strain-gage tests made by the NACA to verify the theory. Three tests published by other investigators are also analyzed by the proposed method. The third part of the paper gives numerical examples illustrating the methods of analysis. An appendix gives comparisons with other methods, particularly with the method of Ebner and Koller.
Theoretical and experimental investigations were made of the distribution of strains in a plywood sheet-stringer combination used as the chord member of a box beam acted upon by bending loads. The theoretical solution was obtained with the help of the principle of minimum potential energy and certain simplifying assumptions. Strain measurements were made on a build-up box beam by means of electrical-resistance strain gages connected with strain indicators. A very satisfactory agreement between the theoretical and experimental strains was obtained.
The Low Speed Vehicle (LSV) is a 14.2% scale remotely piloted vehicle of the revolutionary Blended Wing Body concept. The design of the LSV includes an all composite airframe. Due to internal manufacturing capability restrictions, room temperature layups were necessary. An extensive materials testing and manufacturing process development effort was underwent to establish a process that would achieve the high modulus/low weight properties required to meet the design requirements. The analysis process involved a loads development effort that incorporated aero loads to determine internal forces that could be applied to a traditional FEM of the vehicle and to conduct detailed component analyses. A new tool, Hypersizer, was added to the design process to address various composite failure modes and to optimize the skin panel thickness of the upper and lower skins for the vehicle. The analysis required an iterative approach as material properties were continually changing. As a part of the material characterization effort, test articles, including a proof of concept wing box and a full-scale wing, were fabricated. The proof of concept box was fabricated based on very preliminary material studies and tested in bending, torsion, and shear. The box was then tested to failure under shear. The proof of concept box was also analyzed using Nastran and Hypersizer. The results of both analyses were scaled to determine the predicted failure load. The test results were compared to both the Nastran and Hypersizer analytical predictions. The actual failure occurred at 899 lbs. The failure was predicted at 1167 lbs based on the Nastran analysis. The Hypersizer analysis predicted a lower failure load of 960 lbs. The Nastran analysis alone was not sufficient to predict the failure load because it does not identify local composite failure modes. This analysis has traditionally been done using closed form solutions. Although Hypersizer is typically used as an optimizer for the design process, the failure prediction was used to help gain acceptance and confidence in this new tool. The correlated models and process were to be used to analyze the full BWB-LSV airframe design. The analysis and correlation with test results of the proof of concept box is presented here, including the comparison of the Nastran and Hypersizer results.
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A mathematical procedure is herein developed for obtaining exact solutions of shear-lag problems in flat panels and box beams: the method is based on the assumption that the amount of stretching of the sheets in the direction perpendicular to the direction of essential normal stresses is negligible. Explicit solutions, including the treatment of cut-outs, are given for several cases and numerical results are presented in graphic and tabular form. The general theory is presented in a from which further solutions can be readily obtained. The extension of the theory to cover certain cases of non-uniform cross section is indicated. Although the solutions are obtained in terms of infinite series, the present developments differ from those previously given in that, in practical cases, the series usually converge so rapidly that sufficient accuracy is afforded by a small number of terms. Comparisons are made in several cases between the present results and the corresponding solutions obtained by approximate procedures devised by Reissner and by Kuhn and Chiarito.
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