The use of thin-walled sections for energy absorbing components- a review

Lecture Notes: Thin-walled StructuresEnchun ZhuSchool of Civil EngineeringHarbin Institute of TechnologyFebruary 2014ContentsChapter 1 Introduction to thin-walled structures1 1.1 Examples of thin-walled structures (1)1.2 General remarks on shell structures (1)Chapter 2 Torsion of closed tubes 4 2.1 Single-cell tube (4)2.2 Torsion of tube with multi-cell cross-section (5)2.3 Warping of closed box sections (7)2.3.1 What do we know so far? (7)2.3.2 Analysis of warping of a rectangular cross-section tube when it is twisted..9Chapter 3 Bending and buckling of thin plates13 3.1 Circular plates under axi-symmetric transverse loading (13)3.1.1 Constitutive equations (13)3.1.2 Equilibrium equations (15)3.1.3 Compatibility equations (16)3.1.4 Governing equations (16)3.1.5 An example: the flexibility of a pressure diaphragm (17)3.2 Rectangular plates under transverse loading (19)3.2.1 Constitutive equations (19)3.2.2 Equilibrium equations (20)3.2.3 Compatibility equations (21)3.2.4 Governing equation (21)3.2.5 Boundary conditions (22)3.2.6 An example: a simply-supported plate under pressure loading p (23)3.3 Buckling of a rectangular plate under in-plane loading (25)3.3.1 A look back at the Euler strut (25)3.3.2 Equivalent loading and the governing equation (27)Chapter 4 Shell structures29 4.1 Some characteristics of shell structures (29)4.2 Hooke’s law for a shell element – in local x, y coordinates (33)Ⅰ4.3 Equilibrium of a plane, curved string (33)4.3.1 A plane, flexible string loaded by a uniform normal load (33)4.3.2 A more subtle string problem (34)4.4 Geometry of surfaces (35)4.5 Behaviour of shells under load according to the membrane hypothesis (39)4.6 Bending stress in symmetrically loaded cylindrical shells (45)Chapter 5 Buckling of cylindrical shells under axial compression53 5.1 Eigenvalue buckling of cylindrical shells (53)5.1.1 The two-surface idealisation (53)5.1.2 Buckling of a simple pin-ended column (53)5.1.3 Buckling of flat plates (55)5.1.4 Buckling of cylindrical shells under uniform axial compression (56)5.2 Nonlinear buckling of thin cylindrical shells: some new advances (60)Question Sheets 80ⅡChapter 1 Introduction to thin-walled structures1.1 Examples of thin-walled structuresThis course introduces postgraduate students to some new aspects of the behaviour of thin-walled structures. Let us begin by considering some familiar examples.(i)(ii)(iii)These are all examples of inflated structures. They are made of suitable cloth, which is held taut by internal pressure.Structural analysis of this type of structure is done mainly by consideration of the equilibrium of doubly-curved “free body” elements subject to a pressure difference and bi-axial membrane stress. In some situations the shape of the pressurized membrane is not known, then the problem of determining the membrane stresses is more complicated.(iv)(v) (open top, closed top, cylindrical or spherical, various sorts)In these cases the primary loading on the structure comes from interior pressure. But secondary loading may also be important. E.g. an oil storage tank can collapse under interior vacuum. These thin-walled structures are prone to buckling when the membrane stresses become compressive. The situation is clearly better with sheet metal than with cloth; but the distinction between the two can be blurred.(vi)These structures are made from flat plates. There is no internal pressure loading. The box construction provides both flexural (bending) and torsional (twisting) stiffness, which is advantageous in suspension-bridge and cable-stayed bridge construction. Note that the constituent plates may themselves be stiffened by longitudinal and transverse ribs of various kinds.(vii)(viii)Closed boxes are more rigid than open ones. This principle is used in the construction of ships, cars and other vehicles, and rigidity may depend on having hatches and doors shut.Again, buckling can be a problem, which is why many such structures have complex arrangements of stiffening ribs, etc.All of these examples are genuinely three- , in the Euclidean sense. All real structures are , of course, but some may be thought of, for the purposes of analysis and design, as . With thin-walled structures it is crucial to think in three dimensions. We have to be able to solve some 3D geometrical problems, including the distortion of 3D structures. This will be less straightforward than the corresponding analysis of 2D problems.1.2 General remarks on shell structuresThe word is an old one and is commonly used to describe the hard coverings of eggs, crustaceans, tortoises, etc. The dictionary says that the word shell is derived from , as infish-scale; but to us now there is a clear difference between the tough but flexible scaly covering of a fish and the tough but rigid shell, say, a turtle.We shall be concerned with man-made shell structures as used in various branches of engineering. There are many interesting aspects of the use of shells in engineering, but one alone stands out as being of importance: it is the structural aspect.Now the theory of structures tends to deal with a class of or rarefied structures, stripped of many of the features which make them recognizable as useful objects in engineering. Thus a beam is often represented as a endowed with certain mechanical properties, irrespective of whether it is a large bridge, an aircraft wing or a flat spring inside a weighing machine. In a similar way, the theory of shell structures deals, for example, with ‘the ’ as a single entity: it is a cylindrical surface endowed with certain mechanical properties. This treatment is the same whether the actual structure under consideration is a gas-transmission pipeline, a grain-storage silo or a steam-raising boiler.Before we enter this realm of theory, in which shells are classified by their geometry (cylindrical, , etc.) rather than their function, it is desirable to give a glimpse of the wide range of applications of shell structures in engineering practice. Indeed, a list of familiar examples will be useful in enabling us to pick out some structural features in a way in order to provide an introduction to the main body of theoretical work.It is instructive to assemble a list of applications from a historical point of view, and to take as a connecting theme for a sequence of brief sketches the way in which the introduction of the thin shell as a structural form has made an important contribution to the development of several different branches of engineering. The following is such a list. It is by no means complete. For a more comprehensive treatment, see Sechler (1974).Architecture and building. The development of domes and vaults in the Middle Age made possible the construction of more spacious buildings; and in recent times the development of reinforced concrete has stimulated interest in the use of thin shells for roofing purposes.Power and chemical engineering. The development of steam power during the Industrial Revolution depended to some extent on the construction of suitable boilers. These thin shells were constructed from plates suitably formed and joined by . In recent times the use of in pressure-vessel construction has led to much more efficient designs. Pressure-vessels and associated pipework are key components in thermal and nuclear power plant, and in all branches of the chemical and petroleum industries.Structural engineering. An important problem in the early development of steel for structural purposes was to design compression members against . A advance was the use of tubular members in the construction of the Fourth railway bridge in 1889: steel plates were riveted together to form reinforced tubes as large as 12 feet in diameter, and having a radius/thickness ratio of between 60 and 180, according to .Vehicle body structures. The construction of vehicle bodies in the early days of road transport involved a system of structural ribs and non-structural panelling or sheeting. The modern form of vehicle construction, in which the plays an important structural part, followed the introduction of sheet-metal components, performed into thin doubly-curved shells by large power presses, and firmly connected to each other by welds along the .The use of the curved skin of vehicles as a load-bearing member has similarly revolutionized the construction of railway carriages and . In the construction of all kinds of the idea of a thin but strong skin has been used from the beginning.Boat construction. The introduction of and similar plastic materials has revolutionized the construction of small and medium-sized boats, since the skin of the can be used as a strong, stiff, structural shell.Miscellaneous. Other examples of the impact of shell construction on technology include the development of large economical natural-draught (= draft) water-cooling towers for thermal power stations, using thin reinforced-concrete shells; and the development of various kinds of economical silos for the storage of grain, etc., by use of thin steel shells.This list can easily be extended to include mediaeval armour, cartridge shells, arch dams, etc.Chapter 2 Torsion of closed tubes2.1 Single-cell tubeFig.2.1 A uniform simply-closed tube subject to Torque To.Fig.2.1 shows a uniform simply-closed tube of arbitrary cross-section. The length is L. Thickness t may vary around circumference, hence t = t(s). A torque To is applied, through opposed couples about a longitudinal axis, at the ends. The material is linear-elastic, with shear modulus of elasticity G. “End effects” are to be ignored.As structural engineers we need to know:(i)distribution of shear stress τ =(ii)relative rotation of the ends; or angle of twist per unit length, .In previous years we found for this problem:Shear flow (2-1)Twist (2-2)The first equation comes from the conditions of equilibrium alone: the tube is statically determinate. The “shear flow” q (dimensions F/L) is a useful quantity here because it does not vary around the circumference.The second equation is harder to obtain. Previously it was derived by means of an “energy balance”:total strain energy in the elastic wall=Now usually, with statically determinate elastic structures, we use equilibrium to find the internal stresses, given the external loads; the Hooke’s law to find the internal strains; and finally compatibility to find the external displacement in terms of the internal strains.The reason for using an energy argument for this torsion problem in the past was that the geometrical relationship between (internal) shear strain and (external) rotation involve awkward geometry of 3D distortion of a tube having an arbitrary cross-section.Strain-rotation relation obtained by virtual workVirtual work says:Σexternal force×corresponding displacement=Σbar tension×bar elongationor ∫ d(Volume) or (here) ∫ d(Volume)Here, external force, bar tension, σ, τ, etc. are an “equilibrium set”, satisfying the equations of equilibrium (only); and “corresponding displacement”, bar elongation, ε, γ, etc. are a “compatible set”, satisfying the equations of kinematics, i.e. the geometry of distortion (only).Here we want to find θ, given γ(s): we are trying to solve a geometrical problem.a. Actual compatible set: strain γ, rotation θb. (Dummy) equilibrium set: torque T,shear flow q, shear stress τFig. 2.2 Virtual workT ⋅θ==∫)Ltds (HenceeA dsL2∫=γθ, required formula for geometry of distortion. For this problem, Hooke’s law is ==GτγHence, putting these together, we obtain the previous formula (2-2) for twist in terms of torque, etc.2.2 Torsion of tube with multi-cell cross-sectionFig. 2.3 A two-cell tube in torsionConsider first a 2-cell tube, as sketched schematically above. It has a uniform cross-section, and is stress-free when untwisted. The material is linear-elastic.Equilibrium As before (though not shown above) we find, by considering the equilibrium of an arbitrary longitudinal strip, thatq = constant between “nodes”,irrespective of the applied torque. In this example we have 3 portions that lie between nodes, so we need 3 internal stress variables 1q , 2q , 3q as shown.Fig. 2.4 The equilibrium at a junctionHowever, we can get one equilibrium relation between them by considering the equilibrium of a long T-section strip cut out from the structure:3q = (2-3)Now this is directly analogous to Kirchhoff’s law for currents at the nodes of an electrical network. Following this analogy we can eliminate 3q by defining “circulating shear flows” 1q and 2q , as shown.A convenient feature of this choice of variables is that overall couple equilibrium gives:o T =(Note that the shear flow in the common wall =21q q −)From this analysis we can see that the two-cell tube is statically indeterminate. Thus we have 2 unknown stress variables (q1, q2) but only 1 equation of equilibrium, so the arrangement is statically indeterminate to degree 1. (A 3-cell tube, likewise, would have 2 degrees of indeterminacy).In order to solve the problem, therefore, we need another relation between 1q and 2q . Our previous experience of various kinds of structure suggests that we need to consider the elastic distortion of the arrangement in order to obtain such an equation.Here, the condition that we seek is:φ1 = φ2, LL21θθ=For the sake of simplicity, in the present example let the material be the same as before (G), and let the 3 portions of the cross-section have uniform thickness t 1, t 2, t 3 and arc-lengths s 1, s 2, s 3, respectively, as shown.Fig. 2.5 Cross-sectional dimensions of a two-cell tubeeA ds L2∫=γθ133211111331112/)(/2A Gt s q q Gt s q A s s L −+=+=⎟⎠⎞γγθ233122222332222/)(/2A Gt s q q Gt s q A s s L −+=+=⎟⎠⎞γγθHence, putting θ1 = θ2, we get a relationship between 1q and 2q .ExampleFig. 2.6 A rectangular two-cell tube: thickness t everywhereEquilibrium: T o = 2⋅2a 2q 1 + 2⋅a 2q 2Compatibility: =L1θ=L2θBut θ1 = θ2, so 5q 1+q 1-q 2 = 2(3q 2+q 2-q 1), leading to q 2 =0.8q 1 Hence q 1 in terms of T; and θ.In this example the internal wall has low stress, and contributes relatively little to the torsional stiffness.2.3 Warping of closed box sections 2.3.1 What do we know so far?In the previous two Sections we have found, for both single-cell and multi-cell uniform closed box sections under torsion, the distribution of shear flow q within the cross-section and the overall torsional stiffness (To/φ).Unrestrained warping-flanges undistorted, Restrained warping-flanges bentend-section will not be a plane any more.Fig. 2.7 Warping of an I-sectionIt is also known from an investigation of uniform open cross-sections under torsion that warping would occur, i.e. that plane cross-sections of the member did not remain plane under this type of loading. One implication of this finding is that, when warping is restrained, axial stresses are induced; in the case of an I-section member, it is to interpret the phenomenon in terms of differential bending of the section flanges – see the diagram below. As a corollary, the member becomes rather stiffer in torsion due to the combined action of ‘GJ’ and ‘EI’ effects.The “Teach Yourself Torsion” demonstration on the Mezzanine floor of the Structures Laboratory has an open “channel” section that is easy to twist (i.e. it has low torsional stiffness) and whose cross-sections evidently warp out-of-plane. There is also another specimen of a similar kind, but with ends that are prevented from warping by the presence of solid wooden blocks. This second specimen has much higher torsional stiffness.This raises a question: does warping, and its consequence when restrained, occur when thin-walled tubes are twisted?Consider a thin-walled, closed rectangular box of uniform thickness. From the results of Section 2.1 we know that an applied torque will induce uniform shear flow and hence result in uniform shear stress and uniform shear strain. If the box were frozen in its strained state, then slit along one edge and laid out flat, it would look like this:Fig. 2.8 A ‘frozen’ closed section cut and laid out flatWe can now explore the question of warping experimentally. From A4 flat sheets (297×210 mm), fabricate two paper models with large shear γ = (15/210) ≈ 4o. The two models should have the same total perimeter length of 260 mm but different cross-section, A (65×65 mm) and B (100×30 mm). Insert end diaphragms to maintain the rectangular sections.Observations(i) If diaphragms are not used, what are the observable effects?It is easy to deform the shape of the cross-section; but a diaphragm at each end makes the whole thing rigid.(ii) Is there any detectable difference in either twist or warping between the two models?Twist is greater for the rectangular box – as expected from eA ds L 2∫=γθbecause A e issmaller.There is some warping for rectangular box, but none for square one.(iii) Compare the values of twist and warping obtained experimentally with the theoreticalresults derived in the next Section for small shear strains.2.3.2 Analysis of warping of a rectangular cross-section tube when it is twistedWe have seen in Section 2.3.1 that the shear strains produced in the sides of a tube when it is subjected to torque – as a result of the shear stress and on account of the elasticity of the material – are geometrically compatible with an overall twist of the tube. But in the process of such distortion the ends of the box may go out-of-plane, or “warp”.Suppose that a given tube with free ends warps at its ends when it is twisted. If the same box is now connected to rigid end-fittings, that prevent such warping, then when the tube is twisted some additional stresses will be set up at and near the ends of the tube; and so the tube will be stiffer under torsion than it was when the ends were free.Calculations of the stiffening effect of prevention of the ends from warping is a complicated business, and beyond the scope of these lectures. However, we shall perform the first step of such a calculation. Thus, we shall calculate the warping, or lack-of-planarity, of a rectangular-cross-section tube with free ends when it is twisted.Fig. 2.9 A rectangular tubeLet the cross-section of the tube be a ×b. The applied torque To produces a uniform shear flow q = τt by equilibrium (2-1). Given the wall thickness, which we shall assume to be uniform for eachface of the tube, we can find τ in each face; and hence, by Hooke’s law, we can obtain the shear strain γ in each face: say γ1, γ2, γ3, γ4 in the four faces. Shear strains are measured in radians, although they are sometimes quoted in degrees: take care!We now want to know if these uniform shearing strains in the four faces produce warping at the ends; and if so, how much.The problem is a purely geometrical one; and in principal we could solve it by making a cardboard model as in §2.3.1, and taking suitable measurements. The picture is shown below, with uniform shearing strain γi (i=1,4) in each face. This sheet is to be cut out, folded, and joined along edges AA; and the question is whether the ends ABCD warp out-of -plane; and if so, by how much.Suppose that the model has now been assembled, and that a Cartesian coordinate system has been set up, with the z-axis along the axis of the tube. Let w be the (small) component of displacement in the z direction, measured from a suitable x,y plane; and let w A, w B, w C, w D be the z-direction displacements at the four edges of the tube.Now, since each of the four faces of the tube is subjected only to pure shearing stress, there is no change of longitudinal length anywhere; and so w A has the same value all along the edge AA; and similarly for the other three edges. These displacements have been marked, schematically, on the diagram.Fig.2-10 Shear strain in the rectangular sectionNow it is straightforward to show that any cross-section of the tube would remain plane during the distortion if the edge displacements were to satisfy the equation.(2-4)There are various ways to show this. Perhaps the simplest is to demonstrate that the following three separate patterns of displacements of an imaginary rigid end-plate all satisfy Eq. (2-4):(i) (w A, w B, w C, w D) = (α, α, α, α) ( )(ii) (w A, w B, w C, w D) = (0, 0, β, β) ( )(ii) (w A, w B, w C, w D) = (0, δ, δ, 0) ( ) Since any rigid-body motion can be made up by superposition of three such components, Eq. (2-4) must be a general condition for zero warping. And indeed the expressionW=gives any non-zero warping W, this being defined as the amount by which corner A moves out of the plane defined by the corners B, C and D (or indeed the amount by which any one corner moves away from the plane defined by the other three).How, then, can we calculate W? Let us use the principle of virtual work.Now the shearing strains γ1, γ2, γ3, γ4 are geometrically compatible with the required warping W. For the (dummy) equilibrium set let us impose a longitudinal force of magnitude 2 per unit length along each of the four edges, as shown above, so that each of the four faces is in a state of pure shear flow of magnitude 1, as shown here for faces 2 and 3. Note particularly that this kind of (dummy) loading provides uniform shear flow of magnitude 1, but of opposite sense in adjacent faces.Note also that shear forces of magnitude 1 must be provided on the end edges of the four faces, in order to be in static equilibrium with the uniform shear flow in each face. Now these external forces on opposite edges provide couples about the axis of the tube; but the corresponding forces on the other two edges provide opposite couples of the same magnitude, as may readily be shown. Thus the resultant of the external forces on each end of the tube is precisely zero.Fig. 2-11 Shear stress in the rectangular sectionApplying virtual work, with external forces/displacements on the left and internal ones on the right, we have:2L(w A - w B + w C - w D) ==L(-aγ1 + bγ2 - aγ3 + bγ4)hence(2-5)In the special case γ1 = γ2 = γ3 = γ4 = γ, corresponding to uniform thickness in all four faces, and the same elastic material everywhere – and as in the cardboard model in §2.3.1.W = (2-6)This result immediately shows that there is no warping in the cardboard model square box (since b = a); and it enables us to predict easily the warping in the rectangular-section tube that we have made.Notes.(1)Use Eq. (2-6) to predict the end-warping of your rectangular-section tube.(2)We have assumed throughout that cross-sections of the tube remain rectangular. In practicediaphragms are used to ensure this.(3)If warping of the tube is restrained at the ends, then axial stresses are induced there, and thetorsional stiffness of the tube is thereby increased. These induced stresses are self-equilibrating; but as mentioned above, their computation is not trivial and is not attempted here. However, St Venant’s Principle tells us that such induced stresses will not extend into the tube more than a distance of the order of the width of the tube.(4)We have analysed here the simple case of a rectangular tube. The same general method maybe adapted to tubes having other cross-sectional shapes.Chapter 3 Bending and buckling of thin platesIn the previous chapter we have considered some situations in which a thin-walled box or tube is subjected to . As we saw, the primary structural action at a local level was pure in-plane shear stress, leading to shear strain in the wall of the tube, and twisting of the tube overall. Many kinds of structure involve thin-plate structural elements that are loaded , by pressure or concentrated forces. Obvious examples are the bottom plate of a ship, a simple floor slab in (say) a multi-storey car-park, and a pressure-diaphragm in a piping system. In such cases the primary structural action is : the plate is like a sort of “ .” Engineers need to be able to analyse the stress and deformation of flat plates under transverse loading.In other types of structure, thin-plate elements are loaded primarily by in-plane tensile and compressive forces. Thus the top and bottom plates of a box-girder act like the “flanges” of a beam. But in-plane compressive loading of a plate may precipitate . Qualitatively this is like “Euler” buckling of an axially compressed column – except that with plates there is obviously a “two-way” action, as noted above.In this chapter we shall first analyse some simple circular and rectangular plates under transverse loading, and in the absence of in-plane forces. Then we shall investigate the buckling of plates on account of in-plane compression. For simplicity we shall assume that the material is elastic throughout, which may actually be somewhat artificial in the context of many practical structures.In the following chapter we will consider the behaviour of shell structures, in which the thin-walled structural components are curved. This raises a whole lot of problems that we do not encounter in the present section, throughout which the plate elements are always plane.3.1 Circular plates under axi-symmetric transverse loading3.1.1 Constitutive equationsIn the analysis of a beam element in bending, we assumed that plane sections remain plane and perpendicular to the deformed axis ( ); and deduced that the strain at a distance y the “neutral axis” is given by ε = yκ, where κ is the local change of curvature from the unstressed state. We then assumed a linear-elastic stress/strain relationship, σ = Eε, and hence derived the constitutive relationship between bending moment and curvature in a general beam element:(3-1)We now consider the corresponding situation for a typical element of an elastic plate. We must obviously consider two-way action, which is clearly more complicated than for a beam. We shall consider only a uniform plate, and so we shall avoid here problems of integrating over the arbitrary cross-section of a beam. We start with a small, square element cut from a plate, and loaded simultaneously by independent moments in the two directions, as shown below.Since we are dealing with circular plates at present, we label the two directions r ( ) and θ( ); and these symbols appear as subscripts. rM a bendingmoment in the radial direction (as in a radial strip cut out and acting as a beam); but it is a bending moment per unit width of beam, with width measured in the direction, here θ. Similarly for θM . As we shall find later on, in a circular plate under axi-symmetric loading, there is no twisting of the element; and so r M and θM are the only bending moment components to be considered at present.Fig. 3.1 Bending moments following the RH screw conventionThese bending moments per unit width cause the initially flat element to curve in two directions at once. r κ and θκare directly analogous to κ in the beam example described above; and they are, strictly, changes of with respect to the initial, unstressed state.As before, we assume a simple kinematic model change of curvature corresponding to the “railway track” analogy, in which normals to the central surface of the plate remain normal to this surface as the plate deforms: “Kirchhoff’s Hypothesis”. This immediately leads to simple expressions for the components of strain, r εand θε, in terms of the distance z of a thin slice of material from the central surface of the plate. Thus; θθκεz = (3-2)So much for the kinematics of distortion of the plate element. Next we quote Hooke’s law for an isotropic elastic material in 2D stress – σz being assumed negligible here in comparison with σr and σθ for the slice:⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡−−=⎭⎬⎫⎩⎨⎧θθσσννεεr r E 111 (3-3)Here E is Young’s elastic modulus for the (isotropic) material and νis Poisson’s ratio.Solving Eq.(3-3) for σr , σθ in terms of εr and εθ we have:⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡−=⎭⎬⎫⎩⎨⎧θθεενννσσr r E 1112 (3-4)Since we know how r εand θεvary with z (Eq.(3-2)) through the thickness of the plate, from z = -t/2 to z = +t/2, we may integrate for a unit area of plate to get:=⋅=∫−2/2/)1(t t r r z dz M σ。

PAVEMENT PROBLEMS CAUSEDBY COLLAPSIBLE SUBGRADESBy Sandra L. Houston,1 Associate Member, ASCE(Reviewed by the Highway Division)ABSTRACT: Problem subgrade materials consisting of collapsible soils are -mon in arid environments, which have climatic conditions and depositional and weathering processes favorable to their formation. Included herein is a discussion of predictive techniques that use commonly available laboratory equipment and testing methods for obtaining reliable estimates of the volume change for these problem soils. A method for predicting relevant stresses and corresponding collapse strains for typical pavement subgrades is presented. Relatively simple methods ofevaluating potential volume change, based on results of familiar laboratory tests, are used.INTRODUCTIONWhen a soil is given free access to water, it may decrease in volume,increase in volume, or do nothing. A soil that increases in volume is calleda swelling or expansive soil, and a soil that decreases in volume is called a collapsible soil. The amount of volume change that occurs depends on thesoil type and structure, the initial soil density, the imposed stress state, and the degree and extent of wetting. Subgrade materials comprised of soils that change volume upon wetting have caused distress to highways since the be- ginning of the professional practice and have cost many millions of dollarsin roadway repairs. The prediction of the volume changes that may occur inthe field is the first step in making an economic decision for dealing with these problem subgrade materials.Each project will have different design considerations, economic con-straints, and risk factors that will have to be taken into account. However, with a reliable method for making volume change predictions, the best design relative to the subgrade soils becomes a matter of economic comparison, anda much more rational design approach may be made. For example, typical techniques for dealing with expansive clays include: (1) In situ treatments with substances such as lime, cement, or fly-ash; (2) seepage barriers and/or drainage systems; or (3) a computing of the serviceability loss and a mod- ification of the design to "accept" the anticipated expansion. In order to make the most economical decision, the amount of volume change (especially non- uniform volume change) must be accurately estimated, and the degree of road roughness evaluated from these data. Similarly, alternative design techniques are available for any roadway problem.The emphasis here will be placed on presenting economical and simplemethods for: (1) Determining whether the subgrade materials are collapsible; and (2) estimating the amount of volume change that is likely to occur in the 'Asst. Prof., Ctr. for Advanced Res. in Transp., Arizona State Univ., Tempe, AZ 85287.Note. Discussion open until April 1, 1989. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on February 3, 1988.This paper is part of the Journal of Transportation.Engineering, Vol. 114, No. 6,November, 1988. ASCE, ISSN 0733-947X/88/0006-0673/$1.00 + $.15 per page. Paper No. 22902.673field for the collapsible soils. Then this information will place the engineer in a position to make a rational design decision. Collapsible soils are fre- quently encountered in an arid climate. The depositional process and for- mation of these soils, and methods for identification and evaluation of the amount of volume change that may occur, will be discussed in the following sections.COLLAPSIBLE SOILSFormation of Collapsible SoilsCollapsible soils have high void ratios and low densities and are typically cohesionless or only slightly cohesive. In an arid climate, evaporation greatly exceeds rainfall. Consequently, only the near-surface soils become wettedfrom normal rainfall. It is the combination of the depositional process andthe climate conditions that leads to the formation of the collapsible soil. Although collapsible soils exist in nondesert regions, the dry environment in which evaporation exceeds precipitation is very favorable for the formationof the collapsible structure.As the soil dries by evaporation, capillary tension causes the remainingwater to withdraw into the soil grain interfaces, bringing with it soluble salts, clay, and silt particles. As the soil continues to dry, these salts, clays, and silts come out of solution, and "tack-weld" the larger grains together. This leads to a soil structure that has high apparent strength at its low, natural water content. However, collapse of the "cemented" structure may occurupon wetting because the bonding material weakens and softens, and the soilis unstable at any stress level that exceeds that at which the soil had been previously wetted. Thus, if the amount of water made available to the soilis increased above that which naturally exists, collapse can occur at fairly low levels of stress, equivalent only to overburden soil pressure. Additional loads, such as traffic loading or the presence of a bridge structure, add to the collapse, especially of shallow collapsible soil. The triggering mechanism for collapse, however, is the addition of water.Highway Problems Resulting from Collapsible SoilsNonuniform collapse can result from either a nonhomogeneous subgradedeposit in which differing degrees of collapse potential exist and/or fromnonuniform wetting of subgrade materials. When differential collapse of subgrade soils occurs, the result is a rough, wavy surface, and potentially many miles of extensively damaged highway. There have been several re-ported cases for which differential collapse has been cited as the cause of roadway or highway bridge distress. A few of these in the Arizona and New Mexico region include sections of 1-10 near Benson, Arizona, and sectionsof 1-25 in the vicinity of Algadonas, New Mexico (Lovelace et al. 1982; Russman 1987). In addition to the excessive waviness of the roadway sur-face, bridge foundations failures, such as the Steins Pass Highway bridge,1-10, in Arizona, have frequently been identified with collapse of foundation soils.Identification of Collapsible SoilsThere have been many techniques proposed for identifying a collapsiblesoil problem. These methods range from qualitative index tests conducted on674disturbed samples, to response to wetting tests conducted on relatively un- disturbed samples, to in situ meausrement techniques. In all cases, the en- gineer must first know if the soils may become wetted to a water contentabove their natural moisture state, and if so, what the extent of the potential wetted zone will be. Most methods for identifying collapsible soils are only qualitative in nature, providing no information on the magnitude of the col- lapse strain potential. These qualitative methods are based on various func- tions of dry density, moisture content, void ratio, specific gravity, and At- terberg limits.In situ measurement methods appear promising in some cases, in that many researchers feel that sample disturbance is greatly reduced, and that a more nearly quantitative measure of collapse potential is obtainable. However,in situ test methods for collapsible soils typically suffer from the deficien- cy of an unknown extent and degree of wetting during the field test. This makes a quantitative measurement difficult because the zone of materialbeing influenced is not well-known, and, therefore, the actual strains, in- duced by the addition of stress and water, are not well-known. In addition,the degree of saturation achieved in the field test is variable and usually unknown.Based on recently conducted research, it appears that the most reliablemethod for identifying a collapsible soil problem is to obtain the best quality undisturbed sample possible and to subject this sample to a response to wet- ting test in the laboratory. The results of a simple oedometer test will indicate whether the soil is collapsible and, at the same time, give a direct measureof the amount of collapse strain potential that may occur in the field. Potential problems associated with the direct sampling method include sample distur-bance and the possibility that the degree of saturation achieved in the field will be less than that achieved in the laboratory test.The quality of an undisturbed sample is related most strongly to the arearatio of the tube that is used for sample collection. The area ratio is a measure of the ratio of the cross-sectional area of the sample collected to the cross- sectional area of the sample tube. A thin-walled tube sampler by definitionhas an area ratio of about 10-15%. Although undisturbed samples are best obtained through the use of thin-walled tube samplers, it frequently occursthat these stiff, cemented collapsible soils, especially those containing gravel, cannot be sampled unless a tube with a much thicker wall is used. Samplers having an area ratio as great as 56% are commonly used for Arizona col-lapsible soils. Further, it may take considerable hammering of the tube todrive the sample. The result is, of course, some degree of sample distur- bance, broken.bonds, densification, and a correspondingly reduced collapse measured upon laboratory testing. However, for collapsible soils, which are compressive by definition, the insertion of the sample tube leads to local shear failure at the base of the cutting edge, and, therefore, there is less sample disturbance than would be expected for soils that exhibit general shear failure (i.e., saturated clays or dilative soils). Results of an ongoing study of sample disturbance for collapsible soils indicate that block samples some- times exhibit somewhat higher collapse strains compared to thick-walled tube samples. Block samples are usually assumed to be the very best obtainable undisturbed samples, although they are frequently difficult-to-impossible to obtain, especially at substantial depths. The overall effect of sample distur- bance is a slight underestimate of the collapse potential for the soil.675译文：湿陷性地基引起的路面问题...摘要：在干旱环境中，湿陷性土壤组成的路基材料是很常见的，干旱环境中的气候条件、沉积以与风化作用都有利于湿陷性土的形成。

材料力学中英对照词汇A安全因数safety factorB半桥接法half bridge闭口薄壁杆thin-walled tubes比例极限proportional limit边界条件boundary conditions变截面梁beam of variable cross section 变形deformation变形协调方程compatibility equation 标距gage length泊松比Poisson’s ratio补偿块compensating blockC材料力学mechanics of materials冲击荷载impact load初应力，预应力initial stress纯剪切pure shear纯弯曲pure bending脆性材料brittle materialsD大柔度杆long columns单位荷载unit load单位力偶unit couple单位荷载法unit-load method单向应力，单向受力uniaxial stress等强度梁beam of constant strength低周疲劳low-cycle fatigue电桥平衡bridge balancing电阻应变计resistance strain gage电阻应变仪resistance strain indicator 叠加法superposition method叠加原理superposition principle 动荷载dynamic load断面收缩率percentage reduction in area多余约束redundant restraintE二向应力状态state of biaxial stressF分布力distributed force复杂应力状态state of triaxial stress复合材料composite materialG杆，杆件bar刚度stiffness刚架，构架frame刚结点rigid joint高周疲劳high-cycle fatigue各向同性材料isotropical material功的互等定理reciprocal-work theorem工作应变计active strain gage工作应力working stress构件structural member惯性半径radius of gyration of an area惯性积product of inertia惯性矩，截面二次轴距moment of inertia广义胡克定律generalized Hook’s lawH横向变形lateral deformation胡克定律Hook’s law滑移线slip-linesJ基本系统primary system畸变能理论distortion energy theory畸变能密度distortional strain energydensity极惯性矩，截面二次极矩polar moment of inertia极限应力ultimate stress极限荷载limit load挤压应力bearing stress剪力shear force剪力方程equation of shear force剪力图shear force diagram剪流shear flow剪切胡克定律Hook’s law for shear剪切shear交变应力，循环应力cyclic stress截面法method of sections截面几何性质geometrical properties ofan area截面核心core of section静不定次，超静定次数degree of astatically indeterminate problem静不定问题，超静定问题staticallyindeterminate problem静定问题statically determinate problem静荷载static load静矩，一次矩static moment颈缩neckingK开口薄壁杆bar of thin-walled open crosssection抗拉强度ultimate stress in tension抗扭截面系数section modulus in torsion抗扭强度ultimate stress in torsion抗弯截面系数section modulus inbendingL拉压刚度axial rigidity拉压杆，轴向承载杆axially loaded bar理想弹塑性假设elastic-perfectly plasticassumption力法force method力学性能mechanical properties连续梁continuous beam连续条件continuity condition梁beams临界应力critical stress临界荷载critical loadM迈因纳定律Miner’s law名义屈服强度offset yielding stress莫尔强度理论Mohr theory of failure敏感栅sensitive gridN挠度deflection挠曲轴deflection curve挠曲轴方程equation of deflection curve挠曲轴近似微分方程approximatelydifferential equation of the deflectioncurve内力internal forces扭力矩twisting moment扭矩torsional moment扭矩图torque diagram扭转torsion扭转极限应力ultimate stress in torsion扭转角angel of twist扭转屈服强度yielding stress in torsion扭转刚度torsional rigidityO欧拉公式Euler’s formulaP疲劳极限，条件疲劳极限endurancelimit疲劳破坏fatigue rupture疲劳寿命fatigue life偏心拉伸eccentric tension偏心压缩eccentric compression平均应力average stress平面弯曲plane bending平面应力状态state of plane stress平行移轴定理parallel axis theorem平面假设plane cross-section assumptionQ强度strength强度理论theory of strength强度条件strength condition切变模量shear modulus切应变shear strain切应力shear stress切应力互等定理theorem of conjugate shearing stress屈服yield屈服强度yield strength全桥接线法full bridgeR热应力thermal stressS三向应力状态state of triaxial stress三轴直角应变花three-element rectangular rosette三轴等角应变花three-element delta rosette失稳buckling伸长率elongation圣维南原理Saint-Venant’s principle实验应力分析experimental stress analysis塑性变形，残余变形plastic deformationductile materials塑性材料，延性材料塑性铰plastic hingeT弹簧常量spring constant弹性变形elastic deformation弹性模量modulus of elasticity体积力body force体积改变能密度density of energy ofvolume change体应变volume strainW弯矩bending moment弯矩方程equation of bending moment弯矩图bending moment diagram弯曲bending弯曲刚度flexural rigidity弯曲正应力normal stress in bending弯曲切应力shear stress in bending弯曲中心shear center位移法displacement method位移互等定理reciprocal-displacementtheorem稳定条件stability condition稳定性stability稳定安全因数safety factor for stabilityX细长比，柔度slenderness ratio线性弹性体linear elastic body约束扭转constraint torsion相当长度，有效长度equivalent length相当应力equivalent stress小柔度杆short columns形心轴centroidal axis形状系数shape factor许用应力allowable stress许用应力法allowable stress method许用荷载allowable load许用荷载法allowable load methodY应变花strain rosette应变计strain gage应变能strain energy应变能密度strain energy density应力stress应力速率stress ratio应力比stress ratio应力幅stress amplitude应力状态state of stress应力集中stress concentration应力集中因数stress concentration factor应力－寿命曲线，S-N曲线stress-cyclecurve应力－应变图stress-strain diagram应力圆，莫尔圆Mohr’s circle for stressesZ正应变normal strain正应力normal stress中面middle plane中柔度杆intermediate columns中性层neutral surface中性轴neutral axis轴shaft轴力axial force轴力图axial force diagram轴向变形axial deformation轴向拉伸axial tension轴向压缩axial compression主平面principal planes主应力principal stress主应力迹线principal stress trajectory主轴principal axis主惯性矩principal moment of inertia主形心惯性矩principal centroidalmoments of inertia主形心轴principal centroidal axis转角angel of rotation转轴公式transformation equation自由扭转free torsion组合变形combined deformation组合截面composite area最大切应力理论maximum shear stresstheory最大拉应变理论maximum tensile straintheory最大拉应力理论maximum tensile stresstheory最大应力maximum stress最小应力minimum stress。

1、Besides oil being present in sufficient quantities ,and water and gas available to help move it to surface, three other factors must be present in the reservoir to make it economically worthwhile. These are pressure gradient ,gtavity, and capillary action.除了足够石油存在，水和可以帮助把它到地面的天然气之外，在储层中必须存在其他三个因素使其具有经济开采价值，他们是压力梯度重力和毛细作用。

2、Pressure gradient means that a difference exists between pressures measured at two different points.If the pressure is lower at the wellthan at other points in the reservoir,the areas of higher pressure will exert energy that will force the fluids to and up the wellbore resulting in the well flowing .If such pressure is not present in sufficient quantities ,then a pump or other artificial means will have to be used to lift the oil to the surface.压力梯度意味着测量的两点之间压力存在差异，如果经的压力比油藏的其他点底，那高压区会释放能量，驱使流体到达井眼，何在井眼中爬升从而实现油在井中流动，如果不存在足够这样的压力，那么就必须用泵或者其他人工方法来把油举升到地面。

薄壳承重书实验英文作文回答例子1:Title: "Experimental Study on Thin-shell Structures Bearing Loads"Abstract:Thin-shell structures have gained significant attention due to their lightweight nature and efficient use of materials. This experimental study focuses on investigating the load-bearing capacity of thin-shell structures, specifically exploring the principles behind their design and their application in various engineering fields. Through a series of experiments, the structural behavior of thin shells under different loading conditions is analyzed, providing insights into their performance and potential applications. This article aims to contribute to the understanding of thin-shell structures and their optimization for practical use in engineering projects.Introduction:Thin-shell structures represent a class of engineering回答例子2:Title: The Experimental Study on Shell Structures Bearing LoadsAbstract:Shell structures have been widely utilized in various engineering applications due to their efficient use of materials and aesthetic appeal. However, their structural behavior under load, especially thin shell structures, remains a subject of intense study. This article presents an experimental investigation into the load-bearing capacity of thin shell structures, aiming to provide insights into their structural performance and potential applications.Introduction:Shell structures, characterized by their curved surfaces and minimal use of material, have been employed in architectural, aerospace, and civil engineering projects. The efficiency of shell structures lies in their ability to distribute loads uniformly across their surfaces, leading to optimal use of materials and enhanced structural performance. Thin shell structures, in particular, offer unique advantages such as lightweight construction and architectural flexibility. However, theirload-bearing behavior requires thorough examination to ensure structural integrity and safety.Experimental Setup:The experimental study was conducted using acustom-designed apparatus capable of applying controlled loads to thin shell specimens. The specimens were fabricated from various materials commonly used in shell construction, including reinforced concrete, steel, and composite materials. Each specimen was carefully manufactured to precise dimensions and geometries, ensuring consistency in the experimental setup. Strain gauges and load cells were strategically placed to monitor the deformation and load distribution during testing.Experimental Procedure:The experimental procedure involved subjecting the thin shell specimens to progressively increasing loads until failure occurred. Load increments were applied systematically, and measurements of deflection, strain, and load were recorded at regular intervals. Visual inspections were conducted to identify any signs of structural distress or failure modes. Theexperiments were repeated multiple times to ensure the reliability and repeatability of the results.Results and Analysis:The experimental results revealed significant insights into the load-bearing behavior of thin shell structures. It was observed that the ultimate load-carrying capacity varied depending on several factors, including material properties, shell geometry, and support conditions. In general, shell structures exhibited nonlinear load-deflection responses, characterized by initial stiffness followed by gradual softening under increasing loads. Failure modes ranged from local buckling to global collapse, with the specific mode depending on the material and geometric properties of the specimen.Discussion:The experimental findings underscore the importance of considering various factors in the design and analysis of thin shell structures. Material selection, geometric configuration, and support conditions play crucial roles in determining the structural performance and load-bearing capacity of shell elements. Furthermore, the observed nonlinear behaviornecessitates advanced computational techniques for accurate prediction and analysis of shell structures under load.Conclusion:In conclusion, the experimental study provides valuable insights into the load-bearing behavior of thin shell structures. By systematically investigating the response of shell specimens to applied loads, this research contributes to the understanding of shell structural mechanics and informs the design and analysis of shell-based engineering systems. Future research directions may include further exploration of different materials, geometric configurations, and loading scenarios to enhance the performance and applicability of shell structures in diverse engineering fields.。

八年级英语建筑术语单选题50题1. The wall of this building is made of _____.A.woodB.steelC.plasticD.paper答案：A。

本题考查常见建筑材料。

选项B 钢铁通常用于建筑结构而非墙体；选项 C 塑料一般不用于建筑墙体；选项 D 纸不能作为建筑墙体材料。

而木材是常见的建筑墙体材料之一。

2. The roof of this house is covered with _____.A.glassB.clothC.tileD.leaf答案：C。

选项A 玻璃一般不作为屋顶覆盖材料；选项B 布不适合做屋顶材料；选项D 树叶不能用于屋顶覆盖。

瓦片是常见的屋顶覆盖材料。

3. This bridge is mainly constructed with _____.A.iceB.sandC.cementD.cotton答案：C。

选项A 冰不能用于建造桥梁；选项B 沙子一般不能单独作为桥梁的主要建筑材料；选项D 棉花不能用于建筑桥梁。

水泥是建造桥梁常用的材料。

4. The floor of this room is made of _____.A.waterB.airC.wooden boardD.fire答案：C。

选项 A 水和选项B 空气不能作为地板材料；选项D 火不是建筑材料。

木板是常见的地板材料。

5. The pillar of this building is made of _____.A.cakeB.stoneC.butterD.feather答案：B。

选项A 蛋糕、选项C 黄油和选项D 羽毛都不能作为建筑的柱子材料。

石头是常见的建筑柱子材料。

6.The building has pointed arches and high ceilings. This is characteristic of _____ architecture.A.GothicB.RomanesqueC.BaroqueD.Modernist答案：A。

海口2024年09版小学4年级上册英语第二单元寒假试卷[含答案]考试时间：100分钟（总分:110）B卷考试人：_________题号一二三四五总分得分一、综合题(共计100题共100分)1. 填空题：The _____ (环境保护) initiatives promote planting more trees.2. 听力题：The __________ is a region known for its biodiversity.3. 听力题：We are going ________ a trip.4. 选择题：What is the main ingredient in guacamole?A. TomatoB. AvocadoC. PepperD. Onion答案: B5. 听力题：The chemical symbol for scandium is ______.6. 填空题：Birds make _______ (巢) in the trees.7. 填空题：The ancient Romans drew inspiration from ________ mythology.8. 听力题：A shadow is formed when light is ______.9. 选择题：What do we use to cut paper?A. RulerB. ScissorsC. GlueD. Tape答案: B10. 听力题：Metals are typically good conductors of _____.11. 听力题：The element with the atomic number is ______.12. 填空题：In my opinion, being _______ (形容词) is a valuable trait in life. It helps us connect with others.13. 听力题：The Sahara Desert is one of the hottest _______ on Earth.14. 听力题：The children are _______ (jumping) on the trampoline.15. 选择题：What is the primary color of the sun?A. BlueB. YellowC. GreenD. Red答案: B16. 填空题：The __________ (历史的见证者) recount significant moments.17. 填空题：Do you like to watch _____ (蝴蝶) in the garden?18. 选择题：What do you call a place where you can see many animals?A. ZooB. FarmC. ParkD. Garden答案:A19. 听力题：A saturated solution is in ______ equilibrium.What do we call the area of land that is covered by forests?A. Wooded areaB. TimberlandC. ForestD. All of the above答案: D. All of the above21. 听力题：A chemical change usually cannot be __________ back easily.22. 听力题：We will _______ (attend) the school play.23. 填空题：During the holidays, I visit my ________ (祖父母). They live in the ________ (乡村).24. 听力题：The chemical formula for lactic acid is _______.25. 填空题：A ____(social equity) ensures fair treatment for all.26. 选择题：What is the capital city of Afghanistan?A. KabulB. KandaharC. HeratD. Mazar-i-Sharif答案: A27. 填空题：The ______ (自然环境) supports diverse plant life.28. 填空题：The __________ (历史的开放性) invites exploration.29. 填空题：My cousin is a wonderful __________ (舞者).30. 填空题：I like to collect _______ (玩具) from different countries.31. 填空题：I saw a _________ (小鸟) in my garden.The __________ is a vast grassland area in Africa. (草原)33. 选择题：What do we call the process of plants converting sunlight into food?A. RespirationB. PhotosynthesisC. TranspirationD. Germination答案: B. Photosynthesis34. 填空题：My favorite stuffed _______ is a bear (我最喜欢的毛绒_______是熊).35. 听力题：The _______ can help brighten up your home.36. 听力题：The process of osmosis involves the movement of __________.37. 选择题：What is the name of our nearest star?A. Alpha CentauriB. Proxima CentauriC. SiriusD. Betelgeuse38. 听力题：She is _____ (cooking) dinner.39. 听力题：I love to eat ________ for breakfast.40. 听力题：The capital of Honduras is __________.41. 听力题：The cake is ___. (yummy)42. 听力题：We will go ______ for a picnic tomorrow. (out)43. 选择题：How many months are there in a year?A. TenB. ElevenC. TwelveD. Thirteen答案:C44. 填空题：She is a _____ (设计师) who focuses on sustainability.45. 填空题：My friend is very __________ (友好的) and open-minded.46. 听力题：My brother likes to play ____ (basketball) with his friends.47. 选择题：Which fruit is red and often associated with teachers?A. OrangeB. BananaC. AppleD. Grape答案: C48. 填空题：My favorite thing to learn is ______.49. 选择题：What is the primary ingredient in a peanut butter sandwich?A. JellyB. BreadC. Peanut butterD. Honey50. 选择题：What is the capital of Zimbabwe?A. HarareB. BulawayoC. MutareD. Gweru答案:A51. 选择题：What is the opposite of cheerful?A. GloomyB. JoyfulC. HappyD. Bright答案:AThe process of oxidation involves __________ losing electrons.53. 选择题：What do we call a group of butterflies?A. FlutterB. SwarmC. KaleidoscopeD. Flight答案:C. Kaleidoscope54. 听力题：Acidic solutions have a pH less than _______.55. 填空题：My friend has a lot of __________ (梦想) for the future.56. 听力题：A solar system consists of a star and all its ______.57. 填空题：My sister enjoys __________ (参加) community events.58. 听力题：The main gas involved in the greenhouse effect is __________.59. 听力题：My dad is a ______. He enjoys fishing.60. 选择题：What is 2 x 3?A. 5B. 6C. 7D. 861. 选择题：What do we call a story that is made up?a. Biographyb. Fictionc. Non-fictiond. History答案:B62. 听力题：I like to ______ my friends' houses. (visit)My uncle gave me a rare _________ (玩具) from his collection.64. 听力题：My dad helps me with my ____ (projects) for school.65. 选择题：What is the name of the famous painting by Vincent van Gogh?A. The Last SupperB. Starry NightC. Mona LisaD. The Scream答案:B66. 听力题：Chemical equations must be balanced to follow the law of _____ of mass.67. 填空题：The parrot's bright colors help it blend in with tropical ________________ (植物).68. 填空题：Certain plants have ______ that allow them to survive in cold climates.(某些植物有耐寒的特性，使它们能够在寒冷的气候中生存。