Uniaxial Loading

4 Key excerpts on "Uniaxial Loading"

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  eBook - ePub

  Practical Engineering Failure Analysis

  • Hani M. Tawancy, Anwar Ul-Hamid, Nureddin M. Abbas(Authors)
  • 2004(Publication Date)
  • CRC Press

    (Publisher)

  ß in the preceding equation are given by

  Different states of stress defined in the preceding discussions can be mechanically developed by one or more types of three loading conditions: (i) axial, (ii) flextural, and (iii) torsional. Stresses can also be produced by changing the temperature of the part (thermal stresses). In the next section, the axial loading is distinguished from the flextural and torsional loading in terms of the stresses produced.

  5.5 Loading Conditions and Stress

  From a mechanical point of view, a part may be subjected to an axial load, flextural load, torsional load, or any combination of these during service. To safely design a part, it is essential to (i) define and evaluate the stresses developed by each type of loading and (ii) evaluate the stresses developed by a combined loading condition. In this section, the stresses developed by each type of loading condition are distinguished from each other. Evaluation of stresses developed by a combined loading condition is explained with solved problems at the end of this chapter in Sec. 5.14.

  By definition an axial load acts along a geometric axis of the part and can be either uniaxial, biaxial, or triaxial. In general, the stress produced by an axial load whether it is normal or shear is given by

  When one part is resting on another and transfers a force to it, a bearing or compressive stress is developed at the surface of contact. Similar to direct compressive stress, the bearing stress is a measure of the tendency of the applied force to crush the supporting part, and is given by

  For flat surfaces in contact, the bearing area is simply the area over which the force is transferred from one part to another. If the two pars have different areas, the smaller area is used in the calculation. Bearing stresses are developed in such parts as riveted joints.

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  Higher Engineering Science

  • William Bolton(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  1 Structural analysis

  1.1 Introduction

  Load-bearing structures can take many forms. For example, for the building column shown in Figure 1.1(a) the load of the floors and structure above it are applying forces which tend to axially squash the column. For the simple beam bridge in Figure 1.1(b) the load arising from a car crossing it will tend to bend the beam. For the aeroplane in Figure 1.1(c) , the lift forces on the wings will tend to bend the wings. For the electric motor in Figure 1.1(d) , the load will cause the shaft to become twisted as the motor rotates it, the loading being said to be torsional. To analyse structures so that we can predict their behaviour when loaded it is usual to consider certain basic forms of loading, namely axial tension or compression, bending and torsion.

  In this chapter, axial loading is considered, in chapter 2 bending and in chapter 3 torsional loading. Such analysis is necessary for the safe design of structures, whether they be buildings, bridges, aeroplanes, or motors rotating loads.

  1.2 Axial loading

  Consider a straight bar of constant cross-sectional area when external axial forces are applied at its ends. If the forces stretch the bar (Figure 1.2(a) ) then the bar is said to be in tension, if they compress it (Figure 1.2(b) ) in compression. In structures, a member that is in tension is called a tie while a member in compression is called a strut.

  Figure 1.1 Examples of loading

  Application

  Tensile members of structures can be steel cables or ropes which have no compression resistance. Examples of compressive members are concrete and brickwork columns which are weak in tension but strong in compression

  Figure 1.2 Bar (a) in tension, (b) in compression

  1.2.1 Direct stress

  It is necessary in analysing structures to be able to distinguish between external forces, such as applied loads, and internal forces which are produced in structural members as a result of applying the loads. Consider an axially loaded bar (Figure 1.3(a) ). We can think of the bar as being like a spring which, when stretched by external forces F, sets up internal forces which resist the external forces extending it. Consider a plane in the bar which is at right angles to its axis and suppose we make an imaginary cut along that plane (Figure 1.3(b) ). Equal forces F are required at the break to maintain equilibrium of the two lengths of the bar. This is true for any section across the bar and hence there is a force F acting on any imaginary section perpendicular to the axis of the bar. Thus we can consider, in this case, that there are internal forces F across any section at right angles to the axis. Internal forces are responsible for what is termed stress.

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  Introduction to Composite Materials

  • Hong T. Hahn, Stephen W. Tsai, StephenW. Tsai(Authors)
  • 2018(Publication Date)
  • Routledge

    (Publisher)

  We will limit the composites of this book to the linearly elastic materials. The response of materials under stress or strain follows a straight line up to failure. With assumed linearity we can use superposition which is a very powerful tool. For example, the net result of combining two states of stress is precisely the sum of the two states—no more and no less. The sequence of the stress application is immaterial. We can assemble or disect components of stress and strain in whatever pattern we choose without affecting the result. Combined stresses are the sum of simple, uniaxial stresses. The addition is done component by component.

  Figure 1.9 Unit strain vectors resulting from uniaxial stresses.

  (a)  Biaxial strain (1,−v,0) resulting from uniaxial stress (1,0,0).

  (b)  Biaxial strain (−v,1,0) resulting from uniaxial stress (0,1,0).

  Secondly, elasticity means full reversibility. We can load, unload and reload a material without incurring any permanent strain or hysteresis. Elasticity also means that the material’s response is instantaneous. There is no time lag, no time or rate dependency.

  Experimentally observed behavior of composites follows closer to linear elasticity than nearly all metals and nonreinforced plastics. The assumed linear elasticity for composites appears to be reasonable. If we are to go beyond the linearity assumption, such as the incorporation of nonlinear elasticity, plasticity and viscoelasticity, the increased complexity is beyond the scope of this book.

  For unidirectional composites, the stress-strain relations can be derived by the superposition method. We must recognize that two orthogonal planes of symmetry exist for unidirectional composites: one plane is parallel to the fibers; and the other is transverse to the fibers. Symmetry exists when the structure of the material on one side of the plane is the mirror image of the structure on the other side. The two orthogonal planes are shown in Figure 1.10 , where the x-axis is along the longitudinal direction of the fiber while the y-axis is in the transverse direction. When the reference axes x-y coincide with the material symmetry axes, we call this the on-axis orientation. The stress-strain relation in this chapter is limited to this special case. The off-axis orientation will be discussed in Chapter 3

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  Computational Methods for Reinforced Concrete Structures

  • Ulrich Häußler-Combe(Author)
  • 2014(Publication Date)
  • Ernst & Sohn

    (Publisher)

  This is a diffuse failure – in contrast to local failure under tensile loading – as failure spreads throughout a whole specimen and is not localized. Actually, a bunch of cracks occur with crack directions aligned to the load direction. But this requires an adequate experimental setup with minimized lateral constraints on the loaded faces of a plain concrete specimen.

  Figure 2.6 : Simplified model for force transfer in composition of aggregates and mortar.

  While uniaxial material modeling of concrete is sufficient for structural elements like bars and beams a multiaxial approach is necessary for plates, slabs, and shells. Multiaxial material modeling of concrete is discussed in Section 5.1.1.

  2.2 Long-Term Behavior – Creep and Imposed Strains

  Creep occurs as a delayed response of a material specimen after load application. A concrete specimen exposed to a loading within minutes will have increasing strains within months while its loading is hold constant. The complementary phenomenon to creep is relaxation. Deformations imposed to a concrete specimen within minutes will lead to immediate stresses but these stresses will decrease within months while the imposed deformation is hold constant.

  Mechanisms of creep and relaxation have to be treated in the microscale of materials and are connected to a relatively slow redistribution in the arrangement of microstructures and, in the case of mortar, water. All solids undergo creep and relaxation but its extent is different for different materials. Its extent is relatively large for, e.g., mortar and thus for concrete. A first approach to describe the development of uniaxial strain with time t for a constant uniaxial stress σ 0 is given by

  (2.8)

  with a creep function J (t ). The creep function is specific for every material. Creep strain is proportional to the applied stress with this approach. Such a linear characteristic with respect to stress or linear creep is valid for moderate stress levels relative to strength. A qualitative course of a uniaxial creep strain derived from experimental data is shown in Fig. 2.7 . Equation (2.8)

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