Stress Distribution

3 Key excerpts on "Stress Distribution"

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  eBook - ePub

  Practical Guide to the Packaging of Electronics

  Thermal and Mechanical Design and Analysis, Third Edition

  • Ali Jamnia(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  11 Mechanical and Thermomechanical Concerns

  Introduction

  An important aspect of electronics packaging is developing an understanding of the stresses that its components undergo and their relationship to the system’s failure and/or reliability. The cause of these stresses may be temperature and its variations, vibration, or physical properties such as weight. It may occur at the board and component level, enclosure levels, and up to the system itself.

  Stresses are internal distributed forces, which are caused by external applied loads. Strains are changes in the form under the same loads. Consider a rod of length L and diameter A . One may intuitively recognize that the displacement of the end of this rod depends directly on the magnitude of the applied force—very similar to the force–deflection relationship of a spring–mass system as shown in Figure 11.1 .

  Now consider what happens inside of this rod in Figure 11.2 . The concentrated load is (internally) developed over the area of the cross section. Thus, one may express this distributed force as follows:

  σ = F A

  Similarly, a distributed (average) displacement may also be calculated.

  ε = Δ L

  It turns out that σ and ε have a relationship similar to a force–deflection curve in a spring–mass system. The slope of this line (E ) is called tensile modulus, Young’s modulus, or modulus of elasticity.

  Figure 11.1 Force–deflection relationship.

  Figure 11.2 Internal forces.

  Consider another scenario. A block under a shear force will also deflect. In shear, the force–deflection relationships are defined as follows:

  F = τA

  where A is the area and τ is the shear stress. Furthermore, there is a relationship between the shear stress and shear strain (γ) similar to that of the stress–strain relationship.

  τ = G γ

  where G is shear modulus and γ is shear strain.

  In general, both normal and shear stresses develop in solids under a general loading. For example, a cantilever beam under a simple load at the free end exhibits both normal and shear stresses, as shown in Figure 11.3

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  eBook - ePub

  Building Structures

  understanding the basics

  • Malcolm Millais(Author)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  Chapter 6 .

  Part of the skill of designing structures is the prediction of the Stress Distribution in each element as it acts as part of a load path (or paths). The accuracy of prediction will vary depending on the stage of the design process. For instance the exact size of elements may not need to be calculated at preliminary stages. However, it should be clear to the structural designers that the proposed types of elements, shells, slabs, I beams etc., will act effectively as their part of the load path. This is clarified if the Stress Distribution is known in principle. For instance if load bearing walls are used at different levels and they cross at angles, then the whole wall will not be effective (see Fig. 3.31 ).

  Fig 4.91

  Or again if an element is acting as a beam, then an I section is better than a + section (see

  page 73

  ), and it might be worthwhile to vary the depth (see Fig. 3.57 ).

  The central point is that structural design is not the result of a logical process but the result of an imaginative concept. For this concept to be successful it must be informed by a conceptual understanding of how the imagined structure will behave.

  References— Chapter 4

  1 J. Schlaich, K.Schäfer, M. Jennewein—M Towards a consistent design of structural concrete

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  eBook - ePub

  Geotechnics of Roads: Fundamentals

  • Bernardo Caicedo(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  176 ]. Both results indicate that a minimum integration length of at least ten times the maximum particle size is required to obtain a coefficient of variation of stress of less than 10%.

  1.1.2 Representation of stresses in a continuum media

  As described above, from the micromechanical point of view stresses appear as the result of contact forces between particles. Despite recent advances in modeling soils as discrete sets of particles, limitations remain in terms of the number of particles, their shapes and complex interactions within fine-grained materials. Because of these limitations, the abstract concept of a continuous medium is still extensively used for analysis and design of road structures.

  Continuum mechanics allows mathematical treatment of a broad range of stress-strain problems. Defining stress at a point is the first step in treating it mathematically. Stress is the relationship between a force and the area of the surface upon which the load acts as the area of that surface approaches a point.

  Figure 1.4 illustrates the definition of stress at a point. In geotechnical engineering, compressive stresses are usually positive because this is the typical situation in most geotechnical problems. Shear stresses are denoted with a double index: the first index indicates the axis that coincides with the normal direction of the face upon which the shear stress acts while the second index indicates the direction of the axis of the shear stress. Shear stresses are considered positive if they are acting on the positive face identified by the first index and simultaneously pointing in the positive direction identified by the second index. For example, τ xy is positive if it acts on the +x face and it points in the +y

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