Bending Stress
Bending stress refers to the internal stress that occurs in a structural component when subjected to bending moments. It is caused by the distribution of forces and moments acting on the material, resulting in tension on one side and compression on the other. Understanding bending stress is crucial in designing and analyzing the strength and stability of beams, columns, and other structural elements.
8 Key excerpts on "Bending Stress"
eBook - ePub- Ken Wyatt, Richard Hough(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
...However, for each small tensional force we would expect to find a corresponding compressive force, and those two forces together would constitute a couple, producing a moment. The total resistance to bending of these forces at the section m–m would therefore be the sum of all these internal force couples. To satisfy the equilibrium of the left-hand portion of the beam, this internal resistance moment must equal the Bending Moment M caused by the external forces. 9.2 THE STRESS FORMULA When one considers that people have been using beams in their dwellings and bridges for a period longer than all recorded history, it is interesting that the theory for the distribution of stresses was not developed until 1826. Navier, a French engineer, based his theory upon three principal assumptions, as follows: 1. Strain is proportional to stress for the material of the beam. 2. Cross-sections that are plane and normal to the beam axis before bending remain plane and normal after bending. 3. The beam is assumed to bend in a plane containing the vertical centroidal axis and the beam axis. By considering the beam to be composed of a bundle of ‘fibres’ parallel to the axis of the beam (as in a solid timber beam), he was able to derive a relationship between applied bending moment, the properties of the beam, and the internal stress caused by the Bending Moment: f y = M I = E R where f = Bending Stress (tension or compression in a particular fibre) y = distance of fibre from centroid of cross-sectional area M = applied B.M. at the cross-section being considered I = Second Moment of Area of the cross-section E = Young’s Modulus of the material R = Radius of Curvature of the beam The derivation of this relationship is shown opposite. 9.3 DEDUCTIONS There are several very important conclusions that may be drawn from this derivation: (i) In every beam there will be a neutral surface (i.e...
eBook - ePubDesign-Tech
Building Science for Architects
- Thomas Leslie, Robert Whitehead(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...In this chapter we will deal with the simplest two conditions—pushing and pulling. We will see how these types of stresses affect the behavior of building materials, we will size components to respond to considerations of strength and stiffness, and finally, we will look at form-resistant systems to see how a particular classification of structural forms can efficiently respond to states of tension and compression with basic formal and material design strategies. 21.1 The five states of stress. Basic States of Stress and Element Design Stress (f) is a refection of how materials will behave when subjected to a force. There are different types of stresses that occur in materials, depending on the type of loading to which they are subjected. Objects that are pushed and pulled by loads parallel to their axes are in compression and tension. Shear stress causes adjacent planes to try to slide past each other, distorting an object's shape instead of length. If an object is subjected to torsion, it is twisted or rotated across its section. Finally, when a material is subjected to loads perpendicular to the axis of the member, it is in bending. This chapter will focus on the more easily understandable behaviors found in objects subjected to the axial stresses of tension and compression. Because these forces are either pushing or pulling, and not bending or twisting, the object under stress employs the entire area of its cross section to resist the loading (quite unlike objects under torsion or bending in which cross sectional shape matters) (Figure 21.2)...
eBook - ePubStructure for Architects
A Primer
- Ramsey Dabby, Ashwani Bedi(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
...When bowed, the body compresses along one edge and stretches along the other. The bending of a simply supported beam under load produces tension along the bottom of the beam and compression along the top (Figure 4.6). Moving away from the tension and compression edges toward the centerline of the beam, the tensile and compressive stresses gradually diminish until they reach zero. The imaginary plane passing through the centerline of a beam along its length, at which no tension or compression occurs, is called the neutral plane or neutral axis. Although bending is of paramount importance in beams, it may occur in any structural member, including columns. Bending is also referred to as flexure. Figure 4.6 Tension and Compression in a Beam In addition to producing tensile and compressive stresses from bending, a beam under load will produce shear stress, both perpendicular (i.e., vertical) and parallel (i.e., horizontal) to the length of the beam (Figures 4.7 and 4.8). Figure 4.7 Vertical Shear Perpendicular to the Length of a Beam Figure 4.8 Horizontal Shear Parallel to the Length of a Beam With a load placed at the center of a span, the maximum Bending Stresses in a beam occur at the center of the span (Figure 4.9), with the maximum compressive stress at the top edge and the maximum tensile stress at the bottom edge (4.10). Figure 4.9 Maximum Bending Stresses at the Center of a Span Figure 4.10 Maximum Compressive and Tensile Stresses at the Edges of a Beam Since tensile, compressive, and shear stresses vary for any point along the length of a beam, their computation in beams is more complex than in members under direct stress. Beam analysis, as well as the analysis of members under direct stress, will be examined more closely in subsequent chapters....
eBook - ePub- Paul McMullin, Jonathan Price, Paul W. McMullin, Jonathan S. Price(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
...length). Steel and aluminum are materials that can be treated as isotropic, while timber and reinforced concrete must be considered anisotropic. 7.2 Stress and Strain First, we must understand the terms stress and strain. Stress f is a measure of the internal force per unit area acting within a structural element. Strain ε is the change in length Δ l divided by the original length l. It is unitless, as we divide length by length. Think of it like percentage change in length—assuming you multiply it by 100. Stress and strain are represented in Figure 7.3, showing a solid bar with three loads. When cut at a point along its length, we see the internal stress and strain pattern. Both stress and strain are internal to the member, but are caused by external forces or the member’s self-weight. Figure 7.3 Stress and strain in a solid bar Stress is related to strain through the modulus of elasticity —a material stiffness parameter—by the following equation: where: f = stress (lb/in 2, k/in 2, psi, kN/m 2, MN/m 2, N/mm 2, MPa) E = modulus of elasticity (lb/in 2, k/in 2, psi, kN/m 2, MN/m 2, N/mm 2, MPa) ε = strain (unitless) In this book we use units in the form of k/in 2 (MN/m 2) instead of ksi (MPa). We do this to help keep clarity, such as when dividing a force by an area. Derived from simple tests, the materials scientist plots stress and strain to gain an understanding of material behavior. The longer curve in Figure 7.4 is a material that can deform significantly before losing strength—like steel and aluminum. The shorter curve represents a material that deforms little after reaching its maximum strength—like wood, masonry and concrete. We use larger safety factors when designing materials that do not have significant deformation ability, as there is little warning of a problem. The area under the stress–strain curve is the energy the material can absorb (see Figure 7.5)...
eBook - ePubStructure for Architects
A Case Study in Steel, Wood, and Reinforced Concrete Design
- Ashwani Bedi, Ramsey Dabby(Authors)
- 2019(Publication Date)
- Routledge(Publisher)
...3 Stress, Strain, and Material Behavior 3.1 Stress, Strain, and Material Properties Many of a material’s structural properties are defined by their behavior under stress. A good understanding of these properties is needed to properly design for them. Stress If we apply an axial tensile or compressive force (P) on a material having a length (L) and a cross-sectional area (A) (Figure 3.1), the uniform stress (F) in the material in tension or compression is given by: F = P/A Figure 3.1 Stress in a Material Strain Under the applied load, the material will deform—i.e., it will elongate under tension, or shorten under compression (ΔL) (Figure 3.2). The amount of deformation divided by the original length is known as strain and is given by: STRAIN = ΔL/L Figure 3.2 Strain in a Material Stress-Strain Curves A stress-strain curve is a plot of stress (along the vertical axis) vs. strain (along the horizontal axis) that helps to visualize important behavioral characteristics of a material under load (Figure 3.3). Every material has a unique stress-strain curve, in tension as well as in compression. In a sense, a stress-strain curve can be considered a material’s ‘signature’ behavior under load. Figure 3.3 Axes of a Stress-Strain Curve Material Strength Material strength is the ability of a material to withstand applied loads. Loads on a member cause various internal stresses that tend to produce deformations in its material. Depending on the type of load (transverse, axial, or torsional), the member may experience compression, tension, shear, or a combination of these stresses. Steel’s material strength is represented by various values, most importantly its yield stress (F y). Wood’s material strength is dependent upon several different criteria such as the direction of force, the direction of grain, and its species and grade...
eBook - ePubPractical 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 = τ A 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...
eBook - ePubBuilding Structures
understanding the basics
- Malcolm Millais(Author)
- 2017(Publication Date)
- Routledge(Publisher)
...Because the lines of principal tensile stress curve there are curved lines of principal compressive stress near the hole. Using the idea of principal tensile and compressive stresses diagrams can be drawn showing how a column ‘changes’ into a beam. Both a column and beam are considered to be one-dimensional elements, but the intermediate stages are two-dimensional elements. Fig. 4.23A shows the stresses for a column, a one-dimensional element, and Fig. 4.23F shows the stresses for a beam, also a one-dimensional element. Figs. 4.23B to E show the stresses for various intermediate stages, and all these are two-dimensional elements. Whilst bending moment, axial and shear force diagrams make sense for the column and beam they cannot usefully be drawn for the intermediate structures. Fig 4.23 Without the use of a computer program it is very laborious to calculate numerical values for the magnitude and direction of principal stresses. However, it is possible to make relatively simple calculations by introducing fictitious compressive and tensile members – struts and ties. Fig. 4.23D shows a structure that resembles a deep beam – see Fig. 3.58. By introducing a tie, shown dashed, and struts, 1 shown as full lines, it is possible to approximate the two-dimensional element with a simpler structure. Fig. 4.24 Principal stresses and strut and tie model What the actual cross-section is for each strut and tie obviously cannot be determined precisely so engineering judgement has to be used. However, even without needing to make numerical calculations, drawing possible strut and tie models can lead to an increased conceptual understanding of these two-dimensional structures. Principal stresses can also be drawn for curved elements like a shell. If a curved shell is spanning between end supports and loaded laterally the pattern of principal stresses will be similar to those for the beam shown in Fig...
eBook - ePubSteel Structures
Design using FEM
- Rolf Kindmann, Matthias Kraus(Authors)
- 2012(Publication Date)
- Ernst & Sohn(Publisher)
...If both internal forces and stresses are sketched in, this is not really correct. However, sometimes it can be useful to aid clearity. The sign or the direction of action of the stresses then results from the “replacement” of the internal forces by the stresses. For the common case of uniaxial bending with axial force Figure 7.2 shows the correct approach. In a beam section, the internal forces N, My and Vz are replaced by positive stresses σ x and τ xz at the positive intersection. When using the equilibrium conditions (Figure 7.2 at the bottom), dx → 0 is assumed for the length of the beam section, since the internal forces or the stresses, respectively, are considered at one specific position x. Figure 7.2 Stresses σ and τ xz due to N, M y and V z Material behaviour In this chapter, the determination of the stresses is performed with the theory of elasticity, i.e the universal validity of Hooke’s law is assumed. According to Eqs. (1.5) – (1.7), the relationship of strains or shearing strains, respectively, and stresses is then: (7.1a) (7.1b) For details on Hooke’s law refer to [25]. 7.2 Axial Stresses due to Biaxial Bending and Axial Force If it is assumed that the cross section does not rotate about its longitudinal axis (= 0), according to Section 1.6, Eq. (1.1), we have (7.2) for the displacements in the longitudinal direction of the beam. Since y and z only occur linearly for the rotations φ y and φ z, a plane area is described by Eq. (7.2). It complies with the Bernoulli hypothesis about the cross sections remaining plane. With σ x = E · ε x and ε x =u' follows that (7.3) The stress is described by the elongation in the centre of gravity u' S, the derivatives of the rotations φ' z and φ' y, the ordinates y and z in the principal coordinate system and the modulus of elasticity. The stress distribution also corresponds to a plane area, as can be seen from Eq...
