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Bending Stresses in Beams
Published in B. Raghu Kumar, Strength of Materials, 2022
In Previous chapters we considered the stresses in prismatic bars subjected to axial loads and Twisting moment. In this chapter we consider the stresses in beams subjected to third fundamental loading, bending. A beam is a structural member that is subjected to loads acting transversely to the longitudinal axis, as explained in the preceding chapter. Internal loads develop in beams in the form of shear forces and bending moments to resist the external loads. Shear stresses and Bending moments develop within the cross section due to internally develop shear forces and bending moments respectively. In this chapter we will restrict ourselves to study the bending stresses due to bending moment. The shear stresses due to shear forces will discuss in the next chapter. Before stating the discussion on bending stresses we need to know the concept of Pure bending. Pure bending means a beam or a portion of the beam under a constant bending moment, which means that the shear force is zero.
Deformation Strengthening and Formability
Published in Joseph Datsko, Materials Selection for Design and Manufacturing, 2020
Analysis of the strain history. Since the neutral axis remains at the mid-thickness during pure bending, all points below the mid-thickness are in compression and all points above the neutral axis are in tension. However, some of the metal that originally was below the mid-thickness (area JEFK) is displaced during bending to a position above the mid-thickness (area J′E′F′K′). These elements are first compressed and then stretched so that they have a final residual tensile strain. And there is one element, JK, below the mid-thickness that is first compressed and then displaced upward until it lies at the mid-thickncss at which time its strain is zero.
Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
Pure bending refers to bending of a beam under a constant bending moment M, which means that the shear force V is zero (because V=dM/dx). Nonuniform bending refers to bending in the presence of shear forces, in which case the bending moment varies along the axis of the beam. The sign convention for bending moments is shown in Figure 5.2; note that positive bending moment produces tension in the lower part of the beam and compression in the upper part.
An elasticity solution of functionally graded beams with different moduli in tension and compression
Published in Mechanics of Advanced Materials and Structures, 2018
Xiao-Ting He, Wei-Min Li, Jun-Yi Sun, Zhi-Xiang Wang
Now, let us discuss further such a mechanical model that a bimodular FGMs shallow beam with rectangular section dimension h×b and length l (h < <l) is subjected to the action of uniformly distributed loads q on its upper layer, as shown in Figure 2. The left end of the beam is free and the right end is fixed. Similarly, this causes a downward bending of the beam in plane coordinate system xoz and the coordinate y represents the direction of the beam width, as shown in Figure 2. Contrasting to the pure bending mentioned above, there exists the bending moment and the shear force on any cross-section of the beam, thus developing the bending stress and shearing stress. Besides, there exists a normal stress along the direction of z because of the action of uniformly distributed loads, which is referred to as extrusion stress later. Due to the combined action of the bending stress, shearing stress, and extrusion stress, any point in the beam is in the state of complex stress. If Ambartsumyan's constitutive law defined in principal stress direction is strictly followed, it is very difficult to judge the existence of the neutral layer free of tension and compression.
Dual-horizon peridynamic study of the mode-I J-integral and crack opening displacement in single-edge notched beams
Published in Mechanics of Advanced Materials and Structures, 2023
Figure 2 illustrates the diagrams of the shear force and bending moment in a beam under three- and four-point bend. Member between two concentrated forces in the beam under four-point bend is in pure bending because shear force equals zero in cross-sections of this member. The bending moment M in the central cross-section of the beam under three- and four-point bend yields: