bending

In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically 1/10 or less, of the other two. When the length is considerably longer than the width and the thickness, the element is called a beam. For example, a closet rod sagging under the weight of clothes on clothes hangers is an example of a beam experiencing bending. On the other hand, a shell is a structure of any geometric form where the length and the width are of the same order of magnitude but the thickness of the structure (known as the 'wall') is considerably smaller. A large diameter, but thin-walled, short tube supported at its ends and loaded laterally is an example of a shell experiencing bending. In the absence of a qualifier, the term bending is ambiguous because bending can occur locally in all objects. Therefore, to make the usage of the term more precise, engineers refer to a specific object such as; the bending of rods, the bending of beams, the bending of plates, the bending of shells and so on.

In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically 1/10 or less, of the other two. When the length is considerably longer than the width and the thickness, the element is called a beam. For example, a closet rod sagging under the weight of clothes on clothes hangers is an example of a beam experiencing bending. On the other hand, a shell is a structure of any geometric form where the length and the width are of the same order of magnitude but the thickness of the structure (known as the 'wall') is considerably smaller. A large diameter, but thin-walled, short tube supported at its ends and loaded laterally is an example of a shell experiencing bending. In the absence of a qualifier, the term bending is ambiguous because bending can occur locally in all objects. Therefore, to make the usage of the term more precise, engineers refer to a specific object such as; the bending of rods, the bending of beams, the bending of plates, the bending of shells and so on. \cfrac{\mathrm{d}^2 w(x)}{\mathrm{d} x^2}=\frac{M(x)}{E(x) I(x)} where the second derivative of its deflected shape with respect to x is interpreted as its curvature, E is the Young's modulus, I is the area moment of inertia of the cross-section, and M is the internal bending moment in the beam. If, in addition, the beam is homogeneous along its length as well, and not tapered (i.e. constant cross section), and deflects under an applied transverse load q(x) , it can be shown that: EI~\cfrac{\mathrm{d}^4 w(x)}{\mathrm{d} x^4} = q(x) This is the Euler–Bernoulli equation for beam bending. After a solution for the displacement of the beam has been obtained, the bending moment ( M ) and shear force ( Q ) in the beam can be calculated using the relations M(x) = -EI~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} ~;~~ Q(x) = \cfrac{\mathrm{d}M}{\mathrm{d}x}. Simple beam bending is often analyzed with the Euler–Bernoulli beam equation. The conditions for using simple bending theory are: The beam is subject to pure bending. This means that the shear force is zero, and that no torsional or axial loads are present. The material is isotropic (or orthotropic) and homogeneous. The material obeys Hooke's law (it is linearly elastic and will not deform plastically). The beam is initially straight with a cross section that is constant throughout the beam length. The beam has an axis of symmetry in the plane of bending. The proportions of the beam are such that it would fail by bending rather than by crushing, wrinkling or sideways buckling. Cross-sections of the beam remain plane during bending. Compressive and tensile forces develop in the direction of the beam axis under bending loads. These forces induce stresses on the beam. The maximum compressive stress is found at the uppermost edge of the beam while the maximum tensile stress is located at the lower edge of the beam. Since the stresses between these two opposing maxima vary linearly, there therefore exists a point on the linear path between them where there is no bending stress. The locus of these points is the neutral axis. Because of this area with no stress and the adjacent areas with low stress, using uniform cross section beams in bending is not a particularly efficient means of supporting a load as it does not use the full capacity of the beam until it is on the brink of collapse. Wide-flange beams ( -beams) and truss girders effectively address this inefficiency as they minimize the amount of material in this under-stressed region. The classic formula for determining the bending stress in a beam under simple bending is: \sigma_x = \frac{M_z y}{I_z} = \frac{M_z}{W_z} where {\sigma_x} is the bending stress M_z – the moment about the neutral axis y – the perpendicular distance to the neutral axis I_z – the second moment of area about the neutral axis z. W_z - the Resistance Moment about the neutral axis z. W_z = I_z / y ===Extensions of Euler-Bernoulli beam bending theory=== The equation \sigma = \tfrac{M y}{I_x} is valid only when the stress at the extreme fiber (i.e., the portion of the beam farthest from the neutral axis) is below the yield stress of the material from which it is constructed. At higher loadings the stress distribution becomes non-linear, and ductile materials will eventually enter a plastic hinge state where the magnitude of the stress is equal to the yield stress everywhere in the beam, with a discontinuity at the neutral axis where the stress changes from tensile to compressive. This plastic hinge state is typically used as a limit state in the design of steel structures. ====Complex or asymmetrical bending==== The equation above is only valid if the cross-section is symmetrical. For homogeneous beams with asymmetrical sections, the maximum bending stress in the beam is given by \sigma_x(y,z) = - \frac {M_z~I_y + M_y~I_{yz}} {I_y~I_z - I_{yz}^2}y + \frac {M_y~I_z + M_z~I_{yz}} {I_y~I_z - I_{yz}^2}z where y,z are the coordinates of a point on the cross section at which the stress is to be determined as shown to the right, M_y and M_z are the bending moments about the y and z centroid axes, I_y and I_z are the second moments of area (distinct from moments of inertia) about the y and z axes, and I_{yz} is the product of moments of area. Using this equation it is possible to calculate the bending stress at any point on the beam cross section regardless of moment orientation or cross-sectional shape. Note that M_y, M_z, I_y, I_z, I_{yz} do not change from one point to another on the cross section. ====Large bending deformation==== For large deformations of the body, the stress in the cross-section is calculated using an extended version of this formula. First the following assumptions must be made: Assumption of flat sections – before and after deformation the considered section of body remains flat (i.e., is not swirled). Shear and normal stresses in this section that are perpendicular to the normal vector of cross section have no influence on normal stresses that are parallel to this section. Large bending considerations should be implemented when the bending radius \rho is smaller than ten section heights h: \rho With those assumptions the stress in large bending is calculated as: \sigma = \frac {F} {A} + \frac {M} {\rho A} + {\frac {M} {{I_x}'}}y{\frac {\rho}{\rho +y}} where F is the normal force A is the section area M is the bending moment \rho is the local bending radius (the radius of bending at the current section) {{I_x}'} is the area moment of inertia along the x-axis, at the y place (see Steiner's theorem) y is the position along y-axis on the section area in which the stress \sigma is calculated. When bending radius \rho approaches infinity and y\ll\rho , the original formula is back: \sigma = {F \over A} \pm \frac {My}{I} . EI~\cfrac{\mathrm{d}^4 w}{\mathrm{d} x^4} = q(x) - \cfrac{EI}{k A G}~\cfrac{\mathrm{d}^2 q}{\mathrm{d} x^2} where I is the area moment of inertia of the cross-section, A is the cross-sectional area, G is the shear modulus, k is a shear correction factor, and q(x) is an applied transverse load. For materials with Poisson's ratios ( \nu ) close to 0.3, the shear correction factor for a rectangular cross-section is approximately k = \cfrac{5 + 5\nu}{6 + 5\nu} The rotation ( \varphi(x) ) of the normal is described by the equation \cfrac{\mathrm{d}\varphi}{\mathrm{d}x} = -\cfrac{\mathrm{d}^2w}{\mathrm{d}x^2} -\cfrac{q(x)}{kAG} The bending moment ( M ) and the shear force ( Q ) are given by M(x) = -EI~ \cfrac{\mathrm{d}\varphi}{\mathrm{d}x} ~;~~ Q(x) = kAG\left(\cfrac{\mathrm{d}w}{\mathrm{d}x}-\varphi\right) = -EI~\cfrac{\mathrm{d}^2\varphi}{\mathrm{d}x^2} = \cfrac{\mathrm{d}M}{\mathrm{d}x} EI~\cfrac{\partial^4 w}{\partial x^4} + m~\cfrac{\partial^2 w}{\partial t^2} = q(x,t) where E is the Young's modulus, I is the area moment of inertia of the cross-section, w(x,t) is the deflection of the neutral axis of the beam, and m is mass per unit length of the beam. EI~\cfrac{\partial^4 w}{\partial x^4} + m~\cfrac{\partial^2 w}{\partial t^2} = 0 Free, harmonic vibrations of the beam can then be expressed as w(x,t) = \text{Re}[\hat{w}(x)~e^{-i\omega t}] \quad \implies \quad \cfrac{\partial^2 w}{\partial t^2} = -\omega^2~w(x,t) and the bending equation can be written as EI~\cfrac{\mathrm{d}^4 \hat{w}}{\mathrm{d}x^4} - m\omega^2\hat{w} = 0 The general solution of the above equation is \hat{w} = A_1\cosh(\beta x) + A_2\sinh(\beta x) + A_3\cos(\beta x) + A_4\sin(\beta x) where A_1,A_2,A_3,A_4 are constants and \beta := \left(\cfrac{m}{EI}~\omega^2\right)^{1/4} = {} & q(x,t) + \frac{J}{k A G}~\frac{\partial^2 q}{\partial t^2} - \frac{EI}{k A G}~\frac{\partial^2 q}{\partial x^2} \end{align} where J = \tfrac{mI}{A} is the polar moment of inertia of the cross-section, m = \rho A is the mass per unit length of the beam, \rho is the density of the beam, A is the cross-sectional area, G is the shear modulus, and k is a shear correction factor. For materials with Poisson's ratios ( \nu ) close to 0.3, the shear correction factor are approximately \begin{align} k &= \frac{5 + 5\nu}{6 + 5\nu} \quad \text{rectangular cross-section}\\[6pt] &= \frac{6 + 12\nu + 6\nu^2}{7 + 12\nu + 4\nu^2} \quad \text{circular cross-section} \end{align} EI~\cfrac{\mathrm{d}^4 \hat{w}}{\mathrm{d} x^4} + m\omega^2\left(\cfrac{J}{m} + \cfrac{E I}{k A G}\right)\cfrac{\mathrm{d}^2 \hat{w}}{\mathrm{d} x^2} + m\omega^2\left(\cfrac{\omega^2 J}{k A G}-1\right)~\hat{w} = 0 This equation can be solved by noting that all the derivatives of w must have the same form to cancel out and hence as solution of the form e^{kx} may be expected. This observation leads to the characteristic equation \alpha~k^4 + \beta~k^2 + \gamma = 0 ~;~~ \alpha := EI ~,~~ \beta := m\omega^2\left(\cfrac{J}{m} + \cfrac{E I}{k A G}\right) ~,~~ \gamma := m\omega^2\left(\cfrac{\omega^2 J}{k A G}-1\right) The solutions of this quartic equation are k_1 = +\sqrt{z_+} ~,~~ k_2 = -\sqrt{z_+} ~,~~ k_3 = +\sqrt{z_-} ~,~~ k_4 = -\sqrt{z_-} where z_+ := \cfrac{-\beta + \sqrt{\beta^2 - 4\alpha\gamma}}{2\alpha} ~,~~ z_-:= \cfrac{-\beta - \sqrt{\beta^2 - 4\alpha\gamma}}{2\alpha} The general solution of the Timoshenko-Rayleigh beam equation for free vibrations can then be written as \hat{w} = A_1~e^{k_1 x} + A_2~e^{-k_1 x} + A_3~e^{k_3 x} + A_4~e^{-k_3 x} u_\alpha(\mathbf{x}) & = - x_3~\frac{\partial w^0}{\partial x_\alpha} = - x_3~w^0_{,\alpha} ~;~~\alpha=1,2 \\ u_3(\mathbf{x}) & = w^0(x_1, x_2) \end{align} where \mathbf{u} is the displacement of a point in the plate and w^0 is the displacement of the mid-surface. The strain-displacement relations are \begin{align} \varepsilon_{\alpha\beta} & = - x_3~w^0_{,\alpha\beta} \\ \varepsilon_{\alpha 3} & = 0 \\ \varepsilon_{33} & = 0 \end{align} The equilibrium equations are M_{\alpha\beta,\alpha\beta} + q(x) = 0 ~;~~ M_{\alpha\beta} := \int_{-h}^h x_3~\sigma_{\alpha\beta}~dx_3 where q(x) is an applied load normal to the surface of the plate. In terms of displacements, the equilibrium equations for an isotropic, linear elastic plate in the absence of external load can be written as w^0_{,1111} + 2~w^0_{,1212} + w^0_{,2222} = 0 In direct tensor notation, \nabla^2\nabla^2 w = 0 u_\alpha(\mathbf{x}) & = - x_3~\varphi_\alpha ~;~~\alpha=1,2 \\ u_3(\mathbf{x}) & = w^0(x_1, x_2) \end{align} where \varphi_\alpha are the rotations of the normal. The strain-displacement relations that result from these assumptions are \begin{align} \varepsilon_{\alpha\beta} & = - x_3~\varphi_{\alpha,\beta} \\ \varepsilon_{\alpha 3} & = \cfrac{1}{2}~\kappa\left(w^0_{,\alpha}- \varphi_\alpha\right) \\ \varepsilon_{33} & = 0 \end{align} where \kappa is a shear correction factor. The equilibrium equations are \begin{align} & M_{\alpha\beta,\beta}-Q_\alpha = 0 \\ & Q_{\alpha,\alpha}+q = 0 \end{align} where Q_\alpha := \kappa~\int_{-h}^h \sigma_{\alpha 3}~dx_3 M_{\alpha\beta,\alpha\beta} - q(x,t) = J_1~\ddot{w}^0 - J_3~\ddot{w}^0_{,\alpha\alpha} where, for a plate with density \rho = \rho(x) , J_1 := \int_{-h}^h \rho~dx_3 ~;~~ J_3 := \int_{-h}^h x_3^2~\rho~dx_3 and \ddot{w}^0 = \frac{\partial^2 w^0}{\partial t^2} ~;~~ \ddot{w}^0_{,\alpha\beta} = \frac{\partial^2 \ddot{w}^0}{\partial x_\alpha\, \partial x_\beta} The figures below show some vibrational modes of a circular plate. ==See also== Bending moment Bending Machine (flat metal bending) Brake (sheet metal bending) Brazier effect Bending of plates Bending (metalworking) Continuum mechanics Contraflexure Deflection (engineering) Flexure bearing List of area moments of inertia Pipe bending Shear and moment diagram Shear strength Sandwich theory Vibration Vibration of plates ==External links== Flexure formulae Beam stress & deflection, beam deflection tables What is Bending Stress?

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