Bending

Element of a bent beam: the fibers form concentric arcs, the top fibers are compressed and bottom fibers stretched.

Bending moments in a beam

In the Euler–Bernoulli theory of slender beams, a major assumption is that 'plane sections remain plane'. In other words, any deformation due to shear across the section is not accounted for (no shear deformation). Also, this linear distribution is only applicable if the maximum stress is less than the yield stress of the material. For stresses that exceed yield, refer to article plastic bending. At yield, the maximum stress experienced in the section (at the furthest points from the neutral axis of the beam) is defined as the flexural strength.

Consider beams where the following are true:

• The beam is originally straight and slender, and any taper is slight
• The material is isotropic (or orthotropic), linear elastic, and homogeneous across any cross section (but not necessarily along its length)
• Only small deflections are considered

In this case, the equation describing beam deflection (${\displaystyle w}$) can be approximated as:

${\displaystyle {\cfrac {\mathrm {d} ^{2}w(x)}{\mathrm {d} x^{2}}}={\frac {M(x)}{E(x)I(x)}}}$

where the second derivative of its deflected shape with respect to ${\displaystyle x}$ is interpreted as its curvature, ${\displaystyle E}$ is the Young's modulus, ${\displaystyle I}$ is the area moment of inertia of the cross-section, and ${\displaystyle M}$ 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 ${\displaystyle q(x)}$, it can be shown that:[1]

${\displaystyle EI~{\cfrac {\mathrm {d} ^{4}w(x)}{\mathrm {d} x^{4}}}=q(x)}$

This is the Euler–Bernoulli equation for beam bending.

After a solution for the displacement of the beam has been obtained, the bending moment (${\displaystyle M}$) and shear force (${\displaystyle Q}$) in the beam can be calculated using the relations

${\displaystyle M(x)=-EI~{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}~;~~Q(x)={\cfrac {\mathrm {d} M}{\mathrm {d} x}}.}$

Simple beam bending is often analyzed with the Euler–Bernoulli beam equation. The conditions for using simple bending theory are:[4]

1. The beam is subject to pure bending. This means that the shear force is zero, and that no torsional or axial loads are present.
2. The material is isotropic (or orthotropic) and homogeneous.
3. The material obeys Hooke's law (it is linearly elastic and will not deform plastically).
4. The beam is initially straight with a cross section that is constant throughout the beam length.
5. The beam has an axis of symmetry in the plane of bending.
6. The proportions of the beam are such that it would fail by bending rather than by crushing, wrinkling or sideways buckling.
7. Cross-sections of the beam remain plane during bending.

Deflection of a beam deflected symmetrically and principle of superposition

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 (I-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:[5]

${\displaystyle \sigma _{x}={\frac {M_{z}y}{I_{z}}}={\frac {M_{z}}{W_{z}}}}$

where

• ${\displaystyle {\sigma _{x}}}$ is the bending stress
• ${\displaystyle M_{z}}$ – the moment about the neutral axis
• ${\displaystyle y}$ – the perpendicular distance to the neutral axis
• ${\displaystyle I_{z}}$ – the second moment of area about the neutral axis z.
• ${\displaystyle W_{z}}$ - the Resistance Moment about the neutral axis z. ${\displaystyle W_{z}=I_{z}/y}$

The equation ${\displaystyle \sigma ={\tfrac {My}{I_{x}}}}$ 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.

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

${\displaystyle \sigma _{x}(y,z)=-{\frac {M_{z}~I_{y}+M_{y}~I_{yz}}{I_{y}~I_{z}-I_{yz}^{2}}}y+{\frac {M_{y}~I_{z}+M_{z}~I_{yz}}{I_{y}~I_{z}-I_{yz}^{2}}}z}$[6]

where ${\displaystyle y,z}$ are the coordinates of a point on the cross section at which the stress is to be determined as shown to the right, ${\displaystyle M_{y}}$ and ${\displaystyle M_{z}}$ are the bending moments about the y and z centroid axes, ${\displaystyle I_{y}}$ and ${\displaystyle I_{z}}$ are the second moments of area (distinct from moments of inertia) about the y and z axes, and ${\displaystyle I_{yz}}$ 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 ${\displaystyle M_{y},M_{z},I_{y},I_{z},I_{yz}}$ do not change from one point to another on the cross section.

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:

1. Assumption of flat sections – before and after deformation the considered section of body remains flat (i.e., is not swirled).
2. 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 ${\displaystyle \rho }$ is smaller than ten section heights h:

${\displaystyle \rho <10h.}$

With those assumptions the stress in large bending is calculated as:

${\displaystyle \sigma ={\frac {F}{A}}+{\frac {M}{\rho A}}+{\frac {M}{{I_{x}}'}}y{\frac {\rho }{\rho +y}}}$

where

${\displaystyle F}$ is the normal force

${\displaystyle A}$ is the section area

${\displaystyle M}$ is the bending moment

${\displaystyle \rho }$ is the local bending radius (the radius of bending at the current section)

${\displaystyle {{I_{x}}'}}$ is the area moment of inertia along the x-axis, at the ${\displaystyle y}$ place (see Steiner's theorem)

${\displaystyle y}$ is the position along y-axis on the section area in which the stress ${\displaystyle \sigma }$ is calculated.

When bending radius ${\displaystyle \rho }$ approaches infinity and ${\displaystyle y\ll \rho }$, the original formula is back:

${\displaystyle \sigma ={F \over A}\pm {\frac {My}{I}}}$.

Deformation of a Timoshenko beam. The normal rotates by an amount ${\displaystyle \theta }$ which is not equal to ${\displaystyle dw/dx}$.

In 1921, Timoshenko improved upon the Euler–Bernoulli theory of beams by adding the effect of shear into the beam equation. The kinematic assumptions of the Timoshenko theory are:

• normals to the axis of the beam remain straight after deformation
• there is no change in beam thickness after deformation

However, normals to the axis are not required to remain perpendicular to the axis after deformation.

The equation for the quasistatic bending of a linear elastic, isotropic, homogeneous beam of constant cross-section beam under these assumptions is[7]

${\displaystyle EI~{\cfrac {\mathrm {d} ^{4}w}{\mathrm {d} x^{4}}}=q(x)-{\cfrac {EI}{kAG}}~{\cfrac {\mathrm {d} ^{2}q}{\mathrm {d} x^{2}}}}$

where ${\displaystyle I}$ is the area moment of inertia of the cross-section, ${\displaystyle A}$ is the cross-sectional area, ${\displaystyle G}$ is the shear modulus, ${\displaystyle k}$ is a shear correction factor, and ${\displaystyle q(x)}$ is an applied transverse load. For materials with Poisson's ratios (${\displaystyle \nu }$) close to 0.3, the shear correction factor for a rectangular cross-section is approximately

${\displaystyle k={\cfrac {5+5\nu }{6+5\nu }}}$

The rotation (${\displaystyle \varphi (x)}$) of the normal is described by the equation

${\displaystyle {\cfrac {\mathrm {d} \varphi }{\mathrm {d} x}}=-{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}-{\cfrac {q(x)}{kAG}}}$

The bending moment (${\displaystyle M}$) and the shear force (${\displaystyle Q}$) are given by

${\displaystyle M(x)=-EI~{\cfrac {\mathrm {d} \varphi }{\mathrm {d} x}}~;~~Q(x)=kAG\left({\cfrac {\mathrm {d} w}{\mathrm {d} x}}-\varphi \right)=-EI~{\cfrac {\mathrm {d} ^{2}\varphi }{\mathrm {d} x^{2}}}={\cfrac {\mathrm {d} M}{\mathrm {d} x}}}$

The Euler–Bernoulli equation for the dynamic bending of slender, isotropic, homogeneous beams of constant cross-section under an applied transverse load ${\displaystyle q(x,t)}$ is[7]

${\displaystyle EI~{\cfrac {\partial ^{4}w}{\partial x^{4}}}+m~{\cfrac {\partial ^{2}w}{\partial t^{2}}}=q(x,t)}$

where ${\displaystyle E}$ is the Young's modulus, ${\displaystyle I}$ is the area moment of inertia of the cross-section, ${\displaystyle w(x,t)}$ is the deflection of the neutral axis of the beam, and ${\displaystyle m}$ is mass per unit length of the beam.

For the situation where there is no transverse load on the beam, the bending equation takes the form

${\displaystyle EI~{\cfrac {\partial ^{4}w}{\partial x^{4}}}+m~{\cfrac {\partial ^{2}w}{\partial t^{2}}}=0}$

Free, harmonic vibrations of the beam can then be expressed as

${\displaystyle w(x,t)={\text{Re}}[{\hat {w}}(x)~e^{-i\omega t}]\quad \implies \quad {\cfrac {\partial ^{2}w}{\partial t^{2}}}=-\omega ^{2}~w(x,t)}$

and the bending equation can be written as

${\displaystyle EI~{\cfrac {\mathrm {d} ^{4}{\hat {w}}}{\mathrm {d} x^{4}}}-m\omega ^{2}{\hat {w}}=0}$

The general solution of the above equation is

${\displaystyle {\hat {w}}=A_{1}\cosh(\beta x)+A_{2}\sinh(\beta x)+A_{3}\cos(\beta x)+A_{4}\sin(\beta x)}$

where ${\displaystyle A_{1},A_{2},A_{3},A_{4}}$ are constants and ${\displaystyle \beta$ :=\left({\cfrac {m}{EI}}~\omega ^{2}\right)^{1/4}}

The mode shapes of a cantilevered I-beam
1st lateral bending	1st torsional	1st vertical bending
2nd lateral bending	2nd torsional	2nd vertical bending

In 1877, Rayleigh proposed an improvement to the dynamic Euler–Bernoulli beam theory by including the effect of rotational inertia of the cross-section of the beam. Timoshenko improved upon that theory in 1922 by adding the effect of shear into the beam equation. Shear deformations of the normal to the mid-surface of the beam are allowed in the Timoshenko–Rayleigh theory.

The equation for the bending of a linear elastic, isotropic, homogeneous beam of constant cross-section under these assumptions is[7][13]

${\displaystyle {\begin{aligned}&EI~{\frac {\partial ^{4}w}{\partial x^{4}}}+m~{\frac {\partial ^{2}w}{\partial t^{2}}}-\left(J+{\frac {EIm}{kAG}}\right){\frac {\partial ^{4}w}{\partial x^{2}~\partial t^{2}}}+{\frac {Jm}{kAG}}~{\frac {\partial ^{4}w}{\partial t^{4}}}\\[6pt]={}&q(x,t)+{\frac {J}{kAG}}~{\frac {\partial ^{2}q}{\partial t^{2}}}-{\frac {EI}{kAG}}~{\frac {\partial ^{2}q}{\partial x^{2}}}\end{aligned}}}$

where ${\displaystyle J={\tfrac {mI}{A}}}$ is the polar moment of inertia of the cross-section, ${\displaystyle m=\rho A}$ is the mass per unit length of the beam, ${\displaystyle \rho }$ is the density of the beam, ${\displaystyle A}$ is the cross-sectional area, ${\displaystyle G}$ is the shear modulus, and ${\displaystyle k}$ is a shear correction factor. For materials with Poisson's ratios (${\displaystyle \nu }$) close to 0.3, the shear correction factor are approximately

${\displaystyle {\begin{aligned}k&={\frac {5+5\nu }{6+5\nu }}\quad {\text{rectangular cross-section}}\\[6pt]&={\frac {6+12\nu +6\nu ^{2}}{7+12\nu +4\nu ^{2}}}\quad {\text{circular cross-section}}\end{aligned}}}$

For free, harmonic vibrations the Timoshenko–Rayleigh equations take the form

${\displaystyle EI~{\cfrac {\mathrm {d} ^{4}{\hat {w}}}{\mathrm {d} x^{4}}}+m\omega ^{2}\left({\cfrac {J}{m}}+{\cfrac {EI}{kAG}}\right){\cfrac {\mathrm {d} ^{2}{\hat {w}}}{\mathrm {d} x^{2}}}+m\omega ^{2}\left({\cfrac {\omega ^{2}J}{kAG}}-1\right)~{\hat {w}}=0}$

This equation can be solved by noting that all the derivatives of ${\displaystyle w}$ must have the same form to cancel out and hence as solution of the form ${\displaystyle e^{kx}}$ may be expected. This observation leads to the characteristic equation

${\displaystyle \alpha ~k^{4}+\beta ~k^{2}+\gamma =0~;~~\alpha$ :=EI~,~~\beta :=m\omega ^{2}\left({\cfrac {J}{m}}+{\cfrac {EI}{kAG}}\right)~,~~\gamma :=m\omega ^{2}\left({\cfrac {\omega ^{2}J}{kAG}}-1\right)}

The solutions of this quartic equation are

${\displaystyle k_{1}=+{\sqrt {z_{+}}}~,~~k_{2}=-{\sqrt {z_{+}}}~,~~k_{3}=+{\sqrt {z_{-}}}~,~~k_{4}=-{\sqrt {z_{-}}}}$

where

${\displaystyle z_{+}:={\cfrac {-\beta +{\sqrt {\beta ^{2}-4\alpha \gamma }}}{2\alpha }}~,~~z_{-}:={\cfrac {-\beta -{\sqrt {\beta ^{2}-4\alpha \gamma }}}{2\alpha }}}$

The general solution of the Timoshenko-Rayleigh beam equation for free vibrations can then be written as

${\displaystyle {\hat {w}}=A_{1}~e^{k_{1}x}+A_{2}~e^{-k_{1}x}+A_{3}~e^{k_{3}x}+A_{4}~e^{-k_{3}x}}$

The assumptions of Kirchhoff–Love theory are

• straight lines normal to the mid-surface remain straight after deformation
• straight lines normal to the mid-surface remain normal to the mid-surface after deformation
• the thickness of the plate does not change during a deformation.

These assumptions imply that

${\displaystyle {\begin{aligned}u_{\alpha }(\mathbf {x} )&=-x_{3}~{\frac {\partial w^{0}}{\partial x_{\alpha }}}=-x_{3}~w_{,\alpha }^{0}~;~~\alpha =1,2\\u_{3}(\mathbf {x} )&=w^{0}(x_{1},x_{2})\end{aligned}}}$

where ${\displaystyle \mathbf {u} }$ is the displacement of a point in the plate and ${\displaystyle w^{0}}$ is the displacement of the mid-surface.

The strain-displacement relations are

${\displaystyle {\begin{aligned}\varepsilon _{\alpha \beta }&=-x_{3}~w_{,\alpha \beta }^{0}\\\varepsilon _{\alpha 3}&=0\\\varepsilon _{33}&=0\end{aligned}}}$

The equilibrium equations are

${\displaystyle M_{\alpha \beta ,\alpha \beta }+q(x)=0~;~~M_{\alpha \beta }:=\int _{-h}^{h}x_{3}~\sigma _{\alpha \beta }~dx_{3}}$

where ${\displaystyle q(x)}$ 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

${\displaystyle w_{,1111}^{0}+2~w_{,1212}^{0}+w_{,2222}^{0}=0}$

In direct tensor notation,

${\displaystyle \nabla ^{2}\nabla ^{2}w=0}$

The special assumption of this theory is that normals to the mid-surface remain straight and inextensible but not necessarily normal to the mid-surface after deformation. The displacements of the plate are given by

${\displaystyle {\begin{aligned}u_{\alpha }(\mathbf {x} )&=-x_{3}~\varphi _{\alpha }~;~~\alpha =1,2\\u_{3}(\mathbf {x} )&=w^{0}(x_{1},x_{2})\end{aligned}}}$

where ${\displaystyle \varphi _{\alpha }}$ are the rotations of the normal.

The strain-displacement relations that result from these assumptions are

${\displaystyle {\begin{aligned}\varepsilon _{\alpha \beta }&=-x_{3}~\varphi _{\alpha ,\beta }\\\varepsilon _{\alpha 3}&={\cfrac {1}{2}}~\kappa \left(w_{,\alpha }^{0}-\varphi _{\alpha }\right)\\\varepsilon _{33}&=0\end{aligned}}}$

where ${\displaystyle \kappa }$ is a shear correction factor.

The equilibrium equations are

${\displaystyle {\begin{aligned}&M_{\alpha \beta ,\beta }-Q_{\alpha }=0\\&Q_{\alpha ,\alpha }+q=0\end{aligned}}}$

where

${\displaystyle Q_{\alpha }:=\kappa ~\int _{-h}^{h}\sigma _{\alpha 3}~dx_{3}}$

The dynamic theory of plates determines the propagation of waves in the plates, and the study of standing waves and vibration modes. The equations that govern the dynamic bending of Kirchhoff plates are

${\displaystyle M_{\alpha \beta ,\alpha \beta }-q(x,t)=J_{1}~{\ddot {w}}^{0}-J_{3}~{\ddot {w}}_{,\alpha \alpha }^{0}}$

where, for a plate with density ${\displaystyle \rho =\rho (x)}$,

${\displaystyle J_{1}:=\int _{-h}^{h}\rho ~dx_{3}~;~~J_{3}:=\int _{-h}^{h}x_{3}^{2}~\rho ~dx_{3}}$

and

${\displaystyle {\ddot {w}}^{0}={\frac {\partial ^{2}w^{0}}{\partial t^{2}}}~;~~{\ddot {w}}_{,\alpha \beta }^{0}={\frac {\partial ^{2}{\ddot {w}}^{0}}{\partial x_{\alpha }\,\partial x_{\beta }}}}$

The figures below show some vibrational modes of a circular plate.

• 
  mode k = 0, p = 1
• 
  mode k = 0, p = 2
• 
  mode k = 1, p = 2