1.2.1 The Euler-Bernoulli beam equation

Given below are the Euler-Bernoulli beam governing differential equations:

Figure 1.9: A differential frame element

$$\frac{\mathrm{d}w}{\mathrm{d}x} - \varphi_{y}$$

$$\frac{\mathrm{d^{2}}w}{\mathrm{dx^{2}}} -\frac{\mathrm{d}\varphi_{y}}{\mathrm{dx}} = - \frac{1}{EI_{y}}M_{y}$$ (1.14)

$$\frac{\mathrm{d^{3}}w}{\mathrm{dx^{3}}} - \frac{1}{EI_{y}} \frac{\mathrm{d}M_{y}}{\mathrm{d}x} = - \frac{1}{EI_{y}} Q_{z} \qquad\qquad\qquad$$

$$\frac{\mathrm{d^{4}}w}{\mathrm{dx^{4}}} - \frac{1}{EI_{y}}\frac{\mathrm{d}Q_{z}}{\mathrm{d}x} = \frac{1}{EI_{y}}q_{z}$$

where ${E}$ is the elastic modulus and ${I_{y}}$ is the area moment of inertia.

The last formula of Eq. (1.14) is a non-homogeneous differential equation of 4th order ( ${\mathrm{d^{4}}w}/{\mathrm{dx^{4}}} = {q_{z}}/{EI_{y}}$). We are looking for the general solution of the non-homogeneous differential equation in the form

$$w = w_{0}w_{1} + w_{0}^{\prime}w_{2} + w_{0}^{\prime\prime}w_{3} + w_{0}^{\prime\prime\prime}w_{4} + w_{e}\left(x\right)$$ (1.15)

where
$w_{0},\enspace w_{0}^{\prime}, \enspace w_{0}^{\prime\prime}$, and $w_{0}^{\prime\prime\prime}$ are the values of the sought-for function at $x = x_{0}$;
$w_{1}, \enspace w_{2}, \enspace w_{3},\enspace and \enspace w_{4}$ are a normed fundamental set of solutions to the associated homogeneous differential equation;
$w_{e}\left(x\right)$ is the particular solution of the non-homogeneous differential equation.

Next we consider a set for fundamental solutions for the associated homogeneous differential equation:

$$w_{1} = 1, \quad w_{2} = x, \quad w_{3} = \frac{x^{2}}{2}, \quad \quad w_{4} = \frac{x^{3}}{6}$$ (1.16)

The Wronskian^1.1 $W$ of this set of solutions

$$W\left(x\right) = \left\vert \begin{array}{cccc} 1 & x & {x^{2}}/... ... & {x} & {x^{2}}/{2} \\ 0 & 0 & 1 & x \\ 0 & 0 & 0 & 1 \end{array}\right\vert$$ (1.17)

and at $x = 0$

$$W\left(x=0\right) = \left\vert \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right\vert = 1$$ (1.18)

Thus, the last equation is unit Wronskian and the set of fundamental solutions of Eq. (1.16) is a normed fundamental set of solutions.

By Sign Convention 1, the initial parameters of Eq. (1.15) are

$$w_{0} = w_{0}, \enspace w_{0}^{\prime} = - \varphi_{0}, \enspace ... ... = - \frac{M_{0}}{EI}, \enspace w_{0}^{\prime\prime\prime} = - \frac{Q_{0}}{EI}$$ (1.19)

and by Sign Convention 2

$$w_{0} = w_{0}, \enspace w_{0}^{\prime} = - \varphi_{0}, \enspace ... ...ime} = \frac{M_{0}}{EI}, \enspace w_{0}^{\prime\prime\prime} = \frac{Q_{0}}{EI}$$ (1.20)

Substituting Eqs. (1.16) and (1.19) into Eq. (1.15), we get the solution for the homogeneous differential equation (Sign Convention 1):

$$w = \left[\begin{array}{cccc} 1 & - x & - {x^{2}}/{2EI} & -{x^{3}... ...t[\begin{array}{c} w_{0} \\ \varphi_{0} \\ M_{0} \\ Q_{0} \end{array}\right]$$ (1.21)

From Eqs. (1.16) and (1.20) (Sign Convention 2),

$$w = \left[\begin{array}{cccc} 1 & - x & {x^{2}}/{2EI} & {x^{3}}/{... ...t[\begin{array}{c} w_{0} \\ \varphi_{0} \\ M_{0} \\ Q_{0} \end{array}\right]$$ (1.22)

The general non-homogeneous differential equation of the Euler-Bernoulli beam subjected to an external load and equivalent generalized loads is

$$\mathrm{d^{4}}{w}/{\mathrm{dx^{4}}} = {q_{z}}/{EI_{y}} + \mathcal... ...a\right)/{EI_{y}} + \mathcal{M}_{x}{\delta}^{\prime}\left(t - a\right)/{EI_{y}}$$ (1.23)

where
$\mathcal{F}_{z}{\delta}\left(x - a_{F}\right)/{EI_{y}}$ is the equivalent distributed force $q_{F}$ of a concentrated force of magnitude $\mathcal{F}_{z}$,
$\mathcal{M}_{x}{\delta}^{\prime}\left(x - a_{M}\right)/{EI_{y}}$ is the equivalent distributed force $q_{M}$ of a concentrated moment of magnitude $\mathcal{M}_{x}$,
${\delta}\left(x - a\right)$ is the Dirac delta function.

The particular solution of Eq. (1.15) we are looking for is given by the Cauchy formula

$$w_{e}\left(x\right) = \int_{x_{0}}^{x}K\left(x,t\right)f\left(t\right)dt$$ (1.24)

or, to be more precise,

$$w_{e}\left(x\right) = \int_{x_{0}}^{x}K_{4}\left(x,t\right)f_{4}\... ...3}\left(t\right)dt + \int_{x_{0}}^{x}K_{2}\left(x,t\right)f_{2}\left(t\right)dt$$ (1.25)

where, using Eq. (1.16),

$$K_{4}\left(x,t\right) w_{4}\left(x - t\right) = \frac{\left(x - t\right)^{3}}{6}$$ (1.26)

$$K_{3}\left(x,t\right) w_{3}\left(x - t\right) = \frac{\left(x - t\right)^{2}}{2}$$ (1.27)

$$K_{2}\left(x,t\right) w_{2}\left(x - t\right) = {\left(x - t\right)}$$ (1.28)

and the load function $f\left(t\right)$ is

$$f_{4}\left(t\right) = {q\left(t\right)}/{EI}, \quad f_{3}\left(t\... ... \mathcal{F}_{z}/{EI_{y}}, \quad f_{2}\left(t\right) = \mathcal{M}_{x}/{EI_{y}}$$ (1.29)

The particular solution $w_{e}\left(x\right)$ at a constant load $q_{z}$

$$w_{e}\left(x\right) = - \frac{q_{z}}{EI}\int_{a_{q}}^{x}\frac{\le... ...ht\vert^{x}_{a_{q}} = \frac{q_{z}}{EI}\frac{\left(x - a_{q}\right)^{4}_{+}}{24}$$ (1.30)

where $\left(x -a \right)_{+}$ is the Heaviside step function:

$$\left(x -a \right)_{+} = \left\{ \begin{array}{ccc} 0, & if & {\l... ...< 0} \\ {x - a} , & if & {\left( x -a \right) \geq 0} \end{array}\right. \quad$$ (1.31)

In case of the point load $\mathcal{F}_{z}$ and moment $\mathcal{M}_{y}$, the functions , $K\left(x,t\right)$ in the Cauchy formula (1.24) are respectively

$$f_{3}\left(t\right) = \frac{\mathcal{F}_{z}}{EI}, % \label{difflahendho16a} \\ %%{\delta}\left(t - a\right) \\ f_{2}\left(t\right) = \frac{\mathcal{M}_{y}}{EI}$$ (1.32)

$$K_{3}\left(x,t\right) = w_{3}\left(x - t\right) = \frac{\left(x - t\right)^{2}}{2}$$ (1.33)

$$K_{2}\left(x,t\right) = w_{2}\left(x - t\right) = \left(x - t\right)$$ (1.34)

The particular solution $w_{e}\left(x\right)$ in case of the point load $\mathcal{F}_{z}$ and moment $\mathcal{M}_{y}$ is respectively

$$w_{e}\left(x\right) - \frac{\mathcal{F}_{z}}{EI}\int_{a_{F}}^{x}\frac{\left(x - t\rig... ...} = \frac{\mathcal{F}_{z}}{EI}\frac{\left(x - {a_{F}}\right)^{3}_{+}}{6} \qquad$$ (1.35)

$$- \frac{\mathcal{M}_{y}}{EI}\int_{a_{M}}^{x}{\left(x - t\right)}\... ...M}} = \frac{\mathcal{M}_{y}}{EI}\frac{\left(x - a_{M}\right)^{2}_{+}}{2} \qquad$$ (1.36)

The general solution of the non-homogeneous differential equation

$$\mathrm{d^{4}}{w}/{\mathrm{dx^{4}}} = {q_{z}}/{EI_{y}} + \mathcal... ...a\right)/{EI_{y}} + \mathcal{M}_{x}{\delta}^{\prime}\left(t - a\right)/{EI_{y}}$$ (1.37)

is the sum of the solutions of the homogeneous differential equation (1.22) (Sign Convention 2) and of the particular solutions of Eqs. (1.30), (1.35), and (1.36):

$$w_{x} = w_{A} - \varphi_{A}\cdot x + Q_{A}\frac{x^{3}}{6EI} + M_{A}\frac{x^{2}}{2EI}$$

$$\frac{q_{z}}{EI}\frac{\left(x - a_{q}\right)^{4}_{+}}{24} + \frac... ...{6} + \frac{\mathcal{M}_{y}}{EI}\frac{\left(x - a_{M}\right)^{2}_{+}}{2} \qquad$$ (1.38)

Let us take the derivatives of displacement from Eq. (1.38) and apply those to the governing differential equation (1.14). We obtain beam governing equations with the transfer matrix

$$\underbrace{\left[\begin{array}{c} w_{x} \\ \varphi_{x} \\ \ldots \\ Q_{x} \\ M_{x} \end{array}\right]}_{\mathbf{Z_{x}}} \underbrace{\left[ \begin{array}{ccccc} 1 & - x & \vdots & {x^{3}... ...\varphi_{A} \\ \ldots \\ Q_{A} \\ M_{A} \end{array}\right]}_{\mathbf{Z_{A}}}$$

$${ \large { \underbrace{ \left[\begin{array}{cccccc} + & \frac{q_{... ...ray}\right]}_{\mathbf{\stackrel{\rm\circ}{Z}}} }} % \qquad \qquad \qquad \qquad$$ (1.39)

where the effect of the shear force ( $-{x}/{GA_{red}}$) from Eq. (1.3) is added to the deformation.

In symbolic matrix notation, the formulae of Eq. (1.39) can be written as

$${\mathbf{Z_{x}}} = {\mathbf{U_{x}}}{\mathbf{Z_{A}}} + {\mathbf{\stackrel{\rm\circ}{Z}}}$$ (1.40)

Figure 1.10: External reactions and restraints on displacements