# B2.A

## Video 

Loads acting on a beam are $$F_1$$ and $$F_2$$. The beam is suffering from the deformations $$w_1, \psi_1, w_2, \psi_2$$.

Given symbols: $$a, EI, F_1, F_2.$$

Proceed as follows.

## Steps

### 1. Elements and Nodes

Define two beam elements as shown in Balken-Element B2.

Solution

### 2. Augmented Element Stiffness Matrices

• $$k_1^|$$: The augmented element stiffness matrix of Element 1 in “element shape”.

• $$k_2^|$$: The augmented element stiffness matrix of Element 2 in “element shape”.

• $$K_1^|$$: The augmented element stiffness matrix of Element 1 in “system shape”.

• $$K_2^|$$: The augmented element stiffness matrix of Element 2 in “system shape”.

Solution

\begin{align*} k_1^| \!&=\! \tfrac{EI}{a^3} \! \left[ \begin{array}{cccc|c} 4 a^2 & -6 a & 2 a^2 & 6 a & \psi_0 \\ -6 a & 12 & -6 a & -12 & w_0 \\ 2 a^2 & -6 a & 4 a^2 & 6 a & \psi_1 \\ 6 a & -12 & 6 a & 12 & w_1 \end{array} \right] \\ k_2^| \!&=\! \tfrac{EI}{a^3} \! \left[ \begin{array}{cccc|c} 4 a^2 & -6 a & 2 a^2 & 6 a & \psi_1 \\ -6 a & 12 & -6 a & -12 & w_1 \\ 2 a^2 & -6 a & 4 a^2 & 6 a & \psi_2 \\ 6 a & -12 & 6 a & 12 & w_2 \end{array} \right] \\ K_1^| \!&=\! \tfrac{EI}{a^3} \! \left[ \begin{array}{cccccc|c} 4 a^2 & -6 a & 2 a^2 & 6 a & 0 & 0 & \psi_0 \\ -6 a & 12 & -6 a & -12 & 0 & 0 & w_0 \\ 2 a^2 & -6 a & 4 a^2 & 6 a & 0 & 0 & \psi_1 \\ 6 a & -12 & 6 a & 12 & 0 & 0 & w_1 \\ 0 & 0 & 0 & 0 & 0 & 0 & \psi_2 \\ 0 & 0 & 0 & 0 & 0 & 0 & w_2 \end{array} \right] \\ K_2^| \!&=\! \tfrac{EI}{a^3} \! \left[ \begin{array}{cccccc|c} 0 & 0 & 0 & 0 & 0 & 0 & \psi_0 \\ 0 & 0 & 0 & 0 & 0 & 0 & w_0 \\ 0 & 0 & 4 a^2 & -6 a & 2 a^2 & 6 a & \psi_1 \\ 0 & 0 & -6 a & 12 & -6 a & -12 & w_1 \\ 0 & 0 & 2 a^2 & -6 a & 4 a^2 & 6 a & \psi_2 \\ 0 & 0 & 6 a & -12 & 6 a & 12 & w_2 \end{array} \right] \end{align*}

### 3. System Stiffness Matrix

Use the following symbols:

• $$K$$: The system stiffness matrix.

• $$u$$: The deformations.

• $$f$$: The external nodal loads.

$\begin{split}u = \begin{bmatrix} \psi_0 \\ w_0 \\ \psi_1 \\ w_1 \\ \psi_2 \\ w_2 \end{bmatrix} \qquad f = \begin{bmatrix} M_0 \\ F_0 \\ M_1 \\ F_1 \\ M_2 \\ F_2 \end{bmatrix}\end{split}$

Find $$K$$. And write down the following equation:

$K u = f$

Solution

The system stiffness matrix is the “sum” of all element stiffness matrices in “system shape”:

\begin{align*} \underbrace{ \tfrac{EI}{a^3} \! \left[ \begin{array}{cccccc} 4 a^2 & -6 a & 2 a^2 & 6 a & 0 & 0 \\ -6 a & 12 & -6 a & -12 & 0 & 0 \\ 2 a^2 & -6 a & 8 a^2 & 0 & 2 a^2 & 6 a \\ 6 a & -12 & 0 & 24 & -6 a & -12 \\ 0 & 0 & 2 a^2 & -6 a & 4 a^2 & 6 a \\ 0 & 0 & 6 a & -12 & 6 a & 12 \end{array} \right] }_K \! \underbrace{ \left[ \begin{array}{c} \psi_0 \\ w_0 \\ \psi_1 \\ w_1 \\ \psi_2 \\ w_2 \end{array} \right] }_u \!&=\! \underbrace{ \left[ \begin{array}{c} M_0 \\ F_0 \\ M_1 \\ F_1 \\ M_2 \\ F_2 \end{array} \right] }_f \tag{1} \end{align*}

### 4. Boundary Conditions

Adjust $$u$$ and $$f$$ so that all boundary conditions are met.

$\begin{split}u = \begin{bmatrix} \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \end{bmatrix} \qquad f = \begin{bmatrix} \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \end{bmatrix}\end{split}$

Solution

### 5. Deformations

Find $$\psi_1, w_1, \psi_2, w_2$$. Show that:

\begin{align*} \left[ \begin{array}{c} \psi_1 \\ w_1 \\ \psi_2 \\ w_2 \end{array} \right] \!&=\! \tfrac{a^2}{6 EI} \!\left[ \begin{array}{c} - 3 F_{1} -9 F_{2} \\ a( 2 F_{1} + 5 F_{2} ) \\ - 3 F_{1} -12 F_{2} \\ a ( 5 F_{1} + 16 F_{2} ) \end{array} \right] \end{align*}

Solution

The following equation describes the system:

$\begin{split}\tfrac{EI}{a^3} \begin{bmatrix} 4 a^2 & -6 a & 2 a^2 & 6 a & 0 & 0 \\ -6 a & 12 & -6 a & -12 & 0 & 0 \\ 2 a^2 & -6 a & 8 a^2 & 0 & 2 a^2 & 6 a \\ 6 a & -12 & 0 & 24 & -6 a & -12 \\ 0 & 0 & 2 a^2 & -6 a & 4 a^2 & 6 a \\ 0 & 0 & 6 a & -12 & 6 a & 12 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ \psi_1 \\ w_1 \\ \psi_2 \\ w_2 \end{bmatrix} &= \begin{bmatrix} M_0 \\ F_0 \\ 0 \\ F_1 \\ 0 \\ F_2 \end{bmatrix}\end{split}$

These are six equations for the six unknowns $$\psi_1, w_1, \psi_2, w_2, M_0, F_0$$. The last four equations can be used to solve for $$\psi_1, w_1, \psi_2, w_2$$:

\begin{align*} \tfrac{EI}{a^3} \left[ \begin{array}{cccc} 8 a^2 & 0 & 2 a^2 & 6 a \\ 0 & 24 & -6 a & -12 \\ 2 a^2 & -6 a & 4 a^2 & 6 a \\ 6 a & -12 & 6 a & 12 \end{array} \right] \left[ \begin{array}{c} \psi_1 \\ w_1 \\ \psi_2 \\ w_2 \end{array} \right] \!&=\! \left[ \begin{array}{c} 0 \\ F_1 \\ 0 \\ F_2 \end{array} \right] \end{align*}

Solving for $$\psi_1, w_1, \psi_2, w_2$$ leads to:

\begin{align*} \left[ \begin{array}{c} \psi_1 \\ w_1 \\ \psi_2 \\ w_2 \end{array} \right] \!&=\! \tfrac{a^2}{6 EI} \!\left[ \begin{array}{c} - 3 F_{1} -9 F_{2} \\ a( 2 F_{1} + 5 F_{2} ) \\ - 3 F_{1} -12 F_{2} \\ a ( 5 F_{1} + 16 F_{2} ) \end{array} \right] \end{align*}

### 5. Forces

Solution

$$M_0$$ and $$F_0$$ are calculated from $$\psi_1, w_1, \psi_2, w_2$$ by using the first two equations of the system equation:

\begin{align*} \tfrac{EI}{a^3} \! \left[ \begin{array}{cccccc} 2 a^2 & 6 a \\ -6 a & -12 \\ \end{array} \right] \left[ \begin{array}{c} \psi_1 \\ w_1 \\ \end{array} \right] \!&=\! \left[ \begin{array}{c} M_0 \\ F_0 \\ \end{array} \right] \end{align*}

Plugging in $$\psi_1, w_1, \psi_2, w_2$$ leads to:

\begin{align*} \left[ \begin{array}{c} M_{0} \\ F_0 \\ \end{array} \right] \!=\! \left[ \begin{array}{c} a \left(F_{1} + 2 F_{2}\right)\\ - F_{1} - F_{2}\\ \end{array} \right] \end{align*}

### 7. Solution for Given Quantities

Fine $$w_2$$ in $$\mathrm{mm}$$ (millimeters) for the following given quantities:

$\begin{split}F_1 &= 10 \,\mathrm{kN} \\ F_2 &= 10 \,\mathrm{kN} \\ a &= 1 \,\mathrm{m} \\ E &= 210 \,\mathrm{GPa} \\ I &= 318 \,\mathrm{cm}^4\end{split}$

Round to $$0.01$$. Show that

$w_2 \stackrel{0.01}{\approx} 52.41 \,\mathrm{mm}$

Solution

Plugging in the quantities leads to:

$w_2 \stackrel{0.01}{\approx} 52.41 \,\mathrm{mm}$

Real Life

• $$E= 210 \,\mathrm{GPa}$$ is the Young’s Modulus of Steel.

• $$I = 318 \,\mathrm{cm}^4$$ is $$I_{yy}$$ of a IPE 120.

• $$F_1 = F_2 = 10 \,\mathrm{kN}$$ is the weight of a Volkswagen Polo.

Note

Solution with Python here.