The problem asks us to show that the element stiffness matrix for a pin-jointed structure is given by the provided matrix. The equation is of the form: $\begin{bmatrix} F_{x1} \\ F_{y1} \\ F_{x2} \\ F_{y2} \end{bmatrix} = \frac{AE}{L} \begin{bmatrix} l^2 & lm & -l^2 & -lm \\ lm & m^2 & -lm & -m^2 \\ -l^2 & -lm & l^2 & lm \\ -lm & -m^2 & lm & m^2 \end{bmatrix} \begin{bmatrix} u_1 \\ v_1 \\ u_2 \\ v_2 \end{bmatrix}$ where $l$ and $m$ are the direction cosines of the line 1-2.

1. Problem Description

The problem asks us to show that the element stiffness matrix for a pin-jointed structure is given by the provided matrix. The equation is of the form:

$\begin{bmatrix} F_{x1} \\ F_{y1} \\ F_{x2} \\ F_{y2} \end{bmatrix} = \frac{AE}{L} \begin{bmatrix} l^2 & lm & -l^2 & -lm \\ lm & m^2 & -lm & -m^2 \\ -l^2 & -lm & l^2 & lm \\ -lm & -m^2 & lm & m^2 \end{bmatrix} \begin{bmatrix} u_1 \\ v_1 \\ u_2 \\ v_2 \end{bmatrix}$

where $l$ and $m$ are the direction cosines of the line 1-

2.

2. Solution Steps

The element stiffness matrix for a pin-jointed structure is derived based on the axial stiffness and transformation of coordinates. The axial stiffness of the element is given by $k = \frac{AE}{L}$, where A is the cross-sectional area, E is Young's modulus, and L is the length of the member.

The axial deformation, $\delta$, of the member can be expressed in terms of the global displacements $u_1, v_1, u_2, v_2$ as follows:

$\delta = (u_2 - u_1)l + (v_2 - v_1)m$

The axial force, $F$, in the member is given by:

$F = k\delta = \frac{AE}{L} [(u_2 - u_1)l + (v_2 - v_1)m]$

Now, we resolve this axial force into its x and y components at nodes 1 and

2.

$F_{x1} = -Fl = -\frac{AE}{L}l [(u_2 - u_1)l + (v_2 - v_1)m] = \frac{AE}{L} [-l^2 u_2 + l^2 u_1 -lm v_2 + lm v_1 ] = \frac{AE}{L} [l^2u_1 + lmv_1 -l^2u_2 - lmv_2]$

$F_{y1} = -Fm = -\frac{AE}{L}m [(u_2 - u_1)l + (v_2 - v_1)m] = \frac{AE}{L} [-lm u_2 + lmu_1 -m^2 v_2 + m^2 v_1 ] = \frac{AE}{L} [lmu_1 + m^2v_1 -lmu_2 - m^2v_2]$

$F_{x2} = Fl = \frac{AE}{L}l [(u_2 - u_1)l + (v_2 - v_1)m] = \frac{AE}{L} [l^2 u_2 - l^2 u_1 +lm v_2 - lm v_1 ] = \frac{AE}{L} [-l^2u_1 - lmv_1 +l^2u_2 + lmv_2]$

$F_{y2} = Fm = \frac{AE}{L}m [(u_2 - u_1)l + (v_2 - v_1)m] = \frac{AE}{L} [lm u_2 - lmu_1 +m^2 v_2 - m^2 v_1 ] = \frac{AE}{L} [-lmu_1 - m^2v_1 +lmu_2 + m^2v_2]$

Writing these equations in matrix form, we get:

$\begin{bmatrix} F_{x1} \\ F_{y1} \\ F_{x2} \\ F_{y2} \end{bmatrix} = \frac{AE}{L} \begin{bmatrix} l^2 & lm & -l^2 & -lm \\ lm & m^2 & -lm & -m^2 \\ -l^2 & -lm & l^2 & lm \\ -lm & -m^2 & lm & m^2 \end{bmatrix} \begin{bmatrix} u_1 \\ v_1 \\ u_2 \\ v_2 \end{bmatrix}$

3. Final Answer

The element stiffness matrix for a pin-jointed structure is given by:

$\begin{bmatrix} F_{x1} \\ F_{y1} \\ F_{x2} \\ F_{y2} \end{bmatrix} = \frac{AE}{L} \begin{bmatrix} l^2 & lm & -l^2 & -lm \\ lm & m^2 & -lm & -m^2 \\ -l^2 & -lm & l^2 & lm \\ -lm & -m^2 & lm & m^2 \end{bmatrix} \begin{bmatrix} u_1 \\ v_1 \\ u_2 \\ v_2 \end{bmatrix}$