2.2.1 Compatibility conditions at beam joints

The beam considered is exposed to a shear force and bending moment, and accordingly has the deflection $w$ and slope $\varphi$. The two beam elements in Fig. 2.7 have a rigid connection:

Figure 2.7: Compatibility at rigid joint T2

$$\left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] \lef... ... 2\right)}_{A} \\ \varphi^{\left( 2\right) }_{A} \end{array}\right] = 0 \qquad$$ (2.26)

or

$$\mathbf{I_{2\times 2}}\cdot \mathbf{v^{\left( 1\right) }_{L}} - \mathbf{I_{2\times 2}}\cdot \mathbf{v^{\left( 2\right) }_{A}} = 0$$ (2.27)

where $\mathbf{I_{2\times 2}}$ is a unit matrix.

In Fig. 2.8, the two beam elements have a hinged connection:

Figure 2.8: Compatibility at pin joint T1

$$\left[ \begin{array}{cc} 1 & 0 \end{array} \right] \left[\begin{a... ... 2\right)}_{A} \\ \varphi^{\left( 2\right) }_{A} \end{array}\right] = 0 \qquad$$ (2.28)

or

$$\mathbf{T^{o}_{1}}\cdot \mathbf{v^{\left( 1\right) }_{L}} - \mathbf{T^{o}_{2}}\cdot \mathbf{v^{\left( 2\right) }_{A}} = 0$$ (2.29)