where $k=\frac{\text{I}}{L}$ is the stiffness, $=\frac{}{L}$ is the span's cord rotation, and $($FEM$){?}_{AB}$ is the fixed end moment at A due to any loads applied on the span $AB.$ A similar equation can be written for the moment at $B.$

$24$

$M_{BA}=2Ek(2{}_B+{}_A-3)+(FEM{)}_{BA}$

The two equations are very similar. By using $N$ to represent the near end and $F$ for the far end, they can both be written as

$M_N=2Ek(2{}_N+{}_F-3)+(FEM{)}_N.$

The sign convention for all these equations is that a positive moment is clockwise and positive angles are measured clockwise.

The previous equations are typically applied twice to each span for internal spans or end spans that are fixed supported. When the end is pin or roller supported, the internal moment at the end must be zero. In this case, the equation below can be derived for the moment at the other $($near$)$ support, and only one equation needs to be used for the span,

$M_N=3Ek({}_N-)+(FEM{)}_N$

Figure

$2$ of $4$

Part A $-$ Internal spans

The internal span $CD$ on a continuous beam is deflected as shown $($Figure $2)$ as a result of loads on other portions of the beam, where $L=20ft,=0\mathrm{.}75,=0\mathrm{.}7,E=2\mathrm{.}9{10}^4$ ksi, and $I=2000i{n}^4.$ There is no vertical deflection at either end. What is the internal moment at $D?$

Express your answer with appropriate units to three significant figures.

View Available Hint$($s$)$

Part B $-$ End spans

The end span $EF$ on a continuous beam is deflected as shown $($Figure $3)$ as a result of support settlement and loads on

PLEASE USE THE SLOPE DEFLECTION EQUATION AND SOLVE FOR MD USING THE FIGURE IN THE BOTTOM LEFT OF PICTURE