# Continuous Beam With a Gap and a Zero Moment in Interior Support

**Situation**

A beam of uniform cross section whose flexural rigidity *EI* = 2.8 × 10^{11} N·mm^{2}, is placed on three supports as shown. Support *B* is at small gap Δ so that the moment at *B* is zero.

1. Calculate the reaction at *A*.

A. 4.375 kN | C. 5.437 kN |

B. 8.750 kN | D. 6.626 kN |

2. What is the reaction at *B*?

A. 4.375 kN | C. 5.437 kN |

B. 8.750 kN | D. 6.626 kN |

3. Find the value of Δ.

A. 46 mm | C. 34 mm |

B. 64 mm | D. 56 mm |

**Answer Key**

Part 2: [ B ]

Part 3: [ D ]

**Solution**

$3.5R_A - 2.5(3.5) \left( \dfrac{3.5}{2} \right) = 0$

$R_A = 4.375 ~ \text{kN}$ ← Answer for Part (1)

By symmetry:

$R_C = R_A$

$R_C = 4.375 ~ \text{kN}$

$\Sigma; F_V = 0$

$R_B + R_A + R_C = 7(2.5)$

$R_B = 8.75 ~ \text{kN}$ ← Answer for Part (2)

$\Delta = \Delta_\text{uniform load} - \Delta_{\text{reaction at }B}$

$\Delta = \dfrac{5wL^4}{384EI} - \dfrac{PL^3}{48EI}$

$\Delta = \dfrac{5(2.5)(7^4)(1000^4)}{384(2.8 \times 10^{11})} - \dfrac{8.75(7^3)(1000^4)}{48(2.8 \times 10^{11})}$

$\Delta = 55.83 ~ \text{mm}$ ← Answer for Part (3)