# Hollow Circular Beam with Known Cracking Moment

**Situation**

A concrete beam with cross section in Figure CO4-2B is simply supported over a span of 4 m. The cracking moment of the beam is 75 kN·m.

1. Find the maximum uniform load that the beam can carry without causing the concrete to crack, in kN/m.

A. 35.2 | C. 33.3 |

B. 37.5 | D. 41.8 |

2. Find the modulus of rapture of the concrete used in the beam.

A. 4.12 MPa | C. 3.77 MPa |

B. 3.25 MPa | D. 3.54 MPa |

3. If the hallow portion is replaced with a square section of side 300 mm, what is the cracking moment of the new section in kN·m?

A. 71.51 | C. 78.69 |

B. 76.58 | D. 81.11 |

**Answer Key**

Part 1: [ B ]

Part 2: [ C ]

Part 3: [ A ]

Part 2: [ C ]

Part 3: [ A ]

**Solution**

**Part 1**

$75 = \dfrac{w(4^2)}{8}$

$w = 37.5 ~ \text{kN/m}$

**Part 2**

$f_b = \dfrac{75(1000^2)(300)}{\frac{1}{64}\pi (600^4 - 300^4)}$

$f_b = 3.7726 ~ \text{MPa}$

**Part 3**

$3.7726 = \dfrac{M(1000^2)(300)}{\frac{1}{64}\pi (600^4) - \frac{1}{12}(300^4)}$

$M = 71.5117 ~ \text{kN} \cdot \text{m}$

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