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6th Edition

KASSIMALI + 1 other

Publisher: Cengage,

ISBN: 9781337630931

Chapter 8, Problem 32P

Textbook Problem

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Draw the influence lines for the shears and bending moments at points *B* and *E* of the beam shown in Fig. P8.31.

FIG. P8.31, P8.32

To determine

Draw the influence lines for the shear and bending moment at point *B* and *E*.

**Calculation:**

Use Muller-Breslau's principle.

**Influence line ordinate for shear at B.**

Cut the section at *B* to obtain the released structure of the beam. Next apply a small relative displacement in positive direction of *B* of the portion *BC* upward by *B* of the portion *AB* can neither translate nor rotate due to fixed support of *A* to obtain the deflected shape.

Sketch the deflected shape of the released beam when the beam cut at *B* as shown in Figure 1.

The general shape of the influence line for shear *b* is the similar of deflected shape.

Sketch the general shape of the influence line for shear *B* as shown in Figure 2.

Find the numerical value of influence line ordinate of *B*.

Place the 1 kN load first just to the left of *B* and then just to the right of *B*.

Sketch the free body diagram of beam as shown in Figure 3.

Refer Figure 3.

Find the shear at *B**B* using the equation:

The portion *AB* can neither translate nor rotate. Hence, the shear at *B* just left of *B* is 0.

Thus, the value of influence line ordinate just left of *B* is

Find the shear at *B**B* using the equation:

The influence line ordinate of portion *BC* is 1 kN/kN and the shear at *B* is 1 kN when 1 kN load placed at just right of *B*.

Thus, the value of influence line ordinate just right of *B* is

The numerical values of influence line ordinate in different points on the beam for shear

x (m) | Points | Influence line ordinate of |

0 | A | 0 |

5 | ||

5 | ||

10 | ||

16 | D | 0 |

19 | E | |

22 |

Sketch the influence line diagram for shear at point *B* using Table 1 as shown in Figure 4.

**Influence line ordinate for moment at B.**

Insert a hinge at *B* to obtain the released structure of the beam. Next apply a small rotation *B* in the released structure. The rotation *BC* in counterclockwise and the portion *AB* can neither translate nor rotate due to fixed support of *A* to obtain the deflected shape.

Sketch the deflected shape of the released beam when insert a hinge at *B* as shown in Figure 5.

The general shape of the influence line for bending moment *B* is the similar of deflected shape.

Sketch the general shape of the influence line for bending moment *B* as shown in Figure 6.

Find the numerical value of influence line ordinate for *B*.

Refer Figure 6.

Find the moment at *C* from *B*.

Apply 1 kN load at *B*.

Consider clockwise moment as positive and anticlockwise moment as negative.

Thus, the value of influence line ordinate of *C* is

The remaining values of influence line ordinate are determined by geometry (similar triangle) of the influence line and summarize the values as in Table 2.

x (m) | Points | Influence line ordinate of |

0 | A | 0 |

5 | 0 | |

10 | ‑5 | |

16 | D | |

19 | E | |

22 | 0 |

Sketch the influence line diagram for moment at point *B* as shown in Figure 7.

Structural Analysis

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