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6th Edition

KASSIMALI + 1 other

Publisher: Cengage,

ISBN: 9781337630931

Chapter 8, Problem 30P

Textbook Problem

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Draw the influence lines for the shears and bending moments at points *C* and *F* of the beam shown in Fig. P8.29.

FIG. P8.29, P8.30

To determine

Draw the influence lines for the shear and bending moment at point *C* and *F*.

**Calculation:**

Use Muller-Breslau's principle.

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

Cut the beam at *C* to obtain the released structure of the beam. Next apply a small relative displacement in positive direction of *C* of the portion *BC* downward by *C* of the portion *CD* upward by

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

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

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

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

Place the 1 k load first just to the left of *C* and then just to the right of *C*.

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

Find the reaction *B* and *D*.

The point *C* is the center of support *B* and *D*. Hence, the reaction on the hinge *B* and *D* is half of the load. Then,

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

Substitute

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

Find the shear at *F**C* using the equation:

Substitute

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

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

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

0 | A | 0 |

20 | ||

30 | ||

30 | ||

40 | D | 0 |

60 | E | |

75 | ||

90 | G |

Sketch the influence line diagram for shear at point *C* as shown in Figure 4.

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

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

Insert a hinge at *C* to obtain the released structure of the beam. Next apply a small rotation *C* in the released structure. The rotation *BC* in counterclockwise and the portion *CD* in clockwise to obtain the deflected shape.

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

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

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

Refer Figure 3.

Find the moment at *C* from *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 (ft) | Points | Influence line ordinate of |

0 | A | 0 |

20 | ||

30 | ||

40 | D | |

60 | E | |

75 | ||

90 | G | 0 |

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

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

Cut the beam at *F* to obtain the released structure of the beam. Next apply a small relative displacement in positive direction of *F* of the portion *EF* downward by *F* of the portion *FG* upward by

Structural Analysis

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