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6th Edition

KASSIMALI + 1 other

Publisher: Cengage,

ISBN: 9781337630931

Chapter 8, Problem 1P

Textbook Problem

8.1 through 8.3 Draw the influence lines for vertical reactions at supports *A* and *C* and the shear and bending moment at point *B* of the beams shown in Figs. P8.1 through P8.3.

FIG.P8.1, P8.59

Expert Solution

To determine

Draw the influence lines for the vertical reactions at supports *A* and *C*.

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

**Calculation:**

Apply a 1 k unit moving load at a distance of *x* from left end *A*.

Sketch the free body diagram of beam as shown in Figure 1.

Refer Figure 1.

Find the equation of support reaction *A* using equilibrium equation:

Take moment about point *C*.

Consider moment equilibrium at point *C*.

Consider clockwise moment as positive and anticlockwise moment as negative

Sum of moment at point *C* is zero.

Find the equation of support reaction *C* using equilibrium equation:

Apply vertical equilibrium equation of forces.

Consider upward force as positive

Substitute

Consider Equation (1).

Find the value of influence line ordinate of reaction *A*.

Substitute 0 for *x* in Equation (1).

Similarly calculate the influence line ordinate of reaction *x* and summarize the result in Table 1.

x | |

0 | 1 |

15 | 0.5 |

30 | 0 |

Draw the influence line diagram for the vertical reactions at support *A* using Table 1 as shown in Figure 2.

Consider Equation (2).

Find the influence line ordinate of reaction *C*.

Substitute 30 for *x* in Equation (2).

Similarly calculate the influence line ordinate of reaction *x* and summarize the result in Table 2.

x | |

0 | 0 |

15 | 0.5 |

30 | 1 |

Draw the influence line diagram for the vertical reactions at support *C* using Table 2 as shown in Figure 3.

Find the equation of shear force at *B* of portion *AB*

Sketch the free body diagram of the section *AB* as shown in Figure 4.

Refer Figure 4.

Apply equilibrium equation of forces.

Consider upward force as positive

Substitute

Find the equation of shear force at *B* of portion *BC*

Sketch the free body diagram of the section *BC* as shown in Figure 5.

Refer Figure 5.

Apply equilibrium equation of forces.

Consider upward force as positive

Substitute

Thus, the equations of the influence line for

Find the value of influence line ordinate of shear force at various points of *x* using the Equations (3) and (4) and summarize the value as in Table 3.

x | |

0 | 1 |

30 | 0 |

Draw the influence lines for the shear force at point *B* using Table 3 as shown in Figure 6.

Refer Figure 4.

Consider clockwise moment as positive and anticlockwise moment as negative.

Find the equation of moment at *B* of portion *AB*

Substitute

Refer Figure 5.

Consider clockwise moment as negative and anticlockwise moment as positive.

Find the equation of moment at *B* of portion *BC*

Substitute

Thus, the equations of the influence line for

Find the value of influence line ordinate of moment at various points of *x* using the Equations (5) and (6) and summarize the value as in Table 4.

x | |

0 | 0 |

30 | 0 |

Draw the influence lines for the moment at point *B* using Table 4 as shown in Figure 7.

Therefore, the influence lines for the vertical reactions at supports *A* and *C* and the influence lines for the shear and bending moment at point *B* are drawn.

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Structural Analysis

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