Chapter3 - Chapter 3 Introduction to Statically...

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© K. H. Ha - BCEE 343 Chapter 3 - v2.0 3.1 Chapter 3 Introduction to Statically Indeterminate Systems 1. Characteristics of Statically Indeterminate Systems. .................................... 2 1.1 Example Systems. ............................................................................. 2 1.2 Advantages of Indeterminate Systems. ................................................. 3 1.3 Disadvantages of Indeterminate Systems . ............................................. 4 2. Analysis of Indeterminate Systems . .......................................................... 4 2.1 The Force Method of Analysis. .............................................................. 4 2.2 The Displacement Method of Analysis. ................................................... 7 2.3 Implications on Design. ....................................................................... 8
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© K. H. Ha - BCEE 343 Chapter 3 - v2.0 3.2 1. Characteristics of Statically Indeterminate Systems 1.1 Example Systems Consider the two beams in Fig.3.1. k b k c a b c EI d F b F c w P a b c EI d w P (a) Statically determinate system (b) Statically indeterminate system V a V d L L V a V d Figure 3.1 ± The beam in (a) is statically determinate provided that the applied forces such as w , P , F b and F c are known. ± The beam in (b) is statically indeterminate because the springs’ forces F b and F c are unknown 1 . The number of reaction components acting on this beam is 5, including the two spring forces, while there are only three available equilibrium equations. This system (b) is therefore statically indeterminate to the second degree. Vertical equilibrium The following vertical equilibrium equation applies to both beams: V a + F b + F c + V d = w L + P ( 3 . 1 ) In addition to Eq.3.1, we can use two more equations of moment equilibrium. E.g. = = 0 0 d a M M . Note that, for system (b): ± The sum of the four forces (in the left-hand side of 3.1) is constant and equal to the right-hand side; ± Their magnitudes may change depending on the spring stiffnesses. 1 The springs may represent footings resting on soil, or structural members such as columns.
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© K. H. Ha - BCEE 343 Chapter 3 - v2.0 3.3 1.2 Advantages of Indeterminate Systems. To study the characteristics of the statically indeterminate system (b), let us vary the spring stiffnesses k b , k c from zero to infinity. ± When k b = k c = 0, the springs effectively do not exist, and the system is statically determinate with F b = F c = 0. ± As the springs stiffen, the spring forces F b , F c increase while the reactions V a , V d decrease in order to satisfy Eq.3.1. The springs provide more effective support to the beam, thus reducing both deflections and stresses in the beam. ± When
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This note was uploaded on 10/21/2008 for the course BCEE 343 taught by Professor Dr.ha during the Winter '08 term at Concordia Canada.

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Chapter3 - Chapter 3 Introduction to Statically...

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