6.9 Redundant Structures and the Unit Load Method

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1 6.9 Redundant Structures and the Unit Load Method
: 두 Equation 이 독립적 이중에서 2개의 Equation으로 Unknown force를 구할 수 있다. • Single redundancy – Stability 조건보다 하나의 미지수가 많은 경우, One deflection constraint 필요 • Multiple redundancy – Same No. of deflection equations Static Problem(2-D) Redundant Structure 힘의 평형조건만으로 Unknown force를 구할 수 없다 → Geometric constraint 필요 순간중심 ① ② ③ ①, ②,③ 중에서 하나의 부재가 없어도 이 system은 유지된다. – Redundant structure

2 6.10 Structures with Single Redundancy
(Pin-jointed) ; determined undetermined Pin-jointed Bar S’, 각 부재에 걸리는 힘 (Redundant structure with single redundancy) Note! :외력 에 의하여 부재에 생기는 힘의 비율 즉 일 때 부재에 생기는 내력

3 Elastic truss with single redundancy 에 적용
(the deflection in the direction of the redundant is zero) 부재(1)에 걸리는 전체 내력 Since , 일때 생기는 부재 (1)의 힘 Unknown Force: ① 힘 P에 의하여 부재(1)에 생기는 내력 ①

4 If removing the redundancy from eq.(1)
Remind Castigliano’s theorem For the unit value of the redundant (3) (4) positive direction From eq.(2) The value to give zero deflection

5 Example 6.8 Find the reactions.
① ② Members AB BC AD DC BD Total -56.6 44.8 40.0 1.414 -2.233 -2.0 36.3 -57.3 -51.3 -20.3 -12.5 -11.3

6 Example 6.9 In example 6.8 to the right
decrease Superposition ·temperature decrease ·The point C is displaced an additional (힘을 준 상태에서) ·Example 6.8에서 외력 40kip에 의한 (Example 6.8, ) (Reduce ) (힘을 준 상태에서)

7 Example 6.10 Find the bending moment. Bending deformation only
Strain energy : Bending moment only for statically determinate structure

8 비교 Analogy 에 의한 Bending moment For (unit load of ) 일 때의 bending moment

9 Use conventional method Castigliano’s Theorem

10 Example 6.11 Find 1)Tension in the wire 2) bending moment torsion
: Applied force when statically determinate : without the applied force Sol) put the wire a redundant (invoke ) i) Statically determinate (remove the wire)

11 (-)sign : deflection in the apposite direction to th unit load
ii) (Unknown tension ) 일 때 생기는 방향의 deflection Deflection 에 contribution

12 Problem – at class

13 6.11 Structures with Multiple Redundancy
• Very similar to that used for a structure with ① remove redundant members (statically determinate structure) ② deflection  equal to zero (a) Externally : statically determinate (has 3 reactions) Internally : 2 more members than are required for stability Remove the two redundant  Statically determinate structure : applied load 에 의하여 부재에 생긴 내력(內力) (b)

14 A redundant  unit value of unknown
 이 부재에 생김 Another redundant  unit value of unknown  각 부재에 유발 Total force in any member 은 한 member의 내부력 – Relative deflection = 0 Where

15 So, Thus, can be obtained. Put, For 1,2 redundant The applied loads : Unit value of : Total deflection, : usually zeros.

16 Generalization (for structures with n redundants)
Where, for truss structure For rigid-frame structure (where only bending deformation is considered)

17 Example 6.14 Find the bending moment diagram

18 Variation ① ②