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

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

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

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

Example 6.8 Find the reactions. ① ② Members AB BC AD DC BD Total -56.6 44.8 40.0 0.001414 0.001245 0.000500 1.414 -2.233 -2.0 -0.1130 -0.1245 -0.0400 -0.5150 0.00283 0.00621 0.00200 0.02008 36.3 -57.3 -51.3 -20.3 -12.5 -11.3

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)

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

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

Use conventional method Castigliano’s Theorem

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)

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

Problem – at class

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)

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

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

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

Example 6.14 Find the bending moment diagram

Variation ① ②