Eng. 6002 Ship Structures 1 Hull Girder Response Analysis

1. Eng. 6002 Ship Structures 1Hull Girder Response Analysis Lecture 9: Review of Indeterminate Beams

2. Overview • The internal forces in indeterminate structures cannot be obtained by statics alone. • This is most easily understood by considering a similar statically determinate structure and then adding extra supports • This way also suggests a general technique for analyzing elastic statically indeterminate structures

3. Statically Indeterminate Beams • A uniformly loaded beam is shown with three simple supports. • If there had been only two supports the beam would have been statically determinate • So we imagine the same beam with one of the supports removed and replaced by some unknown force X

4. Statically Indeterminate Beams • If the center support were removed the beam would sag as illustrated • The sag at the centre is counteracted by the reaction force X1, providing an upward displacement • Note: the subscript 0 is used to denote displacements generated by the original external load on the statically determinate structure

5. Statically Indeterminate Beams • In the original statically indeterminate structure there is no vertical displacement of the centre due to the support • Thus, the force X1 must have a magnitude that exactly counteracts the sag of the beam without the centre support

6. Statically Indeterminate Beams • There are two approaches for solving indeterminate systems. Both approaches use • the principle of superposition, by dividing the problem into two simpler problems, • soling the simpler problems and adding the two solutions. • The first method is called the Force Method (also called the Flexibility Method). • The idea for the force method is;

7. Statically Indeterminate Beams • The idea for the force method is: • Step 1. Reduce the structure to a statically determinate structure. This step allows the structure to displace where it was formerly fixed. • Step 2. solve each determinate system, to find all reactions and deflections. Note all incompatible deflections • Step 3. re-solve the determinate structures with only a set of self-balancing internal unit forces at removed reactions. This solves the system for the internal or external forces removed in 1. • Step 4. scale the unit forces to cause the opposite of the incompatible deflections • Step 5. Add solutions (everything: loads, reactions, deflections…) from 2 and 4.