Eng. 6002 Ship Structures 1 Hull Girder Response Analysis

Eng. 6002 Ship Structures 1Hull Girder Response Analysis Lecture 11: Discussion of midterm, stiffness matrix, and Maxwell’s Reciprocal Theorem

Overview • In the last lecture we derived a stiffness matrix that can be applied with the displacement method.

Overview • We noticed that the matrix was symmetrical • For example, k35=k53 • Today we will prove this

Overview • Recall the movement of all 6 degrees of freedom of a beam • We can write the following matrix equation:

Maxwell’s Reciprocal Theorem • the displacement at point i due to a unit load at another point j is equal to the displacement at j due to a unit load at I • Or kij=kji

Proof of Maxwell’s Theorem • According to the theory, δ21= δ12 • Consider two forces, P1 and P2 • First, apply P1 with P2 = 0 . At the final value of P1 , the displacement of point 1 is δ11and the displacement of point 2 is δ21. • Find work done • Now apply second force • Find work done • (see class notes)

Flexibility • If we divide δ by F, we get ‘flexibility’ • This is similar to stiffness (kij) • Note: since δij = δji, kij = kji

Assignment • Solve question #2 on the midterm using Maple (see website)

Eng. 6002 Ship Structures 1 Hull Girder Response Analysis