16.21 Techniques of Structural Analysis and sig Spring 2003 Unit #1 In this course we are going to focus on energy and variational methods for structural analysis. To understand the overall approach we start by con- trasting it with the alternative vector mechanics approach Example of vector mechanics formulation Consider a simply supported beam subjected to a uniformly-distributed load go,(see Fig. 1). To analyze the equilibrium of the beam we consider the free body diagram of an element of length A as shown in the figure and apply Newton's second law ∑F=0:V-△r=(V+△V)=0 ∑MB=0:-V△x-M+(M+△M)+(w△a) Dividing by△ r and taking the limit△x→0: dM� 16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #1 In this course we are going to focus on energy and variational methods for structural analysis. To understand the overall approach we start by con￾trasting it with the alternative vector mechanics approach: Example of Vector mechanics formulation: Consider a simply supported beam subjected to a uniformly-distributed load q0, (see Fig.1). To analyze the equilibrium of the beam we consider the free body diagram of an element of length Δx as shown in the figure and apply Newton’s second law: Fy = 0 : V − q0Δx − (V + ΔV ) = 0 (1) � Δx MB = 0 : −V Δx − M + (M + ΔM) + (q0Δx) = 0 (2) 2 Dividing by Δx and taking the limit Δx → 0: dV = −q0 (3) dx dM dx = V (4) 1