MECH4450 Introduction to Finite Element Methods Chapter 3 FEM of 1-D Problems: Applications

Plane Truss Problems Example 1: Find forces inside each member. All members have the same length L. E = 10 GPa, A = 1 cm 2, L = 1 m, F = 10 kN F

Arbitrarily Oriented 1-D Bar Element on 2-D Plane P1 u1P1 u1 P2 u2P2 u2

x y 

Stiffness Matrix of 1-D Bar Element on 2-D Plane Q 2, v 2  P 2, u 2 Q 1, v 1 P 1, u 1

Matrix Assembly of Multiple Bar Elements Element I I I

Matrix Assembly of Multiple Bar Elements Element I I I

Matrix Assembly of Multiple Bar Elements Apply known boundary conditions

Solution Procedures u 2 = 4FL/5AE, v 1 = 0

Recovery of Axial Forces Element I I I

Stresses inside members Element I I I

Governing Equation and Boundary Condition Governing Equation Boundary Conditions <x<L q(x) x y

Weak Formulation for Beam Element Weighted-Integral Formulation for one element V(x 2 ) x = x 1 M(x 2 ) q(x) y x x = x 2 V(x 1 ) M(x 1 ) L = x 2 -x 1 Weak Form from Integration-by-Parts

Weak Formulation Weak Form Q3Q3 x = x 1 Q4Q4 q(x) y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1

Ritz Method for Approximation Let w(x)=  i (x), i = 1, 2, 3, 4 Q3Q3 x = x 1 Q4Q4 q(x) y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1 where

Derivation of Shape Function for Beam Element In the global coordinates:

Element Equations of 4 th Order 1-D Model u3u3 x = x 1 u4u4 q(x) y(v) x x = x 2 u1u1 u2u2 L = x 2 -x 1 x=x 2 x=x 1 1 3 3 2 2 4 4

Element Equations of 4 th Order 1-D Model u3u3 x = x 1 u4u4 q(x) y(v) x x = x 2 u1u1 u2u2 L = x 2 -x 1

Finite Element Analysis of 1-D Problems Example 1. Finite element model: P 1, v 1 P 2, v 2 P 3, v 3 P 4, v 4 M 1,  1 M 2,  2 M 3,  3 M 4,  4 I II III Discretization: F L L L

Matrix Assembly of Multiple Beam Elements Element I I

Matrix Assembly of Multiple Beam Elements Element I I

Solution Procedures Apply known boundary conditions

Solution Procedures

Shear Resultant & Bending Moment Diagram

Plane Frame Frame: combination of bar and beam E, A, I, L Q 1, v 1 Q 3, v 2 Q 2,  1 P 1, u 1 Q 4,  2 P 2, u 2

Finite Element Model of an Arbitrarily Oriented Frame  x y  x y

local global

Plane Frame Analysis - Example Rigid Joint Element II Element I F F

Plane Frame Analysis P 1, u 1 P 2, u 2 Q 2,  1 Q 4,  2 Q 1, v 1 Q 3, v 2

Plane Frame Analysis P 1, u 2 Q 3, v 3 Q 2,  2 Q 4,  3 Q 1, v 2 P 2, u 3

Plane Frame Analysis