1. One-Dimensional Problems
MCE 561 Computational Methods in Solid Mechanics
One-Dimensional Problems

2. One-Dimensional Bar Element
Axial Deformation of an Elastic Bar x
f(x) = Distributed Loading
A = Cross-sectional Area E = Elastic Modulus
Typical Bar Element W
(i) L
(j)
(Two Degrees of Freedom)
Virtual Strain Energy = Virtual Work Done by Surface and Body Forces
For One-Dimensional Case

3. One-Dimensional Bar Element

4. Linear Approximation Scheme
x (local coordinate system)
(1)
(2) L x
(1)
(2)
u(x)
y1(x)
y2(x) 1 x
(1)
(2)
yk(x) – Lagrange Interpolation Functions

5. Element Equation Linear Approximation Scheme, Constant Properties

6. Quadratic Approximation Scheme
(1)
(3)
(2) L
u(x) x
(1)
(3)
(2)
y2(x)
y3(x)
y1(x) 1 x
(1)
(2)
(3)

7. Lagrange Interpolation Functions Using Natural or Normalized Coordinates
(1)
(2)
(1)
(2)
(3)
(1)
(2)
(3)
(4)

8. Simple Example P
A1,E1,L1
A2,E2,L2
(1)
(3)
(2) 1 2

9. Simple Example Continued
A1,E1,L1
A2,E2,L2
(1)
(3)
(2) 1 2

10. One-Dimensional Beam Element
Deflection of an Elastic Beam
I = Section Moment of Inertia E = Elastic Modulus
f(x) = Distributed Loading x W
(1)
(2)
Typical Beam Element L
(Four Degrees of Freedom)
Virtual Strain Energy = Virtual Work Done by Surface and Body Forces

11. Beam Approximation Functions
To approximate deflection and slope at each node requires approximation of the form
Evaluating deflection and slope at each node allows the determination of ci thus leading to

12. Beam Element Equation

13. FEA Beam Problem f
Uniform EI a b
(1)
(3)
(2) 1 2

14. FEA Beam Problem (1) (3) (2) 1 2
Solve System for Primary Unknowns U3 ,U4 ,U5 ,U6
Nodal Forces Q1 and Q2 Can Then Be Determined

15. Special Features of Beam FEA
Analytical Solution Gives Cubic Deflection Curve
Analytical Solution Gives Quartic Deflection Curve
FEA Using Hermit Cubic Interpolation Will Yield Results That Match Exactly With Cubic Analytical Solutions

16. Truss Element Generalization of Bar Element With Arbitrary Orientation
k=AE/L x

17. Frame Element
Generalization of Bar and Beam Element with Arbitrary Orientation W
(1)
(2) L
Element Equation Can Then Be Rotated to Accommodate Arbitrary Orientation