1. AERSP 301Energy Methods The Stationary Principle Jose Palacios July 2008

2. Today • Due dates - Reminders • HW 4 due Tuesday, July 22, by 2:00 pm • HW 5 uploaded, due Thursday July 24 by 2:00 pm • No Class on Friday July 25 • Class Next Sat. July 26, 8:00 am • Exam Tuesday July 29: • Stationary Principle • Torsion of Cells • Structure Idealization • Shear of beams (Open – Closed Sections) • Bending of beams (Open – Closed Sections) • Aircraft Loads (Plane Stress) • Vocabulary Definitions • Energy Methods – Stationary Principle of Total Potential Energy • Ch. 5.7

3. Energy Methods & The Stationary Principle • Energy Methods (Lagrangian Methods) vs. Newtonian Methods (based on Force/Moment Equilibrium) • Here we define Strain Energy and External Work (also Kinetic Energy, for dynamic problems) • What is the difference between rigid and elastic bodies? • No Strain in rigid body (idealization, no body is rigid) • Strain in elastic body • Is there strain energy associate with “rigid” bodies? … “elastic” structures? • What is Kinetic Energy? • How doe a rigid body behave under the application of loads? • Can it undergo translation? Rotation? Elastic deformation? • How does the behavior of an elastic body under the application of loads differ?

4. Energy Methods & The Stationary Principle • When a force is applied to an elastic body, work is done. That work is stored as energy (Strain Energy) • Consider the following case: • Work done by force, F, as u (instantaneous displacement) goes from 0  q.

5. Stationary Principle Stationary Principle, or Principle of Minimum Total Potential Energy • The external work potential is defined as: • Define a scalar function (q) – Total Potential Energy • For the spring problem The Stationary Principle states that among all geometrically possible displacements, q, (q) is a minimum for the actual q.

6. Stationary Principle • For the spring problem, minimize : • The force equilibrium equation obtained, Kq = F, as a result of using Energy Methods is the same as what you would have obtained using Newtonian Methods. So the two methods are equivalent. • Now examine a 2-Spring System, and develop the equilibrium equations using the two different (Newtonian and Lagrangian) Methods

7. Stationary Principle • Newtonian Method – Basic Force Equilibrium • Junction 1: • Junction 2:

8. Stationary Principle • Lagrangian Method •  = U – W

9. Stationary Principle • Use Stationary Principle: • As with the single-spring example, the equations are identical using either method. • What are the advantages, then, of using Energy Methods? • Energy being a scalar … • Advantageous for larger systems …

10. Continuum systems – bars • Consider a bar under an uni-axial load, undergoing uni-axial displacement, u(x). • Boundary Conditions? • The bar is a continuous structure (how many degrees of freedom does it have? Compare to the single-spring and the two spring examples covered) • Note the difference between a: • bar – loaded axially • beam – loaded transversely

11. Continuum systems – bars • To determine the strain energy, start by considering a small segment of the bar of length dx • Force Equilibrium:  Force equilibrium relation

12. Continuum systems – bars Stress – Strain Relation Strain Displacement Relation • Consider an increment in external work by the applied force associated with a displacement increment, du. • Increment in external work dW Note that:

13. Continuum systems – bars • Therefore, increment in external work: • Thus, increment in external work simply reduces to: P

14. Continuum systems – bars • Comparing expressions A and B, it can be seen that: Increment in external work by applied force, dW Increment in stored strain energy dU Increment in strain energy per unit volume, dU*

15. Continuum systems – bars • dU and dU* are due to a small (incremental) strain dxx (or displacement du) = strain energy per unit volume

16. Continuum systems – bars • The strain energy stored in the entire bar: • Strain energy, U, for a uni-axial bar in extension • Recall, for a spring • For rigid body translation

17. Continuum systems – bars • External Work: • Total Potential:

18. Sample Problem • Simply supported beam with stiffness EI. Determine the deflection of the mid-span point using the stationary principle: • The assumed displacement must satisfy the boundary conditions. • Polynomial functions are the most convent to use. • Simpler assumed solutions are less precise. • Step 1: Assume a displacement • Where vB is the displacement of the mid span. v = 0 @ z = 0, z = L v = vB @ z = L/2 dv/dz = 0 @ z = L/2

19. Sample Problem • The strain energy, U, due to bending of a beam is given by (Given in the problem) From Chapter 16, beam bending lectures

20. Sample Problem

21. Sample Problem • The potential energy is given by: • From the stationary principle of TPE: From Beam Bending Theory