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

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

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?

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.

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.

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

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

Stationary Principle Lagrangian Method –  = U – W

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 …

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

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

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

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

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*

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

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

Continuum systems – bars External Work: Total Potential:

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 v B is the displacement of the mid span. v = z = 0, z = L v = v z = L/2 dv/dz = z = L/2

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

Sample Problem

The potential energy is given by: From the stationary principle of TPE: From Beam Bending Theory