AERSP 301

Energy Methods Energy Methods The Stationary PrincipleThe Stationary Principle

Jose Palacios

July 2008

TodayToday

• 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 PrincipleEnergy 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 PrincipleEnergy 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 PrincipleStationary 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 PrincipleStationary 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 PrincipleStationary Principle

• Newtonian Method – Basic Force Equilibrium

– Junction 1:

– Junction 2:

–

Stationary PrincipleStationary Principle

• Lagrangian Method

= U – W

Stationary PrincipleStationary 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 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 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 – barsContinuum 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 Relation Strain Displacement Relation

Note that:

Continuum systems – barsContinuum systems – bars

• Therefore, increment in external work:

• Thus, increment in external work simply reduces to:P

Continuum systems – barsContinuum 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 – barsContinuum systems – bars

• dU and dU* are due to a small (incremental) strain dxx (or displacement du)

= strain energy per unit volume

Continuum systems – barsContinuum 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 – barsContinuum systems – bars

• External Work:

• Total Potential:

Sample ProblemSample 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.

L

zvv B

sin

v = 0 @ z = 0, z = Lv = vB @ z = L/2

dv/dz = 0 @ z = L/2

Sample ProblemSample Problem

• The strain energy, U, due to bending of a beam is given by (Given in the problem)

2

2

2

2

1

dz

vdEIM

dzEI

MUL

From Chapter 16, beam bending lectures

Sample ProblemSample Problem

L

zvv B

sin

2

2

2

2

1

dz

vdEIM

dzEI

MUL

3

24

0

24

42

4

sin2

L

EIvU

dzL

z

L

vEIU

B

LB

L

z

L

v

dzLz

vd

dz

vd BB

sinsin

2

2

2

2

2

2

Sample ProblemSample Problem

• The potential energy is given by:

• From the stationary principle of TPE:

BB

B WvL

EIvvVUTPE

3

24

4)(

0

2 3

4

W

L

EIv

v

VU B

B

EI

WL

EI

WLv

EI

WL

EI

WLv

B

pBs

33

3

4

3

02083.048

02053.02

From Beam Bending Theory