Date post: | 01-Jan-2016 |
Category: |
Documents |
Upload: | sibyl-armstrong |
View: | 217 times |
Download: | 3 times |
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