CHAP 6 Finite Elements for Plane Solids
307
3. Using two CST elements, solve the simple shear problem described in the figure and
determine whether the CST elements can represent the simple shear condition
accurately or not. Material properties are given
310
Finite Element Analysis and Design
5. A structure shown in the figure is approximated with one triangular element. Plane
strain assumption is used.
(a) Calculate the straindisplacement matrix [B].
(b) When nodal displacements are given by cfw_u1, v1,
CHAP 6.
FINITE ELEMENTS FOR PLANE SOLIDS
1. Repeat Example 6.2 with the following element connectivity:
Element 1: 124
Element 2: 234
Does the different element connectivity change the results?
Solution:
(1) Element 1: Nodes 124
Using nodal coordinates, w
216
Finite Element Analysis and Design
4. The right end of a cantilevered beam is resting on an elastic foundation that can be
represented by as spring with spring constant k = 1,000 N/m. A force of 1,000 N acts
at the center of the beam as shown. Use the
CHAP 4 Finite Element Analysis for Beams and Frames
213
2. The deflection of the simply supported beam shown in the figure is assumed as
v(x ) cx (x 1) , where c is a constant. A force is applied at the center of the
beam. Use the following properties: EI
214
Finite Element Analysis and Design
3. Use the Rayleigh-Ritz method to determine the deflection v(x), bending moment
M(x), and shear force Vy(x) for the beam shown in the figure. The bending moment
and shear force are calculated from the deflection as:
CHAP 3 Weighted Residual and Energy Methods
191
14. A vertical rod of elastic material is fixed at both ends with constant cross-sectional
area A, Youngs modulus E, and height of L under the distributed load f per unit
length. The vertical displacement u(
CHAP 3 Weighted Residual and Energy Methods
189
13. Consider a finite element with three nodes, as shown in the figure. When the solution
is approximated using u(x ) N 1(x )u1 N 2 (x )u2 N 3 (x )u3 , calculate the
interpolation functions N1(x ), N2 (x ),
CHAP 3 Weighted Residual and Energy Methods
197
16. Consider a tapered bar of circular cross section. The length of the bar is 1 m, and the
radius varies as r (x ) 0.050 0.040x , where r and x are in meters. Assume
Youngs modulus = 100 MPa. Both ends of t
CHAP 3 Weighted Residual and Energy Methods
187
3
12. Repeat Problem 11 by assuming w(x )
ci i (x ) c1x 2 c2x 3 c3x 4
i 1
Solution:
Using the second derivatives of the two trial functions, 1 2, 2 6x, 3 12x 2 , we
can calculate the coefficient matrix and
170
Finite Element Analysis and Design
4. One-dimensional heat conduction problem can be expressed by the following
differential equation:
k
d 2T
Q 0,
dx 2
0x L
where k is thermal conductivity, T (x ) temperature, and Q heat source per unit
length. Q , t
168
Finite Element Analysis and Design
3. Using the Galerkin method, solve the following differential equation with the
approximate solution in the form of u(x ) c1x c2x 2 . Compare the approximate
solution with the exact one by plotting them on a graph.
CHAP 3 Weighted Residual and Energy Methods
185
11. The boundary value problem for a cantilevered beam can be written as
d 4w
dx 4
p(x ) 0,
w(0)
0x 1
dw
d 2w
d 3w
(0) 0,
(1) 1,
(1) 1: boundary condtions
dx
dx 2
dx 3
p(x ) x . Assuming the approximate de
124
Finite Element Analysis and Design
22. Determine the normal stress in each member of the truss structure. All joints are
balljoint and the material is steel whose Youngs modulus is E = 210 GPa.
Y
A
b
8 kN
a
a = b = 10 cm
6m
C
X
B
Z
5m
3m
D
6m
Solution
116
Finite Element Analysis and Design
18. Use FEM to solve the plane truss shown below. Assume AE = 106 N, L = 1 m.
Determine the nodal displacements, forces in each element and the support reactions.
1
y
x
1
L
2
L
2
L
4
3
10,000 N
3
Solution:
Connectivi
118
Finite Element Analysis and Design
19. The plane truss shown in the figure has 2 elements and 3 nodes. Calculate the 44
element stiffness matrices. Show the row addresses clearly. Derive the final
equations (after applying boundary conditions) for the
92
Finite Element Analysis and Design
6. In the structure shown, rigid blocks are connected by linear springs. Imagine that
only horizontal displacements are allowed. Write the structural equilibrium equations
[K]cfw_Q cfw_F after applying displacement bo
CHAP 2 Uniaxial Bar and Truss Elements
101
12. A stepped bar is clamped at one end, and subjected to concentrated forces as shown.
Note: the node numbers are not in usual order!
3
5 kN
1m
2
1
2 kN
1m
Assume: E=100 GPa, Small area of cross section =1 cm2,
98
Finite Element Analysis and Design
10. The stepped bar shown in the figure is subjected to a force at the center. Use FEM to
determine the displacement at the center and reactions RL and RR.
Assume: E = 100GPa, area of cross sections of the three porti
AE 429 Performance and Flight Mechanics
Pier Marzocca Pier CAMP 234, MAE Department MW 1:00 - 3:00, CAMP 234 or by appointment (315) 268-3875
Electronic Information
pmarzocc@clarkson.edu http:/www.clarkson.edu/~pmarzocc/
Detailed Outline
Introduction Stan
AE 429 - Aircraft Performance and Flight Mechanics
Takeoff and Landing
Takeoff Performance
V=0 (s=0) V=Vstall V=Vmcg (min control speed on the ground) V=Vmca (min control speed in the air (w/o landing gear in contact with ground) V=V1 decision speed (or c
Finite wings
Infinite wing (2d) versus finite wing (3d)
Definition of aspect ratio:
AR b2 S AR = b c
For rectangular platform
Symbol changes:
Cl CL Cd CD Cm CM
Vortices and wings
What the third dimension does
Difference between upper and lower pressure re
AE 429 - Aircraft Performance and Flight Mechanics
Level Turn, Pull Up and Pull Down
Turning Performance
What is a turn?
r
R
Center of curvature
a turn is a change in flight path direction turn rate is the time rate of change in heading
Lim
t 0 t
1
Tu
AE 429 - Aircraft Performance and Flight Mechanics
Rate of Climb Time to Climb
Rate of Climb R/C
Now lets analyze a steady climb
Forces include a gravity component now
dV m = T cos D W sin dt
m
2 V = L cos + T sin cos W cos r1
T = D + W sin L = W cos
V
T
AE 429 - Aircraft Performance and Flight Mechanics
Range and Endurance
Range and Endurance
Definitions
Range
Total ground distance traversed on a full tank of fuel -Anderson book Distance an airplane can fly on a given amount of fuel -alternative definiti
AE 429 - Aircraft Performance and Flight Mechanics
Equations of Motion
Performance equations
Forces considered
Lift, drag, thrust, and weight
Drag polar known Weight known Thrust available known
Lift is perpendicular to flight path Drag is parallel to f
AE 429 - Aircraft Performance and Flight Mechanics
Power Required and Available
Power
Power is energy per unit time
POWER = P FORCE * DISTANCE = FV if V is constant TIME
For an airplane in level, unaccelerated flight, power PR = TRV required is
PR = TRV =
AE 429 - Aircraft Performance and Flight Mechanics
Steady Flight
Aerodynamic efficiency Lift/drag ratio is a measure of aerodynamic efficiency
It indicates the ability to produce lift without generating L excessive drag L/D
MAX L/D 0.3-0.4 0.312.8 26 VEHI
AE 429 - Aircraft Performance and Flight Mechanics
Aerodynamics of the Airplane
Airfoils
An airfoil is a section of a wing (or a fin, or a stabilizer, or a propeller, etc.)
Cambered
Symmetrical Laminar Flow
Reflexed
Supercritical
1
Airfoil Nomenclature