A method for the development and control of stiffness matrices for the calculation of beam and shell structures using the symbolic programming language MAPLE
N. Gebbeken, E. Pfeiffer, I. Videkhina
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering Institute of Engineering Mechanics and Structural Mechanics Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Relevance of the topic
In structural engineering the design and calculation of beam and shell structures is a daily practice. Beam and shell elements can also be combined in spatial structures like bridges, multi-story buildings, tunnels, impressive architectural buildings etc.
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Truss structure, Railway bridge Firth of Forth (Scotland)
Folded plate structure, Church in Las Vegas
Calculation methods
In the field of engineering mechanics, structural mechanics and structural informatics the calculation methods are based in many cases on the discretisation of continua, i.e. the reduction of the manifold of state variables to a finite number at discrete points.
Type of discretisation e.g.:
- Finite Difference Method (FDM)
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
X
Y
x
i,j
i,j+1 i+1,j+1i-1,j+1
i-1,j
i-1,j-1 i,j-1 i+1,j-1
i+1,j
x
y
y
Inside pointsof grid
Outside pointsof grid
Boundary of continuum
Centerpoint
ΔΔ
ΔΔ
i 1 , j i,j
i,j
i , j 1 i,j
i,j
O( x)x x
O( y)y y
f ff
f ff
Differential quotients are substitutedthrough difference quotients
Continuum
u1
v1
u3
v3
u2
v2
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Calculation methods - type of discretisation
- Finite Element Method (FEM)
Static calculation of a concrete panel
First calculation step: Degrees of freedom in nodes.Second calculation step: From the primary unknowns the state variables at the edges of the elements and inside are derived.
Calculation methods - type of discretisation
- Meshfree particle solvers (e.g. Smooth Particle Hydrodynamics (SPH)) for high velocity impacts, large deformations and fragmentation
Aluminiumplate Fragment cloud
Experimental und numeric presentation of a high velocity impact:a 5 [mm] bullet with 5.2 [km/s] at a 1.5 [mm] Al-plate.
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
PD Dr.-Ing. habil. Stefan Hiermaier
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
FEM-Advantages:
Continua can easily be approximated with different elementgeometries (e.g. triangles, rectangles, tetrahedrons, cuboids)
The strict formalisation of the method enables a simple implementation of new elements in an existing calculus
The convergence of the discretised model to the real systembehaviour can be influenced with well-known strategies,e.g. refinement of the mesh, higher degrees of elementformulations, automated mesh adaptivity depending on stress gradients or local errors
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Aspects about FEM
Extensive fundamentals in mathematics (infinitesimal calculus, calculus of variations, numerical integration, error estimation, error propagation etc.) and mechanics (e.g. nonlinearities of material and the geometry) are needed. Unexperienced users tend to use FEM-programmes as a „black box“.
Teaching the FEM-theory is much more time consuming as other
numerical methods, e.g. FDM At this point it is helpful to use the symbolic programming language MAPLE as an eLearning tool: the mathematical background is imparted without undue effort and effects of modified calculation steps or extensions of the FEM-theory can be studied easier!
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
The Finite Element Method (FEM) is mostly used for the analysis of structures.
Basic concept of FEM is a stiffness matrix R which implicates the vector U of node displacements with vector F of forces.
R U F
Of interest are state variables like moments (M), shear (Q) and normal forces (N), from which stresses (, ) and resistance capacities (R) are derived. It is necessary to assess the strength of structures depending on stresses. [ ] [ ]cal allowable
F
A
R
F
l
l l
l
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Static System
Actions
Reaction forces
Deformation of System
M
V
MH
V
H
F
M
V
MH
V
H
1
T
A S F
A U
B S
R U F
Vector S of forces results from the strength of construction.
Vector U of the node displacements depends on the system stiffness.
Structures should not only be resistant to loads, but also limit deformations and be stable against local or global collapse.
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
In the design process of structures we have to take into account not only static actions, but different types of dynamic influences.
Typical threat potentials for structures:
- The stability against earthquakes
- The aerodynamic stability of filigran structures
- Weak spot analysis, risk minimisation
Collapse of the Tacoma Bridge at a wind velocity of 67 [km/h]
Consequences of an earthquake Consequences of wind-inducedvibrations on a suspension bridge
Citicorp Tower NYC
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
( )M U C U R U F t
dynamic problem static problem
- mass (M)- damping (C)- stiffness (R)
The most static and dynamic influences are represented in thefollowing equation:
Mercedes-multistoreyin Munich
wind loading
FEM for the solution of structural problems
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Research goals:
1. The basic purpose of this work is the creation of an universal method for the development of stiffness matrices which are necessary for the calculation of engineering constructions using the symbolic programming language MAPLE.
2. Assessment of correctness of the obtained stiffness matrices.
Short overview of the fundamental equations for the calculation of beam and shell structures
ui ujj
wj
i
wi
Beam structures Shell structures
Differential equation for a single beam4
4
d w q
dx EJ
with w- deflection, EJ- bending stiffness (E- modul of elasticity, J- moment of inertia), x- longitudinal axis, q- line load
Beams with arbitrary loads and complex boundary conditions
2. Theory of second order3 2
3 2,
d q dw d
dx EJ dx dx
44 4
44 , with
4
d w q kbn w n
dx EJ EJ
1. Beam on elastic foundation
with n- relative stiffness of foundation, k- coefficient of elastic foundation, b- broadness of bearing
with - shearing strain
3. Biaxial bending4 2
4 2
d w N d w q
dx EJ dx EJ
with N- axial force
Differential equations for a disc (expressed in displacements)
2 2 2
2 2
2 2 2
2 2
1 10
2 2
1 10
2 2
u u v
x y x y
v v u
y x x y
Differential equation for a plate
4 4 4
4 2 2 42
w w w p
x x y y D
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Calculation of beam structures
For the elaboration of the stiffness matrix for beams the following approach will be suggested:
1. Based on the differential equation for a beam the stiffness matrix is developed in a local coordinate system.
2. Consideration of the stiff or hinge connection in the nodes at the end of the beam.
3. Extension of element matrix formulations for beams with different characteristics, e.g. tension/ compression.
4. Transforming the expressions from the local coordinate system into the global coordinate system.
5. The element matrices are assembled in the global stiffness matrix.
FEM equations Equations from the strength of materials
Type of the development
Beam structure Tension-
compression
Bending without consideration of the
transverse strain
Bending with consideration of the transverse strain
Equation of equlibrium
A S F A
N dA A
M ydA A
M ydA 1
Geometrical relations
TA U du
dx
2
2
du d wy
dx dx mit
du d dwy
dx dx dx
2
Material law 1B S E E E 3
(2) (3) 1 TB A U S du
Edx
2
2
d wE y
dx mit
d dwE y
dx dx
4
1 TA B A U F du
N EAdx
2
2
d wM EJ
dx mit
d dwM EJ
dx dx
5
(4) (1) 1 1( )TA B A F U
du N
dx EA
2
2
d w M
dx EJ mit
M d dw
EJ dx dx
6
(6) (4) 1 1 1( )T TB A A B A F S N
A
My
J
My
J 7
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
R
Development of differential equations of beams with or without consideration of the transverse strain
4
2 3
31' 2
2''
23'''
4
241
0 1 2 36
0 0 2 6
0 0 0 62
0 0 0 0
1
IV
x
w w x x xC xw x xC qw M xC EJ x
w QC
w qx
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Basic equations:4
4
d w q
dx EJ
2
2
d wM EJ
dx
3
3
d wQ EJ
dx
Solution:4
2 31 2 3 4( )
24
qxw x C C C x C x
EJ
Solution and derivatives in matrix form:
Algorithm for the elaboration of a stiffness matrixfor an ordinary beam
D
homogeneous particular
i
j
u qu L C L
u EJ
4
2 3
2 3
1
2
3
4
01 0 0 0 00 1 0 0
1 240 1 2 3
6
i
i
j
j
w C
C qw C EJ
C
ll l l
l l l
L u
LC
4
2 3
31' 2
2''
23'''
4
241
0 1 2 36
0 0 2 6
0 0 0 62
0 0 0 0
1
IV
x
w w x x xC xw x xC qw M xC EJ x
w QC
w qx
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Unit displacements of nodes
Substituting in the first two rows of the matrix D the coordinates for the nodes with x = 0 and x = l we get expressions corresponding to unit displacements of the nodes:
D
or
L1
4
2 3
31' 2
2''
23'''
4
241
0 1 2 36
0 0 2 6
0 0 0 62
0 0 0 0
1
IV
x
w w x x xC xw x xC qw M xC EJ x
w QC
w qx
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Substituting in the second two rows of the matrix D the coordinates for the nodes with x = 0 and x = l follow the shear forces and moments at the ends of a beam corresponding with the reactions:
f i
l
fwi
f j
fwj
Mi
l
Qi
Mj
Qj
Reaction forces and internal forces
22
-6
1
2
3
4
00 0 0 0
00 0 2 0
0 0 0
0 0 2 62
w i i
i i
w j j
j j
f Q C
f M CEJ q
f Q C
f M C
l
ll
f
C
1 1i
j
ff EJ L C q L
f
1L
or
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
1 1qC L u L L
EJ
We express the integration constants by the displacements of the nodes:
i
j
u qu L C L
u EJ
Replacing with deliversC
1 1f EJ L C q L
1 1 1 11 1 1 1 1q
f EJ L L u L L q L EJ L L u q L L L LEJ
or in simplified form:
qf EJ r u q f
rqf
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Within meansqf EJ r u q f
r the relative stiffness matrix with EJ = 1
the relative load column with q = 1 qr
The final stiffness matrix r and the load column for an ordinary beam:qf
wii wj
j
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
44 4
44 , with
4
d w q kbn w n
dx EJ EJ
Solution:
1 2 3 4 4( ) cos( ) sin( ) cos( ) sin( )
4nx nx nx nx q
w x C e nx C e nx C e nx C e nxn EJ
Elaboration of the stiffness matrix for a beam on an elastic foundation
In analogous steps the development of the stiffness matrix for a beam on an elastic foundation leads to more difficult differential equations:
Basic equations:
n relative stiffness of foundation k coefficient of elastic foundationb broadness of bearing
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Elaboration of the stiffness matrix for a beam on an elastic foundation
The final stiffness matrix r and the load column :qf
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Basic equations:
3
3
2
2
d q
dx EJ
dw d
dx dx
dM EJ
dx
2
2
dQ EJ
dx
Solution:
32
1 2 3
4 2
0 1 2 3
( )6
2 3( ) ( 2 )
2 3 24 2
qxx C C x C x
EJ
x x EJ qx qxw x C C x C C x
GF EJ GF
Algorithm for the elaboration of a stiffness matrix for abeam element following the theory of second order
Considering transverse strain the algorithm changes substantially. Instead of only one equation two equations are obtained with the two unknowns bending and nodal distortion:
withEJ
GF (shearing strain)
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
The final stiffness matrix r and the load column for a beam element following the theory of second order:
qr
wii wj
j
resultmatr_r :=
12 E J
l ( )12 l26 E J
12 l2
12 E J
l ( )12 l26 E J
12 l2q l2
6 E J
12 l24 E J ( )3 l2
l ( )12 l2
6 E J
12 l22 E J ( ) 6 l2
l ( )12 l2q l2
12
12 E J
l ( )12 l2
6 E J
12 l212 E J
l ( )12 l2
6 E J
12 l2q l2
6 E J
12 l22 E J ( ) 6 l2
l ( )12 l2
6 E J
12 l24 E J ( )3 l2
l ( )12 l2q l2
12
x
z
m
n
o
m1
o1
n1
x
xz
x
z
m
n
o
m1
o1
n1
Axis of beam(bended)
x
x
Axis of beam(unformed)
Theory of first order Theory of second order
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Single beam Beam on elastic
foundation Harmonic oscillation Biaxial bending Theory of second order
4
4
d w q
dx EJ
4414
41
4 ,
mit4
d w qn w
dx EJ
kbn
EJ
g
4424
2
42
,
mit
d W qn W
dx EJ
Fn
EJ
4 2234 2
3mit
d w d w qn
dx dx EJ
Nn
EJ
3
3
2
2
d q
dx EJ
dw d
dx dx
The formulas of the moment (M) and the shear force (Q)
2
2
d wM EJ
dx
3
3
d wQ EJ
dx
4
4
d wq EJ
dx
2
2
d wM EJ
dx
3
3
d wQ EJ
dx
4414
4d w
q EJ n wdx
2
2
d WM EJ
dx
3
3
d WQ EJ
dx
4424
d Wq EJ n W
dx
2
2
d wM EJ
dx
3 2233 2
d w d wQ EJ n
dx dx
4 2234 2
d w d wq EJ n
dx dx
mitd dw
M EJdx dx
2
2
dQ EJ
dx
3
3
dq EJ
dx
Fundamental equations for the calculation of beam structures used in the development of the stiffness matrix
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Assessment of correctness of the stiffness matrices
Derivations of stiffness matrices are sometimes extensive and sophisticated in mathematics. Therefore, the test of the correctness of the mathematical calculus for this object is an important step in the development process of numerical methods.
There are two types of assessment:
1. Compatibility condition
2. Duplication of the length of the element
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
1. Compatibility condition
i j
x
Element 1 i jElement 2O-x
x
x
0r z r r z r z Fox xji jj ii ij
Equation of equilibrium at point О:
The displacement vectors and can be expressed as Taylor rows: zx
z
x
'
wzo w
in the centre point O
' '' '''2 3 45
' '' 2! ''' 3! 4!
' '' '''2 3 45
' '' 2! ''' 3! 4!
IVw w w w wx x xz x o x
x IV vw w w w w
IVw w w w wx x xz x o x
x IV vw w w w w
' '' '''2 3 4
5 0' '' 2 ''' 6 24
IVw w w w wx x xr r r r x r r r r r r r r o x Fji ii jj ij ji ij ji ij ji ij ji ijIV vw w w w w
After transformation:
rij
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
2. Duplication of the length of the element
x x
i jElement 1 i jElement 2O-x x
Equation of equilibrium at point -x, О, x :
0( )
( ) 0( ) ( )
0( )
r z r z roxii ij iq x
r z r r z r z r rox xji jj ii ij jq o iq o
r z r z ro xji jj jq x
Or in matrix form:
r r rii ij iq x
r r r r r rji jj ii ij jq o iq o
r r rji jj jq x
Rearrangement of rows and columns
Application of Jordan’s method with - new value of element and - initial value of element.
*r r r rij ij i j *rij
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Calculation of shell structures
Wall- like girder Loaded plate Hall roof- like folded plate structure
Panel Plate Folded plate structure
x
y
p – Boundary load in plane
A BA and B – Reaction force in plane
y
x
P
Reaction force Plane
Load Plane
x
zy
Boundary of panel
+ =+ =
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Systematic approach for the development of differential equations for a disc
Type of the development
Equation of equilibrium 0 0yx xy yxxy xyx y x y
1
Geometrical relations x y
u v u v
x y x y
2
Material law
2
2
1
1
1
1
x x y x x y
y y x y y x
E
E
E
E
3
(2) (3)
2 21 1
2 1
x y
E u v E u v
x y x y
E u v
y x
4
(4) (1)
2 2 2
2 2
2 2 2
2 2
1 10
2 2
1 10
2 2
u u v
x y x y
v v u
y x x y
5
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
The system of partial differential equations for discs changes to a system of ordinary differential equations if the displacements are approximated by trigonometric rows:
Inserting the results of this table into equation (5) from the previous tablewe get a system of ordinary differential equations:
2 1 12 02 2 2
21 12 022 2
d V dUV
dydy
d U dVU
dydy
( ) cos withn
u U y xL
( ) sin with
nv V y x
L
( ) sinu
U y xx
( ) cos
vV y x
x
( )cos
u dU yx
y dy
( )
sinv dV y
xy dy
22 ( ) cos
uU y x
x
2
22
( ) sinv
V y xx
2 2
2 2
( )cos
u d U yx
y dy
2 2
2 2
( )sin
v d V yx
y dy
2 ( )sin
u dU yx
x y dy
2 ( )
cosv dV y
xx y dy
Type of the development Bending of plate
Equation of equilibrium 2
2
h
x xh
m zdz
2
2
h
y yh
m zdz
2
2
h
xy xyh
m zdz
1
Geometrical relations w w
u z v zx y
2 2 2
2 22x y xy
w w wz z z
x y x y
2
Material law 21x x y
E
21y y x
E
2 1xy xy
E
3
(2) (3) 2 2
2 2 21x
E w wz
x y
2 2
2 2 21y
E w wz
y x
2
1xy
E wz
x y
4
2 2
2 2x
w wm D
x y
2 2
2 2y
w wm D
y x
2
1xy
wm D
x y
5
(4) (1) 2 2
2 2xmw w
x y D
2 2
2 2
ymw w
y x D
2
1xymw
x y D
6
(6) (4) 3
12 xx
mz
h
3
12 xy
mz
h
3
12 xyxy
mz
h 7
3
2with
12 1
E hD
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Systematic approach for the development of differential equations for a plate
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Systematic approach for the development of differential equations for a plate
Stress and internal force in plate element
x
dxdy
xy
dxdy
mxmy
dxdy
yz
xz
dxdy
qx
qy
dxdy
xyyx
dxdy
mxy
myx
Shearing stress
Torsion with shear
Shear force
Torsional moment
dx
dy
x
z, w(x,y)
p(x,y)
h/2h/2
Equation of equilibriumBalanced forces in z-direction:
yxqq
px y
Balanced moments for x- and y-axis:
xy yxx xy x
m mm mq q
y x x y
Equation of equilibrium after transformations: 2 22
2 22 (1)xy yx
m mmp
x x y y
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
Partial differential equation for a plate:4 4 4
4 2 2 42
w w w p
x x y y D
This changes to an ordinary differential equation if the displacements are approximated by trigonometric rows.
Inserting the results of the table in the above equation we get the ordinary differential equation:
4 4 42 4
4 2 42
d W d W d W p
dx dy dy D
( ) sin withn
w W y xL
( ) cosw
W y xx
( )sin
w dW yx
y dy
22
2( ) sin
wW y x
x
2 2
2 2
( )sin
w d W yx
y dy
2 ( )cos
w dW yx
x y dy
2 ( )
cosw dW y
xx y dy
33
3( ) cos
wW y x
x
3 3
3 3
( )sin
w d W yx
x dy
32
2
( )sin
w dW yx
x y dy
3 2
2
( )cos
w d W yx
y x dy
44
4( ) sin
wW y x
x
4 4
44 4
( )sin
w d W yx
y dy
4 22
2 2 2
( )sin
w d W yx
x y dy
4 2
22 2 2
( )sin
w d W yx
y x dy
University of the German Armed Forces Munich Faculty of Civil and Environmental EngineeringInstitute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering InformaticsUniv.-Prof. Dr.-Ing. habil. N. Gebbeken
- MAPLE permits a fast calculation of stiffness matrices for different element
types in symbolic form
- Elaboration of stiffness matrices can be automated
- Export of the results in other computer languages (C, C++, VB, Fortran) can help to implement stiffness matrices in different environments
- For students‘ education an understanding of algorithms is essential to test different FE-formulations
- Students can develop their own programmes for the FEM
Conclusion: