Institute of Structural Engineering Page 1
Method of Finite Elements I
Chapter 2
The Direct Stiffness Method
Method of Finite Elements I
Institute of Structural Engineering Page 2
Method of Finite Elements I
Direct Stiffness Method (DSM)
• Computational method for structural analysis
• Matrix method for computing the member forces
and displacements in structures
• DSM implementation is the basis of most commercial
and open-source finite element software
• Based on the displacement method (classical hand
method for structural analysis)
• Formulated in the 1950s by Turner at Boeing and
started a revolution in structural engineering
Institute of Structural Engineering Page 3
Method of Finite Elements I
Goals of this Chapter
• DSM formulation
• DSM software workflow for …
• linear static analysis (1st order)
• 2nd order linear static analysis
• linear stability analysis
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Method of Finite Elements I
Computational Structural Analysis
Modelling is the most important step in
the process of a structural analysis !
X
Y
Physical problem Continuous
mathematical modelDiscrete
computational model
strong form weak form
Institute of Structural Engineering Page 5
Method of Finite Elements I
System Identification (Modelling)
Global Coordinate System
Nodes
Elements
Boundary conditions
Loads
X
Y
1
3 4
2
5
6
1 2
3
4
Element numbers
and orientation
Node numbers
5 6
Institute of Structural Engineering Page 6
Method of Finite Elements I
Deformations
System Deformations
System identification
Nodal Displacements
nodes, elements, loads and supports
deformed shape
(deformational, nodal)
degrees of freedom = dofs
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Method of Finite Elements I
Degrees of Freedom
7 * 2 = 14 dof
Frame Structure
8 * 3 = 24 dof
Truss Structure
ui = ( udx , udy )
uiui
ui = ( udx , udy , urz )dof per node
dof of structure
Institute of Structural Engineering Page 8
Method of Finite Elements I
Elements: Truss
compatibility
ux = displacement in direction
of local axis X
e =𝐷𝑋
𝐿
s = 𝐸 econst. equation
equilibrum
𝑁 = ʃ 𝐸 s = 𝐸𝐹 s =𝐸𝐹
𝐿𝐷𝑋
𝑃2 = −𝑃1 = 𝑁
𝐷𝑋 = (u2−u1)
P2P1
𝑃1 =𝐸𝐹
𝐿(u1−u2)
𝑃2 =𝐸𝐹
𝐿(−u1 + u2)
p = k u
p : (element) stiffness matrix
k : (element) nodal forces
u : (element) displacement vector
1 dof per node
DX
𝐿, 𝐸, 𝐹N
P1 P2
ux
X/Y = local coordinate system
DX = displacement of truss end
Institute of Structural Engineering Page 9
Method of Finite Elements I
Elements: Beam3 dof per node
DX
DY
RZ
𝐿, 𝐸, 𝐹
uy
uy
ux
k u
ux = displacement in direction
of local axis X
uy = displacement in direction
of local axis Y
Institute of Structural Engineering Page 10
Method of Finite Elements I
Elements: Global Orientation
local
global
uglob = u = R uloc
𝑅 𝜃 =
cos 𝜃 − sin 𝜃 0 0 0sin 𝜃 cos 𝜃 0 0 00 0 1 0 00 0 cos 𝜃 − sin 𝜃 00 0 sin 𝜃 cos 𝜃 00 0 0 0 1
𝜃
kglob = k = RT kloc R
Institute of Structural Engineering Page 11
Method of Finite Elements I
Beam Stiffness Matrixe.g. k24 =
reaction
in global direction Y
at start node S
due to a
unit displacement
in global direction X
at end node E
UXE=1
FYS
S
EFXS =
FYS =
MZS =
FXS =
FYS =
MZE =
UXS UYS UZS UXE UYE UZE
k14 k15 k16
k24 k25 k26
k34 k35 k36
k44 k45 k46
k55 k56
k66
k11 k12 k13
k22 k23
k33
symm.
Element stiffness matrix
in global orientation
iE
iS
EEES
SESS
iE
iS
u
u
p
p
ii
ii
kk
kk
p = k u
Institute of Structural Engineering Page 12
Method of Finite Elements I
Nodal Equilibrum
3 4
2
5
6
f4
r4: Vector of all forces acting at node 4
r4 = - k6ES u3 + contribution of element 6 due tostart node displacement u3
- k6EE u4 + contribution of element 6 due toend node displacement u4
- k5EE u4 + contribution of element 5 due tostart node displacement u4
- k5ES u2 + contribution of element 5 due tostart node displacement u2
f4 external load
Equilibrum at node 4: r4 = - k5SE u2 -k6ES u3 - k5EE u4 - k6EE u4 + f4 = 0
Institute of Structural Engineering Page 13
Method of Finite Elements I
Global System of Equations
r1 = -
u1
r2 = -
r3 = -
r4 = -
u2 u3 u4
k5ES k6ES k5EE+
k6EE
1
3 4
2
5
6
1 2
3
4
k1EE+
k3SS+
k4SS
k3SE k4SE
k3ES k2EE+
k3EE+
k5SS
k5SE
k4ES k4EE+
k6SS
k6SE
+ f1 = 0
+ f2 = 0
+ f3 = 0
+ f4 = 0
- K U + F = 0 F = K U
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Method of Finite Elements I
K = global stiffness matrix = Assembly of all ke
F = K U
Global System of Equations
= equilibrium at every node of the structure
F = global load vector = Assembly of all fe
U = global displacement vector = unknown
Institute of Structural Engineering Page 15
Method of Finite Elements I
Solving the Equation System
K U = F
U = K-1 F
What are the nodal displacements for
a given structure (= stiffness matrix K )
due to a given load (= load vector F ) ?
K-1left multiply
K-1 K U = K-1 F
Inversion possible only if K is non-singular
(i.e. the structure is sufficiently supported = stable)
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Method of Finite Elements I
Beam Element Results
2. Element end forces
Calculate element end forces = p = k u
4. Element deformations along axis
1. Element nodal displacements
Disassemble u from resulting global displacements U
3. Element stress and strain along axis
Calculate moment/shear from end forces (equilibrium equation)
Calculate curvature/axial strain from moments/axial force
Calculate displacements from strain (direct integration)
Institute of Structural Engineering Page 17
Method of Finite Elements I
Lateral Load
1. Adjust global load vector
f = local load vector => add to global load vector F
2. Adjust element stresses
M due to u
M due to f M diagrame.g. bending moment M:
Institute of Structural Engineering Page 18
Method of Finite Elements I
Linear Static Analysis (1st order)
Workflow of computer program
1. System identification: Elements, nodes, support and loads
2. Build element stiffness matrices and load vectors
3. Assemble global stiffness matrix and load vector
4. Solve global system of equations (=> displacements)
5. Calculate element results
Exact solution for displacements and stresses
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Method of Finite Elements I
2nd Order Effectsor the influence of the axial normal force
Normal forces change the stiffness of the structure !
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Method of Finite Elements I
Geometrical Stiffness Matrix
kG = geometrical stiffness matrix of a truss element
p = ( k + kG ) u
Very small element rotation
=> Member end forces (=nodal forces p )
perpendicular to axis due to initial N
Truss
NOTE:
It’s only a
approximation
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Method of Finite Elements I
Beams: Geometrical Stiffness
kG = geometrical stiffness matrix of a beam element
kG =
Institute of Structural Engineering Page 22
Method of Finite Elements I
Linear Static Analysis (2nd order)
Global system of equations
( K + KG ) U = F U = ( K + KG )-1 F
Inversion possible only if K + KG is non-singular, i.e.
- the structure is sufficiently supported (= stable)
- initial normal forces are not too big
What are the 2nd order nodal displacements for
a given structure due to a given load ?
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Method of Finite Elements I
Linear Static Analysis (2nd order)
Workflow of computer program
1. Perform 1st order analysis
2. Calculate resulting axial forces in elements (=Ne)
3. Build element geometrical stiffness matrices due to Ne
4. Add geometrical stiffness to global stiffness matrix
5. Solve global system of equations (=> displacements)
6. Calculate element results
NOTE: Only approximate solution !
Institute of Structural Engineering Page 24
Method of Finite Elements I
Stability Analysis
How much can a given load be increased until a
given structure becomes unstable ?
(K + λmax KG0) U = F
Nmax = λmax N0
KG = f(Nmax)KG(Nmax) = λmax KG(N0) = λmax KG0
2nd order analysis No additional load possible
(K + λmax KG0) ΔU = ΔF = 0
linear algebra
(A - λ B) x = 0 Eigenvalue problem
Institute of Structural Engineering Page 25
Method of Finite Elements I
Stability Analysis
Eigenvalue problem
(A - λ B) x = 0
λ = eigenvalue
x = eigenvector
(K - λ KG0) x = 0
λ = critical load factor
x = buckling mode
e.g. Buckling of a column
λ N0
λ F
x
Solution
Institute of Structural Engineering Page 26
Method of Finite Elements I
Stability Analysis
Workflow of computer program
1. Perform 1st order analysis
2. Calculate resulting axial forces in elements (=N0)
3. Build element geometrical stiffness matrices due to N0
4. Add geometrical stiffness to global stiffness matrix
5. Solve eigenvalue problem
NOTE: Only approximate solution !