3. Direct approach – truss analysis

Truss analysis

•Bar – a member in a truss structure• Forces in the bars are either tensile

or compressive – like a spring

3.1 Simple structure consisting of springs- a systematic solution

• Structure

• Element mesh

• Structure + mesh

• Elastic springs• External forces, (N)• Displacements, (m)

• ki spring stiffness, (N/m) • Elements• Nodal points

• Degrees of freedom (DOF)(displacements u1, u2, u3)

1. Element stiffness relation

• Definitions of one element

• Introduce element forces, P1 and P2

Normal force, N(positive in tension) du = u2 - u1

F=k du

P1 = -NP2 = N

, Eq. 1

, Eq. 2

Insert Eq.1 in Eq. 2: In matrix form:

1. Element stiffness relation

• Matrix form

or

where

Ke : element stiffness matrixae : nodal displacement vectorfe : element force vector

From the CALFEM-manual

From the CALFEM-manual

Analogous physical problems

2. Compatibility

Global definitions

Local definitions

Element 1:

Element 2:

Introduce global definitions,(compatibility):

Element 1:

Element 2:

Expanded element formulation:

3. Equilibrium conditions

• Forces acting on each nodal point

Cut the structure at nodes and insert section forces:

Force equilibrium: In matrix form:

4. Assembling

Force equilibrium in matrix form:

Expanded element form:

Element 1:

Element 2:

, Eq. 1

, Eq. 2

Insert Eq.2 in Eq. 1:

4. Assembling

or

where

K : global stiffness matrixa : global displacement vectorf : external force vector

Wich results in the system of linear equations:

5. Boundary conditions

No unique solution:

Introduce boundary conditions:Example: u1 = 0

0

=0

6. Solution

• Reduced system:

• Solution:

Direct assembling

Example element 4:Between dof 2 and 4

Direct assembling.From the CALFEM-manual

• Transformation of displacement vector

• Transformation of force vector

• where

• which gives

3.2 Truss analysis- transformation to global coordinates (see book for details)

1u

2u

3u

4u

x

1u

2u 3u

4u

x

y

Local system

Global system

eee faK

eeefaK

1u

2u1u2u

ILLT

From the CALFEM-manual

Steps in the FE method