Truss Elements (TYPE2)

Truss elements are simple two node linear members that only take axial extension or compression. 図 1 shows a truss element.
図 1. Truss Element


Property Input

The only property required by a truss element is the cross-sectional area. This value will change as the element is deformed. The cross sectional area is computed using:
図 2.
A r e a ( t ) = A r e a ( t − Δ t ) ( 1 − v ε ˙ x Δ t ) 2
Where,
ν
Poisson's ratio defined in the material law

Stability

Determining the stability of truss elements is very simple. The characteristic length is defined as the length of the element, that is, the distance between N1 and N2 nodes.
図 3.
Δ t ≤ L ( t ) C
Where,
L ( t )
Current truss length
C = E ρ
Sound speed

Rigid Body Motion

The rigid body motion of a truss element as shown in 図 2 shows the different velocities of nodes 1 and 2. It is the relative velocity difference between the two nodes that produces a strain in the element, namely ex.
図 4. Truss Motion


Strain

The strain rate, as shown in 図 2, is defined as:
図 5.
ε ˙ x = ∂ v x ∂ x = ∂ ( V → ⋅ e → x ) ∂ x

Material Type

A truss element may only be assigned two types of material properties. These are TYPE1 and TYPE2, elastic and elasto-plastic properties, respectively.

Force Calculation

The calculation of forces in a truss element is performed by explicit time integration:
図 6.
Area( t )= Area( t−Δt ) ( 1−v ε ˙ x Δt ) 2
A generalized force-strain graph can be seen in 図 3. The force rate under elastic deformation is given by:
図 7.
F ˙ x = E A ε ˙ x
Where,
E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36C0@
Elastic modulus
A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36C0@
Cross-sectional area
In the plastic region, the force rate is given by:
図 8.
F ˙ x = E t A ε ˙ x
Where,
E t
Gradient of the material curve at the deformation point
図 9. Force-Strain Relationship. (a) without gap; (b) with gap




In a general case, it is possible to introduce a gap distance in the truss definition. If gap is not null, the truss is activated when the length of the element is equal to the initial length minus the gap value. This results a force-strain curve shown in 図 3(b).