13.40. COMBIN40 - Combination

Matrix or Vector	Shape Functions	Integration Points
Stiffness, Mass, and Damping Matrices	None (nodes may be coincident)	None

13.40.1. Characteristics of the Element

The force-deflection relationship for the combination element under initial loading is as shown below (for no damping).

Figure 13.10: Force-Deflection Relationship

where:

F₁ = force in spring 1 (output as F1)

F₂ = force in spring 2 (output as F2)

K₁ = stiffness of spring 1 (input as K1 on R command)

K₂ = stiffness of spring 2 (input as K2 on R command)

u_gap = initial gap size (input as GAP on R command) (if zero, gap capability removed)

u_I = displacement at node I

u_J = displacement at node J

F_S = force required in spring 1 to cause sliding (input as FSLIDE on R command)

13.40.2. Element Matrices for Structural Applications

The element mass matrix is:

(13–72)

(13–73)

(13–74)

where:

M = element mass (input as M on R command)

If the gap is open during the previous iteration, all other matrices and load vectors are null vectors. Otherwise, the element damping matrix is:

(13–75)

where:

c = damping constant (input as C on R command)

The element stiffness matrix is:

(13–76)

where:

and the element Newton-Raphson load vector is:

(13–77)

F₁ and F₂ are the current forces in the element.

13.40.3. Determination of F1 and F2 for Structural Applications

If the gap is open,
(13–78)
If no sliding has taken place, F₁ = F₂ = 0.0. However, if sliding has taken place during unidirectional motion,
(13–79)
and thus
(13–80)
where:
u_s = amount of sliding (output as SLIDE)
If the gap is closed and the slider is sliding,
(13–81)
and
(13–82)
where:
u₂ = u_J - u_I + u_gap = output as STR2
If the gap is closed and the slider is not sliding, but had slid before,
(13–83)
where:
u₁ = u₂ - u_s = output as STR1

and
(13–84)

13.40.4. Thermal Analysis

The above description refers to structural analysis only. When this element is used in a thermal analysis, the conductivity matrix is [K_e], the specific heat matrix is [C_e] and the Newton-Raphson load vector is , where F₁ and F₂ represent heat flow. The mass matrix [M] is not used. The gap size u_gap is the temperature difference. Sliding, F_slide, is the element heat flow limit for conductor K₁.