Internal Forces Computation



Figure 1.

Nodes are numbered from 1 to n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36CE@ , and strands are numbered from 1 to n-1 (strand k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36CE@ goes from node Nk to node Nk+1).

Averaged Force

The averaged force in the multistrand is computed as:

Linear spring F = K L 0 δ + C L 0 δ ˙

Nonlinear spring F = f ( ε ) g ( ε ˙ ) + C L 0 δ ˙

or F = f ( ε ) + C L 0 δ ˙ if g function identifier is 0

or F = g ( ε ˙ ) + C L 0 δ ˙ if f function identifier is 0

Where, ε is engineering strain: ε = L L 0 L 0

Where, L 0 is the reference length of element.

Force Into Each Strand

The force into each strand k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36CE@ is computed as:

F k = F + Δ F k

Where, Δ F k is computed an incremental way:(1)
Δ F k ( t ) = Δ F k ( t 1 ) + K l k 0 δ ε k K L 0 δ ε

with l k 0 the length of the unconstrained strand k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36CE@ , δ ε = ε ( t ) ε ( t 1 ) and δ ε k = δ t u k ( v k + 1 v k ) .

Where, u k is the unitary vector from node Nk to node Nk+1.

Assuming:(2)
l k l k 0 = L L 0

Where, l k is the actual length of strand k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36CE@ .

Therefore, Equation 1 reduces to:(3)
Δ F k ( t )=Δ F k ( t1 )+ K l 0 ( δ ε k L l k δε )

Friction

Friction is expressed at the nodes: if μ is the friction coefficient at node k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaaaa@36E2@ , the pulley friction at node Nk is expressed as:(4)
Δ F k ( t ) = Δ F k ( t 1 ) + K l k 0 δ ε k K L 0 δ ε

When equation Equation 4 is not satisfied, | Δ F k 1 Δ F k | is reset to ( 2 F + Δ F k 1 + Δ F k ) tanh ( β μ 2 ) .

All the Δ F k (k=1, n-1) are modified in order to satisfy all conditions upon Δ F k 1 Δ F k (k=2, n-1), plus the following condition on the force integral along the multistrand element:(5)
k = 1 , n 1 l k ( F + Δ F k ) = L F
This process could fail to satisfy Equation 4 after the Δ F k ( k = 1 , n 1 ) modification, since no iteration is made. However, in such a case one would expect the friction condition to be satisfied after a few time steps.
Note: Friction expressed upon strands (giving a friction coefficient μ along strand k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36CE@ ) is related to pulley friction by adding a friction coefficient μ / 2 upon each nodes Nk and Nk+1.

Time Step

Stability of a multistrand element is expressed as:(6)
Δ t C k 2 + ρ l k K k C k K k , k
with K k = M a s s o f t h e m u l t i s t r a n d L 0 and (assuming Equation 2):(7)
K k = max ( K l k 0 , F l k l k 0 ) = max ( K L l k L 0 , F L l k ( L L 0 ) )
(8)
C k = ( f ( ε ) d g d ε ˙ ( ε ˙ ) + C ) l k 0 = ( f ( ε ) d g d ε ( ε ˙ ) + C ) L l k L 0