3.7 Kinematic Constraints

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As in the previous sections, rectangular coordinates are assumed throughout and the spatial

reference system coincides with the material reference system.

3.7.1 Points on a straight line

Occasionally, one comes across the condition that points must stay on a straight line. An

example of such a case is a hinge consisting of nodes on a line (Figure 3.21). The line

itself can move in space. A node p lies on the straight line defined by distinct nodes a and

b if

p = a + λ(b − a), λ ∈ R (3.114)

which is equivalent to

xp − xa = λ(xb − xa) (3.115)

yp − ya = λ(yb − ya) (3.116)

zp − za = λ(zb − za) (3.117)

172 GEOMETRIC NONLINEAR EFFECTS

where (x, y, z) are coordinates in the deformed configuration. Since a and b do not coincide,

at least one of the right-hand sides in Equations (3.115) to (3.117) is nonzero. Let xb = xa

and a and b be such that xa = xb is highly improbable throughout the complete deformation,

then, one obtains by solving for λ in Equation (3.115) and substituting into Equation (3.116)

and (3.117),

(yp − ya)(xb − xa) = (xp − xa)(yb − ya) (3.118)

(zp − za)(xb − xa) = (xp − xa)(zb − za). (3.119)

Since

xa = Xa + ua (3.120)

where Xa is the x-coordinate of node a in the undeformed configuration and ua is its

displacement in x-direction and similarly for the other coordinates,

ya = Ya + va (3.121)

za = Za + wa. (3.122)

Equations (3.118) and (3.119) are a set of two nonlinear equations in ua, va,wa, ub, vb,wb

and up, vp,wp. Denoting Equation (3.118) by

f (vp, up, va, ua, vb, ub) = 0 (3.123)

and since (Equation (3.120))

∂f

∂ua

= ∂f

∂xa

∂xa

∂ua

= ∂f

∂xa

(3.124)

and similarly for the other variables, linearization of Equation (3.123) at (v0p

, u0

p, v0

a, u0a

,

v0

b, u0

b) yields

f (v0p

, u0

p, v0

a, u0a

, v0

b, u0

b) + ∂f

∂vp

____

0

(vp − v0p) + ∂f

∂up

____

0

(up − u0

p) + ∂f

∂va

____

0

(va − v0p

)

+ ∂f

∂ua

____

0

(ua − u0

p) + ∂f

∂vb

____

0

(vb − v0p

) + ∂f

∂ub

____

0

(ub − u0

p) ≈ 0 (3.125)

where

∂f

∂vp

____

0

= x0

b

− x0

a (3.126)

∂f

∂up

____

0

= −(y0

b

− y0

a ) (3.127)

GEOMETRIC NONLINEAR EFFECTS 173

∂f

∂va

____

0

= −(x0

b

− x0p

) (3.128)

∂f

∂ua

____

0

= −(y0p

− y0

b ) (3.129)

∂f

∂vb

____

0

= −(x0p

− x0

a ) (3.130)

∂f

∂ub

____

0

= y0p

− y0

a (3.131)

and

x0

a := Xa + u0a

(3.132)

(similarly for the other coordinates). An analogous procedure can be applied to

Equation (3.119). In the present case, vp and wp are suitable selections for the dependent

variables since x0

b

= x0

a is assumed. It is advantageous to select a and b at an appreciable

distance from each other in order to improve the accuracy. Each node p constrained to lie

on the line defined by the nodes a and b will lead to two of the above equations.

3.7.2 Points in a plane

The treatment of points constrained to lie in a plane is somewhat similar to the derivation

in the previous section. Let the plane α be defined by three nodes a, b and c, which are

not colinear, that is,

m := (b − c) × (a − c) = 0. (3.133)

A node p lies in the plane if

m ・ (p − c) = 0. (3.134)

Introducing spatial coordinates (x, y, z), Equation (3.134) is equivalent to

f =

______

xp − xc yp − yc zp − zc

xa − xc ya − yc za − zc

xb − xc yb − yc zb − zc

______

= 0. (3.135)

The vertical lines denote the determinant of the 3 ×3 matrix. f is a nonlinear equation in

ua, va,wa, ub, vb,wb, uc, vc,wc, up, vp and wp since

xa = Xa + ua (3.136)

and similarly for the other coordinates. The derivatives of f at (u0a

, v0

a,w0

a, u0

b, . . . ,w0p

)

with respect to ua, va,wa, ub, vb,wb, up, vp and wp are the corresponding cofactors,

174 GEOMETRIC NONLINEAR EFFECTS

that is,

∂f

∂ua

____

0

= −_

_

__

y0p

− y0

c z0p

− z0

c

y0

b

− y0

c z0

b

− z0

c

____

(3.137)

since (Equation (3.136))

∂f

∂ua

____

0

= ∂f

∂xa

____

0

(3.138)

and the derivatives with respect to uc, vc and wc are sums of cofactors,

∂f

∂uc

____

0

= −_

_

__

y0

a

− y0

c z0

a

− z0

c

y0

b

− y0

c z0

b

− z0

c

____

+_

_

__

y0p

− y0

c z0p

− z0

c

y0

b

− y0

c z0

b

− z0

c

____

−_

_

__

y0p

− y0

c z0p

− z0

c

y0

a

− y0

c z0

a

− z0

c

____

. (3.139)

Consequently, denoting the elements of the matrix at (u0a

, v0

a, . . . ,w0p

) by a11, a12, . . . , a33

and the corresponding cofactors by A11,A12, . . . ,A33, the linearization of Equation (3.135)

yields

f 0 + A11(up − u0

p) + A12(vp − v0p

) + A13(wp − w0p

)

+ A21(ua − u0a

) + A22(va − v0

a) + A23(wa − w0

a) + A31(ub − u0

b)

+ A32(vb − v0

b) + A33(wb − w0

b) − (A11 + A21 + A31)(uc − u0c)

− (A12 + A22 + A32)(vc − v0

c ) − (A13 + A23 + A33)(wc − w0

c ) ≈ 0. (3.140)

Since m = 0, A11,A12 and A13 cannot all be zero. The variable with the largest coefficient

in size should be taken as the dependent degree of freedom. Accordingly, if

|A12| ≥ |A13| ≥ |A11| (3.141)

take vp as the dependent degree of freedom, unless it is already used in another multiple

point constraint. Notice that the nonlinearity only arises because of the fact that the plane

defined by a, b and c is not fixed in space. If the plane is fixed, xa, ya, za, xb, . . . , zc are

constants and Equation (3.135) reduces to a linear equation in xp, yp, and zp.

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