Constraint Methods

Three-Dimensional Modeling

4.1 Constraint Methods

The individual frames need to be connected to the master nodes in some manner.

The average displacement of each frame's horizontal translations at each story is constrained to move as a rigid body consistent with the master node motions. This

allows beams to have some axial deformation that is more realistic than a truly rigid diaphragm assumption. The axial deformation of beams is especially important for the nonlinear inelastic analyses performed in this work. Even if there is only one frame in the model, it needs to be constrained to act with the master node which has additional geometric and stiffness effects included.

Consider the simplest form of this constraint in which a two dimensional analysis is being performed with one frame plus a master node at each floor. The average of the frame nodal horizontal displacements is constrained to be equal to the master node displacement at each floor. This can be written for floor j with n frame horizontal dofs as

(4.1) - (x1 n ¹

+ .. · +

Xn) =X· J ^MN

where Xi is the global horizontal frame displacement at frame joint i and xrN is the global horizontal master node displacement. This can be rewritten for the augmented vector xj of story displacements which includes the frame horizontal displacements and the master node displacement at floor j as

(4.2)

[-~

which is of the form C jxj

=

^{0 where C j} is a constraint matrix.

Several methods exist for including constraints. The method that seems best suited to the iterative method used in this program is the penalty method. Lagrange multipliers, for example, add unknowns and result in a stiffness matrix with some zeroes on the diagonals. The penalty method adds large stiffness penalties to degrees of freedom that are constrained, and they can be chosen so that sufficient accuracy is obtained.

Assume that all of the story constraints for each frame are assembled into a global constraint matrix C operating on all of the degrees of freedom x so that the

desired constraints can be expressed as

(4.3) Cx=O.

If the constraint equations are not exactly satisfied there will be some non-zero entries on the right side of the equation, so the actual expression will be

(4.4) Cx=t.

By choosing a set of penalty numbers ai which form a diagonal matrix [a], the penalty function ~tT[a]t can be defined.1 This penalty function can be added to the potential of the structural system ITp giving the augmented potential

(4.5)

Note that if the residuals t = 0, the potential is unchanged. If t

-:f.

^0, there is a large penalty for violating the constraints as the ai increase. By substituting (4.4) and minimizing the potential {8IIp/8x}

=

0 the result is

(4.6)

[K +

CT[o:]C]x =f.

Since all the degrees of freedom being constrained are horizontal displacements, it is reasonable to choose one penalty number a so that a:

=

al. The previous equation can now be rewritten in incremental form as

(4.7)

In the above derivation, the stiffness matrix Kr was assumed not to be a function of the displacements x. While this is not true for nonlinear behavior in general, the constraint methodology still holds for each instantaneous tangent stiffness matrix.

It also works for dynamic problems. The constraint matrix for the three-dimensional

1The following derivation is adapted from (Cook, Malkus, and Plesha 1989).

formulation is considered in the next section.

A large value of a is desired to accomplish the constraint to a high degree, but too large a value will cause ill conditioning and subsequent loss of accuracy when the equations are solved. However, experience with structural problems has shown that, except in unusual cases, the use of double precision allows a to be selected to accomplish the constraint sufficiently while avoiding excessive loss of digits due to ill conditioning.

The penalty method was selected over other common methods of constraint for its ease of implementation in the iterative nonlinear solution technique. The constraints being imposed are prescribed interactions of certain degrees of freedom, and there are three common methods for including this type of constraint - the penalty method, the transformation method, and Lagrange multipliers.

The transformation method (McGuire and Gallagher 1979; Cook, Malkus, and Plesha 1989) uses the constraint equations to eliminate degrees of freedom from the global stiffness matrix. If there are r equations of constraint, then the method reduces the total number of dofs by r. This method is not suitable for this analysis for several reasons. The number of constraint equations equals the number of elim- inated dofs that could require peculiar non-intuitive choices for constraints. The method also requires a substantial amount of effort for, "reordering, partitioning and matrix multiplications.²" The condensed system uses constraint equations that replace some equilibrium equations instead of retaining all of the equilibrium equa- tions. This can cause certain equilibrium equations of the original system to not be met even though the condensed system satisfies its equilibrium equations. (Cook, Malkus, and Plesha 1989).

The method of Lagrange multipliers forms the scalar product of the homogeneous constraint equations with an equal number of unknown multipliers,

>.i.

This zero sum is added to the system's potential in the same way as the penalty method

2(Cook, Malkus, and Plesha 1989, p. 273)

(equation 4.5):

(4.8) lip = ^{1 T T T}

2

^{x Krx - x f}

+

A { Cx - t} . Minimizing the potential results in the equation

(4.9)

where the homogeneous constraint equations are the latter equations above. Note that there are zeros on the diagonal. If the equations are arranged as shown, Gaus- sian elimination can still be used since the zeros will be filled in before they are reached, although the handedness of the stiffness matrix is affected. If the con- straint equations are interspersed in such a way that whenever one is reached, all the dofs that it couples appear before it, then the system can maintain its smaller bandwidth and the zeros on the diagonal will not halt the solution process (Cook, Malkus, and Plesha 1989). The Lagrange multiplier method and the penalty method are closely related and Bathe presents an augmented Lagrange multiplier method that adds both a Lagrange multiplier zero-value term and a penalty zero-value term to the system's potential. The method is used for iterative procedures and gives reasonable results for all values of penalty factor a (Bathe 1996). The augmented potential is

(4.10) lip= ^{1 T T a T T}

2

^{x Krx-x f +}

2

^{Cx-t} {Cx- t} +A {Cx- t}.

The Lagrange multiplier method can be used if a reordering scheme were developed to avoid zero diagonals during Gaussian elimination. Additional dofs corresponding to the number of constraint equations would be required.

The penalty method is simpler because no reordering or additional dofs are required. The penalty "stiffness" is assembled just like any other stiffness contri- bution. If desired, the penalty stiffness could approximate the in-plane stiffness of

a structure's floor diaphragm providing an intuitive physical interpretation of the method. The only potential drawback of the penalty method is ill conditioning. As discussed earlier, the penalty factor can usually be chosen in a range to avoid ill conditioning if double precision is employed.