Lagrangian Multipliers

The W-W algorithm introduced in Section 3.3 has been developed extensively to obtain the natural frequencies or critical load parameters of an increasing range of geometries, load and boundary conditions. One of the most important developments introduced by Williams and Anderson (1983) extended the W-W algorithm using Lagrangian Multipliers. This extended algorithm enabled the coupling of component structures connected to each other which could then be calculated as a single structure. It also enabled the coupling of different wavelengths of sinusoidal response to satisfy the desired boundary conditions for anisotropic and shear loaded plate assemblies (Williams and Anderson 1983). Whilst retaining the generality and capability of the previous algorithm this extension can be applied to more complicated cases such as plates attached to beamtype transverse supports (Figure 3-3).

To introduce the basic concept of using Lagrangian Multipliers in the ESM and in the VICONOPT software, suppose 𝑗∗ _{independent structures are connected at a number of}

discrete points. If the equation for the 𝑗∗th _{structure is denoted by the subscript 𝑗}∗_{, Eq.}

3-11 can be written as (Williams and Anderson 1983):

𝐊𝑗∗ ∗ 𝐃𝑗∗ = 𝐏𝑗∗， (𝑗∗ = 1, 2, … , 𝑗∗′) Eq. 3-20

where 𝐏𝑗∗ is not null because of the connections between the independent structures. The

constraint equation denoting the connections between the degrees of freedom of the structures can be represented by (Williams and Anderson 1983):

𝑗∗′

∑ 𝐄𝑗∗ 𝐃𝑗∗ = 0 Eq. 3-21

𝑗∗₌₁

Eq. 3-21 is a general formulation that could denote any linear combination of any degrees of freedom of the whole system. Assume the total energy of the 𝑗∗′ _{components including}

the inertia effects of vibration problems is 𝑉.

𝑗∗′ 𝑗∗′

𝑉 Eq. 3-22

𝑗∗_{=1 𝑗}∗₌₁

Chapter 3---Exact Strip Plate Analysis

61

(a)

62

(b)

Figure 3-3 Comparison of mode shapes when Lagrangian Multipliers are included and not included in an analysis.

(a) without Lagrangian Multipliers; (b) including Lagrangian Multipliers for analysis of an infinitely long structure, where the is a typical half-wavelength, 𝐿 is

the length over which the mode shape repeats, 𝑙 is the plate length, and )

is given as: = 2𝑛/𝑀 where 𝑛 and 𝑀 are integers (Zhang 2018).

where 𝐏L is a vector of Lagrangian Multipliers and superscript 𝑡 expresses the transpose

of a matrix. To minimise 𝑉, the partial derivatives of 𝑉 with respect to the elements of the displacement vector are set to zero (Williams and Anderson 1983), giving

𝐊𝑗 , … , 𝑗 Eq. 3-23

where H denotes the Hermitian transpose of a matrix. The values in Eq. 3-20 to Eq. 3-23 can obtained from Eq. 3-18. Using a simply supported plate as an example, the eigenvalue problem of Eq. 3-23 and Eq. 3-21 can be written as:

𝑙𝐊1 𝟎 𝟎 𝟎 𝐄 𝟎 𝑙𝐊2 𝟎 𝟎 𝐄 𝟎 𝟎 𝑙𝐊 𝟎 𝐄 Eq. 3- 24 𝟎 𝟎 𝟎 𝟎 𝑙𝐊𝑗 𝐄𝑗𝐻 [ 𝐄1 𝐄2 𝐄 𝐄𝑗

where 𝑙 is the length of the plate, and the displacement and force variables 𝐃𝑗 and 𝐏𝑗

will be introduced in section 3.5. From Eq. 3-24, a small modification −𝜖𝐈 is introduced which leads to a zero for the initial rigid connections to obtain the original Lagrangian Multiplier form (Williams and Anderson 1983). To solve the complex transcendental

Chapter 3---Exact Strip Plate Analysis

63

eigenvalue problem, the extended W-W algorithm is introduced with Lagrangian Multipliers:

𝑗′

𝐽 𝑟 Eq. 3-25

𝑗

In Eq. 3-25, the first two items count the number of eigenvalues exceeded for each halfwavelength. 𝑠{𝐑} is the sign count of the matrix which replaces −𝜖𝐈 after partial triangulation of the preceding rows in Eq. 3-24, and 𝑟 denotes the number of constraints. Whilst Eq. 3-25 gives the general form of the extended W-W algorithm, here it is simplified by omitting the 𝐽 terms since only the lowest natural frequencies are required.

Lagrangian Multipliers have been introduced in the form of the extended W-W algorithm and successfully applied in the VICONOPT software over the last four decades. In this study, the method used to model damaged plates combines ESM to model the intact part of the structure with FEM to model the damaged part. Hence the intact part needs to couple the stiffness matrices for different assumed wavelengths of sinusoidal response, subject to constraint conditions to equate displacements at the boundaries with the damaged part. The application of Lagrangian Multipliers to this type of problem uses complex arithmetic and will be explained in section 5.1.3.

3.5 VICONOPT

For the solution of the direct problem of damage detection in plate structures, FEM, FSM and DSM have been studied extensively by other researchers in the last few decades. In this work the use of the exact strip software VICONOPT to determine the effect of delaminations and cracks on the vibration behaviour of plates will be explored. In this section the VICONOPT software will be discussed.

The ESM and the W-W algorithm have been applied in the software VICONOPT to solve critical buckling or free vibration problems in isotropic or anisotropic prismatic plate components for many years. They were first incorporated into the VIPASA analysis software developed by Wittrick and Williams (1974). Extension of this code with the introduction of Lagrangian Multipliers (Williams and Anderson 1983) led to the development of VICON (Anderson. et al. 1983; Williams and Anderson 1985). Building on this, Williams et al. (1990) released VICONOPT (VIpasa with CONstraints and

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OPTimisation), a 23,000 line Fortran 77 computer program combining the advantages of VIPASA and VICON (Williams et al. 1991). The program was developed by Cardiff University in collaboration with NASA and British Aerospace (Williams et al. 1990) in order to study the behaviour of structures with cross sections such as those shown in Figure 3-4. VICONOPT can be used to determine critical buckling load factors or undamped natural frequencies of any prismatic structure comprising isotropic (Figure 3- 4 (a)) or anisotropic plates loaded by any combination of in-plane longitudinal (NL),

transverse (NT) and shear (NS) load, see Figure 3-4 (b).

VIPASA

VIPASA, proposed by Wittrick and Williams (1974) is a program based on exact flat plate theory. The program determines the critical buckling load factors or natural frequencies of assemblies of thin prismatic plates using the W-W algorithm. It significantly reduces the computational time, data preparation and memory usage required to solve this type of problem (Williams et al. 1991) compared to finite element analysis. The modes of buckling or vibration are assumed to vary sinusoidally along the longitudinal direction x, either with the far ends simply supported or with the halfwavelength of the mode being much smaller than the overall length 𝑙 of the structure (Wittrick and Williams 1974).

Chapter 3---Exact Strip Plate Analysis 65 (a) (b) Figure 3-4 Plate assemblies in VICONOPT analysis (Williams et al. 1991).

(a) Cross-sections of typical plate assemblies and (b) loading conditions on an individual plate.

Figure 3-5 shows the perturbation edge forces and displacements, which vary according to (𝑖𝜋𝑥/ ) where n is the frequency, 𝑖 and 𝑡 is time (Wittrick and Williams 1974). If divides exactly into 𝑙, the sinusoidal variation enforce simply supported end conditions if the nodal lines run in the transverse direction 𝑦, but otherwise only approximate them (Figure 3-5). For isotropic and orthotropic plates the out-of-plane and in-plane behaviours are uncoupled (Wittrick and Williams 1974).

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The force and displacement terms in general contain complex quantities, and there are phase differences between them (Wittrick and Williams 1974). As shown in Figure 3-5, each nodal line has four degrees of freedom, where 𝑢 and 𝑣 are the in-plane displacements in the x and y directions, respectively, W is the out-of-plane displacement and is the rotation about the x-axis. The perturbation force vector 𝐩𝑞 and displacement vector 𝐝𝑞

are defined at edge q (q = 1 or 2) as:

𝐩𝑞 = {𝑚𝑞 𝑝𝑧𝑞 𝑝𝑦𝑞 𝑝𝑥𝑞}

Eq. 3-26 𝐝𝑞 = { 𝑞 𝑤𝑞 𝑣𝑞 𝑢𝑞}

Figure 3-5 A component plate showing the edge forces, displacements, and nodal lines (Wittrick and Williams 1974).

These are related by stiffness matrices 𝐤𝑖𝑗 as follows:

𝐩1 = 𝐤11𝐝1 + 𝐤12𝐝2

Eq. 3-27 𝐩2 = 𝐤21𝐝1 + 𝐤22𝐝2

which can be simplified to:

𝐩 = 𝐤 ∗ 𝐝 Eq. 3-28

For the uncoupled systems, the elastic properties are defined individually by the following two equations:

Chapter 3---Exact Strip Plate Analysis 67 𝑚𝑥 𝐷11 𝐷12 𝐷13 𝜅𝑥 Out-of-plane: [ 𝑚𝑦 ] = − [𝐷12 𝐷22 𝐷23] [ 𝜅𝑦 ] 𝑚𝑥𝑦 𝐷13 𝐷23 𝐷33 2𝜅𝑥𝑦 Eq. 3-29 𝑛𝑥 𝐴11 𝐴12 0 𝑥 In-plane: [ 𝑛𝑦 ] = [𝐴12 𝐴22 0 ] [ 𝑦 ] 𝑛𝑥𝑦 0 0 𝐴33 𝛾𝑥𝑦 Eq. 3-30 Eq. 3-29 and Eq. 3-30 define the elastic properties of these two systems, where 𝑚𝑥, 𝑚𝑦,

𝑚𝑥𝑦 are the bending and twisting moments per unit length; 𝑛𝑥, 𝑛𝑦, 𝑛𝑥𝑦 are the membrane

forces per unit length; κ𝑥, κ𝑦, κ𝑥𝑦 are the curvatures and twist and 𝑥, 𝑦, 𝛾𝑥𝑦 are the

membrane strains (Wittrick and Williams 1974). D and A are the out-of-plane bending and membrane stiffness matrices respectively. A comparison between VIPASA and two other codes (Smith’s (1968) program and BUCLASP2 (Viswanathan et al. 1973)) is given in Table 3-2. The main advantage of VIPASA is its computational speed which is 1100 times faster than FEM with less than 1% error (Butler and Williams 1992).