# Constraints

When loads are applied to the structure, naturally displacements occur at each degree of freedom. These displacements are a function of the applied loads, which are known, and the stiffness matrix of the damaged structure, which is not known. Using Eq(1), the displacements can be solved for in terms of the known and unknown values by:

u = K_1* P. Eq(8)

If every displacement were measured this equation could easily be solved to find the damaged stiffness matrix. However, as stated earlier, the objective is to develop a damage detection
procedure that is easy, accurate and practical. It is impractical to measure every displacement as one, it is time consuming, and two, often impossible due to obstructions. What remains is an expression with more unknowns than equations. Fortunately, strategic displacement measurements coupled with the correct objective function will provide a good basis that will determine the damaged stiffness matrix and therefore the cross-sectional properties of each member.

 K_1 * P = u

The number of constraints is equal to the number of measured displacements for all load cases. Using the measured displacements of the structure and Eq(8), equality constraints can be created that must be satisfied by the optimization procedure given by:

umeasured = vector of measured displacements K = global stiffness matrix

P = Load vector used to produce measured displacements

Next, in order to write the constraints in a form suitable for optimization, i. e. Eq(3b), Eq(9) can be written as:

(K-1* P) – u measured = 0.

Thus, Eq(10) provides a set of constraint equations that must be satisfied at the optimum set of design variables.

Side Constraints

In order to avoid convergence to a physically meaningless point, it was desired to include side constraints on each design variable. First, as explained earlier side constraints were applied to each design variable preventing negative values. Second, it is physically impossible for damage to create additional moment of inertia for each structural element, therefore an upper bound equal to the value of the healthy moment of inertia was placed on each design variable.

Even though complete damage or collapse of a member would reduce its moment of inertia to zero, it was not desirable to set the lower side constraint to zero in this process. This is because if the optimization process follows a path that forces one of the design variables to pass through zero, as will be seen, the recursive formula which predicts the design values for each subsequent iteration will be unable to change any design variable once it has been set to zero. To solve this apparent problem, the lower bound was set to a value that is small enough to be insignificant in reality but large enough to carry mathematical weight in the optimization process. Therefore, the lower side constraint was set to 5in4 for every member.

Problem Statement

Using the above formulations the engineering problem becomes:

si. {u} – {umeasured } = 0

s. t. 5 ^ Ii ^ Ii _healthy

As before, the optimality criterion must be developed for the current engineering problem. To do this the Lagrangian must be written using the objective function and constraint equations. With the inclusion of lagrange multipliers for each constraint, the Lagrangian becomes:

n д I m

L = Ъ~Т + ХЛГ (Uj ~ Uj_measured К i=1 Li j=1

where Uj are the calculated displacements of the healthy structure. From this, the necessary condition for optimality can be found by:

Eq(13)

In turn, a local minimum of Eq(12) occurs when

1 m dU:

—– 1-/1.———- = 0 (for i=1,2,…,n)

Li j dlt 1 )

Eq(14) can also be used as a barometer for how close each intermediate point within the optimization routine is to the optimal solution. Thus, a recursive formula can be created to determine the design variables for the next iteration based on the optimality criterion of the previous iteration (Cheng et al.). Manipulation of Eq(14) gives,

m du.

T = ц —.

j dIi Eq(15)

When the optimality criterion is satisfied Ti is equal to 1.

In order to solve for each lagrange multiplier, gradients of each constraint equation must be calculated in order to provide a system of equations sufficient to solve for all unknowns (Haug et al. 1979). From the definition of a derivative, the gradient of the constraint equations can be written as such:

Simple manipulation of Eq(16) yields:

The AI term that appears in Eq(17) is determined using Eq(15). The change in I can be calculated using the recursive formula:

Ik+1=Ik+- (Ti – 1)Ik

r

Solving for AI yields:

m=ik+i – it = -(Tt -1) ik r

Substituting Eq(19) in to Eq(17) yields:

« dg. 1

-g'(1)-5ж-r(T -1)I-,

where r is a convergence control parameter that restricts the step size for each iteration (Cheng et al.). The control parameter is assumed to be 2 for all problems covered in this paper. Manipulation of Eq(20) yields:

Eq(21) provides n equations for n unknown Lagrange multipliers, which can be used in Eq(15) and Eq(19) to determine the design values for the next iteration. The cycle is repeated until the optimality criterion is met.

Example

A two-bay frame is pictured in Figure 2. It is used to demonstrate the importance of the location of the applied loads in the damage detection process. Each structural element is assumed to be made of steel with modulus of elasticity 29,000 ksi and dimensions shown in Figure 2.

Table 1 shows the initial and final design variables of each structural element. For this example, all elements will experience damage. All initial values are assumed to be known and will serve as the starting point for each member size. The final design values are not known, however, and will be determined by the optimization routine.

Table 1. Initial and Final Design Values

 Member iinitiJin4) ifinJin4) 1 3500 1500 2 3500 1000 3 3500 1000 4 3500 1000 5 3500 1000 6 3500 700 7 3500 700

Case 1

For the first instance, 5 different load cases are used to measure displacements of the damaged structure. Loads are applied individually at each of the first 5 DOF and the resulting displacement at the same DOF is measured. The applied load and resulting displacement for each of the 5 DOF is shown in Table 2.

Table 2. Load Cases and Resulting Displacements for Case 1 & 2

 Load Case DOF Loaded LOAD DOF Measured Displacement 1 1 10 k 1 0.10743 in 2 2 10 k 2 0.28251 in 3 3 10 k 3 0.30796 in 4 4 10 k-in 4 0.000017438 in/in 5 5 10 k-in 5 0.000012803 in/in

Each measured displacement is used as a constraint equation given by Eq(10). Table 3 shows the results of the optimization process using 5 displacements as constraint equations. As can be seen, the optimization routine converged to the damage state in 5 iterations. The first 7 columns of the last row are identical to the final design values shown in Table 1. It is also observed that all Tt values converged to 1. From the previous formulation (Eq.14-15) this signifies that the optimality criterion has been met and the current design variables are those that minimize the objective function and satisfy the constraint equations. The results are promising in that only 5 measured displacements were required for convergence to the correct damage state rather than all 8 measured displacements. However, two of the five measured displacements required not only rotation measurements but also concentrated moments to be applied, which can be difficult. In order for the damage detection procedure to be practical, these complications must be eliminated.

Table 3. Optimization Results for Case 1

 Iteration I1 12 I3 I4 I5 I6 I7 Obj i-1 X2 ^3 X4 ^5 T1 T2 T3 T4 T5 T6 T7 1 3500 3500 3500 3500 3500 3500 3500 88.83 -45.864 -7.6083 -29.748 -23246 -920240 1.9243 1.2886 1.2886 1.2881 1.2881 0.6796 0.9914 2 1221 955.6 955.6 955.4 955.4 701.3 831.5 85.44 22.267 -6.6294 -27.925 -268290 -1E+06 1.3802 1.0788 1.0788 1.0795 1.0795 0.9998 0.7224 3 1453 993.3 993.3 993.4 993.4 701.2 716.1 84.16 6.2744 -10.394 -27.727 -238190 -1E+06 1.0627 1.0132 1.0132 1.013 1.013 0.9966 0.9562 4 1499 999.8 999.8 999.8 999.8 700 700.4 83.9 1.1466 -11.446 -27.548 -232980 -1E+06 1.0017 1.0004 1.0004 1.0003 1.0003 0.9999 0.9987 5 1500 1000 1000 1000 1000 700 700 83.89 0.9853 -11478 -27.543 -232820 -1 E+06 1 1 1 1 1 1 1

Case 2

If the number of displacement measurements is reduced to 4, the final design values are not close to the actual damage state. Case 2 uses the same initial and final design values and the same loading cases, however only the first 4 DOF measurements are used. The results of reducing the number of measured displacements to 4 are shown in Table 4. The structural members spanning each bay, namely I2,13,14, and I5, converged to values within 8% of their actual values and I6 to within 2.1%. However, It and I7 were grossly misjudged with error in excess of 50% for It and 70% for I7.

Table 4. Optimization Results for Case 2

 Iteration i, І2 Із 14 І5 16 І7 Obj T, ^2 Тз Ті T, T2 Tj Ti T5 T6 T7 1 3500 3500 3500 3500 3500 3500 3500 88.83 -167.66 -24.429 -25.749 224600 1.044 1.3504 1.3504 1.2011 1.2011 0.6952 1.5845 2 853.5 981.4 981.4 919.1 919.1 707.9 1079 86.16 -146.39 -27.873 -21.749 -37047 0.8936 1.085 1.085 1.0261 1.0261 1.0154 1.1011 3 808.1 1023 1023 931.1 931.1 713.3 1134 85.631 -147.38 -27.955 -21.3 -41909 0.9229 1.0127 1.0127 0.995 0.995 1.0014 1.0453 4 776.9 1030 1030 928.7 928.7 713.8 1159 85.625 -147.44 -28.131 -21.118 -42553 0.9534 1.0059 1.0059 0.9967 0.9967 1.0008 1.0259 5 758.9 1033 1033 927.2 927.2 714.1 1174 85.627 -147.39 -28.253 -21.013 -42938 0.9724 1.0035 1.0035 0.9981 0.9981 1.0005 1.0149 6 748.4 1034 1034 926.3 926.3 714.2 1183 85.628 -147.35 -28.325 -20.952 -43182 0.9839 1.002 1.002 0.9989 0.9989 1.0003 1.0086 7 742.3 1036 1036 925.8 925.8 714.3 1188 85.629 -147.32 -28.367 -20.917 -43329 0.9906 1.0012 1.0012 0.9994 0.9994 1.0002 1.0049 8 738.9 1036 1036 925.5 925.5 714.4 1191 85.629 -147.3 -28.392 -20.896 -43416 0.9946 1.0007 1.0007 0.9996 0.9996 1.0001 1.0028 9 736.9 1036 1036 925.3 925.3 714.4 1193 85.629 -147.29 -28.406 -20.885 -43467 0.9969 1.0004 1.0004 0.9998 0.9998 1.0001 1.0016 10 735.7 1037 1037 925.2 925.2 714.4 1194 85.629 -147.28 -28.414 -20.878 -43496 0.9982 1.0002 1.0002 0.9999 0.9999 1 1.0009

Case 3

At this point several alterations to the method can be made, two of which are to find an alternate objective function to be minimized or to try alternate loading cases from which to measure displacements. The former has been explored, using the minimization of strain energy, and minimizing the reduction in bending stiffness. However results have not shown to improve beyond Case 2. The latter alternative has proven to produce favorable results with a similar number of measured displacements. Table 5 shows an alternate loading condition with the resulting displacements using the final design values in Table 1.

Table 5. Load Cases and Resulting Displacements for Case 3

 Load Case DOF Loaded LOAD DOF Measured Displacement 1 2 10 k 1 -0.024894 in 2 2 10 k 2 0.28251 in 3 2 10 k 3 -0.055233 in 4 2 10 k 4 0.00025174 in/in

For Case 3, a 10 kip load is applied at only degree of freedom 2 and displacements are measured at the first 4 DOF. Again the optimization routine is executed in an identical manner as the previous two cases and the results are shown in Table 6.

Table 6. Optimization Results for Case 3

 Iteration h h Із Ii І5 І6 І7 Obj T, І2 h Ті T, T2 Tj Ti T5 T6 T7 1 3500 3500 3500 3500 3500 3500 3500 100.54 -345.81 -19.46 207.68 -18825 3.1223 1.9186 1.9186 1.067 1.067 0.8183 1.8332 2 997.1 705.9 705.9 499.9 499.9 439.8 685.3 94.828 -924.66 -27.458 524.62 -36304 2.2241 1.5161 1.5161 1.4795 1.4795 1.2379 1.1959 3 1607 888.1 888.1 619.8 619.8 492.1 752.4 91.626 -1708.4 -16.893 790.29 -85194 1.3868 1.4052 1.4052 1.9392 1.9392 1.7721 1.1523 4 1691 941.4 941.4 802.9 802.9 601.2 713.7 85.433 -2055.6 -21.505 734.75 -144990 0.8756 1.1042 1.1042 1.3687 1.3687 1.2473 0.978 5 1586 990.5 990.5 950.9 950.9 675.6 705.8 84.161 -1889.3 -28.551 572.51 -159140 0.9451 1.0229 1.0229 1.0792 1.0792 1.0499 0.9987 6 1535 997 997 983.8 983.8 689.1 702 83.92 -1829.8 -30.415 533.78 -159880 0.988 1.0028 1.0028 1.0119 1.0119 1.0105 0.999 7 1525 998 998 989.2 989.2 692.5 701.3 83.893 -1818.7 -30.74 527.84 -159730 0.9977 1.0003 1.0003 1.0016 1.0016 1.0016 0.9998 8 1523 998.1 998.1 990 990 693 701.2 83.892 -1816.7 -30.793 526.99 -159660 0.9997 1 1 1.0002 1.0002 1.0002 1 9 1523 998.2 998.2 990.1 990.1 693.1 701.2 83.892 -1816.4 -30.799 526.9 -159650 1 1 1 1 1 1 1

The optimization routine did not identically match the actual damage state given in Table 1, however the found values are all within 2% of the actual damaged values. By adjusting the loading condition when measurements were taken, results improved compared to Case 2 and one rotational degree of freedom measurement and both concentrated moment loads were eliminated compared to Case 1. Utilizing a loading condition with only vertical loads also provides a process better suited for implementation on real structures.

Conclusion

As shown from examples presented, the damage detection process is highly dependent on the load cases used to produce the measured displacements utilized by the optimization routine. In order for the damage detection process to be easily implemented it is desired that only vertical loads and vertical displacements be required for the damage detection procedure. The example presented is just one example of the possible load cases that can be used to measure displacements. Future work will include the exploration of additional load cases coupled with strategic displacements measurements in order to determine the effectiveness with regards to damage detection.

The optimization routine lends itself well to such engineering problems as all design variables are continuous. Furthermore, the results show quick convergence to a local minimum as signified by the satisfaction of the optimality criterion. Since so few iterations are required computational effort is kept to a minimum. Minimizing computing effort will prove beneficial for damage detection in larger structures.