Multihazard physical theory (MPT) states that hazards interact through the system. We will investigate ways to explore the ways such interactions reveal themselves and how stakeholders can understand, explore, quantify, and utilize such interactions in the field of civil infrastructure. A natural place to start this exploration is analysis. As taught in engineering school, analysis, in particular structural analysis, is the gateway to handling most infrastructures, whether new or existing. At first, we will explore how to isolate and use multihazard interactions, simply by following the basics of current structural analysis techniques to show that we can understand the behavior of structural systems by observing the results from common analysis techniques such as finiteelement methods (Reddy 2005, Yang 1985, Zienkiewicz and Taylor 2011).
Multihazard physical theory (MPT) states that hazards interact through the system. We will investigate ways to explore the ways such interactions reveal themselves and how stakeholders can understand, explore, quantify, and utilize such interactions in the field of civil infrastructure. A natural place to start this exploration is analysis. As taught in engineering school, analysis, in particular structural analysis, is the gateway to handling most infrastructures, whether new or existing. At first, we will explore how to isolate and use multihazard interactions, simply by following the basics of current structural analysis techniques to show that we can understand the behavior of structural systems by observing the results from common analysis techniques such as finiteelement methods (Reddy 2005, Yang 1985, Zienkiewicz and Taylor 2011).
Some authors explored affinities or interactions between hazards in the form of sensitivity analysis. Sensitivity analysis is an important component of any analysis, including risk assessment, to ensure that the variability in the input factors is considered and appropriate decisions are made. Tsai et al. (2011) state two purposes of a sensitivity analysis as to quantify the uncertainty in the calibrated model caused by uncertainty in the estimates of parameters, stresses, and boundary conditions and to identify the model inputs that have the most influence on model calibration and predictions (Albrecht and Miquel 2010, ASTM 2002, Gribb et al. 2002). If the sensitivity analysis shows that the model outputs are highly sensitive to a particular input factor than the associated uncertainty or variability, it will significantly affect the model’s ability to make consequential interpretations and predictions (Seuntjens et al. 2002). Through sensitivity analysis, it is possible to decompose the model output variation back to the input factors and to identify those that need to be measured or estimated accurately to achieve a precision improvement in the model output. With this understanding, authors conducted a sensitivity analysis to evaluate the effectiveness of input parameters on the calculated risks at a petroleumhydrocarbon contaminated site. Under a riskbased corrective action approach, as developed by ASTM (2004), they evaluated the risks to human health and the environment associated with this site and risk management.
A different approach for studying affinities between hazards was investigated by Fujikura et al. (2007, 2008), who presented and experimentally investigated a multihazardresistant multicolumn pier–bent concept relying on concretefilled steel tube columns, to provide both seismic and blast resistance. Analysis of structures under multihazard conditions is not always straightforward due to the fact that a structure may behave differently under different hazard loading conditions and the inherent assumptions built into the analysis methods. Fujikura and Bruneau (2012) investigated the appropriate value of the shape factor (which reduces blast pressures when applied to a circular column) that must be used with two different analysis methods—a singledegreeoffreedom dynamic analysis and a fiberbased dynamic analysis. In the fiberbased model that assumes that a plane section remains plane, a member section is divided into fibers in which the unidirectional stress–strain relationships of materials are assigned to represent the section characteristics. Their study showed that different values of shape factor must be used with different analytical methods due to the differences in assumptions and conditions behind these two models investigated.
The main goal of this chapter is to explore methods to quantify interactions between hazards using the results of conventional structural analysis results. We offer methods for MH interaction matrices for some popular analysis types: static, dynamic, linear, and nonlinear behavior. We also look at common analysis response metrics such as displacements and internal forces. Throughout this chapter, we offer both theoretical background of the methods and pertinent case studies that highlight the analytical techniques (Figure 3.1).
Figure 3.1 Contents of this chapter.
Let us consider the popular equilibrium equation
with [K], {U}, and {P} representing the stiffness matrix, the displacement vector, and the force vectors, respectively. The order of Equation 3.1 is N, and its solution is
This can be rewritten as
Let us also define ith demand/hazard as
For the ith hazard/demand, we express the ith structural response as
Given the response vector {U}_{i} due to the ith hazard, we define the average interaction coefficient between the ith hazard and the jth hazard as
Note that α_{ij} has units and this might limit its utility in practical situations. Hence, it is always desirable to introduce a dimensionless and normalized expression. Hence, we now define a generalpurpose MH interaction coefficient (MHIC) between the ith and the jth hazards as
Note the similarity of Equation 3.8 and the modal assurance criterion used in vibration analysis. The MHIC is a nondimensional scalar in the range of
If the ith and the jth hazards do not affect the system in a consistent manner, then MHIC→0.0. For hazards that affect the system in a consistent manner, MHIC→1.0.
We can now write the MH interaction matrix (MHIM) as
where NH is the number of hazards of interest. Note that [MHIM] is a symmetric matrix with the order of NH.
In order to illustrate the use of [MHIM], we study a case of three loads/hazards applied on the 2D frame shown in Figure 3.2. This figure shows the basic dimensions of the frame. The nodal and element designations of the finiteelement model (FEM) that was used to analyze the building are also shown in Figure 3.2. The coordinate system shown in Figure 3.2 uses displacement (forces) x_{1}, x_{2} and rotations (moments) θ_{3}. The modulus of elasticity of the frame is assumed to be 3.0E^{+6} psi. The beams (horizontal elements) and columns (vertical elements) are assumed to have rectangular cross sections with dimensions shown in Table 3.1. The frame is assumed to be fixed at the two foundation nodes. For this example, we consider dead load (DL), live load (LL), and wind load (W). Each of these is assumed to be a concentrated load as shown in Table 3.2.
Figure 3.2 2D model of framed building.
Depth (in.) 
Width (in.) 
Moment of Inertia (in.^{4}) 
Area (in.^{2}) 


Beams 
24 
12 
13,824 
288 
Columns 
12 
12 
1,728 
144 
Locations 
Hazard Loads (lb) 


Node Number 
Direction 
Dead Load (DL) 
Live Load (LL) 
Wind (W) 
2 
x_{2} 
−0.5 
−0.1  
2 
x_{1} 
0.25 

3 
x_{2} 
−0.5 
−0.25  
3 
x_{1} 
0.5 

4 
x_{2} 
−0.5 
−0.35  
4 
x_{1} 
1.0 

5 
x_{2} 
1.0 
−2.0  
6 
x_{2} 
1.0 
−3.0  
7 
x_{2} 
1.0 
−4.0  
9 
x_{2} 
−0.5 
−0.075  
9 
x_{1} 
0.25 

10 
x_{2} 
−0.5 
−0.05  
10 
x_{1} 
0.5 

11 
x_{2} 
−0.5 
−0.25  
11 
x_{1} 
1.0 
The solution for the displacements of the frame due to the three hazards is a fairly trivial process. The resulting displacements and rotations are shown in Appendix 3A. With a simple operation, we can identify MHIM of this problem, using the displacements as
The order of the hazards corresponding to this matrix is shown in Table 3.3. The MHIM matrix helps in understanding the affinity between responses of a given structural system to different hazards. Such an understanding would ultimately aid the analysis/designer in producing an optimal structural system that is safe and costeffective.
Order (ith Row/Column) 
Hazard 

i = 1 
DL 
i = 2 
LL 
i = 3 
W 
Upon studying the MHIM of Equation 3.11, we can deduce the following:
Although the notes given earlier are not surprising (due to the simplicity of the example and the loading conditions), it is expected that when the structural system and the loading conditions are more complex, the MHIM matrix would produce more revealing and less predictable results. Such results could then be of more value to the analyst/designer if utilized judicially. It is worth noting that there is no reason to limit the formation of MHIM to displacements; it can be formed using internal forces from hazard or even a specific type of internal forces (bending moments only or shearing forces only). We leave such developments to the reader as an exercise.
Equation 3.7 implies summation of all displacement fields within a given system. This means that we sum linear displacements in both x_{1} and x_{2} directions (for a twodegreeoffreedom [2DOF] geometry) and rotations in the θ_{3} direction. This generality might lead to some bias in the results. In some situations, it might be preferable to perform the sum on only one direction, say, x_{2} (vertical displacements) or on one type of displacements both x_{1} and x_{2} directions. For example, if we limit the computations only to x_{1} direction, we obtain
We observe that the interaction between DL and LL has been reduced to 48% from a high of 93%. The interaction between LL and W is now much higher at 47%. The interaction between DL and W is still negligible.
Now if we limit the computations to the x_{2} direction (vertical displacements) only, we obtain
If we limit the computations to the nodal rotations, θ_{3}, only, we get
Matrices in (3.11) through (3.14) show how sensitive MHIM is to the type and direction of displacements used in the computations. Studying those results can aid immensely in understanding, designing, and controlling the structure of interest in an optimal manner. It also shows that one has to be very careful in how the results are computed, analyzed, and understood.
There are some disadvantages of using MHIM in understanding the behavior and MH interactions. These disadvantages are as follows:
We address some of these limitations in the next few sections.
Using displacements as a basis to compute MHIM would give great insight into the behavior of the structure that is subjected to different hazards. Sometimes, the analyst/designer might have to study how hazards interact through internal forces. Such an understanding might help in obtaining optimal structural design/behavior. Let us assume that for a given structural member, m (or a finite element within the whole model), the member end displacement vector due to the ith hazard is
If the member stiffness is [k]_{m}, then we obtain the member internal forces as
The order of the matrix equation (3.16) is nm, the number of nodal degrees of freedom of the system of interest. The average interaction coefficient between the ith hazard and the jth hazard is now defined as
MH force interaction coefficient (MHIC) between the ith and the jth hazards can now be obtained using Equation 3.8 and the force MHIM (FMHIM) is formed using Equation 3.10, subject to conditions (3.9).
In order to form FMHIM for a single column (element number 1 in Figure 3.2), we first compute the six end forces due to the three hazards. The results are shown in Table 3.4.
Hazard 
DL 
LL 
W 

x_{1} 
0.15237 
0.289298 
−1.74991 
x_{2} 
3 
5.200142 
−2.18605 
θ_{3} 
−6.11174 
−11.5961 
116.4794 
x_{1} 
−0.15237 
−0.2893 
1.749906 
x_{2} 
−3 
−5.20014 
2.186046 
θ_{3} 
−12.1727 
−23.1197 
93.5093 
We then apply Equations 3.17 and 3.10. The resulting FMHIM, if we consider all six force measures, is
For bendingonly (θ_{3}), the FMHIM is
Similarly, the respective internal forces of the first floor beam (element number 4 in Figure 3.2) due to the three hazards are shown in Table 3.5.
Hazard 
DL 
LL 
W 

x_{1} 
−0.11292 
−0.33069 
0 
x_{2} 
0.5 
1.00011 
−1.00794 
θ_{3} 
28.83756 
59.42586 
181.429 
x_{1} 
0.112915 
0.330688 
0 
x_{2} 
−0.5 
−1.00011 
1.007938 
θ_{3} 
61.16245 
120.5939 
−1.6E−05 
As before, we apply Equations 3.17 and 3.10. The resulting FMHIM, if we consider all six force measures of the beam, is
If we only consider the beam linear forces (axial, x_{2}, and shear, x_{1}), the resulting FMHIM is
For beam bendingonly (θ_{3}), the FMHIM is
Studying Equations 3.18 through 3.23 reveals that there is great affinity between DL and LL, which is expected given the loading distributions of both the hazards. There is relatively less and varied affinity between wind hazards and both of the other two vertical hazards. Such affinity is much less for beams than columns.
We only studied a single element in this example. However, more than one element can be used to develop FMHIM. At the limit, all elements of the structural system can be included in the evaluation of FMHIM. Evaluating the number and type of elements as well as the type of internal force measures that need to be included in such evaluations is a subject that is still in its infancy and needs further research. As such, it is beyond the scope of this book.
When all hazards are dynamic and with linear elastic systems, we use Equation 3.45 (see Appendix 3B) to define the response to different hazards and Equation 3.7 to define MH components such as
Applying Equation 3.24 to Equations 3.8 through 3.10 will yield the desired DMHIC and DMHIM.
A practical simplification of Equation 3.24 is to use the maxima of {D_{i}(t)} that might be considered as a spectral expression of d_{i}(t). As such, we can replace {D_{i}(t)} in Equation 3.24 by {S_{i}}, such that its components s_{i} satisfy
We can now write the timeindependent expressions of α_{ij} as
Applying Equation 3.26 to Equations 3.8 through 3.10 will yield a timeindependent form of DMHIC and DMHIM.
Vibrating loads are one of the important loading conditions (hazards) in the field of civil infrastructure. The sources of these hazards vary from vibrating machineries to moving trains or cars. Understanding how these hazards interact through the physical structural system is necessary for producing an optimally behaving system. Since the dynamics of the structure form the major basis of structural response, we have to include dynamics while exploring the physical MH interactions of these dynamic loads. To illustrate the process of developing MH dynamic interaction matrix DMHIM, we will continue 2D frame example of Figure 3.2. We will assume a lumped mass matrix of the structure for this example, as shown in Table 3.14 (see Appendix 3E). A mass density of 2.25E–04 lb s^{2}/in.^{4} was used for the material. The locations and amplitudes of applied loads are shown in Figure 3.3 and Table 3.15 (see Appendix 3E), respectively. A conventional frequency analysis of the system produces the set of natural frequencies shown in Table 3.16 (see Appendix 3E). The first 10 natural modes are shown in Table 3.17 (see Appendix 3E).
Figure 3.3 Locations of vibrating loads. (Note: For node numbers, see Figure 3.2.)
Let us first explore the situation where the spectral frequency distributions of the vibrating loads are well separated as shown in Figure 3.4. This set of spectra would produce dynamic displacements according to Equations 3.25 and 3.45, as shown in Table 3.18 (see Appendix 3E). Using Equation 3.26, the corresponding DMHIM (accounting for all DOFs) is computed as
Figure 3.4 Spectra of three vibrating loads.
As expected, there is hardly any affinity or interaction between the three vibrating loads. This is not too surprising since the three loading spectra are well separated in the frequency domain as shown in Figure 3.4. If on the other hand we use only rotational DOF while executing Equation 3.26, the DMHIM is
Let us investigate the effects of frequency distributions on the interactions between vibrating loads. We change the spectrum of VL_3 to a widebanded spectrum as shown in Figure 3.5.
Figure 3.5 Spectrum of widebanded VL_3.
Again, using Equation 3.26, the corresponding DMHIM (accounting for all DOFs) is now computed as
Also, if we use only rotational DOF while executing Equation 3.26, the DMHIM is
For both results, we see no changes in the interaction between VL_1 and VL_2, which is expected since neither of the two spectra has changed. However, the interactions between VL_3 and both VL_1 and VL_2 have both increased greatly. There is 95%–99% affinity between VL_1 and VL_3, whereas there is ~4%–7% affinity between VL_2 and VL_3: small affinity, but much higher than before. The reasons for the higher affinities between VL_1 and VL_3 as compared to affinity between VL_2 and VL_3 are now completely due to how mode shapes ϕ_{i} and loading patterns {P} interact for each of the three hazards.
In March 2013, a truss bridge over the Skagit River in Washington state (United States) was struck by a truck. Because of the limited redundancy of the bridge, it failed subsequent to the impact accident. The accident and subsequent bridge failure highlighted an interesting issue, that is, on how to treat a structure (in this situation, a bridge) due to two hazards: an impact load (in this situation, a truck impact) and a seismic hazard (the Skagit River is located in a highdemand seismic area). Our immediate goal in this section is to address the affinity of truss behavior, which might result from impact load/seismic load interaction through the structural system. Given the essential dynamic nature of both hazards, the methodology of this section is well suited to address the problem via the establishment and ensuing study of DMHIM.
In order to explore the DMHIS even further, we study impact load and earthquakes as applied to a simplified 2Dtruss. The problem as offered is fairly simplified; however, it can be used with appropriate higher resolution for any reallife practical situation. Figure 3.6 shows a typical simple supported truss bridge over a river inlet.
Figure 3.6 Simple supported truss bridge over an inlet.
Consider the simple 2Dtruss FEM of Figure 3.7. The elements are simple 2D beam elements (with rigid connections). We inserted a middistance node at each of the truss’s diagonals so as to be able to simulate the impact load at the midheight of the diagonal member. We assumed a simple roller support and hinged support at each end of the simple supported truss structure. In order to obtain reasonable natural frequencies of the model, we assumed a set of hypothetical mechanical properties of the model. The modulus of elasticity is assumed to be 3.00E 07 lb/in^{2}. The areas and moment of inertias of all beams in the model were assumed to be 30 in.^{2} and 1400 in.^{4}, respectively. We assumed a fairly simple frequency spectra for the two hazards as shown in Figure 3.8. The two spectra represent the basic properties of their respective hazards. The impact affects a wide frequency range, whereas the seismic spectrum affects only on a limited and low frequency range. This information, in addition to lengths and element connectivity, is sufficient to perform the dynamic analysis needed for the computations of DMHIM. Appendix 3E includes all the intermediate results: lumped mass vector, {M}, in Table 3.19 (see Appendix 3E); amplitudes of the two hazards, {P_{i}}, with i = 1 for impact load vector and i = 2 for seismic load vector in Table 3.20 (see Appendix 3E); the natural frequency set of the structure, {ω}, in Table 3.21 (see Appendix 3E); the first 10, i = 1, 2, …, 10, natural modal amplitudes {ϕ}_{i} in Table 3.22 (see Appendix 3E); and the dynamic displacements vectors for the two hazards {U}_{i} = [Φ]⟨Γ⟩{S_{i}} with i = 1 for impact load vector and i = 2 for seismic load vector in Table 3.23 (see Appendix 3E).
Figure 3.7 2D simple bridge truss model.
Figure 3.8 Frequency spectra of impact and seismic (vertical) hazards.
Using Equation 3.26, the twohazard DMHIM (accounting for all DOFs) is computed as
Civil infrastructures are usually designed for static as well as dynamic hazards. We can produce an MHIC and MHIM for such a mix of dynamic–static hazards, which we will name DSMHIC and DSMHIM, respectively. By studying Equations 3.7, 3.24, and 3.26, we can define an interaction coefficient between the static hazards i and the dynamic hazard j as
Applying Equation 3.32 to Equations 3.8 through 3.10 will yield a timeindependent form of DSMHIC and DSMHIM.
A fairly common analysis situation in the building community is analyzing buildings for wind and horizontal seismic hazards. In vast majority of situations, wind hazards are analyzed statically while seismic hazards are analyzed dynamically (using a simple modal analysis technique similar to that of Appendix 3B). For simplicity, we still use the geometry and properties of the simple 2D Frame FEM shown in Figure 3.2. The static Ws are assumed to be horizontally distributed, as shown in Figure 3.9. We also use a dynamic horizontally applied seismic load with the frequency spectrum of Figure 3.8, but we apply it to the building in the horizontal direction as shown in Figure 3.9.
Figure 3.9 Amplitudes of wind and horizontal seismic loads.
Since the structural system of the building is the same as in Figure 3.2, the lumped mass vector, {M}, is the same as in Table 3.14 (see Appendix 3E). Also, the natural frequency set of the structure, {ω}, is shown in Table 3.16 (see Appendix 3E). The first 10, i = 1, 2, …, 10, natural modal amplitudes {ϕ}_{i} are shown in Table 3.17 (see Appendix 3E). The amplitudes of the two hazards, {P_{i}}, with i = 1 for Ws and i = 2 for horizontal seismic load vector are shown in Table 3.24 (see Appendix 3F). The static displacement vector for the wind hazard, i = 1, and the dynamic displacement vector for horizontal seismic hazard with i = 2 are shown in Table 3.25 (see Appendix 3F).
Using Equations 3.26 and 3.32, the threehazard DSMHIM (accounting for all DOFs) is computed as
The affinity between the static W and the dynamic horizontal seismic load is fairly high at 96%. This is an expected result given the fairly similar load amplitude distributions and the resulting displacement distributions of Tables 3.24 and 3.25 (see Appendix 3F), respectively. We need to add here that such high affinity might be misleading since it is based on the assumption that the structural system is linear. It is customary to proportion structural systems in most seismic analysis situations such that they behave in a nonlinear manner, whereas wind designs presume linear behavior, ASCE 710 (2013). In such situations, when a hazard leads to a nonlinear structural behavior, the use of Equation 3.17, 3.26, or 3.32 could lead to a meaningless result if used without care. We address MH in the nonlinear structural range in Section 3.6.
Another popular dynamic horizontal hazard that might affect buildings is an impact or blast hazard (Figure 3.10). Sometimes, the stakeholder might add that loading condition as a third hazard to the previous example to investigate affinities between the three hazards: wind, seismic, and blast (impact) hazards. A simplified assumption of the dynamic nature of the hazard would be a frequency spectrum similar to the spectrum of Figure 3.5 and with a blast (impact) spatial load distribution as in Table 3.26 (see Appendix 3F). Upon repeating the example of Section 4.5.1 with the third hazard, we obtain the new blast (impact) displacements as in Table 3.27 (see Appendix 3F). Finally, using Equations 3.26 and 3.32, the threehazard DSMHIM (accounting for all DOFs) is computed as
Figure 3.10 Blast (impact) load distribution.
We consider another situation where dynamic and static hazards affect another type of infrastructure. We reuse the geometry and properties of the simple 2Dtruss FEM of Figure 3.7. Also, we still use the impact hazard as applied to the truss (Figure 3.11) with the frequency spectrum of Figure 3.8. In this example, we use a dynamic horizontally applied seismic load with the frequency spectrum of Figure 3.8, but we apply it in the horizontal direction as shown in Figure 3.12. For the purpose of this example, we assume that the LLs are statically applied to the truss system.
Figure 3.11 Impact and seismic (vertical) hazards.
Figure 3.12 Amplitudes of live loads and horizontal seismic loads.
We can now perform all needed computations to form DSMHIM. Since the structural system is the same as in Figure 3.7, the lumped mass vector, {M}, is the same as in Table 3.19 (see Appendix 3E). Also, the natural frequency set of the structure, {ω}, is shown in Table 3.21 (see Appendix 3E). The first 10, i = 1, 2, …, 10, natural modal amplitudes {ϕ}_{i} are shown in Table 3.22 (see Appendix 3E). The amplitudes of the three hazards, {P_{i}}, with i = 1 for LLs, i = 2 for impact load vector, and i = 3 for seismic load vector, are shown in Table 3.28 (see Appendix 3G). The static displacement vector for the LL hazard, i = 1; the dynamic displacement vector for impact load, with i = 2; and the horizontal seismic load, i = 3, are shown in Table 3.29 (see Appendix 3G).
Using Equations 3.26 and 3.32, the threehazard DSMHIM (accounting for all DOFs) is computed as
We note that there are negligible affinities (interactions) between the LL hazard and both the impact and horizontal seismic loads. This result is expected since the LLs are vertical and both seismic and impact loads are horizontal. We also note that the affinity (interaction) between impact load and horizontal seismic loads is still relatively small at ~9.07%, almost three times the affinity between the same impact load and the vertical seismic loads shown in Equation 3.31, even though the frequency spectra of the two hazards have not changed. This increase is due to the fact that the two hazards of this example are applied in the same horizontal direction. This shows that, as expected, the affinity between hazards within a structural system is affected by both frequency spectra distributions and the spatial distribution of their amplitudes.
Consider the nonlinear analysis metric a_{ij}. This metric is a convenient metric that is a result of the ith hazard and has the capability of describing the nonlinear state of the jth point or location in the structural space. Some examples of a_{ij} are shown in Table 3.6.
Type of Metric a_{ij} 
Physical Locations, j 
Advantages/Disadvantages 

Strains at a given point 
A given point in any solid 

Crosssectional deformations 
Beams/trusses 

Ductility 
Solids, beams, trusses 

Plasticity 
A given point in any solid 

Principal strains 
A given point in any solid 

As usual, we define the entry MHI_{kℓ}_{j} as the interaction between the kth hazard and the ℓh hazard at the desired jth point/location; see Equation 3.8. It can be evaluated as
subject to the conditions
and
with a_{ELASTIC} representing the elastic limit of a. If conditions (3.37) and (3.38) are not satisfied, then
Reflecting on Equations 3.36 through 3.39, we make the following observations:
and
respectively. Averaging will result in lesser data points. On the other hand, by averaging, we might end with some unexpected or unrealistic results. Hence, the application of Equations 3.40 and 3.41 should be done carefully.
We show an example for developing MH nonlinear behavior interaction matrix, NLMHIM, using the same threestory framed building shown in Figure 3.2. We consider the five hazards: DL, LL, W, S, and B. All of these hazards were considered in some of the examples earlier in this chapter. There is one difference in this example. Recall that all types of MHIM we have considered so far were interested in the distribution of displacement profiles due to different hazards, that is, the relative magnitudes of responses. In the current example, in order to accurately consider nonlinear behavior, we are interested in absolute response of the different components of the building. This can be accomplished by applying the absolute hazards loads and determining the actual responses including nonlinear effect, if any, rather than the relative profile responses. Figure 3.13 shows a simulated heavily damaged building.
Figure 3.13 Heavily damaged building (simulated), Wonderworks Tourist Attraction in Orlando, FL.
For simplicity, we will use maximum ductility in each of the 12 elements of the building as the metric for computing NLMHIM (see Table 3.6 for the description of different nonlinear metrics). Let us assume that the building is analyzed for the five hazards, and the maximum ductility in each element is calculated as shown in Table 3.7. The following can be observed from the data shown in Table 3.7:
Element No. 
Ductility Measure for Each Element for Each Hazard 


DL 
LL 
W 
S 
B 

1 
0.250 
0.375 
0.400 
5.500 
8.100 
2 
0.200 
0.300 
0.300 
3.500 
1.500 
3 
0.100 
0.150 
0.200 
4.200 
1.200 
4 
0.150 
0.225 
0.600 
5.100 
3.400 
5 
0.150 
0.225 
0.400 
4.500 
1.200 
6 
0.150 
0.225 
0.300 
1.500 
0.800 
7 
0.150 
0.225 
0.600 
5.100 
3.400 
8 
0.150 
0.225 
0.400 
4.500 
1.200 
9 
0.150 
0.225 
0.300 
1.500 
0.800 
10 
0.250 
0.375 
0.400 
5.500 
0.400 
11 
0.200 
0.300 
0.300 
3.500 
0.200 
12 
0.100 
0.150 
0.200 
4.200 
0.100 
We compute NLMHIM using only the computed ductility of element number 1 (the column that is exposed to direct blast effects) using Equations 3.36 through 3.39. The resulting matrix is shown in Table 3.8. We can make the following observations after studying the matrix:
DL 
LL 
W 
S 
B 


DL 
1.00 
0.67 
0.63 
−0.05 
−0.03 
LL 
0.67 
1.00 
0.94 
−0.07 
−0.05 
W 
0.63 
0.94 
1.00 
−0.07 
−0.05 
S 
−0.05 
−0.07 
−0.07 
1.00 
−0.68 
B 
−0.03 
−0.05 
−0.05 
−0.68 
1.00 
We now compute the NLMHIM matrix for element number 3 (column of the third floor) as shown in Table 3.9. We see the matrix reflect the changes in the column position. Some observations are as follows:
DL 
LL 
W 
S 
B 


DL 
1.00 
0.67 
0.50 
−0.02 
−0.08 
LL 
0.67 
1.00 
0.75 
−0.04 
−0.13 
W 
0.50 
0.75 
1.00 
−0.05 
−0.17 
S 
−0.02 
−0.04 
−0.05 
1.00 
−0.29 
B 
−0.08 
−0.13 
−0.17 
−0.29 
1.00 
If we are interested in the average NLMHIM for the whole building, we have to apply Equations 3.40 and 3.41 using all 12 elements in the model. The resulting matrix is shown in Table 3.10. Some observations of these results are as follows:
DL 
LL 
W 
S 
B 


DL 
1.00 
0.67 
0.49 
−0.05 
0.20 
LL 
0.67 
1.00 
0.73 
−0.07 
0.16 
W 
0.49 
0.73 
1.00 
−0.10 
0.12 
S 
−0.05 
−0.07 
−0.10 
1.00 
−0.37 
B 
0.20 
0.16 
0.12 
−0.37 
1.00 
These results indicated the need to have a proper mix of local and global NLMHIM analyses in order to arrive at accurate and useful conclusions regarding structural behavior and affinity/interactions between hazards.
We explored a method of developing MH interactions for nonlinear problems. It should be noted that there are many other approaches where we can develop interaction matrices for nonlinear situations. Some of these methods are given as follows:
Another way of considering MH for nonlinear problems is to use design metrics. We discuss this approach in Chapter 4.
One of the disadvantages of the MH formation methods so far is that they are an averaging process of the whole response field. We can reduce this limitation by not using {U}_{i} fully. A truncated vector {Ū}_{i}. might be used, where {Ū}_{i}. is a subset of {U}_{i} that includes only important entries that are of interest to the user. One practical way to establish a truncated vector {Ū}_{i} is to keep the maximum n displacements, (or any other metric of interest), with n ≤ N. The order of the truncated vector is $\overline{N}$ where $\overline{N}\le n\left(NH\right)$.
A limit of {Ū}_{i} is that it includes only a single nonzero cell, that is, no averaging in MHIC.
We repeat the example in Section 4.2.1, with n = 3. The resulting truncated displacement vector, {Ū}_{i}, is shown in Table 3.12 (see Appendix 3C) (Section 4.12). The resulting MHIM is
We repeat the example of Section 4.2.1, with n = 1 maxima points. The resulting truncated displacement vector, {Ū}_{i}, is shown in Table 3.13 (see Appendix 3D). The resulting MHIM is
Note that even though we chose n = 1, we have $\overline{N}=2$. Again, the results are selfevident from Tables 3.11 and 3.13 (see Appendices 3A and 3D) for the three hazards of interest.
The methods of this chapter can be extended to accommodate other forms of MH analysis matrices. For example, in problems that include initial stresses (prestressed components) or initial strains (temperature), we can easily modify the approach, for example, Equations 3.1 through 3.10, to build the required MH matrices.
We also note that there are several other forms for considering the interactions. For example, one can follow the methods offered by Allemang (2003) to study interactions between computed natural modes to produce the desired MH interaction matrices. One of the potentially more beneficial improvements to the processes discussed in this chapter would be introducing a weighting vector to emphasize the importance of some responses in {U}_{i}, or any other response measure, over some other responses. Equations 3.1 through 3.10 can be modified accordingly to produce a weighted average MHIM.
Hazard Displacements/Rotations 


Node 
Direction 
Dead Load (DL) 
Live Load (LL) 
Wind (W) 
1 
x_{1} 
0 
0 
0 
x_{2} 
0 
0 
0 

θ_{3} 
0 
0 
0 

2 
x_{1} 
−2.35289E−08 
−3.35928E−08 
6.45599E−05 
x_{2} 
−8.33333E−07 
−1.44448E−06 
6.07238E−07 

θ_{3} 
−7.01495E−08 
−1.33375E−07 
−2.65857E−07 

3 
x_{1} 
−1.32312E−08 
−1.01798E−08 
0.000135245 
x_{2} 
−1.38889E−06 
−2.58338E−06 
9.34493E−07 

θ_{3} 
−5.28393E−08 
−1.54046E−07 
−2.17823E−07 

4 
x_{1} 
6.84789E−08 
4.49827E−07 
0.000182085 
x_{2} 
−1.66667E−06 
−3.23616E−06 
1.03494E−06 

θ_{3} 
−1.00722E−07 
−4.16253E−07 
−9.9917E−08 

5 
x_{1} 
−4.90764E−12 
3.53005E−08 
6.45599E−05 
x_{2} 
−1.30062E−05 
−2.52251E−05 
7.21792E−12 

θ_{3} 
−8.86358E−15 
−6.3135E−10 
1.27868E−07 

6 
x_{1} 
−9.06117E−12 
1.13841E−07 
0.000135245 
x_{2} 
−1.20038E−05 
−3.40927E−05 
1.1285E−11 

θ_{3} 
−3.33611E−15 
−7.38808E−10 
1.01124E−07 

7 
x_{1} 
−1.0917E−11 
1.96642E−07 
0.000182085 
x_{2} 
−1.6591E−05 
−6.41951E−05 
1.59074E−12 

θ_{3} 
−1.80851E−15 
−6.53679E−10 
4.13341E−08 

8 
x_{1} 
0 
0 
0 
x_{2} 
0 
0 
0 

θ_{3} 
0 
0 
0 

9 
x_{1} 
2.35191E−08 
1.04194E−07 
6.45599E−05 
x_{2} 
−8.33334E−07 
−1.66663E−06 
−6.07233E−07 

θ_{3} 
7.01495E−08 
1.32198E−07 
−2.65857E−07 

10 
x_{1} 
1.32131E−08 
2.37862E−07 
0.000135245 
x_{2} 
−1.38889E−06 
−2.84718E−06 
−9.34482E−07 

θ_{3} 
5.28393E−08 
1.52605E−07 
−2.17823E−07 

11 
x_{1} 
−6.85007E−08 
−5.65429E−08 
0.000182085 
x_{2} 
−1.66667E−06 
−3.47218E−06 
−1.03493E−06 

θ_{3} 
1.00722E−07 
4.14934E−07 
−9.99169E−08 
Note: Displacement units are in inches, while rotational units are in radians.
The equation of motion of forced vibration of a linear elastic structural system (see Clough and Penzien 1975 or Biggs 1964) is expressed by
The mass, damping, and stiffness matrices are [M], [C], and [K], respectively. The timedependent displacements, velocity, and acceleration vectors of the system are {U(t), $\left\{\dot{U}\left(t\right)\right\}$, and {Ü(t}}, respectively. The spatial distribution of the dynamic force is {P} and the time dependency is expressed by f(t).
Following Clough and Penzien (1975), or Biggs (1964), the displacement vector can be expanded in a series form such that
The ith component of {D(t)} is d_{i}(t), which is the dynamic solution of the modal equation
where ω_{i} is the ith eigenvalue of the linear eigenvalue problem, such that
where β_{i} is the modal damping of the system. We also recognize the eigenvectors {ϕ}_{i} as the ith column of [Φ] in Equation 3.45. The ith component in the diagonal matrix (Γ) contains modal participation factor Γ_{i}, which are expressed as
We note that the solutions in (3.46) and (3.47) are valid only for an undamped system ([C] = [0]) or for proportional damping. Such assumptions are fairly acceptable in most civil infrastructure applications. If any of these assumptions are not valid, a quadratic eigenvalue problem has to be assembled in place of Equation 3.47. These situations are beyond the scope of this book.
Two helpful properties of [K], [M], and [Φ] are
and
Hazard Displacements/Rotations 


Node 
Direction 
Dead Load (DL) 
Live Load (LL) 
Wind (W) 
4 
x_{1} 
6.84789E−08 
4.49827E−07 
0.000182085 
5 
x_{2} 
−1.30062E−05 
−2.52251E−05 
7.21792E−12 
6 
x_{2} 
−1.20038E−05 
−3.40927E−05 
1.1285E−11 
7 
x_{1} 
−1.0917E−11 
1.96642E−07 
0.000182085 
7 
x_{2} 
−1.6591E−05 
−6.41951E−05 
1.59074E−12 
11 
x_{1} 
−6.85007E−08 
−5.65429E−08 
0.000182085 
Note: Displacement units are in inches, while rotational units are in radians.
Hazard Displacements/Rotations 


Node 
Direction 
Dead Load (DL) 
Live Load (LL) 
Wind (W) 
7 
x_{2} 
−1.6591E−05 
−6.41951E−05 
1.59074E−12 
11 
x_{1} 
−6.85007E−08 
−5.65429E−08 
0.000182085 
Note: Displacement units are in inches, while rotational units are in radians.
Node 
Direction 
Mass (lb s^{2}/in.) 

1 
x_{1} 
1.94 
x_{2} 
1.94 

θ_{3} 
25.88 

2 
x_{1} 
9.70 
x_{2} 
9.70 

θ_{3} 
168.22 

3 
x_{1} 
9.70 
x_{2} 
9.70 

θ_{3} 
168.22 

4 
x_{1} 
7.76 
x_{2} 
7.76 

θ_{3} 
142.34 

5 
x_{1} 
11.65 
x_{2} 
11.65 

θ_{3} 
232.92 

6 
x_{1} 
11.65 
x_{2} 
11.65 

θ_{3} 
232.92 

7 
x_{1} 
11.65 
x_{2} 
11.65 

θ_{3} 
232.92 

8 
x_{1} 
1.94 
x_{2} 
1.94 

θ_{3} 
25.88 

9 
x_{1} 
9.70 
x_{2} 
9.70 

θ_{3} 
168.22 

10 
x_{1} 
9.70 
x_{2} 
9.70 

θ_{3} 
168.22 

11 
x_{1} 
7.76 
x_{2} 
7.76 

θ_{3} 
142.34 
Node 
Direction 
Amplitudes of Vibrating Loads (lb) 


VL_1 
VL_2 
VL_3 

1 
x_{1} 
0.00 
0.00 
0.00 
x_{2} 
0.00 
0.00 
0.00 

θ_{3} 
0.00 
0.00 
0.00 

2 
x_{1} 
0.00 
0.00 
0.00 
x_{2} 
0.00 
0.00 
0.00 

θ_{3} 
0.00 
0.00 
0.00 

3 
x_{1} 
0.00 
0.00 
0.00 
x_{2} 
0.00 
0.00 
0.00 

θ_{3} 
0.00 
0.00 
0.00 

4 
x_{1} 
0.00 
0.00 
0.00 
x_{2} 
0.00 
0.00 
0.00 

θ_{3} 
0.00 
0.00 
0.00 

5 
x_{1} 
0.00 
0.00 
0.00 
x_{2} 
−1.00 
0.00 
0.00 

θ_{3} 
0.00 
0.00 
0.00 

6 
x_{1} 
0.00 
0.00 
0.00 
x_{2} 
0.00 
−1.00 
0.00 

θ_{3} 
0.00 
0.00 
0.00 

7 
x_{1} 
0.00 
0.00 
0.00 
x_{2} 
0.00 
0.00 
−1.00 

θ_{3} 
0.00 
0.00 
0.00 

8 
x_{1} 
0.00 
0.00 
0.00 
x_{2} 
0.00 
0.00 
0.00 

θ_{3} 
0.00 
0.00 
0.00 

9 
x_{1} 
0.00 
0.00 
0.00 
x_{2} 
0.00 
0.00 
0.00 

θ_{3} 
0.00 
0.00 
0.00 

10 
x_{1} 
0.00 
0.00 
0.00 
x_{2} 
0.00 
0.00 
0.00 

θ_{3} 
0.00 
0.00 
0.00 

11 
x_{1} 
0.00 
0.00 
0.00 
x_{2} 
0.00 
0.00 
0.00 

θ_{3} 
0.00 
0.00 
0.00 
Order 
Frequency (Hz) 

1 
2.89 
2 
8.59 
3 
11.20 
4 
12.50 
5 
13.26 
6 
13.35 
7 
45.87 
8 
46.97 
9 
112.27 
10 
113.10 
11 
125.50 
12 
125.87 
13 
126.23 
14 
177.17 
15 
177.37 
16 
182.85 
17 
183.04 
18 
191.24 
19 
315.61 
20 
331.41 
21 
347.35 
22 
415.89 
23 
439.32 
24 
459.99 
25 
521.80 
26 
531.84 
27 
541.58 
Node 
Direction 
Modal Order 


Mode # 1 
Mode # 2 
Mode # 3 
Mode # 4 
Mode # 5 
Mode # 6 
Mode # 7 
Mode # 8 
Mode # 9 
Mode # 10 

1 
x_{1} 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
x_{2} 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 

θ_{3} 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 

2 
x_{1} 
5.38E−02 
−1.28E−01 
3.62E−04 
5.16E−04 
4.83E−04 
−1.13E−01 
−7.31E−04 
4.56E−04 
1.54E−01 
1.67E−01 
x_{2} 
4.61E−04 
7.81E−04 
−1.82E−03 
1.09E−03 
5.45E−03 
−1.39E−04 
−8.07E−02 
−8.05E−02 
2.18E−03 
−2.28E−03 

θ_{3} 
−2.17E−04 
1.75E−04 
−3.44E−04 
−1.30E−03 
1.16E−03 
−2.51E−04 
2.65E−04 
4.81E−04 
−3.32E−04 
2.68E−04 

3 
x_{1} 
1.10E−01 
−6.56E−02 
−6.96E−04 
8.70E−04 
−4.58E−04 
1.26E−01 
−8.28E−04 
4.12E−04 
1.67E−01 
−1.54E−01 
x_{2} 
6.93E−04 
1.75E−03 
−3.21E−03 
4.16E−03 
8.47E−03 
−5.44E−04 
−1.43E−01 
−1.43E−01 
7.59E−04 
−9.24E−04 

θ_{3} 
−1.60E−04 
−4.92E−04 
8.19E−04 
1.14E−03 
1.08E−03 
−1.24E−04 
4.44E−04 
8.44E−04 
2.63E−04 
3.22E−04 

4 
x_{1} 
1.41E−01 
1.14E−01 
4.87E−04 
−7.98E−04 
−5.55E−04 
−6.23E−02 
2.23E−03 
−1.09E−03 
4.88E−03 
−5.98E−03 
x_{2} 
7.59E−04 
2.19E−03 
−5.42E−03 
5.33E−03 
9.06E−03 
−1.07E−03 
−1.75E−01 
−1.74E−01 
−1.08E−03 
2.38E−03 

θ_{3} 
−6.49E−05 
−4.10E−04 
−1.91E−03 
6.47E−04 
1.39E−04 
4.88E−04 
7.27E−04 
1.26E−03 
3.21E−04 
−3.51E−04 

5 
x_{1} 
5.38E−02 
−1.28E−01 
−8.71E−09 
−1.40E−08 
−7.16E−09 
−1.14E−01 
−8.12E−04 
−3.98E−10 
−1.01E−10 
1.06E−10 
x_{2} 
2.36E−09 
−9.87E−09 
−4.95E−02 
−2.00E−01 
2.08E−01 
2.77E−08 
−2.72E−08 
7.53E−03 
8.41E−04 
−6.52E−04 

θ_{3} 
1.04E−04 
−9.39E−05 
1.21E−12 
2.12E−11 
3.66E−12 
1.27E−04 
5.46E−04 
1.82E−09 
−1.55E−10 
−1.66E−10 

6 
x_{1} 
1.10E−01 
−6.58E−02 
4.70E−09 
1.81E−08 
−2.89E−09 
1.27E−01 
−9.21E−04 
9.81E−10 
−1.03E−10 
5.34E−12 
x_{2} 
4.85E−09 
−6.28E−09 
1.07E−01 
1.84E−01 
2.01E−01 
−3.36E−08 
−4.85E−08 
1.36E−02 
−7.41E−04 
−8.40E−04 

θ_{3} 
7.42E−05 
2.31E−04 
9.26E−12 
7.10E−12 
1.93E−11 
6.65E−05 
9.84E−04 
3.30E−09 
−5.86E−11 
3.61E−11 

7 
x_{1} 
1.41E−01 
1.15E−01 
3.87E−09 
−1.37E−08 
−7.40E−11 
−6.29E−02 
2.48E−03 
1.23E−08 
1.04E−10 
−8.66E−11 
x_{2} 
6.40E−09 
1.12E−08 
−2.68E−01 
1.10E−01 
4.10E−02 
−4.89E−09 
−4.32E−08 
1.23E−02 
−8.45E−04 
8.72E−04 

θ_{3} 
2.61E−05 
1.87E−04 
−5.56E−12 
−3.43E−11 
1.35E−11 
−2.35E−04 
1.10E−03 
3.74E−09 
1.53E−10 
5.16E−11 

8 
x_{1} 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
x_{2} 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 

θ_{3} 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 

9 
x_{1} 
5.38E−02 
−1.28E−01 
−3.62E−04 
−5.16E−04 
−4.83E−04 
−1.13E−01 
−7.31E−04 
−4.56E−04 
−1.54E−01 
−1.67E−01 
x_{2} 
−4.61E−04 
−7.81E−04 
−1.82E−03 
1.09E−03 
5.45E−03 
1.39E−04 
8.07E−02 
−8.05E−02 
2.18E−03 
−2.28E−03 

θ_{3} 
−2.17E−04 
1.75E−04 
3.44E−04 
1.30E−03 
−1.16E−03 
−2.51E−04 
2.65E−04 
−4.81E−04 
3.32E−04 
−2.68E−04 

10 
x_{1} 
1.10E−01 
−6.56E−02 
6.96E−04 
−8.70E−04 
4.58E−04 
1.26E−01 
−8.28E−04 
−4.12E−04 
−1.67E−01 
1.54E−01 
x_{2} 
−6.93E−04 
−1.75E−03 
−3.21E−03 
4.16E−03 
8.47E−03 
5.44E−04 
1.43E−01 
−1.43E−01 
7.59E−04 
−9.24E−04 

θ_{3} 
−1.60E−04 
−4.92E−04 
−8.19E−04 
−1.14E−03 
−1.08E−03 
−1.24E−04 
4.44E−04 
−8.44E−04 
−2.63E−04 
−3.22E−04 

11 
x_{1} 
1.41E−01 
1.14E−01 
−4.87E−04 
7.98E−04 
5.55E−04 
−6.23E−02 
2.23E−03 
1.09E−03 
−4.88E−03 
5.98E−03 
x_{2} 
−7.59E−04 
−2.19E−03 
−5.42E−03 
5.33E−03 
9.06E−03 
1.07E−03 
1.75E−01 
−1.74E−01 
−1.08E−03 
2.38E−03 

θ_{3} 
−6.49E−05 
−4.10E−04 
1.91E−03 
−6.47E−04 
−1.39E−04 
4.88E−04 
7.27E−04 
−1.26E−03 
−3.21E−04 
3.51E−04 
Node 
Direction 
Hazard Displacements/Rotations 


Dead Load (DL) 
Live Load (LL) 
Wind (W) 

1 
x_{1} 
0.0E+00 
0.0E+00 
0.0E+00 
x_{2} 
0.0E+00 
0.0E+00 
0.0E+00 

θ_{3} 
0.0E+00 
0.0E+00 
0.0E+00 

2 
x_{1} 
−1.3E−04 
3.2E−07 
−3.2E−12 
x_{2} 
6.0E−04 
6.2E−05 
7.5E−15 

θ_{3} 
−3.3E−06 
−4.0E−07 
−1.5E−14 

3 
x_{1} 
−1.4E−04 
−1.8E−06 
5.6E−12 
x_{2} 
1.1E−03 
2.0E−05 
−2.4E−14 

θ_{3} 
−6.6E−06 
−1.5E−07 
−9.6E−15 

4 
x_{1} 
4.1E−06 
1.3E−05 
7.2E−14 
x_{2} 
1.3E−03 
−5.6E−05 
2.3E−14 

θ_{3} 
−9.8E−06 
4.1E−07 
2.0E−14 

5 
x_{1} 
3.1E−12 
−8.7E−12 
5.2E−12 
x_{2} 
−5.7E−05 
−6.2E−07 
−6.2E−21 

θ_{3} 
−1.4E−11 
−5.8E−12 
8.8E−15 

6 
x_{1} 
−7.3E−12 
−2.1E−12 
−9.4E−12 
x_{2} 
−1.0E−04 
−1.3E−07 
8.0E−21 

θ_{3} 
−2.5E−11 
4.0E−12 
6.0E−15 

7 
x_{1} 
−9.3E−11 
3.6E−12 
−1.2E−13 
x_{2} 
−9.2E−05 
4.5E−07 
−2.2E−21 

θ_{3} 
−2.8E−11 
−6.7E−13 
−1.2E−14 

8 
x_{1} 
0.0E+00 
0.0E+00 
0.0E+00 
x_{2} 
0.0E+00 
0.0E+00 
0.0E+00 

θ_{3} 
0.0E+00 
0.0E+00 
0.0E+00 

9 
x_{1} 
1.3E−04 
−3.2E−07 
−3.2E−12 
x_{2} 
6.0E−04 
6.2E−05 
−7.5E−15 

θ_{3} 
3.3E−06 
4.0E−07 
−1.5E−14 

10 
x_{1} 
1.4E−04 
1.8E−06 
5.6E−12 
x_{2} 
1.1E−03 
2.0E−05 
2.4E−14 

θ_{3} 
6.6E−06 
1.5E−07 
−9.6E−15 

11 
x_{1} 
−4.1E−06 
−1.3E−05 
7.2E−14 
x_{2} 
1.3E−03 
−5.6E−05 
−2.3E−14 

θ_{3} 
9.8E−06 
−4.1E−07 
2.0E−14 
Note: Displacement units are in inches, while rotation units are in radians.
Node 
Direction 
Mass (lb s^{2}/in.) 

1 
x_{1} 
192.05 
x_{2} 
192.05 

θ_{3} 
5,418.61 

2 
x_{1} 
141.47 
x_{2} 
141.47 

θ_{3} 
2,749.74 

3 
x_{1} 
404.26 
x_{2} 
404.26 

θ_{3} 
12,292.96 

4 
x_{1} 
384.10 
x_{2} 
384.10 

θ_{3} 
13,586.96 

5 
x_{1} 
141.47 
x_{2} 
141.47 

θ_{3} 
2,749.74 

6 
x_{1} 
525.57 
x_{2} 
525.57 

θ_{3} 
16,336.70 

7 
x_{1} 
384.10 
x_{2} 
384.10 

θ_{3} 
13,586.96 

8 
x_{1} 
141.47 
x_{2} 
141.47 

θ_{3} 
2,749.74 

9 
x_{1} 
384.10 
x_{2} 
384.10 

θ_{3} 
13,586.96 

10 
x_{1} 
404.26 
x_{2} 
404.26 

θ_{3} 
12,292.96 

11 
x_{1} 
141.47 
x_{2} 
141.47 

θ_{3} 
2,749.74 

12 
x_{1} 
192.05 
x_{2} 
192.05 

θ_{3} 
5,418.61 
Node 
Direction 
Amplitudes of Vibrating Loads (lb) 


Impact Load 
Seismic Load 

1 
x_{1} 
0 
0 
x_{2} 
0 
0 

θ_{3} 
0 
0 

2 
x_{1} 
1 
0 
x_{2} 
0 
0.5 

θ_{3} 
0 
0 

3 
x_{1} 
0 
0 
x_{2} 
0 
0.75 

θ_{3} 
0 
0 

4 
x_{1} 
0 
0 
x_{2} 
0 
4 

θ_{3} 
0 
0 

5 
x_{1} 
0 
0 
x_{2} 
0 
0.5 

θ_{3} 
0 
0 

6 
x_{1} 
0 
0 
x_{2} 
0 
4 

θ_{3} 
0 
0 

7 
x_{1} 
0 
0 
x_{2} 
0 
1 

θ_{3} 
0 
0 

8 
x_{1} 
0 
0 
x_{2} 
0 
0.5 

θ_{3} 
0 
0 

9 
x_{1} 
0 
0 
x_{2} 
0 
4 

θ_{3} 
0 
0 

10 
x_{1} 
0 
0 
x_{2} 
0 
0.75 

θ_{3} 
0 
0 

11 
x_{1} 
0 
0 
x_{2} 
0 
0.5 

θ_{3} 
0 
0 

12 
x_{1} 
0 
0 
x_{2} 
0 
0 

θ_{3} 
0 
0 
Order 
Frequency (Hz) 

1 
1.74 
2 
3.72 
3 
4.33 
4 
4.39 
5 
4.81 
6 
5.85 
7 
5.92 
8 
7.88 
9 
13.88 
10 
15.78 
11 
16.90 
12 
19.24 
13 
19.56 
14 
20.92 
15 
23.74 
16 
25.76 
17 
30.10 
18 
44.56 
19 
46.00 
20 
47.35 
21 
48.19 
22 
48.71 
23 
51.84 
24 
53.72 
25 
71.90 
26 
73.33 
27 
74.67 
28 
81.29 
29 
81.70 
30 
133.94 
31 
135.47 
32 
137.83 
33 
138.13 
Node 
Direction 
Modal Order 


Mode # 1 
Mode # 2 
Mode # 3 
Mode # 4 
Mode #5 
Mode # 6 
Mode # 7 
Mode # 8 
Mode # 9 
Mode # 10 

1 
x_{1} 
−8.07E−03 
−9.17E−03 
1.74E−03 
−4.93E−04 
1.68E−02 
−4.03E−03 
−1.07E−02 
5.68E−03 
2.03E−02 
1.76E−02 
x_{2} 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 

θ_{3} 
6.17E−05 
−2.02E−04 
−3.42E−04 
−4.54E−05 
−2.06E−04 
−1.78E−04 
−4.47E−05 
−2.97E−05 
−2.91E−05 
−1.82E−05 

2 
x_{1} 
−4.05E−03 
−2.54E−02 
−2.83E−02 
−7.49E−03 
−1.13E−04 
−1.97E−02 
−1.92E−02 
−4.29E−05 
4.93E−03 
1.60E−02 
x_{2} 
9.00E−03 
−3.00E−02 
−5.16E−02 
−8.39E−03 
−3.03E−02 
−2.38E−02 
−8.50E−03 
−6.42E−03 
−3.75E−03 
−1.02E−02 

θ_{3} 
4.50E−05 
−1.73E−05 
1.19E−05 
1.19E−05 
−1.14E−06 
1.20E−04 
8.20E−05 
3.66E−05 
−4.90E−05 
−3.34E−05 

3 
x_{1} 
−1.88E−03 
−1.18E−02 
1.37E−03 
1.99E−03 
1.54E−02 
1.33E−04 
−6.56E−03 
6.97E−03 
−7.89E−03 
1.62E−02 
x_{2} 
1.49E−02 
−1.03E−02 
−3.61E−03 
1.06E−02 
−5.69E−03 
1.12E−02 
1.72E−02 
8.83E−03 
−7.19E−04 
−7.38E−03 

θ_{3} 
3.16E−05 
9.64E−05 
2.38E−04 
1.75E−04 
1.15E−04 
−9.99E−05 
7.37E−06 
2.49E−05 
−3.65E−05 
2.88E−05 

4 
x_{1} 
−6.27E−03 
−1.08E−02 
9.98E−04 
2.13E−03 
1.46E−02 
−1.56E−03 
−5.29E−03 
7.30E−03 
2.71E−02 
9.37E−04 
x_{2} 
1.51E−02 
−1.13E−02 
−4.39E−03 
1.20E−02 
−6.70E−03 
1.35E−02 
2.10E−02 
1.35E−02 
5.92E−03 
1.62E−02 

θ_{3} 
2.71E−05 
1.04E−05 
2.99E−05 
2.80E−05 
3.08E−05 
5.17E−05 
−3.15E−05 
−1.68E−05 
−3.03E−05 
3.23E−05 

5 
x_{1} 
−4.78E−03 
−1.56E−02 
−1.06E−02 
−2.17E−02 
1.30E−02 
3.45E−02 
1.62E−03 
2.52E−03 
−1.61E−04 
−1.67E−04 
x_{2} 
2.07E−02 
−2.39E−03 
1.90E−02 
4.71E−02 
1.94E−03 
−4.75E−02 
−3.83E−03 
5.50E−03 
−1.90E−05 
−2.82E−04 

θ_{3} 
2.05E−05 
−8.09E−06 
−3.23E−05 
−2.28E−05 
2.80E−05 
−6.81E−06 
−1.46E−04 
−7.82E−05 
−1.29E−05 
1.20E−04 

6 
x_{1} 
−4.36E−03 
−1.16E−02 
3.38E−04 
4.69E−03 
1.07E−02 
1.13E−03 
1.06E−03 
6.64E−03 
7.55E−03 
−1.69E−02 
x_{2} 
2.10E−02 
−6.98E−03 
3.65E−03 
−2.09E−04 
8.65E−03 
6.40E−03 
−1.03E−02 
−1.28E−02 
7.76E−04 
7.65E−03 

θ_{3} 
−9.26E−07 
−1.61E−05 
7.25E−06 
−2.67E−04 
2.60E−05 
4.76E−05 
1.04E−04 
−2.80E−05 
−2.68E−05 
1.30E−05 

7 
x_{1} 
−4.33E−03 
−9.56E−03 
7.16E−04 
2.10E−03 
1.20E−02 
−6.22E−04 
−3.11E−03 
1.03E−02 
−2.86E−02 
8.21E−03 
x_{2} 
2.14E−02 
−7.54E−03 
4.53E−03 
−2.18E−04 
9.79E−03 
7.72E−03 
−1.26E−02 
−1.94E−02 
−6.58E−03 
−1.69E−02 

θ_{3} 
2.50E−07 
−1.75E−06 
−7.79E−07 
−6.01E−05 
−3.50E−05 
−2.91E−05 
−6.28E−05 
1.43E−05 
−2.63E−05 
5.08E−05 

8 
x_{1} 
−4.28E−03 
−1.67E−02 
1.19E−02 
−2.11E−02 
1.13E−02 
−2.00E−02 
2.76E−02 
2.83E−03 
−5.63E−03 
−1.46E−02 
x_{2} 
2.04E−02 
−1.07E−02 
2.11E−02 
−4.64E−02 
7.53E−03 
−3.07E−02 
3.35E−02 
−6.91E−03 
3.84E−03 
8.98E−03 

θ_{3} 
−1.89E−05 
4.13E−05 
3.33E−05 
−1.78E−05 
−7.95E−05 
−6.71E−05 
−1.57E−05 
1.21E−04 
−4.67E−05 
1.65E−06 

9 
x_{1} 
−2.19E−03 
−6.08E−03 
2.61E−04 
2.39E−03 
5.64E−03 
6.03E−04 
5.90E−04 
3.92E−03 
7.31E−03 
−2.28E−02 
x_{2} 
1.57E−02 
1.13E−02 
−1.16E−03 
−1.20E−02 
−8.22E−04 
−1.58E−03 
−1.35E−02 
2.89E−02 
3.54E−03 
−1.13E−02 

θ_{3} 
−2.62E−05 
3.07E−05 
−2.45E−05 
2.72E−05 
1.43E−05 
−4.89E−05 
3.45E−05 
2.82E−05 
−2.84E−05 
−8.01E−06 

10 
x_{1} 
−6.71E−03 
−6.66E−03 
−5.65E−06 
2.23E−03 
7.17E−03 
−1.29E−03 
8.51E−04 
1.05E−02 
−2.15E−02 
−1.01E−02 
x_{2} 
1.55E−02 
1.03E−02 
−7.83E−04 
−1.06E−02 
−1.12E−03 
−1.30E−03 
−1.11E−02 
1.89E−02 
−3.92E−04 
5.09E−03 

θ_{3} 
−2.61E−05 
1.49E−04 
−2.08E−04 
1.64E−04 
1.17E−04 
8.55E−05 
−5.56E−05 
4.66E−05 
−6.23E−05 
3.41E−06 

11 
x_{1} 
−4.88E−03 
−1.96E−02 
2.46E−02 
−5.47E−03 
−2.24E−02 
4.24E−03 
−1.50E−02 
1.27E−02 
−8.70E−03 
−2.73E−03 
x_{2} 
1.03E−02 
3.23E−02 
−4.14E−02 
5.65E−03 
4.28E−02 
−8.83E−03 
2.01E−02 
−2.29E−03 
−5.75E−03 
−2.19E−03 

θ_{3} 
−4.75E−05 
−3.48E−05 
−1.31E−05 
9.60E−06 
6.15E−05 
−3.89E−05 
9.34E−05 
−5.97E−05 
−3.39E−05 
−5.33E−05 

12 
x_{1} 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
x_{2} 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 

θ_{3} 
−7.04E−05 
−2.21E−04 
2.72E−04 
−2.75E−05 
−2.83E−04 
7.74E−05 
−1.47E−04 
6.42E−06 
1.22E−05 
4.31E−05 
Node 
Direction 
Hazard Displacements (In.)/Rotations (rad) 


Impact Load 
Seismic Load 

1 
x_{1} 
1.44E−03 
1.78E−03 
x_{2} 
0.00E+00 
0.00E+00 

θ_{3} 
2.23E−05 
5.19E−06 

2 
x_{1} 
3.94E−03 
1.44E−03 
x_{2} 
1.89E−03 
6.76E−04 

θ_{3} 
−7.47E−06 
7.11E−06 

3 
x_{1} 
8.53E−04 
1.30E−03 
x_{2} 
−1.03E−04 
1.80E−03 

θ_{3} 
−6.24E−08 
−1.14E−05 

4 
x_{1} 
−2.04E−04 
2.31E−03 
x_{2} 
7.37E−05 
2.67E−03 

θ_{3} 
−1.31E−05 
−2.04E−06 

5 
x_{1} 
−1.42E−03 
2.73E−03 
x_{2} 
−7.52E−04 
−1.54E−03 

θ_{3} 
9.65E−08 
−7.86E−06 

6 
x_{1} 
2.80E−04 
1.71E−03 
x_{2} 
1.45E−04 
−9.05E−04 

θ_{3} 
5.73E−06 
−7.57E−07 

7 
x_{1} 
−1.17E−04 
8.92E−04 
x_{2} 
8.20E−05 
−1.84E−03 

θ_{3} 
−1.73E−05 
−7.33E−07 

8 
x_{1} 
−5.60E−04 
1.05E−04 
x_{2} 
4.07E−04 
−1.63E−03 

θ_{3} 
−4.68E−06 
5.78E−06 

9 
x_{1} 
−3.91E−05 
1.05E−03 
x_{2} 
6.64E−05 
2.93E−03 

θ_{3} 
−1.35E−05 
5.74E−07 

10 
x_{1} 
1.15E−04 
9.17E−04 
x_{2} 
2.64E−05 
1.69E−03 

θ_{3} 
7.23E−06 
8.87E−06 

11 
x_{1} 
−1.73E−04 
8.62E−04 
x_{2} 
−5.11E−05 
1.81E−04 

θ_{3} 
−3.27E−06 
−5.92E−06 
Amplitudes of Loads (lb) 


Node 
Direction 
Wind Loads i = 1 
Horizontal Seismic Load i = 2 
1 
x_{1} 
0.00 
0.00 
x_{2} 
0.00 
0.00 

θ_{3} 
0.00 
0.00 

2 
x_{1} 
0.00 
0.00 
x_{2} 
1.00 
1.00 

θ_{3} 
0.00 
0.00 

3 
x_{1} 
0.00 
0.00 
x_{2} 
2.00 
1.00 

θ_{3} 
0.00 
0.00 

4 
x_{1} 
0.00 
0.00 
x_{2} 
3.00 
1.00 

θ_{3} 
0.00 
0.00 

5 
x_{1} 
0.00 
0.00 
x_{2} 
0.00 
1.00 

θ_{3} 
0.00 
0.00 

6 
x_{1} 
0.00 
0.00 
x_{2} 
0.00 
1.00 

θ_{3} 
0.00 
0.00 

7 
x_{1} 
0.00 
0.00 
x_{2} 
0.00 
1.00 

θ_{3} 
0.00 
0.00 

8 
x_{1} 
0.00 
0.00 
x_{2} 
0.00 
0.00 

θ_{3} 
0.00 
0.00 

9 
x_{1} 
0.00 
0.00 
x_{2} 
1.00 
1.00 

θ_{3} 
0.00 
0.00 

10 
x_{1} 
0.00 
0.00 
x_{2} 
2.00 
1.00 

θ_{3} 
0.00 
0.00 

11 
x_{1} 
0.00 
0.00 
x_{2} 
3.00 
1.00 

θ_{3} 
0.00 
0.00 

12 
x_{1} 
0.00 
0.00 
x_{2} 
0.00 
0.00 

θ_{3} 
0.00 
0.00 
Node 
Direction 
Hazard Displacements/Rotations 


Wind Loads i = 1 
Horizontal Seismic Load i = 2 

1 
x_{1} 
0.00E+00 
0.00E+00 
x_{2} 
0.00E+00 
0.00E+00 

θ_{3} 
0.00E+00 
0.00E+00 

2 
x_{1} 
9.32E−12 
3.24E−04 
x_{2} 
1.67E−06 
6.46E−02 

θ_{3} 
−3.42E−14 
1.38E−04 

3 
x_{1} 
1.89E−11 
2.15E−04 
x_{2} 
3.06E−06 
1.15E−01 

θ_{3} 
−2.63E−14 
−1.65E−04 

4 
x_{1} 
2.44E−11 
3.64E−04 
x_{2} 
3.89E−06 
1.39E−01 

θ_{3} 
−8.53E−15 
−3.72E−04 

5 
x_{1} 
9.32E−12 
−1.20E−09 
x_{2} 
1.67E−06 
8.62E−02 

θ_{3} 
2.28E−14 
1.50E−11 

6 
x_{1} 
1.89E−11 
−4.85E−09 
x_{2} 
3.06E−06 
8.59E−02 

θ_{3} 
1.68E−14 
1.07E−11 

7 
x_{1} 
2.44E−11 
−6.02E−09 
x_{2} 
3.89E−06 
8.58E−02 

θ_{3} 
1.01E−14 
−2.40E−11 

8 
x_{1} 
0.00E+00 
0.00E+00 
x_{2} 
0.00E+00 
0.00E+00 

θ_{3} 
0.00E+00 
0.00E+00 

9 
x_{1} 
9.32E−12 
−3.24E−04 
x_{2} 
1.67E−06 
6.46E−02 

θ_{3} 
−3.60E−14 
−1.38E−04 

10 
x_{1} 
1.89E−11 
−2.15E−04 
x_{2} 
3.06E−06 
1.15E−01 

θ_{3} 
−2.49E−14 
1.65E−04 

11 
x_{1} 
2.44E−11 
−3.64E−04 
x_{2} 
3.89E−06 
1.39E−01 

θ_{3} 
−8.98E−15 
3.72E−04 
Note: Displacement units are in inches, while rotation units are in radians.
Node 
Direction 
Blast (Impact) Load (Displacements are in inches and rotations are in radians) 

1 
x_{1} 
0.00 
x_{2} 
0.00 

θ_{3} 
0.00 

2 
x_{1} 
0.00 
x_{2} 
1.00 

θ_{3} 
0.00 

3 
x_{1} 
0.00 
x_{2} 
0.00 

θ_{3} 
0.00 

4 
x_{1} 
0.00 
x_{2} 
0.00 

θ_{3} 
0.00 

5 
x_{1} 
0.00 
x_{2} 
0.00 

θ_{3} 
0.00 

6 
x_{1} 
0.00 
x_{2} 
0.00 

θ_{3} 
0.00 

7 
x_{1} 
0.00 
x_{2} 
0.00 

θ_{3} 
0.00 

8 
x_{1} 
0.00 
x_{2} 
0.00 

θ_{3} 
0.00 

9 
x_{1} 
0.00 
x_{2} 
0.00 

θ_{3} 
0.00 

10 
x_{1} 
0.00 
x_{2} 
0.00 

θ_{3} 
0.00 

11 
x_{1} 
0.00 
x_{2} 
0.00 

θ_{3} 
0.00 

12 
x_{1} 
0.00 
x_{2} 
0.00 

θ_{3} 
0.00 
Node 
Direction 
Blast (Impact) Hazard Displacements/Rotations 

1 
x_{1} 
0.00E+00 
x_{2} 
0.00E+00 

θ_{3} 
0.00E+00 

2 
x_{1} 
−2.74E−05 
x_{2} 
1.03E−01 

θ_{3} 
−5.97E−04 

3 
x_{1} 
−6.66E−05 
x_{2} 
1.64E−05 

θ_{3} 
5.18E−05 

4 
x_{1} 
−6.21E−05 
x_{2} 
−3.59E−06 

θ_{3} 
−6.88E−06 

5 
x_{1} 
−2.48E−05 
x_{2} 
6.26E−05 

θ_{3} 
−3.13E−04 

6 
x_{1} 
−5.01E−05 
x_{2} 
−1.26E−05 

θ_{3} 
3.07E−05 

7 
x_{1} 
−6.49E−05 
x_{2} 
2.23E−06 

θ_{3} 
−1.15E−06 

8 
x_{1} 
0.00E+00 
x_{2} 
0.00E+00 

θ_{3} 
0.00E+00 

9 
x_{1} 
−2.27E−05 
x_{2} 
8.49E−07 

θ_{3} 
1.08E−04 

10 
x_{1} 
−4.48E−05 
x_{2} 
6.30E−06 

θ_{3} 
−4.14E−05 

11 
x_{1} 
−6.70E−05 
x_{2} 
−2.05E−06 

θ_{3} 
7.43E−06 

12 
x_{1} 
0.00E+00 
x_{2} 
0.00E+00 

θ_{3} 
0.00E+00 
Note: Displacement units are in inches, while rotation units are in radians.
Node 
Direction 
Amplitudes of Vibrating Loads (lb) 


Live Load, i = 1 
Impact Load, i = 2 
Seismic Load, i = 3 

1 
x_{1} 
0 
0 
0.5 
x_{2} 
0 
0 
0 

θ_{3} 
0 
0 
0 

2 
x_{1} 
0 
1 
0.5 
x_{2} 
0 
0 
0 

θ_{3} 
0 
0 
0 

3 
x_{1} 
0 
0 
0.75 
x_{2} 
0 
0 
0 

θ_{3} 
0 
0 
0 

4 
x_{1} 
0 
0 
4 
x_{2} 
1 
0 
0 

θ_{3} 
0 
0 
0 

5 
x_{1} 
0 
0 
0.5 
x_{2} 
0 
0 
0 

θ_{3} 
0 
0 
0 

6 
x_{1} 
0 
0 
4 
x_{2} 
1 
0 
0 

θ_{3} 
0 
0 
0 

7 
x_{1} 
0 
0 
1 
x_{2} 
0 
0 
0 

θ_{3} 
0 
0 
0 

8 
x_{1} 
0 
0 
0.5 
x_{2} 
0 
0 
0 

θ_{3} 
0 
0 
0 

9 
x_{1} 
0 
0 
4 
x_{2} 
1 
0 
0 

θ_{3} 
0 
0 
0 

10 
x_{1} 
0 
0 
0.75 
x_{2} 
0 
0 
0 

θ_{3} 
0 
0 
0 

11 
x_{1} 
0 
0 
0 
x_{2} 
0 
0 
0 

θ_{3} 
0 
0 
0 

12 
x_{1} 
0 
0 
0 
x_{2} 
0 
0 
0 

θ_{3} 
0 
0 
0 
Node 
Direction 
Hazard Displacements/Rotations 


Live Load, i = 1 (Static Displacement) 
Impact Load, i = 2 
Seismic Load, i = 3 (Dynamic Displacement) (Dynamic Displacement) 

1 
x_{1} 
−3.29E−06 
1.44E−03 
7.87E−03 
x_{2} 
0.00E+00 
0.00E+00 
0.00E+00 

θ_{3} 
2.75E−08 
2.23E−05 
−5.13E−06 

2 
x_{1} 
−1.55E−06 
3.94E−03 
5.48E−03 
x_{2} 
3.98E−06 
1.89E−03 
−8.38E−04 

θ_{3} 
2.11E−08 
−7.47E−06 
−2.11E−06 

3 
x_{1} 
−5.58E−07 
8.53E−04 
4.56E−03 
x_{2} 
6.71E−06 
−1.03E−04 
1.32E−03 

θ_{3} 
1.08E−08 
−6.24E−08 
3.52E−06 

4 
x_{1} 
−2.47E−06 
−2.04E−04 
8.69E−03 
x_{2} 
7.10E−06 
7.37E−05 
2.56E−03 

θ_{3} 
1.21E−08 
−1.31E−05 
−1.21E−06 

5 
x_{1} 
−1.35E−06 
−1.42E−03 
4.60E−03 
x_{2} 
8.37E−06 
−7.52E−04 
2.34E−03 

θ_{3} 
8.88E−09 
9.65E−08 
−2.02E−07 

6 
x_{1} 
−1.64E−06 
2.80E−04 
5.51E−03 
x_{2} 
9.22E−06 
1.45E−04 
1.68E−03 

θ_{3} 
−1.06E−14 
5.73E−06 
−6.00E−06 

7 
x_{1} 
−1.64E−06 
−1.17E−04 
1.23E−03 
x_{2} 
9.20E−06 
8.20E−05 
4.12E−04 

θ_{3} 
8.47E−15 
−1.73E−05 
−7.10E−06 

8 
x_{1} 
−1.94E−06 
−5.60E−04 
3.46E−03 
x_{2} 
8.37E−06 
4.07E−04 
1.48E−03 

θ_{3} 
−8.88E−09 
−4.68E−06 
−1.44E−05 

9 
x_{1} 
−8.18E−07 
−3.91E−05 
3.36E−03 
x_{2} 
7.10E−06 
6.64E−05 
1.40E−03 

θ_{3} 
−1.21E−08 
−1.35E−05 
−3.47E−06 

10 
x_{1} 
−2.73E−06 
1.15E−04 
7.30E−04 
x_{2} 
6.71E−06 
2.64E−05 
−3.29E−05 

θ_{3} 
−1.08E−08 
7.23E−06 
−4.01E−06 

11 
x_{1} 
−1.74E−06 
−1.73E−04 
−2.44E−04 
x_{2} 
3.98E−06 
−5.11E−05 
8.07E−04 

θ_{3} 
−2.11E−08 
−3.27E−06 
4.65E−06 

12 
x_{1} 
0.00E+00 
0.00E+00 
0.00E+00 
x_{2} 
0.00E+00 
0.00E+00 
0.00E+00 

θ_{3} 
−2.75E−08 
4.82E−06 
−7.86E−06 
Note: Displacement units are in inches, while rotation units are in radians.