Analysis in MH Environment

Authored by: Mohammed M. Ettouney , Sreenivas Alampalli

Multihazard Considerations in Civil Infrastructure

Print publication date:  December  2016
Online publication date:  November  2016

Print ISBN: 9781482208320
eBook ISBN: 9781315373959

10.1201/9781315373959-4

Abstract

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 finite-element methods (Reddy 2005, Yang 1985, Zienkiewicz and Taylor 2011).

3.1.1  Overview

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 finite-element 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 petroleum-hydrocarbon contaminated site. Under a risk-based 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 multihazard-resistant multicolumn pier–bent concept relying on concrete-filled 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 single-degree-of-freedom dynamic analysis and a fiber-based dynamic analysis. In the fiber-based 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.

3.1.2  This chapter

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.

3.2  MH Interaction Matrix

Let us consider the popular equilibrium equation

3.1()$[ K ] { U } = { P }$

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

3.2()${ U } = [ K ] − 1 { P }$

This can be rewritten as

3.3()${ U } = [ ℑ ] { P }$
where the flexibility matrix is defined as
3.4()$[ ℑ ] = [ K ] − 1$

Let us also define ith demand/hazard as

3.5()${ D } i = { P }$

For the ith hazard/demand, we express the ith structural response as

3.6()${ U } i = [ ℑ ] { D } i$

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

3.7()$α i j = { U } i T { U } i$

Note that αij 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 general-purpose MH interaction coefficient (MHIC) between the ith and the jth hazards as

3.8()$M H I C = M H I i j = ( α i j ) 2 ( α i i ) ( α j j )$

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

3.9()$0.0 ≤ M H I C ≤ 1.0$

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

3.10()$[ M H I M ] = [ M H I 11 M H I 1 j M H I 1 N H M H I j 1 M H I i j M H I j N H M H I N H 1 M H I N H j M H I N H N H ]$

where NH is the number of hazards of interest. Note that [MHIM] is a symmetric matrix with the order of NH.

3.2.1  Case Study 3.1: MH Statics of Building Frames

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 finite-element 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) x1, x2 and rotations (moments) θ3. 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.

Table 3.1   Dimensions of Beams and Columns

Depth (in.)

Width (in.)

Moment of Inertia (in.4)

Area (in.2)

Beams

24

12

13,824

288

Columns

12

12

1,728

144

Table 3.2   Hazards Magnitudes

Locations

Node Number

Direction

Wind (W)

2

x2

−0.5

−0.1

2

x1

0.25

3

x2

−0.5

−0.25

3

x1

0.5

4

x2

−0.5

−0.35

4

x1

1.0

5

x2

1.0

−2.0

6

x2

1.0

−3.0

7

x2

1.0

−4.0

9

x2

−0.5

−0.075

9

x1

0.25

10

x2

−0.5

−0.05

10

x1

0.5

11

x2

−0.5

−0.25

11

x1

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

3.11()$[ M H I M ] = [ 1.000 0.931 1.2 E − 12 0.931 1.000 2.6 E − 05 1.2 E − 12 2.6 E − 05 1.000 ]$

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 cost-effective.

Table 3.3   Order of Hazards in the MH Interaction Matrix

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:

• The matrix is symmetric.
• The diagonal of the matrix is unity.
• The interaction (affinity) between DL and LL is fairly high at 0.93. This is an expected result given the magnitudes and direction of the loads.
• The interaction (affinity) between W and LL is fairly low at 2.6E-05. This is an expected result, since most of the deformations of LL are vertical (x2 direction) while most of the deformation due to W is lateral (x1 direction). The same can be said in regard to the affinity between W and DL.

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 x1 and x2 directions (for a two-degree-of-freedom [2-DOF] geometry) and rotations in the θ3 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, x2 (vertical displacements) or on one type of displacements both x1 and x2 directions. For example, if we limit the computations only to x1 direction, we obtain

3.12()$[ M H I M ] = [ 1.000 0.479 6.2 E − 8 0.479 1.000 0.471 6.2 E − 8 0.471 1.000 ]$

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 x2 direction (vertical displacements) only, we obtain

3.13()$[ M H I M ] = [ 1.000 0.931 3.2 E − 11 0.931 1.000 1.4 E − 05 3.2 E − 11 1.4 E − 05 1.000 ]$

If we limit the computations to the nodal rotations, θ3, only, we get

3.14()$[ M H I M ] = [ 1.000 0.92 4.3 E − 15 0.92 1.000 2.7 E − 06 4.3 E − 15 2.7 E − 06 1.000 ]$

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.

3.2.3  Limitations of MHIM

There are some disadvantages of using MHIM in understanding the behavior and MH interactions. These disadvantages are as follows:

• All hazards need to be static.
• It handles only liner problems.
• It is an average, and in some situations, using “average” can be misleading.
• The normalization process ignores magnitude of forces/responses.

We address some of these limitations in the next few sections.

3.3  MHIM for Internal Forces: Force MH Interaction Matrix

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

3.15()${ u } i m ∈ { U } i$

If the member stiffness is [k]m, then we obtain the member internal forces as

3.16()${ p } i m = [ k ] i m { u } i m$

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

3.17()$α i j = { p } i m T { p } j m$

MH force interaction coefficient (MHIC) between the ith and the jth hazards can now be obtained using Equation 3.8 and the force MHIM (F-MHIM) is formed using Equation 3.10, subject to conditions (3.9).

3.3.1  Case Study 3.2: MH Static Analysis of Internal Forces

In order to form F-MHIM 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.

Table 3.4   Column End Forces

Hazard

DL

LL

W

x1

0.15237

0.289298

−1.74991

x2

3

5.200142

−2.18605

θ3

−6.11174

−11.5961

116.4794

x1

−0.15237

−0.2893

1.749906

x2

−3

−5.20014

2.186046

θ3

−12.1727

−23.1197

93.5093

We then apply Equations 3.17 and 3.10. The resulting F-MHIM, if we consider all six force measures, is

3.18()$[ F - M H I M ] = [ 1.000 0.99 0.76 0.99 1.000 0.77 0.76 0.77 1.000 ]$
If we want to study the F-MHIM for only that single column linear forces (axial, x2, and shear, x1) the resulting F-MHIM is
3.19()$[ F - M H I M ] = [ 1.000 0.99 0.66 0.99 1.000 0.65 0.66 0.65 1.000 ]$

For bending-only (θ3), the F-MHIM is

3.20()$[ F - M H I M ] = [ 1.000 1.000 0.82 1.000 1.000 0.82 0.82 0.82 1.000 ]$

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.

Table 3.5   Beam End Forces

Hazard

DL

LL

W

x1

−0.11292

−0.33069

0

x2

0.5

1.00011

−1.00794

θ3

28.83756

59.42586

-181.429

x1

0.112915

0.330688

0

x2

−0.5

−1.00011

1.007938

θ3

61.16245

120.5939

−1.6E−05

As before, we apply Equations 3.17 and 3.10. The resulting F-MHIM, if we consider all six force measures of the beam, is

3.21()$[ F - M H I M ] = [ 1.000 0.99 0.18 0.99 1.000 0.20 0.18 0.20 1.000 ]$

If we only consider the beam linear forces (axial, x2, and shear, x1), the resulting F-MHIM is

3.22()$[ F - M H I M ] = [ 1.000 0.99 0.95 0.99 1.000 0.90 0.95 0.90 1.000 ]$

For beam bending-only (θ3), the F-MHIM is

3.23()$[ F - M H I M ] = [ 1.000 1.000 0.18 1.000 1.000 0.90 0.18 0.90 1.000 ]$

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 F-MHIM. At the limit, all elements of the structural system can be included in the evaluation of F-MHIM. 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.

3.4  Dynamic Hazards: Evaluation for Dynamic Multihazard Interaction Coefficient (D-MHIC) and Multihazard Interaction Matrix (D-MHIM)

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

3.24()$α i j ( t ) = { U } i T { U } j = { D i ( t ) } T ⟨ Γ ⟩ T [ Φ ] T [ Φ ] ⟨ Γ ⟩ { D j ( t ) }$

Applying Equation 3.24 to Equations 3.8 through 3.10 will yield the desired D-MHIC and D-MHIM.

A practical simplification of Equation 3.24 is to use the maxima of {Di(t)} that might be considered as a spectral expression of di(t). As such, we can replace {Di(t)} in Equation 3.24 by {Si}, such that its components si satisfy

3.25()$s i = MAX ( d i ( t ) )$

We can now write the time-independent expressions of αij as

3.26()$α i j = { S i } T ⟨ Γ ⟩ T [ Φ ] T [ Φ ] ⟨ Γ ⟩ { S i }$

Applying Equation 3.26 to Equations 3.8 through 3.10 will yield a time-independent form of D-MHIC and D-MHIM.

3.4.1  Case Study 3.3: MH Dynamics of Building Frames

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 D-MHIM, 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 s2/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 D-MHIM (accounting for all DOFs) is computed as

3.27()$[ D - M H I M ] = [ 1.000 8.60 E − 03 6.06 E − 16 8.60 E − 03 1.000 3.33 E − 16 6.06 E − 16 3.33 E − 16 1.000 ]$

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 D-MHIM is

3.28()$[ D - M H I M ] = [ 1.000 5.30 E − 02 3.62 E − 14 5.30 E − 02 1.000 1.58 E − 13 3.62 E − 14 1.58 E − 13 1.000 ]$
Again, there is scant interaction between the three vibrating loads. The interactions between the three vibrating modes is negligible, even though a small interaction of about 5% still exists between VL_1 and VL_2. Still, for all practical purposes, the interaction between the three vibrating loads is negligible.

Let us investigate the effects of frequency distributions on the interactions between vibrating loads. We change the spectrum of VL_3 to a wide-banded spectrum as shown in Figure 3.5.

Figure 3.5   Spectrum of wide-banded VL_3.

Again, using Equation 3.26, the corresponding D-MHIM (accounting for all DOFs) is now computed as

3.29()$[ D - M H I M ] = [ 1.000 8.60 E − 03 9.51 E − 01 8.60 E − 03 1.000 3.62 E − 02 9.51 E − 01 3.62 E − 02 1.000 ]$

Also, if we use only rotational DOF while executing Equation 3.26, the D-MHIM is

3.30()$[ D - M H I M ] = [ 1.000 5.30 E − 02 9.96 E − 01 5.30 E − 02 1.000 6.50 E − 02 9.96 E − 01 6.50 E − 02 1.000 ]$

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 ϕi and loading patterns {P} interact for each of the three hazards.

3.4.2  Case Study 3.4: MH Dynamics of Truss Bridges

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 high-demand 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 D-MHIM.

In order to explore the D-MHIS even further, we study impact load and earthquakes as applied to a simplified 2D-truss. The problem as offered is fairly simplified; however, it can be used with appropriate higher resolution for any real-life 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 2D-truss FEM of Figure 3.7. The elements are simple 2D beam elements (with rigid connections). We inserted a mid-distance 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/in2. 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 D-MHIM. Appendix 3E includes all the intermediate results: lumped mass vector, {M}, in Table 3.19 (see Appendix 3E); amplitudes of the two hazards, {Pi}, 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⟩{Si} 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 two-hazard D-MHIM (accounting for all DOFs) is computed as

3.31()$[ D - M H I M ] = [ 1.00 3.37 E − 02 3.37 E − 02 1.00 ]$
The affinity between the impact hazard and seismic (vertical) hazard as exemplified in D-MHIM is 3.37%, which is fairly low. This is not surprising given the large differences in the frequency spectra of the two hazards, as shown in Figure 3.8. We observe that we included all DOFs in this computation. A detailed study for a more limited DOF set, as well as force-based (F-MHIM) analysis, would reveal more information and may be helpful in the design/retrofit stages of this structure. Such a detailed study is recommended for practical situations; however, it is beyond the scope of this book.

3.5  Dynamic–Static Hazards: DS-MHIM

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 DS-MHIC and DS-MHIM, 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

3.32()$α i j = { U } i T [ Φ ] ⟨ Γ ⟩ { S j }$

Applying Equation 3.32 to Equations 3.8 through 3.10 will yield a time-independent form of DS-MHIC and DS-MHIM.

3.5.1  Case Study 3.5: Wind–Seismic Hazard Interaction Analysis of Building Frames

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, {Pi}, 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 three-hazard DS-MHIM (accounting for all DOFs) is computed as

3.33()$[ D - M H I M ] = [ 1.00 0.96 0.96 1.00 ]$

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 7-10 (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.

3.5.2  Case Study 3.6: Wind–Seismic–Blast Hazard Interaction Analysis of Building Frames

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 three-hazard DS-MHIM (accounting for all DOFs) is computed as

3.34()$[ D S - M H I M ] = [ 1.00 96.03 E − 02 3.40 E − 02 96.03 E − 02 1.00 4.38 E − 02 3.40 E − 02 4.38 E − 02 1.00 ]$
The affinities between the blast (impact) hazard and both wind and seismic hazards is fairly small at 3.4% and ~4.4%, respectively. This is due to the localized nature of blast (impact) loading as well as its wider frequency spectrum range. Again, the reader should note that blast and seismic loads usually produce large nonlinear effects, so affinities shown in the matrix of Equation 3.34 have to be utilized carefully.

Figure 3.10   Blast (impact) load distribution.

3.5.3  Case Study 3.7: Live Load–Seismic Hazard Interaction Analysis of Truss Bridges

We consider another situation where dynamic and static hazards affect another type of infrastructure. We reuse the geometry and properties of the simple 2D-truss 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.

We can now perform all needed computations to form DS-MHIM. 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, {Pi}, 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 three-hazard DS-MHIM (accounting for all DOFs) is computed as

3.35()$[ DS-MHIM ] = [ 1.00 7.58 E − 05 8.78 E − 06 7.58 E − 05 1.00 9.07 E − 02 8.78 E − 06 9.07 E − 02 1.00 ]$

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.

3.6.1  Theoretical Development

Consider the nonlinear analysis metric aij. 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 aij are shown in Table 3.6.

Table 3.6   MH Nonlinear Metrics

Type of Metric aij

Physical Locations, j

Strains at a given point

A given point in any solid

• Differentiates between tension and compression

Cross-sectional deformations

Beams/trusses

• Differentiates between tension and compression

Ductility

Solids, beams, trusses

• Popular and simple
• Can’t differentiate between tension and compression

Plasticity

A given point in any solid

• Applies popular plasticity measures such as Von Mises and Tresca (see Hill 1989)
• Can’t differentiate between tension and compression

Principal strains

A given point in any solid

• Uses maxima that can be beneficial for design
• Can’t differentiate between tension and compression

As usual, we define the entry MHIkℓ|j as the interaction between the kth hazard and the h hazard at the desired jth point/location; see Equation 3.8. It can be evaluated as

3.36()$M H I k ℓ | j = MIN ( a k j a ℓ j , a ℓ j a k j )$

subject to the conditions

3.37()$| a k j | ≤ | a E L A S T I C |$

and

3.38()$| a ℓ j | ≤ | a E L A S T I C |$

with aELASTIC representing the elastic limit of a. If conditions (3.37) and (3.38) are not satisfied, then

3.39()$M H I k ℓ | j = − MIN ( ( a k j a ℓ j ) n , ( a ℓ j a k j ) n )$

Reflecting on Equations 3.36 through 3.39, we make the following observations:

• Diagonal terms, i = j, are unity, as expected.
• The matrix NL-MHIM is symmetric, MHIkℓ|j = MHIℓk|j.
• A negative sign indicates the prevalence of nonlinear behavior of one or both hazards.
• Similarly, a positive sign indicates the prevalence of linear behavior of one or both hazards.
We can generalize Equations 3.36 and 3.39 to include as many points, NP, as desired by averaging the components
3.40()$M H I k ℓ = 1 N P ∑ j = 1 j = N P MIN ( a k j a ℓ j , a ℓ j a k j )$

and

3.41()$M H I k ℓ = 1 N P ∑ j = 1 j = N P [ − MIN ( ( a k j a ℓ j ) n , ( a ℓ j a k j ) n ) ]$

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.

3.6.2  Case Study 3.8: Using Ductility as a MH Metric

We show an example for developing MH nonlinear behavior interaction matrix, NL-MHIM, using the same three-story 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 NL-MHIM (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:

• The ductility in this example is assumed to be strain ductility. It is the ratio of the maximum strain computed within the element to the yield strain of the structure.
• For simplicity, we disregard the sign of the ductility.
• The blast loading is assumed to be in the neighborhood of element number 1 (first floor column).

Table 3.7   Ductility Measures of a Three-Story Building due to the Five Hazards

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 NL-MHIM 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:

• Affinities/interactions between the lower-magnitude hazards (DL, LL, and W) range between moderate and high (67%–94%).
• Affinities between lower-magnitude hazards (DL, LL, and W) and high-magnitude hazards (S and B) are fairly small in the range of 3%–7%. This is due to the fact that the high-magnitude hazards experience a high level of nonlinearity, whereas the low-magnitude hazards are all below the elastic limit.
• There is a high affinity between the S and B hazards at 68%.

Table 3.8   NL-MHIM of a Column Subjected to Blast Load

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 NL-MHIM 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:

• The affinity between S and B has been reduced to 29%. This is due to the fact that blast ductility demand has been attenuated much faster than the attenuation of the S ductility demands.
• Affinities between W and both DL and LL have been reduced.
• Affinities between DL, LL, and W on one hand and S and B on the other remain fairly small, in the range of 3%–17%, again, reflecting the nonlinear ductility demands of S and B. But there is a marked increase of B interaction with DL, LL, and W from the case of Table 3.8 due to the large attenuation of ductility demands of the B hazard.

Table 3.9   NL-MHIM of a Column away from Blast Load

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 NL-MHIM 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:

• The average affinities between low-demand hazards (DL, LL, and W) for the whole building are similar to those of single-element numbers 1 and 3.
• The average affinities between low-demand hazards (DL, LL, and W) and the S hazard for the whole building are similar to those of single-element numbers 1 and 3: they are still small and indicate high ductility demands of the S hazard.
• Perhaps the main change in NL-MHIM between average and single-element situations is in how the B hazard interacts with other hazards. Its affinity with all other hazards increases. This can be attributed to the fact that the DL, LL, W, and S hazards are all global in nature: they affect all parts of the building. The B hazard on the other hand is localized in nature; its main effects are felt by small portion of the building. Thus, by averaging, the affinities between global DL, LL, W, and S hazards and the localized B hazard will be changed greatly.

Table 3.10   NL-MHIM of All Elements

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 NL-MHIM analyses in order to arrive at accurate and useful conclusions regarding structural behavior and affinity/interactions between hazards.

3.6.3  Remarks

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:

• Use of principal strains as metrics instead of ductility. In this case, we lose the advantage of signs (as in tensile or compressive strains). Of course in many design situations, principal strains are more beneficial to understand than Cartesian strains.
• If the stakeholder requires stress metrics in nonlinear problems, perhaps using Von Mises, Tresca, or any other plasticity metric (Hill 1989) to form MH interaction matrices would offer an insight into the behavior.

Another way of considering MH for nonlinear problems is to use design metrics. We discuss this approach in Chapter 4.

3.7  Partial MH Indices

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 nN. The order of the truncated vector is $N ¯$ where $N ¯ ≤ n ( N H )$.

A limit of {Ū}i is that it includes only a single nonzero cell, that is, no averaging in MHIC.

3.7.1  Revisiting Case Study 3.1 with n = 3

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

3.42()$[ MHIM ] = [ 1.000 0.932 0.66 E − 12 0.932 1.000 1.96 E − 05 0.66 E − 12 1.96 E − 05 1.000 ]$
Note that while n = 3, we have $N ¯ = 6$. This is self-evident if we carefully study Tables 3.11 and 3.12 (see Appendices 3A and 3C) for the three hazards of interest.

3.7.2  Revisiting Case Study 3.1 with n = 1

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

3.43()$[ MHIM ] = [ 1.000 0.999 1.7 E − 05 0.999 1.000 7.76 E − 07 1.7 E − 05 7.76 E − 07 1.000 ]$

Note that even though we chose n = 1, we have $N ¯ = 2$. Again, the results are self-evident from Tables 3.11 and 3.13 (see Appendices 3A and 3D) for the three hazards of interest.

3.8  Other Forms of MH Analysis

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.

Table 3.11   Hazard Displacements/Rotations

Hazard Displacements/Rotations

Node

Direction

Wind (W)

1

x1

0

0

0

x2

0

0

0

θ3

0

0

0

2

x1

−2.35289E−08

−3.35928E−08

6.45599E−05

x2

−8.33333E−07

−1.44448E−06

6.07238E−07

θ3

−7.01495E−08

−1.33375E−07

−2.65857E−07

3

x1

−1.32312E−08

−1.01798E−08

0.000135245

x2

−1.38889E−06

−2.58338E−06

9.34493E−07

θ3

−5.28393E−08

−1.54046E−07

−2.17823E−07

4

x1

6.84789E−08

4.49827E−07

0.000182085

x2

−1.66667E−06

−3.23616E−06

1.03494E−06

θ3

−1.00722E−07

−4.16253E−07

−9.9917E−08

5

x1

−4.90764E−12

3.53005E−08

6.45599E−05

x2

−1.30062E−05

−2.52251E−05

7.21792E−12

θ3

−8.86358E−15

−6.3135E−10

1.27868E−07

6

x1

−9.06117E−12

1.13841E−07

0.000135245

x2

−1.20038E−05

−3.40927E−05

1.1285E−11

θ3

−3.33611E−15

−7.38808E−10

1.01124E−07

7

x1

−1.0917E−11

1.96642E−07

0.000182085

x2

−1.6591E−05

−6.41951E−05

1.59074E−12

θ3

−1.80851E−15

−6.53679E−10

4.13341E−08

8

x1

0

0

0

x2

0

0

0

θ3

0

0

0

9

x1

2.35191E−08

1.04194E−07

6.45599E−05

x2

−8.33334E−07

−1.66663E−06

−6.07233E−07

θ3

7.01495E−08

1.32198E−07

−2.65857E−07

10

x1

1.32131E−08

2.37862E−07

0.000135245

x2

−1.38889E−06

−2.84718E−06

−9.34482E−07

θ3

5.28393E−08

1.52605E−07

−2.17823E−07

11

x1

−6.85007E−08

−5.65429E−08

0.000182085

x2

−1.66667E−06

−3.47218E−06

−1.03493E−06

θ3

1.00722E−07

4.14934E−07

−9.99169E−08

Note: Displacement units are in inches, while rotational units are in radians.

Modal Analysis of Structures

The equation of motion of forced vibration of a linear elastic structural system (see Clough and Penzien 1975 or Biggs 1964) is expressed by

3.44()$[ M ] { U ¨ ( t ) } + [ C ] { U ˙ ( t ) } + [ K ] { U ( t ) } = f ( t ) { P }$

The mass, damping, and stiffness matrices are [M], [C], and [K], respectively. The time-dependent displacements, velocity, and acceleration vectors of the system are {U(t), ${ U ˙ ( t ) }$, 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

3.45()${ U ( t ) } = [ Φ ] ⟨ Γ ⟩ { D ( t ) }$

The ith component of {D(t)} is di(t), which is the dynamic solution of the modal equation

3.46()$d ¨ i + 2 β i ω i d ˙ i ( t ) + ω i 2 d i ( t ) = f ( t )$

where ωi is the ith eigenvalue of the linear eigenvalue problem, such that

3.47()$[ − ω 2 [ M ] + [ K ] ] { ϕ } = { 0 }$

where βi 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 Γi, which are expressed as

3.48()$Γ i = { ϕ } i T { P }$

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

3.49()$[ Φ ] T [ M ] [ Φ ] = ⟨ I ⟩$

and

3.50()$[ Φ ] T [ K ] [ Φ ] = ⟨ ω 2 ⟩$
I⟩ is the identity matrix. The ith component of the diagonal matrix ⟨ω2⟩ is $ω i 2$, which are also recognized as the squares of the ith natural frequencies of the system.

Table 3.12   Hazards Displacements/Rotations

Hazard Displacements/Rotations

Node

Direction

Wind (W)

4

x1

6.84789E−08

4.49827E−07

0.000182085

5

x2

−1.30062E−05

−2.52251E−05

7.21792E−12

6

x2

−1.20038E−05

−3.40927E−05

1.1285E−11

7

x1

−1.0917E−11

1.96642E−07

0.000182085

7

x2

−1.6591E−05

−6.41951E−05

1.59074E−12

11

x1

−6.85007E−08

−5.65429E−08

0.000182085

Note: Displacement units are in inches, while rotational units are in radians.

Table 3.13   Hazards Displacements/Rotations

Hazard Displacements/Rotations

Node

Direction

Wind (W)

7

x2

−1.6591E−05

−6.41951E−05

1.59074E−12

11

x1

−6.85007E−08

−5.65429E−08

0.000182085

Note: Displacement units are in inches, while rotational units are in radians.

Node

Direction

Mass (lb s2/in.)

1

x1

1.94

x2

1.94

θ3

25.88

2

x1

9.70

x2

9.70

θ3

168.22

3

x1

9.70

x2

9.70

θ3

168.22

4

x1

7.76

x2

7.76

θ3

142.34

5

x1

11.65

x2

11.65

θ3

232.92

6

x1

11.65

x2

11.65

θ3

232.92

7

x1

11.65

x2

11.65

θ3

232.92

8

x1

1.94

x2

1.94

θ3

25.88

9

x1

9.70

x2

9.70

θ3

168.22

10

x1

9.70

x2

9.70

θ3

168.22

11

x1

7.76

x2

7.76

θ3

142.34

Node

Direction

VL_1

VL_2

VL_3

1

x1

0.00

0.00

0.00

x2

0.00

0.00

0.00

θ3

0.00

0.00

0.00

2

x1

0.00

0.00

0.00

x2

0.00

0.00

0.00

θ3

0.00

0.00

0.00

3

x1

0.00

0.00

0.00

x2

0.00

0.00

0.00

θ3

0.00

0.00

0.00

4

x1

0.00

0.00

0.00

x2

0.00

0.00

0.00

θ3

0.00

0.00

0.00

5

x1

0.00

0.00

0.00

x2

−1.00

0.00

0.00

θ3

0.00

0.00

0.00

6

x1

0.00

0.00

0.00

x2

0.00

−1.00

0.00

θ3

0.00

0.00

0.00

7

x1

0.00

0.00

0.00

x2

0.00

0.00

−1.00

θ3

0.00

0.00

0.00

8

x1

0.00

0.00

0.00

x2

0.00

0.00

0.00

θ3

0.00

0.00

0.00

9

x1

0.00

0.00

0.00

x2

0.00

0.00

0.00

θ3

0.00

0.00

0.00

10

x1

0.00

0.00

0.00

x2

0.00

0.00

0.00

θ3

0.00

0.00

0.00

11

x1

0.00

0.00

0.00

x2

0.00

0.00

0.00

θ3

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

x1

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

x2

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

θ3

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

x1

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

x2

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

θ3

−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

x1

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

x2

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

θ3

−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

x1

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

x2

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

θ3

−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

x1

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

x2

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

θ3

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

x1

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

x2

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

θ3

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

x1

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

x2

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

θ3

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

x1

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

x2

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

θ3

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

x1

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

x2

−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

θ3

−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

x1

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

x2

−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

θ3

−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

x1

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

x2

−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

θ3

−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

Table 3.18   Dynamic Displacements of a Three−Story Building

Node

Direction

Hazard Displacements/Rotations

Wind (W)

1

x1

0.0E+00

0.0E+00

0.0E+00

x2

0.0E+00

0.0E+00

0.0E+00

θ3

0.0E+00

0.0E+00

0.0E+00

2

x1

−1.3E−04

3.2E−07

−3.2E−12

x2

6.0E−04

6.2E−05

7.5E−15

θ3

−3.3E−06

−4.0E−07

−1.5E−14

3

x1

−1.4E−04

−1.8E−06

5.6E−12

x2

1.1E−03

2.0E−05

−2.4E−14

θ3

−6.6E−06

−1.5E−07

−9.6E−15

4

x1

4.1E−06

1.3E−05

7.2E−14

x2

1.3E−03

−5.6E−05

2.3E−14

θ3

−9.8E−06

4.1E−07

2.0E−14

5

x1

3.1E−12

−8.7E−12

5.2E−12

x2

−5.7E−05

−6.2E−07

−6.2E−21

θ3

−1.4E−11

−5.8E−12

8.8E−15

6

x1

−7.3E−12

−2.1E−12

−9.4E−12

x2

−1.0E−04

−1.3E−07

8.0E−21

θ3

−2.5E−11

4.0E−12

6.0E−15

7

x1

−9.3E−11

3.6E−12

−1.2E−13

x2

−9.2E−05

4.5E−07

−2.2E−21

θ3

−2.8E−11

−6.7E−13

−1.2E−14

8

x1

0.0E+00

0.0E+00

0.0E+00

x2

0.0E+00

0.0E+00

0.0E+00

θ3

0.0E+00

0.0E+00

0.0E+00

9

x1

1.3E−04

−3.2E−07

−3.2E−12

x2

6.0E−04

6.2E−05

−7.5E−15

θ3

3.3E−06

4.0E−07

−1.5E−14

10

x1

1.4E−04

1.8E−06

5.6E−12

x2

1.1E−03

2.0E−05

2.4E−14

θ3

6.6E−06

1.5E−07

−9.6E−15

11

x1

−4.1E−06

−1.3E−05

7.2E−14

x2

1.3E−03

−5.6E−05

−2.3E−14

θ3

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 s2/in.)

1

x1

192.05

x2

192.05

θ3

5,418.61

2

x1

141.47

x2

141.47

θ3

2,749.74

3

x1

404.26

x2

404.26

θ3

12,292.96

4

x1

384.10

x2

384.10

θ3

13,586.96

5

x1

141.47

x2

141.47

θ3

2,749.74

6

x1

525.57

x2

525.57

θ3

16,336.70

7

x1

384.10

x2

384.10

θ3

13,586.96

8

x1

141.47

x2

141.47

θ3

2,749.74

9

x1

384.10

x2

384.10

θ3

13,586.96

10

x1

404.26

x2

404.26

θ3

12,292.96

11

x1

141.47

x2

141.47

θ3

2,749.74

12

x1

192.05

x2

192.05

θ3

5,418.61

Node

Direction

1

x1

0

0

x2

0

0

θ3

0

0

2

x1

1

0

x2

0

0.5

θ3

0

0

3

x1

0

0

x2

0

0.75

θ3

0

0

4

x1

0

0

x2

0

4

θ3

0

0

5

x1

0

0

x2

0

0.5

θ3

0

0

6

x1

0

0

x2

0

4

θ3

0

0

7

x1

0

0

x2

0

1

θ3

0

0

8

x1

0

0

x2

0

0.5

θ3

0

0

9

x1

0

0

x2

0

4

θ3

0

0

10

x1

0

0

x2

0

0.75

θ3

0

0

11

x1

0

0

x2

0

0.5

θ3

0

0

12

x1

0

0

x2

0

0

θ3

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

x1

−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

x2

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

θ3

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

x1

−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

x2

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

θ3

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

x1

−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

x2

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

θ3

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

x1

−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

x2

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

θ3

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

x1

−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

x2

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

θ3

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

x1

−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

x2

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

θ3

−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

x1

−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

x2

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

θ3

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

x1

−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

x2

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

θ3

−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

x1

−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

x2

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

θ3

−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

x1

−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

x2

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

θ3

−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

x1

−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

x2

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

θ3

−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

x1

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

x2

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

θ3

−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

1

x1

1.44E−03

1.78E−03

x2

0.00E+00

0.00E+00

θ3

2.23E−05

5.19E−06

2

x1

3.94E−03

1.44E−03

x2

1.89E−03

6.76E−04

θ3

−7.47E−06

7.11E−06

3

x1

8.53E−04

1.30E−03

x2

−1.03E−04

1.80E−03

θ3

−6.24E−08

−1.14E−05

4

x1

−2.04E−04

2.31E−03

x2

7.37E−05

2.67E−03

θ3

−1.31E−05

−2.04E−06

5

x1

−1.42E−03

2.73E−03

x2

−7.52E−04

−1.54E−03

θ3

9.65E−08

−7.86E−06

6

x1

2.80E−04

1.71E−03

x2

1.45E−04

−9.05E−04

θ3

5.73E−06

−7.57E−07

7

x1

−1.17E−04

8.92E−04

x2

8.20E−05

−1.84E−03

θ3

−1.73E−05

−7.33E−07

8

x1

−5.60E−04

1.05E−04

x2

4.07E−04

−1.63E−03

θ3

−4.68E−06

5.78E−06

9

x1

−3.91E−05

1.05E−03

x2

6.64E−05

2.93E−03

θ3

−1.35E−05

5.74E−07

10

x1

1.15E−04

9.17E−04

x2

2.64E−05

1.69E−03

θ3

7.23E−06

8.87E−06

11

x1

−1.73E−04

8.62E−04

x2

−5.11E−05

1.81E−04

θ3

−3.27E−06

−5.92E−06

Table 3.24   Amplitudes of the Two Hazards Applied to a Three-Story Building

Node

Direction

Horizontal Seismic Load i = 2

1

x1

0.00

0.00

x2

0.00

0.00

θ3

0.00

0.00

2

x1

0.00

0.00

x2

1.00

1.00

θ3

0.00

0.00

3

x1

0.00

0.00

x2

2.00

1.00

θ3

0.00

0.00

4

x1

0.00

0.00

x2

3.00

1.00

θ3

0.00

0.00

5

x1

0.00

0.00

x2

0.00

1.00

θ3

0.00

0.00

6

x1

0.00

0.00

x2

0.00

1.00

θ3

0.00

0.00

7

x1

0.00

0.00

x2

0.00

1.00

θ3

0.00

0.00

8

x1

0.00

0.00

x2

0.00

0.00

θ3

0.00

0.00

9

x1

0.00

0.00

x2

1.00

1.00

θ3

0.00

0.00

10

x1

0.00

0.00

x2

2.00

1.00

θ3

0.00

0.00

11

x1

0.00

0.00

x2

3.00

1.00

θ3

0.00

0.00

12

x1

0.00

0.00

x2

0.00

0.00

θ3

0.00

0.00

Table 3.25   Dynamic and Static Displacements of a Three-Story Building

Node

Direction

Hazard Displacements/Rotations

Horizontal Seismic Load i = 2

1

x1

0.00E+00

0.00E+00

x2

0.00E+00

0.00E+00

θ3

0.00E+00

0.00E+00

2

x1

9.32E−12

3.24E−04

x2

1.67E−06

6.46E−02

θ3

−3.42E−14

1.38E−04

3

x1

1.89E−11

2.15E−04

x2

3.06E−06

1.15E−01

θ3

−2.63E−14

−1.65E−04

4

x1

2.44E−11

3.64E−04

x2

3.89E−06

1.39E−01

θ3

−8.53E−15

−3.72E−04

5

x1

9.32E−12

−1.20E−09

x2

1.67E−06

8.62E−02

θ3

2.28E−14

1.50E−11

6

x1

1.89E−11

−4.85E−09

x2

3.06E−06

8.59E−02

θ3

1.68E−14

1.07E−11

7

x1

2.44E−11

−6.02E−09

x2

3.89E−06

8.58E−02

θ3

1.01E−14

−2.40E−11

8

x1

0.00E+00

0.00E+00

x2

0.00E+00

0.00E+00

θ3

0.00E+00

0.00E+00

9

x1

9.32E−12

−3.24E−04

x2

1.67E−06

6.46E−02

θ3

−3.60E−14

−1.38E−04

10

x1

1.89E−11

−2.15E−04

x2

3.06E−06

1.15E−01

θ3

−2.49E−14

1.65E−04

11

x1

2.44E−11

−3.64E−04

x2

3.89E−06

1.39E−01

θ3

−8.98E−15

3.72E−04

Note: Displacement units are in inches, while rotation units are in radians.

Table 3.26   Amplitudes of the Blast (Impact) Applied to a Three-Story Building

Node

Direction

Blast (Impact) Load (Displacements are in inches and rotations are in radians)

1

x1

0.00

x2

0.00

θ3

0.00

2

x1

0.00

x2

1.00

θ3

0.00

3

x1

0.00

x2

0.00

θ3

0.00

4

x1

0.00

x2

0.00

θ3

0.00

5

x1

0.00

x2

0.00

θ3

0.00

6

x1

0.00

x2

0.00

θ3

0.00

7

x1

0.00

x2

0.00

θ3

0.00

8

x1

0.00

x2

0.00

θ3

0.00

9

x1

0.00

x2

0.00

θ3

0.00

10

x1

0.00

x2

0.00

θ3

0.00

11

x1

0.00

x2

0.00

θ3

0.00

12

x1

0.00

x2

0.00

θ3

0.00

Table 3.27   Dynamic Displacements of a Blast Hazard

Node

Direction

Blast (Impact) Hazard Displacements/Rotations

1

x1

0.00E+00

x2

0.00E+00

θ3

0.00E+00

2

x1

−2.74E−05

x2

1.03E−01

θ3

−5.97E−04

3

x1

−6.66E−05

x2

1.64E−05

θ3

5.18E−05

4

x1

−6.21E−05

x2

−3.59E−06

θ3

−6.88E−06

5

x1

−2.48E−05

x2

6.26E−05

θ3

−3.13E−04

6

x1

−5.01E−05

x2

−1.26E−05

θ3

3.07E−05

7

x1

−6.49E−05

x2

2.23E−06

θ3

−1.15E−06

8

x1

0.00E+00

x2

0.00E+00

θ3

0.00E+00

9

x1

−2.27E−05

x2

8.49E−07

θ3

1.08E−04

10

x1

−4.48E−05

x2

6.30E−06

θ3

−4.14E−05

11

x1

−6.70E−05

x2

−2.05E−06

θ3

7.43E−06

12

x1

0.00E+00

x2

0.00E+00

θ3

0.00E+00

Note: Displacement units are in inches, while rotation units are in radians.

Node

Direction

1

x1

0

0

0.5

x2

0

0

0

θ3

0

0

0

2

x1

0

1

0.5

x2

0

0

0

θ3

0

0

0

3

x1

0

0

0.75

x2

0

0

0

θ3

0

0

0

4

x1

0

0

4

x2

1

0

0

θ3

0

0

0

5

x1

0

0

0.5

x2

0

0

0

θ3

0

0

0

6

x1

0

0

4

x2

1

0

0

θ3

0

0

0

7

x1

0

0

1

x2

0

0

0

θ3

0

0

0

8

x1

0

0

0.5

x2

0

0

0

θ3

0

0

0

9

x1

0

0

4

x2

1

0

0

θ3

0

0

0

10

x1

0

0

0.75

x2

0

0

0

θ3

0

0

0

11

x1

0

0

0

x2

0

0

0

θ3

0

0

0

12

x1

0

0

0

x2

0

0

0

θ3

0

0

0

Table 3.29   Dynamic and Static Displacements of a Simple Truss Bridge

Node

Direction

Hazard Displacements/Rotations

Live Load, i = 1 (Static Displacement)

Seismic Load, i = 3 (Dynamic Displacement) (Dynamic Displacement)

1

x1

−3.29E−06

1.44E−03

7.87E−03

x2

0.00E+00

0.00E+00

0.00E+00

θ3

2.75E−08

2.23E−05

−5.13E−06

2

x1

−1.55E−06

3.94E−03

5.48E−03

x2

3.98E−06

1.89E−03

−8.38E−04

θ3

2.11E−08

−7.47E−06

−2.11E−06

3

x1

−5.58E−07

8.53E−04

4.56E−03

x2

6.71E−06

−1.03E−04

1.32E−03

θ3

1.08E−08

−6.24E−08

3.52E−06

4

x1

−2.47E−06

−2.04E−04

8.69E−03

x2

7.10E−06

7.37E−05

2.56E−03

θ3

1.21E−08

−1.31E−05

−1.21E−06

5

x1

−1.35E−06

−1.42E−03

4.60E−03

x2

8.37E−06

−7.52E−04

2.34E−03

θ3

8.88E−09

9.65E−08

−2.02E−07

6

x1

−1.64E−06

2.80E−04

5.51E−03

x2

9.22E−06

1.45E−04

1.68E−03

θ3

−1.06E−14

5.73E−06

−6.00E−06

7

x1

−1.64E−06

−1.17E−04

1.23E−03

x2

9.20E−06

8.20E−05

4.12E−04

θ3

8.47E−15

−1.73E−05

−7.10E−06

8

x1

−1.94E−06

−5.60E−04

3.46E−03

x2

8.37E−06

4.07E−04

1.48E−03

θ3

−8.88E−09

−4.68E−06

−1.44E−05

9

x1

−8.18E−07

−3.91E−05

3.36E−03

x2

7.10E−06

6.64E−05

1.40E−03

θ3

−1.21E−08

−1.35E−05

−3.47E−06

10

x1

−2.73E−06

1.15E−04

7.30E−04

x2

6.71E−06

2.64E−05

−3.29E−05

θ3

−1.08E−08

7.23E−06

−4.01E−06

11

x1

−1.74E−06

−1.73E−04

−2.44E−04

x2

3.98E−06

−5.11E−05

8.07E−04

θ3

−2.11E−08

−3.27E−06

4.65E−06

12

x1

0.00E+00

0.00E+00

0.00E+00

x2

0.00E+00

0.00E+00

0.00E+00

θ3

−2.75E−08

4.82E−06

−7.86E−06

Note: Displacement units are in inches, while rotation units are in radians.

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