The spectrum analysis capability is a separate analysis type (ANTYPE,SPECTR) and it must be preceded by a mode-frequency analysis. If mode combinations are needed, the required modes must also be expanded, as described in Mode-Frequency Analysis.

The four options available are the single-point response spectrum method (SPOPT,SPRS), the dynamic design analysis method (SPOPT,DDAM), the random vibration method (SPOPT,PSD) and the multiple-point response spectrum method (SPOPT,MPRS). Each option is discussed in detail subsequently.

Both excitation at the support (base excitation) and excitation away from the support (force excitation) are allowed for the single-point response spectrum analysis (SPOPT,SPRS). The table below summarizes these options as well as the input associated with each.

Damping is evaluated for each mode and is defined as in Equation 14–26.

Note that the material dependent damping contribution (MP,DMPR) is computed in the modal expansion phase with MXPAND,,,,YES, so that this damping contribution must be included there.

The participation factors for the given excitation direction are defined as:

(15–157)

(15–158)

where:

= participation factor for the ith mode

= eigenvector normalized using Equation 15–51 (Nrmkey on the MODOPT command has no effect). If the unsymmetric eigensolver is used, the left eigenvectors are used to calculate the participation factors.

= vector describing the excitation direction (see Equation 15–159)

= input force vector

The vector describing the excitation direction has the form:

(15–159)

where:

= excitation at DOF j in direction a; a may be either X, Y, Z, or rotations about one of these axes.

X, Y, Z = global Cartesian coordinates of a point on the geometry

Xo, Yo, Zo = global Cartesian coordinates of point about which rotations are done (reference point)

= six possible unit vectors

We can calculate the statically equivalent actions at j due to rigid-body displacements of the reference point using the concept of translation of axes [T] (Weaver and Johnston([280])).

For spectrum analysis, the D^a values may be determined in one of two ways:

R is defined by:

(15–166)

where:

In a modal analysis, the ratio of each participation factor to the largest participation factor (output as RATIO) is printed out.

The displacement, velocity or acceleration vector for each mode is computed from the eigenvector by using a “mode coefficient”:

(15–167)

where:

The mode coefficient is computed in five different ways, depending on the type of excitation (SVTYP command).

When , , , or are needed between input frequencies, log-log interpolation is done in the space as defined.

The spectral values and the mode coefficients output in the RESPONSE SPECTRUM CALCULATION SUMMARY table are evaluated at the input curve with the lowest damping ratio, not at the effective damping ratio .

The modal displacements, velocity and acceleration (Equation 15–167) may be combined in different ways to obtain the response of the structure. For all excitations but the PSD this would be the maximum response, and for the PSD excitation, this would be the 1-σ (standard deviation) relative response. The response includes DOF response as well as element results and reaction forces if computed in the expansion operations (Elcalc = YES on the MXPAND command).

In the case of the single-point response spectrum method (SPOPT,SPRS) or the dynamic-design analysis method (SPOPT,DDAM) options of the spectrum analysis , it is possible to expand only those modes whose significance factor exceeds the significant threshold value (SIGNIF value on MXPAND command). Note that the mode coefficients must be available at the time the modes are expanded.

Only those modes having a significant amplitude (mode coefficient) are chosen for mode combination. A mode having a coefficient of greater than a given value (input as SIGNIF on the mode combination commands SRSS, CQC, GRP, DSUM, NRLSUM, ROSE and PSDCOM) of the maximum mode coefficient (all modes are scanned) is considered significant.

These methods generate coefficients for the combination of mode shapes. This combination is done by a generalization of the method of the square root of the sum of the squares which has the form:

(15–172)

where:

= total modal response

= total number of expanded modes

= coupling coefficient. The value of = 0.0 implies modes i and j are independent and approaches 1.0 as the dependency increases

= modal response in the ith mode (Equation 15–167)

= modal response in the jth mode

= mode coefficient for the ith mode

= mode coefficient for the jth mode

= the ith mode shape

= the jth mode shape

and may be the DOF response, reactions, or stresses. The DOF response, reactions, or stresses may be displacement, velocity or acceleration depending on the user request (Label on the mode combination commands SRSS, CQC, DSUM, GRP, ROSE or NRLSUM).

The mode combination instructions are written to File.MCOM by the mode combination command. Inputting this file in POST1 automatically performs the mode combination.

15.7.6.1. Complete Quadratic Combination Method

This method (accessed with the CQC command), is based on Wilson, et al.([65]).

(15–173)

where:

15.7.6.2. Grouping Method

This method (accessed with the GRP command), is from the NRC Regulatory Guide([41]). For this case, Equation 15–172 specializes to:

(15–174)

where:

Closely spaced modes are divided into groups that include all modes having frequencies lying between the lowest frequency in the group and a frequency 10% higher. No one frequency is to be in more than one group.

15.7.6.3. Double Sum Method

The Double Sum Method (accessed with the DSUM command) also is from the NRC Regulatory Guide([41]). For this case, Equation 15–172 specializes to:

(15–175)

where:

= damped natural circular frequency of the ith mode

= undamped natural circular frequency of the ith mode

= modified damping ratio of the ith mode

The damped natural frequency is computed as:

(15–176)

The modified damping ratio is defined to account for the earthquake duration time:

(15–177)

where:

= earthquake duration time, fixed at 10 units of time

15.7.6.4. Square Root of the Sum of the Squares (SRSS) Method

The SRSS method, accessed with the SRSS command, is taken from the NRC Regulatory Guide([41]). For this case, Equation 15–172 reduces to:

(15–178)

15.7.6.5. Naval Research Laboratory Sum (NRL-SUM) Method

The NRL-SUM method (O'Hara and Belsheim([107])), accessed with the NRLSUM command, calculates the maximum modal response as:

(15–179)

where:

= absolute value of the largest modal displacement, stress or reaction at the point

= displacement, stress or reaction contributions of the same point from other modes.

15.7.6.6. Closely Spaced Modes (CSM) Method

The CSM method, accessed with the NRLSUM,,,CSM command, calculates the combined effect of the modal responses of pairs of closely spaced modes (NAVSEA [412]). Modes are close if their frequencies are within 10% of the mean frequency. Once each CSM pair combination is determined, it is used in the NRL-SUM as a single effective mode. The modified NRL (or CSM) sum is:

(15–180)

where

ranges from 1 to N, excluding the closely spaced modes j and k.

is the maximum amplitude of a mode pair response. It is computed as follows.

This procedure first requires a modal response correction to account for the effect of damping during the first quarter cycle:

(15–181)

where:

= modal response of the ith mode including the quarter cycle correction.

= damping ratio (constant for all modes)

The closed form treatment can be expressed as the envelope of the sum of two decaying sinusoids (modes j and k). This closed form is a slightly conservative approach, as it is based on determining the peak of the envelope rather than the peak of the superposed values.

(15–182)

where:

= combined response of two closely spaced modes j and k

The times at which this function reaches a minimum or a maximum are t = 0, and:

(15–183)

where:

and

Equation Equation 15–183 has multiple solutions only if S is smaller or equal to 1. If S is greater than 1, the CSM cannot be used to reduce the NRL sum.

15.7.6.7. Rosenblueth Method

The Rosenblueth Method ([376]) is accessed with the ROSE command.

The equations for the Double Sum method (above) apply, except for Equation 15–175. For the Rosenblueth Method, the sign of the modal responses is retained:

(15–184)

The effective mass (output as EFFECTIVE MASS) for the i^th mode (which is a function of excitation direction) is (Clough and Penzien([80])):

(15–185)

Note from Equation 15–51 that

(15–186)

so that the effective mass reduces to . This does not apply to:

If the unsymmetric eigensolver is used, the left and right eigenvectors are used in Equation 15–185 and Equation 15–186. For acoustic fluid-structure interaction, the effective mass is expressed as ([426]):

(15–187)

where:

is the vector of excitation direction, structural degrees of freedom only

is the stiffness matrix of the structural part (see Equation 8–143)

is the ith right eigenvector, structural degrees of freedom only.

The cumulative mass fraction for the i^th mode is:

(15–188)

where N is the total number of modes.

For the DDAM (Dynamic Design Analysis Method) procedure (SPOPT,DDAM) (O'Hara and Belsheim([107])), modal mass in thousands of units are computed from the participation factor:

(15–189)

with

(15–190)

and

(15–191)

For units system BIN or BFT, the modal weights are computed by:

(15–192)

For other unit systems, it is calculated as:

(15–193)

where:

= modal weight and/or modal mass

= generalized mass of the ith mode

= acceleration due to gravity

= ith mode shape normalized to unity

The mode coefficients are computed by:

(15–194)

where:

= the greater of or

= minimum acceleration (input as AMIN on the ADDAM command) defaults to 6g)

= the lesser of or

= spectral acceleration

= spectral velocity

= acceleration spectrum computation constants (input as AF, AA, AB, AC, AD on the ADDAM command)

= velocity spectrum computation constants (input as VF, VA, VB, VC on the VDDAM command)

The acceleration response, which is required for the nodal acceleration check (refer to Mode Selection Based on the DDAM Procedure), is computed by:

(15–195)

where:

= acceleration response for the ith mode

The DDAM procedure is generally used with the NRL-SUM method of mode combination, which was described in Single-Point Response Spectrum. Note that unlike Equation 15–51, O'Hara and Belsheim ([107]) normalize the mode shapes to the largest modal displacements. As a result, the NRL-1396 participation factors and mode coefficients will be different.

The random vibration method (SPOPT,PSD) allows multiple power spectral density (PSD) inputs (up to two hundred) in which these inputs can be:

The procedure is based on computing statistics of each modal response and then combining them. It is assumed that the excitations are stationary random processes.

For partially correlated nodal and base excitations, the complete equations of motions are segregated into the free and the restrained (support) DOF as:

(15–196)

where are the free DOF and are the restrained DOF that are excited by random loading (unit value of displacement on D command). Note that the restrained DOF that are not excited are not included in Equation 15–196 (zero displacement on D command). {F} is the nodal force excitation activated by a nonzero value of force (on the F command). The value of force can be other than unity, allowing for scaling of the participation factors.

The free displacements can be decomposed into pseudo-static and dynamic parts as:

(15–197)

The pseudo-static displacements may be obtained from Equation 15–196 by excluding the first two terms on the left-hand side of the equation and by replacing by :

(15–198)

in which . Physically, the elements along the i^th column of [A] are the pseudo-static displacements due to a unit displacement of the support DOFs excited by the i^th base PSD. These displacements are written as load step 2 on the .RST file. Substituting Equation 15–198 and Equation 15–197 into Equation 15–196 and assuming light damping yields:

(15–199)

The second term on the right-hand side of the above equation represents the equivalent forces due to support excitations.

Using the mode-superposition analysis of Mode-Superposition Method and rewriting Equation 14–104 as:

(15–200)

the above equations are decoupled yielding:

(15–201)

where:

= number of mode shapes chosen for evaluation (input as NMODE on SPOPT command)

= generalized displacements

and = natural circular frequencies and modal damping ratios

The modal loads are defined by:

(15–202)

The modal participation factors corresponding to support excitation are given by:

(15–203)

and for nodal excitation:

(15–204)

Note that, for simplicity, equations for nodal excitation problems are developed for a single PSD table. Multiple nodal excitation PSD tables are, however, allowed in the program.

These factors are calculated (as a result of the PFACT action command) when defining base or nodal excitation cases and are written to the .psd file. Mode shapes should be normalized with respect to the mass matrix as in Equation 15–51. If the unsymmetric eigensolver is used, the left eigenvectors are used to calculate the participation factors.

The relationship between multiple input spectra are described in Cross Spectral Terms for Partially Correlated Input PSDs.

Using the theory of random vibrations, the response PSD's can be computed from the input PSD's with the help of transfer functions for single DOF systems and by using mode-superposition techniques (RPSD command in POST26). The displacement response PSD's for i^th DOF are given by:

Dynamic Part

(15–205)

Pseudo-Static Part

(15–206)

Covariance Part

(15–207)

where:

= participation factor for mode j (respectively k) corresponding to force excitation l (respectively m), see Equation 15–204

= participation factor for mode j (respectively k) corresponding to base excitation l (respectively m), see Equation 15–203

= input force PSD

= input acceleration PSD

= number of mode shapes chosen for evaluation (input as NMODE on SPOPT command)

and = number of nodal (away from support) and base PSD tables, respectively

The transfer functions for the single DOF system assume different forms depending on the type (Type on the PSDUNIT command) of the input PSD and the type of response desired (Lab and Relkey on the PSDRES command). The forms of the transfer functions for displacement as the output are listed below for different inputs.

where:

= forcing frequency

= natural circular frequency for jth mode

Now, random vibration analysis can be used to show that the absolute value of the mean square response of the i^th free displacement (ABS option on the PSDRES command) is:

(15–211)

where:

| |Re = denotes the real part of the argument

= variance of the ith relative (dynamic) free displacements (REL option on PSDRES)

= variance of the ith pseudo-static displacements

= covariance between the static and dynamic displacements

The general formulation described above gives simplified equations for several situations commonly encountered in practice. For fully correlated nodal excitations and identical support motions, the subscripts and m would drop out from the Equation 15–205 thru Equation 15–207. When only nodal excitations exist, the last two terms in Equation 15–211 do not apply, and only the first term within the large parentheses in Equation 15–205 needs to be evaluated. For uncorrelated nodal force and base excitations, the cross PSD's (i.e. ≠ m) are zero, and only the terms for which = m in Equation 15–205 thru Equation 15–207 need to be considered.

Equation 15–205 thru Equation 15–207 can be rewritten as:

(15–212)

(15–213)

(15–214)

where:

= modal PSD's, terms within large parentheses of Equation 15–205 thru Equation 15–207

Closed-form solutions for piecewise linear PSD in log-log scale are employed to compute each integration in Equation 15–211 (Chen and Ali([194]) and Harichandran([195])) .

Subsequently, the variances become:

(15–215)

(15–216)

(15–217)

The modal covariance matrices are available in the .psd file. Note that Equation 15–215 thru Equation 15–217 represent mode combination (PSDCOM command) for random vibration analysis.

The variance for stresses, nodal forces or reactions can be computed (Elcalc = YES on SPOPT (if Elcalc = YES on MXPAND)) from equations similar to Equation 15–215 thru Equation 15–217. If the stress variance is desired, replace the mode shapes () and static displacements with mode stresses and static stresses . Similarly, if the node force variance is desired, replace the mode shapes and static displacements with mode nodal forces and static nodal forces . Finally, if reaction variances are desired, replace the mode shapes and static displacements with mode reaction and static reactions . Furthermore, the variances of the first and second time derivatives (VELO and ACEL options respectively on the PSDRES command) of all the quantities mentioned above can be computed using the following relations:

(15–218)

(15–219)

Finally, the square root of Equations Equation 15–215, Equation 15–216, and Equation 15–217 are taken to yield the 1-σ results (i.e., the standard deviation) for the i^th displacement (or strain, stress, or force) quantity, and the they are written to the .rst file.

15.7.11.1. Equivalent Stress Mean Square Response

The equivalent stress (SEQV) mean square response is computed as suggested by Segalman et al([355]) as:

(15–220)

where:

= vector of component "stress shapes" for mode j at node nd

= quadratic operator

Note that the probability distribution for the equivalent stress is neither Gaussian nor is the mean value zero. However, the"3-σ" rule (multiplying the RMS value by 3) yields a conservative estimate on the upper bound of the equivalent stress (Reese et al([356])). Since no information on the distribution of the principal stresses or stress intensity (S1, S2, S3, and SINT) is known, these values are set to zero.

For excitation defined by more than a single input PSD, cross terms which determine the degree of correlation between the various PSDs are defined as:

(15–221)

where:

= input PSD spectra which are related. (Defined by the PSDVAL command and located as table number (TBLNO) n)

= cospectra which make up the real part of the cross terms. (Defined by the COVAL command where n and m (TBLNO1 and TBLNO2) identify the matrix location of the cross term)

= quadspectra which make up the imaginary part of the cross terms. (Defined by the QDVAL command where n and m (TBLNO1 and TBLNO2) identify the matrix location of the cross term)

The normalized cross PSD function is called the coherence function and is defined as:

(15–222)

where:

Although the above example demonstrates the cross correlation for 3 input spectra, this matrix may range in size from 2 x 2 to 200 x 200 (i.e., maximum number of tables is 200).

For the special case in which all cross terms are zero, the input spectra are said to be uncorrelated. Note that correlation between nodal and base excitations is not allowed.

The degree of correlation between excited nodes may also be controlled. Depending upon the distance between excited nodes and the values of R _MIN and R _MAX (input as RMIN and RMAX on the PSDSPL command), an overall excitation PSD can be constructed such that excitation at the nodes may be uncorrelated, partially correlated or fully correlated. If the distance between excited nodes is less than R _MIN , then the two nodes are fully correlated; if the distance is greater than R _MAX , then the two nodes are uncorrelated; if the distance lies between R _MIN and R _MAX , excitation is partially correlated based on the actual distance between nodes. The following figure indicates how R _MIN , R _MAX and the correlation are related. Spatial correlation between excited nodes is not allowed for a pressure PSD analysis (PSDUNIT,PRES).

Figure 15.7:  Sphere of Influence Relating Spatially Correlated PSD Excitation

Node i excitation is fully correlated with node j excitation

Node i excitation is partially correlated with node k excitation

Node i excitation is uncorrelated with node excitation

For two excitation points 1 and 2, the PSD would be:

(15–223)

where:

= distance between the two excitation points 1 and 2

= basic input PSD (PSDVAL and PSDFRQ commands)

To include wave propagation effects of a random loading, the excitation PSD is constructed as:

(15–224)

where:

= delay

= separation vector between excitations points and m

= velocity of propagation of the wave (input as VX, VY and VZ on PSDWAV command)

= nodal coordinates of excitation point

More than one simultaneous wave or spatially correlated PSD inputs are permitted, in which case the input excitation [S(ω)] reflects the influence of two or more uncorrelated input spectra. In this case, partial correlation among the basic input PSD's is not currently permitted. Wave propagation effects are not allowed for a pressure PSD analysis (PSDUNIT,PRES).

The response spectrum analysis due to multi-point support and nodal excitations (SPOPT,MPRS) allows up to three hundred different excitations (PFACT command) based on up to two hundred different spectrum tables (SPUNIT, SPFREQ and SPVAL commands). The input spectrum are assumed to be uncorrelated to each other.

Most of the ingredients for performing multi-point response spectrum analysis are already developed in the previous subsection of the random vibration method. As with the PSD analysis, the static shapes corresponding to equation Equation 15–198 for base excitation are written as load step #2 on the *.rst file, Assuming that the participation factors, , for the ^th input spectrum table have already been computed (by Equation 15–203, for example), the mode coefficients for the ^th table are obtained as:

(15–225)

where:

= interpolated input response spectrum for the th table at the jth natural frequency (defined by the SPFREQ, SPVAL and SPUNIT commands)

For each input spectrum, the mode shapes, mode stresses, etc. are multiplied by the mode coefficients to compute modal quantities, which can then be combined with the help of any of the available mode combination techniques (SRSS, CQC, Double Sum, Grouping, NRL-SUM, or Rosenblueth method), as described in the previous section on the single-point response spectrum method. The Absolute Sum method (AbsSumKey = yes on the SRSS command) can also be used as described in the section that follows.

Finally, the response of the structure is obtained by combining the responses to each spectrum using the SRSS method.

The mode combination instructions are written to the file Jobname.MCOM by the mode combination command. Inputting the file in POST1 (/INPUT command) automatically performs the mode combination.

15.7.15.1. Absolute Sum Combination Method

In a multi-point response spectrum analysis with base excitation, the square root of sum of square (SRSS) combination method does not take into account the phase relationship between the modal responses when same direction of excitation is applied at each support. The Absolute Sum combination method [[421]] takes into account that the peak values for one excitation direction are occurring at the same time so that the absolute values are summed. The remaining combination of modes and directions is performed with the SRSS method.

The total response with missing mass included is expressed as:

Where:

is the excitation direction index. The excitation direction is specified using SEDX, SEDY, or SEDZ on the SED command.

is the total number of excitation directions.

is the support (node group) index. The node group is defined with a component based on nodes (Cname on the SED command).

is the total number of supports.

is the modal response for the j^th mode, the direction, and the support.

is the missing mass response for the direction, and the support (MMASS command). It is optional.

---

Note:  All supports are not necessarily excited in all directions. For example, if the support is not excited in the direction then is zero for all modes j.

---

The spectrum analysis is based on a mode-superposition approach where the responses of the higher modes are neglected; therefore, part of the mass of the structure is missing in the dynamic analysis. The missing-mass response method ([375] and [425]) permits inclusion of the missing mass effect in a single-point response spectrum (SPOPT, SPRS) or multiple-point response spectrum analysis (SPOPT,MPRS) when base excitation is considered

Considering a rigid structure, the inertia force due to ground acceleration is:

(15–226)

where:

= total inertia force vector

= spectrum acceleration at zero period (also called the ZPA value), input as ZPA on the MMASS command.

Mode-superposition can be used to determine the inertia force. For mode j, the modal inertia force is:

(15–227)

where:

= modal inertia force for mode j.

Using equations Equation 15–167 and Equation 15–170, this force can be rewritten:

(15–228)

The missing inertia force vector is then the difference between the total inertia force given by Equation 15–226 and the sum of the modal inertia forces defined by Equation 15–228:

(15–229)

The expression within the parentheses in the equation above is the fraction of degree of freedom mass missing:

(15–230)

The missing mass response for displacement results is the static shape due to the inertia forces defined by the equation:

(15–231)

where:

is the missing mass response

For acceleration results, the missing mass corresponds to the residual acceleration load from Equation 15–229 and is written:

(15–232)

The application of these equations can be extended to flexible structures because the higher truncated modes are supposed to be mostly rigid and exhibit pseudo-static responses to an acceleration base excitation.

In a single-point response spectrum analysis, the missing mass response is written as load step 2 in the .RST file. In a multiple-point response spectrum analysis, it is written as load step 3.

Combination Method

Because the missing-mass response is a pseudo-static response, it is in phase with the imposed acceleration but out of phase with the modal responses; therefore, the missing-mass response and the modal responses defined in Equation 15–172 are combined using the square root of sum of the squares (SRSS) method.

The total response including the missing mass effect is:

(15–233)

For frequencies higher than the amplified acceleration region of the spectrum, the modal responses consist of both periodic and rigid components. The rigid components are considered separately because the corresponding responses are all in phase. The combination methods listed in Combination of Modes do not apply

The rigid component of a modal response is expressed as:

(15–234)

where:

= the rigid component of the modal response of mode i

= rigid response coefficient in the range of values 0 through 1. See the Gupta and Lindley-Yow methods below.

= modal response of mode i

The corresponding periodic component is then:

(15–235)

where:

= periodic component of the modal response of mode i

Two methods ([376]) can be used to separate the periodic and the rigid components in each modal response. Each one has a different definition of the rigid response coefficients .

Gupta Method

(15–236)

where:

= i^th frequency value.

and = key frequencies. is input as Val1 and is input as Val2 on RIGRESP command with Method = GUPTA.

Lindley-Yow Method

(15–237)

where:

= spectrum acceleration at zero period (ZPA). It is input as Val1 on RIGRESP command with Method = LINDLEY

= spectrum acceleration corresponding to the i^th frequency

Combination Method

The periodic components are combined using the square root of sum of squares (SRSS), the complete quadratic (CQC) or the Rosenblueth (ROSE) combination methods.

Since the rigid components are all in phase, they are summed algebraically. When the missing mass response (MMASS) is included in the analysis, since it is a rigid response as well, it is summed with those components. Finally, periodic and rigid responses are combined using the SRSS method.

The total response with the rigid responses and the missing mass response included is expressed as:

(15–238)