Response Spectrum Analysis

Structural analysis method utilized to predict the maximum response of a structure subjected to dynamic loading, typically transient events such as earthquakes or blasts.

Response Spectrum Analysis (RSA) provides estimates of the maximum displacements, stresses, and forces experienced by the structure under such conditions. RSA combines predefined response spectra for specific dynamic loadings with the outcomes of a normal modal analysis. Unlike time-history analyses, RSA doesn't generate the time evolution of responses.

Response spectra illustrate the maximum response against the natural frequency of a single-degree-of-freedom (1-DOF) system under specified dynamic loads. These spectra facilitate the computation of maximum modal responses for each structural mode. Various combination methods, like the absolute sum (ABS), square root of sum of squares (SRSS), naval research laboratory (NRL) or complete quadratic combination (CQC), amalgamate these modal maxima to estimate the peak structural response.

Compared to traditional transient analysis methods, RSA offers a simpler and computationally efficient means of approximating peak responses. The primary computational demand lies in acquiring an adequate number of normal modes to accurately represent the entire frequency range of input excitation and resultant response. Typically, design specifications provide response spectra, enabling quick calculations of peak responses under different dynamic excitations. Consequently, RSA serves as a prevalent design tool, particularly in seismic analysis for buildings.

When performing RSA on a structure, selecting the appropriate modal combination method is crucial for obtaining accurate results. This method determines how the software consolidates the raw results from individual vibration modes (modal responses) into a single set of values for displacement, reactions, internal forces, and other parameters for each degree of freedom. These combined results serve as the basis for designing the structure, underscoring the significance of choosing the modal combination method carefully. The following section explores several modal combination methods commonly used in RSA.

Modal Combinations

Absolute sum (ABS)

The Absolute Sum modal combination method calculates the absolute value of the result (such as displacement or internal force) for each vibration mode and sums up these absolute values. This method assumes that all peak modal responses occur simultaneously, resulting in a conservative estimation. Consequently, it is not widely favoured in structural design applications due to its overly conservative nature.

The formula for calculating the peak value of the total response using the absolute sum method is:

Peak Response Total = Σ_{i} |R_{n}|

R_{n} = A_{i n} ψ_{i} X

Where,

$R_{n}$: Represents the transient response at a single degree of freedom from each vibration mode
$A$: Eigen vector
$ψ$: Modal participation factor
$X$: Input response spectrum

This straightforward formula sums up the absolute values of the responses from all modes, disregarding their signs.

Square root of sum of squares (SRSS)

The SRSS modal combination method calculates the square root of the sum of squares of the results for each vibration mode, offering an approximation of the peak of the total response. This method is particularly effective for structures with distinct natural frequencies. However, if the natural frequencies of the structure are closely spaced, SRSS may not yield accurate results and should be avoided.

Formally, the peak total response using the SRSS method can be expressed as:

Peak Response Total = \sqrt{Σ_{i} R_{n}^{2}}

R_{n} = A_{i n} ψ_{i} X

Where,

$R_{n}$: Represents the transient response at a single degree of freedom from each vibration mode
$A$: Eigen vector
$ψ$: Modal participation factor
$X$: Input response spectrum

This formula calculates the square root of the sum of the squares of the responses from all modes, providing an estimate of the peak total response.

Complete quadratic combination (CQC)

The CQC method indeed addresses the limitations of the SRSS method when combining modal responses in structures with closely spaced natural frequencies. The peak total response using the CQC method is obtained through the formula:

Peak Response Total = \sqrt{\sum_{i} Σ_{j} R_{i} ρ_{i j} R_{j}}

Where,

$R_{i} R_{j}$: Peak modal responses for the i, j vibration modes, respectively
$ρ_{i j}$: Modal correlation coefficient for the two modes being combined at each summation step

The modal correlation coefficient

ρ_{i j}

is calculated using the equation:

ρ_{i j} = \frac{8 \sqrt{ξ_{i} ξ_{j}} (ξ_{i} + β_{j i} ξ_{j}) β_{j i}^{1 \cdot 5}}{{(1 - β_{j i}^{2})}^{2} + 4 ξ_{i} ξ_{j} β_{j i} (1 + β_{j i}^{2}) + 4 (ξ_{i}^{2} + ξ_{j}^{2}) β_{j i}^{2}}

Where,

$β_{j i}$: Ratio between the natural frequencies of the i and j modes ( $λ_{j} / λ_{i}$ )
$ξ_{i}$ and $ξ_{j}$: Modal damping values of the two modes

This equation accounts for the relationship between the damping ratio, the natural frequencies of the modes, and their correlation, ensuring a more accurate estimation of the peak total response compared to the SRSS method.

Naval research laboratory (NRL)

The NRL modal combination method integrates aspects of both the square root of the sum of the squares (SRSS) and absolute sum (ABS) methods to achieve a balanced approach in response spectrum analysis. It adds the maximum peak modal response by ABS method and the rest of the modal contributions to the structures with responses calculated having distinct natural frequencies which is evaluated using SRSS method.

Formulation for the NRL modal combination method is as follows:

Peak Total Response = |A_{i k}| |Ψ_{i} X| + \sqrt{\sum_{j \neq i} {(A_{j k} Ψ_{j} X)}^{2}}