An interval framework for uncertain frequency response of multi-cracked beams with application to vibration reduction via tuned mass dampers

Abstract

The paper addresses the frequency response of beams in presence of open cracks with interval parameters. On adopting the standard Euler–Bernoulli beam theory, every crack is modelled as a linearly-elastic rotational spring whose stiffness and position are treated as uncertain-but-bounded parameters. A two-step method is proposed to calculate the bounds of all response variables. First, the sensitivity functions of the response are calculated as every uncertain parameter varies within the respective interval. Next, the bounds of the response are computed by either a sensitivity-based method or a global optimization technique, the former if the response is monotonic with respect to all uncertain parameters and the latter if the response is non-monotonic with respect to even one parameter only. The method relies on analytical forms for all response variables and the associated sensitivity functions. The applications focus on the frequency response of multi-cracked beams equipped with tuned mass dampers, showing potential and accuracy of the method.

Appendices

Appendix A: Analytical form of the frequency response

Terms in Eq. (3) were derived in closed analytical form in previous studies [56, 57]. For convenience, here they are reported for a general case where cracks (linearly-elastic rotational springs) and TMDs (mass-spring-damper subsystems) always occur at the same position $x_{k}$, for k = 1,2…N. Changes are straightforward when either a crack or a TMD are located at a given abscissa, as explained below.

Terms in Eq. (3) are:

$${\mathbf{W}}\left(x \right) = {\varvec{\Omega}}\left(x \right) + \sum\limits_{j = 1}^{N} {{\mathbf{J}}\left({x,x_{j}} \right){\varvec{\Phi}}_{{\varvec{\Omega}}} \left({x_{j}} \right)} + \sum\limits_{j = 2}^{N} {{\mathbf{J}}\left({x,x_{j}} \right)\sum\limits_{2 \le q \le j}^{{}} {\sum\limits_{{(\underbrace {j,m,n,\ldots,r,s}_{q}) \in {\mathbb{N}}_{q}^{\left(j \right)}}}^{{}} {{\varvec{\Phi}}_{{\mathbf{J}}} \left({x_{j},x_{m}} \right){\varvec{\Phi}}_{{\mathbf{J}}} \left({x_{m},x_{n}} \right) \cdot \cdot {\varvec{\Phi}}_{{\mathbf{J}}} \left({x_{r},x_{s}} \right){\varvec{\Phi}}_{{\varvec{\Omega}}} \left({x_{s}} \right)}}}$$

(A.1)

$${\mathbf{Y}}^{\left(f \right)} \left(x \right) = {\mathbf{F}}\left(x \right) + \sum\limits_{j = 1}^{N} {{\mathbf{J}}\left({x,x_{j}} \right){\varvec{\Phi}}^{\left(f \right)} \left({x_{j}} \right)} + \sum\limits_{j = 2}^{N} {{\mathbf{J}}\left({x,x_{j}} \right)\sum\limits_{2 \le q \le j}^{{}} {\sum\limits_{{\left({j,m,n,\ldots,r,s} \right) \in {\mathbb{N}}_{q}^{\left(j \right)}}}^{{}} {{\varvec{\Phi}}_{{\mathbf{J}}} \left({x_{j},x_{m}} \right) \cdots {\varvec{\Phi}}_{{\mathbf{J}}} \left({x_{r},x_{s}} \right){\varvec{\Phi}}^{\left(f \right)} \left({x_{s}} \right)}}}$$

(A.2)

where ${\mathbb{N}}_{q}^{\left(j \right)} = \{(\underbrace {j,m,n,\ldots r,s}_{q}):j > m > n >\cdots > r > s;m,n,\ldots r,s = 1,2,\ldots,\left({j - 1} \right)\}$ is the set including all possible q-ples of indexes $(\underbrace {j,m,n,\ldots r,s}_{q})$ such that $j > m > n >\cdots > r > s$, being $2 \le q \le j$.

In Eq. (A.1), matrices ${\varvec{\Omega}}\left(x \right)$ and ${\varvec{\Phi}}_{{\varvec{\Omega}}} \left({x_{j}} \right)$ depend on the solutions to the homogeneous differential equation associated with Eq. (4), i.e.

$${\varvec{\Omega}}\left(x \right) = \left[{\begin{array}{*{20}c} {{\text{e}}^{- \beta x}} & {{\text{e}}^{\beta x}} & {\cos \left({\beta x} \right)} & {\sin \left({\beta x} \right)} \\ {- \beta {\text{e}}^{- \beta x}} & {\beta {\text{e}}^{\beta x}} & {- \beta \sin \left({\beta x} \right)} & {\beta \cos \left({\beta x} \right)} \\ {- EI\beta^{2} {\text{e}}^{- \beta x}} & {- EI\beta^{2} {\text{e}}^{\beta x}} & {EI\beta^{2} \cos \left({\beta x} \right)} & {EI\beta^{2} \sin \left({\beta x} \right)} \\ {EI\beta^{3} {\text{e}}^{- \beta x}} & {- EI\beta^{3} {\text{e}}^{\beta x}} & {- EI\beta^{3} \sin \left({\beta x} \right)} & {EI\beta^{3} \cos \left({\beta x} \right)} \\ \end{array}} \right]$$

(A.3)

$${\varvec{\Phi}}_{{\varvec{\Omega}}} \left({x_{j}} \right) = \left[{\begin{array}{*{20}c} {- \kappa_{{meq_{j}}} \left(\omega \right)\left({{\varvec{\Omega}}\left({x_{j}} \right)} \right)_{1}} \\ {- \kappa_{{{\Delta}\Theta_{j}}}^{- 1} \left({{\varvec{\Omega}}\left({x_{j}} \right)} \right)_{3}} \\ \end{array}} \right]$$

(A.4)

being $\beta = \left({EI} \right)^{{{{- 1} \mathord{\left/{\vphantom {{- 1} 4}} \right. \kern-0pt} 4}}} \rho^{{{1 \mathord{\left/{\vphantom {1 4}} \right. \kern-0pt} 4}}} \omega^{{{1 \mathord{\left/{\vphantom {1 2}} \right. \kern-0pt} 2}}}$ and $\left({{\varvec{\Omega}}\left({x_{j}} \right)} \right)_{i}$ row vectors coinciding with the ith row of matrix ${\varvec{\Omega}}\left(x \right)$.

In Eqs. (A.1) and (A.2), matrices ${\mathbf{J}}\left({x,x_{j}} \right)$ and ${\varvec{\Phi}}_{{\mathbf{J}}} \left({x_{j},x_{k}} \right)$ depend on the particular solutions of Eq. (4) associated with a unit Dirac’s delta and its second formal derivative, representing respectively a unit transverse force and a unit relative rotation between adjacent cross sections. Specifically,

$${\mathbf{J}}\left({x,x_{j}} \right) = \left[{\begin{array}{*{20}c} {J_{V,P}^{{}}} & {J_{V,\Delta \Theta}^{{}}} \\ {J_{\Theta,P}^{{}}} & {J_{\Theta,\Delta \Theta}^{{}}} \\ {J_{M,P}^{{}}} & {J_{M,\Delta \Theta}^{{}}} \\ {J_{S,P}^{{}}} & {J_{S,\Delta \Theta}^{{}}} \\ \end{array}} \right]$$

(A.5)

$${\varvec{\Phi}}_{{\mathbf{J}}}^{{}} \left({x_{j}^{{}},x_{k}^{{}}} \right) = \left[{\begin{array}{*{20}c} {- \kappa_{{meq_{j}}} \left(\omega \right)\left({{\mathbf{J}}\left({x_{j}^{{}},x_{k}^{{}}} \right)} \right)_{1}} \\ {- \kappa_{{\Delta \Theta_{j}}}^{- 1} \left({{\mathbf{J}}\left({x_{j}^{{}},x_{k}^{{}}} \right)} \right)_{3}} \\ \end{array}} \right]$$

(A.6)

where $\left({{\mathbf{J}}\left({x_{j},x_{k}} \right)} \right)_{i}$ denotes the i^th row of matrix ${\mathbf{J}}\left({x_{j},x_{k}} \right)$. In Eq. (A.5), $J_{V,P}^{{}}$ and $J_{V,\Delta \Theta}^{{}}$ are the particular solutions of Eq. (4) associated with a unit Dirac’s delta and its second formal derivative at $x_{j}$,

$$J_{V,P}^{{}} \left({x,x_{j}} \right) = \alpha \left({\sinh \left({\beta \left({x - x_{j}} \right)} \right) - { \sin }\left({\beta \left({x - x_{j}} \right)} \right)} \right)H\left({x - x_{j}} \right)$$

(A.7)

$$J_{V,\Delta \Theta}^{{}} \left({x,x_{j}} \right) = EI\alpha \beta^{2} \left({\sinh \left({\beta \left({x - x_{j}} \right)} \right) + \sin \left({\beta \left({x - x_{j}} \right)} \right)} \right)H\left({x - x_{j}} \right)$$

(A.8)

being $\alpha = 2^{- 1} \left({EI} \right)^{{{{- 1} \mathord{\left/{\vphantom {{- 1} 4}} \right. \kern-0pt} 4}}} \rho^{{- {3 \mathord{\left/{\vphantom {3 4}} \right. \kern-0pt} 4}}} \omega^{{- {3 \mathord{\left/{\vphantom {3 2}} \right. \kern-0pt} 2}}}$. Terms on the first and second columns of Eq. (A.5) are derived using the beam equations for shear force, bending moment, rotation and deflection under a unit transverse force and a unit relative rotation between adjacent cross sections at $x_{j}$ (see Appendix A in Ref. [19]).

Finally, in Eq. (A.2) matrix ${\varvec{\Phi}}^{\left(f \right)} \left({x_{j}} \right)$ and vector ${\mathbf{F}}\left(x \right)$ are load-dependent terms given as

$${\mathbf{F}}\left(x \right) = {\mathbf{J}}\left({x,\xi} \right)P$$

(A.9)

$${\varvec{\Phi}}^{\left(f \right)} \left({x_{j}} \right) = \left[{\begin{array}{*{20}c} {- \kappa_{{meq_{j}}} \left(\omega \right)\left({{\mathbf{F}}\left({x_{j}} \right)} \right)_{1}} \\ {- \kappa_{{{\Delta}\Theta_{j}}}^{- 1} \left({{\mathbf{F}}\left({x_{j}} \right)} \right)_{3}} \\ \end{array}} \right]$$

(A.10)

where P is the point force in Eq. (4). It is noteworthy that Eq. (3) can readily be generalized for arbitrary distributed loads $f\left(x \right)e^{{{\text{i}}\omega t}}$. In this case, Eq. (A.9) takes the form

$${\mathbf{F}}\left(x \right) = \int_{0}^{L} {{\mathbf{J}}\left({x,y} \right)f\left(y \right)dy}$$

(A.11)

Using simple integration rules of generalized functions, closed analytical solutions to Eq. (A.11) can be obtained for typical loading functions $f\left(x \right)$ as, e.g., polynomial ones [56, 57].

Changes if a single crack or a TMD occurs at a given $x_{j}^{{}}$ are straightforward. Specifically, $\kappa_{{\Delta \Theta_{j}}}^{{}} = \infty$ shall be set if no crack occurs at $x_{j}^{{}}$ and, correspondingly, all terms in the 2^nd row of matrices ${\varvec{\Phi}}_{{\varvec{\Omega}}}^{{}} \left({x_{j}^{{}}} \right)$, ${\varvec{\Phi}}_{{\mathbf{J}}}^{{}} \left({x_{j}^{{}},x_{k}^{{}}} \right)$ will vanish. In addition, terms in the 2^nd column of matrix ${\varvec{\Phi}}_{{\mathbf{J}}}^{{}} \left({x_{m}^{{}},x_{j}^{{}}} \right)$, shall be set equal to zero for all $x_{m}^{{}} > x_{j}^{{}}$. Likewise, if no TMD occurs at $x = x_{j}^{{}}$, $\kappa_{{meq_{j}}}^{{}} \left(\omega \right) = 0$ shall be set at $x_{j}^{{}}$ and, correspondingly, terms in the 1st row of matrices ${\varvec{\Phi}}_{{\varvec{\Omega}}}^{{}} \left({x_{j}^{{}}} \right)$, ${\varvec{\Phi}}_{{\mathbf{J}}}^{{}} \left({x_{j}^{{}},x_{k}^{{}}} \right)$ will vanish and terms in the 1st column of matrix ${\varvec{\Phi}}_{{\mathbf{J}}}^{{}} \left({x_{m}^{{}},x_{j}^{{}}} \right)$ shall be set equal to zero for all $x_{m}^{{}} > x_{j}^{{}}$.

Appendix B: Application of monotonicity test

The present Appendix reports an explicative application of the M-test discussed in Sect. 3.1. The analysis is focused on the multi-cracked clamped-supported beam equipped with a TMD. The data concerning the beam parameters as well as the TMD parameters are the same reported in the “Numerical applications” Sect. 4. Let us remind the examined beam is loaded by a transversal 3 Hz point force applied at $\xi = {L \mathord{\left/{\vphantom {L 4}} \right. \kern-0pt} 4} = 3{\text{m}}$.

The damaged beam present N_s= 3 springs modelling the cracks located at the nominal positions $x_{0,1} = {L \mathord{\left/{\vphantom {L 6}} \right. \kern-0pt} 6} = 2{\text{m}}$,${\kern 1pt} {\kern 1pt} x_{0,2} = {L \mathord{\left/{\vphantom {L 3}} \right. \kern-0pt} 3} = 4{\text{m}}$ and $x_{0,3} = {{2L} \mathord{\left/{\vphantom {{2L} 3}} \right. \kern-0pt} 3} = 8{\text{m}}$ with a nominal stiffness value $\kappa_{{{\Delta \Theta}_{0,j}}} = \kappa_{{{\Delta \Theta}_{0}}} = 5.27 \times 10^{6} {\text{Nm}}_{{}} ( ={{ 6EI} \mathord{\left/{\vphantom {{ 6EI} {L )}}} \right. \kern-0pt} {L )}}$ for j = 1,2,3.

To illustrate the results of the M-test, the attention is focused on the real part $\text{Re} \left[{V^{I} (x)} \right]$ of the deflection response function $R(x,{\boldsymbol{\upkappa}}_{\Delta \Theta}^{I},{\mathbf{x}}^{I}) = V(x,{\boldsymbol{\upkappa}}_{\Delta \Theta}^{I},{\mathbf{x}}^{I})$. All the considerations can be extended to the imaginary part $\text{Im} \left[{V^{I} (x)} \right]$ as well as to the other variable responses, in real and imaginary parts.

First the sensitivity functions in Eq. (15a, 15b) related to the deflection response $V(x)$ are calculated with respect to the uncertain parameters characterizing the 1st crack, namely $\sigma_{{\kappa_{{\Delta \Theta_{1}}}}}^{\left(V \right)} (x,\kappa_{{\Delta \Theta_{1}}})$ for the stiffness $\kappa_{{{\Delta \Theta}_{1}}}$ and $\sigma_{{x_{1}}}^{\left(V \right)} (x,x_{1})$ for the position $x_{1}$.

Figure 17a represents the real part of the deflection sensitivity $\text{Re} \left[{\sigma_{{\kappa_{{\Delta \Theta_{1}}}}}^{\left(V \right)} (x,\kappa_{{\Delta \Theta_{1}}})} \right]$ along the beam axis and for the spring stiffness $\kappa_{{{\Delta \Theta}_{1}}}$ varying in the related interval being $\underline{\kappa}_{{\Delta \Theta_{1}}} \le \kappa_{\Delta \Theta} \le \overline{\kappa}_{{\Delta \Theta_{1}}}$(see Eq. (15a)). In particular considering a deviation $\Delta \kappa_{{{\Delta \Theta}_{1}}} = 0.2\kappa_{{{\Delta \Theta}_{0}}}$ with $\kappa_{{{\Delta \Theta}_{0}}} = {{ 6EI} \mathord{\left/{\vphantom {{ 6EI} L}} \right. \kern-0pt} L}$ we obtain $\underline{\kappa}_{{\Delta \Theta_{1}}} = {{4.8EI} \mathord{\left/{\vphantom {{4.8EI} {L = 4.2 \times 10^{6}}}} \right. \kern-0pt} {L = 4.2 \times 10^{6}}}{\text{Nm}}$ and $\overline{\kappa}_{{\Delta \Theta_{1}}} = {{7.2EI} \mathord{\left/{\vphantom {{7.2EI} {L = 6.3 \times 10^{6}}}} \right. \kern-0pt} {L = 6.3 \times 10^{6}}}{\text{Nm}}$.

Fig. 17

Multi-cracked clamped-supported beam in Fig. 3b: real part of the deflection sensitivity function $\text{Re} [\sigma_{{\kappa_{{\Delta \Theta_{1}}}}}^{(V)} (x,\kappa_{{\Delta \Theta_{1}}})]$ for spring stiffness $\kappa_{{{\Delta \Theta}_{1}}}$ varying in $[\underline{\kappa}_{{\Delta \Theta_{1}}},\overline{\kappa}_{{\Delta \Theta_{1}}}]$, computed at a all abscissas x; b x = $x_{0,1}$ = L/6; c x = L/2

As explained in Sect. 3.1, the M-test requires the sign analysis of $\text{Re} \left[{\sigma_{{\kappa_{{\Delta \Theta_{1}}}}}^{\left(V \right)} (x,\kappa_{{\Delta \Theta_{1}}})} \right]$ at every abscissa x of interest as $\kappa_{{{\Delta \Theta}_{1}}}$ varies in the interval $\left[{\underline{\kappa}_{{\Delta \Theta_{1}}},\overline{\kappa}_{{\Delta \Theta_{1}}}} \right]$. As example let us consider as abscissas of interest $x = x_{0,1} = L/6$ at the spring position and $x = L/2$ at the beam mid-span. Figures 17b and c show that both $\text{Re} \left[{\sigma_{{\kappa_{{\Delta \Theta_{1}}}}}^{\left(V \right)} ({L \mathord{\left/{\vphantom {L 6}} \right. \kern-0pt} 6},\kappa_{{\Delta \Theta_{1}}})} \right]$ (always positive at $x = x_{0,1} = L/6$) and $\text{Re} \left[{\sigma_{{\kappa_{{\Delta \Theta_{1}}}}}^{\left(V \right)} ({L \mathord{\left/{\vphantom {L 2}} \right. \kern-0pt} 2},\kappa_{{\Delta \Theta_{1}}})} \right]$ (always negative at $x = L/2$) do not change their sign as $\kappa_{{{\Delta \Theta}_{1}}}$ varies in the interval $\left[{\underline{\kappa}_{{\Delta \Theta_{1}}},\bar{\kappa}_{{\Delta \Theta_{1}}}} \right]$: we can deduce that the real part of the deflection function $\text{Re} \left[{V(x)} \right]$ is monotonic with respect to $\kappa_{{{\Delta \Theta}_{1}}}$ at the selected abscissas $x = L/6$ and $x = L/2$.

Similarly, let us consider the uncertainty in the spring position $x_{1}$. Figure 18 shows the real part of the deflection sensitivity function $\text{Re} \left[{\sigma_{{x_{1}}}^{\left(V \right)} (x,x_{1})} \right]$ with respect to the uncertain parameter $x_{1}$. For a deviation $\Delta x_{1} = 0.2$(see Eq. (9b)) the interval parameter $x_{1} = 2{m}$ is bounded by $\underline{x}_{1} = 1.8{\text{m}}$ and $\overline{x}_{1} = 2.2{\text{m}}$: in Fig. 18a the interval $\left[{1.8,2.2} \right]$ is singled out from the whole beam axis. Aiming to establish the monotonic or non-monotonic behaviour of $\text{Re} \left[{V(x)} \right]$ with respect to $x_{1}$, the M-test requires again an exam on the sign of $\text{Re} \left[{\sigma_{{x_{1}}}^{\left(V \right)} (x,x_{1})} \right]$ which is conducted, for the sake of simplicity, at the same previous abscissas of interest, as $x_{1}^{{}}$ varies within the pertinent interval $\left[{\underline{x}_{1},\overline{x}_{1}} \right]$. In detail, Fig. 18b shows that at $x = x_{0,1} = L/6$ the deflection function $\text{Re} \left[{V(x)} \right]$ in its real part is non-monotonic with respect to the parameter $x_{1}$ as evident by the change in sign of $\text{Re} [\sigma_{{x_{1}}}^{\left(V \right)} (L/6,x_{1})]$ for varying $x_{1}$. At the beam mid-span $x = L/2$ it is confirmed a monotonic behaviour of $\text{Re} \left[{V(x)} \right]$ with respect to the parameter $x_{1}$ ($\text{Re} [\sigma_{{x_{1}}}^{\left(V \right)} (L/6,x_{1})]$ always negative for varying $x_{1}$ as shown in Fig. 18c).

Fig. 18

Damaged clamped-supported beam in Fig. 3b: real part of the deflection sensitivity function $\text{Re} [\sigma_{{x_{1}}}^{(V)} (x,x_{1})]$ for spring stiffness $x_{1}$ varying in $[\underline{x}_{1},\overline{x}_{1}]$, computed at a all abscissas x; b x = $x_{0,1}$ = L/6; c x = L/2

It is worth to note that in the M-test application analogous graphics shall be evaluated at any abscissa x of interest, to conclude on monotonicity/non-monotonicity of the real part of the deflection function with respect to $\kappa_{{{\Delta \Theta}_{1}}}$ and $x_{1}$ of the 1st crack, at that x. Moreover, the M-test described for the uncertain parameters of the 1st crack shall be implemented for the uncertain parameters of all the other cracks and for other frequency values.

Appendix C: Bounds of the stress resultants

Mirroring the steps followed in the main text to calculate the bounds of the deflection $V^{I} (x)$, in this Appendix LBs and UBs of bending moment $M^{I} (x) \equiv M\left({x,{\boldsymbol{\upkappa}}_{{\Delta {\Theta}}}^{I},{\mathbf{x}}^{I}} \right)$ and shear force $S^{I} (x) \equiv S\left({x,{\boldsymbol{\upkappa}}_{{\Delta {\Theta}}}^{I},{\mathbf{x}}^{I}} \right)$ are similarly calculated for the multi-cracked beams in Fig. 3. As in the main text, the forcing frequency is 3 Hz.

For brevity, the uncertainty is considered simultaneously in stiffness $\kappa_{{\Delta {\Theta}_{j}}}^{I}$ and position $x_{j}$ of all springs (j = 1,2,3).

3.1 Multi-cracked simply-supported beam equipped with a TMD

The M-test conducted for the simply-supported beam in Fig. 3a reveals that, for the selected forcing frequency = 3 Hz, real and imaginary parts of bending moment $M\left({x,{\boldsymbol{\upkappa}}_{{\Delta {\Theta}}}^{I},{\mathbf{x}}^{I}} \right)$ and shear force $S\left({x,{\boldsymbol{\upkappa}}_{{\Delta {\Theta}}}^{I},{\mathbf{x}}^{I}} \right)$ exhibit both a monotonic behaviour with respect to all the parameters collected in the vectors ${\boldsymbol{\upkappa}}_{{\Delta {\Theta}}}^{I}$ and ${\mathbf{x}}^{I}$ characterizing the cracks. It is noteworthy that this holds at every abscissa x.

Therefore, the SB method (Sect. 3.2) can be applied to calculate the bounds of both stress resultants, taking full advantage of the sensitivities provided by Eq. (16a, b) and the explicit expressions in Eq. (3).

Figures 19c and 20c show the bounds of real and imaginary parts of the bending moment function $M\left({x,{\boldsymbol{\upkappa}}_{{\Delta {\Theta}}}^{I},{\mathbf{x}}^{I}} \right)$, along with the sensitivities with respect to stiffness $\kappa_{{\Delta {\Theta}_{j}}}^{I}$ (Figs. 19a, 20a) and position $x_{j}$ (Figs. 19b, 20b) for all springs. Analogous results are reported in Figs. 21 and 22 for the shear force function $S\left({x,{\boldsymbol{\upkappa}}_{{\Delta {\Theta}}}^{I},{\mathbf{x}}^{I}} \right)$.

Fig. 19

Multi-cracked simply-supported beam in Fig. 3a, considering uncertain stiffness $\kappa_{{{\Delta \Theta}_{j}}}$ and position $x_{j}$ of all springs ($\Delta \kappa_{{{\Delta \Theta}_{j}}} = 0.2\kappa_{{{\Delta \Theta}_{0}}}$,$\Delta x_{j} = 0.2$ with j = 1,2,3): a real parts of the sensitivity functions $\text{Re} [\sigma_{{\kappa_{{\Delta {\Theta}_{j}}}}}^{\left(M \right)} (x)]$, b real parts of the sensitivity functions $\text{Re} [\sigma_{{x_{j}}}^{\left(M \right)} (x)]$ and c LB and UB of the real part of the bending moment function $\text{Re} [M(x)]$ provided by the proposed method (SB method) and VM

Fig. 20

Multi-cracked simply-supported beam in Fig. 3a, considering uncertain stiffness $\kappa_{{{\Delta \Theta}_{j}}}$ and position $x_{j}$ of all springs ($\Delta \kappa_{{{\Delta \Theta}_{j}}} = 0.2\kappa_{{{\Delta \Theta}_{0}}}$,$\Delta x_{j} = 0.2$ with j = 1,2,3): a imaginary parts of the sensitivity functions $\text{Im} [\sigma_{{\kappa_{{\Delta {\Theta}_{j}}}}}^{\left(M \right)} (x)]$, b imaginary parts of the sensitivity functions $\text{Im} [\sigma_{{x_{j}}}^{\left(M \right)} (x)]$ and c LB and UB of the imaginary part of the bending moment function $\text{Im} [M(x)]$ provided by the proposed method (SB method) and VM

Fig. 21

Multi-cracked simply-supported beam in Fig. 3a, considering uncertain stiffness $\kappa_{{{\Delta \Theta}_{j}}}$ and position $x_{j}$ of all springs ($\Delta \kappa_{{{\Delta \Theta}_{j}}} = 0.2\kappa_{{{\Delta \Theta}_{0}}}$,$\Delta x_{j} = 0.2$ with j = 1,2,3): a real parts of the sensitivity functions $\text{Re} [\sigma_{{\kappa_{{\Delta {\Theta}_{j}}}}}^{\left(S \right)} (x)]$, b real parts of the sensitivity functions $\text{Re} [\sigma_{{x_{j}}}^{\left(S \right)} (x)]$ and c LB and UB of the real part of the shear force function $\text{Re} [S(x)]$ provided by the proposed method (SB method) and VM

Fig. 22

Multi-cracked simply-supported beam in Fig. 3a, considering uncertain stiffness $\kappa_{{{\Delta \Theta}_{j}}}$ and position $x_{j}$ of all springs ($\Delta \kappa_{{{\Delta \Theta}_{j}}} = 0.2\kappa_{{{\Delta \Theta}_{0}}}$,$\Delta x_{j} = 0.2$ with j = 1,2,3): a imaginary parts of the sensitivity functions $\text{Im} [\sigma_{{\kappa_{{\Delta {\Theta}_{j}}}}}^{\left(S \right)} (x)]$, b imaginary parts of the sensitivity functions $\text{Im} [\sigma_{{x_{j}}}^{\left(S \right)} (x)]$ and c LB and UB of the imaginary part of the shear force function $\text{Im} [S(x)]$ provided by the proposed method (SB method) and VM

As explained in detail in Sect. 3.2, the study on the signs of sensitivities (see Figs. 19a, b; 20a, b; 21a, b; 22a, b) provides the fundamental information to evaluate the vectors in Eq. (18a–d) and build the interval solution.

Efficacy and the accuracy of the proposed approach is confirmed by the excellent agreement between the proposed solutions and those obtained by the VM.

Additionally, notice Eq. (14a, b) allows to straightforwardly evaluate the amplitude bounds for both the stress resultants, reported in Fig. 23.

Fig. 23

Multi-cracked simply-supported beam in Fig. 3a: LB and UB of the amplitude of a the bending moment function $\left| {M^{I} (x)} \right|$ and b the shear force function $\left| {S^{I} (x)} \right|$ provided by the proposed method considering uncertain stiffness $\kappa_{{{\Delta \Theta}_{j}}}$ and position $x_{j}$ of all springs ($\Delta \kappa_{{{\Delta \Theta}_{j}}} = 0.2\kappa_{{{\Delta \Theta}_{0}}}$,$\Delta x_{j} = 0.2$ with j = 1,2,3)

3.2 Multi-cracked clamped-supported beam equipped with a TMD

For the clamped-supported beam in Fig. 3b, the M-test implemented at the forcing frequency = 3 Hz reveals that there are ranges of abscissas x where both bending moment function $M\left({x,{\boldsymbol{\upkappa}}_{{\Delta {\Theta}}}^{I},{\mathbf{x}}^{I}} \right)$ and shear force $S\left({x,{\boldsymbol{\upkappa}}_{{\Delta {\Theta}}}^{I},{\mathbf{x}}^{I}} \right)$ are non-monotonic functions with respect to uncertain parameters related to the positions of the springs; this occurs for both real and imaginary parts. At these abscissas x, the bounds of the stress resultant functions are evaluated via the GO technique (Sect. 3.3) while, at those abscissas where bending moment and shear force functions have monotonic behaviour with respect to all the uncertain parameters, the SB method (Sect. 3.2) can be applied once again. These considerations hold true for both real and imaginary parts.

Figures 24c and 25c depict the bounds of real and imaginary part of bending moment function $M\left({x,{\boldsymbol{\upkappa}}_{{\Delta {\Theta}}}^{I},{\mathbf{x}}^{I}} \right)$; the sensitivities with respect to stiffness $\kappa_{{\Delta {\Theta}_{j}}}^{I}$ and position $x_{j}$ for all springs are reported in Fig. 24a, b for the real part and in Fig. 25a, b for the imaginary parts

Fig. 24

Multi-cracked clamped-supported beam in Fig. 3b, considering uncertain stiffness $\kappa_{{{\Delta \Theta}_{j}}}$ and position $x_{j}$ of all springs ($\Delta \kappa_{{{\Delta \Theta}_{j}}} = 0.2\kappa_{{{\Delta \Theta}_{0}}}$,$\Delta x_{j} = 0.2$ with j = 1,2,3): a real parts of the sensitivity functions $\text{Re} [\sigma_{{\kappa_{{\Delta {\Theta}_{j}}}}}^{\left(M \right)} (x)]$, b real parts of the sensitivity functions $\text{Re} [\sigma_{{x_{j}}}^{\left(M \right)} (x)]$ and (b) LB and UB of the real part of the bending moment function $\text{Re} [M(x)]$ provided by the proposed method (SB method/GO technique) and VM/MCS

Fig. 25

Multi-cracked clamped-supported beam in Fig. 3b, considering uncertain stiffness $\kappa_{{{\Delta \Theta}_{j}}}$ and position $x_{j}$ of all springs ($\Delta \kappa_{{{\Delta \Theta}_{j}}} = 0.2\kappa_{{{\Delta \Theta}_{0}}}$,$\Delta x_{j} = 0.2$ with j = 1,2,3): a imaginary parts of the sensitivity functions $\text{Im} [\sigma_{{\kappa_{{\Delta {\Theta}_{j}}}}}^{\left(M \right)} (x)]$, b imaginary parts of the sensitivity functions $\text{Im} [\sigma_{{x_{j}}}^{\left(M \right)} (x)]$ and c LB and UB of the imaginary part of the bending moment function $\text{Im} [M(x)]$ provided by the proposed method (SB method/GO technique) and VM/MCS

Corresponding results are reported for the shear force $S\left({x,{\boldsymbol{\upkappa}}_{{\Delta {\Theta}}}^{I},{\mathbf{x}}^{I}} \right)$ in Figs. 26 and 27.

Fig. 26

Multi-cracked clamped-supported beam in Fig. 3b, considering uncertain stiffness $\kappa_{{{\Delta \Theta}_{j}}}$ and position $x_{j}$ of all springs ($\Delta \kappa_{{{\Delta \Theta}_{j}}} = 0.2\kappa_{{{\Delta \Theta}_{0}}}$,$\Delta x_{j} = 0.2$ with j = 1,2,3): a real parts of the sensitivity functions $\text{Re} [\sigma_{{\kappa_{{\Delta {\Theta}_{j}}}}}^{\left(S \right)} (x)]$, b real parts of the sensitivity functions $\text{Re} [\sigma_{{x_{j}}}^{\left(S \right)} (x)]$ and c LB and UB of the real part of the shear force function $\text{Re} [S(x)]$ provided by the proposed method (SB method/GO technique) and VM/MCS

Fig. 27

Multi-cracked clamped-supported beam in Fig. 3b, considering uncertain stiffness $\kappa_{{{\Delta \Theta}_{j}}}$ and position $x_{j}$ of all springs ($\Delta \kappa_{{{\Delta \Theta}_{j}}} = 0.2\kappa_{{{\Delta \Theta}_{0}}}$,$\Delta x_{j} = 0.2$ with j = 1,2,3): a imaginary parts of the sensitivity functions $\text{Im} [\sigma_{{\kappa_{{\Delta {\Theta}_{j}}}}}^{\left(S \right)} (x)]$, b imaginary parts of the sensitivity functions $\text{Im} [\sigma_{{x_{j}}}^{\left(S \right)} (x)]$ and c LB and UB of the imaginary part of shear force function $\text{Im} [S(x)]$ provided by the proposed approach (SB method/GO technique) and VM/MCS

Again, an excellent agreement is noticed between the proposed solutions and those obtained by the VM combined with the MCS, for both real and imaginary parts of the stress resultants. For completeness, Fig. 28 shows the bounds of the amplitude of bending moment and shear force frequency responses, confirming the remarkable accuracy of the proposed approach.

Fig. 28

Multi-cracked clamped-supported beam equipped with a TMD in Fig. 3b: LB and UB of the amplitude of a the bending moment function $\left| {M^{I} (x)} \right|$ and b the shear force function $\left| {S^{I} (x)} \right|$ provided by the proposed method considering uncertain stiffness $\kappa_{{{\Delta \Theta}_{j}}}$ and position $x_{j}$ of all springs ($\Delta \kappa_{{{\Delta \Theta}_{j}}} = 0.2\kappa_{{{\Delta \Theta}_{0}}}$,$\Delta x_{j} = 0.2$ with j = 1,2,3)