Static damage identification in beams by minimum constitutive relation error

Figures & data

Figure 1. A Bernoulli-Euler beam with damage.

Table 1. A two-step substitution algorithm for min-CRE approach under multiple sets of static measurements w.

Given the initial stiffness $k^0$ and the initial bending moment $M^{\left(1\right)0},M^{\left(2\right)0},\dots$, i.e.

$\begin{array}{l}{M}^{\left(1\right)0}=Forc{e}_{-}Method(k^0,{\mathrm{\Im }}^{\left(1\right)},{\mathcal{U}}^{\left(1\right)}),\\ M^{\left(2\right)0}=Forc{e}_{-}Method(k^0,{\mathrm{\Im }}^{\left(2\right)},{\mathcal{U}}^{\left(2\right)}),\\ \dots \end{array}$

Successively determine $\left(k^{n+1},M^{\left(1\right)n+1},M^{\left(2\right)n+1},\dots \right)$ from $\left(k^n,M^{\left(1\right)n},M^{\left(2\right)n},\dots \right)$ by the following two steps,

step 1: $k^{n+1}=\{\sqrt{\frac{{\int }_{X_i}\sum _j{r}_j{\left(M^{\left(j\right)n}\right)}^{2}dx}{{\int }_{X_i}\sum _j{r}_j{\left({w^{\left(j\right)}}^{\mathrm{\prime \prime }}\right)}^{2}dx}},x\in X_i,i=1,2,\dots ,N\}$,

step 2: $\begin{array}{l}{M}^{\left(1\right)n+1}=Forc{e}_{-}Method(k^{n+1},{\mathrm{\Im }}^1,{\mathcal{U}}^1),\\ M^{\left(2\right)n+1}=Forc{e}_{-}Method(k^{n+1},{\mathrm{\Im }}^2,{\mathcal{U}}^2),\\ \dots \end{array}$

Test of convergence with given error tolerance $\text{TOL}$,

$\sqrt{\frac{{\int }_X{(k^{n+1}-k^n)}^{2}dx}{{\int }_X{\left(k^0\right)}^{2}dx}}+\sqrt{\frac{{\int }_X{(M^{\left(1\right)n+1}-M^{\left(1\right)n})}^{2}dx}{{\int }_X{\left(M^{\left(1\right)0}\right)}^{2}dx}}+\sqrt{\frac{{\int }_X{(M^{\left(2\right)n+1}-M^{\left(2\right)n})}^{2}dx}{{\int }_X{\left(M^{\left(2\right)0}\right)}^{2}dx}}+\dots \le \text{TOL}$

Table 2. A two-step substitution algorithm for modified min-CRE approach under multiple sets of static measurements w.

Given the initial stiffness $k^0$, the initial displacements $w^{\left(1\right)0},w^{\left(2\right)0},\dots$ and the initial bending moments $M^{\left(1\right)0},M^{\left(2\right)0},\dots$, i.e.

$\begin{array}{l}(M^{\left(1\right)0},w^{\left(1\right)0})=recovery(k^0,{\mathrm{\Im }}^{\left(1\right)},{\mathcal{U}}^{\left(1\right)},\{{{\hat{w}}_i}^{\left(1\right)},i\in \mathrm{\wp }\left\}\right),\\ (M^{\left(2\right)0},w^{\left(2\right)0})=recovery(k^0,{\mathrm{\Im }}^{\left(2\right)},{\mathcal{U}}^{\left(2\right)},\{{{\hat{w}}_i}^{\left(2\right)},i\in \mathrm{\wp }\left\}\right),\\ \dots \dots \end{array}$

Successively determine $\left(k^{n+1},M^{\left(j\right)n+1},w^{\left(j\right)n+1}\right)$ from $\left(k^n,M^{\left(j\right)n},w^{\left(j\right)n}\right)$ by the following two steps,

step 1: $k^{n+1}=\left\{\sqrt{\frac{{\int }_{X_i}\sum _j{r}_j{\left(M^{\left(j\right)n}\right)}^{2}dx}{{\int }_{X_i}\sum _j{r}_j{\left({w^{\left(j\right)n}}^{\mathrm{\prime \prime }}\right)}^{2}dx}},x\in X_i,i=1,2,\dots ,N\right\}$,

step 2: $\begin{array}{l}(M^{\left(1\right)n+1},w^{\left(1\right)n+1})=recovery(k^{n+1},{\mathrm{\Im }}^{\left(1\right)},{\mathcal{U}}^{\left(1\right)},\{{{\hat{w}}_i}^{\left(1\right)},i\in \mathrm{\wp }\left\}\right),\\ (M^{\left(2\right)n+1},w^{\left(2\right)n+1})=recovery(k^{n+1},{\mathrm{\Im }}^{\left(2\right)},{\mathcal{U}}^{\left(2\right)},\{{{\hat{w}}_i}^{\left(2\right)},i\in \mathrm{\wp }\left\}\right),\\ \dots \dots \end{array}$

Test of convergence with given error tolerance $\text{TOL}$,

$\begin{array}{l}\sqrt{\frac{{\int }_X{(k^{n+1}-k^n)}^{2}dx}{{\int }_X{\left(k^0\right)}^{2}dx}}+\sqrt{\frac{{\int }_X{(M^{\left(1\right)n+1}-M^{\left(1\right)n})}^{2}dx}{{\int }_X{\left(M^{\left(1\right)0}\right)}^{2}dx}}+\sqrt{\frac{{\int }_X{(M^{\left(2\right)n+1}-M^{\left(2\right)n})}^{2}dx}{{\int }_X{\left(M^{\left(2\right)0}\right)}^{2}dx}}+\dots \\ +\sqrt{\frac{{\int }_X{(w^{\left(1\right)n+1}-w^{\left(1\right)n})}^{2}dx}{{\int }_X{\left(w^{\left(1\right)0}\right)}^{2}dx}}+\sqrt{\frac{{\int }_X{(w^{\left(2\right)n+1}-w^{\left(2\right)n})}^{2}dx}{{\int }_X{\left(w^{\left(2\right)0}\right)}^{2}dx}}+\dots \le \text{TOL}\end{array}$

Figure 2. Geometry and loads for a propped cantilever beam.

Figure 3. Damage identifications in propped cantilever beam: (a) 25 elements under single load pattern and two load patterns, (b) 50 elements under two load patterns, (c) 100 elements under two loads pattern and three load patterns and (d) 200 elements under three load patterns.

Figure 4. Convergence of bending moment at left end of propped cantilever beam.

Figure 5. Geometry and experimental displacements of simply supported beam.

Figure 6. Second moment of area identified by modified min-CRE approach comparing to the theoretical second moment of area of simply supported beam.

Figure 7. Geometry, damage location and load cases of three-span continuous beam (D1-damage case 1, D2-damage case 2).

Figure 8. Mean values and standard deviations of DAI with 0.1 mm noise for: (a) small damage case D1 by min-CRE approach, (b) small damage case D1 by modified min-CRE approach, (c) large damage case D2 by min-CRE approach, (d) large damage case D2 by modified min-CRE approach.

Figure 9. Convergence of the damage indices of three-span continuous beam under D2.

Figure 10. Displacement fields rebuilt by modified min-CRE approach comparing to the displacement boundary.

Figure 11. Geometry and finite element model of the cracked beam with two-fixed ends.

Figure 12. DAI from the min-CRE approach vs. NDE from referred approach [58].

Table 3. DAI obtained by modified min-CRE approach comparing with accurate results and results in Reference [58].

		DAI
Method	Possible damaged element	1st load case	2nd load case
Accurate	9	0.125	0.125
Reference	9	0.594	0.580
Min-CRE	8	0.366	0.403
	9	0.290	0.326
	10	0.287	0.315
	11	0.377	0.396

Wang X, Hu N, Fukunaga H, et al. Structural damage identification using static test data and changes in frequencies. Eng Struct. 2001;23:610–621. doi: 10.1016/S0141-0296(00)00086-9

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