Analytical solution of deflection of multi-cracked beams on elastic foundations under arbitrary boundary conditions using a diffused stiffness reduction crack model

Abstract

Crack can significantly affect the performance of structures and is one of the crucial indicators of damage in structural health monitoring. In this paper, the deflection behaviors of Euler–Bernoulli beams with arbitrary open edge cracks under arbitrary elastic boundary conditions are investigated. A continuous diffused stiffness reduction crack model is implemented to simulate the cracks in beams, which can incorporate multiple cracks and consider the stiffness reduction effect in the vicinity of a crack. With the proposed diffused stiffness reduction model, the fourth-order differential equation governing the deflection behavior of the multi-cracked Euler–Bernoulli beam is constructed. The powerful variational iteration method is applied to obtain the analytical solution of the multi-cracked beams on elastic foundations. Five shape functions are introduced, based on which the deflection of the multi-cracked beam is proposed. Both the solutions corresponding to the general elastic boundary conditions and the conventional boundary conditions are presented explicitly. The deflection solution is benchmarked and verified against the literature, and encouraging agreements are obtained. Parametric studies are carried out to investigate the influences of crack position, crack ratio, stiffness of the elastic foundation, and boundary conditions on the deflection of the cracked beams. The proposed crack model and the deflection solution overcome some of the limitations in the literature.

Appendices

Appendix A

$$\begin{aligned}&{\cosh }\left( {\frac{{\sqrt{2}}}{2} {{\beta }} {{\xi }}} \right) {\cos }\left( {\frac{{\sqrt{2}}}{2} {{\beta }} {{\xi }}} \right) ={{S}}_{0} ({{\xi }}) \end{aligned}$$

(A.1)

$$\begin{aligned}&\frac{{{{\sin }}\left( {\frac{{\sqrt{2}}}{2}{{\beta }} {{\xi }}} \right) {{\cosh }}\left( {\frac{{\sqrt{2}}}{2}{{\beta }} {{\xi }}} \right) + {{\sinh }}\left( {\frac{{\sqrt{2}}}{2}{{\beta }} {{\xi }}} \right) {{\cos }}\left( {\frac{{\sqrt{2}}}{2}{{\beta }} {{\xi }}} \right) }}{{\sqrt{2} {{\beta }}}}={{S}}_{1} ({{\xi }}) \end{aligned}$$

(A.2)

$$\begin{aligned}&\frac{{{{\sin }}\left( {\frac{{\sqrt{2}}}{2}{{\beta }} {{\xi }}} \right) {{\sinh }}\left( {\frac{{\sqrt{2}}}{2}{{\beta }} {{\xi }}} \right) }}{{{{\beta }}^{2} }}={{S}}_{2} ({{\xi }}) \end{aligned}$$

(A.3)

$$\begin{aligned}&\frac{{{{\sin }}\left( {\frac{{\sqrt{2}}}{2}{{\beta }} {{\xi }}} \right) {{\cosh }}\left( {\frac{{\sqrt{2}}}{2}{{\beta }} {{\xi }}} \right) - {{\sinh }}\left( {\frac{{\sqrt{2}}}{2}{{\beta }} {{\xi }}} \right) {{\cos }}\left( {\frac{{\sqrt{2}}}{2}{{\beta }} {{\xi }}} \right) }}{{\sqrt{2} {{\beta }}^{3} }}={{S}}_{3} ({{\xi }}) \end{aligned}$$

(A.4)

$$\begin{aligned}&\frac{{{{\gamma }}_{0} }}{{{{\beta }}^{4} }}\left[ {1 - {{S}}_{0} ({{\xi }})} \right] ={{S}}_{4} ({{\xi }}) \end{aligned}$$

(A.5)

$$\begin{aligned} {{{{\varPsi }}}_0}({{\xi }})= & {} {{{S}}_0}({{\xi }}) + {{{\beta }}^4}\int \limits _0^{{\xi }} {{{S}}_1}({{{\tau }} - {{\xi }}}){{{r}}_{{c}}}({{\tau }}) ({1 + {{{r}}_{{c}}}({{\tau }})}){{{S}}_2} ({{\tau }}){{d}}{{\tau }} \nonumber \\&+ {{{\beta }}^8}\int \limits _0^{{\xi }} {{{S}}_1}({{{\tau }} - {{\xi }}}){{{r}}_{{c}}} ({{\tau }})\int \limits _0^{{\tau }} {{{S}}_3}({{{u}} - {{\tau }}}){{{r}}_{{c}}} ({{u}}){{{S}}_2}({{u}}) {{\mathrm{d}u\mathrm{d}}}{{\tau }} \end{aligned}$$

(A.6)

$$\begin{aligned} {{{\varPsi }}_1}({{\xi }})= & {} {{{S}}_1}({{\xi }}) + {{{\beta }}^4}\int \limits _0^{{\xi }} {{{S}}_1}({{{\tau }} - {{\xi }}}){{{r}}_{{c}}}({{\tau }})({1 + {{{r}}_{{c}}}({{\tau }})}) {{{S}}_3}({{\tau }})\mathrm{d}{{\tau }} \nonumber \\&+ {{{\beta }}^8}\int \limits _0^{{\xi }} {{{S}}_1}({{{\tau }} - {{\xi }}}){{{r}}_{{c}}} ({{\tau }})\int \limits _0^{{\tau }} {{{S}}_3}({{{u}} - {{\tau }}}){{{r}}_{{c}}} ({{u}}){{{S}}_3}({{u}}) {{\mathrm{d}u\mathrm{d}}}{{\tau }} \end{aligned}$$

(A.7)

$$\begin{aligned} {{{\varPsi }}_2}({{\xi }})= & {} {{{S}}_2}({{\xi }}) - \int \limits _0^{{\xi }} {{{S}}_1}({{{\tau }} - {{\xi }}}){{{r}}_{{c}}} ({{\tau }})({1 + {{{r}}_{{c}}} ({{\tau }})}){{{S}}_0}({{\tau }}) \mathrm{d}{{\tau }} \nonumber \\&- {{{\beta }}^4}\int \limits _0^{{\xi }} {{{S}}_1}({{{\tau }} - {{\xi }}}){{{r}}_{{c}}} ({{\tau }})\int \limits _0^{{\tau }} {{{S}}_3}({{{u}} - {{\tau }}}){{{r}}_{{c}}}({{u}}) {{{S}}_0}({{u}}){{\mathrm{d}u\mathrm{d}}}{{\tau }} \end{aligned}$$

(A.8)

$$\begin{aligned} {{{\varPsi }}_3}({{\xi }})= & {} {{{S}}_3}({{\xi }}) - \int \limits _0^{{\xi }} {{{S}}_1}({{{\tau }} - {{\xi }}}){{{r}}_{{c}}}({{\tau }}) ({1 + {{{r}}_{{c}}}({{\tau }})}) {{{S}}_1}({{\tau }})\mathrm{d}{{\tau }} \nonumber \\&- {{{\beta }}^4}\int \limits _0^{{\xi }} {{{S}}_1}({{{\tau }} - {{\xi }}}){{{r}}_{{c}}}({{\tau }}) \int \limits _0^{{\tau }} {{{S}}_3}({{{u}} - {{\tau }}}){{{r}}_{{c}}}({{u}}) {{{S}}_1}({{u}}){{\mathrm{d}u\mathrm{d}}}{{\tau }} \end{aligned}$$

(A.9)

$$\begin{aligned} {{{\varPsi }}_4}({{\xi }})= & {} \frac{{{{{\gamma }}_0}}}{{{{{\beta }}^4}}} [{1 - {{{S}}_0}({{\xi }})}] - {{{\gamma }}_0}\int \limits _0^{{\xi }} {{{S}}_1}({{{\tau }} - {{\xi }}}){{{r}}_{{c}}} ({{\tau }})({1 + {{{r}}_{{c}}} ({{\tau }})}){{{S}}_2}({{\tau }}) \mathrm{d}{{\tau }} \nonumber \\&- {{{\gamma }}_0}{{{\beta }}^4}\int \limits _0^{{\xi }} {{{S}}_1}({{{\tau }} - {{\xi }}}){{{r}}_{{c}}} ({{\tau }})\int \limits _0^{{\tau }} {{{S}}_3}({{{u}} - {{\tau }}}){{{r}}_{{c}}} ({{u}}){{{S}}_2}({{u}}) {{\mathrm{d}u\mathrm{d}}}{{\tau }} \end{aligned}$$

(A.10)

Appendix B

Boundary conditions	Equation
P–P	Coefficient equation	$\left( \frac{{{\phi }}({1})}{{{\phi }}^{\prime \prime }({1})} \right) =\left[ {\begin{array}{*{20}c} {{{\varPsi }}}_{{1}}({1}) &{} {{{\varPsi }}}_{{3}}({1})\\ {{{\varPsi }}}_{{1}}^{\prime \prime }({1}) &{} {{{\varPsi }}}_{{3}}^{\prime \prime }({1})\\ \end{array} } \right] \left( \frac{{{\phi }}^{\prime }(0)}{{{\phi }}^{\prime \prime \prime }(0)} \right) =\left( {\begin{array}{l} -{{{\varPsi }}}_{{4}}({1}) \\ -{{{\varPsi }}}_{{4}}^{\prime \prime }({1}) \\ \end{array}} \right)$	(B. 1)
	Deflection equation	${{\phi }}( {{\xi }})={{\phi }}^{\prime }(0){{{\varPsi }}}_{{1}}( {{\xi }})+{{\phi }}^{\prime \prime \prime }(0){{{\varPsi }}}_{{3}}( {{\xi }})$	(B. 2)
C–C	Coefficient equation	$\left( \frac{{{\phi }}({1})}{{{\phi }}^{\prime }({1})} \right) =\left[ {\begin{array}{*{20}c} {{{\varPsi }}}_{{2}}({1}) &{} {{{\varPsi }}}_{{3}}({1})\\ {{{\varPsi }}}_{{2}}^{\prime }({1}) &{} {{{\varPsi }}}_{{3}}^{\prime }({1})\\ \end{array} } \right] \left( \frac{{{\phi }}^{\prime \prime }(0)}{{{\phi }}^{\prime \prime \prime }(0)} \right) =\left( {\begin{array}{l} -{{{\varPsi }}}_{{4}}({1}) \\ -{{{\varPsi }}}_{{4}}^{\prime }({1}) \\ \end{array}} \right)$	(B. 3)
	Deflection equation	${{\phi }}( {{\xi }})={{\phi }}^{\prime \prime }(0){{{\varPsi }}}_{{2}}( {{\xi }})+{{\phi }}^{\prime \prime \prime }(0){{{\varPsi }}}_{{3}}( {{\xi }})$	(B. 4)
C–F	Coefficient equation	$\left( \frac{{{\phi }}^{\prime \prime }({1})}{{{\phi }}^{\prime \prime \prime }({1})} \right) =\left[ {\begin{array}{*{20}c} {{{\varPsi }}}_{{2}}^{\prime \prime }({1}) &{} {{{\varPsi }}}_{{3}}^{\prime \prime }({1})\\ {{{\varPsi }}}_{{2}}^{\prime \prime \prime }({1}) &{} {{{\varPsi }}}_{{3}}^{\prime \prime \prime }({1})\\ \end{array} } \right] \left( \frac{{{\phi }}^{\prime \prime }(0)}{{{\phi }}^{\prime \prime \prime }(0)} \right) =\left( {\begin{array}{l} -{{{\varPsi }}}_{{4}}^{\prime \prime }({1}) \\ -{{{\varPsi }}}_{{4}}^{\prime \prime \prime }({1}) \\ \end{array}} \right)$	(B. 5)
	Deflection equation	${{\phi }}( {{\xi }})={{\phi }}^{\prime \prime }(0){{{\varPsi }}}_{{2}}( {{\xi }})+{{\phi }}^{\prime \prime \prime }(0){{{\varPsi }}}_{{3}}( {{\xi }})$	(B. 6)
C–P	Coefficient equation	$\left( \frac{{{\phi }}({1})}{{{\phi }}^{\prime \prime }({1})} \right) =\left[ {\begin{array}{*{20}c} {{{\varPsi }}}_{{2}}({1}) &{} {{{\varPsi }}}_{{3}}({1})\\ {{{\varPsi }}}_{{2}}^{\prime \prime }({1}) &{} {{{\varPsi }}}_{{3}}^{\prime \prime }({1})\\ \end{array} } \right] \left( \frac{{{\phi }}^{\prime \prime }(0)}{{{\phi }}^{\prime \prime \prime }(0)} \right) =\left( {\begin{array}{l} -{{{\varPsi }}}_{{4}}({1}) \\ -{{{\varPsi }}}_{{4}}^{\prime \prime }({1}) \\ \end{array}} \right)$	(B. 7)
	Deflection equation	${{\phi }}( {{\xi }})={{\phi }}^{\prime \prime }(0){{{\varPsi }}}_{{2}}( {{\xi }})+{{\phi }}^{\prime \prime \prime }(0){{{\varPsi }}}_{{3}}( {{\xi }})$	(B. 8)