Mode Shape Analysis of Multiple Cracked Functionally Graded Timoshenko Beams

Lien, Tran Van; Đuc, Ngo Trong; Khiem, Nguyen Tien

doi:10.1590/1679-78253496

Tran Van Lien ^** Corresponding author

National University of Civil Engineering; Vietnam, LienTV@nuce.edu.vn

Ngo Trong Đuc

Design Consultant and Investment of Construction, Vietnam trongduc.4s@gmail.com.

Nguyen Tien Khiem

Institute of Mechanics, Vietnam Academy of Science and Technology, Vietnam ntkhiem@imech.vast.vn.

* Corresponding author

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Figure 1
A multiple cracked FGM beam.

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Figure 2
Comparison of three lowest mode shapes for simple support Timoshenko beam with 2 cracks at position 0.2m and 0.4m, crack depth a/h=30%.

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Figure 3
Comparison of the first three mode shapes with (Su & Banerjee 2015H. Su, J.R. Banerjee (2015), “Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beam”, Computers & Structures, 147, 107-116.) for intact cantilever FGM Timoshenko beam.

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Figure 4
Change in first three mode shapes of simple support FGM Timoshenko beam which has 0 to 6 equidistant cracks (∆=0.15m) of the same depth a/h=30%.

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Figure 5
Change in first three mode shapes of simple support FGM beam with 2 cracks at 0.4m and 0.6m and depth varying from a/h=0% to 50%.

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Figure 6
Change in first three mode shapes of simple support FGM beam with a crack at 0.2m and depth varying from a/h=10% to 30%.

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Figure 7
Change in first three mode shapes of clamped end FGM beam with a crack at 0.2m and depth varying from a/h=10% to 30%.

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Figure 8
Change in first three mode shapes of cantilever FGM beam with a crack at 0.2m and depth varying from a/h=10% to 30%.

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Figure 9
Change in first three mode shapes of simple support FGM beam with 4 equidistant cracks and depth a/h=10%, 20%, 30%.

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Figure 10
Change in first three mode shapes of clamped end FGM beam with 4 equidistant cracks of the depth a/h=10%, 20%, 30%.

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Figure 11
Change in first three mode shapes of cantilever FGM beam with 4 equidistant cracks of the depth a/h=10%, 20%, 30%.

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Figure 12
Change in first three mode shapes of simple support FGM beam with various number of cracks (from 1 to 4) and depth a/h=30%.

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Figure 13
Change in first three mode shapes of clamped end FGM beam with various number of cracks (from 1 to 4) and depth a/h=30%.

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Figure 14
Change in first three mode shapes of cantilever FGM beam with various number of cracks (from 1 to 4) and depth a/h=30%.

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Table 1
Natural frequencies corresponding to the mode shapes illustrated in the -.

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Figs	Cases	Freq. 1 (rad/s)	Freq. 2 (rad/s)	Freq. 3 (rad/s)
2	Present	1316.9	5131.4	10743
2	Lien & Hao	1330.7	5162.5	10803
3	Present	708.5	4248.7	11086
3	Su & Banerjee	719.0	4314.0	11220
4	0 crack	990.53	3782.6	7963.2
	1 crack	972.22	3584.8	7464.5
	2 cracks	919.77	3403.7	7463.9
	3 cracks	856.15	3402.4	7016.1
	4 cracks	808.42	3274.1	6971.5
	5 cracks	785.25	3073.8	6662.5
	6 cracks	781.0	3016.6	6439.2
5	a/h=0%	990.53	3782.6	7963.2
	a/h=10%	973.11	3758.3	7918.2
	a/h=20%	927.29	3689.6	7806.4
	a/h=30%	858.76	3572.1	7656.0
	a/h=40%	771.73	3393.3	7491.0
	a/h=50%	669.14	3131.8	7328.9
6	a/h=10%	987.12	3751.1	7903.8
	a/h=20%	977.39	3664.6	7752.3
	a/h=30%	960.34	3526.3	7540.6
7	a/h=10%	2141.4	5472.1	9855.6
	a/h=20%	2140.5	5430.4	9702.7
	a/h=30%	2139.0	5365.0	9481.5
8	a/h=10%	352.26	2131.6	5548.5
	a/h=20%	342.07	2131.3	5502.9
	a/h=30%	325.72	2130.8	5418.2
9	a/h=10%	966.69	3697.1	7798.0
	a/h=20%	906.19	3478.2	7367.6
	a/h=30%	820.88	3165.0	6736.2
10	a/h=10%	2120.6	5417.1	9758.9
	a/h=20%	2064.6	5237.8	9357.3
	a/h=30%	1979.2	4982.5	8781.0
11	a/h=10%	350.39	2095.6	5477.2
	a/h=20%	335.52	2002.4	5234.9
	a/h=30%	312.86	1866.2	4874.6
12	1 crack	960.34	3526.3	7540.6
	2 cracks	892.98	3478.9	7256.2
	3 cracks	839.06	3353.7	7185.8
	4 cracks	820.88	3165.0	6736.2
13	1 crack	2139.0	5365.0	9481.5
	2 cracks	2052.0	5247.7	9229.8
	3 cracks	1987.4	5054.7	9181.3
	4 cracks	1979.2	4982.5	8781.0
14	1 crack	325.72	2130.8	5418.2
	2 cracks	315.48	2014.1	5297.4
	3 cracks	313.05	1889.2	5093.8
	4 cracks	312.86	1866.2	4874.6

{\begin{matrix} E (z) \\ G (z) \\ ρ (z) \end{matrix}} = {\begin{matrix} E_{b} \\ G_{b} \\ ρ_{b} \end{matrix}} + {\begin{matrix} E_{t} - E_{b} \\ G_{t} - G_{b} \\ ρ_{t} - ρ_{b} \end{matrix}} {(\frac{z}{h} + \frac{1}{2})}^{n}, - h / 2 \leq z \leq h / 2

\tilde{A} = [\begin{matrix} A_{11} & - A_{12} & 0 \\ - A_{12} & A_{22} & 0 \\ 0 & 0 & A_{33} \end{matrix}]; \tilde{Π} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & A_{33} \\ 0 & - A_{33} & 0 \end{matrix}]; \tilde{C} (ω) = [\begin{matrix} ω^{2} I_{11} & - ω^{2} I_{12} & 0 \\ - ω^{2} I_{12} & ω^{2} I_{22} - A_{33} & 0 \\ 0 & 0 & ω^{2} I_{11} \end{matrix}]

{z_{0}} = [\begin{matrix} \begin{matrix} d_{11} & d_{12} \end{matrix} & ... & d_{16} \\ \begin{matrix} d_{21} & d_{22} \end{matrix} & ... & d_{26} \\ \begin{matrix} d_{31} & d_{32} \end{matrix} & ... & d_{36} \end{matrix}] \cdot {\begin{matrix} e^{λ_{1} x} \\ ⋮ \\ e^{λ_{6} x} \end{matrix}}

[\begin{matrix} \begin{matrix} d_{11} & d_{12} \end{matrix} & ... & d_{16} \\ \begin{matrix} d_{21} & d_{22} \end{matrix} & ... & d_{26} \\ \begin{matrix} d_{31} & d_{32} \end{matrix} & ... & d_{36} \end{matrix}] = [\begin{matrix} \begin{matrix} α_{1} C_{1} & α_{2} C_{2} \end{matrix} & ... & α_{6} C_{6} \\ \begin{matrix} C_{1} & C_{2} \end{matrix} & ... & C_{6} \\ \begin{matrix} β_{1} C_{1} & β {}_{2}C_{2} \end{matrix} & ... & β_{6} C_{6} \end{matrix}]

G_{1} (x, ω) = [\begin{matrix} e^{k_{1} x} & e^{k_{2} x} & e^{k_{3} x} \\ α_{1} e^{k_{1} x} & α_{2} e^{k_{2} x} & α_{3} e^{k_{3} x} \\ β_{1} e^{k_{1} x} & β_{2} e^{k_{2} x} & β_{3} e^{k_{3} x} \end{matrix}]; G_{2} (x, ω) = [\begin{matrix} e^{- k_{1} x} & e^{- k_{2} x} & e^{- k_{3} x} \\ α_{1} e^{- k_{1} x} & α_{2} e^{- k_{2} x} & α_{3} e^{- k_{3} x} \\ - β_{1} e^{- k_{1} x} & - β_{2} e^{- k_{2} x} & - β_{3} e^{- k_{3} x} \end{matrix}]

\begin{matrix} U (e_{j} + 0) = U (e_{j} - 0); Θ (e_{j} + 0) = Θ (e_{j} - 0) + M (e_{j}) / K_{j}; W (e_{j} + 0) = W (e_{j} - 0) \\ N (e_{j}) = N (e_{j} + 0) = N (e_{j} - 0); Q (e_{j} + 0) = Q (e_{j} - 0); M (e_{j} + 0) = M (e_{j} - 0) = M (e_{j}) \end{matrix}

\begin{matrix} U (e_{j} + 0) = U (e_{j} - 0); Θ (e_{j} + 0) = Θ (e_{j} - 0) + γ {}_{j}{Θ^{'}}_{x} (e_{j}); W (e_{j} + 0) = W (e_{j} - 0) \\ {U^{'}}_{x} (e_{j} + 0) = {U^{'}}_{x} (e_{j} - 0); {Θ^{'}}_{x} (e_{j} + 0) = {Θ^{'}}_{x} (e_{j} - 0); {W^{'}}_{x} (e_{j} + 0) = {W^{'}}_{x} (e_{j} - 0) + γ_{j} {Θ^{'}}_{x} (e_{j}) \\ γ_{j} = A_{22} / K_{j}; j = 1,2,3,..., n \end{matrix}

\begin{matrix} γ_{j} = γ_{b} θ_{2} (R_{E}, n) \\ γ_{b} = E_{b} I / R; I = b h^{3} / 12; θ_{2} (R_{E}, n) = 12 (\frac{3 R_{E} + n}{3 (3 + n)} - \frac{2 R_{E} + n}{(2 + n)} α + \frac{R_{E} + n}{(1 + n)} α^{2}) \end{matrix}

\begin{matrix} S_{1} (x) = δ_{1} \cosh k_{1} x + δ_{2} \cosh k_{2} x + δ_{3} \cosh k_{3} x \\ S_{2} (x) = δ_{1} α_{1} \cosh k_{1} x + δ_{2} α_{2} \cosh k_{2} x + δ_{3} α_{3} \cosh k_{3} x \\ S_{3} (x) = δ_{1} β_{1} \sinh k_{1} x + δ_{2} β_{2} \sinh k_{2} x + δ_{3} β_{3} \sinh k_{3} x \end{matrix}

\begin{matrix} δ_{1} = [k_{2} β_{2} - k_{3} β_{3} + (α_{3} - α_{2})] / δ; δ_{2} = [k_{3} β_{3} - k_{1} β_{1} + (α_{1} - α_{3})] / δ; \\ δ_{3} = [k_{1} β_{1} - k_{2} β_{2} + (α_{2} - α_{1})] / δ; δ = k_{1} β_{1} (α_{3} - α_{2}) + k_{2} β_{2} (α_{1} - α_{3}) + k_{3} β_{3} (α_{2} - α_{1}) . \end{matrix}

\begin{matrix} U_{j} (e_{j}) = U_{j - 1} (e_{j}); Θ_{j} (e_{j}) = Θ_{j - 1} (e_{j}) + γ_{j} {Θ^{'}}_{j - 1} (e_{j}); W_{j} (e_{j}) = W_{j - 1} (e_{j}) \\ {U^{'}}_{j} (e_{j}) = {U^{'}}_{j - 1} (e_{j}); {Θ^{'}}_{j} (e_{j}) = {Θ^{'}}_{j - 1} (e_{j}); {W^{'}}_{j} (e_{j}) = {W^{'}}_{j - 1} (e_{j}) + γ_{j} {Θ^{'}}_{j} (e_{j}) \end{matrix}

\begin{matrix} z_{c} (x) = z_{0} (x) + \sum_{j = 1}^{n} μ_{j} K (x - e_{j}) \\ μ_{j} = γ_{j} [{Θ^{'}}_{0} (e_{j}) + \sum_{k = 1}^{j - 1} μ_{k} {S^{'}}_{2} (e_{j} - e_{k})], j = 1,2,3,..., n \end{matrix}

\begin{matrix} B_{01} C_{1} + B_{02} C_{2} = 0 \\ B_{01} (ω) = B_{0} {G_{1} (x, ω)} | {}_{x = 0}; B_{02} (ω) = B_{0} {G_{2} (x, ω)} | {}_{x = 0} \end{matrix}

\begin{matrix} [B_{L 0} (ω)] {D} + \sum_{j = 1}^{n} μ_{j} {b (e_{j})} = 0 \\ B_{L 0} (ω) = B_{L} {G_{0} (x, ω)} | {}_{x = L}; b_{c} (e_{j}) = B_{L} {S (x - e_{j})} | {}_{x = L} \end{matrix}

\begin{matrix} μ_{j} = (γ_{j} / L_{0}) \sum_{k = 1}^{n} μ_{k} [L_{0} (ω) {K^{'}}_{2} (e_{j} - e_{k}) - {g^{'}}_{j k}], j = 1,2,3,..., n \\ {g^{'}}_{j k} = {g^{'}}_{21} (e_{j}, ω) {\bar{b}}_{c 1} (e_{k}) + {g^{'}}_{22} (e_{j}, ω) {\bar{b}}_{c 2} (e_{k}) + {g^{'}}_{23} (e_{j}, ω) {\bar{b}}_{c 3} (e_{k}) \end{matrix}

\begin{matrix} [L_{0} (ω) I - Γ (γ) A (e, ω)] {μ} = 0 \\ Γ (γ) = d i a g {γ_{1},..., γ_{n}}; A (e, ω) = [a_{j k} = L_{0} (ω) . {K^{'}}_{2} (e_{j} - e_{k}) - {g^{'}}_{j k}; j, k = 1,2,..., n] \\ μ = {μ_{1},..., μ_{n}}^{T}; γ = {γ_{1},..., γ_{n}}^{T}; e = {e_{1},..., e_{n}}^{T} \end{matrix}

\begin{array}{l} (A_{11}, A_{12}, A_{22}) = \int_{A} E (z) (1, z - h_{0}, {(z - h_{0})}^{2}) d A; A_{33} = κ \int_{A} G (z) d A; \\ (I_{11}, I_{12}, I_{22}) = \int_{A} ρ (z) (1, z - h_{0}, {(z - h_{0})}^{2}) d A . \end{array}

\begin{matrix} A_{11} = b h \frac{E_{t} + n E_{b}}{1 + n}; A_{12} = b h^{2} (\frac{2 E_{t} + n E_{b}}{2 (2 + n)} - \frac{E_{t} + n E_{b}}{1 + n} α); α = \frac{1}{2} + \frac{h_{0}}{h} \\ A_{22} = b h^{3} (\frac{3 E_{t} + n E_{b}}{3 (3 + n)} - \frac{2 E_{t} + n E_{b}}{2 + n} α + \frac{E_{t} + n E_{b}}{1 + n} α^{2}); A_{33} = κ b h \frac{G_{t} + n G_{b}}{1 + n}; I_{11} = b h \frac{ρ_{t} + n ρ_{b}}{1 + n} \\ I_{12} = b h^{2} (\frac{2 ρ_{t} + n ρ_{b}}{2 (2 + n)} - \frac{ρ_{t} + n ρ_{b}}{1 + n} α); I_{22} = b h^{3} (\frac{3 ρ_{t} + n ρ_{b}}{3 (3 + n)} - \frac{2 ρ_{t} + n ρ_{b}}{2 + n} α + \frac{ρ_{t} + n ρ_{b}}{1 + n} α^{2}) \end{matrix}

\begin{matrix} a = ω^{2} [\frac{I_{11}}{A_{33}} + \frac{I_{11} A_{22} + I_{22} A_{11}}{A_{11} A_{22}}]; b = ω^{4} [\frac{I_{11} I_{22} - I_{12}^{2}}{A_{11} A_{22}} + \frac{I_{11}}{A_{33}} \frac{I_{11} A_{22} + I_{22} A_{11}}{A_{11} A_{22}}] - ω^{2} \frac{I_{11}}{A_{22}} \\ c = ω^{4} [\frac{ω^{2} I_{11}}{A_{33}} \frac{I_{11} I_{22} - I_{12}^{2}}{A_{11} A_{22}} - \frac{I_{11}^{2}}{A_{11} A_{22}}] \end{matrix}

[1] * Corresponding author

Brasil

Brasil

Mode Shape Analysis of Multiple Cracked Functionally Graded Timoshenko Beams

Abstract