Free vibration analysis of a single edge cracked symmetric functionally graded stepped beams

Abstract

Free vibration analysis of a single edge cracked multi-layered symmetric sandwich stepped Timoshenko beams, made of functionally graded materials, is studied using finite element method and linear elastic fracture mechanic theory. The cantilever functionally graded beam consists of 50 layers, assumed that the second stage of the beam (step part) is created by machining. Thus, providing the material continuity between the two beam stages. It is assumed that material properties vary continuously, along the thickness direction according to the exponential and power laws. A developed MATLAB code is used to find the natural frequencies of three types of the stepped beam, concluding a good agreement with the known data from the literature, supported also by ANSYS software in data verification. In the study, the effects of the crack location, crack depth, power law gradient index, different material distributions, different stepped length, different cross-sectional geometries on natural frequencies and mode shapes are analysed in detail.

Keywords

cracked stepped beam finite element method free vibration functionally graded materials Timoshenko beam

Introduction

The idea of the functionally graded materials (FGMs) was first revealed and developed in the 1980s, by a group of Japanese scientists during the preparation of thermal barrier materials for space plane needs (Koizumi, 1993). The FGMs are a specific type of material composites, where material properties vary smoothly and continuously through the surfaces and thereby avoiding delamination problem occurred at the laminated composites. Typically, the ceramic and the metal mixture compose FGMs, where the ceramic acts as an excellent resistant to high-temperature environment surroundings and of corrosion while the metal offers structural strength and toughness (Reddy and Chin, 1998). The use of FGMs finds its application in various engineering areas such as mechanical, aerospace, biomedical industries, defence industries and so on. For example, Gayen and Roy (2014) studied vibration and stability analysis of functionally graded (FG) shaft systems. Khosravi et al. (2017) presented numerical study on FGM in stent design, with an objective of reducing undesirable deformations of the stent. Carvalho et al. (2015) presented a study on using FGM for engine piston rings.

There are plenty of studies on free vibration behaviour and characteristics of uniform cross-section homogeneous and composite intact structures (Erdurcan and Cunedioğlu, 2020; Jiang et al., 2018; Kim and Shin, 2008; Mohanty et al., 2013; Zahedinejad, 2016; Zhang et al., 2020) and cracked structures (Cunedioglu, 2015; Ferezqi et al., 2010; Gayen and Chakraborty, 2016; Gayen et al., 2017a, 2017b, 2018; Han et al., 2020; Kisa and Brandon, 2000; Kisa et al., 1998; Liu et al., 2017; Papadopoulos, 2004; Papadopoulos and Dimarogonas, 1987; Shabani and Cunedioglu, 2019).

Furthermore, the authors extended the studies also on intact stepped beams (Bambill et al., 2015; Su et al., 2018; Suddoung et al., 2014; Wattanasakulpong and Charoensuk, 2015), cracked homogeneous isotropic stepped beams (Al-Said, 2008; Attar, 2012; Kisa and Arif Gurel, 2007; Mao and Pietrzko, 2010; Naguleswaran, 2002; Nandwana and Maiti, 1997) and only one study on cracked FG stepped beams (Khiem et al., 2019).

As one can observe from the late literature review, there is no study available regarding the cracked stepped symmetric FG sandwich beams. Therefore, the purpose of this study is to analyse vibration problems of these three most common cracked FG stepped beams with cantilever boundary condition using finite element methods (FEMs), within Timoshenko first-order shear deformation beam theory, and employing mixture rules and laminate theory for the material properties. The research is carried out to understand the effects of different parameters such as crack location, crack depth, power index (n), different material distributions, different step length and different cross-sectional geometries on the dynamic behaviour of the structure, thus being able to choose which factors should be taken into consideration when performing modal analysis.

Sandwich FG stepped beam

Consider three types of rectangular symmetric sandwich FG stepped beam as shown in Figure 1. The cantilever FG beam consists of 50 layers, assumed that the second stage of the beam (step part) is created by machining. Thus, providing the material continuity between the two beam stages. Figure 2 shows the change of the elasticity modulus (E) and mass density (ρ) along the beam thickness for the exponential law and different power-index values of power law functions. The strain between the layers is linear assumed. Each layer is composed of a mixture of aluminium (Al) and alumina phases (Al₂O₃), and layers are prepared symmetrically to the neutral axis of the beam.

Figure 1.

Three types of symmetric sandwich FGM stepped beam.

Figure 2.

Variations of the modulus of elasticity (a) and the density (b) through the thickness of the beam for exponential law and different power-index values of power law functions.

For the mixture ratio, a polynomial or an exponential function is chosen, varying continuously from the upper and lower part towards the beam neutral plane. Three types of the stepped beam are as follows: type A is with a step change in thickness; type B is with a step change in width; and type C is with a step change in thickness and width. The parameters of the considered beam are as follows: length L, step location Ls, thickness b, width d, thickness b1 and width d1 of the stepped part, respectively.

The material properties are first calculated for the upper half using the exponential and power law equations (Demir et al., 2013)

E (y) = E_{c} e^{(- δ (1 - 2 y))}, δ = \frac{1}{2} \ln (\frac{E_{c}}{E_{m}})

(1)

E (y) = (E_{c} - E_{m}) {(y + \frac{1}{2})}^{n} + E_{m}

(2)

where E_c, E_m, y and n are Young’s modulus of ceramic, metal, the coordinate axis and power law index, respectively. The above equations are adopted for the density

ρ (y) = ρ_{c} e^{(- δ (1 - 2 y))}, δ = \frac{1}{2} \ln (\frac{ρ_{c}}{ρ_{m}})

(3)

ρ (y) = (ρ_{c} - ρ_{m}) {(y + \frac{1}{2})}^{n} + ρ_{m}

(4)

where $ρ_{c}$ and $ρ_{m}$ are the mass density for the ceramic and the metal, respectively.

The variable y is defined as

y = - \frac{1}{2}, - \frac{1}{2} + \frac{1}{η}, - \frac{1}{2} + \frac{2}{η}, \dots, \frac{1}{2}

where $η = m / 2 - 1$ and m is the number of layers.

The effective Young’s modulus and mass density of the whole beam are calculated using laminate theory, by following formulas (Gibson, 1994)

E_{ef} = \frac{8}{b^{3}} \sum_{i = 1}^{N / 2} {(E_{y})}_{i} (y_{i}^{3} - y_{i - 1}^{3})

(5)

ρ_{ef} = \frac{8}{b^{3}} \sum_{i = 1}^{N / 2} {(ρ_{y})}_{i} (y_{i}^{3} - y_{i - 1}^{3})

(6)

Stiffness and mass matrices of the Timoshenko beam element

Stiffness and mass matrix derivation for a beam element having two nodes with 2 degrees of freedom {v, θ} at each node, and for a beam element having two nodes with 1 degree of freedom, local axial displacement, in the x-direction, is given by Petyt (1990) as

[K_{1}] = \frac{E I_{z}}{2 k^{3} (1 + 3 β)} [\begin{matrix} 3 & 3 k & - 3 & 3 k \\ 3 k & (4 + 3 β) k^{2} & - 3 k & (2 - 3 β) k^{2} \\ - 3 & - 3 k & 3 & - 3 k \\ 3 k & (2 - 3 β) k^{2} & - 3 k & (4 + 3 β) k^{2} \end{matrix}]

(7)

[M_{1}] = \frac{ρ Ak}{210 {(1 + 3 β)}^{2}} [\begin{matrix} m_{1} & m_{2} & m_{3} & m_{4} \\ m_{2} & m_{5} & - m_{4} & m_{6} \\ m_{3} & - m_{4} & m_{1} & - m_{2} \\ m_{4} & m_{6} & - m_{2} & m_{5} \end{matrix}] + \frac{ρ I_{z}}{30 k {(1 + 3 β)}^{2}} [\begin{matrix} m_{7} & m_{8} & - m_{7} & m_{8} \\ m_{8} & m_{9} & - m_{8} & m_{10} \\ - m_{7} & - m_{8} & m_{7} & - m_{8} \\ m_{8} & m_{10} & - m_{8} & m_{8} \end{matrix}]

(8)

[K_{2}] = \frac{EA}{k} [\begin{matrix} \frac{1}{2} & - \frac{1}{2} \\ - \frac{1}{2} & \frac{1}{2} \end{matrix}]

(9)

[M_{2}] = ρ \cdot A \cdot k [\begin{matrix} \frac{3}{2} & \frac{1}{3} \\ \frac{1}{3} & \frac{2}{3} \end{matrix}]

(10)

where E and G represent the elastic and shear modulus, the cross-sectional area with A, the moment of inertia with I, while k represents the half-length, k = 0.5 L. The matrix elements m₁–m₉ in equation (8) and the shear deformation factor β are as follows

β = \frac{E I_{z}}{κ AG k^{2}}

(11)

\begin{matrix} m_{1} = 156 + 882 β + 1260 β^{2} \\ m_{2} = (44 + 231 β + 315 β^{2}) k \\ m_{3} = 54 + 378 β + 630 β^{2} \\ m_{4} = (- 26 - 189 β - 315 β^{2}) k \\ m_{5} = (16 + 84 β + 126 β^{2}) k^{2} \\ m_{6} = (- 12 - 84 β - 126 β^{2}) k \\ m_{7} = 18 m_{8} = (3 - 45 β) k \\ m_{9} = (8 + 30 β + 180 β^{2}) k^{2} \\ m_{10} = (- 2 - 30 β + 90 β^{2}) k \end{matrix}

(12)

where κ is the shear correction factor depending on the shape.

Using the above-obtained K₁, K₂, M₁ and M₂ matrices, the total stiffness and mass matrices for 3 degrees of freedom at each node become

K_{el} = {[\begin{matrix} K_{2} 11 & K_{2} 12 \\ K_{1} 11 & K_{1} 12 & K_{1} 13 & K_{1} 14 \\ K_{1} 21 & K_{1} 22 & K_{1} 23 & K_{1} 24 \\ K_{2} 21 & K_{2} 22 \\ K_{1} 31 & K_{1} 32 & K_{1} 33 & K_{1} 34 \\ K_{1} 41 & K_{1} 42 & K_{1} 43 & K_{1} 44 \end{matrix}]}_{(6 \times 6)}

(13)

M_{el} = {[\begin{matrix} M_{2} 11 & M_{2} 12 \\ M_{1} 11 & M_{1} 12 & M_{1} 13 & M_{1} 14 \\ M_{1} 21 & M_{1} 22 & M_{1} 23 & M_{1} 24 \\ M_{2} 21 & M_{2} 22 \\ M_{1} 31 & M_{1} 32 & M_{1} 33 & M_{1} 34 \\ M_{1} 41 & M_{1} 42 & M_{1} 43 & M_{1} 44 \end{matrix}]}_{(6 \times 6)}

(14)

The stiffness matrix for the crack

Figure 3 shows the schematic diagram of a cracked cantilever beam with a rectangular cross-section, subjected to axial force P1, shearing force P2 and bending moment P3. The corresponding displacements are given as u, v and θ. While the geometric parameters are as follows: length L, width d, height b, crack location Lc and crack depth a.

Figure 3.

Schematic diagram of a cracked cantilever beam with loading conditions.

The cracked node is considered as a cracked element of zero length and zero mass (Gounaris and Dimarogonas, 1988). The strain energy because (due to) of crack leads to flexibility coefficients and is completely derived by Kisa and Brandon (2000). Using the flexibility coefficient according to the displacement vector $δ = {u, v, θ}$ , compliance coefficient matrix is as follows

C = {[\begin{matrix} c_{11} & 0 & c_{13} \\ 0 & c_{22} & 0 \\ c_{31} & 0 & c_{33} \end{matrix}]}_{(3 \times 3)}

(15)

The inverse of the compliance matrix is the stiffness matrix due to crack. As a result, the crack stiffness matrix is as follows

K_{cr} = {[\begin{matrix} {[C]}^{- 1} & - {[C]}^{- 1} \\ - {[C]}^{- 1} & {[C]}^{- 1} \end{matrix}]}_{(6 \times 6)}

(16)

The eigenvalue problem of free vibration of an intact beam given by Cunedioglu (2015) and cracked beam given by Akbaş (2013) is as follows

([K] - ω^{2} [M]) {\bar{d}} = 0

(17)

(([K] + [K_{cr}]) - ω^{2} [M]) {\bar{d}} = 0

(18)

where $[K]$ is the global stiffness matrix, $[M]$ is the global mass matrix, $ω$ is the natural angular frequency in radians per second, and ${\bar{d}}$ is the mode shape.

Verification of the cracked beam

In order to verify the accuracy of the finite element MATLAB code, results are compared with a study from the literature (Kisa et al., 1998). The dimensions of the considered beam with a crack are as follows: length L = 0.2 m, width d = 0.025 m and height b = 0.0078 m. Material properties are given as follows: Young’s modulus E = 216 GPa, shear stiffness modulus G = 3E/8 and density ρ = 7850 kg/m³. Various crack locations (Lc/L) and crack depth ratios (a/b), together with the ANSYS software results, are considered for the purpose of the verification (Table 1).

Table 1.

The first natural frequencies for the cracked Timoshenko cantilever beam (rad/s).

Crack location		Crack depth ratio (a/b)				Intact beam
Crack location		0.2	0.4	0.6	0.8	Intact beam
Lc/L = 0.2	Present	1020.046	966.847	842.088	550.888	1036.932
	Kisa et al. (1998)	1020.137	966.9525	842.2205	551.0463	1037.0189
	ANSYS	1024.4105	972.3858	842.7760	556.686	1037.0397
Lc/L = 0.4	Present	1030.004	1006.750	942.585	724.027
	Kisa et al. (1998)	1030.095	1006.856	942.7322	724.2739
	ANSYS	1031.824	1011.844	957.557	728.842
Lc/L = 0.6	Present	1035.196	1029.166	1010.741	920.525
	Kisa et al. (1998)	1035.284	1029.262	1010.864	920.7848
	ANSYS	1035.657	1030.505	1015.3	926.248
Lc/L = 0.8	Present	1036.798	1036.326	1034.850	1026.646
	Kisa et al. (1998)	1036.884	1036.414	1034.943	1026.769
	ANSYS	1036.851	1036.411	1035.218	1029.186

Verification of the FG beam

Table 2 shows the accordance between the present results and the obtained results from available literature study (Demir et al., 2013). The considered beam is characterized by 50 layered elements, where material properties change along the thickness of the beam. The geometric dimension of the beam is as follows: Length L = 0.2 m, thickness b = 0.005 m and width d = 0.02 m. While the material properties are as follows: E = 70 GPa and ρ = 2700 kg/m³ for aluminium, and E = 380 GPa and ρ = 3950 kg/m³ for ceramic. 100 elements are used to generate finite element model, since this satisfies convergence as shown in Table 3 for power index n = 0.1.

Table 2.

The first four natural frequencies (Hz) of a simply supported symmetric FG sandwich Timoshenko beam.

Function			Mode 1	Mode 2	Mode 3	Mode 4
Polynomial	n = 0.1	Present	550.94	2196.71	4916.51	8677.10
	n = 0.1	Demir et al. (2013)	550.91	2196.89	4917.36	8679.65
	n = 0.5	Present	535.00	2133.13	4774.23	8425.98
	n = 0.5	Demir et al. (2013)	534.98	2133.31	4775.06	8428.46
	n = 1	Present	518.09	2065.69	4623.29	8159.59
	n = 1	Demir et al. (2013)	518.2	2065.88	4624.09	8162

Table 3.

Convergence analysis of a cantilever symmetric FG sandwich Timoshenko beam.

Function type	Natural frequency (Hz)	30 elements	40 elements	50 elements	60 elements	70 elements	80 elements	90 elements	100 elements
Polynomial (n = 0.1)	First mode	486.431	486.426	486.423	486.423	486.422	486.422	486.422	486.422
	Second mode	1939.604	1939.516	1939.479	1939.463	1939.454	1939.450	1939.447	1939.445
	Third mode	4341.656	4341.146	4340.935	4340.836	4340.784	4340.755	4340.737	4340.726
	Fourth mode	7664.383	7662.503	7661.716	7661.340	7661.139	7661.021	7660.948	7660.899

Furthermore, ANSYS is used to support presented results for a cracked FG stepped beam using the FEM (Table 4). Three types A, B and C of the stepped beam are being considered to verify the results. The 50-layer stepped cantilever beam is modelled in INVENTOR program and exported to ANSYS. Material properties have been assigned to each layer using exponential and power law functions from equations (1) to (4), while ν = 0.3. The beam is meshed by the SOLID186 element which is defined by 20 nodes having 3 degrees of freedom per node, while the crack is defined as a zero thickness surface. The block Lanczos method is used for the eigenvalue extractions to calculate frequencies. The material properties are the same as in verification of FG beam, while the geometric dimensions are as follows: L = 0.2 m, d = 0.02 m, d1 = 0.015 m, b = 0.005 m, b1 = 0.004 m, Lc/L = 0.2, a = 1 mm, Ls/L = 0.25, 0.5 and 0.75, and n = 5.

Table 4.

The first four natural frequencies (Hz) of a single edge cracked cantilever FG sandwich Timoshenko stepped beam.

Natural frequencies	Ls/L	Type A		Type B		Type C
		a = 1 mm		a = 1 mm		a = 1 mm
		MATLAB	ANSYS	MATLAB	ANSYS	MATLAB	ANSYS
First mode	0.25	139.757	139.838	172.012	172.080	147.112	147.129
Second mode	0.25	766.052	772.843	1022.432	1057.900	804.690	810.705
Third mode	0.25	1958.737	1985.942	2764.055	2834.500	1988.493	2015.651
Fourth mode	0.25	3757.157	3796.495	5308.997	5338.900	3732.529	3764.856
First mode	0.5	172.927	172.925	177.489	177.340	193.021	192.986
Second mode	0.5	773.097	788.482	998.656	1031.600	769.835	783.598
Third mode	0.5	2223.890	2270.731	2741.994	2744.600	2239.101	2277.718
Fourth mode	0.5	4096.456	4142.745	5295.727	5324.800	4081.510	4118.975
First mode	0.75	174.261	174.231	171.834	171.930	189.378	189.450
Second mode	0.75	989.568	1016.279	1022.612	1058.500	1022.114	1046.500
Third mode	0.75	2420.373	2472.758	2761.725	2824.500	2393.553	2438.600
Fourth mode	0.75	4627.849	4690.143	5308.530	5348.700	4642.640	4700.000

Numerical results

Consider a cracked cantilever FG sandwich stepped Timoshenko beam, symmetrically with respect to the neutral axis, with different cross-sections. It is assumed that the cantilever beam consists of 50 layers. Furthermore, it is assumed that the second stage of the beam is created by machining, providing material continuity between the beam stages. The variation of the material properties is done along the beam thickness according to the power and the exponential laws. Dimensions of the beam geometry are as follows: L = 200 mm, d = 20 mm, d1 = 15 mm, b = 5 mm and b1 = 4 mm. In the study, the effects of crack location, crack depth, power index (n), different material distributions, different step length and different cross-sectional geometries on the first four natural frequencies are analysed.

Effect of the crack location and the crack depth

The analyses are carried out to investigate the effects of the crack location (Lc/L) and the crack depth (a) on the first four natural frequencies, for different types of a cracked cantilever FG stepped beam and step location Ls/L = 0.25. The graphs for the polynomial function are given only for the power law index n = 5.

As can be seen from Figure 4(a), for both functions, the minimum first natural frequency values for types A and C occurred at the Lc/L = 0.4, then an increase in the first natural frequencies is observed. However, in the type B, as the crack location changes from a fixed end to a free end, an increase in the first natural frequencies is observed over all the way. As it is shown in Figure 4(b), the minimum second natural frequency values for all types A, B and C occurred at the Lc/L = 0.6. From Figure 4(c), for the exponential and the polynomial functions, one can notice decreases-increases in the third natural frequency values, for the types A and C. While for the type B, there is a continuous decrease in the third natural frequencies as the crack location changes towards the free end of the beam. As shown in Figure 4(d), for A and C types, the fourth natural frequency values decrease after gaining the maximum value at the Lc/L = 0.4, while for the B type, the increase of the fourth natural frequencies goes up to the Lc/L = 0.6, being followed by a decrease. From Figure 4, one can observe, for all three types and both functions, that all first four natural frequencies decrease as the crack depth increases, while the largest and the smallest frequency values are observed in types B and A, respectively. Also from Figure 4, one can notice that natural frequency values are higher in the exponential function than the polynomial one (n = 5).

Figure 4.

The effects of the crack location (Lc/L) and crack depth (a) on the first four natural frequency values for Ls/L = 0.25.

Effect of the step location and the crack depth

In this case, the analyses are carried out to see the effects of the step location (Ls/L) and crack depth (a) on the first four natural frequencies for different cross-sections, and for exponential and polynomial (n = 5) functions, by considering the crack location constant Lc/L = 0.2.

As shown in Figure 5(a), the maximum first natural frequency values for all three types A, B and C, and for the exponential function occurred at the Ls/L = 0.5, followed by a decrease. However, the decrease in the type C is larger than the one of the type A. On the contrary, for the n = 5, the first natural frequencies of type A are observed to decrease after gaining maximum value at Ls/L = 0.75, while for the types B and C, the decrease is observed after Ls/L = 0.5. The maximum first natural frequency difference in the type A in Figure 5(a) is due to the power index (n) as it is shown in Figure 7(a). Furthermore, in all types defined by both functions, the first natural frequency values decrease as the crack depth increases. As one can see from Figure 5(b), the second natural frequency values, for the type A, show an increase as it approaches the end of the beam at Ls/L = 0.75, and then a decrease is observed, whereas for types B and C, the second frequency values decrease-increase. Furthermore, exploring the Figure 5(b), one can notice that for all three types and both functions, the depth of the crack does not have much effect on the second frequency values. In Figure 5(c), in relation to types A and C, one can observe an increase in the third natural frequency values as the step location increases; however, the effect of the crack depth on third natural frequency values is quite small. On the contrary, for the type B, there is a decrease-increase in the third natural frequencies as the step location increases. However, for the type B, there is a significant decrease in the third natural frequency values as a crack depth increases. As shown in Figure 5(d), for types A and C, there is an increase in the fourth natural frequency values as the step location increases. On the other side, for type B, with increasing the step location, one can see decreases-increases in fourth natural frequencies. In addition, the fourth natural frequencies are significantly affected by increasing the crack depth. Similarly to previous case, the natural frequencies are higher for the exponential function than the polynomial one (n = 5).

Figure 5.

The effects of the step location (Ls/L) and crack depth (a) on first four natural frequency values for Lc/L = 0.2.

Effect of the crack location and the power index

Analyses were conducted to see the effects of changing the crack location (Lc/L) and the power index (n) on the first four natural frequencies, considering the step location (Ls/L = 0.25) and crack depth (a = 2 mm) as constant values.

In Figure 6(a), for the types A and C, the minimum values of the first natural frequency are observed at the Lc/L = 0.4, and after that point, there is an increase in these values, while in the type B, there is a continuous increase in the first frequencies as the crack location increases. As shown in Figure 6(b), for all three types, there is a decrease in the second natural frequency values up to the Lc/L = 0.6, being followed by an increase. In Figure 6(c), a decrease-increase is observed for the types A and C, while in the case of type B, there is a permanent decrease in the third frequency values. In Figure 6(d), in types A and C, fourth natural frequency attains the maximum value at the Lc/L = 0.4, and for the type B, the maximum value of fourth natural frequency occurred at the Lc/L = 0.6. Examining the graphs from Figure 6, one can notice that the first four natural frequency values decrease as the values of the power index increase. This drop in natural frequencies can be explained by examination of Figure 2. An increase of the n parameter decreases the elastic modulus and density. This decrease affects the global stiffness and mass matrix of the beam in its turn decreasing the natural frequencies of the beam.

Figure 6.

The effects of the crack location (Lc/L) and different power index (n) in the first four natural frequency values for Ls/L = 0.25 and a = 2 mm.

Effect of the step location and the power index

In this case, the effects of the step location (Ls/L) and the power index (n) on natural frequencies are analysed by considering a crack location (Lc/L = 0.2) and crack depth (a = 2 mm) as constant values. As shown in Figure 7(a), for the types B and C, there is a decrease in first natural frequency values, after maximum value is observed at the Ls/L = 0.5. In the type A, the first natural frequency maximum values occurred at the Ls/L = 0.5 for parameters n = 0.1, n = 0.5 and n = 1, while for parameters n = 5 and n = 10, the maximum value occurred at the Ls/L = 0.75. In Figure 7(b), for types B and C, the values of the second natural frequency decrease-increase, with the maximum value at the Ls/L = 0.75. However, in type A, the second natural frequency maximum value is at Ls/L = 0.75, followed by a decrease in all parameters except for n = 10. In Figure 7(c) and (d), for the types A and C, one can observe an increase in the third and fourth natural frequencies as the step location increases, while for the type B, one can observe that the third and fourth natural frequencies are not affected a lot by the step location changes. Exploring the graphs from Figure 7, one can notice that the natural frequency values decrease as power index increases.

Figure 7.

The effects of the step location (Ls/L) and power index (n) in the first four natural frequency values for Lc/L = 0.2 and a = 2 mm.

Mode shapes

From Figure 8, one can observe the effects of the power index (n), step location (Ls/L), crack location (Lc/L), and crack depth (a) in the first three mode shapes, for the three types of the cantilever cracked FG stepped beam. As shown in Figure 8, there is a similar trend of data developing for each same mode. The mode shape 1 values are observed to increase monotonically, while the mode shapes 2 and 3 decrease/increase along the length of the beam. It can be seen from Figure 8(a) that the change of the power index affects all three mode shapes of types A and C, while it has almost no effect in the type B. However, from Figure 8(b), we see that the mode shapes are affected much more in all types, by the change of step location, while as shown in Figure 8(c) and (d), the change in crack location and crack depth affects slightly all mode shape behaviours.

Figure 8.

The effects of the power index (n), step location (Ls/L), crack location (Lc/L) and crack depth (a) in the first three mode shapes.

The variation of the graphs and mode shapes of Figures 7 and 8 can be explained as follows. Natural frequency values are a function of stiffness and mass matrix as seen from equation (18). Cracks in sections decrease this stiffness and thus cause decreasing frequency values. When looking at the effect of step location, it can be seen that while for type A and type C beams, the elastic modulus and density are not the same before and after the step, for type B these remain the same. However, since for all beam types the cross-sectional area, moment of inertia and mass change after the step, this affects the global stiffness and mass matrix. Since a variation of the n parameter causes the elastic modulus and density to vary, the natural frequencies will change.

Conclusion

In this study, modal analysis was carried out on the FG cracked beam with various cross-sections. Three types of cantilever stepped beam were investigated, with varying material properties along the thickness according to the power and exponential functions. In detail, the effects of the crack location, crack depth, power index (n), different material distributions, different step length, different cross-sectional geometries on natural frequencies and mode shapes are investigated. Some of the main results obtained are listed as follows:

The beam whose material properties are assumed to vary according to the exponential function has higher frequencies than that with polynomial function.

An increase of the crack depth decreases all four natural frequency values; however, when the crack depth is shallow (a < 1 mm), it has insignificant effects on natural frequencies.

A variation in crack location and step location causes the natural frequencies to increase or decrease, respectively.

An increasing polynomial degree n decreases all the natural frequency values. However, for n = 5 and n = 10, the effects are more significant evident.

The effects of step location in mode shapes are more profound than crack depth (a) and crack location (Lc/L) changes, while for the parameter n, the effect is slightly visible for types A and C.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Shkelzen Shabani

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