# Effect of Crack Severity on a Curved Cantilever Beam Using Differential Quadrature Element Method

Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)

## Abstract

In the present paper, investigation of a crack in a curved cantilever beam is studied. The crack in the beam is modeled using a rotational spring with stiffness defined in terms of depth of the crack. The equation of motion for the cracked cantilever beam is solved using Differential Quadrature Element Method (DQEM). The Finite Element Analysis (FEA) using ANSYS is also carried out for the cracked curved beam. The severity of the crack with its location on cross-section of the beam is studied using DQEM and FEA. Many numerical examples are demonstrated to verify the aforementioned effects. It is concluded that both the approaches are very close to each other for different crack locations and severities.

## Keywords

Crack Differential quadrature element method Curved beam FEA

## Nomenclature

$$\overrightarrow {\tau } (\phi ,t)$$

Shear force

$$\overrightarrow {{F_{n} }} (\phi ,t)$$

Normal force

$$\overrightarrow {M} (\phi ,t)$$

Bending moment of curved beam

t

Time period

A1

Cross-sectional area of curved beam

I

Moment of inertia of the curved beam

R

Mean radius of the circular arc

$$\gamma$$

Mass per unit volume of curved beam

E

Modulus of elasticity of the curved beam

$$\alpha$$

Bending slope

$$v$$

x

Tangential displacement

$$\phi_{t}$$

Total angle between deformed and undeformed neutral axis

$$K_{o}$$

Rotational spring stiffness

k

Numerical share factor of cross section

G

Modulus of rigidity

## References

1. 1.
Takahashi S (1963) Vibration of circular arc bar in its plane. Jpn Soc Mech Eng (JSME) 6(9)Google Scholar
2. 2.
Davis R, Henshel RD, Warburton GB (1972) Constant curvature beam finite element for in-plane vibration. J Sound Vib 25(4):561–576
3. 3.
Issa MS, Wang TM, Hsiao BT (1987) Extensional vibration of the continuous circular curved beam with rotational inertia and shear deformation. J Sound Vib 144(2):297–308
4. 4.
Pandey AK, Biswas M, Samman MM (1991) Damage detection from changes in curvature mode shapes. J Sound Vib 145(2):321–332
5. 5.
Kang K, Bert CW, Striz AG (1995) Vibration analysis of shear deformable circular arches by the differential quadrature method. J Sound Vib 181(2):353–360
6. 6.
Cerri MN, Rutta GC (2004) Detection of localized damage in-plane circular arches by frequency data. J Sound Vib 270:39–59
7. 7.
Oz HR, Das MT (2006) In-plane vibrations of circular curved beams with a transverse open crack. Math Comput Appl 11(1):1–10
8. 8.
Chang-New C (2008) DQEM analysis of out-of-plane vibration of nonprimitive curved beam structures considering the effect of shear deformation. Adv Eng Software 39:466–672
9. 9.
Byoung KL, Kwang P, Tae EL, Hee MY (2014) Free vibration of horizontally curved beams with constant volume. KSCE J Civ Eng 18(1):199–212
10. 10.
Ebrahim S, Mojtaba S, Abdolreza O (2016) Free vibration analysis of a debonded curved sandwich beam. Euro J Mech A/Solid 57:71–84
11. 11.
Das MT, Ayse Y (2018) Experimental modal analysis of curved composite beam with transverse open crack. J Sound Vib 436:155–164
12. 12.
Wang X, Zhangxian Y (2018) Three-dimensional vibration analysis of curved and twisted beams with irregular shapes of the cross-section by sub-parametric quadrature element method. Comput Math Appl 76:1486–1499
13. 13.
Matahari H, Hossein R, Ali K (2018) Static and dynamic analysis of crack curved beams using experimental study and finite element analysis. Periodica Polytech Civil Eng 62(2):337–345Google Scholar
14. 14.
Prawin J, Lakshmi K, Rao AM (2019) A novel vibration based breathing crack localization technique using a single sensor measurement. Mech Syst Sign Process 122:117–138

© Springer Nature Singapore Pte Ltd. 2020

## Authors and Affiliations

• Baharul Islam
• 1
• Prases K. Mohanty
• 1
Email author
• Dayal R. Parhi
• 2
1. 1.National Institute of Technology Arunachal PradeshYupiaIndia
2. 2.National Institute of Technology RourkelaRourkelaIndia