FATIGUE ANALYSIS OF A BEAM SUBJECTED TO DYNAMIC LOADING. Cheng-Shun Chen * I-Chih Wang **

Transcription

1 FATIGUE ANALYSIS OF A BEAM SUBJECTED TO DYNAMIC LOADING ABSTRACT Cheng-Shun Chen * I-Chih Wang ** Department of Mechanical, Industrial, and Nuclear Engineering University of Cincinnati, Ohio, U.S.A. * Graduate student. Former faculty of National Taipei Institute of Technoloav, Taiwan, R.O.C. ** Professor -. Classical analyses of strucfures subjected fo fatigue loading are based on the assumption that fhe loads and the resulfing stresses are independent of time. However, in fhe case of a suddenly applied constant load, nof only higher displacemenfs and stresses, but also transient vibration, will be generated within the structure. As a result, some fatigue life could be consumed durfng the vibrational motion. Therefore, it is important that the fatigue analysis must, in part, take dynamic effecfs into accoun1. The objeciive of this study is to investigate the fatigue life of a beam under this kind of dynamic loading. For fhis study, the finite element rransienr analysis was used to determine the time-varying stresses and strains at dffferenf /eve/s of loads and damping values. The strain-life based fatigue analysis method was used fo predict the crack initiation life. When designing a lightly damped structure subjected fo dynamic loading, ii is important to study fhe dynamic response and estimate the fatigue life of fhe structure at the design stage to make the design satisfactory. NOMENCLATURE [Ml, [ml = structure and element mass matrices [Cl. [c] = structure and eiement damping matrices VI. PI = structure and element stiffness matrices {R}, {r} = force vectors applied on nodal d.o.f VJI *b> = nodal d.o.f of structure and element [B] PI [N] = strain-displacement matrix = elasticity = stress = matrix of shape function = strain vector A0 - = true stress amplitude 2 A& - = total strain amplitude 2 2Nf = reversals to failure ( 1 rev = 112 cycle) (3,, b = fatigue strength coefficient and exponent &/ c = fatigue ductility coefficient and exponent k = cyclic strength coefficient n = cyclic strain hardening exponent 0 o = mean stress p E = mass density = elastic modulus W, = natural frequency c = damping ratio 1. INTRODUCTION Fatigue is a failure mode of a structure element subjected to repeated applications of load even though the resulting stress levels that are considerably lower than those permitted for static loading. It is an important design consideration because most of structures fail in this category. A very large amount of efforts and researches in fatigue analysis and design had been done. Among these researches, one widely used fatigue loading was the so called variable amplitude or irregular loading like SAE fatigue load histories [l]. The fatigue analysis of this kind of loading was usually based on the assumption that the loads and the resulting stresses ware independent of time, that is, the fatigue loading was considered as a series of static loads. The elastic-plastic static analyses were directly applied to these loads on a reversal by reversal basis (21. However, in the cass of dynamic loading like suddenly applied constant force, the transient vibration will be generated before the response of the structure reaches the final static equilibrium position. 1544

2 As a structure experiences the vibration, an alternating dynamic stress and strain with cedain period will be induced within the structure. From the fatigue point of view, these alternating dynamic stress and strain could be considered as a kind of fatigue loading. Once the amplitude of this dynamic stress exceeds the endurance limit of material and the repeated cycles of motion reach a critical value, a fatigue damage could be initiated. It happens easier for the lightly damped structure in which the dynamic stiffness can be much small than the static stiffness. This vibration induced cumulative fatigue is the primary concern of our study. The causes of vibration induced fatigue could be arise from the random vibration caused by nondeterministic loading [3], or forced vibration caused by periodic dynamic loading [4], or transient vibration caused by nonperiodic dynamic loading. In the present paper, we concentrate our study on the topic of transient vibration and the resulting fatigue. Currently, very little can be found in the literature dealing with this topic. In 1983, Verdonck and Snoeys [5] suggested a method to predict lifetime of structures based on the combined use of Finite Element (FE) and modal analysis data. The paper demonstrated an example of lifetime prediction of a tennis racket subjected to repeated impact. Vis Devis, Snoeys, and Sas [6] recognized an important fact that lifetime estimation of structures should cover the dynamic behavior for some types of structures. They developed a similar procedure to estimate fatigue lifetime of structures based on experimental modal parameters. A T- plate subjected to hammer impact was used to illustrate the procedure. Today, the progress of the dynamic FE technique and the power of computer provide us a new environment to handle this problem easier and more effective at the early design stage. Chen and Wang [7] took advantage of this and started the investigation of fatigue life of undamped linear structure subjected to different types of dynamic loading. The fatigue life prediction was based on the dynamic stresses and strains obtained from FE dynamic analysis. In this paper, we continue previous study and take into account some important parameters like material nonlinearity and damping of the structure. We also pay more attention on the understanding of structural fatigue problem under transient vibration that will die out in a short time rather than the predicting of exact value of fatigue life. 2. BACKGROUNDS In this section, basic theoretical materials required for the fatigue analysis are briefly reviewed. 2.1 Transient response When a dynamical system is excited by a suddenly applied nonperiodic excitation, the response to such excitation is called transient response. For a single degree of freedom (DOF) linear system with mass m, spring constant k, and damping ratio6, the response x to step excitation of magnitude F, with the initial conditions of zero initial displacement and velocity can be shown as [El: F x=0 k where It is evident that the peak response to the step excitation of magnitude F, is less than 2 F,/k when damping is present and is equal to twice the static deflection for undamped system. 2.2 Displacement-based finite ekment method The basis of the displacement-based finite element solution is the principle of virtual work. Applying this principle, the dynamic equilibrium equation for one element can be shown as [9]: where the element mass and damping matrices are defined a*: The basic procedures and approaches used for this study are as follows: (1). Find the transient response of the structure by FE dynamic analysis. (2). Determine the critical place where the crack is likely to start and compute the strain history of this point. (3). Divide the strain history into individual events (cycles) by the simple rainflow counting method. (4).Calculate the fatigue damage caused by each event by strain-life equation. (5). Accumulate the damage by linear damage rule. where [N] is the matrix of shape function and Kd is a material damping parameter analogous to viscosity. The element internal force in Eq. (2) represents loads at nodes caused by straining of material and is defined as: 1545

3 The Eq. (2) and (5) are valid for both linear and nonlinear material behavior. The element external force in Eq. (2) may include the body forces, surface tractions, and concentrated loads that act on the element. For linearly elastic material behavior, {ol= [Dl{El= [DI[Bl{ul ( 3) where [D]IS a matrix of elastic coefficients. Combining this with Equation (5) yields {rin }=[k]{u} (7) where [kl= pltpipl~v (8) is called element stiffness matrix. For the entire structure, the structural mass, dampimg, and stiffness matrices are constructed by the conceptual expansion of element matrices to structure size followed by addition of overlapping coefficients. For multi-dof linearly elastic structure, the dynamic equilibrium equation becomes: [M]{ti}+ [C]{ir}+ [KXU} = {R (t)} (9) In a time-history or dynamic response problem, we solve Eq. (9) for {(I}, {e}, and {u}as functions of time. In practice, the two principal techniques used for the solution of equilibrium equation are direct numerical integration and mode superposition method. In order to solve the equation, a suitable boundary conditions, initial conditions, and the applied load must be known. The primary solution data are the nodal DOF displacements, and the element strains and stresses at integration point are recovered and computed by following = [BIbI (10) 10 I= PIGI (11) and these element results are average at the nodes to become nodal solution data. 2.3 Crack initiation life prediction method Three primary fatigue analysis methods are the stress-life, strain-life, and the fracture mechanics methods. These methods have their own region of application with some degree of overlap. In general, traditional stress-life method concentrates on the long life of the component based on the concept of the fatigue limit, the strain-life method is widely used to predict life to crack initiation at notch under complex load sequence, and the fracture mechanics approach deals directly with the propagation of fatigue cracks. Because our major concern is the crack initiation life, only strain-life method will be discussed. The strain-life relationship is the basis of the strain-life method. The analytical representation of this strain-life relationship is expressed by the following equation [lo]: AE s+ace, -0, -= - - T(2N,)h +E, (2N,) (12) where A&, A&e, and A&, are the total strain, elastic strain, and plastic strain ranges, respectively. If accounting for mean stress effects in elastic term, the equation becomes [ll]: -= AE of *, E (2N,)h +&,I (2N,) 2 This equation can be numerically solved for N, for the given material fatigue property, the strain range of interest, and the mean stress. 2.4 Damage summing method for initiation A commonly used damage rule is the Palmgren-Miner s linear damage rule which states that the damage fraction at stress level s, is equal to the cycle ratio, ni / N, The failure criterion for variable amplitude loading can be stated ES: where ni is the number of cycles at stress level s, and N, is the fatigue life in cycles at stress level,??, The life to failure can be estimated by summing the percentage of life used up at each stress level. 3. NUMERICAL EXAMPLE 3.1 Dynamic response analysis In this section, we will briefly present the dynamic response of a thin cantilever beam as shown in Fig. la with a cross section of 51x3.2 iw?? and a length of m subjected to suddenly applied constant force. The dynamic response was determined by using a general finite element program with nonlinear capability. In this paper. the nonlinear effect 1546

4 V (a) ok----d2 time (S, (b) Figure 1: Cantilever beam and the forcing function lies only in the nonlinear stress-strain relation. The material selected is a high strength, roller quenched, and tempered steel (WC-100). The basic smooth specimen material properties of this material can be found in the appendix C of reference 1. In order to perform the analysis, a FE analysis model was built to represent the continuous beam and required information. Some features of this model are as follows: (1). The element type used for analysis was a Z-dimensional plastic beam element with 3 DOF in each node. (2). The mass matrix was constructed by consistent mass formulation method. (3). The popular Reyleigh damping, [C] = CL[K] + p [M], was used to calculate the damping matrix of the system. The two constants (a and 0 ) were obtained by solving two simultaneous equations from given damping ratio and frequency range of interest. Four levels of damping ratios were selected and their values were , 0.025, and 0.1, respectively. (4). The multilinear kinematic hardening plasticity behavior was used to describe and approximate the cyclic stressstrain relationship of the material. (5). The suddenly applied constant force shown in Fig. l(b) was acted at the free end of the beam. In order to cover the elastic and plastic responses, several levels of applied force from 50 to 150 N were used for study. (6). The initial conditions were assumed to be zero initial displacement and velocity. Figure 2: Responses of the free end node for different damping ratio. (A)6 =0.125, (B) (C) 0.05, (D) 0.1, and (E) static. analysis was set which was about l/800 of the period of the first bending mode of vibration. Because a wide range of applied force was used in analysis, the response of the structure may be elastic or plastic. Fig. 2 shows a plot of the displacement responses of the free end node of the beam under the action of suddenly applied constant force F(t) = 50 N with damping ratio as a parameter. In this elastic case. from the response cwves we see that the free end of the beam vibrates, as expected, about the static equilibrium position (curve E). It should also be observed that the amplitude curves for different damping all decay with time. However, the response pattern is slightly different for the case of plastic response. Fig. 3 shows a series of displacement-time cures for the case of force F(t)= 150 N with damping ratio as a parameter. It was found 25 Before transient analysis, a modal analysis was performed to understand the dynamic characteristics of the structure. The first three natural frequencies obtained from FE analysis were , and Hz respectively, which had a good agreement with exact values. Since the load was applied suddenly and the displacements and strains were assumed to be small, we calculated the response of the beam by using a dynamic analysis with material nonlinearfties only. A FE transient analysis with the selection of full solution method was performed. This full solution method solves the equilibrium equation directly based on the procedure of Newmark time integration technique. In a nonlinear analysis, the Newton-Raphson method is employed along with the Newmark assumption. For the Newmark time integration scheme, it has been found that using approximately twenty points per cycle of the highest frequency of interest results in a reasonably accurate solution. The integration time step chosen for Figure 3: Responses of the free end node with different damping ratio. (A)c =0.125, (8) 0.025, (C) 0.05, (D) 0.1, and (E) static. 1547

5 Figure 4: Deformed shapes of the cantilever beam at several instants of time. Curve (A) original position, (6) first maximum, and(c) first minimum. in this numerical solution that after a maximum deflection (taken as positive in the direction of the loading) the beam moved backward to a deflection on the negative side, the beam then vibrated elastically with certain minimum and maximum, and died out to the final permanent deflection. It should be noted that the final permanent deflection position did not coincide with the equivalent static position (curve E). The difference between them increased with the decreasing of damping ratio. For fatigue analysis it is important to find the critical fatigue location where a crack is likely to start. In the present paper, we assume that there is no stress concentration due to the geometrical or material discontinuity. Therefore, the determination of the maximum strain location is straight forward. Fig. 4 is a plot of a series of deformed shapes at several instants of time of the beam considered for the case of F(t)= 125 N and < = It shows that most of the response is due to the first normal mode for the given initial conditions. This is to be expected because the first mode shape is similar to the static deflection shape of the cantilever beam under a free end load. Therefore, the maximum stress and strain would be developed at the upper fiber of the fixed end. However, attention should be paid to the fact that other initial conditions cause other critical points, and other modes can be found to be dominant. Fig. 5 is a plot of strain response for the case of F(t)= 150 N and c =0.05. As can be seen, the amplitude curve of elastic strain (curve A) vibrates and decays with time, and the amplitude of plastic strain (curve B) remains constant after it reaches the maximum value. This is to be expected because the permanent deformation is unrecoverable. The total strain response (curve C) is obtained by adding the ordinates of the elastic and plastic components. Fig 6 shows a series of total strain-time cul~es with damping ratio as a parameter. Figure 5: Strain response of the fixed end node. Curve (A) elastic, (El) plastic, and (C) total strain. It should be observed that an irregular strain history with mean stress was generated for each damping case Crack initiation life prediction Once the strain history of the critical fatigue location was determined, the next step for fatigue life estimation is to define a cycle count procedure to separate the irregular loading history into a number of events (cycles) which can be compared to the available constant amplitude test data, and then assume a damage rule to measure the fatigue damage caused by each single event (cycle). The rainflow cycle count method had been widely used [12]. In our study, the simple rainflow count algorithm [13] which determines load cycles for closed hysteresis loops in a loading histoly was used to count the events for the strain Figure 6: Strain responses for different damping ratio. (A)c =0.0125, (6) 0.025, (C) 0.05, (D) 0.1, and(e) static. 1548

6 Table 1: Damage of each cycle and accumulated damage strain mean stress damage per total amplitude (Mpa) CYCk damage 3.683e e e e O.l837e s e e e z O.l709e e Results and Discussion Figure 7: Example of a strain history An example of a typical strain history for the case of F(t)= 100 N and c ~0.025 was shown in Fig. 7. As can be seen, the reversals after H-l were discarded because the resulting fatigue life of these individual reversals was infinite based on their corresponding stress amplitude. The history from A to Al was defined as one repetition or block of the fatigue loading. Damage for each cycle was found by using the strain-life approach with accounting for mean stress effects in elastic term (Eq. 13). The mean stress used to calculate the damage was based on the mean value of the alternating stress curve. Table 1 gives the damage for each cycle and the accumulated damage of strain history shown in Fig. 7. A total damage of e-6 was found. According to the Miner s rule, the fatigue life was about 198,170 repetitions of the applied load for the case considered. Fig. 8 shows the predicted fatigue life versus load curves for four different levels of damping ratio. The given results display some impodant general features. The fatigue life decreased with the increasing of the applied force. It also decreased with the decreasing of the damping ratio. These results are reasonable. Table 2 summarizes the numerical data of three types of analysis for two different load levels. As can be seen, when the constant force was applied slowly to the structure, the maximum stress generated was smaller than the yielding strength of the material (620/ 550 Mpa). It meant that the structure was statically safe. However, if the constant force was suddenly applied, the results from dynamic analysis show that a transient vibration will be generated and plastic strains will also be built. If a fatigue analysis was conducted for this transient vibration, the result shows that the fatigue crack will be initiated after some repetitions of the loading. Therefore, It is important for a designer to recognize this potential failure mode. 4. CONCLUSIONS Dynamic loading will cause the vibration of the structure and such repeated oscillatory motions could be considered as a kind of fatigue loading. Attempt has been made to study how important of this type of fatigue loading. A procedure had been developed to estimate the crack initiation life based on the finite element dynamic analysis and strain-based life prediction technique. The procedure enables the simulating Table 2: Numerical data for three types of analysis Figure 8: Repetitions to cracking versus load data for four different levels of damping ratio (A) 0.1, (B) 0.05, (C) and(d) F(t) =loo N. F(t) =125 N. c ~ = Static 0 ~291.8 Mpa, 0 =364.6 Mpa, 6 = In, 6 = m, EC =O.l438e-2, & I =o. 1797e-2. E, =o E, =o Dynamic rs ~538.0 Mpa. 0 =595.8 Mpa, 6 = m, 6 = m. E e =0.2650e-2, E c =0.2935e-2, 6 p =O.O704e-2 E, =O.lSSle-2 Transient vibration Transient vibration Fatigue N, =198,170 N, =13,

7 of fatigue life at the early design stage and provides numerical values for engineering judgment. A thin cantilever beam under suddenly applied constant force with several levels of forces and damping ratio had been numerically studied. In the present paper, we also assumed that the strain-life data from axial push-pull tests can be applied to dynamic strains. It meant that the strains obtained from FE dynamic analysis can be directly applied to fatigue analysis equation. When designing a lightly damped structure subjected to a cedain level of dynamic load, it is important to extend the study of the problem from traditional dynamic analysis to fatigue analysis because the structures may fail by fatigue after some repeated applications of the dynamic load. This investigation can be easily performed at the early design stage by the application of finite element dynamic analysis and commonly used fatigue analysis method like strain-life method. [9] Cook, R. D., Malkus, D. S., and Plesha, M. E., Concepts and Application of Finite Element Analysis, 3rd ed.. John Wiley & Sons, pp , [lo] Morrow J., Cyclic Plastic Strain Energy and Fatigue of Metals, ASTM STP 378, pp , [ll] Morrow, J., Fatigue Design Handbook, Advances in Engineering, Vol. 4, Published by the Society of Automotive Engineers, Warrendale, Pa., sec. 3.2, pp , [12] Dowling, N. E.. Fatigue Prediction for Complicated Stress-Strain Histories, Journal of Material Vol. 7, pp , [13] Downing, S. D., and Socie, D. F., Simple Ftainflow counting Algorithms, International Journal of Fatigue, Vol. 4, No. 1, pp , REFERENCES [l] Tucker, L., and Bussa, S., The SAE Cumulative Fatigue Damage Test Program, Fatigue Under Complex Loading: Analyses and Experiments, published by The Society of automotive engineers, inc., pp [2] Hsieh, H. D.. Fatigue Life Prediction of Notched Plates Subjected to Complex Loading -- Finite Element Approach. PhD. Dissertation, University of Cincinnati, pp , [3] Sherratt, F., Vibration and Fatigue: Basic Life Estimation Methods, Journal of the Society of Environmental Engineers, pp. 12-l 7, Dec [4] Bolton, A., Structural Dynamics in practice: a Guide for Professional Engineers, McGraw-Hill, pp , [5] Verdonck, E. and Snoeys, R., Life time Prediction based on the Combined Use of Finite Element and Modal Analysis Data, Proceedings of the 6th International Seminars on Modal Analysis, Katholicke Universiteit Leuven, Belgium, sep [6] De Vi% D., Snoeys. R., and Sas P.. Fatigue Lifetime Estimation of Structures Subjected to Dynamic Loading, AIAA Journal. Vol. 24, pp , August [7] Chen, C. S., and Wang, I. C., Fatigue Life Prediction of Structures Based on Finite Element Dynamic Analysis, Proceedings of the 14th International Modal Analysis Conference, Michigan, pp , Feb [6] Thomson, W. T., Theory of Vibration with Applications, 4th ed., Prentice-Hall. pp ,