PSI - Issue 17

C P Okeke et al. / Procedia Structural Integrity 17 (2019) 589–595 C P Okeke et al / Structural Integrity Procedia 00 (2019) 000 – 000

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2.1. Stiffness matrix The construction of the stiffness matrix is normally based on initial elastic modulus, however, the use of secant modulus is recommended for non-linear elastic materials such as polymers. Here, the initial elastic modulus and secant modulus are described. The value of initial elastic modulus is obtained from small strain of 0.05% to 0.25%, BS EN ISO 527-1(2012). The secant modulus is defined as the ratio of the nominal stress to corresponding strain at any chosen point on the stress-strain curve. The equations for the initial elastic modulus and secant modulus are given below in equation (2): = 2 − 1 2 − 1 , = Ɛ (2) σ 1 is the stress in megapascals (MPa), measured at the strain value ε 1 = 0.05% σ 2 is the stress in megapascals (MPa), measur ed at the strain value ε 2 = 0.25% The fatigue analysis enables us to understand how long a structure will last in service before experiencing failure. In this study a random vibration fatigue analysis was performed numerically using ANSYS software. A specimen with an end mass of 0.0098kg attached is clamped to a fixture with a fixed support boundary condition. The ISO 16750-3 random vibration input loading shown in fig 1 (a) (acceleration vs frequency) is applied to the base of the fixture to excite the system. The root mean square (RMS) of input acceleration which is the total energy into the system is 27.8m/s 2 . The set-up is shown in fig 1 (b). A damping ratio of 4.2% measured experimentally was used for the analysis. ANSYS calculates the frequency based fatigue life by taking the ratio of exposure time to the total damage (T e /D). The fatigue life evaluation of variable amplitude loading is generally performed with Palmgren Miner rule. For example, in the variable amplitude loading spectrum, there will be different levels of stress amplitudes with their corresponding number of cycles. Each level of stress amplitude will produce a fraction of damage. The Palmgren-Miner rule states that the fatigue failure occurs when the damages from the entire stress amplitudes sum to unity – in effect, failure occurs when 100% of life is wasted, Palmgren (1924), Miner (1945). 3. Random vibration numerical fatigue analysis

ISO (16750-3) Random Vibration Test Profile

10

Fixture

27.8m/s2 RMS

Mass

Specimen

1

(a)

(b)

0.1

10

100

1000

PSD Acceleration ((m/s²)²/Hz)

Frequency (Hz)

27.8m/s 2 RMS

Figure 1: (a): ISO (16750-3) loading profile / (b): Simulation model

There are currently three formulations for frequency based random vibration fatigue damage estimation supported by ANSYS, these are Steinberg, Narrow band and Wirsching. 3.1. Frequency based methods for fatigue life analysis Steinberg formulation: Steinberg formulation, Steinberg (2000), makes use of all three stress standard deviation spans (1σ, 2σ, 3σ) and their rate of occurrence together with Miner’s rule to calculate the total fatigue damage of a system: