Allowable Fully Reversed Load of an Actuating Rod A round rod is subjected to an axial tensile force Fm. Calculate the limiting value of the completely reversed load Fa that can be applied with a ½ in. eccentricity without causing fatigue failure at 10^6 cycles Given: The diameter of the rod

Question

Allowable Fully Reversed Load of an Actuating Rod

A round rod is subjected to an axial tensile force [latex]F_{m}[/latex]. Calculate the limiting value of the completely reversed load [latex]F_{a}[/latex] that can be applied with a ½ in. eccentricity without causing fatigue failure at [latex]10^{6}[/latex] cycles

Given: The diameter of the rod [latex]D[/latex] = 2 in. and [latex]F_{m}[/latex] = 40 kips.

Assumptions: The rod is made of ASTM-A36 steel having [latex]S_{y}[/latex] = 36 ksi, [latex]S_{u}[/latex] = 58 ksi (Table B.1), and [latex]S_{n}[/latex] = 30 ksi at [latex]10^{6}[/latex] cycles.

Design Decisions: Apply the Soderberg and Goodman criteria.

Accepted Answer

The mean and maximum alternating stresses are

[latex] \begin{array}{l} \sigma_m=\frac{F_m}{A}=\frac{40}{\pi(1)^2}=12.73 ksi \ \sigma_a=\frac{M_a c}{I}=\frac{0.5 F_a(1)}{\pi(1)^4 / 4}=0.637 F_a \end{array} [/latex]

Soderberg criterion. Substituting these and the given numerical values into Table 7.4,

[latex] \frac{0.637 F_a}{30 imes 10^3}+\frac{12.73 imes 10^3}{36 imes 10^3}=1 \quad ext { or } \quad F_a=30.4 kips [/latex]

Goodman criterion. Similar to the preceding, we now have

[latex] \frac{0.637 F_a}{30 imes 10^3}+\frac{12.73 imes 10^3}{58 imes 10^3}=1 \quad ext { or } \quad F_a=36.8 kips [/latex]

Comments: According to the Goodman theory, the eccentric load that can be carried by the rod is thus about 17% larger than on the basis of the Soderberg theory.

Table 7.4 Failure Criteria for Fatigue
Fracture Theory	Goodman	Gerber	SAE
Equation	[latex] \frac{\sigma_a}{S_e}+\frac{\sigma_m}{S_u}=1 [/latex]	[latex] \frac{\sigma_u}{S_e}+\left\lgroup \frac{\sigma_m}{S_u} ight group ^2=1 [/latex]	[latex] \frac{\sigma_a}{S_e}+\frac{\sigma_m}{S_f}=1 [/latex]
Yield Theory	Soderberg	Modified Goodman
Equation	[latex] \frac{\sigma_a}{S_e}+\frac{\sigma_m}{S_y}=1 [/latex]	[latex] \frac{\sigma_a}{S_e}+\frac{\sigma_m}{S_u}=1 [/latex]	[latex] \left\lgroup ext { for } \frac{\sigma_a}{\sigma_m} \geq \beta ight group [/latex]
		[latex] \frac{\sigma_a+\sigma_m}{S_y}=1 [/latex]	[latex] \left\lgroup ext { for } \frac{\sigma_a}{\sigma_m} \leq \beta ight group [/latex]
Notes: [latex]a[/latex], alternating; [latex]y[/latex], static tensile yield; [latex]m[/latex], mean; [latex]u[/latex], static tensile ultimate; [latex]e[/latex], modified endurance limit; [latex]f[/latex], fracture. Material constant, [latex] \beta=S_e\left(S_u-S_y ight) / S_u\left(S_y - S_e ight) [/latex]

Question 7.3: Allowable Fully Reversed Load of an Actuating Rod A round ro...

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