5. A brittle steel bar has the dimensions shown. Determine the maximum axial force P that can be applied not to exceed the allowable tensile stress of allow = 100 MPa. (Make sure to determine and account for the stress concentration factors). K 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 P W h 30 mm -4.0 I W=3.0 h W Jang W h W || TH 24 mm P ht 2.0- 1.5. T LAL W W h II 12 mm 1.2- || = 1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K 60 mm 3.2 3.0 2.8 2.6 2.4 2.2 2.0 0.1 r = 15 mm P 0.2 2r o avg 0.3 2r P (w - 2r)t 0.4 0.
Design Against Fluctuating Loads
Machine elements are subjected to varieties of loads, some components are subjected to static loads, while some machine components are subjected to fluctuating loads, whose load magnitude tends to fluctuate. The components of a machine, when rotating at a high speed, are subjected to a high degree of load, which fluctuates from a high value to a low value. For the machine elements under the action of static loads, static failure theories are applied to know the safe and hazardous working conditions and regions. However, most of the machine elements are subjected to variable or fluctuating stresses, due to the nature of load that fluctuates from high magnitude to low magnitude. Also, the nature of the loads is repetitive. For instance, shafts, bearings, cams and followers, and so on.
Design Against Fluctuating Load
Stress is defined as force per unit area. When there is localization of huge stresses in mechanical components, due to irregularities present in components and sudden changes in cross-section is known as stress concentration. For example, groves, keyways, screw threads, oil holes, splines etc. are irregularities.
![**Problem 5: Stress Concentration in a Brittle Steel Bar**
**Description:**
A brittle steel bar has the dimensions shown in the provided diagram. Determine the maximum axial force \( P \) that can be applied so that the tensile stress does not exceed the allowable stress of \( \sigma_{\text{allow}} = 100 \, \text{MPa} \). Be sure to include calculations for stress concentration factors.
**Diagram Details:**
- The steel bar includes a central section with a necked region and a hole.
- Dimensions:
- Total width: 60 mm
- Neck width: 30 mm
- Height: 12 mm
- Hole radius: 15 mm
- Distance from hole's center to edge: 24 mm
**Graphs and Stress Concentration Factors:**
1. **Left Graph:**
- **Horizontal Axis:** Ratio of notch radius to fillet radius, \( \frac{r}{h} \).
- **Vertical Axis:** Stress concentration factor, \( K \).
- The graph includes several curves for different ratios \( \frac{w}{h} \) (width to height ratio), ranging from 1.1 to 4.0.
- Interpretation: As \( \frac{r}{h} \) increases, \( K \) decreases, indicating less concentration of stress.
2. **Right Graph:**
- **Horizontal Axis:** Ratio of cut width to hole diameter, \( \frac{2r}{w} \).
- **Vertical Axis:** Stress concentration factor, \( K \).
- Includes a single curve showing \( K \) values which decrease with increasing \( \frac{2r}{w} \).
- Interpretation: Larger hole diameters relative to the width result in lower stress concentration.
**Stress Calculation Equations:**
- **Left Graph Equation:**
\[
\sigma_{\text{avg}} = \frac{P}{ht}
\]
where \( P \) is the applied force, \( h \) the height, and \( t \) the thickness.
- **Right Graph Equation:**
\[
\sigma_{\text{avg}} = \frac{P}{(w - 2r)t}
\]
where \( P \) is the applied force, \( w \) the width, \( r \) the radius of the hole](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9496fb66-15f0-4d2e-bc7e-ca3bb9b70577%2Fd4e619c0-9b39-4d85-801a-1424cb986e54%2Fw0x2nq_processed.png&w=3840&q=75)
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