Three wooden planks are fastened together by a series of...

Three wooden planks are fastened together by a series of bolts to form a column. The diameter of each bolt is $12 \mathrm{mm}$ and the inner diameter of each washer is $16 \mathrm{mm}$, which is slightly larger than the diameter of the holes in the planks. Determine the smallest allowable outer diameter $d$ of the washers, knowing that the average normal stress in the bolts is $36 \mathrm{MPa}$ and that the bearing stress between the washers and the planks must not exceed $8.5 \mathrm{MPa}$.

Key Concepts

Washer Design and Geometry

Washers are used to distribute loads more evenly over a larger area to reduce the chance of material damage and to ensure structural integrity in bolted joints. Proper washer design involves selecting an appropriate outer diameter, relative to the inner diameter, so that the bearing stress remains within acceptable limits while accounting for the dimensions of the bolt and the holes in the connected elements.

Stress Limitation and Safety Factors

In mechanical design, the allowable stress for both the fastener material and the bearing surfaces is predetermined based on material properties and design safety requirements. The design process ensures that the anticipated stresses, including both normal stress in the bolt and bearing stress on the contact surfaces, are maintained below these allowable limits to ensure structural safety and reliability.

Bearing Stress

Bearing stress refers to the pressure exerted on a surface, such as where a washer contacts a plank. It is calculated by dividing the applied load by the contact area between the fastener component (e.g., washer) and the supporting material. Maintaining the bearing stress within allowable limits is important to avoid local material failure or excessive deformation.

Normal Stress in Fasteners

Normal stress, defined as force divided by cross-sectional area, is a critical parameter in assessing the load-carrying capacity of fasteners such as bolts. In engineering design, ensuring that the average normal (tensile) stress does not exceed the material limits of the bolt is essential to prevent yielding or rupture under applied loads.

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