The bracket is held to the wall using three A-36 steel bolts...

The bracket is held to the wall using three A-36 steel bolts at $B, C,$ and $D .$ Each bolt has a diameter of 0.5 in. and an unstretched length of 2 in. If a force of 800 lb is placed on the bracket as shown, determine how far, $s$, the top bracket at bolt $D$ moves away from the wall. For the calculation, assume that the bolts carry no shear; rather, the vertical force of 800 lb is supported by the toe at $A$. Also, assume that the wall and bracket are rigid. A greatly exaggerated deformation of the bolts is shown.

Key Concepts

Cross-sectional Area Calculation

In deformation problems, especially those involving round members like bolts or rods, accurate calculation of the cross-sectional area is crucial. The area, typically calculated using the geometry of the section (for instance, the area of a circle for circular cross-sections), directly affects the stress distribution and the subsequent deformation under an applied force.

Young's Modulus

Young's modulus is a material property that measures a material's stiffness by defining the ratio of tensile stress to tensile strain in the elastic deformation region. It is essential for estimating the deformation of a member under load, as it quantifies how resistant the material is to being deformed elastically.

Structural Assumptions in Deformation Analysis

Structural assumptions, such as treating connections as rigid and neglecting shear deformations, simplify the analysis by focusing solely on axial forces. These assumptions allow for the use of straightforward formulas to calculate displacement and deformation, by ensuring that the entire load is transferred through axial stress without the complexities introduced by bending or other forms of deformation.

Hooke's Law

Hooke's Law explains the linear relationship between the force applied to a material and the resulting deformation, provided the material remains within its elastic limit. This principle is fundamental for calculating the extension or compression of a component, where the deformation is directly proportional to the applied force and the original length, and inversely proportional to the cross?sectional area and the material's stiffness (as expressed by Young’s modulus).

Axial Deformation

Axial deformation refers to the change in length experienced by a structural element when subjected to a force along its length. This concept involves determining how much a member will lengthen or shorten based on its material properties, geometry, and the magnitude of the applied force, typically using the formula delta = (F*L)/(A*E).

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