Final answer:
To find the maximum thickness of a plate for a 100 mm hole, we equate the shear force needed for punching to the ultimate shear stress times the plate thickness. The smallest diameter hole for a 10 mm thick plate can be calculated by rearranging the formula and equating the force due to compressive stress with that due to shear stress.
Step-by-step explanation:
To determine the maximum thickness of a plate from which a 100 mm diameter hole can be punched given an ultimate shear stress of 300 MPa and a compressive stress in the punch limited to 400 MPa, we need to equate the shear stress developed in the plate to the ultimate shear stress. The shear force needed to punch out a hole is equivalent to the perimeter of the hole × shear stress × thickness (t) of the plate. So the shear force (F) is given by:
F = π × diameter × shear stress × t
For the maximum compressive stress of 400 MPa allowed in the punch, we can relate this to the force by considering the surface area (A) of the punch (which is the cross-sectional area of the hole), leading to the following:
F = 400 MPa × A
By combining the expressions for F, and solving for t, we can find the maximum thickness:
t = (400 MPa × π × (diameter/2)^2) / (π × diameter × 300 MPa)
For a 10mm thick plate, to find the smallest diameter hole that can be punched, we rearrange the formula to solve for the diameter:
diameter = (400 MPa × π × (thickness/2)^2) / (π × thickness × 300 MPa)