The C-clamp in more detail (Figure E8). Exercise E4.4 introduced the C-clamp for processing of electronic components. The clamp has a square cross-section of width x and given depth b. It is essential that the clamp have low thermal inertia so that it reaches temperature quickly. The time t it takes a component of thickness x to reach thermal equilibrium when the temperature is suddenly changed (a transient heat flow problem) is

x2 t

Figure E8

The time to reach thermal equilibrium is reduced by making the section x thinner, but it must not be so thin that it fails in service. Use this constraint to eliminate x in the equation above, thereby deriving a material index for the clamp. Use the fact that the clamping force F creates a moment on the body of the clamp of M = F L , and that the peak stress in the body is given by

= x M

2 I

where I =b x³ / 12 is the second moment of area of the body. The table summarizes the requirements.

Function		C-clamp of low thermal inertia
Constraints		Depth b specified
		Must carry clamping load F without failure
Objective		Minimize time to reach thermal equilibrium
Free variables		Width of clamp body, x
		Choice of material