3 032 Problem Set 1 Fall 2006 Due Start of lecture 9 15 06 1 Thin lm silicon nitride cantilevers such as the one shown in Figure 1 are used in scanning probe microscopy resonant frequency measurements and electrostatic ac tuation Let us approximate this cantilever as a clamped free beam with a length of 200 m a width of 30 m and a thickness of 0 8 m Take the Young s modulus of thin lm silicon nitride to be 210 GN m2 The weight of the cantilever can be neglected Image by OpenCourseWare Adapted from work by KristianMolhave Wikimedia Commons Figure 1 Thin lm silicon nitride cantilever a If the free end of the beam contacts a surface represented by a point load in Figure 1 and is de ected upwards by 1 0 m nd the magnitude of the point load b Draw a free body diagram of the beam Sketch the shear and bending moment in the beam along its length labeling maximum and minimum values Use the following sign convention for positive shear and bending moment Beer and Johnson Mechanics of Materials 1992 V M M V Figure by MIT OpenCourseWare c In your scanning probe experiments you need to be able to measure point loads that are only one tenth of the value you calculated above Assuming that a de ection of 1 m produces a suitable signal for detection how much longer does the cantilever need to be 2 You are assembling a structure in a heavy wind which acts as a distributed load q force per length on one of the wide boards you are erecting Figure 2 Your friend 1 weight 150 lb can just barely keep the board upright by hanging from a exible cord connected to the board The board is connected to the ground by a pin joint and the mass of the board is 30 kg m Ignore the e ect of the wind on your friend Figure 2 Upright board subject to a distributed load and a point load a Draw a free body diagram of the board and nd the wind load q and the reaction forces at the pin joint b Sketch the shear and bending moment in the board 3 The Harvard Bridge on Mass Ave Figure 3 a b was rebuilt in the late 1980s due to the fact that the pin and hanger assemblies of the expansion joints on the bridge were the same as those of the Mianus River Bridge in Connecticut The Mianus River Bridge collapsed in 1983 when a single pin became overloaded and caused the death of three people and serious injury of three others Your task is to nd the normal and shear forces and moment associated with that failure which occurred at the position marked x the midspan between pins C and D Figure 3 c a Draw a free body diagram of the entire pin and hanger assembly The weight of the assembly can be neglected b Determine the reaction forces about joint B c Determine the forces along and normal to the horizontal beam and the moment at the point of failure in the steel beam 4 Nanowires of amorphous silica SiO2 Figure 4 are considered as possible optical waveguides in miniaturized electronics For handling and assembly purposes it is necessary to know and be able to measure the critical buckling load Pcr of such nanowires 2 Image courtesy flickr user afagen Figure 3 a b Harvard Bridge c Pin and hanger assembly adapted from Hibbeler 2005 Fig 4b in Tong Limin et al Subwavelength Diameter Silica Wires for Low Loss Optical Guiding Nature 426 December 2003 816 819 Figure 4 Silica nanowire a Amorphous silica bers act as optical waveguides Tong et al Nature 426 816 2003 b Critical buckling load can be measured via compression with instruments such as atomic force microscopes a For the above nanowire of 400 nm diameter and 1 mm length determine the critical buckling load Assume the value of the Young s modulus E for bulk silica which is 70 GN m2 b Using the cantilever in Problem 1 what is the de ection of that cantilever that you would need to achieve to impose this elastic instability 3