2010 Year Month 11 Day 19
Beam Deflection of Cantilever Beam
Aerospace Engineering Laboratory II
Name : Martin Suhartono Student ID : 20106182
1. Objective A cantilever is basically a beam that is supported on one end. Cantilever facilitates one to build an overhanging structure without external bracing and thus it is widely used in construction of bridge, aircraft wing, balcony and automobiles. A cantilever beam usually has variable cross sectional area along its axes and is usually subjected to various loads. However, for our experiment, the cross sectional area is constant and the load is put only on the free-end. The load on the cantilever beam will produce a shear stress on its cross sectional area. This shear stress may be expressed as Where M is bending moment due to the load, V is the shear force in Newton, and x is the discretionary distance from the loading point in meter. The axial stress along the axes, in contrast, can be expressed as Where c is the distance between the surface of the beam and the neutral axis, I is the moment inertia of the cross sectional area of the beam in m 4, P is load in Newton, b is the beam width, t is the beam thickness, and Z is the section modulus of the beam in m3. The strain of the beam's surface can then be expressed according to Hooke's law as Where is the deformation rate (called strain) and E is the modulus of elasticity in N/m2. Based on the above equations, we can get the longitudinal strain at a particular discretional distance x expressed as In this experiment, the strain gauges are placed at different discretionary distances from the point of loading. Based on the assumption that the strain varies linearly the strain formula can be expressed as This, in turn, gives us the following equation Which, when applied to a pair of strain gauges give us
The shear strength as expressed by the above formulas can be regarded as the load imposed on the beam. Nonetheless, there are experimental errors that render this calculation to be rather inaccurate. Using average values to calculate the load is then preferable. Another method to obtain the strain rate is by plotting the values of individual strain as measured by the three strain gauges. The gradient of the graph will then give us /x which can be used to confirm the value of the calculated load. Consequently, the stress at point x1 can be found using the calculated load with the formulas
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On the other hand, a strain gauge functions to measure strain by detecting a change in the resistance of the strain gauge's resistor. Any deformation of the resistor will result in the change of its resistance and this electrical signal is measured accordingly by the strain indicator. 2. Equipment and material a. b. c. d. e. f. g. Flexor, cantilever flexure frame Aluminum alloy beam Electronic strain gauge (FLA-5-11-1L, Tokyo Sokki Kenkyujo Co. Ltd.) Portable strain indicator Strain gauge application kit Micrometer SB-10 switch and balance unit
3. Method Differential strain measurement 1. Set up the half bridge circuit where two strain gauges are connected to one circuit. To do this connect gauge 1 to p+, gauge 2 to p- and common terminal to s. 2. Input 2.11 as the strain gauge factor into the indicator amplifier and adjust the zero point using the balance control. In other words, ensure that the initial strain detected by the indicator for 1 2 is 0. 3. Turn the loading screw for exactly 13 mm deformation. This value should also be applied to the individual strain measurement method later. 4. Read the indicator and record the value after the load is applied. 5. Disconnect all the cables and unload the loading screw. 6. Connect gauge 2 to p+ and gauge 3 to p-. Adjust the zero point for this case too. 7. Turn the loading screw so that the deformation is 13 mm again. Record the value of strain as indicated by the indicator. Individual strain measurement 1. Repeat the procedures for differential strain measurement but in this case, use the use the quarter bridge To method. do this, connect gauge 1 with p+, common cable with s- and another common cable with D120.
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4. Results and Discussion Beam dimension: Width : 25.037 mm Thickness : 5.875 mm Length : 255 mm Gauge Location: X1 : 150 mm X1 X2 : 50 mm X2 X3 : 50 mm Values Initial Final Final Initial Differential Strain Measurement: 1 2 2 3 0 0 123 127 123 127 Computation of Load:
Individual Strain Measurement: Values 1 2 Initial 0.0 1145 Final 365 1385 Final Initial 365 240 Discretionary Length 150 100 (mm)
3 1008 894 114 50
Slope of best straight line through individual strain readings
Stress at station 1
Discussion of different measurement methods A strain gauge consists of a metallic foil pattern that deforms when it is subjected to a load or force. The deformation results in the change of the foil electrical resistance. This resistance change is then measured using either full, half or quarter bridge. A full bridge uses four strain gauges to measure the strain while half uses two strain gauges and two fixed resistors and quarter uses one strain gauge with three fixed resistors. Basically, regardless of the method used, the strain gauges and the fixed resistors should form a Wheatstone bridge. The unbalanced resistance of the strain gauge subjected to pressure in a Wheatstone bridge will cause the voltage across the
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bridge to be non-zero and this is the electrical signal that the indicator will measure as strain. The full bridge advantage is that it can still be used when temperature of the specimen (or environment) changes. Two strain gauges are placed on the specimen while the other two are left unattached, unstrained. This configuration will nullify the effect of temperature change because the two strain gauges will register the change in resistance due to temperature variation with the same proportion. Hence the bridge will still be balanced and yield zero voltage reading when there is no load. The half bridge can also be utilized in the same way by putting the two strain gauges on the specimen and the fixed resistors unattached. In contrast, a quarter bridge will register a non-zero voltage across the bridge when there is temperature variation even though there is no load because, the single strain gauge will deform while the fixed resistor will not, resulting in unbalanced voltage. Hence a quarter bridge method is exposed to temperature-induced error. Moreover, the full bridge and half bridge configurations are more sensitive than the quarter bridge. This is so because any deformation in the full and half bridge is registered by more strain gauges and thus the resulting resistance imbalance across the bridge is larger, yielding higher voltage reading for the same deformation in quarter bridge. The full and half bridge are then more responsive than the quarter one. Additionally, a full bridge is completely linear with the strain rate while the other two bridges are not. Half and quarter bridges give output (imbalance) signals which are only approximately proportional with the strain gauge force. This is so because in quarter and half bridges, the voltage change is affected by the relative change of the strain gauge resistance with the fixed resistors. In contrast, the full bridge has no fixed resistors and all four gauges can deform accordingly, yielding a linear result as long as the change in resistance for all the strain gauges are the same. Nonetheless, it is not always possible to position all the four strain gauges on the specimen to form a full bridge. Sometimes, it is also impossible to put two strain gauges so that they can form a half bridge. Hence, a quarter bridge is still widely used to measure strain even though it is not as sensitive or accurate as the other bridges.
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5. References Strain Gauges, All About Circuits http://www.allaboutcircuits.com/vol_1/chpt_9/7.html date accessed: 20 November 2010
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