Derivation of :.
Projection on the x-z plane.
At a distance z along the beam between the mid-span C and the support B,.
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Projection on the y-z plane.
At a distance z along the beam between the mid-span C and the support B,.
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Now,.
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Hence,.
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The equations describing the deflections are,.
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We are left with the final equations (1) & (2),.
Horizontal delfection - Vert. defl. - .
The reciprocal theorem states that: The displacement caused at point 1 by a UNIT force at point 2 (12) ie equal to the displacement caused at point 2 by a UNIT force at point 1 (21). In this experiment this basically states that,.
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du/dV = dv/dU {as shown in lab. Sheet}.
* The reciprocal theorem is often be used to extract average measures of deformation or stress in an elastic state.
RESULTS:.
∂u/∂V ∂v/∂V ∂u/∂U ∂v/∂U θ.
Theory 2.6*10-5 7.6*10-5 2.71*10-5 2.6*10-5 23.37o.
Expt. 2.63*10-5 7.4*10-5 2.7*10-5 2.5*10-5 22.78o.
∆ (%) 1.14 2.63 0.37 3.85 2.52 {Table.A} .
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{The rest of the results/calculations are attached (hand-written)}.
Experimental arrangements:.
There were 2 weight hangers, one for horizontal loading and the other for vertical, subsequently each had a Dial gauge fixed about it to measure horizontal and vertical deflection respectively. One person will conduct the loading and unloading operation while 2 other observers viewed the readings. Initially, with U(horizontal load) kept at 0 and V(vertical load) increased from zero, until 8kg(all the while the u&v values are being read). Then the V hanger would be unloaded one by one with the unloaded weights being consecutively placed on the U hanger until we arrive at U=8, V=0. Then to finish off the U loads would be removed (while off course taking u/v readings) with [V=constant=0], until U=0, V=0. All the u/v readings had to have the initial u/v values (at U=0, V=0) deducted from them to obtain the deflection values for the subsequent readings.
