RECIPROCAL THEOREM

EXPERIMENT NO.1

M E 2 1 8 : S O L I D M E C H A N I C S L A B O R A T O R Y

B A T C H : A 8

0 7 0 1 0 0 4 2 - 0 7 0 1 0 0 4 6

2

R E C I P RO C A L T H E O R E M EXPERIMENT NO.1

AIM

To understand the reciprocal theorem apparatus to possibly illustrate:

1. Measurement of beam deflections. 2. Verify reciprocal theorem.

APPARATUS

1. Simply supported beam.

2. Weights.

3. Dial gauge.

4. Ruler.

THEORY

When loads are applied to a beam, their original straight axes become curved. The amount of flexural deflection in a beam is related to the beams’ area moment of inertia, load, length of the beam, modulus of elasticity and the position of the applied load.

RECIPROCAL THEOREM

1. Choose 2 non-symmetrical reference points on beam.

2. Apply a concentrated load P1 at one point and measure the deflection d21 at the other point.

3. Remove the load from the first reference point and place a load P2 on the second reference

point. Measure the resulting deflection d12 at the first reference point. The loads and the

deflections should satisfy the following relationship:

P1.d12=P2.d21

3

PROCEDURE

Mark points A, B using a ruler at a distance of 20cm, 40 cm respectively from one end of the beam. Start applying variably increasing loads at A and measure the corresponding deflections at B using dial guage. After achieving an optimum high weight, start removing the weights one by one and measure the deflections. Now repeat the above process by loading at B and measuring deflections at A. Tabulate the results.

TABLE

Table 1.1

S No. P(=P1=P2) d21 (0.01mm) d21 (avg) d12 (0.01mm) d12 (avg)

(grams) Reading 1 Reading 2 (0.01mm) Reading 1 Reading 2 (0.01mm)

1. 1936 39 37 38 42 42 42

2. 3909 79 77 78 83 81 82

3. 5915 119 118 118 123 123 123

4. 7934 159 160 160 163 164 164

5. 9958 201 201 201 204 204 204

Table 1.2

S No. P1 d12 P2 d21 P1.d12 P2.d21 error

1 1936 42 1936 38 81312 73568 5

2 3909 82 3909 78 320538 304902 2.5

3 5915 123 5915 118 727545 697970 2.074689

4 7934 164 7934 160 1301176 1269440 1.234568

5 9958 204 9958 201 2031432 2001558 0.740741

INFERENCE

From the table 1.2, we observe that the columns 6 and 7 are approximately equal within the condonable error range. This verifies the Reciprocal theorem.

PRECAUTIONS

1. Make sure that the zero error of the dial guage is taken care of.

2. Do not disturb the table while taking readings as this is a common source of error.

3. Gently add and remove the weights.

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DETAILS OF STUDENTS

Table 1.3

ROLL NUMBER NAME

07010042 Repaka Jyoti Swaroop

07010043 Sharath Chandra Pawar

07010044 Puvvada N V Prudhvi Teja

07010045 J G M V Pramod

07010046 Vishnu Narayanan Suresh