Expt 1 - Reciprocal Theorem

ME 218: SOLID MECHANICS LABORATORY

R E C I PRO C A L TH E OR E M EXPERIMENT NO.1

BATCH: A8 07010042-07010046

RECIPROCAL THEOREM EXPERIMENT NO.1

AIM

To understand the reciprocal theorem apparatus to possibly illustrate: 1. Measurement of beam deflections. 2. Verify reciprocal theorem.

A P PA R A T U S 1. 2. 3. 4.

Simply supported beam. Weights. Dial gauge. Ruler.

T H E O RY

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.

R E C I P R OC A L T H E O R E M

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

2

P RO C E D U R E

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.

TA B L E

Table 1.1 S No. 1. 2. 3. 4. 5.

P(=P1=P2) (grams) 1936 3909 5915 7934 9958

d21 (0.01mm) Reading 1 Reading 2 39 37 79 77 119 118 159 160 201 201

d21 (avg) (0.01mm) 38 78 118 160 201

d12 (0.01mm) Reading 1 Reading 2 42 42 83 81 123 123 163 164 204 204

d12 (avg) (0.01mm) 42 82 123 164 204

Table 1.2 S No.

P1

d12

P2

d21

P1.d12

P2.d21

error

1 2 3 4 5

1936 3909 5915 7934 9958

42 82 123 164 204

1936 3909 5915 7934 9958

38 78 118 160 201

81312 320538 727545 1301176 2031432

73568 304902 697970 1269440 2001558

5 2.5 2.074689 1.234568 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. P R E C AU T I ON S

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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D E TA I L S OF S T U D E N T S

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

4