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    Experimental Verification of Principle of Superposition For An Indeterminate Structure

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    Rupinder Singh
    The document describes an experiment to verify the principle of superposition for an indeterminate beam structure. Strain gauges and a dial gauge were used to measure the deflection of a beam loaded at two points. The measured deflection values matched reasonably well with calculated theoretical values based on the principle of superposition. Some error was observed, particularly towards the ends of the beam. The experiment demonstrated that the principle of superposition can be applied to analyze indeterminate structures, though with greater accuracy towards the structure's center.

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    Experimental Verification of Principle of Superposition For An Indeterminate Structure

    Uploaded by

    Rupinder Singh
    21 views10 pages
    The document describes an experiment to verify the principle of superposition for an indeterminate beam structure. Strain gauges and a dial gauge were used to measure the deflection of a beam loaded at two points. The measured deflection values matched reasonably well with calculated theoretical values based on the principle of superposition. Some error was observed, particularly towards the ends of the beam. The experiment demonstrated that the principle of superposition can be applied to analyze indeterminate structures, though with greater accuracy towards the structure's center.

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    structures in aerodynamics experiment

    Original Title

    exp_4-

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    The document describes an experiment to verify the principle of superposition for an indeterminate beam structure. Strain gauges and a dial gauge were used to measure the deflection of a beam loaded at two points. The measured deflection values matched reasonably well with calculated theoretical values based on the principle of superposition. Some error was observed, particularly towards the ends of the beam. The experiment demonstrated that the principle of superposition can be applied to analyze indeterminate structures, though with greater accuracy towards the structure's center.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    21 views10 pages

    Experimental Verification of Principle of Superposition For An Indeterminate Structure

    Uploaded by

    Rupinder Singh
    The document describes an experiment to verify the principle of superposition for an indeterminate beam structure. Strain gauges and a dial gauge were used to measure the deflection of a beam loaded at two points. The measured deflection values matched reasonably well with calculated theoretical values based on the principle of superposition. Some error was observed, particularly towards the ends of the beam. The experiment demonstrated that the principle of superposition can be applied to analyze indeterminate structures, though with greater accuracy towards the structure's center.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 10

    AE451A

    Experimental Verification of Principle of Superposition


    for an Indeterminate Structure

    Prepared by: Rupinder Singh (160593)


    Group: S9
    OBJECTIVES:

    (i) To compare the experimentally obtained beam deflection values


    from dial gauge with theoretical values calculated from principle of
    superposition for a given indeterminate beam which is transversely
    loaded at two points.

    (ii) To verify the principle of superposition by the strain values


    obtained from strain gauges mounted on indeterminate beam which
    is transversely loaded at two points.

    INTRODUCTION:

    (i)Definitions

    • Indeterminacy: The aim of structural analysis is to evaluate the


    external reactions, the deformed shape and internal stresses in the
    structure. If this can be accomplished by equations of equilibrium,
    then such structures are known as determinate structures. If no. of
    unknown reactions were more than no. of equilibrium equation, then
    it is known as Indeterminacy.

    • Redundancy: Redundancy means the structure's forces and stresses


    cannot be determined by just the basic equilibrium equations. A truss
    having more than (2j – 3) members is statically indeterminate or
    redundant, the degree of indeterminacy or redundancy being equal
    to the number of extra members.

    • Basics of principle of superposition and its validity: The Principle


    of Superposition Theorem states that in any linear network
    containing two or more sources, the response in any element is equal
    to the algebraic sum of the responses caused by individual sources
    acting alone, while the other sources are non-operative.
    EXPERIMENTAL SETUP:

    FIGURE 1:
    Free Body Diagram of beam-

    CALCULATIONS:
    E (Aluminium) = 69GPa

    Moment of Inertia (I) = (1/12) bh3 = (1/12)*25*(12)3 = 36*10-10 m4

    In similar for the other two cases:


    Now for deformation at each point:--

    TABLE 1: Deflection Reading on Dial Gauge.

    Gauge Position Zero Load Reading (mm) P1= 5N P1= 5N P1= 0N


    no. (mm) P2 = 0N P2 = 5N P2 = 5N

    Initial Final Avg. Actual Raw Actual Raw Actual Raw Actual
    Data Data Data Data Data Data
    (mm) (mm) (mm) (mm) (mm) (mm)

    1 45 0 0 0 0.00 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01

    2 225 0 0 0 0.00 -0.05 -0.05 -0.12 -0.12 -0.06 -0.06

    3 360 0 -0.01 -0.005 0.00 -0.07 -0.065 -0.19 -0.185 -0.11 -0.105

    4 495 -0.01 0 -0.005 0.00 -0.07 -0.065 -0.21 -0.205 -0.13 -0.125

    5 685 -0.01 0 -0.005 0.00 -0.04 -0.035 -0.11 -0.105 -0.08 -0.075
    TABLE 2: Dial Gauge Deflection value for the four loading cases and their
    Theoretical deflection.

    Theoretical Deflection vs Dial Gauge Deflection

    P1 = 0 N P1= 5N P1= 5N P1= 0N


    Gauge Position P2 = 0N P2 = 0N P2 = 5N P2 = 5N
    no. (mm)
    Theory Dial Theory Dial Theory Dial Theory Dial
    Gauge Gauge Gauge Gauge

    1 45 0.00 0.00 -0.00052 -0.01 -0.00097 -0.01 -0.00043 -0.01

    2 225 0.00 0.00 -0.00816 -0.05 -0.01612 -0.12 -0.00795 -0.06

    3 360 0.00 0.00 -0.01205 -0.065 -0.02714 -0.185 -0.01510 -0.105

    4 495 0.00 0.00 -0.01144 -0.065 -0.03004 -0.205 -0.01861 -0.125

    5 685 0.00 0.00 -0.00550 -0.035 -0.01622 -0.105 -0.01074 -0.075

    FIGURE 2:

    Deflection vs position
    0
    0 100 200 300 400 500 600 700 800
    -0.05
    Deflection (mm)

    -0.1
    Theory
    -0.15 Dial Gauge

    -0.2

    -0.25
    Position (mm)
    TABLE 3: Data records for the strain gauges in all 4 loading conditions. And
    strain values obtained from the principle of superposition.

    Strain Distance Loading Condition and corresponding strain values(in 10-6) Error (in %)
    Gauge (in cm)
    Number
    P1 = 0 N P1 = 5 N P1 = 0 N P1 = 5 N P1 = 5 N
    P2 = 0N P2 = 0 N P2 = 5 N P2 = 5 N P2 = 5 N
    (P1+ P2) Exp.

    3 76 -1.9445 -2.9504 -6.3943 -9.3447 -8.6787 7.6739

    4 70 -1.9250 -4.0208 -10.5656 -14.5864 -14.2646 2.2559

    5 64 -1.6379 -5.1371 -14.6713 -19.8084 -19.5865 1.1329

    6 58 -1.5356 -6.1297 -19.0981 -25.2278 -25.2559 0.1112

    7 52 -1.6388 -7.2719 -22.0456 -29.3175 -29.5757 0.8730

    8 46 -1.6418 -9.1819 -17.6623 -26.8442 -26.9582 0.4310

    9 40 -1.5673 -10.8647 -13.6829 -24.5476 -25.1760 2.4960

    10 34 -1.9824 -13.0532 -10.0146 -23.0678 -23.2595 0.8241

    11 28 -1.7013 -14.6048 -5.2305 -19.8353 -19.6705 0.8378

    12 22 -2.1416 -8.8017 -1.5248 -10.3265 -9.5581 0.0803

    13 16 -1.9276 -2.0421 3.0094 0.9673 1.5993 39.5172

    14 10 -2.1252 4.2464 6.7036 10.9502 12.2331 10.4887

    15 4 -2.1110 10.5594 22.5478


    10.4874 21.0468 6.6592
    FIGURE 3:

    SOURCES OF ERROR:

    1. Error can come while taking the reading from the dial gauges.
    2. Effect of air can disturb the values as well.
    3. Temperature conditions can alter the results as well.
    4. There could be a permanent deflection in beam after a long use

    Conclusion:

    • The dial gauges provided a fair idea of the distribution of deflection


    throughout the beam, but the readings of the deflections were
    inaccurate with respect to the theoretical data.

    • Principle of Superposition can be applied to analyse with


    indeterminate structure with fair accuracy. But it is observed that the
    error is higher towards the ends of the beam and accurate with
    (|error|<5%) towards the centre of the beam.

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