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Elements of Structures MIT 2.02

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This is my last chance and I need to see if my calculus is correct.

Is it correct ?

Q2_2_4

vA=-5.82 cm ??

  • Elements of Structures MIT 2.02 -

    Huh?

  • Elements of Structures MIT 2.02 -

    Yes I have lost 3 chances and I almos sure this is the correct answer but I hve a doubt with the minus sign.

    The problem is this.


    The composite beam AB, of length L=2m, is free at A (x=0) and fixed at B (x=L) and is composed of a round cylindrical core of constant radius R0=1cm bonded inside a sleeve of thickness R0 (outer radius 2R0=2cm). The beam is loaded, as indicated, by a downward linearly varying distributed load per unit length of magnitude
    q(x)=qxL,with
    q0=2.76kN/m.
    The material moduli are:

    For the core, EC=70GPa=E0
    For the sleeve, ES=210GPa=3E0
    This is I want to know.

    Q2_2_4 : 70.0 POINTS

    Obtain the numerical value (in cm) for the displacement at the free end, vA=v(x=0):

    vA= ....cm


    Is it correct ?

    Q2_2_4

    vA=-5.82 cm ?

  • Elements of Structures MIT 2.02 -

    "
    q(x)=qxL,with
    q0=2.76kN/m.
    "

    Most of the time, x is measured from the fixed end (of a cantilever). Is this the case?

  • Elements of Structures MIT 2.02 -

    I guess I did not read that x=0 at the free end (A), and the fixed end (B) is x=L.

    Also, do you mean
    q(x)=q0*x*L?

    What did you get for EI of the composite beam?

  • Elements of Structures MIT 2.02 -

    Do you get 8050π for the EI of the composite beam? I get 8050π

    For some reason, I get δ=-0.1164, which is exactly double your number.

  • Elements of Structures MIT 2.02 -

    no x=0 at the free end

  • Elements of Structures MIT 2.02 -

    is q(x)=q0*x
    or is
    q(x)=q0*x*L (as you had it above?)

  • Elements of Structures MIT 2.02 -

    No (EI)eff=350ð for the composite beam, remember the radius is in cm, E_0 in GPa.

  • Elements of Structures MIT 2.02 -

    (EI)eff=350*pi

  • Elements of Structures MIT 2.02 -

    I have for the core
    I0=2.5π*10^-9
    and for the sheath
    I1=3.75π*10^-8

    Multiplied by the corresponding E gives me
    EI0=175π (core) and
    EI1=7875π (sheath).

    Total(effective)=8050π

  • Elements of Structures MIT 2.02 -

    Did you use
    Ix=Iy=πd^4/64
    ?

  • Elements of Structures MIT 2.02 -

    Ok (EI)eff= 1080*pi is correct

  • Elements of Structures MIT 2.02 -

    I=pi*R^4/2

  • Elements of Structures MIT 2.02 -

    I got a new delta=-43.38 cm but I'm not sure, I see it to high.

  • Elements of Structures MIT 2.02 -

    my delta equation is
    delta=-q_o(x-5xL^5+4L^5)/(120LEI)

    en x=0 at the free end

    delta=(-q_0*L^4)/(30EI)

    where EI=1080*pi

    thus

    delta= (-q_0*L^4)/(32400*pi)

    so

    delta=-0,4338 m =-43,38 cm

  • Elements of Structures MIT 2.02 -

    1. I suggest you check your EI.

    2. You have not confirmed
    q(x)=q0*x*L (as you have written).
    I think you mean q(x)=q0*(x/L)
    If that's the case, I also get δ=-0.0582 as you did.

    I think the large δ comes from the erroneous EI.
    If you use EI=8050π, you'd get δ=-0.0582 as I have, and as you had before.

  • Elements of Structures MIT 2.02 -

    my delta equation is
    delta=-q_o(x-5xL^5+4L^5)/(120LEI)

    en x=0 at the free end

    delta=(-q_0*L^4)/(30EI)

    where EI=8050*pi

    thus

    delta= (-q_0*L^4)/(241500*pi)

    so

    delta=-0,0582 m =5,82 cm

  • Elements of Structures MIT 2.02 -

    Ok, thanks a lot MathMate. I'm sure the answer is -5,82cm

  • Elements of Structures MIT 2.02 -

    Good luck!

  • Elements of Structures MIT 2.02 -

    sigma max en core and sigma max I sleeve
    I got 47 MPa in core and 35 MPa in sleeve are this correct ?

  • Elements of Structures MIT 2.02 -

    In this problem

    The composite beam AB, of length L=2m, is free at A (x=0) and fixed at B (x=L) and is composed of a round cylindrical core of constant radius R0=1cm bonded inside a sleeve of thickness R0 (outer radius 2R0=2cm). The beam is loaded, as indicated, by a downward linearly varying distributed load per unit length of magnitude
    q(x)=qxL,with
    q0=2.76kN/m.
    The material moduli are:

    For the core, EC=70GPa=E0
    For the sleeve, ES=210GPa=3E0



    Now i got

    Q2_2_5

    max STRESS in CORE=9 MPa

    and max stress in sleeve= 73 MPa

    Are this values correct ? Ples help me this are the last values to finish and I have only more chance and I will pass the course.

  • Elements of Structures MIT 2.02 -

    @11YearsOldMITStudent

    They're not right.

  • Elements of Structures MIT 2.02 -

    I have quite different values as you have. It would help if you show your work so we can compare notes.

    My approach would be:

    Since the beam is composite, there is only one value of 1/r at each cross section x, which is given by
    M(x)/EI.

    For a cantilever beam, M(x) is evidently at the fixed end, equal to
    (q0*L/2)*(L/3)=q0*L^2/6=1840 N-m

    EI had been calculated before and is equal to 8050π
    Thus 1/r=M(L)/EI=0.22857/π=0.07276 (approx.)

    Recall that
    σ=Ey/r
    where r is the radius of curvature and 1/r approximately equals M/EI for large r.

    So for the core,
    σ0=E0*y0*(1/r)
    where E0=70 Gpa
    y0=0.01=distance from neutral axis
    =70*10^9*0.01*(1/r)
    =50.9 MPa

    For the sheath,
    σ1=E1*y1*(1/r)
    where E1=210 GPa
    y1=0.02 = distance from neutral axis
    =210*10^9*0.02*(1/r)
    = 305.6 MPa (approx. 44 ksi)


    Since this is going to be your last life, I would like you to compare my work with yours and be completely convinced of any number before you make your last attempt.

  • Elements of Structures MIT 2.02 -

    Can someone update the answers for the other questions?

  • Elements of Structures MIT 2.02 -

    Please clarify what are the "other" questions.

  • Elements of Structures MIT 2.02 -

    2_1_1
    2_1_2
    2_1_3
    2_1_4
    Please?

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