Elements of Structures MIT 2.02
posted by ElementarySchoolStudent .
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 
bobpursley
Huh?

Elements of Structures MIT 2.02 
11YearsOldMITStudent
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 
MathMate
"
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 
MathMate
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 
MathMate
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 
ElementarySchoolStudent
no x=0 at the free end

Elements of Structures MIT 2.02 
MathMate
is q(x)=q0*x
or is
q(x)=q0*x*L (as you had it above?) 
Elements of Structures MIT 2.02 
ElementarySchoolStudent
No (EI)eff=350ð for the composite beam, remember the radius is in cm, E_0 in GPa.

Elements of Structures MIT 2.02 
ElementarySchoolStudent
(EI)eff=350*pi

Elements of Structures MIT 2.02 
MathMate
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 
MathMate
Did you use
Ix=Iy=πd^4/64
? 
Elements of Structures MIT 2.02 
ElementarySchoolStudent
Ok (EI)eff= 1080*pi is correct

Elements of Structures MIT 2.02 
ElementarySchoolStudent
I=pi*R^4/2

Elements of Structures MIT 2.02 
ElementarySchoolStudent
I got a new delta=43.38 cm but I'm not sure, I see it to high.

Elements of Structures MIT 2.02 
ElementarySchoolStudent
my delta equation is
delta=q_o(x5xL^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 
MathMate
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 
ElementarySchoolStudent
my delta equation is
delta=q_o(x5xL^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 
ElementarySchoolStudent
Ok, thanks a lot MathMate. I'm sure the answer is 5,82cm

Elements of Structures MIT 2.02 
MathMate
Good luck!

Elements of Structures MIT 2.02 
ElementarySchoolStudent
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 
11YearsOldMITStudent
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 
fuubo
@11YearsOldMITStudent
They're not right. 
Elements of Structures MIT 2.02 
MathMate
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 Nm
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 
anonymous
Can someone update the answers for the other questions?

Elements of Structures MIT 2.02 
MathMate
Please clarify what are the "other" questions.

Elements of Structures MIT 2.02 
Ash
2_1_1
2_1_2
2_1_3
2_1_4
Please?
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