The statically-indeterminate composite beam AB of length 2L is fixed at A (x=0) and is simply supported at B (x=2L). The beam is composed of a core of modulus E0 of constant square cross section of dimensions 2h0×2h0 bonded inside a sleeve of modulus 2E0 and constant square cross section of outer dimensions 4h0×4h0, as indicated in the figure. The beam is loaded by a downward concentrated load of magnitude P applied at the midspan of the beam (x=L) as indicated.

Question

The KNOWN quantities in this problem are P[N], L[m], h0[m], and E0[Pa]. In symbolic expressions, enter P, L, h0 and E0 as P, L, h_0 and E_0, respectively.

1)Consider the SD companion problem obtained by selecting the roller at B as the redundant support, and obtain the bending moment resultant M(x) in terms of x, L, the unknown redundant reaction RBy (enter this as R_y^B), and P.

for 0≤x≤L,M(x)=
for L≤x≤2L,M(x)=


2)Obtain a symbolic expression for the redundant reaction RBy in terms of P.

RBy=


3)Obtain symbolic expressions for the curvature of the beam at the fixed support A, (1ρ(x=0)), and at the midspan of the beam, (1ρ(x=L)), in terms of P, L, h0, and E0.

1ρ(x=0)=
1ρ(x=L)=


4)Obtain symbolic expressions in terms of the known quantities for the maximum tensile stresses in the core and in the sleeve on the x=0 cross section, and indicate at what y each of them occurs. Express your answers in terms of P, L, and h0.

σmaxcore=
at y= -
σmaxsleeve=
at y=

Answer 1

Did anyone found a full solution to this?

I found:
For 1):
for 0≤x≤L,M(x)=P*(x-L) +(2*L -x) * R_y^B
for L≤x≤2L,M(x)= R_y^B *( 2*L- x)
For 2)
RBy= P/4, BUT THIS IS INCORRECT !

And I am stuck here! Which I think is weird, My Matlab graphs show everything to conform with RBy= P/4

Answer 2

UF did you figure out any more of the second problem(other parts). I'm majorly stuck here

Answer 3

i found Q2_2_4 the two y's.the first one is h_0 and the second one is 2*h_0

Answer 4

Thanks guys! Anything u have on the the other problem helps too. I'm looking for fill. I need a 70 to pass class and am close but not there.....this is the first problem:

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Homework Help: Physics
Posted by GaryG on Monday, August 25, 2014 at 11:47am.

CONICAL SHAFT WITH DISTRIBUTED TORSIONAL LOAD

The conical shaft AB is made of steel, with shear modulus G0. The shaft has length L and is free at A (where x=0) and fixed at B (where x=L). The shaft is solid, with a linear taper so that the radius of the generic (circular) cross section is defined by the equation R(x)=R0(xL), where R0 is the radius at the wall, B. The shaft is subjected to an applied distributed torque per unit length tx(x)=t0(xL)4, where t0 is the magnitude of the applied distributed torque at B (x=L).

The KNOWN quantities in this problem are:

L=2m
G0=70GPa
R0=2cm
t0=2kN⋅m/m
In symbolic expressions, do NOT substitute ANY of the given numerical values of the known quantities, enter L, G0, R0, and t0 as L, G_0, R_0, and t_0, respectively, and enter π as pi.

1)Obtain a symbolic expression for the axial torque resultant in the bar T(x) in terms of x, t0, and L.

T(x)=

2)Compute the numerical value of the reaction TBx at support B, in units of kN·m. (Note: You will need to provide your answer to the second decimal digit.)

TBx=

3)Obtain a symbolic expression for the twist rate along the shaft, dφdx, in terms of x, G0, t0, and R0.

dφdx(x)=

4)Compute the numerical magnitude (absolute value) of the maximum shear stress in the shaft, τmax, in units of MPa:

τmax=

Enter symbolic expressions for the location (r,x) at which this maximum shear stress occurs, in terms of R0 and L:

r=
x=

5)Obtain a symbolic expression for the angle of rotation of the free end A of the shaft in terms of t0, G0, R0, and L:

φ(x=0)=

Compute the numerical value of the angle of rotation of the free end, in units of radians. (Note: You will need to provide your answer to the third decimal digit.):

φ(x=0)=