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.
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=
2 answers
4 answers