Question 6.2: A sandwich beam having aluminum-alloy faces enclosing a plas......

Use a four-step problem-solving approach. Combine steps as needed for an efficient solution.
1. Conceptualize: Use the general theory of flexure for composite beams.
2. Categorize:
Neutral axis: Because the cross section is doubly symmetric, the neutral axis (the z axis in Fig. 6-9) is located at mid-height.
Moments of inertia: The moment of inertia I_1 of the cross-sectional areas of the faces (about the z axis) is
I_1 = \frac{b}{12}(h^3 – h_c^3) = \frac{200 ~mm}{12}[(160~ mm)^3 – (150~ mm)^3]
\quad\quad\quad = 12.017 × 10^6 ~mm^4
and the moment of inertia I_2 of the plastic core is
I_2 = \frac{b}{12}(h_c^3) = \frac{200~ mm}{12}(150 ~mm)^3 = 56.250 × 10^6 ~mm^4
As a check on these results, note that the moment of inertia of the entire cross-sectional area about the z axis (I = bh^3 /12) is equal to the sum of I_1 ~ and~ I_2.
3, 4. Analyze, Finalize:
Part (a): Normal stresses calculated from the general theory for composite beams.
To calculate these stresses, use Eqs. (6-7a and b).

\quad\quad\sigma_{x1}=-\frac{M y E_{1}}{E_{1}I_{1}+E_{2}I_{2}}\qquad\sigma_{x2}=-\frac{M y E_{2}}{E_{1}I_{2}+E_{2}I_{2}}\quad\quad (6-7a,b)
As a preliminary matter, evaluate the term in the denominator of those equations (that is, the flexural rigidity of the composite beam):
E_1I_1  +  E_2I_2 = (72 GPa)(12.017 × 10^6 ~ mm^4) + (800 ~MPa)(56.250 × 10^6 ~mm^4)
\quad = 910,200 ~N·m^2
The maximum tensile and compressive stresses in the aluminum faces are found from Eq. (6-7a):
\quad(σ_1)_{max} = ± \frac{M(h/2)(E_1)}{E_1I_1 + E_2I_2}
\quad\quad\quad\quad = ± \frac{(3.0 ~ kN·m)(80 ~mm)(72~ GPa)}{910,200 ~N·m^2} = ±19.0~ MPa
The corresponding quantities for the plastic core (from Eq. 6-7b) are
\quad(σ_2)_{max} = ± \frac{M(h_c/2)(E_2)}{E_1I_1  +  E_2I_2}
\quad\quad\quad\quad = ± \frac{(3.0 ~kN·m)(75 ~mm)(800 ~MPa)}{910,200 ~N·m^2} = ±0.198 ~MPa
The maximum stresses in the faces are 96 times greater than the maximum stresses in the core, primarily because the modulus of elasticity of the aluminum is 90 times greater than that of the plastic.
Part (b): Normal stresses calculated from the approximate theory for sandwich beams:
The approximate theory disregards the normal stresses in the core and assumes that the faces transmit the entire bending moment. Then the maximum tensile and compressive stresses in the faces is found from Eqs. (6-10a and b), as

\quad\quad \sigma_{{top}}=-\frac{M h}{2I_{1}}\quad\sigma_{{bottom}}=\frac{M h}{2I_{1}}\quad\quad (6-10a,b)
\quad(σ_1)_{max} = ± \frac{Mh}{2I_1} = ±\frac{(3.0 ~kN·m)(80 ~mm)}{12.017 × 10^6~ mm^4} = ±20.0 ~MPa
As expected, the approximate theory gives slightly higher stresses in the faces than does the general theory for composite beams.