Question 6.2: A sandwich beam having aluminum-alloy faces enclosing a plas...
A sandwich beam having aluminum-alloy faces enclosing a plastic core (Fig. 6-8) is subjected to a bending moment M = 3.0 kN·m. The thickness of the faces is t = 5 mm and their modulus of elasticity is E_{1} = 72 GPa. The height of the plastic core is h_{c} = 150 mm and its modulus of elasticity is E_{2} = 800 MPa. The overall dimensions of the beam are h = 160 mm and b = 200 mm.
Determine the maximum tensile and compressive stresses in the faces and the core using: (a) the general theory for composite beams, and (b) the approximate theory for sandwich beams.
Neutral axis. Because the cross section is doubly symmetric, the neutral axis (the z axis in Fig. 6-8) is located at midheight.
Moments of inertia. The moment of inertia I_{1} of the cross-sectional areas of the faces (about the z axis) is
and the moment of inertia I_{2} of the plastic core is
I_2=\frac{b}{12}\left(h_c^3\right)=\frac{200 mm }{12}(150 mm )^3=56.250 \times 10^6 mm ^4As a check on these results, note that the moment of inertia of the entire cross-sectional area about the z axis (I = bh³/12) is equal to the sum of I_{1} and I_{2}.
(a) Normal stresses calculated from the general theory for composite beams. To calculate these stresses, we use Eqs. (6-6a) and (6-6b). As a preliminary matter, we will evaluate the term in the denominator of those equations (that is, the flexural rigidity of the composite beam):
\sigma_{x1} = -\frac{MyE_{1}}{E_{1}I_{1}+E_{2}I_{2}} \sigma_{x2} = -\frac{MyE_{2}}{E_{1}I_{1}+E_{2}I_{2}} (6-6a,b)
E_1 I_1+E_2 I_2=(72 GPa )\left(12.017 \times 10^6 mm ^4\right)+(800 MPa )\left(56.250 \times 10^6 mm ^4\right)= 910,200 N·m²
The maximum tensile and compressive stresses in the aluminum faces are found from Eq. (6-6a):
\begin{aligned}\left(\sigma_1\right)_{\max } &=\pm \frac{M(h / 2)\left(E_1\right)}{E_1 I_1+E_2 I_2} \\\\&=\pm \frac{(3.0 kN \cdot m )(80 mm )(72 GPa )}{910,200 N \cdot m ^2}=\pm 19.0 MPa\end{aligned}The corresponding quantities for the plastic core (from Eq. 6-6b) are
\begin{aligned}\left(\sigma_2\right)_{\max } &=\pm \frac{M\left(h_c / 2\right)\left(E_2\right)}{E_1 I_1+E_2 I_2} \\\\&=\pm \frac{(3.0 kN \cdot m )(75 mm )(800 MPa )}{910,200 N \cdot m ^2}=\pm 0.198 MPa\end{aligned}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.
(b) Normal stresses calculated from the approximate theory for sandwich beams. In the approximate theory we disregard the normal stresses in the core and assume that the faces transmit the entire bending moment. Then the maximum tensile and compressive stresses in the faces can be found from Eqs. (6-9a) and (6-9b), as follows:
\sigma_{top} = -\frac{Mh}{2I_{1}} \sigma_{bottom} = \frac{Mh}{2I_{1}} (6-9a,b)
\begin{aligned}\left(\sigma_1\right)_{\max } &=\pm \frac{Mh}{2 I_1}=\pm \frac{(3.0 kN \cdot m )(80 mm )}{12.017 \times 10^6 mm ^4}=\pm 20.0 MPa\end{aligned}As expected, the approximate theory gives slightly higher stresses in the faces than does the general theory for composite beams.