Question 5.15: A sandwich beam having aluminum-alloy faces enclosing a plas...

Neutral axis. Because the cross section is doubly symmetric, the neutral axis (the z axis in Fig. 5-45) 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

I_1=\frac{b}{12}(h^3  –  h_{c}^{3})=\frac{200  mm}{12}[(160  mm)^3  –  (150  mm)^3]=12.017 \times 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 \times 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.

(a) Normal stresses calculated from the general theory for composite beams. To calculate these stresses, we use Eqs. (5-53a) and (5-53b).

σ_{x1}=-\frac{MyE_1}{E_1I_1  +  E_2I_2}               σ_{x2}=-\frac{MyE_2}{E_1I_1  +  E_2I_2}                (5-53a,b)

As a preliminary matter, we will 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 \times 10^6  mm^4)  +  (800  MPa)(56.250 \times 10^6  mm^4)=910,200  N·m^2

The maximum tensile and compressive stresses in the aluminum faces are found from Eq. (5-53a):

(σ_1)_{max}=\pm \frac{M(h/2)(E_1)}{E_1I_1  +  E_2I_2}

=\pm \frac{(3.0  kN·m)(80  mm)(72  GPa)}{910,200  N·m^2}=\pm  19.0  MPa

The corresponding quantities for the plastic core (from Eq. 5-53b) are

(σ_2)_{max}=\pm \frac{M(h_c/2)(E_2)}{E_1I_1  +  E_2I_2}

=\pm \frac{(3.0  kN·m)(75  mm)(800  MPa)}{910,200  N·m^2}=\pm 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.

(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. (5-56a) and (5-56b), as follows:

σ_{top}=-\frac{Mh}{2I_1}               σ_{bottom}=\frac{Mh}{2I_1}                (5-56a,b)

(σ_1)_{max}=\pm \frac{Mh}{2I_1}=\pm \frac{(3.0  kN·m)(80  mm)}{12.017 \times 10^6  mm^4}=\pm 20.0  MPa

As expected, the approximate theory gives slightly higher stresses in the faces than does the general theory for composite beams.