Consider a composite beam formed by sandwiching a wood beam between two steel plates, whose cross section is shown.
Say the beam is loaded by some arbitrary loading, due to which at any cross section of this composite beam the bending moment is $M$. The bending moment $M$ at that section will be distributed between wood and steel. Let them be $M_w$ and $M_s$, such that
$$M= M_s + M_w$$
is there any way I can determine in what ratio will the bending moment be distributed between wood and steel, i.e.
$$\frac{M_s}{M_w}$$
Edit: I was actually trying to mathematically justify that a composite beam like the one in figure can carry more bending moment than a wooden beam of same cross section and area. I need to get your opinion on its correctness.
Let the allowable stresses in wood and steel be $\sigma_{a,w}$ and $\sigma_{a,s} $
The maximum bending moment that wood will be able to carry will be,
$$(M_{max})_w = \sigma_{a,w} \,Z_w$$
where $Z_w$ is the section modulus of the wood portion.
Similarly,
The maximum bending moment that steel will be able to carry will be,
$$(M_{max})_s = \sigma_{a,s} \,Z_s$$
where $Z_s$ is the section modulus of the steel portion.
At first I thought I can simply sum these values to obtain the maximum bending moment that the composite beam can carry, but after some thought concluded that can't be the case. So,
If the ratio of the bending moments in each portion at any cross section if the composite beam were to be loaded is,
$$\frac{M_s}{M_w} = \frac{E_s I_s}{E_w I_w}$$
then the maximum beding moment which the composite beam can carry will be,
$$M_{max,composite} = (M_{max,})_w + \frac{E_s I_s}{E_w I_w}(M_{max,})_w $$
$$M_{max,composite} = (M_{max,})_w (1+ \frac{E_s I_s}{E_w I_w})$$
$$M_{max,composite} = \sigma_{a,w} \,Z_w(1+ \frac{E_s I_s}{E_w I_w}) = M_2$$
if the entire beam were to made up of wood then the maximum bending moment that the beam would have carried would be,
$$M_{max,homogeneous} = \sigma_{a,w} \,Z = M_1$$
where $Z$ is the section modulus of the beam entirely made of wood (having the same cross sectional area and shape as that of composite beam)
Taking the ratio,
$$\frac{M_2}{M_1} = \frac{Z_w}{Z} (1+ \frac{E_s I_s}{E_w I_w}) $$
Am I correct up to this point? If yes, how should I proceed further to show that $M_2 > M_1$