# A problem on mechanics of materials

I would be appreciate if anyone can solve problems below for this beam.

・Find the all sectional forces $$F_x,F_y,M_x,M_z$$ for both elements① and element②.
・Find elastic strain energy of element① and element②.
Flexural rigidities of two elements are both $$EI$$ and torsional rigidities of them are $$GJ$$.

• Search on here, this, or a very similar problem, has been seen before on here - have you looked? – Solar Mike Jan 12 at 5:44
• @SolarMike Yes, but I couldn't find. – owen Jan 12 at 5:53

Lets start from the point A and name the other end of the beam B and the end at support, where element 1 is connected c, and radius of the beams, R.

Loads are P concentrated at A and q distributed between points A and B over L= a.

For the circular beam ,

$$I = \frac{\pi R^4}{4} ,\ \quad J = \frac{π R^4}{ 2} ,\quad Tmax = (π / 2) τ_{max} R^3 \quad τ = T r / J$$

τ = shear stress (Pa, psi)

T = twisting moment (Nm, in lb)

r = distance from center to stressed surface in the given position (m, in)

J = Polar Moment of Inertia of Area (m4, in4)

The deflection at the point A is sum of two deflections due to loads P and q,

$$\delta = P a^3 / (3 E I) \quad + q a^4 / (8 E I)$$

The deflection on point B is rotation under torsion due to beam AB torque and deflection under P an q, which is

$$(P + q*a)a^3/3EI$$

$$\theta_{rotation} = L T / (J G)= a(P*a + q*a^2/2 )/GJ$$

$$and\quad \delta = (P +qa )a^3 / (3 E I)$$

Elastic, strain, energy of beam AB , U is the work done by the forces which caused deflection.

$$U_{AB} = m^2L/2EI= (a*P+a^2q/2)^2/2EI$$

$$U_{C B}= \theta *T/2 = T^2*a/2GJ = (P*a + q*a^2/2)^2/2GJ$$

$$F_{x, ab} = 0 \quad F_{y, ab} = P +qa\quad \\ M_{z,ab} =aP +qa^2/2$$

$$F{x,bc }=0\quad F{y,bc} =P+aq \\ M{z,bc}= a( P+aq )$$

Note: all coordinate are assumed local WRT the beam.

• I am grateful for your support！ – owen Jan 12 at 20:15