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 $GJ$.
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.
