kill(all);

equilibrium equations for the quarter
(simplified) ladder frame chassis

eqtx : UA=0;
eqty : VD=0;
eqtz : -F+WD+WA=0;

eqrOx : +WD*b-F*c+ThetaD=0 ;

eqrOy : +PhiA-WA*a+F*a=0;

eqrOz : 0=0 ;

blindly solve the equilibrium equations in the parametric reactions

linsolve(
[ eqtx , eqty, eqtz , eqrOx , eqrOy, eqrOz ],
[UA,PhiA, WA, WD,VD,ThetaD]
) ;

there is one dependent equation, so I'm unable to solve 
the problem of obtaining all the reaction forces/moments by
equilibrium alone.

The quarter ladder frame is once statically indeterminate.

The returned is parametric in a %r# parameter introduced by maxima.

let's try removing the WA unknown reaction from the unknown list

the WA unknown reaction force has been removed from
the unknowns list, thus obtaining a parametric
representation of the reaction forces and moments, 
with WA as a parameter (like F)

to obtain the actual value of WA, the deflection in the 
z direction of the A point has to be evaluated and
zeroed. Castigliano's Theorem fits for the goal.

N  : FILLME;
Q1 : FILLME;
Q2 : FILLME;

equilibrium with respect to rotation along the
local Cz axis 

FILLME;

equilibrium with respect to rotation along the
local Gx and Gy axes

FILLME;

FILLME;

Internal energy for the DB beam

U : integrate(
(J22*M1^2+J11*M2^2+2*J12*M1*M2)
 /
(2*E*(J11*J22-J12^2))
 /alpha_flx
+

N^2/2/E/A
 /alpha_axl
+

( chi1 * Q1^2 + chi2 * Q2^2 + chi12 * Q1*Q2 )
 /
(2*G*A)
 /alpha_shr
+

Mt^2/2/G/Kt
 /alpha_trs
,s,0,a),alpha_axl=1$

The strain energy is now known as a quadratic function of F and WA
alone

Castigliano's Theorem is now applied to obtain the 
z deflection at A; the WA skew-symm. reaction is tuned
to make such a deflection vanish, and thus granding
the material continuity at the A-B-L rigid body

FILLME$

the z deflection at the L point is evaluated for the
quarter ladder frame structure, constrained according 
to two skew-symm planes

d : diff(U , F)$

The deflection of the whole chassis, mounted on the
original three supports, may be derived by observing
that its internal energy is four times its one-quarter
counterpart.

dw : diff(4*U , F)$

such a results may be rationalized by superposing
suitable rigid body motions to the 
twice-mirrored deformed quarter ladder frame structure;
such rigid body motions make the chassis comply with the
null z deflection condition a the three supports.  

the torsional stiffness of the chassis, measured in 
torque per unit twist angle, is derived from the dw deflection
as

ts : fullratsimp(
 (F * (2*c)^2) / ( dw )
) ;

such a quantity is independent on the actual axle track

diff(ts,c);

The properties of a reference cross-section are introduced,
in order to numerically evalate the results

Channel section dimensions:
_h : web heigth, measured at the thin wall midsurface
_b : wing width, measured at the thin midsurface
_t : web and wing wall thickness
(e,f) : Gx,Gy position of the shear center, with respect to the
center of gravity
(m,n) : Ox,Oy position of the center of gravity, with respect to
the web midpoint
upsilon: _b/_h
epsilon: _t/_h

Chassis dimensions:
a   : half wheelbase
b   : half the distance between the profiles
c   : half the axle track

dim : 
[
    a=1200,
    b=400-m,
    c=600,
   _h=80         * _h_mult,
    upsilon=40/80,
    epsilon=6/80  * _t_mult / _h_mult
],_t_mult=1,_h_mult=1;

here, the "_h_mult" and the "_t_mult" multipliers
are introduced to straighforwardly scale 
the cross section size and its thickness, for observing
their influence on the torsional stiffness.
They are provisionally set to unity.

detailed section properties

sec :
[
    A       = 2*epsilon*_h^2*upsilon+epsilon*_h^2,
    J11     =(epsilon*_h^4+6*epsilon*_h^4*upsilon)/12,
    J22     =(2*epsilon*_h^4*upsilon^3+epsilon*_h^4*upsilon^4)/(6*upsilon+3),
    J12     =0,
    Kt      =(epsilon^3*_h^3*(_h+2*_h*upsilon))/3, /*OTW profile formula*/
    chi1    =(15+78*upsilon+138*upsilon^2+96*upsilon^3+24*upsilon^4)/(20*upsilon^4+80*upsilon^3+80*upsilon^2),
    chi2    =(6+72*upsilon+300*upsilon^2+480*upsilon^3+240*upsilon^4)/(180*upsilon^2+60*upsilon+5),
    chi12   =0,
    e       = (4*_h*upsilon^2+12*_h*upsilon^3)/(12*upsilon^2+8*upsilon+1),
    f       = 0,
    m       = (_h*upsilon^2)/(2*upsilon+1),
    n       = 0
],dim;

material properties

mat : 
[
    G  = 210000 / 2 / (1 +3/10),
    E  = 210000,
    nu = 3/10    
];

my specific value for the torsional stiffness, as
a function of the various multipliers

myts : ev(ts, [dim, mat, sec]), infeval$

a reference value, obtained for unit multipliers

mytsref:ev(myts,alpha_flx=1,alpha_shr=1,alpha_trs=1);
%,numer;

I speculate the section bending and shear stiffness to vanish,
i.e. the torsional stiffness
of the profile is left alone in elastically connecting the two
rigid bodies

myts/mytsref$
ev(%, alpha_trs=1)$
limit( %, alpha_flx,0)$
limit( %, alpha_shr,0)$
p:%;
p,numer;

the value shown is the fraction of residual stiffness remained.

I suppose the torsional stiffness to vanish, while maintaining 
both the shear and the bending stiffness unchanged.

myts/mytsref$
ev(%, alpha_flx=1,alpha_shr=1)$
notp:limit(%, alpha_trs,0);
notp,numer;

p and notp are complementary

p+notp;

I hypothesize the torsional stiffness to vanish, while
case 1) maintaining the bending stiffness unaltered, with
vanishing shear stiffness
case 2) maintaining the shear stiffness unaltered, with vanishing
bending stiffness

myts/mytsref $
ev(%, alpha_flx=1)$
limit( %, alpha_trs,0)$
limit( %, alpha_shr,0);

myts/mytsref $
ev(%, alpha_shr=1)$
limit( %, alpha_trs,0)$
limit( %, alpha_flx,0);

no residual stiffness for the structure is obtained 
if the flexural and the shear stiffness of the profiles acts alone.

by raising toward infinity the stiffness of one of the two
contributions, 
the remaining compliance is isolated.

myts/mytsref $
ev(%, alpha_flx=1)$
limit( %, alpha_trs,0)$
q:limit( %, alpha_shr,infinity);

myts/mytsref $
ev(%, alpha_shr=1)$
limit( %, alpha_trs,0)$
r:limit( %, alpha_flx,infinity);

the sum of the two isolated compliances returns the overall
bending compliance

1/q+1/r - 1/notp;

moreover, if only one amongst the flexural and the shear stiffness
vanish 
no further stiffening is observed with respect to the torsion alone

myts/mytsref$
ev( % , alpha_trs=1)$
ev( % , alpha_flx=1)$
limit( % , alpha_shr,0);
%,numer;

myts/mytsref$
ev( % , alpha_trs=1)$
ev(%, alpha_shr=1)$
limit( % , alpha_flx,0);
%,numer;

We hence may rationalize the stiffness contributions
due to flexure, shear and torsion as
                              ___
\\|       trs.stiff          |   |
\\|-----/\/\/\/\/\/\/\/\-----|   |   F*2*c
\\|                          |   |=======> action
\\|                          |   |
\\|                          |   |  dw/2/c
\\|  flx.stiff.  shr.stiff.  |   |-------| deflection
\\|---/\/\/\/\----/\/\/\/\---|   |
\\|                          |___|

The torsional stiffness of the profile acts **in parallel**
with the combination **in series** of the flexural stiffness and
of the
shear stiffness of the same member. Pretty funny.