PK >uLBH mimetypetext/x-wxmathmlPK >uL:|_x content.xml
kill(all);dichiaro le quote dimensionali essere ristrette in segno, in modo da avere|x| = xDimensions are declared positive, in order to simplify abs(x) with xassume(b>0,t>0,h>0,c>0);considero per semplicità uno spessore costante tFor the sake of simplicity, wall thickness is assumed constant.two coordinate systems are employed:OXY is centered at the web midpointGxy is centered at the (initially unknown) section center of gravitycoordinate dei punti nodali del profiloprofile nodal point coordinates according to the OXY reference systemXA : b $ YA: h/2$XB : 0 $ YB: h/2$XC : 0 $ YC:-h/2$XD : b $ YD:-h/2$derivo lunghezze dei trattiprofile segment lengthsl_AB : sqrt((XA-XB)^2 + (YA-YB)^2)$l_BC : sqrt((XB-XC)^2 + (YB-YC)^2)$l_CD : sqrt((XC-XD)^2 + (YC-YD)^2)$coordinate, in funzione di un'ascissa curvilinea adimensionale xi in [0,1](x,y) coordinates for points along the profile segments are parametrically definedwith respect to a normalized arlength coordinate xi that spans from 0 to 1 while moving along the segment from the first to the second endpoint.The Gxy centroidal coordinate system is here employed, with unknown XG,YG. x_AB(xi) := (XB-XG)*xi + (XA-XG)*(1-xi)$y_AB(xi) := (YB-YG)*xi + (YA-YG)*(1-xi)$x_BC(xi) := (XC-XG)*xi + (XB-XG)*(1-xi)$y_BC(xi) := (YC-YG)*xi + (YB-YG)*(1-xi)$x_CD(xi) := (XD-XG)*xi + (XC-XG)*(1-xi)$y_CD(xi) := (YD-YG)*xi + (YC-YG)*(1-xi)$calcolo areeArea calculation by integration along the sectionarea : integrate(t * l_AB ,xi,0,1) + integrate(t * l_BC ,xi,0,1) + integrate(t * l_CD ,xi,0,1);momento statico del primo ordineFirst order (or "statical") moments of area, with respect to the Center of GravityMx : integrate(y_AB(xi) * t * l_AB ,xi,0,1) + integrate(y_BC(xi) * t * l_BC ,xi,0,1) + integrate(y_CD(xi) * t * l_CD ,xi,0,1);My : integrate(x_AB(xi) * t * l_AB ,xi,0,1) + integrate(x_BC(xi) * t * l_BC ,xi,0,1) + integrate(x_CD(xi) * t * l_CD ,xi,0,1);derivo baricentro imponendo momento statico nullo se calcolato su sistema di riferimento baricentricothe first order moment of area is known to be nullif the reference system is centroidal; such condition is employedto evaluate the center of gravity coordinates with respect to OXYlinsolve([Mx=0,My=0],[XG,YG]), globalsolve=true;ridefinisco coordinate noto il baricentroMultiple assignments that redefine the (x,y) coordinates along the segmentsaccording to the just evaluated center of gravity position.[x_AB , y_AB ,x_BC ,y_BC ,x_CD ,y_CD ] : ev([x_AB , y_AB ,x_BC ,y_BC ,x_CD ,y_CD ],infeval);momenti del secondo ordinesecond order moments of area, with respect to GxyJxx : fullratsimp( integrate(y_AB(xi)^2 * t * l_AB ,xi,0,1) + integrate(y_BC(xi)^2 * t * l_BC ,xi,0,1) + integrate(y_CD(xi)^2 * t * l_CD ,xi,0,1) );Jyy : fullratsimp( integrate(x_AB(xi)^2 * t * l_AB ,xi,0,1) + integrate(x_BC(xi)^2 * t * l_BC ,xi,0,1) + integrate(x_CD(xi)^2 * t * l_CD ,xi,0,1) );Jxy : integrate(x_AB(xi)*y_AB(xi) * t * l_AB ,xi,0,1) + integrate(x_BC(xi)*y_BC(xi) * t * l_BC ,xi,0,1) + integrate(x_CD(xi)*y_CD(xi) * t * l_CD ,xi,0,1);due to the symmetric nature of the cross section, the product (not "mixed") momentof area is zero.coefficienti alpha, betawe now define the alpha,beta coefficients of the equationd sigma_z--------- = alpha(x,y,...) Sy - beta(x,y,...) *Sx dzMaterial is assumed homogeneous and isotropic along the section.define( alpha(x,y) , fullratsimp ( (-Jxy * x + Jyy*y)/(Jxx*Jyy-Jxy^2) ));define( beta(x,y) , fullratsimp ( (-Jxx * x + Jxy*y)/(Jxx*Jyy-Jxy^2) ));inequilibrio dsigmaz_dzfor each point along the section, the axial stress component variation with increasing z is defined.Such variation causes a disequilibrium that induces the compensating shear actions. define( dsigmaz_dz(x,y) , fullratsimp( alpha(x,y) * Sy - beta(x,y) * Sx ) );tensione tagliante all'estremo Ashear stress at the A end, which is a free surface in the case of an open section.In the case of a closed thin wall section, such value is generally nonzero, and it is to be treated as an unknown parameter.tau_A : 0;tratto AB : tensione tagliante in funzione dell'ascissa curvilinea adimensionalizzata.AB segment : The tau_sz shear stress component is evaluated as a function of thenormalized arc length coordinate xi, based on the axial equilibrium of the portionof section spanning from A (s=0) to the current point s=xi*l_AB along the segmentdefine( tau_AB(xi), fullratsimp( ( tau_A *t + integrate( dsigmaz_dz(x_AB(dummy),y_AB(dummy)) * t * l_AB, dummy,0,xi ) )/t ));
*xi+
3*h3+24*b*h2+36*b2*h
*Sx*xi2
4*b*h3+26*b2*h2+12*b3*h
*ttratto BC : tensione tagliante in funzione dell'ascissa curvilinea adimensionalizzataBC segment: the shear stress component is evaluated as for segment AB; at the B segment end, the shear stress transmitted from the previous segment is applied. The same normalized arc length coordinate is considered, whereas the absolute arclength coordinate (if needed) may be defined ass = l_AB + xi*l_BC, ds = l_BC * dxidefine( tau_BC(xi), fullratsimp( ( tau_AB(1) *t + integrate( dsigmaz_dz(x_BC(dummy),y_BC(dummy)) * t * l_BC, dummy,0,xi ) )/t ));tratto CD : tensione tagliante in funzione dell'ascissa curvilinea adimensionalizzataBC segment: the shear stress component is evaluated as for segments AB and CD.here,s = l_AB + l_BC + xi*l_CD, ds = l_CD * dxidefine( tau_CD(xi), fullratsimp( ( tau_BC(1) *t + integrate( dsigmaz_dz(x_CD(dummy),y_CD(dummy)) * t * l_CD, dummy,0,xi ) )/t ));controllo: è tau_D nulla?check : is tau_sz equal to zero at the D end??tau_D : fullratsimp(tau_CD(1));wow. ok.altro controllo: plot dei grafici per uno specifico dimensionamentofurther check : let's plot the tau_zs variation along the cross section segments.sample dimensions are required for extracting numerical results to be plotted.dim : [h=80,b=40,t=6];applico tagli in direzione x,y t.c. la tensione nominale (taglio/area) sia unitariaShear resultants Sx,Sy are applied for plotting purposes s.t. the nominal shear stress, avaluated as the load divided by the area, is unitary.wxplot2d( ev([tau_AB(xi),tau_BC(xi),tau_CD(xi)],dim,Sy=area*1,Sx=0,infeval), [xi,0,1], [legend,"AB","BC","CD"] );wxplot2d( ev([tau_AB(xi),tau_BC(xi),tau_CD(xi)],dim,Sy=0,Sx=area*1,infeval), [xi,0,1], [legend,"AB","BC","CD"] );ok, pare credibile.ok, it seems reasonable.versore associato alla tau sul trattoin order to evaluate resultants, a unit vector is to be defined for each segmentthat orients the tau_sz actions along the plane.vx = - diff ( x, s) = -diff (x, xi) / l;those vectors are a function of xi (and should be defined accordingly) in thecase of curved segments.vx_AB : - diff(x_AB(xi),xi)/l_AB;vy_AB : - diff(y_AB(xi),xi)/l_AB;vx_BC : - diff(x_BC(xi),xi)/l_BC;vy_BC : - diff(y_BC(xi),xi)/l_BC;vx_CD : - diff(x_CD(xi),xi)/l_CD;vy_CD : - diff(y_CD(xi),xi)/l_CD;risultante delle azioni in direzione xx component of the resultant force of the tau_sx stress distribution. Fx : integrate( vx_AB * tau_AB(xi) * t * l_AB + vx_BC * tau_BC(xi) * t * l_BC + vx_CD * tau_CD(xi) * t * l_CD, xi,0,1 );risultante delle azioni in direzione yy component of the resultant force of the tau_sx stress distribution. Fy : integrate( vy_AB * tau_AB(xi) * t * l_AB + vy_BC * tau_BC(xi) * t * l_BC + vy_CD * tau_CD(xi) * t * l_CD, xi,0,1 )$Fy : fullratsimp(Fy);componente z del momento risultante rispetto al baricentroaxial z component of the resultant moment of the tau_sx stress distribution.MGz : integrate( (vx_AB * (-y_AB(xi)) + vy_AB * (+x_AB(xi))) * tau_AB(xi) * t * l_AB + (vx_BC * (-y_BC(xi)) + vy_BC * (+x_BC(xi))) * tau_BC(xi) * t * l_BC + (vx_CD * (-y_CD(xi)) + vy_CD * (+x_CD(xi))) * tau_CD(xi) * t * l_CD, xi,0,1 )$MGz : fullratsimp(MGz);posizione del centro di taglio in coordinate (e,f) rispetto al baricentroThe evaluation follows of the shear center coordinates (e,f), based on an equal resultant moment condition with respect to the CoG.eq : MGz = Sx * (-f) + Sy * (+e);vale per Sx=0 e Sy=0, e deve mantenere tale valore al variare arbitrario di Sx e Sysuch equation holds for null shear components, we now impose that such null residualcondition is also constant for arbitrary varying Sx, Sylinsolve( [diff(eq,Sx), diff(eq,Sy)],[e,f] ), globalsolve=true;confronto con risultati terzi, tipohttps://theconstructor.org/structural-engg/analysis/shear-centre-with-examples/3677/controllo la differenzacheck based on comparison with known results from literaturefullratsimp( ( -(XG + b^2*h^2*t/4/Jxx) ) - ( e ) );okvaluto ora l'energia potenziale elastica associata alla distribuzione di taglio, per unità di lunghezza di travethe shear contribution to the elastic strain energy per unit beam length is evaluated in the following dU_dz : integrate( tau_AB(xi)^2/2/G * t* l_AB + tau_BC(xi)^2/2/G * t* l_BC + tau_CD(xi)^2/2/G * t* l_CD, xi,0,1 );confronto con la sua controparte nominale corretta dai fattori chia nominal counterpart is defined, that contains the three corrective chi factorseq : dU_dz = (chi_x * Sx^2 + chi_y * Sy^2 + chi_xy *Sx*Sy)/2/G/area;non posso derivare in Sx^2,Sy^2,Sx*Sy, aggiro il problema con un cambio di variabilesince it is not allowed to differentiate any expression with respect to non-atomic terms as Sx^2,Sy^2,Sx*Sy, a workaround is applied in the followingconsisting in a change of variable.eq : ratsubst(Sxq , Sx^2 , eq)$eq : ratsubst(Syq , Sy^2 , eq)$eq : ratsubst(SxSy , Sx*Sy, eq);posso ora derivareI'm now allowed to differentiate with respect to the just defined symbolslinsolve( [diff(eq,Sxq),diff(eq,Syq),diff(eq,SxSy)], [chi_x,chi_y,chi_xy] ), globalsolve=true;per farmi un'idea, valuto nel dimensionamento specificoa numerical value may then be derived for the specific dimensions.ev( [chi_x,chi_y,chi_xy],dim,numer);noto che non sono funzione dello spessoreit may be observed that those coefficients do not depend on the thickness;the only remaining geometrical ratio they may depend on is hence h/b.ev( [chi_x,chi_y,chi_xy],b=1,h=l);wxplot2d(%,[l,0.1,10],[logx],[legend,"chi_x","chi_y","chi_{xy}"],[y,0-1,5]);
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