beam2 - Engineering Beam Theory for the First Order...

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Engineering Beam Theory for the First Order Analysis with Finite Element Method 1998 Winter, Kikuchi Slender structures whose length is much larger than the size of the cross section, are called beams. In such structures, deformation may be decomposed into 1. axial deformation 2. bending deformation in two directions 3. torsional deformation In order to describe the engineering beam theory, we shall introduce the Cartesian coordinate system (x,y,z), where the z axis coincides with the beam axis define by the line formed by the centroid of the cross section, while the x and y axes are the principal axes of the cross section. The origin is set up at the centroid of the left edge cross section of the beam. Based on a) the cross section does not deform, i.e., ε x y = γ xy = 0 b) Bernoulli-Euler assumption that no shearing strain is generated by pure bending c) Saint-Venant torsion theory the displacement of an arbitrary point (x,y,z) of the beam is approximated by Ux , y , z () = u s z ()− y y s θ z Vx , y , z = v s z ()+ x x s θ z Wx , y , z = wz xu s ' z yv s ' z ()+ω ns x , y θ ' z where g’ is the derivative of g in z, u s z , v s s transverse deflections of the shear center x s , y s in the x and y direction, respectively, average axial displacement θ z angle of twist of the cross section at z ω ns x , y normalized Saint-Venant warping function when torque is applied about the shear center axis ( that is the line passing through the shear center x s , y s ), and is defined a property attached to the cross section such that
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ω ns x , y () n x , y y s x + x s y ω ns dA A ns xdA A ns ydA A = 0 ω n x , y normalized warping function when torque is applied at the centroid axis ( that is the beam axis, the line passing through the centroid of the cross section ) The beam theory based on the displacement approximation stated in above is called the engineering beam theory. Using the assumed displacement field, strains are calculated as ε x y = γ xy = ε z = w ' xu s '' yv s ns θ '' γ zx = ∂ω ns x y y s θ ' γ zy = ∂ω ns y + x x s θ Assuming Hook’s low σ z = E ε z , τ zx = G γ zx , τ zy = G γ zy , the total strain energy U e stored in the beam is given by U e = 1 2 σ z ε z zx γ zx zy γ zy dV V = 1 2 σ z ε z zx γ zx zy γ zy dA A 0 l dz = 1 2 EAw ' 2 + EI xx u s 2 + EI yy v s 2 + EI ω s θ 2 + GK θ ' 2 {} dz 0 l where A = dA A cross sectional area
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I xx = x 2 dA A moment of inertia about the y axis I yy = y 2 dA A moment of inertia about the x axis I ω s () ns 2 dA A warping moment K = ∂ω ns x y y s 2 + ∂ω ns y + x x s 2 A dA Saint-Venant torsion constant. Here the x and y axes are the principal axes of the cross section such that xdA A = ydA A = xydA A = 0 .
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beam2 - Engineering Beam Theory for the First Order...

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