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Beam Distortion

A beam distortion can be a displacement discontinuity or a rotation discontinuity. For a displacement discontinuity of vv at xx the rotation is then

θ0=θ1=vl\theta_{0} = \theta_{1} = \frac{v}{l}

For a rotational discontinuity of ϕ\phi at aa the rotation is calculated by defining a displacement of hh at aa then the angle α\alpha and end 0 and β\beta at end 1 gives the set of equations

b=l−ab = l - a
ϕ=α+β\phi = \alpha + \beta
tan⁡α=sin⁡αcos⁡α=ha\tan\alpha = \frac{\sin\alpha}{\cos\alpha} = \frac{h}{a}
tan⁡β=sin⁡βcos⁡β=hb\tan\beta = \frac{\sin\beta}{\cos\beta} = \frac{h}{b}

Substituting for β\beta in the last equation gives

bsin⁡(ϕ−α)=hcos⁡(ϕ−α)b\sin(\phi - \alpha) = h\cos(\phi - \alpha)
b(sin⁡ϕcos⁡α−cos⁡ϕsin⁡α)=h(cos⁡ϕcos⁡α+sin⁡ϕsin⁡α)b\left( \sin\phi\cos\alpha - \cos\phi\sin\alpha \right) = h\left( \cos\phi\cos\alpha + \sin\phi\sin\alpha \right)
b(sin⁡ϕ−cos⁡ϕsin⁡αcos⁡α)=h(cos⁡ϕ+sin⁡ϕsin⁡αcos⁡α)b\left( \sin\phi - \cos\phi\frac{\sin\alpha}{\cos\alpha} \right) = h\left( \cos\phi + \sin\phi\frac{\sin\alpha}{\cos\alpha} \right)

Substituting for terms in α\alpha

b(asin⁡ϕ−hcos⁡ϕ)=h(acos⁡ϕ+hsin⁡ϕ)b\left( a\sin\phi - h\cos\phi \right) = h\left( a\cos\phi + h\sin\phi \right)
sin⁡ϕh2+(a+b)cos⁡ϕh−absin⁡ϕ\sin\phi h^{2} + (a + b)\cos\phi h - ab\sin\phi

Then the rotation angles at end 0 and 1 are

α=tan⁡−1(ha)β=tan⁡−1(hb)\begin{aligned} \alpha = \tan^{- 1}\left( \frac{h}{a} \right) \\ \beta = \tan^{- 1}\left( \frac{h}{b} \right) \end{aligned}