Critical Bending Moment

Critical bending moment

M_{crit} = \frac{(2 \cdot e \cdot c^{*} - a_{z}^{*} \cdot B^{*}) + \sqrt{{(2 \cdot e \cdot c^{*} - a_{z}^{*} \cdot B^{*})}^{2} + 4 \cdot 1.0 \cdot (B^{*} \cdot T^{*} - e^{*} \cdot {c^{*}}^{2})}}{2}

c^{*} = K_{y} \cdot \frac{L^{2}}{π^{2} \cdot n^{2}}

a_{z}^{*} = \frac{a_{z}}{n^{2}}

B^{*} = E_{0.05} \cdot I_{Z} \cdot \frac{π^{2} \cdot n^{2}}{L^{2}} + c^{*}

T^{*} = G_{0.05} \cdot I_{T} + c^{*} \cdot e^{2}

n = 1, 2, 3 . . .

M_crit	Critical bending moment
a_z	Distance of the load application from the shear center
e	Distance of the member elastic foundation from the shear center
K_y	Elastic member foundation per truss member
L	Beam length
n	n^th eigensolution
E_0.05	5 % quantile of the modulus of elasticity
I_z	Second moment of area about the weak axis
G_0.05	5 % quantile of the shear modulus
I_T	Torsional constant

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