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### Channel Section with Left End Free to Twist Right End Fixed Equations and Calculator

**Channel Section with Left end free to twist and warp right end fixed (no twist or warp) Deflection and Stress Equations and Calculator**.

Formulas for the elastic deformations of uniform thin-walled open members under torsional loading.

Per. Roarks Formulas for Stress and Strain - Formulas for torsional properties and stresses in thin-walled open cross sections, Table 10.2, Loading Configuration 1b.

Section Dimensional Definitions

Figure 1

Loading configuration left end free to twist and warp right end fixed (notwist or warp)

Figure 2

Concentrated intermediate torque of Channel Beam Orientation Declarations Image

Figure 3

Preview: Channel section thin wall stress and torsional properties #1b calculator

Channel Section Properties Constants Formulas See: Figure 1

Selected maximum values of stress and torsion

throughout the thickness at corners A and D

throughout the thickness at a distance
from corners A and D |

Left end free to twist and warp, right end free to warp but not twist Formulas:

Boundary values for Loading condition See Figure 2

The following constants and functions are hereby defined in order to permit condensing the tabulated formulas which follow.

Concentrated intermediate torque See Figure 3

Where (when used in equations and this calculator):

Point 0 indicates the shear center se "Concentrated intermediate torque of Channel Beam Orientation Declarations image ";

e = distance from a reference to the shear center (in, m);

K = torsional stiffness constant (in^{4}, m^{4});

C =warping constant (in^{6}, m^{6});

τ_{1} = shear stress due to torsional rigidity of the cross section (lbsf/in^{2}, m^{2});

τ_{2} = shear stress due to warping rigidity of the cross section (lbsf/in^{2}, m^{2});

σ_{x} = bending stress due to warping rigidity of the cross section (lbsf/in^{2}, m^{2});

E = modulus of elasticity (lbs/in^{2}, m^{2});

G = modulus of rigidity (shear modulus) of the material (lbs/in^{2}, m^{2})

T_{o} = applied torsional load (in-lbs, N-m);

t_{o} = applied distributed torsional load (lbsf/in, N/m);

T_{A} and T_{B} are the reaction end torques at the left and right ends, respectively (in-lbs, N-m);

θ = angle of rotation at a distance x from the left end (radians).

θ', θ'', θ''', = are the successive derivatives of y with respect to the distance x.

C_{w} = the warping constant for the cross section;

All rotations, applied torsional loads, and reaction end torques are positive as shown (CCW when viewed from the right end of the member)

Unit step function defined by use of ⟨ ⟩

⟨ x - θ ⟩^{0}

if x < a, ⟨ x - a ⟩^{n} =0;

if x > a, ( x - a) ^{n} ;

The function sinh β

β = ( KG/C_{w}E)^{1/2}

Supplemental selected special cases and maximum values (not included in calculator), See Figure 2.

Reference:

Roarks Formulas for Stress and Strain, 7th Edition, Table 10.2 and 10.3 Loading Configuration 1b. Formulas for torsional properties and stresses in thin-walled open cross sections.