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C-Section No. 2 of 1b Loading Concentrated Intermediate Torque applied Deflection and Stress Equatio

C-Section with Concentrated Intermediate Torque applied Deflection and Stress Equations and Calculator #2 of 1b Loading .
Loading configuration left end free to twist and warp right end fixed (no twist or warp). 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.

Section Dimensional Definitions
Figure 1

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Loading Configuration (Section Shown May not Match Section given in Figure 1)

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

Figure 2

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Loading Declarations Image (Section Shown May not Match Section given in Figure 1)

Concentrated intermediate torque of C-Section Beam Orientation Declarations Image
Figure 3

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Preview: C-Section thin wall stress and torsional properties #2b calculator

C-Section Properties Constants Formulas See: Figure 1

Selected maximum values of stress and torsion

Stress throughout the thickness at corners A and D

Shear stress due to warping rigidity Throughout the thickness on the top and bottom flanges at a distance e from corners C and D.

Shear stress due to torsional rigidity

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Left end free to twist and warp right end fixed (no twist or warp)

Boundary values for Loading condition See Figure 2

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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 (in⁴, m⁴);
C =warping constant (in⁶, m⁶);
τ₁ = shear stress due to torsional rigidity of the cross section (lbsf/in², m²);
τ₂ = shear stress due to warping rigidity of the cross section (lbsf/in², m²);
σ_x = bending stress due to warping rigidity of the cross section (lbsf/in², m²);
E = modulus of elasticity (lbs/in², m²);
G = modulus of rigidity (shear modulus) of the material (lbs/in², m²);
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;
A, B, C, D, E, F, e, h, b1= Dimensional data Figure 1 (in, m).

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⟨ x - a ⟩ is defined as zero if x < a

β = ( KG/C_wE)^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 Formulas for torsional properties and stresses in thin-walled open cross sections.