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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

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

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

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 Left end free to twist and warp right end fixed (no twist or warp) 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);
σx = 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);
To = applied torsional load (in-lbs, N-m);
to = applied distributed torsional load (lbsf/in, N/m);
TA and TB 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;
Cw = 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/CwE)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.