STAAD.Pro EN 1993-1-1:2005 implementation: Calculation of Torsional Capacities of CrossSections as per SCI P057
Introduction:
General:
When a section is subjected to torsional forces, the section resists them by developing pure torsional capcities and the warping torsional capacities. The pure torsional capacity is in actuality a single set of shear stresses developed around the cross section. For warping torsional capacities, the sections develop both shear stresses and longitudinal stress for maintaining internal equilibrium.
The mechanics of behavior of the section in resisting torsional forces is dependent on the following parameters:
1. Torsion Parameter:
The torsion parameter is defined as K = √(π^{2}EI_{w}/GI_{t}L^{2}). Approximately, it gives a ratio of the warping rigidity EI_{w} to the torsional rigidity GI_{t}. The sections having a high torsional rigidity or low warping rigidity or both(lower values of Torsion Parameter) will essentially develop pure or St. Venant’s torsion capacity to resist the torsion forces. Sections with high values of Torsion Parameter will essentially resist the torsion force by developing warping torsion capacity.
Sections having intermediate values of the Torsion Paramter will essentially resist the torsion force by developing acombination of pure and warping torsional capacities.
2. Nature of Torsional Loading:
If the nature of torsional loading is such to create a uniform rate of twist along the length of the member, then thetorsional forces are resisted by the pure torsion capacity. The system is said to be in a state of uniform torsion.
However, if the torsional loading creates a non-unirom rate of twist along the length of the member, thetorsional forces are resisted by developing the wapring shear and longitudinal stresses. The system is said to be in a state of non-uniform torsion.
3. End Restraint:
The nature of end restraint in terms of resisting warping determines the mechanics of the section in resisting thetorsional forces. If the end restraint allow free warping then the section will primarily develop pure torsional stresses even if the section has a higher value of the Torsion Parameter, provided the nature of torsion loading createsa uniform rate of twist along the length of the member.
However, if the end restraint restrains warping, the section will resist the torsion loading by developing warping shear and longitudinal stresses.
P057 specifications:
The SCI publication P057 calculates the torsion capacities based on the type of sections subjected to torsion loading. The section type is classified as below:
1. Closed Sections:
The torsional rigidity of the closed sections with thin walls are very high compared to the warping rigidity. So, irrespective of the nature of loading and the warping restraint, it is reasonably approximated that these sections resist the torsion by developing pure torsional capacity only.
2. Open Sections:
In case of open sections, if the load is so applied to create a state of uniform torsion and the end restraints let thesection to warp freely, the torsion is considered to be resisted by the pure torsion capacity component only.
However, in case of the sections under a state of non-uniform torsion or end restraints preventing the section fromwarping or both, the sections resist the torsion by a combination of the pure and warping torsions. The degree of the development of each component will depend on the torsion parameter. The SCI P057 has coined a term called thetorsional bending stress “a” which defined as √(EI_{w}/GI_{t}), which is qualitatively similar to the Torsion Parameter we have discussed earlier.
Implementation of Torsional Capacity Calculation in STAAD for EN 1993-1-1:2005 implementation:
The EN 1993-1-1:2005 requires the calculation of the torsional capacity to calculate the design ratio for clause 6.2.7(1). The implementation of the Torsional Capacity Calculation was based on SCI P057 as no other documents were available at the point of implementation. Currently, we have P385 which is not yet implemented in STAAD.Pro.
Calculation of Pure Torsion Capacity:
When a section is subjected to Torsion, the stress distribution in it can be found out by solving a differential equation.
The solution of that equation is mathematically identical to that of a thin membrane stretched over a hole and subjected to light air pressure force from one side. The hole should be geometrically similar to the cross section that is being studied. The slope and the volume of the membrane give an indication of the direction and the quantity of the shear stress developed and also the torsion that is carried by the section. This analogy, which is known as Prandtl’s analogy is extremely useful to understand the nature and the quantity of the stress developed for sections in which closed form solutions to the above differential equation are not available.
For thin-walled closed sections, the developed shear stress due to pure torsion, τ_{t }, is given by the relation ,
τ_{t } = T/ (2 x Ac x t) , T = Applied Torsion
The PureTorsional Capacity, T_{t,Rd}, will be given when the section will develop the maximum shear stress τ_{max}.
T_{t,Rd }= 2 x Ac x t x τ_{max}
Here Ac = Area enclosed by the Mean Parameter.
t = maximum thickness of the section.
For Rectangular Sections, the Pure Torsion Capacity is given by the following equation:
T_{t,Rd} = τ_{max }x (J/t)
J = Torsional Constant
Calculation of Warping Torsion Capacity (for I and Channel Sections):
The Warping Torsion Capacity calculations are quite complex.
However, the following standard approximation makes the warping stress calculations simple and is used in standard documents on Torsion like the AISC Design Guide 9. The torsion is decoupled into two equal and opposite forces acting on the flange as shown in the figure below.
(Figure: Courtesy Design Guide 9, AISC)
It is c onsidered that the flange is subjected to bending due to the in-plane forces. The warping normal stresses are the resulting bending stresses and the warping shear stresses are the resulting Shear Stresses from this Approximation.
Thus, the warping shear stress can be approximately calculated as the following:
τ_{w } = T/hb_{f}t_{f}.
The warping normal stresses can approximately calculated as the following:
σ_{w } = 6M_{f}/b_{f }^{2}t_{f}
τ_{w }= Warping Shear Stress
σ_{w }= Longitudinal Stress due to Warping
M_{f } = Flange Moment
b_{f }= width of the flange
t_{f }= Thickness of the flange
Now, the maximum possible flange moment can be calculated using as (fy/γ_{M0}). b_{f }^{2}.tf/6. Here fy = Yield Strength.
This maximum flange moment can be treated as the Torsional Warping Capacity of the section.
Thus, T_{w,Rd }= (fy/γ_{M0}). b_{f }^{2}.tf/6
References:
2. “Engineering Mechanics Of Solids”, Prentice Hall. Egor P. Popov
3. Eurocode 3:Design of Steel Structures – Part 1-1: General Rules and Rules for Building.
4. SCI Publication 057: Design of Members Subject to Combined Bending and Torsion. D. A. Nethercot, P. R. Salter, A. S. Malik
5. Steel Design Guide Series 9: Torsional Analysis of Structural Steel Members, AISC. Paul A. Seaburg, Charles J. Carter.