Can you help me this out?
There is a simple beam resisting a torque as shown in the file attached. The beam is fixed at both ends. Calculated by STAAD, the torsional angle at the loading point is 0.121 rad. But when I do the hand calculation (using formulas in AISC Steel Design Guideline Series 9), the angle is 0.067 rad (about 50% for this calculation case).
I checked the sectional properties of the beam given in STAAD, everyone looks good except the warping constant Cw is not given. Coicidently, when I ignore Cw in the calculation the rotational angle is close to the value from STAAD. Does STAAD consider Cw?
Here is the attachment.
I believe STAAD does not take into account warping stiffness factor Cw.
By default, only St. Venant's torsional deformation is considered in the STAAD analysis. The torsional rigidity is calculated using the assumption that both ends are free to warp for non-circular cross sections.
If the ends of the member are prevented from warping, you can convey that information to STAAD using a command called
SET WARP f
where f can take on a value in the range 0 to 1. 0.0 means no warping restraint which is the default option, 1.0 means full warping restraint. Cw, the warping constant, will be computed and used in the torsional rigidity calculation if f is assigned a non-zero value. Values between 0.0 and 1.0 will result in a partial warping restraint. You can find this described in Section "5.5 Set Command Specification" of the STAAD Technical Reference manual.
SET WARP is a global command, which means it affects every member of the model which has an untapered I-shape. STAAD does not consider this condition for other shapes.
The complete procedure is available in the book
Roark's Formulas for Stress & Strain
Author : Warren C.Young
6th edition, Published by McGraw Hill
Section 9.3 Effect of End Constraint
Look at example 1 in the above section. It shows the calculation of the equivalent stiffness constant K'.
Thanks for the information of SET WARP for end warping restraint.
But there is no warping restraint required in this calculation example (it is Case 3 in AISC Design Guideline Series 9). What I found is that the I-beam rotation under a concentrated torque from STAAD is about 2 times of that from the analytical equation. And it seems STAAD simply ignored the cross-sectional warping constant Cw. I just want to confirm this.
You wrote that "But there is no warping restraint required in this calculation example (it is Case 3 in AISC Design Guideline Series 9)."
Here is my opinion on that.
Case 3 on page 58 of the AISC Design Guide stands for the following torsional end restraints :
At left support : Theta = 0, Theta' (first derivative of Theta) = 0
At right support : Theta' (first derivative of Theta) = 0
In the table on page 108 of the AISC Design Guide 9,
Theta = 0 stands for no rotation,
Theta' (first derivative of Theta) = 0 stands for cross section cannot warp
In other words, case 3 represents a condition where neither end of the member is allowed to warp, which means, warping is restrained at both ends. In this situation, as shown in equation 2.3 on page 3 of the Design Guide, the twist at any point along the longitudinal axis of the member comprises of a component which is associated with the St.Venant's Torsional Constant (J, or IX as it is known in STAAD) and another component associated with Cw, the warping constant.
So, in my view, the AISC Guide does assume that warping restraint is available for case 3.
As I wrote in my previous response, in STAAD, the default condition is that both ends of a member are allowed to warp freely. For this case, the twist in the member is solely a function of the St.venant's Torsional Constant J. That is why STAAD is not using Cw.
To ensure that STAAD uses Cw, you need to specify SET WARP.
Case 3 with alpha=0.3 (Page 60) suits my calculation case better. Where you can find the Torsional End Restraint conditions are Theta=Theta''= 0 (Theta' not equal to 0). As the definition on page 108, the member can warp freely at the both end.
Also on Page 60, it is indicated that "Concentrated torque at alpha=0.3 on member with pinned ends" You can compare this page with Page 70 for the difference of Torsional End Restraints.
On page 61 of the Guide, the plot of the torsional function Theta''*(GJa/T) is zero only at the 2 ends of the beam. At all intermediate locations, it is non-zero, indicating that warping resistance is mobilized at interior points along the span. It is allowed to warp freely only at the ends.
In the STAAD analysis, warping stiffness is not considered at any location along the longitudinal axis of the member. By default, the torsional resistance anywhere along the length of the beam is solely the St.Venant's torsional resistance.
From a broader perspective, the reason why STAAD and probably most other structural analysis programs do not consider the stiffness associated with warping is perhaps the following.
The equations contained in the AISC Design Guide for computing the warping resistance are for idealized end conditions - Free, Fixed or Pinned. In a general framed structure where the type of connections between members determine the amount of fixity at the ends, the guide recommends a more advanced analysis (see the left half of page 108), which usually means modeling every component of the cross section - flanges and webs - using finite elements.
The other aspect is the torsional loading. The exact position of the loading decides the amount of twist the member undergoes. But in a generalized program like STAAD based on the stiffness method, the torsional load is converted to equivalent "fixed end actions" at the nodes of the member. So, the rotation is calculated on the basis of uniform torsion on the full member span, not a torsion that varies from one end of the span to the other.
Now I got it. Thanks!
No warping constant included in the member stiffness sounds like a loss for me. Just as the calculation example shows, the torsional angle increased about 100%. Warping deformation is beyound the 6 degrees-of-freedom of a node in a member, but I don't know if the warping stiffness Cw can be somehow incorprated into the term of pure torsion (St.Venant's Torsion).