The goal of this article is to explain how FAGUS obtains the torsional constant for its cross sections.
Firstly we will make a brief summary of how the program calculate the torsional constant depending on what cross section we have, and later deepening on metallic cross sections (standard profiles, thin-walled cross sections…).
On page C-4 of the FAGUS manual we can see the table below, where it is specified, in summarised way, how the torsional constant is obtained depending on the section.
COMMENTS ON EACH OF THE MODELS PRESENTED IN THE PREVIOUS TABLE
- MODELS 1 AND 2: CIRCULAR AND RECTANGULAR CROSS SECTIONS
The relation between the torque and the derivative of the rotation of the previous element around it longitudinal axis is defined as:
In this case, Ix is the torsional constant, torsional strength or buckling strength. The reinforced cross sections of small dimensions are generally solid and the torsion constant is calculated using St Venant’s theory (membrane analogy). For compact generic sections, generally the exact value can only be obtained solving the differential equation (using a Finite Element program). FAGUS, as we will see along this article, has that tool.
- MODEL 3: THIN WALLED SECTIONS
For polygonal sub-sections (a division of a cross section) that have a big perimeter in relation with the area, ‘I’ and ‘t’ can be estimated with the program using those two values. In any case, if working with a thin-walled section, it is always better to work entering the elements that form it with its dimensions.
- MODEL 4: SOLID CROSS SECTIONS WITH GENERIC SHAPE
A radius r is obtained for a circle with an equivalent area to the cross section we have.
- MODEL 5: STANDARD STEEL PROFILES
The torsional constant Ix will be obtained from the data catalog.
- MODELS 6 AND 7: GENERIC POLYGONAL CROSS SECTIONS.
For generic polygonal cross sections and without any additional information, the program will automatically assign one of the cases shown above to each of the sub-sections. Hence, the necessary dimensions for the previously explained formulae will be estimated from the area and the length of the contour. The value of Ix obtained by FAGUS must be checked and, if necessary, replaced by another value given by the user, especially if the value of the torsional constant strongly affects the rest of the analysis.
For a cross section made up of many sub-sections, the parts of the sub-sections are determined individually and added at the end. Only the parts of the section created with contours contribute to the value of Ix (the reinforcements, for example, don’t).
For cross sections with big holes, the torsional constant is calculated using the Bredt formula:
A0: area enclosed by the equivalent box section
u: perimeter of the enclosed area
t: (assumed as constant) thickness of the wall of the equivalent box section
The parameters A0 and u are estimated by the program from the individual areas of the sides of the polygon and the length of the contour.
The used model is specified in the tabular output of each of the subsections.
- MODEL 8 AND 9: THIN-WALLED CROSS SECTIONS
The torsional constant of St. Venant for thin-walled open sections is obtained as the sum of the wall elements that constitute it. If the wall elements form a closed section with one or more holes, the torsion modulus is obtained basically from the Bredt formula for the ‘outer circumference’. A cross section made up by thin-walled elements gives correct values of Ix so, if the exact value of the torsion modulus of a multi-cell box beam of a bridge is desired, it is suggested to model it by means of thin-walled elements.
OBTAINING THE TORSIONAL CONSTANT Ix BY A FINITE ELEMENT ANALYSIS
For cross sections made of defined contour of a polygon (with the possible inclusion of holes), the torsional constant, according to St. Venant’s theory, can be obtained by a finite element analysis. This option can be selected on the tab ‘Variants>’Properties of the Variant’>’Values of the cross section’.
The numerical calculations are based on the theory explained below:
With the introduction of a stress function Φ, the elastic behaviour under torsion of an homogeneous region is described by the following equation:
Θ: rotation angle by unit length
G: shear modulus
With the following contour conditions:
Outer edge: Φ=0
Holes: Φ=constant along the hole
With the help of the stress function Φ, the components of the shear stress can be specified at any point:
The change of variables is:
The torsional constant comes from the integration over the total area of the cross section:
Taking everything that has been explained into account, let’s focus on the case of standard profile cross section, to end up concluding with some recommendations about how to obtain the torsional constant in the case of a standard steel profile and in the case of a box section made up of two standard profiles.
STUDY OF THE STEEL PROFILE CASE
- STANDARD PROFILE
If we take, for example, a HEB200 profile:
We can see that, in any catalog, its torsional constant is:
Ix = 59.28 cm4
If we compare this value with the one given by FAGUS:
We see that, indeed, the program is taking this value directly from the table.
If we select the option of calculation by St. Venant:
We see that the value does no longer correspond with the previous one, but instead the program is obtaining it with the numerical calculation explained above.
- BOX SECTION MADE UP OF 2xHEB200
In this case we obtain the value of Ix according to 4 different models:
2.a) Cross section entered as a closed polygon + hole -> Ix by Bredt’s formula
Ix = 6023 cm4
In the following table we see how, as explained above, the value is obtained from the values of area and perimeter.
2.b) Cross section entered as closed polygon + hole -> Ix by St. Venant
Ix = 9023 cm4
In the next table we see how, as explained above, the value is obtained numerically by finite elements and not from the values of area and perimeter:
2.c) Cross section entered as thin-walled -> Ix by Bredt’s formula
For thin-walled closed sections like the one shown (or with more than one cell, if that was the case), the torsional constant for each cell or hole is calculated using Bredt’s formula:
For multiple cells, the torsional constant and the shear flow is given by compatibility and equilibrium. This is carried out by FAGUS automatically solving the corresponding system of equations.
For the specific case of the value of Ix of the cross section, the default value given is the sum of the two parts:
In the case of thin-walled cross sections, the first term can be neglected. Bredt’s part, determined by the program, depends on whether there is one or more closed holes or cells (materialised connection joining the end points of each element). The topology recognised by the program can be verified with the tabular results output. In this case:
We see that a torsional constant Ix = 8178 cm4 is obtained, this value can be compared to the one obtained manually and approximately if proceeding as usual:
A0 = (h-tf)*(b/2-tw) = (20-1.5)*(40/2-0.9) = 353.35 cm2
u = [(h-tf)+(b/2-tw)]*2 = [(20-1.5)+(40/2-0.9)]*2 = 75.2 cm2
Ix = [4*353.35^2*(1.5+0.9)/2]/75.2 = 7969.55 cm4
As we can see, we get a very close value to the one given by FAGUS (the manual calculation is only an approximation).
2.d) Cross section entered as thin-walled -> Ix by St. Venant’s formula
Selecting this option we should get the same value that in case c) according to what was explained about not considering St. Venants term in the formula:
In the table above we see that, indeed, it is that way.
When our cross section is made up of one only standard profile, we will enter the cross section directly from the steel library available provided by FAGUS and we will take the default value of Ix without choosing the option of finite element calculation (St. Venant). By doing so the program will give the value from the catalog.
When we have a compound cross section, we will model it with thin wall elements and we will take the value of Ix calculated by Bredt’s formula (without selecting St. Venant’s option, even if, as we have seen, this would not affect the value obtained) since, as explained above, it is the value we can check.
We can always specify a different value of Ix from the one provided by FAGUS, indicating it in the field highlighted below:
Where, entering the torsional stiffness GIx, the program will directly consider the value, ignoring the value of Ix that has being calculated. This means that the most interesting procedure , if it is desired to define a cross section with the exact geometry of the profiles (and not made of rectangular thin-walled elements) to use it in STATIK, for example, it will be done introducing a model of the cross section like in case 2.a), exposed before (cross section entered with a polygonal contour including a hole) and manually entering, in the indicated field on the previous figure, the value of the torsional stiffness from the Ix obtained with an auxiliary thin-walled model. For that purpose, that Ix will have to multiplied by the value of G we are considering for that material: