Eccentrically Loaded Bolt Groups
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The tool above will calculate the C-value for an eccentrically loaded bolt group using an inelastic instantaneous center of rotation method. This page is intended to provide information that may be useful to engineers applying the results of the method. The tables provided in Part 7 of the AISC Manual are in effect a black box. Though sufficient information is provided in the Manual for engineers to replicate the values provided, the tedious and iterative process in my experience deters many engineers from delving into the details in a way that would provide a deeper understanding of what is being done.
By providing both the location of the instantaneous center and the normalized force on each bolt, the tool presented here enables engineers to draw a free-body of the condition. This should provide a deeper understanding of the results. It can also enable engineers to apply the model to more accurately calculate the tearout strength of the individual bolts.
Also as this discussion proceeds other models will be discusses and similar tools will be provided so that the results of various models can be compared.
Do the values produced match the values in the AISC Manual?
No. The processes used to generate the values both in the AISC Manual and here are iterative and criteria is set to determine if a solution has been found. In neither case is the location of the instantaneous center of rotation precisely found. Both the AISC Manual and the programs provided on this page are intended to find a reasonable approximation of the location of the instantaneous center of rotation. If the programs on this page used the same convergence criteria as those used to generate the AISC Manual tables, then presumably the results from the programs on this page would be identical to the those published in the AISC Manual. The results from the programs on this page tend to be slightly (typically 1-4%, sometimes as much as 8%, I think always less than 10%) higher than those in the Manual. No attempt has been made to judge the precision and accuracy of either the values produced here or those in the Manual. In both cases, the results will tend to be “low” because the equation representing the load-deformation relationship, (1-e^-10D)^0.55 = 0.982 – not 1.
Can this be used in design?
The tools and information on this page are intended to be informational and educational. Since more information is provided than is available in the AISC Manual engineers can check the validity of the information provided on this page to determine for themselves whether the information is suitable to their application.
Why is this an inelastic method?
The method is based on an assumed load-deformation curve described in the AISC Steel Construction Manual. Other models could be used. For example, if a linear load-deformation curve is assumed then the method will provide the same results as the "elastic method" described in the Manual. It might also be possible to developed a closed-formed (and non-iterative) solution to the problem by defining a simpler load-deformation curve that generates conservative results. This would also be an inelastic approach.
It may be interesting to note that the Manual describes elastic and inelastic methods. This might be contrasted with other commonly encountered parings such as the elastic and plastic section moduli. The method described in the Manual does not assume a plastic model. In a perfectly-plastic model the individual bolts would achieve their maximum strength and maintain this strength through infinite deformation. The Manual procedure imposes a maximum deformation on the bolts of 0.34 inches based on a 3/4" diameter F3125 Grade A325 bolt. The method then calculates the deformation on each of the bolts based on its distance from the instantaneous center of rotation. In this way all three equations necessary to produce the "correct" solution are satisfied: equilibrium compatibility, and constitutive. Both the elastic and inelastic models described in the Manual satisfy all three constraints. Both solutions are therefore exact solutions consistent with the assumptions made.
The elastic model
The elastic model could also be referred to as the linear model. Rather than assuming the curve shown in the Manual, the elastic model assumes a linear (straight-line) relationship between the load and the deformation. The equation describing the load-deformation relationship can be replaced with the linear relationship used in the inelastic instantaneous center of rotation method to determine the C-value. Due to its simplicity, a closed-form solution as described in the Manual is also possible.
The plastic model
As described above the plastic model is simply the inelastic model described in the Manual without consideration of compatibility. The bolts are assumed to be infinitely plastic.