In this discussion of what is “The 4D Solutions(s)” let’s begin by considering some historical perspectives. The concept of “An Engineers’ Coffee Break” is based on an open dialogue about an observation that results in a solution. This can be traced back to 17th Century England’s early coffee house traditions. An example of this environment is a notable meeting between three Royal Society members, Christopher Wren, Robert Hooke and Edmond Halley discussion regarding an observation of celestial motion(s) as a function of the “inverse square law,” but needed a proof (The 4D Solution). The Royal Society’s motto is Nullius in Verba (Latin for “Take nobody’s word for it”). Of course the solution was Isaac Newton’s “Philosophiae Naturalis Principia Mathematic,” published July 5, 1686 provided this proof.
The importance of this event is that these discussions about an observation which resulted in the “Principal of Mathematic”, calculus, prediction of Halley’s comet, the “General Theory of Relativity” and so much more. The perspective of this discussion group dynamics is they had an archetype (Edmond Halley), an individual who could connect with others to solve a problem. An archetype can also be a solution method that connects with multiple sources for the best solution. I will call this “The 4D Solution” an analysis technique of an object(s) in motion (space and time). This kind of analysis can be applied to any problem such as failure, fatigue, impact, project planning and etc. This approach has an effect of enhancing the visualization and subsequently a better communication of the problem and possible solution(s).
Here is an example, using “The 4D Solution” in assessing the design feature of commercial building exterior panel design to fail during an internal pressure event. This design condition is intended to prevent excessive building structural damage during an internal over pressure event. Therefore, the structural loading path is simply, the panel acting as a wind sail, transferring the load to the channels that distributed to the adjoining columns. Subsequently, the weak link in the structure is the unequal leg clip-angle. How does this work?
The response of this channel is like a plank spanning over an opening, and as you walk across towards the other side, the plank bows and rocks. As you reach mid-span you achieve the maximum deflection and instability for the load applied. In order to prevent this condition, a stiffness criterion is applied that is based on the tension member length (L) divided by the least radius of gyration ( r = (I/A)1/2 ). Therefore, the actual design’s L/r ratio is about 400 as compared to the industry standard of 240 to 300, member is potential unstable. However, the end condition rotation is restrained by the unequal leg clip-angles. The challenge is this condition exceeds the Euler–Bernoulli beam equation since the clip-angles exceed their material yield strength and results in excess material strain. In this consideration we can apply plastic analysis to define the limiting condition that achieves “Equilibrium”, where the mid-span bending moment is relative to the end span bending moments that achieves full plastic bending condition. This is one of the primary criterion per the, “Plastic Design in Steel – A Guide and Commentary”. This concept was applied to the design of structures as early as 1954 in France and adopted by the American Institute of Steel Construction (AISC), 5th edition 23rd printing, by 1958. This reference is supported by a significant amount of research and testing at Lehigh University Fritz Engineering Laboratory. In addition, this reference is based on an accumulation of reports prepared by the Welding Research Council and American Society of Civil. The basic principles of this approach are to determine the structure’s inherent nature to redistribute the applied loading throughout the structural system based on geometry, component capacity and the following conditions (Section 2.2 Plastic Theory):
1. Mechanism – “Is that sufficient plastic hinges form to allow the structure (or part of) to deform as a mechanism which is compatible with the rest of the restraints.”
2. Equilibrium – “[The] initial yield, the limits of usefulness is the attainment of plastic moments at each of sections evolved in the mechanism motion.”
3. Plastic – “[Is] that moments in excess of plastic limit strength cannot be resisted.”
At this point we have only determined the unequal leg clip-angle plastic condition loading, but not the actual deflection and/or the ultimate strength loading capacity. Therefore, we need to determine the channel axial shorting impact on the plastic strain condition of the unequal leg clip-angles.
The channel’s homogenous condition during the period where the angles are in full plastic deformation can be considered symbolic of a plank on a roller support (pure bending) to determine axial shorting. This is based on the end supports (unequal leg clip-angles) developing a plastic hinge, where stresses have a slight increase due to strain hardening. This stage results in an increase in bend moment due to strain hardening and membrane force in the unequal leg clip-angles and channel. This approach demonstrates the importance of considering axial shortening as a tension action versus just applying lateral/rotational strain beyond the material strength proportional limit (yield strength) response in the inelastic range. The effect of axial shorting action is like a plank on rollers, which is an accurate depiction of the tension force that is applied after plastic hinge(s) formation. This condition is representative of forces beyond the capacity of end restraints and shows the structure’s instability condition due to a moderately large deflection that is not sustainable and amplifies this connection’s failure. The final step was to approximate potential fracture planes locations and affect by using “Voronoi-based Interpolants for Fracture Modelling,” prepared by N. Sukumar and J. E. Bolander, Department of Civil and Environmental Engineering at University of California.
This presentation of the sequence of actions to this structural system and unequal leg clip-angle failure can provide a visualization of the transition between flexure and fracture (space and time) in relationship to a stress block going through linear elastic, plastic and fracture behaviors, is “The 4D Solution” that is intended to provide a dialogue for of this solution.