Introduction: The blog attempts to underline a need for more inter-disciplinary technical design interaction between disciplines of Civil and Mechanical engineering of Composites with a view to developing methods of more efficient membrane design and cheaper building construction. Structural steel net membrane design avoiding bending could be taken a step forward starting from state of art of Buckminster Fuller’s domes and Shukov’s towers.
To launch payloads on satellites and rockets structural hardware has to be designed and built for weight optimality. Weight minimization as a design objective ensures low cost per kg of launched payload. In upper stages of a satellite launch vehicle or on board geo-synchronous & low earth orbital spacecraft especially, use of composites has become imperative as these materials have high specific (i.e.,for given density of material) stiffness and strength properties.It enables low spacecraft structural weight realization and so, a part of launch cost reduction. It is the case with aircraft construction also; the Airbus 380 uses 50 percent FRP composites including main wings for saving in unit cost of the commercial passenger plane.
The engineering mechanics principles that are adopted and applied from civil construction are the same whether in building of concrete structures or in aerospace composite structures. In both, the same laws of force equilibrium, in-plane force action, beam bending and strain energy storage operate, but in composites the scale is smaller, stress higher and structural loading more efficient. The approach is same in analysis, design and finite element modeling of structures for stress, strain and deformation.
But there is one important difference. Due to launch cost margin of safety (difference between design operating and failure stress levels) is just a percentage only in composites compared to civil structures. Typically, factors of safety (an indicator of efficiency in material usage) are 1.5 to 2.5. However in civil practice, factors of safety can be as much as 3 in stretching and 10 in bending, due to load uncertainties and long term environmental degradation effect on material properties. Attempts to bring down weight, and so also cost of civil structures by direct general introduction of composites such as fiberglass, carbon, Kevlar can progress only very slowly due to their prohibitive cost. Polymeric matrix and petroleum products are becoming ever expensive. As a compromise, part usage of these materials has been attempted for beams, tie-rod applications, pultruded sections in transmission towers, non-structural rust prevention application etc.
Steel can be used with tremendous advantage for indoor working/living spaces, protecting it with periodic paint or anodic provision maintenance even if steel also is getting expensive. The author has seen hall terrace slabs, supported on a single 8 inch I - beam with unsupported span 30 feet standing still in good condition after 75 years of their making. Chennai Egmore Railway Station stands to this day in usable condition even after a full century. Of late the largest steel constructions are attempted in Japan and China for buildings and bridge construction. By using computer analysis and going through a few design/analysis/fabrication trial cycles, engineers can attempt reduced mass structures carrying huge membrane loads without compromising on safety margin. This may entail some liberty with the Code.
Adoption of design/fabrication from Aerospace Structures Fabrication & Design practices can bring in advantages perhaps not sufficiently exploited before. There are a number of composites structural building concepts that can be brought into direct usage onto a bigger scale. Structural bending as a phenomenon has to be basically avoided, and stress uniformity, as a cost cutting criterion needs to be assiduously adopted, so that higher strength of steel at high cost can be offset using lower permissible safety factor. In geodesic pressure vessels of ‘ Iso-Tensoid’ class for example, the stress uniformity is so good that energy methods can be employed to define performance factors in terms of material strength properties. In contrast when seen from uniform strength design viewpoint, a bent beam has index of stress variation [values of (max-min)/average stress] orders of magnitude large compared to a small percentage changes seen in membrane design. The present structure design is stiffness oriented for beams and reticulated plates, material goes unutilized waste stress wise so cost is prohibitive. Proper stable membrane design is free from buckling and is based on stress rather than stiffness. It appears long drawn out that to a level of membrane design of pressure vessels or geodesic domes can at all be adapted to self-loaded shell design in general. Apprehension is right; this is due to unavailability of general stable trajectories of stress appropriate to loading and geometry to avoid buckling in columns and beams and reticulated plates. These are studied in Form Finding exercises. For example a cantilever truss formed by nodal intersection between two in-plane log spirals provides constant stress. However, some approach concepts have in fact already begun influencing building construction:
In an RC beam, material near the neutral axis is wasted at center of slab from a normal stress point of view during bending, however is retained to carry lateral shear. However shear can be better met with ± 45 steel rebars without extra cost. In FRP sandwich construction the core is hollow, as it need not take large vertical shear loads, the skins take up tensile and compressive stresses. For large shears the core is filled with foamed plastic of suitably higher density. In effect, it is an I-beam spread out as a surface in two dimensions and stiffened distributively. Hollow bricks stacked up vertically as a wall in civil construction employ this principle but it would be more advantageous if entire horizontal slabs could be made with continuity of top and bottom skins using unbroken steel wires or plates welded or secured together by bolted connections. Low-cost homebuilder Laurie Baker attempted to incorporate some concepts of sandwich in brick placement. Steel re-bars carry lateral shear loads easily as diagonal elements.
As against an orthogonal gridding where reinforcements run in 0 and 90-degree directions, the Isogrid (used here in the sense of directional invariance/independence of elastic stiffness) beam construction is stronger due to effective 45-degree or off-axis resistance. For instance if 0/90 slab and Isogrid slab of same weight are supported asymmetrically using only three of four corner supports, the latter withstands more live loads due to active internal resisting twist moments in addition to bending. Unsymmetrical loading and supports are disastrous in earthquakes; Isogrid can be a part of the solution for walls, slabs and the roof. Directional dependence of a plate/laminate can be removed by laying reinforcements or rebars in 3 or more directions. Just as there is equilibrium of 3 or more equal and equally spaced forces in a plane, isotropy can be established as an invariant result of directional randomization for rebars. A theorem in composites gives a smeared stiffness as 3/8 value of the maximum property, for n omni /equi-spaced directions or angles each 180/n degrees, n > = 3.
Such a latticed roof construction (a trapezium for a quarter of a small house 12 feet X16 feet designed for mounting Mangalore tiles) was made at a frame cost of about Rs. 30/- per square foot. It may be used for walls, with ceramic tile/phenol formaldehyde laminate fill-ins. A steel floor is more expensive as webs are positioned to take 600 kg/m^2. A lightweight fabrication method is under development. Arch action also promises to reduce some slab and roof weights/cost as bending is avoided.
Civil Engineering construction being the root discipline,it itself can benefit from methods of optimization when once it is accepted that steel angles/plates/tubes are main materials of structural truss construction. In traditional analysis geometrical shapes/materials are taken from existing State of Art and materials availability as initial design. The configured structure is analyzed, areas of stress concentration found out and structure adequately reinforced locally to reduce stress concentration. Standard Civil Code is in vogue since a good part of a century and has become the industry standard and even sacrosanct. Any departure from custom/habit is associated with a fear of failure and unforeseen consequential cost. Only a few professionals with conviction or those oriented to development are motivated to face or adopt change of methods, the large majority yield to the builder’s easy building suggestions or wishes in deference to their own. An all steel SOTA of acceptable cost is not yet been brought into definition or put into place, that is still a far cry. Using genetic algorithms and neural networks some improvements in design and cost is possible, however in Structural Synthesis one can directly determine configurative geometry satisfying equilibrium.
Fuller domes owe their stability to the achievement of an approximate overall membrane uniform stress character of a spherical shell, avoiding bending. The equilateral triangular small elements chosen (sub-divided into almost-equilateral spherical triangles of geodesic arcs out of a regular icosahedron) impart a quasi-isotropic character to the entire dome, even if a continuous isotropic skin sheet is not used. As mentioned for an Isogrid an isotropic wall allows itself to be idealized / quantized along a few finite directions along which we orient the tubes. This has also been checked using computed Von Mises stress derived from gravity induced stress resultants; they are virtually constant up to tangent rotation or co-latitude 52 degrees from vertex in the Figure below. It may be recalled that at the time of introduction of geodesic domes Buckminster Fuller allayed fears about suggestion that deep domes would fail by base hoop tension. The concern could also arise by consideration of maximum stress theory over shear or distortion energy failure theories. He provided for an essentially isotropic or monolithic or membrane or a single smeared layer character of the entire dome wall, in effect a design choice of isotropic construction, employing only 3 or 4 directions by a simple and ingenious choice of reticulated triangular grids adopted as shown in his first public picture presented alongside.
It can be seen even deep domes can be designed as a pure membranes, base support has be adequately rigidised with thickened tubes to cater to inevitable local shell bending.
Shukov Towers and Funicular Construction
Tall impressive tower structures designed by Shukov before WW II stand even today as reminders of advanced steel design and construction work done in Russia, The tower design was only for lightweight Radio wave transmission equipment, but can be now adopted for loads orders of magnitude larger. The creative work was mentioned in German media. One only hopes the pioneering Shukov tower standing there would not be demolished. The geometry is like the common cane-wood furniture lightweight item Moda and such one groundnut sellers also use on Indian roadsides. Cooling towers of Egmore Thermal Power Station, LeCorbusier's concrete shell on Punjab Secretariat Chandigarh, Carnegie Science Center Pittsburgh rooftop item etc, have this shape.
Hyperboloid type of shells obtained by rotating a skew line about a vertical axis as a mesh impart the necessary membrane structural stability, ruling higher above strength and stiffness considerations for material economy and cost as a centrally important structural design consideration.
Structural Bending is the number one problem source or bane of the structural engineer; it should be removed at source to the maximum extent possible in order to achieve economy of material utilization. The characteristic negative curvature postpones first buckling onset point for loads induced by self-weight or even other external pressure loads.
An ideal membrane design approaching that of a rigid strut should have designed weights on basis of stability and utilization of compressive energy. (Total live load P * Total length of reticulations L / Total weight W) should equal specific strength of steel as performance factor, a pure membrane strut design’s high goal post. Plate bending and shell bending of base tubes in Fuller domes or Shukov towers due to ground or segment fixity are necessarily local modifications lowering this performance.
Fig 1. Schematically shows a section of a tower whose dimensions follow optimal calculations. It is an overall sandwich tube, sandwich core thickness is as large as the full radius, is in fact serving as a diaphragm. Each element is rhombic/diamond shape as in Shukov. There are circumferential rings outside for each floor to balance membrane discontinuity. Stiffened steel floor bases can support standard live loads and even roof loads of helicopters, further transmitting them down below to 3-D lattice trusses of saddle surfaces.
Fig 2 A. shows a one-sheet circular hyperboloid shell with generators the type adopted by Shukov. It is beneficial to impart more torsion to the generators making them as deeper hyperboloid 3D space curves, as shown in Fig 2 B.
The outer rhombic lattice walls can be made in two layers and connected for a sandwich effect as well as for convection heat transfer hollow spaces in between latticed walls. There are no circumferential lines, shown for graphics convenience only. The Sukhov tower can be improved in several other structural ways as well. It had been stated using modern methods, an Eiffel size tower design could be upgraded to build to the same height using only one third of its total steel weight, or even one fourth.
Stability is partially contributed by negative double curvature of the shell wall. Shell mode deformation prevention is by mutually restraining forward/backward-balancing components of beam compressive forces. It is suitable for building residential and commercial complexes. Preliminary cost estimates made are seen to be sufficiently cost-effective for towers. It is possible to design up to 50 percent of compressive yield stress without buckling by structural optimization. Circular sections offer less drag induced bending stress than flat building elevations. Speed of construction (using prefab elements) and considerable savings in structure cost is envisaged.
There is plenty of living/office space annular space between outer reticulated column and the inner ‘strut’. The central hollow shaft can be designed to contain elevators, emergency stairs, and water and power/communication lines. A hollow cylinder carries almost all load on its outside. The outer double shell or multiple shells impart compressive strength and freedom from buckling and convective building cooling by wall webs separated by air columns. Resistance to damage due to severe lateral loads like earthquakes is an order of magnitude higher than what can be procured with framed constructions and concrete pillars. Joints with individual steel structural members between column, beam, wall panel, floor panel have been developed to various degrees.
Large span Roofing
When bending is avoided, and in-plane loads predominate,large spanned roofs are designed and checked to be within permissible limits. In Fig 3, a typical span of 75 m is shown. When supported at lengths less than span size indefinite lengths can be realized for covered spaces.
Large Circular Domes
As there is no lateral load except self weight, the roof truss by equilibrium needs to be of negative Gauss curvature. This is a feature quite in contrast to Fuller domes and has considerable weight saving over it. A reticulated dome structurally synthesized with FormFinding design principles for a size 50 m (Fig 4) has been configured, analyzed and a steel model replicating construction with joints has been assembled.
Large pressure vessels
When size grows, membrane tension loading is to be slightly compromised towards stiffer designs as buckling occurs due to un-vented tank emptying, handling and heavy corrosion of steel left out in the open. A welded spherical tank in Fig 5A and 5B composed of 24 spherical segments stiffened for such purpose. It is also easier to fabricate and install. Analysis shows adequate strength safety factor.
A leaf can be taken out of Nature’s design of the hollow bamboo plant. High slenderness ratio, made of fibrous material of plants, never buckles under self-load. Shear diaphragms compartmentalize hollow space in long tubes and arrest any cracks propagating beyond the diaphragm. Overlapping annular sheaths in a banana plant are curved sandwich segments. A Flamingo’s flimsy legs, the human femur and phalanges bones are shaped as efficient struts in Nature. A continuous curved outer tube wall is more stable than a building design of separated frames and central pillars. Traditional civil design employs compression stabilization rather than let in bending, be it for walls or masonry construction. Incorporation of steel renders the walls resistant to tension also, which is essential component of bending that can be met by rebars as in RCC. Adoption of an all steel structure without concrete is a step forward in membrane design/construction. It however entails a fresh look at design with an eye for material economy and construction methods brought in from composites discipline.
Collaborative development work in realization of the above concepts can be addressed for a new state of art promotion building construction. Such a suggestionl may sound as if coal is carried back to Newcastle (i.e., to conventional Civil Engineering), but this coal now has added cost benefit and value. The purpose of the short blog is to attempt to overcome attitudes against living spaces in steel,demonstratable with lower cost.