E3 2020 Pedro T Nº13 Laminares

Summary

The lecture introduces laminar structures—thin surface structures such as shells and membranes—and explains their geometry, structural behavior, and a practical method for preliminary analysis of reinforced-concrete hyperbolic paraboloids.

1. What Laminar Structures Are

Shells and membranes are curved surfaces whose thickness is small compared with their other dimensions. Unlike structures assembled primarily from linear elements, laminar structures rely on the surface itself to carry forces.

Their behavior depends on both their geometry and how they are supported. A surface’s shape is therefore central to its structural performance.

2. Curvature and Surface Types

A small patch of a shell can be described by two principal curvatures, each associated with a radius of curvature. Their product—the total or Gaussian curvature—distinguishes different surface types:

• Negative total curvature: The principal curvatures have opposite signs. Saddle-shaped surfaces, including hyperbolic paraboloids, are examples.
• Zero total curvature: One curvature is zero, as in a cylindrical surface. These are surfaces of single curvature.
• Positive total curvature: Both curvatures have the same sign, as in a spherical surface.

Examples discussed in the lecture include the Oceanographic in Valencia, the Cristo Obrero church in Atlántida, and a dome in Osaka.

3. Ways Surfaces Are Generated

The lecture describes several geometric methods for producing shell surfaces:

• Surfaces of revolution: A line or curve rotates around an axis. Examples include cylindrical, conical, spherical, and toroidal forms.
• Translational surfaces: A curve moves along another curve, often described as the generatrix and directrix.
• Ruled surfaces: A surface is formed by straight lines. This is useful in construction because straight elements can create a curved overall form, potentially simplifying formwork.
• Free-form surfaces: These can combine regions of different curvature. The lecture connects their increasing use to advances in computer-based analysis and design.

4. Membrane Theory: Efficient Behavior and Its Limits

The lecture uses an egg as an analogy: a shell can resist substantial force when the load is distributed, but it is vulnerable to a concentrated impact or pressure.

For membrane theory to apply, the main assumptions are:

• Loads are distributed over the surface rather than concentrated at a point.
• The shell is thin, and its deformation does not substantially change its shape.
• The surface’s normal direction remains approximately unchanged during deformation.
• Bending and transverse shear effects are neglected; forces are treated primarily as acting within the shell’s middle surface.

When these assumptions hold, the surface can use its material efficiently and cover large spans with relatively little weight. The analysis is also comparatively straightforward because it focuses on the equilibrium of forces within the surface.

Membrane theory becomes less reliable near supports, edges, connections, intersections between shell surfaces, and abrupt changes in thickness. These regions can develop bending and shear stresses, which may require additional material and more complex analysis. Ignoring them can lead to excessive deformation or structural failure.

The lecture suggests that edge effects can be considered separately during preliminary sizing, but they must be checked in a more complete design.

5. Hyperbolic Paraboloids

The hyperbolic paraboloid is a saddle-shaped surface with negative total curvature. It can be generated by moving one parabola along another, or described geometrically using four non-coplanar points and the straight lines joining them.

Important features include:

• The surface is curved but can be formed from straight generatrices, making it a ruled surface.
• It contains two families of parabolic sections with opposite curvature.
• Structurally, one family acts as a compression path and the other as a tension path. The lecture relates these to the anti-funicular and funicular forms of the load.
• Forces transfer toward the edges, where edge beams carry them to the supports.
• Depending on the support arrangement, edge beams or connecting beams may carry tension or compression. Where supports cannot resist horizontal forces, tension members may be needed.
• Several hyperbolic paraboloids can be combined to form larger roofs and more complex structures.

The form was especially popular for roofs in the 1950s and 1960s because it was structurally efficient and could be constructed using straight formwork boards. Arranging those boards still required considerable skilled labor, although newer formwork methods keep the form viable.

6. Preliminary Analysis Method

The lecture presents the following method for a preliminary hyperbolic-paraboloid analysis.

1. Establish the Geometry

• Identify the vertex, axes, spans, rises or deflections, and boundary beams.
• Draw the roof form, including its key dimensions and elevations.
• Determine the lengths of the relevant parabolic sections.
• For an equilateral hyperbolic paraboloid, the lecture gives a specific geometric length relation involving twice the plan diagonal. This relation does not apply to every paraboloid configuration.

2. Identify the Principal Parabolas

Determine which parabolic family acts in compression and which acts in tension. Establish the span and rise of each parabola.

3. Calculate the Service Load

Combine the shell’s self-weight—based on its thickness and the unit weight of reinforced concrete—with any surcharge. Use the resulting load for the preliminary force calculation.

4. Calculate the Horizontal Forces

Treat the surface as a set of one-metre-wide strips. For each parabolic strip, use the familiar relation for horizontal force:

[ H = \frac{qL^2}{8f} ]

where (q) is the load per unit length, (L) is the span, and (f) is the rise.

Calculate the compression and tension cases separately. If following the lecture’s sign convention, indicate compression with a negative sign.

5. Determine Forces in the Edge Beams

Resolve how the forces from the two parabolic families combine and transfer to the edge beams. Multiply the force per unit strip width by the relevant beam length.

For inclined beams, use the true three-dimensional length, found with the Pythagorean theorem, rather than relying only on the plan length. Determine whether each beam is in tension or compression from the geometry and support arrangement.

6. Interpret the Support Reactions

Check whether the forces balance at columns and other supports. Include tension members where the configuration produces horizontal forces that the supports cannot otherwise resist.

In a symmetric example, the forces may balance so that the central support carries a vertical resultant.

7. Recognize the Limits of the Preliminary Model

The method is intended for preliminary sizing and does not fully calculate complex boundary effects. Detailed checks of shell reinforcement, concrete compression, edge beams, bending, shear, and support regions are still required.

7. Practical Assignment and Course Transition

The practical work is divided into two parts:

• Numerical exercise: Draw the roof geometry and its views, analyze a quarter of the structure using descriptive-geometry methods, calculate its load, and determine forces in the tension and compression parabolas and in the edge beams.
• Built-structure analysis: Document the work’s author, location, date, requirements, and available technology; provide concise architectural context; draw dimensioned plans and sections; identify the surface type and materials; and explain the structural system, load paths, principal stresses, and how lateral forces are resisted.

The lecture closes by noting that upcoming classes will consider other large-span systems, including structures that incorporate more explicit bending behavior, such as trusses and folded structures.

Speakers and Sources Featured

• Speaker: One unnamed lecturer; no other speaking participants are identified.
• Architects and designers cited: Félix Candela; Santiago Calatrava, mentioned in relation to the later development of the Valencia project; Mamoru Kawaguchi; and a name transcribed in the subtitles as “Flavio de Este” for the Cristo Obrero church. The subtitle transcription may be inaccurate.
• Works and examples cited: The Oceanographic in Valencia; Cristo Obrero church in Atlántida, Uruguay; the Osaka dome; the Miraculous Medal church in Mexico City; Alicante airport; nuclear-plant cooling towers; the Palmira Chapel in Cuernavaca; the Celestino Fernández warehouse in Vallejo; the Julio Herrera y Obes warehouse in Montevideo; the Zarzuela racetrack marquee in Madrid; and a bus station in Cáceres.