10-12 (pages 135-195) and prepare a synopsis (summary) of the contents about 1 page long for each chapter (Chapters 10thru12 all in one WORD Document)and submit it to the Instructor by midnight Sunday. Please prepare your assignment in a single-spaced WORD Document format and it doesn't have to be in APA Style. You can use charts/figures/visuals to enhance your assignment.
Chapter 10: Grids, Plates, Folded Plates, and Space-Frames
If two equal length beams are placed at right angles to each other on a rectangular opening and load is applied at their intersection, both beams will deflect the same amount. One-quarter of the load will be transferred to each of the four supports. If a short and a long beam, at right angles are placed on a rectangular opening, again loaded at their intersection, the short beam will carry more of the load to the supports then the long one. Beam deflection is proportional to the cube of the length, hence a longer beam deflects easier. To put it another way, more force is needed to deflect the short beam the same amount as the longer one, hence the shorter beam transfers more of the load to the supports. To cover a rectangular area several long and short beams may be arranged into a rectangular grid. Again the shorter beams transfer more of the loads to the supports. The long beams are inefficient. A more efficient arrangement may be obtained by a skewed grid. Here all beams are of the same length with the exception of the ones near the corner of the rectangle. If an infinite number of infinitesimally narrow beams are considered, a solid plate is obtained that behaves similarly to a skewed grid. It carries loads by a combination of beam stresses, shear stresses and additionally by the internal twist. The combination of stresses at every point in a plate results in maxima and minima that may be plotted as a series of isostatics. The thickness of plates is relatively small and consequently, plates have small moments of inertia. To increase their bending capacities ribs may be added to them, similarly to T-beams. Another method for increasing moment inertia is to fold a plate into an accordion shape. In addition, to be used as ceilings and floors plates may be used as vertical or slanted walls either in a flat or folded form. Trusses consisting of triangulated tensile and compressive bars in a plane may also be used in spatial triangles, forming the edges of interconnected pyramids. Large space frames are constructed based on such space trusses.
Summary of Ideas Presented in Chapter 10:
A set of beams laid parallel to each other may be used to cover a rectangular opening. A load applied to one of the beams is carried to its supports, and the other beams are not affected if the beams are not interconnected. If the beams are crisscrossed, a load applied to any beam will be shared by neighboring beams, and the load is carried to the boundaries of the rectangle. Shorter beams in such a rectangular grid are stiffer and carry more loads than longer ones.
To eliminate longer, less effective members in a grid structure, beams may be laid at angles in a skew grid. In this arrangement, most beams have the same length and therefore distribute the load more efficiently.
If the beams of a grid are assumed to be very close to each other, the grid becomes a solid plate.
Every point on a plate is subjected to bending, shear, and also torsion. Since bending stresses consist of tension, compression, and shear, all these stresses are present in the plate.
As discussed in Chapter 9, Tension and compression have principal directions and have maximum and minimum values without shear or torsion in those directions. They form lines of “Isostatics.”
Plates may cover areas of various shapes and may have a variety of supports. When a plate is loaded, depending on the support conditions on its boundaries, it deflects into a barrel or dish shape. In a dish shape, membrane action will develop (see Chapter 9) and help in carrying the load.
Plates have relatively shallow depths and therefore have small moments of inertia. To increase the moment of inertia, ribs may be added to their undersides in a manner analogous to T beams, discussed in Chapter 7. Such ribbed plates have greater load carrying capacities.
To increase the moment of inertia of a plate, it may be folded into a V-shape that forms it into a beam. If several folds are made, a folded plate is obtained. Like ribbed plates, it has a higher moment of inertia than a flat plate and thus has a greater load-carrying capacity.
Folded plates can be made into vertical walls, radial domes, or even into barrel vaults.
To cover large areas, instead of beams in a grid, trusses may be substituted for deep beams. Due to low twisting resistance, such a truss grid carries the most load in bending action.
Space trusses may be arranged three-dimensionally to create space frames by providing interconnecting diagonal members. Suchspace framesare much stiffer than space trusses because they can develop substantial twisting resistance.
Chapter 11: Membranes
In Chapters 11 and 12 curved surfaces are discussed. Membranes, shells, and domes owe their load carrying capacities to their curvatures. A curvature of a surface is defined as the reciprocal of the radius of curvature. If a half cylinder is cut by a plane perpendicular to its axis a half circle is obtained. If cut along the axis a straight-line results. The half circle has a radius while the radius of the straight line is infinite. Here the curvature is zero and at the half circle, the curvature is large. Cut by a diagonal plane parabola results; its curvature is less than that of the half circle. Hence a surface parallel to the axis has a minimum curvature and one perpendicular to it has maximum curvature. These are the two principal curvatures of the surface.
The small square cut by planes parallel to the principal directions has opposite parallel sides; the sides of an element cut by diagonal planes are not. Along these directions, the surface has “twist”. Along with principal directions, there is no twist. A piece of thin flexible material stretched by forces along its edges is a membrane. It is analogous to a set of cables at right angles to each other. Just as cables, membranes carry loads by tensile stresses. When loaded perpendicular to its plane the membrane deforms into a curved surface. If a square piece of paper is held vertically by one of its edges and pulled down along the opposite edge, in-plane shear stresses are developed. A small square element in a curved loaded membrane has two sides at different heights and at different angles. i.e. it is twisted and is consequently sheared. Membranes carry out of plane loads by tension and shear stresses. A typical membrane structure is a circus tent supported on two vertical struts and stretched by stakes in the ground. Between the two poles, the membrane has an upward, negative curvature and downward, positive curvature at right angles. These two are minimum and maximum curvatures of the surface.
Membranes are unstable structures, they change their shape when loads are shifted. To give it stability an umbrella and a dirigible are stiffened by ribs while balloons are stiffened by internal pressure. An inflated longitudinal toy balloon may be loaded as a beam. Its whole surface is in tension but its compressive side will buckle if the membrane tension is reduced by compression.
Summary of Ideas Presented in Chapter 11:
A membrane is a thin sheet of material that has practically no moment of inertia. It can therefore only carry tensile stresses and will buckle under compression.
A curved membrane behaves essentially likes a series of closely placed cables connected in two directions. Though its principal load-carrying mechanism is tension in two directions, there is a small amount of torsion, and hence in-plane shear stress is also present to help in carrying part of the load.
A curved membrane usually has two principal curvatures (a maximum and a minimum). Along these lines, there is no twist or shear stress.
Membranes, like cables, are unstable structures. They change their shape under varying loads: They may be stabilized by internal or external ribs such as umbrellas, by two-directional opposing curvature, or by internal pressure (such as in balloons and air-supported structures).
Chapter 12: Thin Shells and Reticulated Membranes
As an inverted funicular cable becomes a funicular arch, similarly an inverted membrane becomes a thin shell carrying loads mostly by compression. To withstand compressive stresses shells have to be constructed from stiffer materials that can, in addition, carry bending stresses as well as shear and tension. Like arches, shells have to be supported along their edges to prevent them from opening up. Spherical domes are compressed along their meridians and are in tension along the parallels. In high rise domes, parallels near the top are compressed.
The inverted membrane analogy is frequently used to construct shells. An inflated membrane may be covered by a layer of cement, sometimes reinforced by a wire mesh. When the concrete sets, the membrane is deflated and removed. Surface shapes are classified as rotational, translational and ruled surfaces. A plane curve rotated around an axis produces a rotational surface, such as s sphere, a cone or an onion dome. A translational surface is generated by moving a plane curve over another plane curve. Hyperbolic paraboloids, toroids, and half cylinders are translational surfaces. Some of these shapes may be constructed as ruled surfaces when a straight line is moved along a curved path. A hyperboloid of one sheet a cone or a cylinder may be generated in this manner. Surfaces may also be classified as developable
(synclastic) and non-developable (anticlastic). A developable surface may be flattened while a non-developable can only be flattened if cut; they are stronger than developable shapes.
Long cylindrical shells act as long beams with compression in their top layers and tension along the lower edges. They are usually supported on rigid end frames by shear forces. Domes and cylinders may also be constructed with latticed bars covered by some surface material. A large variety of shell structures are illustrated based on the combination of shells discussed. Their uses as roofs, walls and dams are described.
Summary of Ideas Presented in Chapter 12:
Membranes, shells, and domes are form-resistant structures that can carry loads because they are curved surfaces.
If a membrane that carries loads by tension stresses is turned upside down, it becomes a shell or a dome carrying loads by compression just like an upside-down cable has the shape of an arch. The materials of shells, domes, and arches must be able to withstand compressive stresses.
As was the case for membranes, shells and domes have curvatures at every point, with maximum and minimum values in principal directions.
Surfaces can be developable, that is, they may be flattened without cutting them, or undevelopable that have to be cut in order to be flattened.
Curvatures may be downward (positive), dome or cylindrical shaped or upward (negative), bowl-shaped at every point on the surface.
Others may have an upward curvature in one principal direction and downward in the other.
Curved surfaces may be generated by rotating a plane curve around an axis, by moving a line along a curve or a curve along a line.
Shallow circular domes have arch action along both meridians and parallels. They are in compression.
High-rise domes are compressed along meridians, but their parallels are mostly in tension. Just like for arches their base needs buttressing to prevent it from moving outward. As a consequence bending stresses are also present along with shear,
Cylindrical shells have only one curvature. Along the curved surface, it has arch action, but a long cylinder acts similarly to a beam.
Saddle shells have positive curvatures along one principal direction and negative in the other. Along the positive curvatures, arch action and compression prevail while the negative cable action supports the arches in tension.
Reticulated domes are similar to space frames and are constructed of triangular or hexagonal elements.
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