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Frames and grids are structural systems composed of linear elements that distribute loads in a linear pattern. Frames are composed of columns and beams, in the form of portals, that provide paths for the distribution of loads. Frames can be built of steel, or in combinations of steel and reinforced concrete. In cases where steel and reinforced concrete are combined, the beams are usually steel and the reinforced concrete is a secondary slab, cast insitu, of either lightweight concrete on metal decking or precast concrete with an insitu topping spanning between the steel beams. Where the frame and the slab are built entirely of reinforced concrete a number of alternatives are available: one-way reinforced concrete slabs on beams spanning one way (most appropriate for rectangular grids with an aspect ratio of 2:1 or greater); two-way, reinforced-concrete slabs spanning to two-way-spanning beams (gives lower self weight and therefore longer spanning capability on square grids with an aspect ratio between 1:1 and 2:1) and a two-way-spanning, reinforced-concrete, flat slab spanning onto columns (most economical in terms of formwork and results in minimum depth of structure and minimized story height on square grids with an aspect ratio of less than 2:1). System considerations have not specifically addressed resistance to lateral loads. One-way frames, built of steel, timber, or reinforced concrete, direct the primary bending moments along a single axis running the length of a beam resting on two columns at either end. These primary portals stand parallel to one another, with a smaller secondary set of beams spanning between them at closer centers. Slabs can be added to the frame, and they can be designed to act independently or, with the addition of studs, channels or shear connectors, compositely with the supporting beams. In one case the slab spans the short distance, while in the other the compositely acting slab and beam provide the support. Two-way frames direct the primary bending moments along two axes, laterally and longitudinally, in the form of a grid of beams spanning a set of columns. Diagrids are single-layer grids constructed from elements which can be made of diagonal steel, steel-reinforced concrete, or lightweight timber beams that distribute axial loads across the lines of a diamond or triangulated grid. Loads are distributed regularly or irregularly, according to whether the angles that form the diamond geometry are regular or change along the surface of the grid. A diagrid – a grid with diagonal geometry – used in a vertical format can thus eliminate the need for large corner columns by providing lateral support, and produce shapes that are crystalline or smooth, for example in high-rise structures. Grid shells are single-layer grids that distribute loads in two directions. They are constructed from a grid or a lattice, in wood or steel, composed of linear members such as tubes or solid members which are continuous, passing above and below each other at the nodes, or discontinuous, with bolted or welded connections. Grid-shells derive their strength from their double curvature, where two grids or lattices are combined to form a rigid shell, accommodating large spans and many shapes. Space frames are double-layer grids that distribute loads in three directions. They are constructed from interlocking steel struts in a geometric pattern, often square, pentagonal or hexagonal, forming interlocking square pyramids or tetrahedral pyramids. These are composed of diagonal struts that join the upper and lower grids, and have either a single unit length or change the lengths of the struts to curve the overall structure or incorporate other geometrical shapes. Space frames transmit loads in tension and compression along the length of each strut, thereby uniting the two layers into one that has strength along the longitudinal, perpendicular, and diagonal axes. Accordingly, space frames are lightweight yet rigid and multidirectional, accommodating long spans and all types of shapes.
SYSTEM
Grids and Frames
One-Way Frame
Two-Way Frame
Grid-Slab
Diagrid
Double Layer Grid
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50
Grids and frames tessellate along horizontal, vertical and curved axes of growth to produce horizontal, vertical or curved forms. Horizontal tessellation can occur within any subsystem of grids and frames because the protogeometry of all such systems is capable of distributing loads along the horizontal axis. Horizontal tessellation results in shed-like forms that may vary in section and in ground plan or enclosing roof plan. Such variations are a result of changes in the grid or frame as it grows along the horizontal axis. Vertical tessellation can occur within any system of diagrids, two-way frames, and grid-slab frames because they are all capable of distributing loads along more than one axis. The capacity to respond to bending moments in multiple axes is necessary in vertical forms where lateral forces and vertical loads have to be directed simultaneously. The protogeometry of a diagrid is capable of vertical tessellation because the diagonal grid can distribute loads in both vertical and lateral (horizontal) directions. The vertical tessellation of diagrids can result in forms in which variations in the profile and surface geometry can be produced by changes in the scale of the diagrid or in the geometry of the enclosing plan along the vertical axis of growth. The protogeometry of a two-way frame, in the form of a grid of beams resting on columns, enables it to grow both horizontally and vertically because of its capacity to stack vertically while responding to bending moments. Vertical tessellation of the two-way frame can result in forms that vary in profile, ranging from a regular extrusion to staggered vertical growth. The protogeometry of a grid-slab frame, in the form of a slab and grid, enables it to grow both horizontally and vertically because of its capacity to stack vertically while distributing the bending moments in three directions (two of the directions within the gridded slab and a third one through the columns). However, the resulting vertical forms can vary in sectional and surface profile, from a symmetrical extruded structure to an asymmetrical faceted or smooth structure. Curved tessellation can occur within all grid and frame subsystems, ranging from a regular array of portals, resulting in a dome, to the eccentric volumes with irregular curved profiles which can result from a diagrid, a two-way grid, a grid shell, or a three-way space frame. Curved tessellation produces irregular and differentiated forms which in some cases additional structural strength is found when the load-bearing lines of these frames approximate to structural catenaries, which are closer to arches or domes. The protogeometry of a diagrid, a two-way grid, a grid shell and a grid slab can produce ruled or curved gridded surfaces with gradual and changing curvatures that distribute loads in a way that is similar to the distribution of loads in structural catenaries, arches or domes. The protogeometry of a three-way space frame can produce curved forms with differing degrees of curvature by varying the orientation of the individual pyramidal cells.
TESSELLATION
Grids and Frames
Horizontal Tessellation
Vertical Tessellation
Curved Tessellation
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52
The base unit of a one-way frame is composed of a rectangular grid of columns supporting a pair of primary, parallel beams along the column lines which in turn support secondary beams that span between them at much closer intervals than the centers of the columns. One-way frames direct the primary bending moments along the beam lines both primary and secondary. The loads travel from secondary to primary beams, and then to the columns. The purpose of using such a system is to cover the largest possible area with the most uniform and minimal depth of floor structure. This requires a balance between grid aspect ratio, structural depth of primary and secondary elements and the relative spans of each. For uniformity of floor depth the primary beam will usually span the shorter distance of a rectangular grid. This distribution of loads along the lines of the steel, concrete or wooden frame embeds the one-way frame with an optical affect of lightness and hierarchy that remains consistent within any space it defines. The reflection of sound by a slab or surface that is supported by a frame can be modified by the characteristics of the frame. The degree to which the sound is modified corresponds to the degree of openness and lightness of the frame, and to the direction and spectrum of the sound. For example, a flat slab that would otherwise have an affect of specularity can, in conjunction with a frame, have one of diffusion. A frame together with its overlaid surfaces can have an affect of specularity, focusing or diffusion, depending on the degree of its openness and lightness, and the contours of the surfaces. A one-way frame usually modifies the reflection of sound by the supported slab or surface to a far greater extent if the sound is travelling orthogonally in relation to the frame than if it is travelling parallel to it. The one-way frame is flexible in several ways: Span: Because the protogeometry of a one-way frame is not fixed to an exact dimension but works with approximate depths to span ratios of 1:15~20, it allows for a range of spans from column to column. Accordingly, the base unit can produce a variety of plan forms in single structures by adopting different sizes of span. Depth: The structural strength of the beam increases as the depth of its section increases. Increasing the depth of the beam therefore increases the rigidity of the system and allows for longer spans. Flanges: When the frame is made of steel, the flanges of the beam as well as the depth of the web can be adjusted in scale in response to the loading requirements. Profile: One-way frames can tessellate to produce forms that are horizontal (mats or sheds) or curved (domes). Because the base unit of the one-way frame can vary as it tessellates, it can allow for both regular and irregular profiles. Affect: The affective properties of a one-way frame can be multiplied when the base unit imbricates or intertwines with external factors, such as asymmetries that respond to the physical constraints of the site, environmental considerations, programmatic requirements, etc. As a result, in addition to lightness and griddedness, a one-way frame can transmit other optical affects, including openness, freedom, segmentation, bending, porosity, orthogonality, diagonality, enclosure, rippling. A one-way frame can modify or dominate the acoustical affect of an overlaid surface by adding an affect of directional diffusion.
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BASE UNIT
One-Way Frame
primary beams
secondary beams
X
Z
A
B Z = approx. 20
X =
A 15 ~ 20
Ratios assume pin connections for the columns
B
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 AFFECT
Openness, Lightness, Freedom, Horizontality, Specularity
55
TESSELLATION / SYSTEM
Horizontal / One-Way Frame S.R. CROWN HALL, IIT
L. MIES VAN DER ROHE, PACE ASSOCIATES
perpendicular cross-beam
CHICAGO, USA
1950–56
18 ft
secondary column primary column 120 ft
60 ft
One-way portal frame allows the space to be uninterrupted by any structure, transmitting an affect of freedom
The construction of Crown Hall uses the simple repetition of a one-way frame portal. Since the beams of the portal frame appear externally, the interior space transmits an affect of openness and lightness.
Crown Hall transmits an acoustic affect of specularity.
mullions glazing The simple repetition of the one-way frame portal along a single axis creates a form that transmits an affect of horizontality.
120 ft
20 ft
60 ft
60 ft
60 ft
20 ft
220 ft
Crown Hall is formed by the tessellation of a one-way portal frame, utilizing three full bays and two smaller bays one-third of the width of the full bays. In this case the base unit varies in scale only, but variations in section are also possible by changing the height of the portals, which would change the profile of the resulting form. Crown Hall transmits an optical affect of orientation, lightness, freedom and horizontality. The acoustic affect of specularity is removed from the ceiling by the sound-absorbent suspended ceiling. The specularity of the glazing and other non-absorbent surfaces remains.
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 AFFECT
Segmentation, Lightness, Bending, Horizontality, Specularity
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TESSELLATION / SYSTEM
Horizontal / One-Way Frame The tessellation of the one-way frame base units creates a non-extruded horizontal form that is divided internally into compartments defined by the structural columns, transmitting an affect of segmentation and horizontality.
perpendicular cross-beam
longitudinal cross-beam column
2x x 2x x
2x
2x x
diagonal portal
This space defined by the light-framed structure transmits an acoustic affect of specularity with some diffusion from the frame and secondary beams.
orthogonale portal
This form is constructed from a single, one-way, steel-frame base unit which is angular in its protogeometry, transmitting an affect of lightness and bending.
The horizontal tessellation of the two one-way-frame base units creates a corrugated perimeter and an affect of bending.
4x
2x
2x
2x
This horizontal form is produced by the tessellation of a one-way portal frame resting on columns at mid-span, with the one-way orthogonal portal varying, as it repeats, to become diagonal. The combination of orthogonal and diagonal portals changes both the external and the interior perimeter profiles of the space structure. This form of enclosure transmits an optical affect of segmentation, lightness, bending and horizontality, and an acoustical affect of specularity and some diffusion.
x
58
AFFECT
Porosity, Lightness, Horizontality, Specularity , Diffusion
59
TESSELLATION / SYSTEM
Horizontal / One-Way Frame roof aperture perpendicular cross-beam
column 2x
longitudinal cross-beam
2x x
0.5x 0.5x 2x x
This form is constructed using six different one-way-frame base units whose beams and columns are arranged differently so, when assembled, they leave openings in between which act as skylights.
This space defined by the light-framed structure transmits an acoustic affect of specularity with some diffusion from the frame and secondary beams.
The number and size of the openings can vary and also be staggered across the bays of the frame.
2x 0.2x
0.2x
Variations in the position of the column and beam in each base unit create openings in the roof surface which can become skylights or courtyards, bringing natural light or ventilation into deep-plan shed forms.
x
2x
2x x
0.5x
0.5x 2x x
4x
The introduction of skylights within the orthogonal form establishes a unique horizontal reference to the space below, transmitting affects of horizontality, porosity and lightness.
2x 0.5x 0.5x
6x
This horizontal form is produced by the tessellation of six different one-way-frame base units, introducing a staggered arrangement of supporting columns which results in the displacement of the secondary beams and openings on the roof surface. This form of enclosure transmits an optical affect of porosity, orthogonality, lightness and horizontality, and an acoustical affect of specularity and some diffusion.
x
60
AFFECT
Bifurcation, Lightness, Diagonality, Specularity
61
TESSELLATION / SYSTEM
Horizontal / One-Way Frame perpendicular cross-beams
longitudinal cross-beam
The number and size of the openings can vary and also be staggered across the bays of the frame.
2x
column
h=a 2x x
2x h=b 2x x
This space defined by the light-framed structure transmits an acoustic affect of specularity.
The tessellation of six varying base units creates a continuous surface with openings transmitting affects of bifurcation and lightness.
2x h=c 2x
Skylights within the roof surface introduce an oblique orientation within the space and transmit an affect of diagonality.
Skylights transmit an affect of lightness to the space below.
x
The resulting roof surface introduces a topography that can serve for drainage.
2x
h=d 2x x
2x h=e 2x x 4x
2x h=f 2x x
6x
This horizontal form is produced by the tessellation of six different one-way-frame base units that vary in the height and profile of the portals. The tessellation of the six base units leaves gaps in the resulting roof surface that can introduce light in the interior, and a diagonal orientation within  the space. This form of enclosure transmits an optical affect of bifurcation, lightness, and diagonality, and an acoustical affect of specularity.
62
AFFECT
Distortion, Lightness, Segmentation, Specularity , Diffusion
63
TESSELLATION / SYSTEM
Horizontal / One-Way Frame longitudinal cross-beam perpendicular cross-beams
This shed form is produced by the horizontal tessellation of six variations of a one-way-frame base unit that together transmit affects of distortion, lightness and segmentation.
a
b
c x
d x
This space defined by the light-framed structure transmits an acoustic affect of specularity with some directional diffusion from the secondary structure.
Changes in section are produced by changing the heights of the columns. The changes in section can accommodate variable height and volumetric program requirements. The resulting roof surface introduces a topography that can provide drainage.
The changes in section allow for a number of elevation heights to be introduced. These can respond to program requirements such as the need for more or less natural light.
e
Varying disposition of columns and beams of the frame system creates an irregular plan perimeter that can correspond to different contingencies of the physical context around the form or cater for different uses of the interior.
x
f
x
g
x
3x
i
h x x x
x
x
x
g>h>i e<f<g e<d<c
This horizontal form is produced by the tessellation of a one-way-frame portal base unit, with a number of portals changing in section as irregular changes are introduced to the overall height of the perimeter columns, varying the height of the interior space and the topography of the roof. This form of enclosure transmits an optical affect of distortion, lightness, and segmentation, and an acoustical affect of specularity with some diffusion.
x
64
The base unit of a two-way frame is composed of a lattice of beams resting on four or more columns. Bending moments in a two-way frame are distributed in two directions, producing a rigid structure. Two-way frames need to have an aspect ratio of 2:1 or less. The distribution of loads along lines of steel or laminated timber embeds the two-way frame with an optical affect of lightness and striation that remains consistent within all spaces it defines. The reflection of sound by a slab or surface that is supported by a frame can be modified by the characteristics of the frame. The degree to which the sound is modified corresponds to the degree of openness and lightness of the frame, and to the direction and spectrum of the sound. For example, a flat slab that would otherwise have an affect of specularity can, in conjunction with a frame, have one of diffusion. A frame together with its overlaid surfaces can have an affect of specularity, focusing or diffusion, depending on the degree of its openness and lightness, and the contours of the surfaces. The two-way frame is flexible in several ways: Span: As the protogeometry of the two-way frame is not fixed to an exact dimension but works with approximate depths to span ratios of 1:25-30 in a grillage, the frame can be thinner overall because it bridges from column to column. Accordingly, the base unit can use spans of different sizes to produce a variety of non-extruded plan forms in single structures. Depth: The depth of the beam’s web varies according to the size of the span, with the structural strength and stiffness of the beam increasing in proportion to the depth of its section. Profile: Two-way frames can tessellate to produce horizontal forms (mats or sheds), or, when combined with steel-decked thin concrete slabs, they can be stacked to produce vertical forms (towers). As it tessellates, the base unit of a one-way frame can vary to allow for regular or irregular perimeters and profiles. Affect: The affective properties of a two-way frame can be multiplied when the base unit imbricates or intertwines with external factors, such as asymmetries that respond to the physical constraints of the site, environmental considerations, programmatic concerns, etc. As a result, in addition to lightness and striation, a structure based on a two-way frame can transmit other optical affects, including boundlessness, repetition, differentiation, enclosure, centeredness, weightiness, extrusion, stacking, hinging, continuity. A two-way frame can modify or dominate the acoustical affect of the overlaid surface by adding an affect of diffusion.
BASE UNIT
Two-Way Frame
Z
B A
Z=
the greater of A or B 25 ~ 30
A = B so then distribution is even If A = 2B all loads go to A as loads follow shortest route A = approx. (25–30) Z Span (A,B) to Depth (Z) Ratio = 1:25–30
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66
AFFECT
Openness, Coffering, Abstraction, Horizontality, Diffusion
67
TESSELLATION / SYSTEM
Horizontal / Two-Way Frame NEW NATIONAL GALLERY
L. MIES VAN DER ROHE
BERLIN, GERMANY
Openings can be introduced in the roofâ&#x20AC;&#x2122;s surface to increase its porosity. These can respond to programmatic requirements such as natural light or ventilation.
Since the load-bearing columns of the New National Gallery frame structure are placed on the periphery of the form to liberate the interior from divisions, its part (base unit) and whole are one and the same thing.
1962â&#x20AC;&#x201C;68
7.9
50.6
The section is determined by the height of the columns which in this instance are equal.
The exposed beams of the two-way structure are deep to cater for the long span and accordingly transmit an affect of coffering and openness.
The New National Gallery frame structure transmits an acoustic affect of diffusion.
50.6
50.6
The New National Gallery is formed by the simple repetition of a regularly spaced two-way frame. In this case the base unit extends simply across the space, but this type of structure can be varied by changing the height of the columns or the depth of the beams, which would in turn vary the profile of the resulting form. The New National Gallery transmits an optical affect of openness, abstraction, horizontality and coffering, and an acoustical affect of diffusion.
50.6
68
AFFECT
Undulation, Coffering, Diffusion
69
TESSELLATION / SYSTEM
Horizontal / Two-Way Frame ruled surface load-bearing column
a
b
b
Two variations of a simple two-way frame base unit, exposed to the interior, results in a continuous roof surface that transmits affects of undulation and coffering.
x
x
h=a
h=b
a
a
x
x h=b
variable elevation area
Varying the height of the two-way frame beams produces undulation in the perimeter elevation.
This space defined by the framed structure transmits an acoustic affect of diffusion.
The ruled surface defined by the beams of the two-way frame introduces multidirectionality to the space below, which would otherwise be bidirectional on account of the orthogonal grid of the beams and load-bearing columns.
2x
3x
This horizontal form is produced by the horizontal tessellation of two two-way-frame base units that vary in their elevational profiles. Together they create a continuous roof surface which transmits an optical affect of undulation and coffering, and an acoustical affect of diffusion.
70
AFFECT
Variegation, Coffering, Diffusion
71
TESSELLATION / SYSTEM
Horizontal / Two-Way Frame load-bearing column
curv. = d
curvature = b c
The single surface created by the horizontal tessellation of six variations of the two-way-frame base unit transmits affects of variegation and coffering.
curv. = a
x
x curv. = c
x
x
a
a b
roof surface aperture This space defined by the framed structure transmits an acoustic affect of diffusion which dominates any affect of focusing from concave areas of the overlaid surface.
cross beam with variable curvature
x
x
c
The distribution of different curvatures within the grid of the surface results in a varying section and different column heights. x
x
x
x
2x
x
x
3x
This horizontal form is produced by the horizontal tessellation of several base units composed of a regularly spaced grid of beams, the curvature of which varies irregularly as the base unit repeats, producing an increase or decrease in the curvature of the roof, and breaks between each of the curved parcels of which it is composed. These changes in height result in irregularly shaped apertures because of the differences in curvature from one parcel to the next of the surface. The undulating and discontinuous roof surface transmits an optical affect of variegation and coffering, and an acoustical affect of diffusion.
72
AFFECT
Continuity, Coffering, Diffusion
TESSELLATION / SYSTEM
Horizontal / Two-Way Frame
73
curved beams surface column with variable radius aperture radius a
The radial distribution of the beams results in surface continuity from the roof to the load-bearing columns, which are large enough to act as exterior rooms, transmitting affects of coffering and continuity.
a
x
x
x
x
aperture radius b
x
This space defined by the framed structure transmits an acoustic affect of diffusion.
x
aperture radius c
2x
The flexibility which is possible in the shaping of the beams of a two-way frame allows the introduction of courtyards of differing scales. These can be given distinct programmatic uses and can also be used to bring natural light and ventilation into the space surrounding each.
3x
This horizontal form is produced by the tessellation of a base unit composed of a grid of two-way beams arranged in a radial pattern and bent to produce, simultaneously, the roof and its supporting walls. The radial bent frame produces circular void spaces that vary in scale as the base unit tessellates along the horizontal extension, introducing different degrees of porosity to the plan. This two-way frame transmits an optical affect of continuity and coffering, and an acoustical affect of diffusion.
74
AFFECT
Extrusion, Verticality, Lightness, Tactility
TESSELLATION / SYSTEM
Vertical / Two-Way Frame SEAGRAM BUILDING
L. MIES VAN DER ROHE AND P. JOHNSON; KAHN AND JACOBS
NEW YORK CITY, USA
1954â&#x20AC;&#x201C;58
bronze I beam steel structure columns elevator cores stone cladding curtain wall I beams Vertical mullions are made of bronze to trigger an affect of tactility.
floor-to-ceiling glazing 8.5
8.5
Vertical mullions are attached to the vertical members of the two-way frame to give the vertical lines on the elevations more depth, triggering the sensation of verticality. 157
The two-way-frame base unit is simply repeated vertically, transmitting an affect of extrusion and verticality. The frame is left open as the tower meets the ground to give the form a sensation of lightness.
The exterior cladding on the north and south sides of this bay are in stone, which also wraps around the corner to cover the corner and first window bay along the exterior perimeter that faces the elevator core on the east elevation. 34.25 8.5
8.5
44.25
The Seagram Headquarters building is formed by the vertical tessellation of a base unit composed of a single floor made of steel columns and beams that behave like a two-way frame, working in conjunction with the elevator core. The base unit of the Seagram tower simply repeats vertically, but is also capable of being staggered or internally differentiated by the unequal distribution of the columns, which in turn varies the degree of transparency of the resulting vertical tube. The Seagram Headquarters building transmits an optical affect of extrusion, lightness, tactility and verticality.
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76
AFFECT
Cantilevering, Verticality, Stacking, Floating, Rotation, Differentiation
TESSELLATION / SYSTEM
Vertical / Two-Way Frame 111 FIRST STREET
OFFICE FOR METROPOLITAN ARCHITECTURE; WSP CANTOR SEINUK
The rotation of the floors can optimize views available along different heights as well as varying the solar orientation of floor plates which offer different occupancy models.
The rotation of floor plates relative to one another creates a tower form that transmits affects of cantilevering, stacking and rotation.
The base unit varies as it tessellates, creating a form that begins with a stack of square-shaped plans, followed by a stack of rectangularshaped plans, followed by a stack of smaller rectangular plans, thereby transmitting an affect of differentiation. Different plan types also offer a variety of occupancy models.
592 ft
2006
JERSEY CITY, USA
164 ft 72 ft
265 ft 68 ft
164 ft 164 ft
The profile of the three stacked sets of floor plates transmits simultaneously affects of stacking and floating.
401 ft 207 ft
68 ft
164 ft
164 ft
164 ft
265 ft
72 ft
Typical plan: Lower stack
Typical plan: Middle stack
Typical plan: Upper stack
The proposal for 111 First Street in New York is formed by the vertical tessellation of a base unit composed of a single floor made up of steel columns and beams that behave like a two-way frame, working in conjunction with the elevator core. The base unit varies in scale and changes orientation as it tessellates, creating a vertical form that shifts orientation three times, transmitting affects of cantilevering, verticality, stacking, floating, rotation and differentiation.
77