Some trusses have bars with zero force under certain loads.  The example here has zero force in bars HG, LM, and PG under the given  load.  Under asymmetrical loads these bars would not be zero and, therefore, cannot be eliminated.  Bars with zero force have  vectors of zero length in the equilibrium polygon and, therefore, have both letters at the  same location.  

Tension and compression in truss bars can be visually verified by deformed shape (4),  exaggerated for clarity.  Bars in tension will elongate; bars in compression will shorten.  

In the truss illustrated the top chord is in compression; the bottom chord is in tension;  inward sloping diagonal bars in tension; outward sloping diagonal bars in compression.

Since diagonal bars are the longest and, therefore, more likely subject to buckling, they  are best oriented as tension bars.

1 Truss diagram
2 Force polygon
3  Tabulated bar forces (+ implies tension, - compression)
4  Deformed truss to visualize tension and compression bars
A  Bar elongation causes tension
B  Bar shortening causes compression


Graphic truss analysis (Bow’s Notation) is a method to find bar forces using graphic vectors as in the following steps:

A  Draw a truss scaled as large as possible (1) and compute the reactions as for beams (by moment method for asymmetrical trusses).
B  Letter the spaces between loads, reactions, and truss bars.  Name bars by adjacent letters: bar BH between B and H, etc.
C  Draw a force polygon for external loads  and reactions in a force scale, such as  1”=10 pounds (2).  Use a large scale for accuracy.  A closed polygon with head-to-tail arrows implies equilibrium.  Offset the reactions to the right for clarity.
Draw polygons for each joint to find forces in connected bars.  Closed polygons
with head-to-tail arrows are in equilibrium.  Start with left joint ABHG.  Draw a
vector parallel to bar BH   through B in the polygon.  H is along BH.  Draw a vector
parallel to bar HG through G to find H at intersection BH-HG.
E  Measure the bar forces as vector length in the polygon.
F  Find bar tension and compression.  Start with direction of   load  AB  and  follow  polygon ABHGA with head-to-tail arrows.  Transpose arrows to respective bars in  the truss next to the joint.  Arrows pushing toward the joint are in compression; arrows pulling away are in tension.  Since the arrows reverse for adjacent joints,  draw them only on the truss but not on the polygon.

G  Draw equilibrium arrows on opposite bar ends; then proceed to the next joint with  two unknown bar forces or less (3).  Draw polygons for all joints (4), starting with  known loads or bars (for symmetrical  trusses half analysis is needed).

1 Truss diagram
2  Force polygon for loads, reactions, and the first joint polygon
3  Truss with completed tension and compression arrows
4  Completed force polygon for left half of truss
5  Tabulated bar forces (- implies compression)



First used by Leonardo da Vinci, graphic vector analysis is a powerful method to analyze  and visualize the flow of forces through a structure.  However, the use of this method is restricted to statically determinate systems.  In addition to forces, vectors may represent  displacement, velocity, etc.  Though only two-dimensional forces are described here, vectors may represent forces in three-dimensional space as well.  Vectors are defined by  magnitude, line of action, and direction, represented by a straight line with an arrow and  defined as follows:

Magnitude is the vector length in a force scale, like 1” =10 k or 1 cm=50 kN
Line of Action is the vector slope and location in space
Direction is defined by an arrow pointing in the direction of action

1  Two force vectors P1 and P2 acting on a body pull in a certain direction.  The resultant R is a force with the same results as P1 and P2 combined, pulling in the  same general direction.  The resultant is found by drawing a force parallelogram [A]  or a force triangle [B].  Lines in the vector triangle must be parallel to corresponding  lines in the  vector plan [A].  The line of action of the resultant is at the intersection  of P1 / P2 in the vector plan [A].  Since most structures must be at rest it is more  useful to find the  equilibriant E that puts a set of forces in equilibrium [C].  The  equilibriant is equal in magnitude but opposite in direction to the resultant.  The  equilibriant closes a force triangle with all vectors connected head-to-tail.  The line  of action of the equilibriant is also at the intersection of P1/P2 in the vector plan [A].

2  The equilibriant of three forces [D] is found, combining interim resultant R1-2 of  forces P1 and P2 with P3 [E].  This process may be repeated for any number of  forces.  The interim resultants help to clarify the process but are not required [F].  The line of action of the equilibriant  is located at the intersection of all forces in the  vector plan [D].  Finding the equilibriant for any number of forces may be stated as  follows:

The equilibriant closes a force polygon with all forces connected head-to-tail,  and puts them in equilibrium in the force plan.

3  The equilibriant of forces without a common cross-point [G] is found in stages:   First the interim resultant R1-2 of P1 and P2 is found [H] and located at the  intersection of P1/P2 in the vector plan [G].  P3 is then combined with R1-2 to find  the equilibriant for all three forces, located at the intersection of  R1-2 with P3 in the  vector plan.  The process is repeated for any number of forces.



To find reactions for asymmetrical beams:

•  Draw an abstract beam diagram to illustrate computations
•  Use Σ M = 0 at one support to find reaction at other support
•  Verify results for vertical equilibrium

1 Floor framing
2  Abstract beam diagram

Support reactions:

Alternate method (use uniform load directly)

1  Simple beam with point loads

2  Beam with overhang and point loads 

3  Beam with uniform load and point load (wall)


The diagrams show common types of support at left and related symbols at right.  In  addition to the pin and roller support described above, they also include fixed-end  support (as used in steel and concrete moment frames, for example).


For convenience, support types are described  for beams, but apply to other horizontal  elements, like trusses, as well.  The type of support affects analysis and design, as well  as performance.  Given the three equations of statics defined above, ΣH=0, ΣV=0, and  ΣM=0, beams with three unknown reactions are considered determinate  (as described  below) and can be analyzed by the three static equations.  Beams with more than three  unknown reactions are considered  indeterminate and cannot be analyzed by the three  static equations alone.  A beam with two pin supports (1 has four unknown reactions, one  horizontal and one vertical reaction at each support.  Under load, in addition to bending,  this beam would deform like a suspended cable in tension, making the analysis more  complex and not possible with static equations.

By contrast, a beam with one pin and one roller support (2) has only three unknown  reactions, one horizontal and two vertical.  In bridge structures such supports are quite  common.  To simplify analysis, in building structures this type of support may be  assumed, since supporting walls or columns usually are    flexible enough to simulate the  same behavior as one pin and one roller support.  The diagrams at left show for each  support on top the physical conditions and below the symbolic abstraction.

Beam with fixed supports at both ends subject to bending and tension
2  Simple beam with one pin and one roller support subject to bending only
3  Beam with flexible supports, behaves like a simple beam

Simple beams, supported by one pin and one roller, are very common and easy to  analyze.  Designations of roller- and pin supports are used to describe the structural  behavior assumed for analysis, but do not always reflect the actual physical support.  For  example, a pin support is not an actual pin but a support that resists horizontal and  vertical movement but allows rotation.  Roller supports may consist of Teflon or similar  material of low friction that allows horizontal movement like a roller.


Braced frames resist gravity load in bending and axial compression, and lateral load in axial compression and tension by triangulation, much like trusses.  The triangulation results in greater stiffness, an advantage to resist wind load, but increases seismic  forces, a disadvantage to resist earthquakes.  Triangulation may take several  configurations, single diagonals, A-bracing, V-bracing, X-bracing, etc., considering both  architectural and structural criteria.  For example, location of doors may be effected by  bracing and impossible with X-bracing.  Structurally, a single diagonal brace is the  longest, which increases buckling tendency  under compression.  Also the number of  costly joints varies: two for single diagonals, three for A- and V-braces, and five joints for  X-braces.  The effect of bracing to resist load is visualized through amplified deformation  as follows:

1  Single diagonal portal under gravity and lateral loads
2  A-braced portal under gravity and lateral load
3  V-braced portal under gravity and lateral load
4  X-braced portal under gravity and lateral load
5  Braced frame building without and with lateral load

Note: deformations and forces reverse under reversed load


Moment frames resist gravity and lateral load in bending and compression. They are derived from post-and beam portals with moment resisting beam to column connections (for convenience refered to as moment frames and moment joints).  The effect of moment joints is that load applied to the beam will rotate its ends and in turn rotate the attached columns.  Equally, load applied to columns will rotate their ends and in turn  rotate the beam.  This mutual interaction makes moment frames effective to resist lateral load with ductility. Ductility is the capacity to deform without breaking, a good property to resist earthquakes, resulting in smaller seismic forces than in shear walls and braced frames.  However, in areas with prevailing wind load, the greater stiffness of shear walls and braced frames is an advantage,  The effect of moment joints to resist loads is  visualized through amplified deformation as follows:

1  Portal with pin joints collapses under lateral load
2  Portal with moment joints at base under lateral load
3  Portal with moment beam/column joints under gravity load
4  Portal with moment beam/column joints under lateral load
5  Portal with all moment joints under gravity load
6  Portal with all moment joints under lateral load
7  High-rise moment frame under gravity load
8  Moment frame building under lateral load
I  Inflection points (zero bending between negative and positive bending

Note: deformations reverse under reversed load


Cantilevers resist lateral load primarily in bending.  They may consist of single towers or  multiple towers.  Single towers act much like trees and require large footings like tree  roots to resist overturning.  Bending in cantilevers increases from top down, justifying  tapered form in response.

1  Single tower cantilever
2  Single tower cantilever under lateral load
3  Twin tower cantilever
Twin tower cantilever under lateral load
5  Suspended tower with single cantilever
6  Suspended tower under lateral load

Shear Walls Systems

As the name implies, shear walls resist lateral load in shear.  Shear walls may be of wood, concrete or masonry.  In the US the most common material for low-rise  apartments is light-weight wood framing with plywood or particle board sheathing. Framing studs, spaced 16 or 24 inches, support gravity load and sheathing resists lateral  shear.  In seismic areas concrete and masonry shear walls must be reinforced with steel  bars to resist lateral shear.

1  Wood shear wall with plywood sheathing
2  Light gauge steel shear wall with plywood sheathing
3  Concrete shear wall with steel reinforcing
4  CMU shear wall with steel reinforcing
5  Un-reinforced brick masonry (not allowed in seismic areas)
8  Two-wythe brick shear wall with steel reinforcing


Vertical systems transfer the load of horizontal systems from roof to foundation, carrying  gravity and/or lateral load.  Although they may  resist gravity or lateral load only, most  resist both, gravity load in compression, lateral load in shear.  Walls are usually designed  to define spaces and provide support, an appropriate solution for apartment and hotel  buildings.  The four systems are:

1  Shear walls (apartments / hotels)
2  Cantilever (Johnson Wax tower by F L Wright)
Moment frame
4  Braced frame

A  Concrete moment resistant joint Column re-bars penetrate beam and beam re-bars penetrate column)  B  Steel moment resistant joint (stiffener plates between column flanges resist beam flange stress)


Vertical elements

Vertical elements transfer load from roof to foundation, carrying gravity and/or lateral  load.  Although elements may resist only gravity or only lateral load, most are designed to  resist both.  Shear walls designed for both gravity and lateral load may use gravity dead  load to resist overturning which is most important for short walls.  Four basic elements  are used individually or in combination to resist gravity and lateral loads

1  Wall under gravity load
2  Wall under lateral load (shear wall)
3  Cantilever under gravity load
4  Cantilever under lateral load
5  Moment frame under gravity load
6  Moment frame under lateral load
7  Braced frame under gravity load
9  Braced frame under lateral load


The funicular concept can be best described  and visualized with cables or chains,  suspended from two points, that adjust their form for any load in tension.  But funicular  structures may also be  compressed like arches. Yet, although funicular tension  structures adjust their form for pure tension under any load, funicular compression  structures may be subject to bending in addition  to compression since their form is rigid  and not adaptable.  The funicular line for tension and compression are inversely identical;  the form of a cable becomes the form of an arch upside-down.  Thus funicular forms may  be found on tensile elements.

1  Funicular tension triangle under single load
2  Funicular compression triangle under single load
3  Funicular tension trapezoid under twin loads
4  Funicular compression trapezoid under twin loads
5  Funicular tension polygon under point loads
6  Funicular compression polygon under point load
7  Funicular tension parabola under uniform load
8  Funicular compression parabola under uniform load


IBM traveling exhibit by Renzo Piano A series of trussed arches in linear extrusion form a vault space  The trussed arches  consist of wood bars with metal connectors for quick assembly and disassembly as  required for the traveling exhibit.  Plastic  panels form the enclosing skin,  The trussed  arches provide depth and rigidity to accommodate various load conditions

Suspension roof
Exhibit hall Hanover by Thomas Herzog


Trusses support load much like beams, but for longer spans.  As the depth and thus dead weight of beams increases with span they become increasingly inefficient, requiring  most capacity to support their own weight rather than imposed live load.  Trusses replace  bulk by triangulation to reduce dead weight.

1  Unstable square panel deforms under load. Only triangles are intrinsically stable polygons
2  Truss of triangular panels with inward sloping diagonal bars that elongate in tension under load (preferred configuration)
3  Outward sloping diagonal bars compress (disadvantage)
4  Top chords shorten in compression Bottom chords elongate in tension under gravity load
Gable truss with top compression and bottom tension

Warren trusses
Pompidou Center, Paris by Piano and Rogers

Prismatic trusses
IBM Sport Center by Michael Hopkins
(Prismatic trusses of  triangular cross section provide rotational resistance)

Space trusses 
square and triangular plan

Note: Two way space trusses are most effective if  the spans in the principle directions are  about equal, as described for two-way slabs above.  The base modules of trusses should  be compatible with plan configuration (square, triangular, etc.)


Horizontal systems come in two types: one way and two way. Two way systems are only  efficient for spaces with about equal span in both directions; as described below.  The  diagrams here show one way systems at left and two way systems at right

1  Plywood deck on wood joists
2  Concrete slab on metal deck and steel joists
3  One way concrete slab
4  One way beams
5  One way rib slab
6  Two way concrete plate
7  Two way concrete slab on drop panels
8  Two way concrete slab on edge beams
9  Two way beams  
10  Two way waffle slab
11 Deflection ∆ for span length L1
12 Deflection ∆=16 due to double span L2 = 2 L1

Note: Deflection increases with the fourth power of span.  Hence for double span deflection  increase 16-fold.. Therefore two way systems over rectangular plan are ineffective  because elements that span the short way control deflection and consequently have to  resist most load and elements that span the long way are very ineffective.

Rupture Length (material properties, i.e., structural efficiency)

Rupture length is the maximum length a bar of constant cross section area can be  suspended without rupture under its weight in tension (compression for concrete & masonry).

Rapture length defines material efficiency as strength / weight ratio:

R = F / λ

R = rupture length
F = breaking strength
λ = specific gravity (self weight) 

Rupture length, is of particular importance for long-span structures.  The depth of  horizontal span members increases with span.  Consequently the weight also increases  with span.  Therefore the capacity of material to span depends on both its strength and  weight.  This is why lightweight material, such as glass fiber fabrics are good for long- span structures.  For some material, a thin line extends the rupture length to account for  different material grades.

The graph data is partly based on a study of the Light weight Structures Institute, University Stuttgart, German.

Structural design for: Strength, Stiffness, Stability, Synergy

Structures must be designed to satisfy three Ss and should satisfy all four Ss of structural design – as demonstrated on the following examples, illustrated at left.

1  Strength to prevent breaking
Stiffness to prevent excessive deformation
3  Stability to prevent collapse
4  Synergy to reinforce architectural design, described on two examples:

  Pragmatic example: Beam composed of wooden boards
  Philosophical example: Auditorium design

Comparing beams of wooden boards, b = 12” wide and d = 1”deep, each.  Stiffness is  defined by the Moment of Inertia, I = b d^3/12

Note:  The same amount of material is 100 times stiffer and 10 times stronger when glued  together to transfer shear and thereby engage top and bottom fibers in compression and  tension (a system, greater than the sum of its parts).  On a philosophical level, structures  can strengthen architectural design as shown on the example of an auditorium:

•  Architecturally, columns define the circulation
•  Structurally, column location reduces bending in roof beams over 500% !


Expansion or control joints should be installed in both the structural slab portion and the protection layer. Providing for expansion only at the structural portion does not allow for thermal or structural movement of the topping slab.

This can cause the topping slab to crack, leading to membrane deterioration. Refer to Figs. 3.62 and 3.63 for proper detailing. Membranes should be adhered only to the structural deck, not to topping layers, where unnecessary stress due to differential movement between the two layers will cause membrane failure.

Expansion joint detailing for topping slab construction.
FIGURE 3.62 Expansion joint detailing for topping slab construction.
FIGURE 3.63 Expansion joint detailing for topping slab construction.
Waterproof membranes should be adequately terminated into other building enve- lope components before applying topping and protection layers. The topping is also tied into the envelope as secondary protection.

Control or expansion joints are installed along topping slab perimeters where they abut other building components, to allow for adequate movement (Fig. 3.64). Waterproof membranes at these locations are turned up vertically, to prevent water intrusion at the protection layer elevation. Refer to Fig. 3.65 for a typical design at this location.

Perimeter expansion joint detailing for sandwich-slab membranes.
FIGURE 3.64 Perimeter expansion joint detailing for sandwich-slab membranes.
Transition detailing for sandwich-slab membranes.
FIGURE 3.65 Transition detailing for sandwich-slab membranes.

When pavers are installed as the protection layer, pedestals are used to protect the membrane from damage. Pedestals allow leveling of pavers, to compensate for elevation deviations in pavers and structural slabs (Fig. 3.66).

 Pedestals permit the leveling of the walking surface on sloped structural decks using sandwich-slab membranes.
FIGURE 3.66 Pedestals permit the leveling of the walking surface on sloped structural decks
using sandwich-slab membranes.

At areas where structural slabs are sloped for membrane drainage, pavers installed directly over the structural slab would be unlevel and pose a pedestrian hazard.

Pedestals allow paver elevation to be leveled at these locations. Pedestals are manufactured to allow four different leveling applications, since each paver typically intersectsfour pavers, each of which may require a different amount of shimming (Fig. 3.67).

FIGURE 3.67 Pedestal detail.

If wood decking is used, wood blocking should be installed over membranes so that nailing of decking into this blocking does not puncture the waterproofing system. Blocking should runwith the structural drainage design so that the blocking does not prevent water draining.

Tile applications, such as quarry or glazed tile, are also used as decorative protection layers with regular setting beds and thin-set applications applied directly over membranes.

With thin-set tile installations, only cementitious or liquid-applied membrane systems are used, and protection board is eliminated. Tile is bonded directly to the waterproof membrane.

Topping slabs must have sufficient strength for expected traffic conditions. Lightweight orinsulating concrete systems of less than 3000 lb/in2 compressive strength are not recommended.

If used in planting areas, membranes should be installed continuously over a structural deck and not terminated at the planter walls and restarted in the planter. This prevents leakage through the wall system bypassing the membrane. See Fig. 3.68 for the differences in these installation methods. Figure 3.69 represents a typical manufacturer detail for a similar area.

FIGURE 3.68 Planter detailing for split-slab membrane.
FIGURE 3.69 Typical detail for planters on decks.

Using below-grade membranes for above-grade planter waterproofing is very common, especially on plaza decks. While these decks themselves are often waterproofed using the techniques described in this chapter, the planter should in itself be made completely water- proof to protect the building envelope beneath or adjacent to the planter.

Figures 3.70 and 3.71 detail the typical application methods of waterproofing above- grade planter areas. Note that each of these details incorporates the use of drainage board to drain water towards the internal planter drain. Since these areas are watered frequently, drainage is imperative, in this case, not only for waterproofing protection but also for the health of the vegetation planted in the planter.

FIGURE 3.70 Typical detailing for above-grade planter areas.

FIGURE 3.71 Typical detailing for above-grade planter areas.
Figure 3.72 shows the application of liquid membrane to planter walls as does Fig. 3.73. In the latter note how difficult the use of a sheet good system would be in this particular application. Whenever waterproofing above-grade planters with tight and numerous changes-in- plane or direction, liquid applied membranes are preferred over sheet-good systems as the preferred “idiot-proof” application. The con- tinual cutting of sheets in these smaller appli- cations results in a corresponding number of seams that emphasize the 90%/1% principle. Liquid applications are seamless and can prevent the problems associated with sheet-good installation in small planter areas.

Selection of a protected system should be based on the same performance criteria as those for materials used with below-grade applications. For example, cementitious systems are rigid and do not allow for structural movement. Sheet-goods have thickness controlled by premanufacturing but contain seams; liquid-applied systems are seamless but millagemust be controlled.


Protected membranes are used for swimming pool decks over occupied areas, rooftop pedestrian decks, helicopter landing pads, parking garage floors over enclosed spaces, balconies, and walkways. Sandwich membranes should not be installed without adequate pro- vision for drainage at the membrane elevation; this allows water on the topping slab, as well as water that penetrates the protection layer onto the waterproof membrane, to drain (Fig. 3.56). If this drainage is not allowed, water will collect on a membrane and lead to numerous problems, including freeze–thaw damage, disbonding, cracking of topping slabs, and deterioration of insulation board and the waterproof membrane. Refer to Figs. 3.57 and 3.58 for an example of these drainage requirements.

FIGURE 3.56 Prefabricated drainage layer in sandwich application. Note the insulation is spaced to
permit drainage also.

FIGURE 3.57 Dual drain installed for proper drainage of protected membrane level.
FIGURE 3.58 Schematic view of drainage requirements for sandwich-slab membranes.
For the best protection of the waterproofing membrane, a drainage layer should be installed that directs water to dual drains or terminations of the application. Water that infil- trates through the topping slab can create areas of ponding water directly on top of the membrane even if the structural slab is sloped to drains.

This ponding can be created by a variety of causes, including imperfections of the topping slab and protection layer, dirt, and debris.

To prevent this from occurring and to ensure that the water is removed away from the envelope as quickly as possible, a premanufactured drainage mat should be installed on top of the waterproofing membrane. The drainage layer can also be used in lieu of the protection layer.

However, the sandwich membrane drainage systems have one major difference: they are produced with sufficient strength to prevent crushing of the material when traffic, foot or vehicular, is applied after installation. A typical drainage mat is shown in Fig. 3.59.

It is imperative that termination detailing be adequately included to permit the drainage or weeping of water at the edges or perimeter of the sandwich slab installation. This is usually provided by installing an edge-weep system and counter-flashing, as shown in Fig. 3.60. Or if the structural slab is sloped to drain towards the edges of the slab, a drain and gutter system should be provided as shown in Fig. 3.61.

Note that in each of these details the drainage is designed to sweep water directly at the pre- fabricated drainage board level. The drainage board should be installed so that the channels created are all aligned and run towards the intended drainage. The entire purpose of the various drainage systems in a sandwich-slab application (drainage mat, deck drains, and edge drainage systems) can be entirely defeated if the pre-fabricated drainage board is not installed correctly.

FIGURE 3.61 Drainage system detailed into gut-
ter system.


With certain designs, horizontal above-grade decks require the same waterproofing protection as below-grade areas subjected to water table conditions. At these areas, membranes are chosen in much the same way as below-grade applications. These installations require a protection layer, since these materials cannot be subjected to traffic wear or direct expo- sure to the elements. As such, a concrete topping slab is installed over the membrane, sandwiching the membrane between two layers of concrete; hence the name sandwich-slab membrane. Figure 3.51 details a typical sandwich-slab membrane.

FIGURE 3.51 Typical sandwich-slab membrane detailing. (Courtesy of TC MiraDRI)

In addition to concrete layers, other forms of protection are used, including wood decking, concrete pavers (Fig. 3.52), natural stone pavers (Fig. 3.53), and brick pavers (3.54). Protected membranes are chosen for areas subjected to wear that deck coatings are not able to withstand, for areas of excessive movement, and to prevent the need for excess maintenance. Although they cost more initially due to the protection layer and other detailing required, sandwich membranes do not require the in-place maintenance of deck coatings or sealers.

FIGURE 3.52 Protected membrane application using concrete pavers. (Courtesy of American Hydrotech)
FIGURE 3.53 Protected membrane application using stone pavers. (Courtesy of TC MiraDRI)
FIGURE 3.55 Insulation layer in protected membrane application. (Courtesy of American Hydrotech)
Protected membranes allow for installation of insulation over waterproof membranes and beneath the topping layer (Fig. 3.55). This allows occupied areas beneath a deck to be insulated for environmental control. All below-grade waterproofing systems, with the exception of hydros clay and vapor barriers, are used for protected membranes above grade. These include cementitious, fluid-applied, and sheet-good systems, both adhering and loose-laid. Additionally, hydros clay systems have been manufactured attached tosheet-good membranes, applicable for use as protected membrane installations.

FIGURE 3.55 Insulation layer in protected membrane application. (Courtesy of American Hydrotech)