Vierendeels Configurations

Vierendeels may have various configurations, including one-way and two-way spans. One-way girders may be simply supported or continuous over more than two supports. They may be planar or prismatic with triangular or square profile for improved lateral load resistance.  Some highway pedestrian bridges are of the latter type.  A triangular cross-section has added stability, inherent in triangular geometry.  It could be integrated with bands of skylights on top of girders.

When supports are provided on all sides, Vierendeel frames of two-way or three-way spans are possible options.  They require less depth, can carry more load, have less deflection, and resist lateral load as well as gravity load.  The two-way option is well suited for orthogonal plans; the three-way option adapts better to plans based on triangles, hexagons, or free-form variations thereof.

Moment resistant space frames for multi-story or high-rise buildings may be considered a special case of the Vierendeel concept.

1  One-way planar Vierendeel girder
2  One-way prismatic Vierendeel girder of triangular cross-section
3  One-way prismatic Vierendeel girder of square cross-section
4  Two-way Vierendeel space frame
5  Three-way Vierendeel space frame
6  Multi-story Vierendeel space frame

Joist, Beam, Girder

Joists, beams, and girders can be arranged in  three different configurations: joists supported by columns or walls1; joists supported by beams that are supported by columns2; and joists supported by beams, that  are supported by girders, that are supported by columns3.  The relationship between joist, beam, and girder can be either flush or layered framing.  Flush framing, with top of joists, beams, and girders flush with each other, requires less structural depth but may require additional depth for mechanical systems.  Layered framing allows the integration of mechanical systems. With main ducts running between beams and secondary ducts between joists.  Further, flush framing for steel requires more complex joining, with joists welded or bolted into the side of beams to support gravity load. Layered framing with joists on top of beams with simple connection to prevent displacement only

2  Single layer framing: joists supported directly by walls
3  Double layer framing: joists supported by beams and beams by columns
4  Triple layer framing: joists supported by beams, beams by girders, and girders by columns
5  Flush framing: top of joists and beams line up May require additional depth for mechanical ducts
6  Layered framing: joists rest on top of beams Simpler and less costly framing May have main ducts between beams, secondary ducts between joists

A Joists
B Beam
C Girders
D Wall
E Column
F Pilaster
G Concrete slab on corrugated steel deck

Gerber Beam

The Gerber beam is named after its inventor, Gerber, a German engineering professor at Munich. The Gerber beam has hinges at inflection points to reduce bending moments, takes advantage of continuity, and allows settlements without secondary stresses.  The Gerber beam was developed in response to failures, caused by unequal foundation settlements in 19th century railroad bridges.

1.  Simple beams over three spans
2.  Reduced bending moment in continuous beam
3.  Failure of continuous beam due to unequal foundation settlement, causing the span to double and the moment to increase four times
4.  Gerber beam with hinges at inflection points minimizes bending moments and avoids failure due to unequal settlement

BEAM OPTIMIZATION

Optimizing long-span girders can save scares resources.  The following are a few conceptual options to optimize girders.  Optimization for a real project requires careful evaluation of alternate options, considering  interdisciplinary aspects along with purely structural ones.

1  Moment diagram, stepped to reflect required resistance along girder
2  Steel girder with plates welded on top of flanges for increased resistance
3  Steel girder with plates welded below flanges for increased resistance
4  Reinforced concrete girder with reinforcing bars staggered as required
5  Girder of parabolic shape, following the bending moment distribution
1  Girder of tapered shape, approximating bending moment distribution

Structures: Bending, Effect of Overhang

Bending moments can be greatly reduced, using the effect of overhangs.  This can be describe on the example of a beam but applies also to other bending members of horizontal, span subject to gravity load as well.  For a beam subject to uniform load with two overhangs, a ratio of overhangs to mid-span of 1:2.8 (or about 1/3) is optimal, with equal positive and negative bending moments.  This implies an efficient use of material because if the beam has a constant size – which is most common – the beam is used to full capacity on both, overhang and span.  Compared to the same beam with supports at both ends, the bending moment in a beam with two overhangs is about one sixth !  To a lesser degree, a single overhang has a similar effect. Thus, taking advantage of overhangs in a design may result in great savings and economy of resources.

1. Simple beam with end supports and uniform load
2. Cantilevers of about 1/3 the span equalize positive and negative bending moments and reduces them to about one sixth, compared to a beam of equal length and load with but with simple end support.

Portal Method For Rough Moment Frame Design

The Portal Method for rough moment frame design is based on these assumptions:

•  Lateral forces resisted by frame action
•  Inflection points at mid-height of columns
•  Inflection points at mid-span of beams
•  Column shear is based on tributary area
•  Overturn is resisted by exterior columns only

1.    Single moment frame (portal)
2.    Multistory moment frame
3.  Column shear is total shear V distributed proportional to tributary area:

4.   Column moment = column shear x height to inflection point

5.  Exterior columns resist most overturn, the portal method assumes they resist all
6.  Overturn moments per level are the sum of forces above the level times lever arm of each force to the column inflection point at the respective level:

7.  Beam shear = column axial force below beam minus column axial force above beam Level 1 beam shear:

ARCHITECTURAL STRUCTURES - GLOBAL MOMENT AND SHEAR

Global moments help to analyze not only a beam but also truss, cable or arch. They all resist global moments by a couple F times lever arm d:

The force F is expressed as T (tension) and C (compression) for beam or truss, and H (horizontal reaction) for suspension cable or arch, forces are always defined by the global moment and lever arm of resisting couple.  For uniform load and simple support, the maximum moment M and maximum shear V are computed as:

For other load or support conditions use appropriate formulas

Beam

Beams resist the global moment by a force couple, with lever arm of 2/3 the beam depth d; resisted by top compression C and bottom tension T.

Truss 

Trusses resist the global moment by a force couple and truss depth d as lever arm; with compression C in top chord and tension T in bottom chord.  Global shear is resisted by vertical and / or diagonal web bars. Maximum moment at mid-span causes maximum chord forces.  Maximum support shear causes maximum web bar forces.

Cable 

Suspension cables resist the global moment by horizontal reaction with sag f as lever arm.  The horizontal reaction H, vertical reaction R, and maximum cable tension T form an equilibrium vector triangle; hence the maximum cable tension is:

Arch 

Arches resist the global moment like a cable, but in compression instead of tension:

However, unlike cables, arches don’t adjust  their form for changing loads; hence, they assume bending under non-uniform load as product of funicular force and lever arm between funicular line and arch form (bending stress is substituted by conservative axial stress for approximate schematic design).

Seismic Design, Eccentricity

Offset between center of mass and center of resistance causes eccentricity which causes torsion under seismic load.  The plans at left identify concentric and eccentric conditions:

1  X-direction concentric
    Y-direction eccentric

2  X-direction eccentric
    Y-direction eccentric

3  X-direction concentric
    Y-direction concentric

4  X-direction concentric
    Y-direction concentric

5  X-direction concentric
    Y-direction concentric

6  X-direction concentric
    Y-direction concentric

Note: Plan 5 provides greater resistance against torsion than plan 6 due to wider wall spacing Plan 6 provides greater bending resistance because walls act together as core and thus provide a greater moment of inertia.

Structures - Horizontal Floor and Roof Diaphragms

Horizontal floor and roof diaphragms transfer lateral load to walls and other supporting elements.  The amount each wall assumes depends if diaphragms are flexible or rigid.

1.  Flexible diaphragm

Floors and roofs with plywood sheathing are usually flexible; they transfer load, similar to simple beams, in proportion to the tributary area of each wall. Wall reactions R are computed based on tributary area of each wall. Required shear flow q (wall capacity)

2.  Rigid diaphragm

Concrete slabs and some steel decks are rigid; they transfer load in proportion to the relative stiffness of each wall.  Since rigid diaphragms experience only minor deflections under load they impose equal drift on walls of equal length and stiffness. For unequal walls reactions are proportional to a resistance factor r.

Structures - Design Response Spectrum

The IBC Design Response Spectrum correlate time period T and Spectral Acceleration, defining three zones.  Two critical zones are:  

T < TS          governs low-rise structures of short periods
T > TS          governs tall structures of long periods

where

T = time period of structure (T ~ 0.1 sec. per story - or per ASCE 7 table 1615.1.1)
TS = SDS/SD1  (See the following graphs for SDS and SD1)

Seismic Design

Earthquakes are caused primarily by release of shear stress in seismic faults, such as the San Andreas fault, that separates the Pacific plate from the North American plate, two of the plates that make up the earth’s crust according to the plate tectonics theory.  Plates move with respect to each other at rates of about 2-5 cm per year, building up stress in the process.  When stress exceeds the soil’s shear capacity, the plates slip and cause earthquakes.  The point of slippage is called the hypocenter or focus, the point on the surface above is called the epicenter.  Ground waves propagate in radial pattern from the focus.  The radial waves cause shaking somewhat more vertical above the focus and more horizontal far away; yet irregular rock formations may deflect the ground waves in random patterns.  The Northridge earthquake of January 17, 1994 caused unusually strong vertical acceleration because it occurred under the city.

Occasionally earthquakes may occur within plates rather than at the edges.  This was the case with a series of strong earthquakes in New Madrid, along the Mississippi River in Missouri in 1811-1812.  Earthquakes are also caused by volcanic eruptions, underground explosions, or similar man-made events.

Buildings are shaken by ground waves, but their inertia tends to resists the movement which causes lateral forces.  The building mass (dead weight) and acceleration effects these forces.  In response, structure height and stiffness, as well as soil type effect the response of buildings to the acceleration.  For example, seismic forces for concrete shear walls (which are very stiff) are considered twice that of more flexible moment frames.  As an analogy, the resilience of grass blades  will prevent them from breaking in an earthquake; but when frozen in winter they would break because of increased stiffness.

The cyclical nature of earthquakes causes dynamic forces that are best determined by dynamic analysis.  However, given the complexity of dynamic analysis, many buildings of regular shape and height limits, as defined by codes, may be analyzed by a static force method, adapted from Newton’s law F= ma (Force = mass x acceleration).

1  Seismic wave propagation and fault rupture
2  Lateral slip fault
3 Thrust fault
4 Building overturn
5 Building shear
6  Bending of building (first mode)
7  Bending of building (higher mode)
E Epicenter
H Hypocenter