Structural Icons for Architects part.2

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F I Bonded and unbonded prestressing tendons

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Prestressing is a special state of stress and deformations which is induced to improve structural behaviour. Structures can be prestressed either by artificial displacements of the supports or by steel reinforcement that has been pre-strained before load is applied. The forces induced by the former method are sharply reduced by creep and shrinkage and are generally ineffective at ultimate limit state.

Due to these inherent disadvantages, support displacements are rarely used for prestressing.

The forces induced by pre-strained reinforcement can, however, survive the effects of shrinkage and creep, provided the initial steel strain is sufficiently larger than the anticipated shortening in the concrete. The required pre-strains are best achieved using high-strength steel. High-strength steel that has been pre-strained can normally be stressed to its full yield strength at ultimate limit state.

Prestressing can be full, limited, or partial. Full prestressing is designed to eliminate concrete tensile stresses in the direction of the prestressing under the action of design service loads, prestressing, and restrained deformations. In structures with limited prestressing, the calculated tensile stresses in the concrete must not exceed a specified permissible value. Behaviour at ultimate limit state must nevertheless be checked in both cases. Partial prestressing places no restrictions on concrete tensile stresses under service conditions.

Concrete stresses need not, therefore, be calculated.

Partial prestressing encompasses the entire range of possibilities from conventionally reinforced to fully prestressed concrete.

Designs must ensure adequate behaviour at ultimate limit state and under service conditions, both of which must be verified directly.Structures can be prestressed either by pre-tensioning or posttensioning.

Pre-tensioning is used primarily for the prefabrication of concrete components. The prestressing steel is stressed between fixed abutments, forms are installed around the steel, and the concrete is cast. After the concrete has hardened, the prestressing steel is detached from the abutments. Anchorage of the steel, and hence the transfer of prestressing force from steel to concrete, is achieved entirely through bond stresses at the ends of the member.

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Pre-tensioning is usually more economical for large-volume precasting operations, since the costs of anchors and grouting can be eliminated. it is quite common, however, to combine both methods of prestressing in a given structures.

Prestressing

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pre-strained steel reinforcement induces a self-equilibrating state of stress in the cross-section. The tensile force in the steel and the compressive force in the concrete, obtained from integration of the concrete stresses, are equal and opposite. In statically determinate

F3 structures, the two forces act at the same location in the cross-section. The sectional forces in the concrete due to prestressing can thus be easily determined from equilibrium. The following expressions are based on the assumption that the direction of the prestressing force deviates only slightly from a vector normal to the cross-section.

The maximum tensile force in the tendons at tensioning should generally not exceed the lower of the following values after transfer or prestressing to the concrete. rpo,max=0.75 fptk rpo,max = 0.85 fpo,1k

Losses occurring before prestressing (pretensioning) The following losses should be considered in design a) Loss due to friction at the bends (in the case of

curved wires or strands)

b) Losses due to drive-in of the anchoring devices (at the abutments) when anchoring on a prestressing bed. c)

Loss due to relaxation of the pretension tendons during the period which elapses between the tensioning of the tendons and prestressing of the concrete.

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The minimum concrete strength required at the time when tensioning takes place is given in the approval documents for the prestressing system concerned.The initial prestress (at time t = 0) is calculated taking into account the prestressing force and the permanent actions presents at tensioning.

Where particular rules are not given, the time when prestressing takes place should be fixed with due regard to: 1. deformation conditions of the component 2. safety with respect to the compressive strength of the concrete 3. safety with respect to local stresses 4. early application of a part of the prestress, to reduce shrinkage effects.

d) The prestressing force at a given time t is ob- tained by subtracting from the initial prestressing force the value of the time dependent losses at this time t.

e) These losses are due to creep and shrinkage of concrete and relaxation of steel. f) The finale value of the prestressing force is obtained by subtracting from the initial prestressing force the maximum expected value of the time-dependent losses. g) The strength of the anchorage zones should exceed the characteristic strength of the tendon, both under static load and under slow-cycle load.

F4 h) Possible formation of small cracks in the anchorage zone may not impair the permanent efficiency of the anchorage if sufficient transverse reinforcement is provided. i) This condition is considered to be satisfied if stabilization of strains and cracks widths is obtained during testing.

Partial prestressing is generally more economical than full or limited prestressing. Although structures that are partially prestressed require a significant portion of mild reinforcement for crack control and distribution, this steel contributes to the ultimate resistance of the section. Whatever mild steel is added to improve behaviour under service conditions thus reduces the amount of prestressing steel necessary for safety at ultimate limit state.

The prestressing force must always be carefully monitored during construction, since deviations from the prescribed prestressing force can led to cracking, deformations, and fatigue.

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In post-tensioned construction, the prestressing steel is only stressed after the concrete has been cast and hardened. The steel must therefore be enclosed in ducts and anchored using special devices. The ducts are most commonly embedded in the concrete and filled with grout after stressing to bond steel to concrete and to provide protection against corrosion. The ducts can also be located outside of the concrete section and left unbonded for the entire life of the structure. In the such cases, grout is only a means of protecting the steel.

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TT Prestress Beam

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Stree-Strain relationship of prestress tendon and strain diagram of cross-section

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Diagram of prestress tendon

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Prestress cross-section and stress-strain diagram of prestress tendon

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Stress-Strain diagram of prestress tendon

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Prestress cross-section and stress-strain diagram of prestress tendon

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Diagrams of ordinary and prestress reinforcement

F13 Prestressed concrete sections are in fact designed with respect to the serviceability limit state: 1. In bending plane sections remain plane- Hypothesis of Bernouilli.

2. Linear elastic behaviour of concrete in an uncracked section - laws of Navie rand Hooke.

3. In the cross-sectional properties the

reinforcement is not taken into account.

4. In the serviceability limit state, the concrete stresses are limited to permissible values in compression and in tension. In the serviceability limit state, no tension is permitted in any section of fully prestressed concrete structures. In the case of limited prestressing some tensile stresses are permitted. Under the initial prestressing force, requirements with respect to permissible stresses in concrete are less restrictive than under the effective prestressing force

-effictive prestressing force. The magnitude of this force depends on the magnitude of the initial prestressing force taking into account losses of prestress due to time-dependent effects. The effective prestressing force is independent of the loading of the structure

7. Losses of prestress are caused by: -relaxition losses of initial stress in the prestressing steel -shortening of the steel due to creep of concrete, under conditions of sustained load acting on the structure, in the centroid of the area of the prestressing steel.

In the calculation of losses of prestress the cross-section of the reinforcement is not taken into account.

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5. If statically determinate structures are considered prestressing is introduced as compressive force acting on the plain concrete section in the centroid of the area of the prestressing steel.

6. Two magnitudes of prestressing forces are taken into account: -intial prestressing force - directly after transfer of the prestressing force to the concrete section, taking into account effects of friction between tendons and concrete structure

8. In the ultimate limit state, it is assumed that: - the concrete, in the sections with the maximum bending moment, is subjected in the ultimate fibres to the maximum strain and the corresponding concrete strength.

- the depth of the compression zone is limited to a maximum -the stress in the prestressing steel bonded with concrete or in grouted tendons is the characteristic strength.

-the stress in the prestressing steel is the effective prestress, if tendons are used.

After prestressing δp≤0,7.fpk If all prestressing losses are taken in to account, the steel stress δp∞≥0,45.fpk

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Box girder cross-section

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Box girder cross-section

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Positions of einforcement and prestress tendons in box girder cross-section

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Preliminary design aids of box girder cross-section

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Line influence in bridge design

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Line influence in bridge design

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Longitudinal arrangement of the bridge elevation

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Primary effect of prestress tendon

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Secondary effect of prestress tendon

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Determination of sectional forces according to influence lines, dead and imposed load in bridge design

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Influence lines in cantilever method

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Construction stages II

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Determination of sectional forces for vehicle loading

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Inffluence of ideal prestress tendon on sectional forces

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An efficient post-tensioned design can be achieved with a solid flat slab, which is ideally suited to multi-storey construction where there is a regular column grid. These are sometimes referred to as flat plate slabs. The benefits of a solid flat slab are the flush soffit and minimum construction depth, which are suited to rapid construction methods. These provide the maximum flexibility for horizontal service distribution and keep slab weight low and building height down to a minimum. The depth of a flat slab is usually controlled by deflection requirements or by the punching shear capacity around the column. Post-tensioning improves control of deflections and enhances shear capacity.

The latter can be increased further by introducing steel shear heads within the slab depth, or drop panels.

The effect of prestressing on the service behaviour of the slab can best be determined from the direct reactions of the cable on the concrete slab, because they can be considered as external loads referred to as requirement loads. In a tendon element d, the vertical load uv is directly proportional to the prestressing force P and indirectly proportional to the radius of curvature r.

This is adequately accurate for small angels up to 12o, which is usual for most cases. For a parabolic tendon profile, the equivalent load u is evenly distributed and leads to the well-known expression: u = 8 (f P / l2) where f = rise of the parabola P = prestressing force L = span

Post-tension details over supports

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Detail of prestress cables

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Stresses over cross-section and in bridge design, design of prestress cables

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Stresses over cross-section and in bridge design, design of prestress cables, State II

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Prestress Tendon layout in Box girder bridge design- cantilever method

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Box-girder with bonded tendons

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Deflection of prestress beam in the bonded tendon

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Elevation of different continnous prestress slab sections according to load balancing method

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Calculation of equivalent load in prestress RC beam

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Calculation of equivalent load in prestress RC beam

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Calculation of equivalent load in prestress RC beam

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The principle of balance loading method over two-span prestress beam

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Calculation of equivalent load in prestress RC beam

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Calculation of equivalent load in prestress RC beam

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Calculation of equivalent load in prestress RC beam

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Sabah Shawkat © Determination of the equivalent load u for a parabolic tendon profile The cable in the in flex point, where the cable changes the shape of the geometry from concave to convex, must be smooth, and therefore first derivation of the both equation z1(x) and z2(x) must be the same. The coordinate of the in flex point in this case are [0,0].

In the simplest case, for a uniformly loaded simply-supported beam, the bending moment is parabolic, as is the ideal tendon profile. At the supports the tendon has no eccentricity and hence there is no bending due to the tendon forces.

Equivalent load in prestress beam

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Forces between tendon and concrete girder due to prestress tendon

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Prestress load and temperature subjected of RCPB

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Structural system with ultimate load and assumed tendon load

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Analysis of normal stress in prestress concrete cross section

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Calculation of equivalent load in prestress RC beam

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Calculation of equivalent load in prestress RC beam

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Calculation of equivalent load in prestress RC beam

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Calculation of equivalent load in prestress RC beam

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Equivalent load in prestress beam due to prestress tendon

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Internal forces due to prestress cables

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Different cases of equivalent load in prestress beam due to prestress tendon

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The example of The Method of Equivalent Loads.

Figures below illustrates an idealised tendon profile for a two-span member. while the bending moment peak over the supports, it is clear that in practice a tendon cannot do this and some approximation must be made. The balancing loads upwards and downwards due to the tendons can thus be calculated. The effect of prestressing on the service behaviour of the element can best be determined from the direct reactions of the cable on the concrete element, because they can be considered as external loads referred to as

Sabah Shawkat © equivalent loads u.

In a tendon element d, the vertical load uv is directly proportional to the prestressing force P and indirectly proportional to the radius of curvature r. This is adequately accurate for small angles up to 12°. In addition to these equivalent loads, the end support forces, due to prestressing, have to be introduced ( P.sin(α) and P.cos(α)).

Note: the centre gravity of the concrete and the centre gravity of the tendon coincide at the end of member so that no equivalent load moments are applied at the end of member.

Equivalent load in prestress beam due to prestress tendon

Double - T Girder with Unbonded Tendons

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External prestress un-bonded tendons

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F57 Analysis of Externally Prestressed Concrete Beams

External prestressing refers to a posttensioning method in which tendons are placed on the outside of a structural member and attached to the beam at some deviator points along the beam.

It is an attractive method in rehabilitation and strengthening operations because: • It adds little weight to the original structure • Its application poses little disturbance to users • It allows the monitoring, re-stressing and replacement of tendons.

An ideal tendon material should not only have highstrength but it also has to remain in elastic range until relatively high stresses are reached. Furthermore, it has to show sufficient ductility and good bonding properties, low relaxation and high resistance to fatigue and corrosion. There are three main types of tendons used in prestressed concrete: wires, strands (made of several wires), and bars.

The strands are made by several wires.

For example, in the seven-wire strand six spherical wires are helically wound over a central wire which

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The first use with external steel tendons was in the 1950s, but after that it lay dormant for some time. Now external prestressing techniques with steel tendons have been widely used with success to improve existing structures in the United States, Japan and Switzerland.

However, there can be a problem with corrosion in the steel that forces the use of steel protection on the external tendons, for example by plastic sheeting. This problem can be resolved by the use of FRP (Fibre Reinforced Polymer) materials.

Therefore, research in the area has been conducted since the early 1970’s. In the beginning, glass FRP was used but at the moment aramid and carbon are mainly used due to higher modulus of elasticity.

has a larger diameter than the surrounding wires.

Because wires are usually produced with small diameters, the strands show superior properties than single bars due to better quality control. They are also easier to handle due to more flexibility than in a single bar of the same diameter.

Prestressing bars are produced with smooth or ribbed surfaces. The first ones can be mechanically end-treated to be used in anchoring systems, while the second ones can be anchored anywhere along their length. The mechanical properties of prestressing steel must satisfy some requirements, which include the tensile strength and the corresponding strain, the yield point and the modulus of elasticity.

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External prestress un-bonded tendons

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Diagram of stress vs strain of prestress cross-section after the full development of cracks

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N compression

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Unbonded prestress cross-section, diagram of internal forces due to prestress force and external load

F67 In prestressed concrete, prestress forces are introduced by applying tension forces to steel cables (called steel tendons) prior to the application of external loading. When external loads are applied, the undesirable concrete tensile stresses are balanced with the prestressed compressive forces.

This results in the reduction or elimination of both stresses. There are two basic methods of prestressing concrete members - pre-tensioning and posttensioning.

In a reinforced concrete beam, the steel carries the tension forces developed below the neutral axis. The concrete in that region is dead weight and its structural purpose is to provide a Wbond with

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the steel rods. Concrete used for prestressed construction is characterized by high strength, because prestressed concrete is generally subjected to higher forces and high strength concrete results in more economical designs.

Diagram of prestress steel

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Stress-Strain relationship of unbonded prestress cross-section

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Secondary effects from ideal prestress tendon

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Reinforced concrete prestress slab, calculation of equivalent load due to prestress tendon

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Design of prestress concrete

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Design of prestress concrete

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Design of prestress concrete

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Design of prestress concrete

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Design of prestress concrete

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References [1] Picard A., Massicotte B. and Bastien J. (1995) „Relative

[8] Naaman A.E., Burns N., French C, Gamble W.L. and

Efficiency of External Prestressing“ Journal of Structural

Mattock A.H. (2002) „Stresses in Unbounded Prestressing

Engineering, Vol. 121, No. 12, December 1995

Tendons at Ultimate: Recommendation“ ACI Structural Journal, July-August, 2002

[2] Pincherira and Woyak (2001) „Anchorage of Carbon

[9] Nanni A., Bakis C.E., CNeil E.F. and Dixon T.,

Fiber Reinforced Polymer (CFRP) Tendons Using Cold-

(1996) „Performance of FRP tendon anchorsystems for

Swaged Sleeves“, PCI Journal, Vol. 46 no. 6, November-

prestressed concrete structures“ PCI Journal, January-

December 2001

February 1996

[3] Pisani M.A. (1998), „A numerical survey on the

[10] Ng C. (2003) „Tendon Stress and Flexural Strength

behaviour of beams pre-stressed with FRP cables“

of Externally Prestressed Beams“ ACI Structural Journal,

Construction and Building Materials 12, 1998, pp 221-232

September-October 2003, pp 644-653.

[4] McKay K.S. and Erki M.A. (1993) „Flexural behaviour

[11] Tan K. and Ng C. K. (1997) „Effects of Deviators

of Concrete Beams Pretensioned with Aramid Fibre

and Tendon Configuration on Behaviour of Externally

Reinforced Plastic Tendons“, Canadian Journal of Civil

Prestressed Beams“, ACI Structural Journal, v 94, n 1,

Engineering vol 20

January-February 1997.

[5] Meier U (1998), US-Patent 5,713,169, Feb 3, 1998

[12] Tan K., Farooq A. and Ng C. (2001) „Behaviour

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of Simple-Span Reinforced Concrete Beams Locally [6] Meier U., Deuring H. and Schwegler G. (1992)

Strengthened with External Tendons“ ACI Structural

„Strengthening of structures with CFRP laminates:

Journal, March- April 2001, pp 174-183

Research and application in Switzerland“ Advanced composites materials in bridges and structures, Edt.

[13] Tan K. And Tjandra R.A. (2003) „Shear Deficiency in

Neale K.W. and Labossiere P., 1992

Reinforced Concrete Continuous Beams Strengthened with External Tendons“ ACI Structural Journal, September-

[7] Mutsuyoshi H., Machida A. and Sano M., (1991), „Behaviour of pretressed concrete beams using FRP as external cable“ Japan Concrete Institute, Vol. 13, 1991

October 2003, pp 565-572

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G I Steel Structures

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Why this chapter deals with steel construction? because steel is a recyclable material.

A structural steelwork is easily demolished and melted down after use and processed in to a new structures. Demolition is easier if bolted connections were used in the original construction, and the elegance of the welding profiles may be used to construct light and adaptable structures. It is possible to produce a profile for every application with as optimal as possible use of material.

This chapter gives a general overview of structural design and behaviour of steel structures to the structural Euro-code, and includes a set of graphical schemes showing the design of structural elements. The idea behind the creation of this chapter came from trying to find the best way to introduce students, designers in architectural building as an introduction, reference on how to design steel structures. Every step in the design of the steel structure is explained in simple terms.

of structural members. This publication contains comprehensive section property data and member resistances for a wide range of steel sections such as double symmetric hybrid beam, hollow crosssection, and box profiles. It is intended to be of particular help in undergraduate teaching, although it will also provide guidance to practising designers who want to become acquainted with design to the steel structures design.

Graphical schemes have all been evaluated using the values of parameters and design options given in the euro-code. The Structural Euro-codes are a set of structural design standards, which cover the design of all types of structures, however, always be used together with the relevant national application documents (NAD). The designer should separately check the relevant requirements for each country in the national application documents.

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Designers interested in learning more about behaviour of steel members will find a series of 3d pictures which I made them during the realization projects on site that will complement the text therefore demonstrate the concept and building details of the constructions to further the readers understanding. The publication has been produced with the assistance of structural design lecturers, the set of worked examples present the design

G3 Permissible ratio of effective height resp. span to effective depth for simplified analysis of deflection limit.

- Rolled steel of open section (column): d = height / 22.5 - Rolled steel of close hollow section (column): d = height / 27,5 Buckling height / d >20 Closed sections have smaller exposed surface and greater torsional stiffness than open sections of same weight.

- Steel decking (trapezoidal profiles): d = span / 35 - Deep rolled steel section(floors-beam): d = span / 18 - Rolled steel truss: d = span / 12 - Vierendeel girder: d = span / 8 - Composite concrete steel girder: d = span / 22.5 - Deep rolled steel section(roofs-beam): d = span / 22 Irequired =14.9 Mmax (kN.m) L (m) = cm4 - Wide flange rolled steel section (roofs-beam): d = span / 25 -Two-layer space frame (roofs): d = span / 22.5 -Sloping rolled steel truss: d = span / 8 -Single storey rigid frame: d = span / 40, frame is rigid in its own plane.

Typical spacing of frames L/4 – L/6 - Arch: d = span / 45 -Single storey beam and post (frame): d = span / 16 -Multi-storey rigid frame: d = span / 25

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Suggested Deflection Limits Deflection on beam due to unfactored imposed load Cantilevers Beams carrying plaster or other brittle finish All other beams

Length / 180 Span / 360 Span / 200

Horizontal deflection of columns (other than portal frames) due to unfactored imposed and wind load Tops of columns in single storey buildings In each storey of a building with more than one storey

Height / 300 Height of storey / 300

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In slip resistance connections, the preloaded bolts clamp the contact surfaces together and the contact surfaces cannot slip with respect to each other. The resistance of the connection depends on the preloading force in the bolt as well as the slip factor of the contact surfaces in addition to the material properties of the connected parts.

The stiffness of the splice in a compression member has to be at least equal to that of the member with respect to both principal axes even if the member is axially loaded only and abutting ends are provided at the splice.

Furthermore, the parts of the splice must be able to carry 2,5% of the compressive force in any direction perpendicular to the axis of the member.

Column Bracket: The thickness of the bracket is normally 30-50mm. Its width is taken as equal to that of the end-plate of the beam. Its depth is determined by the connection welds. Vertical welds and the lower horizontal weld are taken to be load-bearing. the upper surface has to be flat so that the end-plate of the beam is able to rest on it. The pinned joints are normally used when an I-beam is to be joined to a column. A typical pinned joint between a rafter and column and a rigid joint shown in figures below

Some beam to column joints, some column joints to the foundation, some typical connections beam to beam joints

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Sabah Shawkat © Column-to –Foundation connections if a rigid column-to-foundation is assumed in the structural model the moments transferred into the foundation by the column must be accounted for when designing the holding down bolts and the base plate. If a pinned joint is assumed in the model, moments need not be taken into account.

The holding down bolts must be designed such that they are able to carry the construction loads the column is subjected to. The thickness of the second stage concrete layer is taken into account when calculating the buckling length for the design of the holding down bolts.

Caculate the compression stress of the concrete, verification the plate of the column

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Sabah Shawkat © Welded connections Joints may be classified by either their rigidity of strength characteristics. A joint may be classified according to its rigidity as a nominally pinned, rigid or semi-rigid joint. According to its strength, it may be classified as a nominally pinned, full-strength or partial-strength joint. Nominally pinned joints shall be so designed that they cannot develop significant moments which might adversely

affect members of the structure. The joint shall be capable of transmitting the calculated design forces, and it shall be capable of accepting the resulting rotations. The design of semi-rigid joints for momentresisting steel framed is based on the interaction between members, i.e. on the moment-rotation characteristics. Joints not meeting the criteria for nominally pinned joints of rigid joints are treated as semi-rigid joints.

Design od Y, T, and K joints in hollow square steel cross-sections

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Design od Y, T, and K joints in hollow circular steel cross-sections

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Baseplate connections in simple construction are generally modelled as pins, and designed to transfer either concentric force (compression or tension) or a combination of axial and shear force (usually when the column is part of the bracing system). In some instances, they may, however, be designed to transmit also bending moments due to moderate load eccentricity, or for erection stability. The plate is always attached to the column by means of fillet welds. However, if the column carries only compression loads, direct bearing may be assumed, provided that the contact surfaces are machined or can be considered to be flat. No verification of the welds is then required.

Machining may be omitted if loads are relatively small. Where there are moderate tension forces or no net tension the holding down bolts are usually cast into the foundation.

In the joint of a beam-column to the foundation the tension resistance of the holdingdown bolts has to be checked in addition to the resistance of the steel base plate and the foundation if the holding-down bolts are subject to tensile force. in addition, the effect of the shear force has to be taken account of the thickness of the base plate has to be determined with respect to both compression and tension sides.

Joint of column to foundation, design the joint resistance of a hollow steel section, and calculation of pressure on foundation

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The next step in the analysis is to determine the distribution of forces within the connection, see figure below. It is not necessary and often not feasible to determine the real internal distribution of forces. It is sufficient to assume a realistic distribution, provided that:

1. the internal forces are in equilibrium with the applied loading 2. each component is capable of resisting the forces 3. the deformations implied by the assumed distribution are within the deformation capacity of the fasteners (bolts, welds) and of the connected parts.

Design the resistance of the joints

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Trusses and reticulated structures

Stable assemblages of individual tie and strut elements are known as reticulated structures. A triangulated assemblage of ties and struts in two dimensions is known as a plane truss and consists, in principle, of elements connected by pin joints working in tension or compression see picture below. Trusses are often used in place of rolled steel beams, they make better use of material and are more economical for long spans or heavy loads, in spite of the extra costs of their fabrication.

An essential point is that the elements are always arranged in triangular configurations

which are inherently stable, assuming none of the angles in the triangle is small. Comparing a beam to a rectangular truss, both simply supported, the horizontal top and bottom beam members of the truss carry compression and tension respectively while diagonal and vertical elements serve to carry the shear force which is a maximum near supports.

An important aspect of the design of plane trusses is that the top boom be stabilised to prevent buckling in the plane at right angles to the truss see picture below.

Design of steel trusses, steps in trusses design

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Design the bearing strength of the foundation, determination of bending moments over fixed steel beams

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Sabah Shawkat © This critical stress fixes the upper limit of load that may be applied axially even though ultimate axial stress capacity of the member may be very high.

Yet another related phenomena is the lateral torsional buckling of laterally unrestrained beams or beam columns. When I sections are used

as beams or beam columns the compression flange is under compressive stress and has a tendency to buckle but it is attached to the tension flange which resists the buckling giving rise to torsion within the beam section. This torsion twists and warps the unrestrained part of beam leading to lateral torsional buckling.

Buckling in steel columns and lateral tortional buckling in steel beam

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Sabah Shawkat © To calculate the lateral-torsional buckling moment and the resistance moment to lateraltorsional buckling, the geometrical warping rigidity, the geometrical torsional rigidity as well as the second moment of area about the z-axis are needed.

Lateral-torsional buckling for a beam with the cross-section I

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Sabah Shawkat © The critical load is dependent upon the end restrains. For columns with various combinations of fixed, free, and pinned supports. The behaviour of an ideal column is often represented on a plot of average compressive stress versus slenderness ratio. Such a representation offers a clear rationale for the classification of compression bars. Tests of columns verify each portion of the curve with reasonable accuracy. The range of λ is a function of the material under consideration.

Most structural columns lie in a region between short and long classifications. Such intermediate-length columns do not fail by direct compression or by elastic instability. The failure of an intermediate column occurs by inelastic buckling at stress levels exceeding the proportional limit.

Presented below is one practical approach to determination of inelastic buckling, this approach is known as the tangent-modulus theory.

Average steel stress in steel columns versus slenderness ratio

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Useful thickness of asymmetrical cords

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Shear buckling resistance of square and rectangular hollow section, joint of column to foundation

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Column web in shear, web in compression and web in tension

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Graph, illustrated shearing forces vs depth of the steel beam

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Sabah Shawkat © Shear rupture resistance A bolt group can break down near the end of profile.

This is called a block shear failure, which is caused by tensile rupture along the fastener holes on the tension face of the hole group, accompanied by the gross section yielding in shear at the row of the

fastener holes along the shear face of the hole group.

The block rupture may determine the resistance of the connection where high-grade steels and bolts with small edge distances are used.

Block shear-effective shear area

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Different steel cross-section profiles

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Different steel cross-section profiles

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Dimensions and axes of sections composite beam

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The values of plastic adaptation coefficients for different steel cross-sections

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Sabah Shawkat © Every system in the all existing matter and space considered as a whole has got different stages of equilibrium and the system will get deformed at the minimum possible load. Consider the system shown in figure above. We assume here that the deflections in the system are very small, that is the

column has slightly buckled. Now if you take the equilibrium of the buckled shape of the column you will find that the load required to buckle the column is less that the load that can be actually carried by the full cross sectional area.

Lenghts of buckling in some common cases

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Sabah Shawkat © In 1757, mathematician Leonhard Euler derived a formula that gives the maximum axial load that a long, slender, ideal column can carry without buckling. An ideal column is one that is perfectly straight, homogeneous, and free from initial stress.

The maximum load, sometimes called the critical load, causes the column to be in a state of unstable equilibrium; that is, the introduction of the slightest lateral force will cause the column to fail by

buckling. Buckling is called an instability occurred in a structure because of excessive loading.

But what do we mean by slender? When the longitudinal dimensions of the member are much greater than the cross section of the member then it is called a slender member and remember you should use the word slender while describing the compression in column not when there is any tension.the full cross sectional area.

The values of plastic adaptation coefficients for determination of steel stress to avoid the lateral deformation

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Sabah Shawkat © Have you ever seen Charlie Chaplin stick when he rests on it?

The stick in the picture bellow describes one of the most fundamental characteristic of a column in the field of structural engineering, called „Buckling of Column“. But why did it buckle? What made the stick to bend instead of taking the load straight down the ground? Well, here we are to discuss about the event.

Buckling can be defined as the sudden, large, lateral deflection of a column owing to a small increase in an existing compressive load. This response leads to instability and collapse of the member. In this section we shall describe the critical, or buckling, load for welded and bolted profiles, the compressive load that cases the instability.

Buckling in steel column subjected to high compressive stress

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Bolts working with simple shear

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Different buckling modes, column buckling length values of non-sway and sway frames

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Sabah Shawkat © Hollow section exposed to fire on three sides, on two opposite sides, on two adjacent sides

The fire resistance period required for the building is 15 min. The steel grade used is S355J2H and the buckling length of the column is Lfl. The temperature evolution in the fire compartment is determined with the standard time-temperature curve.

Increase in temperature of unprotected hollow sections protected with mineral wool boards (t = 15 min) of dimensions 190x190x7. Development of temperature in a protected hollow section A column of 190x190x7 is protected with 200 mm mineral wool boards.

The values of plastic adaptation coefficients for determination of steel stress to avoid the lateral deformation

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Sabah Shawkat © Joints between structural line elements, in one plane, may be classified as simple connections, as sliding connections, as pin or pinned connections as fixed connections, also known as rigid connections. A simple connection allows rotation and movement in any direction except downwards and only transmits shear force that acts downwards at a support. A sliding connection allows movement along the line of the element and rotation and only transmits shear force. A pinned connection allows

rotation but no translational movement, it transmits shear and axial forces but not bending moment.

A fixed connection does not allow translational movement or rotation and transmits shear, axial forces, bending and torsional moment. rotation but no translational movement, it transmits shear and axial forces but not bending moment. A fixed connection does not allow translational movement or rotation and transmits shear, axial forces, bending and torsional moment. the full cross sectional area.

Distinction between joint and connection, distribution of forces between bolts

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Connecting the bracing

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Values K1 in graphs depending on slenderness ratio

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Graphs of simple shear and double shear

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Buckling lenghts where the A and B ends are fixed in position, case where one of the ends is free to move laterally

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Sabah Shawkat © In frame structures with rigid joints, the benefits of structural hollow sections can be utilized when determining the column buckling length values. Another factor influencing the buckling length in frames is the lateral support of the frame. A nonsway structure can be stiffened either with lattices or by supporting it with a rigid structural element (a lift shaft or stair well). Generally speaking, a frame structure can be classified non-sway if the following condition is met. vsd ≤ 0,1 vcr

The stiffening of sway structure is based on columns functioning as cantilevers and fixed to foundations with rigid joint, or on the rigidity of the joints. In the case of a continuous column, the buckling length can be determined using figures below. η1 and η2 are distribution factors. The buckling length of columns in rigid joined structures is obtained from figure below for a non-sway frames and for sway frames. The curve values represent the relation of buckling length to the actual column length.

Different buckling modes

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increase in temperature of unprotected hollow sections protected with mineral wool boards, and the limit curve for local buckling due to flange deflection

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Buckling the web of the beam

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Plane truss, top and pinned-jointed model showing buckling mode ,design beam to column joint cinfiguration, column web in shear

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Composite roof slab using sheet metal profile

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Composite roof slab using sheet metal profile

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These figures are demonstrating the design of the composite floor slab. Verification is needed for both the construction stage (non-composite) and constructed stage (composite).

Although generally checks at the noncomposite stage are based on two continuous spans, for simplicity only a single span case will be considered here. Design and evaluate the composite steel-concrete slab on a span L, the distance of the ceiling is L1, the thickness of the concrete slab is h, the imposed load on the ceiling is v.

The roof structure is designed as a roof system „Hoesch“ (supporting plates).

Load - reinforced concrete slab + sheet metal. The floor slab should be designed for both the construction stage and the composite stage. During the construction stage, the metal decking acts as formwork and has to support its own weight, wet concrete, and construction loads. The resistance of the metal decking during the construction stage needs to be verified at the ultimate and serviceability limit state

Composite roof slab using sheet metal profile

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Composite Steel Concrete Ceilings

Design and evaluate the composite steel-concrete ceiling on a span L, including the assessment of the plex-concrete rib profile as a lost formwork. The distance of the ceilings is L1, the thickness of the concrete slab is h, the imposed load on the ceiling v, the ceilings are not supported during the construction.

The total stresses in the bottom of the steel beam are calculated as the sum of the stresses in

the lower part of the steel beam in the construction stage where only the steel beam and the stresses in the lower part of the steel beam where the steel beam is composite.

Then we should check, the number of shear connectors on one half of the beam, and carrying capacity of one shearing connector, and the spacing of the shear connectors will be half the width of the beam

RC panel profile / Axis distance of the shear connectors , Neutral axis position of the cross section

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Steel I beam

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Types of splices - continuity

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Steel shelter structure

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Two layers steel shelter structure using circular hollow sections

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Circular steel stairs using circular hollow cross-section

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Steel staircase using I cross-section

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Steel structure

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Skeleton of steel structure

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Frames in steel structures

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Deformation on column due to horizontal forces (left) Steel Stairs with assignment of cross-sections (right)

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Steel Stairs and Steel frame structure with assignment of cross-sections properties

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Anchoring the steel plate into the reinforced concrete foundation-hinged connection

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Lay-out of steel frame structure

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Welded connections of steel cross-sections

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Joints of steel structures over reinforced concrete column

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Joints of steel structures over reinforced concrete column

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Cross-sections of the steel shelter construction

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Floor plan of the shelter steel structure

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Details of steel structures

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Details of steel structures

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References [1] ENV 1993-1-1: Eurocode 3: Design of steel structures:

[9] ENV 1991-2-1: Euro-code1: Basis of design and

Annex K: Hollow section lattice girder connections, 1994

actions on structures. Part 2-1: Actions on structures, densities, self-weight and imposed load

[2] CIDECT: Design guide for rectangular hollow section joints under predominantly static loading, Verlag TUV

[10] ENV 1991-2-3: Euro-code 1: Basis of design and

Rheinland GmbH, Koln 1992

actions on structures. Part 2-3: Actions on structures, Snow loads, 1995

[3] CIDECT: Design guide for circular hollow section joints under predominantly static loading, Verlag TUV Rheinland

[11] ENV 1991-2-4: Euro-code1: Basis of design and

GmbH, Koln 1992

actions on structures. Part 2-4: Actions on structures, Wind loads, 1995

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[4] CIDECT: Design guide for circular hollow section joints under predominantly static loading, Verlag TUV Rheinland

[12] ENV 1991-1: Euro-code1: Basis of design and actions

GmbH, Koln 1991

on structures. Part 1: Basis of design, 1995

[5] ISO/FDIS 12944-2: Paints and varnishes-corrosion

[13] ENV 1991-2-2: Euro-code1: Basis of design and

protection of steel structures by protective painting

actions on structures. Part 2-2: Actions on structures.

systems. Part 2: Classification of enviroments, 1997

Actions on structures exposed to fire, 1995

[6] ISO/FDIS 12944-3: Paints and varnishes-corrosion

[14] ENV 1993-1-2: Euro-code3: Design of steel structures.

protection of steel structures by protective painting

Part 1-2: General rules. Structural fire design, 1996

systems. Part 3: Design considerations, 1997 [15] ENV 1994-1-1: Design of composite steel and [7] CIDECT: Design guide for fabrication, assembly and

concrete structures: Part 1.2: Structural fire design, 1994

erection of hollow section structures, 1996 [16] ENV 1994-1-1: Design of composite steel and [8] ENV 1090-1: Execution of steel structures-Part 1:

concrete structures: Part 1.1: General rules and rules for

General rules and rules for buildings, 1996

buildings, 1994

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H I Timber Structures

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Behaviour and Conception of Timber Structures

Timber is a structural material which has similar properties to steel and reinforced concrete in the sense that it can carry both tension and compression with almost equal facility.

Generally, bending is critical condition for medium span beams, deflection for long span beams, and shear for heavily loaded short span beams or at notched ends.

It is therefore capable of resisting bending-type load and may be used for all types of structural element. It is significantly less strong than either steel or reinforced concrete, however, with the result that larger cross-sections are required to carry equivalent amounts of load.

For designs based on permissible stress philosophy, bending is checked by appling the basic theory of bending principles. In relation to timber design this must also take into account the relevant modification factors for exposure, load duration, load sharing and so on.

Sabah Shawkat ©

There are no specific differences in the roof construction of a timber framed structure compared to other types of construction. It is however, important to ensure that any additional point loads from the roof (from girder trusses, purlins, etc) are adequately supported by additional studs or posts in the timber wall panels.

The main design considerations for which flexural members should be examined are: 1. Bending (including lateral buckling) 2. Deflection 3. Shear 4. Bearing.

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Structural Design

As with any other type of design, the evolution of the form of a structure is a creative act which involves the making of a whole network of interrelated decisions. It may be thought of as consisting of two broad categories of activity: first, the invention of the overall form and general arrangement of the structure and, secondly, the detailed specification of the precise geometry and dimensions of all of the individual components of the structure and of the junctions between them.

3D Structural system in Timber

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In pitched roofs with insulation at rafter level and in cold flat roofs, a vapour control layer should be included on the warm side, beneath the insulation, to restrict the passage of moisture and air.

Between the insulation and the underlay, a 50mm minimum ventilation space must be provided with vents at eaves and ridge, or at opposite sides of a flat roof. This space, combined with the thickness of the insulation can mean that the depth of the rafters is determined by thermal rather than structural requirements.

3D Timber pitch roof structure

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Support systems of beam structure shown in sections and methods under the effect of the vertical load.

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Lightweight timber structures

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a) Transformation of transverse forces to shear forces, b) transformation of transverse forces to compressive, resp. tensile forces and bending, c) buckling, d) bending tension, e) bending pressure

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Simply supported beam subjected to concentrated load

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Wooden frame constructions

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Representation of stress parallel and perpendicular to the fibers under load

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Primary structural systems

Secondary structural systems

The number of primary structural systems, their spacing and the positions of supports are governed by the plan layout. The design of the grid depends on the utilisation conditions, e.g. movable partions, lighting.

Secondary structural systems give form to the roof and also the interior layout. The loadbearing arrangement is determined by the number of the nature of the supports, the number and interconnection of independent loadbearing elements, and the form of the loadbearing members.

Load distribution from the roof structure to the perimeter columns by means of the main supporting wooden frames

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H14 Permissible ratio of effective span to effective depth for simplified analysis of deflection limit.

- Rafter: - Purlins or joists:

h = span / 24

Irequired = 208.3 Mmax (kN.m) L (m) = cm4

h = span / 16

- Ceiling tie:

Irequired = 26 q (kN/m) L3 (m) = cm4

-Columns or Posts (studs): n = 6, EII= 105(kg/cm2) -Circle posts D≥(H / 18,75)

Irequired = 312.5 Mmax (kN.m) L (m) = cm4

-Square posts a ≥ (H / 21,7)

Irequired = (n.F(kg).H2(cm)) / (π2. EII) = cm4 -Rectangular posts a ≥(H / 21,7)

a is least lateral dimension of cross-section -Glued laminated timber column: a = H / 22.5

Sabah Shawkat ©

-Glued laminated timber beam: h = span / 16

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Boundary conditions of columns and design of reinforcemnt to RC frame

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Wooden structures for roofing larger spans, bound triangular truss frames

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Static scheme of a rafter as a simple beam, Rafter with overhanging ends on the right and left

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Coupling the ceiling, reinforced concrete vs wooden beams

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Coupling the ceiling, reinforced concrete vs wooden beams

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Concept of wooden saddle truss

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Course of forces in wooden structures

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Calculation of stress in the foundation joint due to compressive force from a wooden post

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Load distribution at oblique bending. Laying the rafter on the beam

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Different shapes of wooden roofs

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Construction and computational models, boundary conditions of elements

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Calculation of the value of geometric characteristics of Tie beam

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Expression of different stiffness on the frame structure and transferred load to the values of bending moments acting on the horizontal and vertical elements of the frame system.

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Carrier systems beam structures shown in sections and methods of supports under the effect of vertical load.

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Timber connections

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Cable-stayed bridges

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Sabah Shawkat © Floor depth beams can be fabricated by nailing or bolting floor joists together so that they act structurally as one unit.

When long spans and/or larger loads have to be supported, beams of greater depth may be required. When deep beams are necessary to carry loads over long spans, these may be of solid

timber or a composite structural timber beam. If solid timber is used, it is important that it is dried to the appropriate moisture content to avoid differential shrinkage. It can be difficult to achieve appropriate moisture contents throughout large solid timber sections and it may be preferable to bolt together two or more thinner sections, or to use a structural timber composite.

Collar connections

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Trussed beams

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Trussed beams

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Trussed beams

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Three pin frames

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Three pin frames

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Laminated timber members

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Calculating the necessary area and moment of inertia of the column

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Pitched roofs

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The proposal dimensions pitched roof

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Timber structure

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Trusses

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Physical models of Timber Structures

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Physical models of Timber Structures

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Physical models of Timber Structures

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References [1] Kandemir-Yucel, A., Tavukcuoglu, A., CanerSaltik, E.N. (2007): In situ assessment of structural timber elements of a historic building by infrared thermography and ultrasonic velocity. Elsevier Publishing Co., Oxford, England. Infrared Physics & Technology 49 : pp. 243–248.

[2] Kloiber, M., Kotlínová, M. (2007): Nedestruktivní defektoskopické přístroje používané při provádění stavebně technických průzkumů historických dřevěných konstrukcí. In: Stavební ročenka 2008, Jaga, Bratislava, Slovensko : pp. 39-43.

[7] Michael Hough, “Principles for Regional Design” in Theory in Landscape Architecture, ed. Simon Swaffield, (Philadelphia, 2002), 211.

[8] Avent, R.R., “Design Criteria for Epoxy Repair of Timber Structures,” Journal of Structural Engineering, Vol.112, No.2, Feb, 1986, p.222.

[9] Avent, R.R., “Decay, Weathering and Epoxy Repair of Timber,” Journal of Structural Engineering, Vol.111, No.2, Feb, 1985, p.328.

Sabah Shawkat ©

[3] Lear, G. Ch. (2005): Improving the Assessment of In Situ Timber Members with the Use of Nondestructive and Semi-Destructive Testing Techniques. M.S. Thesis – North Carolina State University, Raleigh, , 137 pp.

[4] Pellerin, R. F., Ross, R.J. (2002): Nondestructive Evaluation of Wood, Forest Products Society Madison: pp. 210 p.

[10] Triantafillou, T.C., “Shear Reinforcement of Wood Using FRP Materials,” Journal of Materials in Civil Engineering, May 1997, p.65.

[11] Hallstrom, S. and J.L. Grenestedt, “Failure Analysis of Laminated Timber Beams Reinforced with Glass Fibre Composites,” Wood Science and Technology, 31 (1997) p.17.

Green

[12] Zombori, B.; 2000 – “In situ” Nondestructive Testing of Built in Wooden Members”. NDT.net – March 2001, Vol. 6, No. 03.

[6] Simon Swaffield, “Integrating Site, Place, and Region” in Theory in Landscape Architecture, ed.

Simon Swaffield, (Philadelphia, 2002), 207.

[13] Rinn, F.; 1994 – “Resistographic Inspection of Construction Timber, Poles and Trees”. Proceedings of Pacific Timber Engineering Conference. Gold Coast, Australia.

[5] Alex Nikolai Steffen, “The Next Revolution”, in WIRED, (May 2006), 139

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I - Creativity in Structural Design

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Architects and engineers are executives of services to reap the intellectual property of previous generations. They are studying, improving and discovering. They are the creators of new spaces, forms and structures that are constantly improving.

This creative activity connects architects and engineers on their way to the art of “Prof. Frei Otto

Lightweight structures used in various forms and variations can be seen in broad spectrum use on the current market. These structures, developed over the years, together with advances in material engineering and technology, continue to progress and are now an integral part of architectural creation.

construction of amphitheaters, sports stadiums, airports, courtyards, building facades, parks, seafront and interiors.

Designing lightweight constructions to meet all criteria is a complex task. Every part is visible and constructive, relying on the proper functioning of all parts. For example, diaphragm fabrics are developed to meet the requirements for high tensile strength, long life with a high modulus of elasticity.

The surface layer applied to the material ensures fabric resistance against weathering and dirt, provides resistance to UV radiation and has noncombustible properties.

Sabah Shawkat ©

Architectural lightweight structures can now be seen in different shapes and sizes. They may be internal, external, permanent, temporary, large, small, supported, membranes filled with air or stretched. New subgroups of these constructions - shell structures, tensile-integrity structures - are also very current. These unique forms have played an important role in contemporary architecture, interior design and various cultural events since the time they first appeared in the 1960s by the worldfamous German architect and engineer Frei Otto.

At present, light constructions are designed and constructed independently of the geographic location. They transform the space and have so subtle and elegant quality. In addition to providing basic functions such as shading and shutdown, they are an important and functional element in the

We think it important to explore new trends in lightweight constructions, among which we can include: 1. Lightweight designs designed with regard to sustainability 2. Modular light constructions 3. Sliding light constructions

Designers see the benefits of mass production of simple structures, such as an “umbrella” that has reproducible components.

Savings from a modular strategy lead to costeffective constructions. The clients’ demands are high nowadays, there is a demand for temporary, transformable solutions that can offer sliding lightweight constructions ranging from simple sliding

I3 marquee solutions to staircases, which disappear by pressing the button (Khalifa International Stadium).

Tension grid systems are being developed today for NASA and form unprecedented options for blame.

The shell structures, once iron-reinforced, are now parameterized (Robotic Fabrications), converted to various non-traditional material concepts (Timber Shells). A great challenge for today’s architects is global warming, membrane structures are increasingly seen in our territory and will be an integral part of sustainable climate solutions for public spaces, parks. We therefore consider this issue to be very topical and important.

analysis (process shape search), the areas are curved in two directions, which are generally positively evaluated aesthetically and represent a very non-traditional form of revitalization of the public architecture in the Central European space.

However, in many practical cases, in the world, lightweight and large-scale structures are the only structural layer of objects. Its role is, besides aesthetic criteria, to combine architectural, static and facade functions and, moreover, to provide a pleasant interior of the building.

This step combines the role of architect and engineer, naturally deducting the requirement for multidisciplinary qualification of a designer. The unification of the role of architect and engineer in designing lightweight constructions is also a top trend in top foreign workplaces and is known, for example, under the Archineer brand. Our team aims to train such specialists, who will be able to apply in artistic and technical practice thanks to the interdisciplinary integration of knowledge.

Sabah Shawkat ©

The ambition of the authors is to prepare the basis for a new training program lightweight construction, which is not yet adequate in Slovakia.

Easy constructions are specific by optimizing the built-in material. This is achieved by minimizing or completely eliminating bending stress. This special case can only be achieved on structures with a specific shape. Thus, lightweight structures, in place of stiffness of the material, derive their resistance from their shape and may therefore be denoted as shape-active.

One of the main design steps of lightweight

constructions is therefore the design of a shape that is controlled by the required stress state.

The specificity in the design of light structures is the fact that the result of the shape

I4

Modern Lightweight Constructions - Tensile Integrity Structures - Membrane Structures - Reciprocal Frame - Grid Shells - Tensairity - Suspension Bridges, Cable Stayed Bridges - Pneumatic Structures

Sabah Shawkat ©

I5

The Art of Tensegrity History of Tensegrity

Sabah Shawkat ©

- Karl Ioganson 1920 - Richard Buckminster Fuller (1895-1983) - Kenneth Snelson 1965 - Rene Motro 2003

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Definitions of Tensegrity – Connected set of tension members and a disconnected set of compression members – Islands of compression in a sea of tension – Free-standing pin-jointed networks

Sabah Shawkat © – Tensile integrity structures – Self-supporting structures

How we can Fuse Art, Science and Engineering?

Tensegrity is a proper way to reach this task

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Characteristics of a Tensegrity Structure: – The structure is free-standing, without any support – The structural members are straight

Sabah Shawkat ©

– There are only two different types of structural members struts carrying compression and cables carrying tension

- The struts do not contact with each other at their ends

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Tensegrity as a Models

Tensegrity as a Models

Sabah Shawkat ©

I9

Tensegrity as Art

The dome is an architectural art element, it may be positive synclastic structures or negative antisynclastic Structures. Domes are said to be those whose structure is in a state of compression.

We know the dome as corbelled dome, compound domes, cross arched dome, onion dome, oval dome, Parabolic dome, geodesic dome, sail dome, saucer dome, umbrella dome, glass domes, transparent plexi glass domes, bamboo domes, fractal dome.

The dome as an art that was made by the author of this book is geodesic dome. The concept of the structures are remarkable construction composed of non-touching compression members embedded in a mesh of tension members, in such a way that the compressed members (usually bars or struts) do not touch each other and the tensioned members (usually cables or tendons) delineate the system spatially. The term geodesic dome was coined by Buckminster Fuller. Shorter columns or struts in compression are stronger than longer ones. This in turn led some, namely Fuller, to make claims that tensegrity structures could be scaled up to cover whole cities. The other pioneer of lightweight structures was also Kenneth Snelson.

that extends back into history and they have been constructed from mud, stone, timber, brick, reinforced concrete, steel, glass, and plastic over the centuries. The symbolism associated with domes as a structural system was an evident case starting from the Mesopotamia.

Tensegrity as an Art

Sabah Shawkat ©

The dome gives the impression of a sail pinned down at each corner and extending upwards. Domes have a long architectural lineage

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Magical Floating Tensegrity Tea Table

Our passion for creating innovative solutions in the field of design of anti-gravity tensegrity models, which bring beauty and elegance to our spaces, led me to design these tea tables.

The structures of these models are made of wood or aluminium gives me a sense of freedom while the models fly in the air thanks to the suspension steel ropes which ensure the stability of entire system, It’s really beauty of physics and the elegancy of tensegrity.

then connect them in place, It works beautifully.

The centre string provides tension and the other strings provide balance it follows that all strings are necessary, not just the centre one but the centre string is the most important one. Take any of them out and the whole thing falls apart.

Tendon is a tension component of a tensegrity structure and can be divided into active tendons and passive tendons. Passive tendons are tension elements that are deployed without any method of modulating their characteristics. Passive tendons are intended to be adjusted, or tuned, periodically. Active tendons are tension elements that have the ability to alter their behaviour. These may include mechanisms to change length, and rigidity.

Sabah Shawkat ©

The optical illusion is my favourite part of the end result when I have replaced the main parameters such as the middle cable with magnets and the best part is that it does not require a lot of time or complex materials, so anyone can start producing something like that. Very nice application of the basic Tensegrity structure; simple in principle, finicky to build.

According to Anti-Gravity models that were made by the author in the studio to serve as tea tables the load carrying capacity of the structure is equal to the tensile capacity of the string. Since only one string is provided at the centre aesthetically it is elegant. It’s just a matter of achieving the right tension strings to centre the structure and

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Antigravity table 1

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Antigravity table 1

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Antigravity table 2

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Antigravity table 2

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Antigravity table 3

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Antigravity table 3

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Antigravity table 4

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Antigravity table 4

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Antigravity table 5

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Antigravity table 5

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Antigravity table 6

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Antigravity table 6

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Tensegrity sculpture

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Tensegrity sculpture

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A Tale of Anti Gravity

Tensegrity Structures

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Anti – gravity floating tensegrity

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Magical anti – gravity table

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Magical anti – gravity floating tensegrity

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Magical anti – gravity floating tensegrity

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Magical anti – gravity floating tensegrity

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Magical anti – gravity floating tensegrity

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Magical anti – gravity floating tensegrity

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Magical anti – gravity floating tensegrity

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Forms in motion - Tensegrity structures

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Form in motion - Tensegrity structures

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Lightweight table

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Lightweight tables

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Anti – gravity floating tensegrity application in interior

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Tensegrity as a Chair

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Tensegrity as a Chair, student work, created by Lenka Kozáková.

Head of project : Sabah Shawkat, Miroslav Debnár

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Tensegrity as a Hammock

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Geodesic Dome Structures Geodesic means the shortest distance between two points. The structure consisting of as many struts of the same length as possible as well as congruent surfaces. It is a network of equal triangles whereby the cross points are always situated on the surface. This triangulation guarantees strength and rigidity of the ball-shaped structure. They act therefore neither as tension nor as compression element. There is no direct contact between the compression elements.

Fuller’s domes have a framework of rigid struts which hold tension and compression. The struts are combined to triangles, pentagons or hexagons, whereby each strut is aligned in a way that each connection point is held in a firm position. This guarantees the stability of the whole structure. Tension is distributed equally to all parts of the whole construction. Increased tension in one part provides increased tension in all parts.

A global increase of tension is balanced by an increase of tension in various parts. Whilst tension is thus distributed evenly in the whole system, only individual parts actually balanced by compression.

The tensegrity model. According to Buckminster Fuller the icosahedron is a basic tensegrity structure (Buckminster Fuller 1975). It is a three dimensional structure consisting of twenty triangle surfaces. Loads applied at any point distribute about the truss as tension or compression. There are no levers within the truss. Only trusses are inherently stable with freely moving hinges.

The only way to fully stabilize and constrain any structure is by triangulating surfaces or cavities in compression and/or tension in all three dimensions. Tensegrity structures on the other hand show the forces acting upon them by differentiating out tension and compression vectors into separate components.

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Fuller was the first to note that tensegrity systems can be constructed as structural hierarchies in which the tension or compression elements that comprise the structure at one level are themselves tensegrity systems composed of multiple components on a smaller scale.

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Tensegrity as a Dome

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Geodesic domes - Polyhedra models

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Geodesic domes

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Geodesic dome student work, created by Kristína Betušová Head of project: Sabah Shawkat

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We created a model of a hyperbolic paraboloid using 15 straight 30cm length wood elements with dimensions 6mm x 6mm. As we know from mathematical diagrams the hyperbolic paraboloid is a ruled surface, which means that we can create it using only straight lines even though it is curved.

First we created an equilateral triangle using three elements of our straight timbers. Then we used this triangle as a template to form two skewers into a 60-degree angle, completing 2/3 of another equilateral triangle. Connecting the first triangle by

one skewer to the second triangle, created together a regular tetrahedron shape. Further, we marked the edges of the tetrahedron in regular 4 cm intervals to connect the skewers between the marks. Then we connected a skewer to the first marks on each of the edges. In another step, we created another layer of skewers so that they would overlap. We kept connecting up skewers by moving one interval mark on each edge. The overlap got greater each time.

To reach the goal we had to use two of the other edges to form the other ruling lines. The process of creating this model is demonstrated in the following figures.

Make a Hyperbolic Paraboloid Model

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Make a Hyperbolic Paraboloid Model

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Tensegrity as a pedestrian bridge

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In recent years there has been a growing trend to design structures of modern lightweight materials for the construction of pedestrian bridges proportional with long spans. Because in these slenderer designs it is obvious that due to the increased flexibility of such structures, dynamic forces (action that cause significant time-dependent force input to the structure or structural members) can cause greater amplitudes. The slenderer the structures, the more attention should be given to vibrational phenomena.

The increase in vibration problems in modern footbridges shows when people use bridges that should no longer be designed for static loads alone. The vibration problem of pedestrian bridges can arise in the horizontal and vertical directions, and in addition, the bridge deck may be twisted.

The design should take in to account the vibration performance of the footbridge due to walking pedestrians. It is important to note that there are currently no code regulations available.

regulations such as euro code, limits the construction of footbridge design: very slender and lightweight structures, such as cable bridges and suspension bridges, may not satisfy these requirements. In addition, the dynamic response is determined not only by natural frequencies, but also by dumping properties, bridge weight and pedestrian load.

The dynamic effect (Caused by the effect of people jumping on the bridge decking, they rock their bodies horizontally, shaking ropes, etc. In this case, comfort is certainly not fulfilled, but the structure must not collapse) or the dynamic effect due to cyclists is negligible compared to actions caused by walking and running individuals. For better understanding we show the calculation of natural frequency.

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In case where a footbridge is susceptible to vibrations that might affect the comfort, additional information is given concerning such as measurement procedures and methods for evaluation of dynamic properties, design modification and introduction of damping devices should be consider.

However, compliance with the natural frequency requirements, which are set out in many

The critical ranges for natural frequencies f1 of footbridges with pedestrian excitation are: -For vertical and longitudinal vibrations: 1,25Hz < f1 < 2,3Hz -For lateral vibrations: 0,5Hz < f1< 1,2Hz

Footbridges with frequencies for vertical or longitudinal vibrations in the range 2,5Hz < f1 < 4,6Hz might be excited to resonance by the 2nd harmonic of pedestrian loads. In that case, the critical frequency ranges for vertical and longitudinal

I51 vibrations expand to: 1,25Hz < f1 < 4,6Hz Lateral vibrations are not affected by the 2 harmonic of pedestrian forces might take place.

Until now there is no hint in the literature that the significant vibration of footbridges due to the second harmonic of pedestrians have occurred.

sorted by the amount of energy that is activated by the oscillation; the first natural frequency is that on the lowest energy level and is the most likely to be activated.

nd

The equation for the natural frequency of a single degree of freedom (SDOF) system is:

Natural frequencies as the original system:

Where K is the stiffness M is the mass.

Sabah Shawkat ©

Where fi´ is the i natural frequency, expressed in Hz ki´ is the modal stiffness of mode i mi´ is the modal mass of mode i. th

Thus the modal can be interpreted to be the mass activated in a specific mode of vibration.

Natural frequency = Eigen frequency: A natural frequency is a frequency of free vibration of a system. For a multiple degree of freedom system, the natural frequencies are the frequencies of the modes of vibration. Each structure has as many natural frequencies and associated modes of vibration as degree of freedom. They are commonly

Hence in modern footbridge designs, the assessment of human-induced vibrations needs to be considered by the designer to ensure that a. Vibrations due to pedestrian traffic is acceptable for the users, b. The lock-in phenomenon (when pedestrian motion and bridge vibration were strongly coupled) does not arise, c. The footbridge does not collapse when subjected to intentional excitation (jumping on the bridge decking, shaking stay cables, swaying body horizontally) The design tools should consider all of these factors. Provided that the vibration behaviour due to

I52 expected pedestrian traffic is checked with dynamic calculations and satisfies the required comfort, any type of footbridge can be designed and constructed.

If the vibration behaviour does not satisfy some comfort criteria, changes in the design or dumping devices could be considered. Footbridges should be designed in such a way that this pedestrianbridge-interaction phenomenon (pedestrian motion and bridge vibration were strongly coupled where self-excited response of the large amplitude may take place and cause discomfort), does not arise.

These lightweight footbridges have lower natural frequencies, resulting in a greater risk of resonance. Resonance occurs if one natural frequency of the bridge coincides with the frequency (The frequency f is the reciprocal of the oscillation period T (f =1/T)) of the excitation, e.g. the step frequency of pedestrians.

When a footbridge is susceptible to vibrations that might affect the comfort, additional information is given concerning such as measurement procedures and methods for evaluation of dynamic properties and design modification and introduction of damping devices should be consider.

A system is at resonance when any change in the frequency of a forced vibration, however small, causes a decrease in the response of the system.

When damping is small, the resonant frequency is approximately equal to the natural frequency of the system (the frequency of free vibration).

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Pedestrian induced excitation is an important source of vibration of footbridges.

Pedestrian loading is by nature unsteady, transient and waddling in a small range of excitation frequency. It is therefore obvious that dynamic responses play a fundamental role in the design of vibration susceptible structures. Vibrations of footbridges may lead to serviceability problems, as effects on the comfort and emotional reactions of pedestrians might occur. Collapse or even damage due to human induced dynamic forces have occurred very rarely.

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Design of cable-stayed bridge

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Design of cable-stayed bridge

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Proposed construction procedure

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Design of cable-stayed bridge

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Design of suspension bridge

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Design of suspended bridge, different elevations

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Tensegrity as a Lighting

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Tensile Fabric Structures What Are Tensile Structures And How Do They Work?

Sabah Shawkat © Types of Tensile Structures Applications And Uses For Tensile Buildings

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Light tensile structures

Light tensile structures belong to membrane structures in so-called “free form shapes”. We can consider them as an expression of modern fine art structural design, architecture and they have large potential to develop in the future. Tensile structures have negligible thickness compared to the other dimensions, it means that their static efficiency is based on their shape.

and erection processes on the one hand, and geometry and load-bearing behaviour on the other.

Finding the minimal surface is defined by three criteria: - Minimum surface is between any boundary - Equal and opposite curvature at any point - Uniform stress throughout the surface

Sabah Shawkat ©

The creation and formation of new space which could not otherwise be accomplished by using conventional methods is common for all such edifices.

Because of the many advantages of this modem style, several permanent membrane structures have been officially approved and constructed around the world. The uses of these structures vary widely from sports facilities to exposition buildings. Such structures are becoming more and more recognized as an independent field of fine art of structure engineering and architecture design.

There is an important and interesting interrelationship amongst the different types of lightweight structure: between manufacture

A minimal surface may be anticlastic or flat. Anticlastic tensile structures are flexible membranes with double curvature and prestress are essential for stability.

A surface of flat or triangular boundaries is always flat. Flat membranes are unstable structures under load. Increased curvature increases stability.

Membrane structures are often referred to as textile structures. However, the actual membrane construction is far removed from the classic tent.

The main difference is its exact geometric shape.

For the functioning of the membrane structure, the exact geometric criteria must be computed.

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The basic criterion is to maintain the concavity and convexity of the main directions of the membrane surface. Following this principle, we can talk about the basic four types of membranes: • saddle-shaped (hyperbolic paraboloid) • conical shape, • Wave shape, • vault shape.

• bars (struts) systems - stable problems of

compressive and bent bars, • cables systems - prestress, stiffness of the

structure, necessity of solving the theory II. order, • membrane systems - prestress, dynamic

resistance, large deformation solutions.

There are a lot of different names for tensile structures.

Sabah Shawkat ©

Another criterion of lightweight membrane construction is its prestress. The correct geometric shape and prestress guarantees its stability, stiffness and dynamic resistance. At the same time, it allows the structure to resist the effects on which it was designed, rain, wind and snow.

Talking about modern systems of steel, wire and membrane has its merit. And that these systems are at the top of the current building options. Limits are given by the physical properties and laws of the material and the construction system. These must be fully taken into account and used in the creation of the modern system.

In practice, we most often encounter the following issues:

– Tension membrane structures – Tensile membrane structures – Tensile fabric structures – Thin - shell structures – Tensile facilities – Tensile buildings

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Small Scale Counts

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Reliability Properties

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Shape coefficient for the bar element

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Introduction to FEM

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Example element types

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Example element shapes

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Shape functions

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Principal stresses - Mohr´s circle of stress

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Plane stress state

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Force density method

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Cable net-membrane

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Form finding

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Triangular membrane element in 3d space

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Hybrid structures in design, form finding and analysis

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Static analysis vs form finding

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Mechanical behaviour of a cable under tension/compression

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Differential Geometry Surfaces

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Differential Geometry Surfaces

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Installation steps of membrane structures

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Membrane structures

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Detail of the arch , hangers and attachments of net cables

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Riyad Stadium - elevation of one unit

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Connection details of membrane structures

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Connection details of membrane structures

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Assemblage Structural Elements y are spatial 1.

They areofspatial Working Uniquely under Tensile Stress y are doubly 2. Theycurved are doubly curved 1. They are spatial y are 1.prestressed 3.

They are prestressed They are 2. They arespatial doubly curved 2. They are curved 3. They aredoubly prestressed y are 3.nice 4.

They are nice They are prestressed 4. They are nice 4. They are nice

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ey are Shaped in a Form of Minim face ! They are Shaped in a Form I107

of Minimal Suface ! surface, that localy minimizes its area

a. It is a surface, that localy minimizes its area

y means under certain b. Localy meansconstraints under certain constraints c. Constraints are represented by boundary conditions supports

raints are represented by boundary conditions - supports

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Form Follows Force For the surface in equilibrium, it have to stand = No stiffness = No loading possible :

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Let´s make it simple

What do we have now ?

That means:

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How do we divide them Membranes

Cable Structures

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Design Process

In opposite direction is the right way

Form Finding

Form Finding Form Finding

Form finding is a process to find the equilibrium state of Form finding is a process Form finding is a processtoto find the equilibrium state of a cable-membrane find the structure equilibrium state of a

Structural Analysis

Structural Analysis Structural Analysis

Process that determines structural response under Process that determines Process that determines structuraland response under given loading supportstructural response under given loading and state supportinggiven conditions.

Stress a cable-membrane structure structure cable-membrane at loading and supporting at a given stress level and ing conditions.

Stress state and structural displacements at a given stress level and a given stress level and with conditions.

Stressdisplacements state and and structural with specified boundary investigated. with specified boundary arestructural displacements are are investigated.

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specified boundary conditions. conditions. conditions.

investigated.

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Detailing

Detailing can have Detailing Detailing canon have significant impact significant impact on can have significant theDetailing global behaviour the global behaviour on thestructure global behaviour of impact the whole of the whole structure of the whole structure and and therefore have to and therefore have to have to be realized Detailing can have betherefore realized precisely be realized precisely Detailing precisely according assumpsignificant impact onassumpaccording to to assumpaccording to tions done indone analysis the global behaviour tionsin in analysis tions done analysis

Detailing

Cutting Patte

Cutting Patter

Cutting Pattern

The production plan of ca The production plan of cable-membrane The production plan of cable-structure i ble-membrane structure is called pattern.This Th membrane structure is called calledcutting cutting pattern. cutting pattern.step This is deals with cutting, stepisstep isdeals deals with cutting, with cutting, developement and developement andcomcomdevelopement and Cutting Pattern The production of cacompensation of fabricplan cloths. pensation of fabric fabric cloths. pensation of cloths

Cutting Pattern

ble-membrane structure is called cutting pattern. This The production of castep isplan deals with cutting, ble-membrane structure is developement and comcalled cutting pattern. This step is deals with cutting, pensation of fabric cloths.

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of the wholecan structure Detailing have significant impact and therefore haveonto the global behaviour be realized precisely of the whole structure according to assumpand therefore have to realized tionsbedone inprecisely analysis according to assumptions done in analysis

developement and compensation of fabric cloths.

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Process of Form Finding - in a Final Equilibrium Known forces

Unknown Geometry

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Analysis

Analysis

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Keep the structure in admissible tensile stress state under Keep the structure in admissible tensile stress state under every load comevery load combination and preserve the reaction forces as bination andas preserve small possiblethe reaction forces as small as possible.

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Detailing - Terminology Corner point

Edge cable pocket

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Cutting lines and seams

Corner plate

Corner reinforcement

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Tensile Structures 1. Prestress Tendon

2. Membrane

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Design Process

Physical Models

Numerical Methods

1. Model

Form Finding

2. Measure

Analyze

3. Build

Details Pattern-Cut-Flatten-Compensate

Typical Shapes of Membrane Structures

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Reciprocal Frame A reciprocal frame is a three-dimensional structure with complex geometry, consist of linear members which are mutually supported and interlocking, forming either a flat, horizontal structure or a pitched three-dimensional frame system.

History has many examples Serlio, da Vinci and Villard de Honnecourt –but these early ones were all planar examples. RFs and structures similar to them have been built by many cultures throughout history.

structure in the UK. Graham used ‘reciprocal’ because of the way the beams mutually support each other.

The reciprocal frame is a three-dimensional grillage structure mainly used as a roof structure, consisting of mutually supporting sloping beams placed in a closed circuit. The inner end of each beam rests on and is supported by the adjacent beam. At the outer end the beams are supported by an external wall, ring beam or by columns. If the reciprocal frame (RF) is used as a roof structure, the inner polygon gives an opportunity of creating a roof light.

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Villard de Honnecourt, provides us with information on how to deal with the problem of beams shorter than the span, but he gives no information on the spans, his solution to this problem was a planar grillage and it adopts similar principles to the RF, for spanning long distances with shorter beams.

In this chapter we trying to present the opportunities the ‘reciprocal frame’ RF offers, defining the geometrical parameters of the structure and its structural behaviour but also describe the most common challenges that arise on side and on other, like any structural form, the RF structure has its limitations too.

The name ‘reciprocal frame’ comes from Graham Brown, who developed this type of

The morphology in models such as the length and number of beams that form the RFs will be used to describe the arrangement of structural members to helps create a particular three-dimensional and different architectural expressions. This allows for higher quality and greater speed of construction We are still investigating for a long time with several research students through both small-scale physical models and computer simulations, the different aspects of RF structures. In the field of RF morphology, and so we present in this book a several small timber models made by authors which simply illustrates the principles, morphology, geometry and structural behaviour of RFs how these structures work and transfer the load.

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We trying to create models as a design tool to contribute a better understanding of how these structures are set up, how they become structurally and how we can use this type of great structures in practice.

exploring the geometrical and structural principles which help us to understand how and why the RF will be integral to the design project.

I involved in scaling the RF of flat grillage structures and using them in real building structures to describe the load transfer and load paths through the structure.

Reciprocal Reciprocal FramesFrames These type of the structures of selfsupporting beams is very powerful. It clearly not only makes the buildings stand up, but affects how the spaces can be used as well as the overall architectural expression, these types of structure have been known for a very long time.Leonardo da Vinci’s proposals for temporary bridges, arched form created by using short timbers are assemblies of simply supported interlocking beams, smallscale physical model we create for teaching and

The multiple RFs can be divided into two basic groups: -multiple RF grids: Leonardo’s proposals of multiple grids -complex RFs: This type consists of more than one RF unit are complex RFs. These are formed by combining single RF units that are inserted in the central opening (the inner polygon).

A reciprocal frame A reciprocal is a three-dimensional frame is a three-dimensional structure withstructure with complex geometry, complex consist geometry, of linear consist members of linear which members are which are mutually supported mutually andsupported interlocking and interlocking

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I119 -Different morphologies of the complex RFs shows the potential and power of the structure, this way of the design gives the designer a unique opportunity for creating a new expression with each different RF configuration and how they may be used in architecture.

If these were to be used in building design they would need to be developed further and the forms would need to be rationalized to achieve

efficient structural design. In addition, depending on the material chosen for the structure.

In practical design, the first step towards designing an RF building would be to think about the architectural requirements for the size of the spaces, which will determine the RF spans; then to consider the cladding materials that would determine the roof slope and influence the roof dead load, because loads influence the type of detailing of joints.

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These type of the structures of self-supporting beams is very powerful. developed by Leonardo da Vinci, Villard de Honnecourt, Graham Brown

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- scale model physical model we create teaching mall-scale small physical we create forforteaching and explorand exploring the geometrical and structural principles ng the geometrical and structural principles which help us to which help us to understand how and why the RF will nderstandbehow and why theproject.

RF will be integral to the design integral to the design roject.

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Physical model of the MORPHOLOGY of Three- and four- beam RF assemblies, flat beam grillage

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Grid Shells The first freeform grid shell was realized in the city of Mannheim in 1975.

Lighten flat shell structures and simplify their construction Parametric surfaces use the display of a planar region 2d into a region in 3d space

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Multihalle in Mannheim, Frei Otto, Carlfried Mutschler and Joachim Langner

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Grid Grid Shell FormShell FindingForm

Finding

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Grid Shell Form Finding

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Catenary Curve An arch consists of two weaknesses which, leaning one against the other, make a strenght.

Leonardo da Vinci

Catenary in Mathematics

Sabah Shawkat © Eero Saarinen´s Gateway Arch in St. Louis has the form of catenary, that is, the form taken by a suspended chain.

y = acosh(x/a)

means a curve that descibes the shape of a flexible hanging chain or cable.

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Lightweight Structure in Architecture, student work, created by Chris Varga Head of project : Sabah Shawkat

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MÖbius Strip From Topology to Virtual Architecture ‘‘Topology is the study of the behaviour of a structure of surfaces that undergo deformation’’ The inside of the object becomes the outside, where floor become wall become floor or where a place is reversed into o non place

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MÖbius rotations of other geometric shapes

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Topology of MÖbius Strip

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A Tale of Möbius strip

The construction of the Moebius strip is complete. The surface has been deformed, without cuts or tears, and the rotation performed on one end of the strip has profoundly altered its properties.

A typical single continuous surface named after astronomer and mathematician in 1858 August Ferdinand Möbius (1790-1868) described for the first time a new surface in three-dimensional space in a work presented to the Academy of Sciences in Paris, a surface now known as the Moebius strip.

Then Moebius had described how it is possible to easily construct the surface that now carry his name: take a rectangular strip of sufficiently long paper, has only one side and one edge, and is made by twisting the band by 180° then joining the two ends.

twisting, stretching, turning and contorting, but no configuration of shapes using cutting, tearing or pasting. Here is the role of topology as seen by ab architect: “Topology is the study of the behaviour of a structure of surfaces that undergo deformation “

In topology, we no longer differentiate between two entities, two spaces, if you can pass from one to the other by means of a continuous deformation with neither leap nor cut this is the case of Möbius geometry which plays a crucial task in stylish design. Several Topological ideas were to be understood by artists and architects through the course of the decades, first by the artists and then much later by architects.

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Geometry or more precisely Topology, which aims to study geometric shapes that remain unchanged when the shape is subjected to deformations so complete that it loses all its metric and projected properties, such as shape and size from the perspective of modern mathematics began to develop in a completely different way in 19th century. And as we know that the meaning of Topology, is a (mathematical) way of conceiving of TOPOS: the place, the space, all space, and everything included in it. Topology theory entails deformation processes such as pulling,

The Möbius strip regulate a metaphorical looping and by its conversion of binary notions something having two parts in architecture, it offers new possibilities for architecture where the inside of the object becomes the outside, where horizontal floor become vertical walls become horizontal floor again or where a space is reversed into a nonspace. These interesting properties do not only mark the non-Euclidean nature of the double Möbius strip, thus making it an ideal polygon for formal and conceptual research in post-Cartesian architecture, they also transfer to a possibility of reversing and even uniting archetypal binary notions of surface/ volume, space/time, inside/outside, etc.

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This is why the Moebius strip has become a modern geometrical structure par excellence of topological architecture, being a relatively adequate representation of dwelling activities.

The Möbius strip is a mathematical construction demonstrating an evolution of a two dimensional plane into a three-dimensional space; by merging the inner with the outer surface, it creates a single continuously curved surface.

One consists of the fact that if you trace along the longest axis with a finger, you realise that you run the whole length and return to exactly where you set off, without having to cross the edge of the strip; the Moebius Strip has therefore only one single side, not two, an inside and an outside like a cylindrical surface has, for example.

Back before geometry was able to make use of the virtual animations of the computer, the various examples of forms were represented by models made of plaster or steel wire which aided the teaching of geometry.

It allows returning to the point of departure after having completed a tour by following a path along its surface. Studying the Möbius strip is a complicated task when using a physical model, but the current technology enables us to model and reconfigure it in a digital environment. if the curved profile of the structure following the path of the Möbius surface has positive direction at the starting point of the translation, it will have a negative direction after having completed the path and revolving to the starting point.

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In this way, many elegant examples of topology are produced; thus, the ensemble of spatial directions becomes a new topological space (topological space), which then resembles a Moebius strip.

On the other hand, for each of the preceding stages – point, knot, surfaces – we considered figures which had a “degree of liberty”: zero for the point, where no movement is possible, one for the knot, and two for the surfaces. This number of degrees of liberty is called dimension;

Space is our common ground, and it’s impossible to break free of it, even in our dreams.

Thegreat Master of the universe is an architect, and ever since Galileo we have been imagining him speaking and communicating in the language of mathematics.

Topology lends itself to comparisons and metaphors, for its flexibility is inscribed in its very structure: we can deform objects, as long as it’s done with gentleness and subtlety.

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Lightweight Structure in Architecture, student work, created by Mirka Grožáková.

Head of project : Sabah Shawkat

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Lookout tower is situated on Bradlo, which is one of the hills in mountain area Slanské vrchy in Eastern Slovakia. Hill is 840 meters above the sea level. Forest on the northern part of the hill is less wooded and offers beautiful views on the surrounding landscape, Slanec village, ruins of the castle of Slanec and Košice city, which is 15km far from the Bradlo. In good visibility, you can see High Tatras mountains from here. Distance between High Tatras and Bradlo is more than 100km. These views are main point of the loocation chosen for this lookout.

The design is inspirated with lissajous curve and spiral. Shape consist of two logaritmic spirals, which are connected at the top and lowest point. This connection of spirals makes ascending and descending path and that provides fluent movement. Base of the tower has diameter of 8 meters and exponcially grows to 12 meters at the top. Height of the tower is 25 meters. Construction is made from wood and glulam. Spiral pathway consist of glulam beams and wooden stairs which are divided into parts and mounted on the place.

Casing is made from wooden beams and is also a skeleton of the construction. Pathway is mounted on the main beams and is supported from the inside and outside.

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Lightweight Structure in Architecture, student work, created by Viliam Jankovič.

Head of project : Sabah Shawkat

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I140 Parametric Architecture or Curving Architecture in Light Weight Structures There is big trend about parametric approach and computation in design and architecture.

Parametric design is a process based on algorithmic thinking that enables the expression of parameters and rules that, together, define, encode and clarify the relationship between design intent and design response.

The fact that it is a process is clear, it is not “just a tool” also not be the “purpose” of a design.

It is a way of thinking. Actually a fundamental shift in thinking of how we go about designing (how we go about doing it). The main shift is one from a high fidelity in the manifestation of design concepts to a high fidelity in the expression of the logic of design concepts.

• parametric design is shapes and forms that have a curving nature, often similar to a parabola or other flowing forms in the shape of arcs.

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Surface plot

• or the entire shape of the structure can be in the form of flowing curves

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Sabah Shawkat © forms with curves and arcs that are atypical from the box-like, rectangular forms of most buildings.

- in conclusion, the overall form or shape of the object represents movement or action - parameters are a key term to this design approach, parabola or other conical cross section, using the mathematical approach to achieve these shapes.

- “Parameter”, which can be defined as, “in mathematics, a variable for which the range of possible values identifies a collection of distinct cases in a problem.

In the mid-20th century, designers and engineers, such as Frei Otto, would create these curving designs by utilizing mathematical equations as “parameters” to engineer these curves and forms.

I142 It is essentially an arc that is defined by a set of parameters, or numbers, whose shapes can be defined in an equation, curves are often given as the image of some function.

parametric form, the curve is all points (cos(t), sin(t)), when t varies over some set of values, like [0, 2π), or (-∞, ∞): (x,y)=(\cos \;t,\sin \;t), where t is the parameter.

- implicit form, the curve is all points (x,y) that satisfy the relation (x2+y2=1)

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I143 Patrik Schumacher, of the Zaha Hadid design group, has said that he, “believes the work of Frei Otto (1925 – 2015) is a precursor of Parametricism, as Frei “used physical processes as simulations and design engines to ‘find’ form rather than to draw conventional or invented forms.”

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References [1] Adriaenssens, S., Block, P., Veenendaal, D., Williams, C. - Shell structures for architecture: Form finding and optimization, Routledge, 2014 [2] Argyris, J. H., Angelopoulos, T., Bichat, B. - A general method for the shape finding of lightweight tension structures, Comput. Meth. Appl. Mech.

Engng 3, 135-149 1974 [3] Barnes, M.R. - Form-finding and analysis of tension space structures by dynamic relaxation: Ph.D. thesis, City University, London, 1977

[8] K. Ishii, Membrane Structures in Japan, SPS Publ. Co., Tokyo, 1995 [9] Kilian, A., Ochsendorf, J. - Particle-spring systems for structural form finding, Journal of the International Association for Shell and Spatial Structures, IASS, Vol. 46, n. 147,2005 [10] K.U. Bletzinger, M. Firl, J. Linhard, R. Wüchner Optimal shapes of mechanically motivated surfaces.

In Computer Methods in Applied Mechanics and Engineering, 2008

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[4] E. Ramm, K.U. Bletzinger, R. Reitinger - Shape optimization of shell structures, IASS Bull. 34, 103–121, 1993

[5] E. Ramm – Shape finding of concrete shell roofs, Journal of International Association for Shell and Spatial Structures, IASS, Vol. 45, n. 144, pages 2939, 2004 [6] G. N. Vanderplaats - Multidisciplinary design optimization. Vanderplaats Research & Development, Inc., 2007 [7] Haber, R.B., Abel, J.F. - Initial equilibrium solution methods for cable reinforced membranes, Part I and II. Computers Methods in Applied Mechanical Engineering, 30, 263–289 et 285–30, 1982

[11] K.U. Bletzinger, R. Wüchner, F. Daoud, N. Camprubi - Computational methods for form finding and optimization of shells and membranes.

In Computer Methods in Applied Mechanics and Engineering, Vol. 194, pages 3438–3452, 2005 [12] K.U. Bletzinger – Structural Optimization, Lecture Notes, TU Munich, 2014 [13] K.-U. Bletzinger, E. Ramm - Structural optimization as tool for shape design, in: C.Hirsch, et al., (Eds.), Numerical Methods in Engineering ‚92, Elsevier, Amsterdam, New York, 1992, pp. 465–477 [14] Balmond, Cecil. 1998. Number 9, The Search for the Sigma Code. Munich, New York: Prestel.

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[15] DECOI. 1999. Aegis Hypo-surface. Birmingham Hippodrome Foyer Artwork Competition (1st prize).

[16] PETERSON, IVARS. 2000. Möbius and his Band, Science News Online 8, 8.

[17] http://www.sciencenews.org [18] PETRESIN, E. 2001. Hypothesis of Fluid Dynamics / Hipoteza gibanja tekocin. University of Maribor.

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[19] SCHULTZE, W. 1978. Zahl, Proportion, Analogie. Muenster: Aschendorff.

[20] SHAWKAT, S. https://pinterest.com/seroomafad [21] SHAWKAT, S. https://issuucom/vsvu

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J I Terminology

J2 Architectural System - Architectural elements including interior spaces, building function, materials, partitions, exterior enclosures, noise control, thermal storage, and safety system.

Allowable Stress: Maximum permissible stress used in design of members of a structure and based on a factor of safety against yielding or failure of any type.

An Determinate Structure: In such a structure, the internal forces acting on any “cut“ taken through the structure can be found simply by using the equations of statical equilibrium and the structure is said to be determinate.

Allowable Stress Design (ASD): Design principle according to which stresses resulting from service or working loads are not allowed to exceed specified allowable values.

An Indeterminate Structure: is one in which the three equations of statics do not give one unique solutions for the unknown support reactions, but many possible selutions.

Balanced Load: Combination of axial force and bending moment that causes simultaneous crushing of concrete and yielding of tension steel.

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A Space Truss: is an assemblage of ties and struts in three dimensions connected by pin joints and works in a similar manner to a plane truss.

A Space Frame: is similar to a space truss but has fixed or moment connections.

Arch Structures or Cable Structures: is that the structures be shaped so that there is no shear or bending moment in them but only axial forces.

Anchorage: means, in post-tensioning, a device used to anchor the tendon to the concrete member, in pretensioning, a device used to anchor the tendon during hardening of concrete.

Balanced Reinforcement: An amount and distribution of flexural reinforcement such that the tensile reinforcement reaches its specified yield strength simultaneously with the concrete in compression reaching its assumed ultimate strain of 0.003.

Bond: Adhesion and grip of concrete or mortar to reinforcement or to other surfaces against which it is placed; to enhance bond strength, ribs or other deformations are added to reinforcing bars.

Bonded Tendon: means prestressing tendon that is bonded to concrete either directly or through grouting.

Building System: The structural, service, and architectural system of the building.

J3 Braced Frames: are those in which the lateral stability of the building, that is its resistance to horizontal forces, is assured by vertical stiffening systems, such as shear wall or core structures, to which the frame is connected. The frame is then only required to resist gravity laods. The combination of the bending and compression is only likely to be significant in end columns. The floor slabs can generally be designed to take vertical load only.

Beam: A structural member subjected primarily to flexure.

Beam-Column: A structural member that is subjected simultaneously to bending and substantial axial forces.

outer surface of the concrete; minimum values are specified to protect the reinforcement against corrosion and to assure sufficient bond strength.

Cast-in-Place Concrete: Concrete poured in its final or permanent location; also called in situ concrete; opposite of precast concrete.

Concrete Cover: means specified distance from the concrete surface to the nearest surface of reinforcement.

Current Systems: are those precast systems that have been used in the construction industry

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Beam Grids: is one particular arrangement of beam which may be used for floor and roofs. It consists of beam elements rigidly connected to each other in a two-way grid. To make efficient use of material the beam grid needs to be supported on all four edges and the plan shape of the supported grid should be as close to a square as is possible.

Concrete: A composite material that consists essentially of a binding medium within which are embedded particles or fragments of aggregate; in portland cement concrete, the binder is a mixture of portland cement and water.

Cover: In reinforced concrete, the shortest distance between the surface of the reinforcement and the

Creep: creep is the slow, continuing deformation of a material with under a constant load and may be contrasted with the instantaneous deflection of the load when it is first applied. For a linear-elastic material in the normal working range of stress, this instantaneous deflection is that calculated using the modulus of elasticity. One simple way of measuring the effects of creep is to use the creep factor which is the ratio of the strain in the material due to creep to that due to the instantenous elastic strain.

Concrete Composite Beam: is popular type in which a rolled steel beam is used with a concrete slab, either precast or cast-in-place, the tension and shear stresses being mostly taken by the steel and compression by the concrete slab.

J4 Composite Concrete Flexural Members: means concrete flextural members of precast or cast-inplace concrete elements, or both, costructed in separate placements but so interconnected that all elements respond to loads as a unit.

Column: Line elements which are substantially in compression are known as columns or, more generally, as struts. Used primarily to support axial compressive load. Or a member used to support primarily axial compression loads with a height of at least three times its least lateral dimensions; the capacity of short columns is controlled by strength; the capacity of long columns is limited by buckling.

Continuous Beam or Slab: A beam or slab that extends as a unit over three or more supports in a given direction and is provided with the necessary reinforcement to develop the negative moments over the interior supports; a redundant structure that requires a statically indeterminant analysis (opposite of simple supported beam or slab).

Continuous Beam: A beam that extends over several supports.

Core Structure: is a develeopment of the shear wall in which a number of shear walls are placed together, usually enclosing lifts, stairs or services to form tubes or other plan shapes.

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Column Strip: The portion of a flat slab over a row of columns consisting of the two adjacent quarter panels on each side of the column centerline.

Compression Member: A member subjected primarily to longitudinal compression; often synonymous with “column”.

Compressive Strength: Strength typically measured on a standard 16cm x 32cm or 15cm x 30cm. cylinder of concrete in an axial compression test, 28 d after casting.

Conjugate-Beam Method: the technique of applying the M/EI diagram as a load and determining the deflection as a bending moment is known as the cojugate-beam method

Combined Footing: are those that incorporate two or more point loads on the same foundation. To even out the bearing pressure as far as possible the combined footing is designed so that the resultant of all the point loads coincides with the centroid, or centre of gravity of the pad foundation.

Cracks: Results of stresses exceeding concrete’s tensile strength capacity; cracks are ubiquitous in reinforced concrete and needed to develop the strength of the reinforcement, but a design goal is to keep their widths small (hairline cracks).

Cracked Section: A section designed or analyzed on the assumption that concrete has no resistance to tensile stress.

J5 Cracking Load: The load that causes tensile stress in a member to be equal to the tensile strength of the concrete.

Deformed Bar: Reinforcing bar with a manufactured pattern of surface ridges intended to prevent slip when the bar is embedded in concrete.

Design Strength: Ultimate load-bearing capacity of a member multiplied by a strength reduction factor.

Diagonal Crack: An inclined crack caused by a diagonal tension, usually at about 45 degrees to the neutral axis of a concrete member.

brittleness is the starin capacity of the material. A material with a low strain capacity, such as glass or concrete, will crack when the strain in the material reaches a relatively low value Durability: The ability of concrete to maintain its qualities over long time spans while exposed to weather, freeze-thaw cycles, chemical attack, abrasion, and other service load conditions.

Effictive Depth of Section (d): means the distance measured from the extreme compression fibre to centroid of the tension reinforcement.

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Deflection: Movement of a structure or parts of a structure under applied loads.

Direct Stress: is any tensile or compressive stress which acts across any of the four faces at any crosssection throught a structural member.

Drop Panel: The portion of a flat slab in the area surrounding a column, column capital, or bracket which is thickened in order to reduce the intensity of stresses.

Ductility: the ductility of a bar in tension is its percentage increase in length at fracture over its original length before loading. Material with high ductility, such as polythene or mild steel, are said to be ductile materials, while materials with low ductility, such as glass, are said to be brittle materials. A property of a material associated with

Effective Span: The lesser of the distance between centers of supports and the clear distance between supports plus the effective depth of the beam or slab.

Embedment Length: means the length of embeded reinforcement provided beyond a critical section.

Equilibrium: If all the loads, whether vertical or horizontal, that act on the building structure are balanced by forces from the ground then the structure is said to be in equilibrium.

Endbearing Piles: endbearing piles transmit the load on them to the ground at the bottom of the pile.

Emerging Systems are those systems that are still under development or awaiting patent approval.

Effective Height: is in principle, the height of a column pinned at each end which would deflect in

J6 a similar way to the column being considered, the effective height depends on the way the column is held at the ends and on whether the column is free to move horizontally at one end.

Earthquakes Action: earthquakes cause accelarations of the ground so that inertial forces then act on the building structure. The resultant inertia force acts through the center of mass.

Or centre of gravity, of the building, therfore in earthquake areas, the centre of mass should as far as possible coincide with the shear centre in order to minimise the torsional moment on the building.

This may be achieved by placing the buildings mass and the buildings vertical stiffening systems symmetrically with one another.

Fixed Connection does not allow translational movements or rotation and transmits shear, axial force, bending and tortional moment.

Fracture Toughness: the fracture toughness of a material gives a direct indication of how likely it is that the material will fail by fast fracture and, more specifically, what sizes of defects may be tolerated in the material when it is in service, high values of fracture toughness indicate good resistance to failure.

Fast Fracture: is a failure of a material in a brittle manner, in which existing but small, perhaps almost invisible, cracks in the material suddenly increase in size, usually causing a complete and sudden failure of the material, in brittle manner, at stresses well below the yield strength of the material

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Elastic Curve: when a beam is loaded, it deflects.

The new position of its longitudinal axis is called the elastic curve Effective Prestress: means stress remaining in prestressing tendons after all losses have occurred.

Friction Piles: friction piles transfer the load in them to the ground, gradually down the length of the pile by friction between the pile and the ground.

Floor System: The structural floor system and the space below the floor (and above the finished ceiling of the level below) that ie required for service systems and architectural systems.

Fatigue: fatigue can occure when a material is given repeated cycles of stress, that is the material undergoes a fluctuating stress, which, for example, may change from tension to compression. One way in which cracks could increase in size is through fatigue.

Fatigue Ratio: the fatigue ratio is the ratio of this stress to the tensile strength of the material. If fatigue does occure then small cracks in the material will slowly increase in size and may be the cause of failure of the material by a fast fracture when the crack reaches a certain critical size.

J7 Flat Plate Floor Structure: in which there are no beams, the columns directly supporting the slab.

Flat plate are almost always continuous over several bays and for preliminary analysis may be sized by considering slab strips running in each direction through the centre of the slab and other slab strip running in each direction over the column positions.

Each slab strip can be analysed as a continuous beam assuming it to carry a uniformly distributed load which is slightly less and slightly more, respectively, than the actual uniformly distributed floor load on these widths of strip.

Hogging Moment: Bending moment causing upward deflection in a beam.

as walls, floors, roof, plumbing, and fixtures that are permanently attached to the structure.

Variable loads are commonly referred to as live or imposed loads which may include those caused by construction operations, wind, rain, earthquakes, snow, blasts, and temperature changes in addition to those that are movable, such as furniture and warehouse materials.

Ponding load is due to water or snow on a flat roof which accumulates faster than it runs off. Wind loads act as pressures on windward surfaces and pressures or suctions on leeward surfaces. Impact loads are caused by suddenly applied loads or by the vibration of moving or movable loads. They are usually taken as a fraction of the live loads.

Earthquake loads are those forces caused by the acceleration of the ground surface during an earthquake.

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Interaction Diagram: Failure curve for a member subjected to both axial force and the bending moment, indicating the moment capacity for a given axial load and vice versa; used to develop design charts for reinforced concrete compression members.

Jacking Force: means temporary force exerted by the device that introduces tension into prestressing tendons.

Loads: Loads may be broadly classified as permanent loads that are of constant magnitude and remain in one position and variable loads that may change in position and magnitude. Permanent loads are also referred to as dead loads which may include the self weight of the structure and other loads such

A structure that is initially at rest and remains at rest when acted upon by applied loads is said to be in a state of equilibrium. The resultant of the external loads on the body and the supporting forces or reactions is zero. If a structure or part thereof is to be in equilibrium under the action of a system Load, Sustained: means the specified dead load plus that portion of the specified live load expected to act over a period of time sufficient to cause significant long-time deflection.

J8 Load, Factored: means the specified load multiplied by appropriate load factors.

Linear-Elastic: For many materials in the normal working range of stress, the stress is proportional to the strain and the material is then said to be linear-elastic. Or a material in which the deflection is directly proportional to the load, at normal working loads, is said to be linear-Elastic.

Loadbearing Wall: is one used to carry gravity load but is known as a shear wall if it also transfers horizontal forces to ground. Whatever structural system is adopted, it must be able to carry the vertical loads associated with gravity forces, and the horizontal loads associated with wind and earthquakes.

elasticity is needed to calculate deflection, althogh a slightly diffrent value of the modulus may be used to calculate deflection due to bending to that used to calculate the extension of a bar in tension, for example.

Member: Any individual component of a structural frame.

Membrane Action: means that the structure is resisting load by direct stresses of tension or compression in the plane of the structural element.

If the deflection of the rectangular plate exceeds about twice the thickness of the plate then almost all the load is carried by the membrane action rather than by bending, for very thin plates this amount of deflection is reached even under light loading.

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Line (strip) Footing: walls or series of point loads may be supported on a strip footing, also known as a line footing.

Limit States: means those conditions of a building structure in which the building ceases to fulfill the function for which it was designed.

Modulus of Elasticity : the modulus of elasticity of a material is the ratio of stress to strain and is low if the material has a large stretch under load. For a linearelastic material, the modulus of elasticity is constant up to a point just below the yield point but, above this, large deformations or strains may occur with only small or no increases of stress. The modulus of

Moment Connection could be discribed as one which although only semi-rigid approaches the behaviour of a fixed connection and transfers a substantial bending moment. A fixed joints, or nearly fixed joints, between two members are said to provide continuity so that there is little or no rotation between the members at the joint and the members bahave as if no joint existed.

The internal forces in each element are split up into component forces which are either rotational forces consisting of bending moments and torsional moments, or translational forces consisting of axial forces and shear forces. These components forces and moments are known as stress resultants.

J9 Modulus of Repture: In general the modulus of rupture is close to but diffrent from the tensile strength of the material. If a material is tested in bending then the calculated tensile stress at which fracture of the material occurs is known as the modulus of rupture.

Neutral Plane: There is no direct stress in the horizontal plane at mid-depth of the beam One-way-Slab: a one-way-slab which is simply supported on two opposite sides and carries a uniformly distributed load may be imagined as a series of beam elements. There is no torsion.

Plastic Section: A cross-section that can develop a plastic hinge with sufficient rotational capacity to allow redistribution of bending moments within the section.

least three piles. The piles should be symmetrically placed under the loads and in general, the spacing between piles should be three times the pile diameter.

Points of Inflection: are transition points between the portions of the beam that are bent downwards and the portions that are bent upwards.

Reinforced Concrete: Concrete containing adequate reinforcement (prestressed or not) and designed on the assumption that the two materials act together in resisting forces.

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Plastic Hinge: Position at which a member has developed its plastic moment of resistance.

Pinned Connection allows rotation but no translational movements, it transmits shear and axial forces but not bending moment.

Pad (spread) or Column Footing: on a good ground, the normal way to support a point load, as from a column, is to use a pad foundation, also known as a spread or column footing.

Pile Foundations: if raft foundations are inadequate or unsuitable, pile foundations may be used, piles are usually used in conjunction with pile caps. To ensure stability, isolated pile caps usually have at

Raft (mat) Foundation: where the building must be founded on poor ground, a raft foundation, also known as a mat foundation, may be used to support the entire building. Another aim is to reduce differential settlement. Rafts are often combined with the bottom slabs of a basement structure. The design of a raft is similar to that of a continuous floor slab, except that now the load is applied by the ground from below and that unlike gravity loads on floors, the distribution of pressure from the ground is not generally known with any degree of accuracy.

Reinforcement: Bars, wires, strands, and other slender members that are embedded in concrete in such a manner that the reinforcement and the concrete act together in resisting forces.

Reaction: The load carried by each support.

J10 Rigid Frame: An indeterminate plane frame consisting of members with fixed end connections.

Slab: A flat, horizontal (or neatly so) molded layer of plain or reinforced concrete, usually of uniform thickness, either on the ground or supported by beams, columns, walls, or other frame work.

Simple Beam: A beam restrained at its end only against vertical movement.

Span: The distance between the supports of a beam or a truss.

Structural Concrete: Concrete used to carry load or to form an integral part of a structure (opposite of, for example, insulating concrete).

Services: Plumbing, HVAC, and electrical systems.

Shaped Line Element – (two-dimensional) such as arches, cables, and frames, arches and cables take axial forces of compression and tension respectively, this time along a curved path, frames take shear, axial forces and bending moments.

Surface Elements: (three-dimensional) such as the surface elements that occur in slabs, shells, or prestressed membranes, surface elements take shear and axial forces acting in the plan of the surface.

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Serviceability Limit States: are those states which restrict the intended use and occupancy of the building and include deflection, vibration, permanent deformation, and cracking.

Static Equilibrium: If there are no dynamic forces on the building such as those due to earthquakes or wind buffeting then the building will be at rest and is said to be in static equilibrium.A building is said to be stable if it is able to withstand the expected range of forces no matter in which direction they act.

Structural System: All structural components of the building, includding precast members, cast-inplace members, cast-in-place conections, welded connections, and bolted connections.

Shear Wall: are efficient at resisting horizontal loads which are applied in their own plane. As horizontal forces may be applied in any direction, the shear walls must be placed in at least two diffrent directions and, in addition, must be placed such that they cannot rotate about a point. Each shear wall should carry as much gravity load as possible in order to prevent tension occuring under horizontal forces.

Shear Force: An internal force acting normal to the longitudinal axis; given by the algebraic sum of all forces to one side of the section chosen.

Simple Connection: allows rotation and movement in any direction except downwards and only trasmits shear force that acts downwards at support.

J11 Sliding Connection: allows movement along the line of the element and rotation and only transmits shear force.

Simply Supported Beam or Truss: has no more than three unknown reactions, or forces, on it from the supports. If there is a vertical and horizontal rection from the support at one end and a vertical reaction from the support at the other end, then there are three unknown quantities which, therefore, may be solved by the three equations of statics.

elements can be joined to make a post and beam system, column elements, also known as struts, and tie elements can be joined to make a truss system, columns and ties take axial forces of compression and tension respectively, beams take shear forces and bending moments.

Strength Reduction Factor: Capacity reduction factor (typically designated as) by which the nominal strength of amember is to be multiplied to obtain the design strength; specified by the ACI Code for different types of members.

Silica Fume: Very fine non crystalline silica produced in electric arc furnaces as a by-product of the production of metallic silicon and various silicon alloys (also know as condensed silica fume); used as a mineral admixture in concrete.

Standard Cylinder: Cylindric specimen of 32cm. height and 16cm. diameter, used to determine standard compressive strength and splitting tensile strength of concrete.

Shear Connectors: are necessary to transfer the longitudinal shear stresses between the steel and concrete, more being needed near supports where shear stresses are greatest.

Split Cylinder Test: Test for tensile strength of concrete in which a standard cylinder is loaded to failure in diametral compression applied along the entire length (also called Brazilian test)

Short Columns: is one types of columns which have little or no tendency to buckle.

Stirrup: A reinforcement used to resist shear and diagonal tension stresses in a structural member; typically a steel bar bent into a U or rectangular shape and installed perpendicular to or at an angle to the longitudinal reinforcement, and properly anchored; the term “stirrup” is usually applied to lateral reinforcement in flexural members and the term“tie” to lateral reinforcement in compression members.

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Silider Column: is another types of columns which are prone to bukle at loads well below that which would exceed the compressive strength of the material.

Straight Line Elements: (one-dimensional) such as beams, columns, and ties, beam and column

J12 Statically Determinate Structure: A structure in which support reactions may be found from the equations of equilibrium.

Space Frame: A three-dimensional structure.

Shear Stress: is one which acts along any of the four “cut“ faces of the small rectangular area.

The Radius of Gyration: is geometrical property of the section which may be calculated for preliminary design, the radius of gyration about the minor or major axis of bending can be approximately related to the least width for some common section shapes

frames resist the horizontal laods, such as wind or earthquake loads, by frame action, that is by the bending action of the column and beam elements.

The floor beams in unbraced frames carry gravity laods in the same way as braced frames. However unbraced frames must also carry horizontal loads and in this case both columns and beam are put in bending in addition to that already present due to the vertical load. Note that, in general, all columns in unbraced frames are put both into bending and compression. The floor slabs can generally be designed to take vertical load only.

If straight line elements, shaped line elements and surface elements only work in this way as we mentioned above then are known as membrane structures.

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Toughness: the toughness of a material is the energy absorbed in making a unit area of a crack in the material.

Two-way-Slab: a two-way-slab which is simply supported on four sides and carries a uniformly distributed load also works in bending and torsion, in this case the torsion in the slab elements would cause the four corners of the slab to lift off from their supports Tension Stiffening Effect: The added stiffness of a single reinforcing bar due to the surrounding uncracked concrete between bond cracks.

Unbraced Frames: are those in which the frame is required to resist both vertical gravity loads as well as the horizontal, or lateral, loads. Unbraced

The lever arm depth, is the distance between the resultant of the compression at the top of the beam and the tension reinforcement at the bottom.

Ultimate Limit States: are those states which concern safety and include exceeding of load carrying capacity, overturning, sliding, fracture, and fatigue.

Ultimate Strength Design: Design principle such that the actual (ultimate) strength of amember or structure, multiplied by a strength factor, is no less than the effects of all service load combinations, multiplied by respective overload factors.

J13 Vierendel Truss: It is an assemblage of members that are not triangulated and must therefore work in bending and shear as well as in axial load, it should therefore be classified as a frame structure.

mements to occur. If possible the shear centre and the centre of mass should coincide throughout the height of the building to prevent twisting movements and oscillations ocurring.

Virtual Work: The virtual work principle may be applied to relate a system of forces in equilibrium to a system of compatible displacements. For example, if a structure in equilibrium is given a set of small compatible displacement, then the work done by the external loads on these external displacements equal to the work done by the internal forces on the internal deformation. In plastic analysis, internal deformations are assumed to be concentrated at plastic hinges.

Yield Stress: the yield stress, is the stress at the yield point, the yield point is the point at which plastic flow of the material begins to occur. Below the yield point the material is elastic and is able to recover its original shape when the load is removed. Above this point the material is said to be plastic and ungergoes a permanent deformation, which remains after the load is removed.

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Waffle Slab: which consist of a beam grid with an integral slab, in which the beam or rib elements are closely spaced, may be treated structuraly like a solid slab of the same overall depth.

Water-cement Ratio: Ratio by weight of water to cement in a mixture; inversely proportional to concrete strength.

Wind Forces: wind forces will cause the building to twist if the centre of pressure of the wind forces, that is the place where the resultant wind force acts, and the shear centre of the building are not close together, this is especially true in high-rise building.

In addition, high-rise buildings in which the plan position of the shear centre changes significantly within the height of the building will cause torsional

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Other books by Sabah Shawkat 1. Design of reinforced concrete members (2000) 2. Železobetónové konštrukčné sústavy (2002) 3. Deformation behaviour of reinforced concrete beams (2011)

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4. Konštrukčné projekty (2014) 5. Structural design I (2014)

6. Structural design II (2015) 7. Architektonika (2015)

8. Structural design III (2015) 9. Structural projects (2016) 10. Element design to shape a structure I. (2016) 11. Element design to shape a structure II. (2017) 12. Inžinierske drevené konštrukcie (2017) 13. The art of structural design (2017) 14. Lightweight steel structures (2018) 15. The art and engineering of Lightweight Structures (2019) 16. Art in/of Nature (2020) 17. Application of structural system in building design (2020) 18. Structural icons for architects (2021)

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1. Edition, Tribun EU, s.r.o.

Brno, Czech republic 2021 ISBN 978-80-263-1624-4