Notes of Design of steel structures

DESIGN OF STEEL STRUCTURES 1-Basis of structural design: Ch,PP 82 .

2- Analysis and design : Ch,PP 105 For statically determinate structures (simply supported beams ,cantilevers, trusses …etc) analysis is done using the laws of statics to determine the forces and moments on each member ,Then members can be designed using an suitable method (e.g.the working stress method ,limit state method,…etc ).

However ,for statically indeterminate structures, the analysis is different and the laws of statics are not suitable .A number of analytical methods have been developed ,which include the slope deflection method, moment distribution method,…etc.

Sizes of structural members should be assumed in advance. For this a preliminary design is required .Thus , analysis and design are interactive processes(have an effect upon each other).

Various computer programs are available for the analysis and design of different types of structures . However, the designer should be aware of any assumptions and limitations of these programs.

3-Codes and Specifications: 3.1 Codes are written for the purpose of protecting the public .They provide guidelines for the design and construction of structures . They are revised at regular intervals to include new developments (in research , materials, construction techniques ,… etc),experience gained from past design practice, behavior of existing structures, and failures of structures.

Codes serve at least four functions: a. They ensure adequate structural safety, by stating minimum requirements for the design. b. Help the designer in the design process; the results of long and tedious (long and difficult) sophisticated analyses are presented in the form of tables or charts ready for use.

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c. Ensure consistency among different designers. d. May protect the designer in case of disputes.

3.2 Project Specifications: They include the following items: a. Materials to be used in the structure. b. size of structural members(normally in the drawings). c. Joints and other details . d. Required quality of work ,tolerances . e.Instructions on how execution of work to be done.

F.Procedure and requirements for materials samples and shop drawings ( working drawings ) to be submitted for approval.

Specifications are provided to the contractor or builder as part of the contract document.

4-Design Philosophy: Various design philosophies have been used with regard to structural steel design.

4.1 The earliest is the working stress method of design (WSM).

It is based on linear elastic theory and although still used in few countries, it is being replaced by the limit state design philosophy, and the ultimate strength design (USD)or plastic design which was developed in the 1950s. It is based on the ultimate strength of steel at ultimate loads.

The limit state method (LSM) was introduced in the British code CP110 (1972)(now BS 8110)for reinforced concrete, and adopted for steel structures in the British codes BS 5950 and BS 5400(bridge code).

Now many British codes are being incorporated or classified within the euro-codesystem.

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In mid 1960s, probabilistic concepts of design were developed based on the theory that the various uncertainties in design could be handled more rationally in the mathematical framework of probability theory.

The risk involved in design was quantified in terms of a probability of failure. The method known as reliability-based method. This was simplified in the code and reduced to a format involving multiple (partial) safety factors rather than probability of failure.

4.2 Ultimate load Design (ULD): The shortcomings of the working stress method (WSM) led to the development of the ULD. This method is also referred to as the load factor method or the ultimate strength method, and for steel structures also known as the plastic design method,(research by prof. Baker in Cambridge University, UK). Plastic design is a special case of limit states design, where the limit state is obtained when the members reach plastic moment strength (Mp) and the structure is transformed into a mechanism, (refer to the plastic theory pp609, ch8). Mp is the moment strength when all the fibers of the cross section of a member are at the yield stress fy,(one side of the neutral axis is in tension and the other side in compression).

The safety measure in the design is introduced by choice of the load factor, which is defined as (the ratio of the ultimate load (design load) to the working load). The (USD) makes it possible to use different load factors for different types of loads, under combined loading conditions, thereby overcoming the related shortcomings of (WSM). ∑(working load X load factor)≤ ultimate load.

Load factors for plastic Design as per IS800:1984 Working

Load factor(min)

Dead load

1.7

Dead load + imposed load

1.7

Dead load + wind load or seismic load

1.7

Dead load + imposed load+ wind load or seismic 1.3 load

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For variables such as loads ,material strength ,and member dimensions ,the structure must be designed for the possibility of overload .There are also several unforeseen factors that influence the prediction of strength and serviceability ,They include construction methods ,workmanship and quality control ,service life of the structure ,human errors ,possible future change of use ,frequency of loading ,‌etc. These uncertainties make it impossible for the designer to guarantee that the structure will be absolutely safe.

4.3 limit States: Is the state beyond which a structure ceases to perform its intended function satisfactorily.

Limit states for structural steel are: a )ultimate (safety)limit state . b)serviceability limit state . ultimate safety limit state (ULS)

Serviceability limit state (SLS)

1.strength (including yielding ,buckling, 1.deformation and Deflection . and transformation into a mechanism.

2.stability against overturning and sway .

2.vibration (e.g. wind-induced ,floor vibration ‌etc).

3.failure due to excessive deformation or 3.Repairable damage due to fatigue rupture. (cracking).

4.fracture due to fatigue.

4.corrosion and durability.

5.Brittle fracture.

5.fire.

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Characteristic load are characteristic strength : In normal design calculations, a single value is used for each load and for each material property, with a margin to take care of all uncertainties .This is the characteristic load ,and characteristic strength ,which is defined as that value of strength below which more than a prescribed percentage of test results will fall,(95%).Thus the characteristic yield strength of steel is the value of yield strength below which not more than 5%of the test values may fall.

Similarly, the characteristic load, is defined as the load which is not expected to be exceeded with more than 5% probability, during the life span of a structure .Thus the characteristic load will not be exceeded 95% of the time.

The values to be used in the design are obtained from the characteristic values by the use of partial safety factors, both for material and for load.

4.4 Limit states Method (LSM): LSM aims for a comprehensive and rational solution to the design problem, by considering safety at ultimate load and serviceability at working loads ,by provision and use of multiple safety factors ,which supposed to have sound probabilistic basis ,involving the seperate consideration of different kinds of failures ,types of materials ,and types of loads.

LOADING AND LOAD COMBINATIONS: 1. loads: Before designing any structural member, loads acting on them has to be determined , (natural or man introduced loads).

Lode combination which will produce the most critical stress has to be determined are used in design.

The used load which has major effect on structural members are:

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1. Dead loads (DL).

2.Imposed loads (IL) or live loads (LL).

3.Wind loads (WL).

Other loads are present in special conditions, snow, earthquake, temperature ,erection and fabrication loads ,impact ,explosion, foundation settlement, hydrostatic ,earth pressure.

For assessing the loads acting on a structure, it is important to refer to a code of practice ,for example British codes ,Euro codes ,Indian codes ,American codes, local codes …etc .

1.1Characteristic Load: It is the load which will not exceed 95% during the life of the structure.

-Design loads are calculated by multiplying the characteristic load by partial safety factor (or load factors).

1.2 Load Combinations : A general expression for the effect of combinations of load effects is given by (Euro code 3-1993): Load effect = ∑¥G,j Gk,j + ¥Q,1Qk,1 + ∑i≥1¥Q,iΩ0,iQk,i Where Gk,j are the characteristic values of the permenant actions (loads) ,Q k,1 is the characteristic value of one of the variable actions ,Q k,i are the characteristic values of other variable actions ,¥G,j are the partial safety factors for the permanent actions G k,j , , ¥Q,I are partial safety factors for the variable actions Q K,i ,and Ω0,i are the factors that are derived statistically using the theory of probability .

For accidental design situations , Load effect = ∑j ¥GA,j Gk,j + Ad + Ω1,1Qk,1 + ∑i≥1Ω2,iQk,i Where Ad are the design values for accidental design situations ,¥ GA,j are partial safety factors for permenant actions along with an accidental design situation .

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The basic concept is that, in any loading case, each of the actions is multiplied by the appropriate load factor ,and the sum of the factored load is used in the design ,when there is more than one variable action, the second variable action is multiplied by a factor ,Ω, as in the above formula. This reduces the loading to account for the fact that the probability of all the variable actions reaching their maximum value at the same time is very low.

If the partial safety factors given in the (Indian code) are to be used, then the following load combinations are obtained: i- 1.5(DL+IL) + 1.05(CL or SL) ii- 1.2(DL+IL) + 1.05(CL or SL) ± 0.6(WL or EL) ii- 1.2(DL + IL ± WL or EL) + 0.53 (CL or SL) iv- 1.5 (DL ± WL or EL) v- 0.9 DL ± 1.5 (WL or EL) vi- 1.2 (DL + ER) vii- 0.9 DL + 1.2ER viii- DL + 0.35 (IL +CL or SL) + AL where CL = crane load, SL = snow load, ER = erection load, EL = earthquake load, AL= accidental load.

If we don’t consider CL, SL and AL, then will have (13) loading combinations to consider for a building in which horizontal (lateral) load is resisted by frames or walls oriented in two orthogonal directions, X and Y. 1. 1.5(DL + IL) 2. 1.2(DL+IL+ELx) 3. 1.2(DL+IL-ELx) 4. 1.2(DL+IL+ELy)

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5. 1.2(DL+IL-ELy) 6. 1.5(DL+ELx) 7. 1.5(DL-ELx) 8. 1.5(DL+ELy) 9. 1.5(DL-ELy) 10. 0.9DL + 1.5ELx 11. 0.9DL – 1.5ELx 12. 0.9DL + 1.5ELy 13. 0.9DL – 1.5ELy Where ELx and ELy are earthquake loads in x,y directions. Since earthquake and wind may not occur together , then wind load(WL) may replace (EL) if wind load is more frequent and predominant.

1.3 Load combinations for plastic Design: a) 1.7(DL +IL) b) 1.7(DL±IL or WL) c) 1.3(DL + IL ±EL or WL) d) 0.9 DL ±1.7EL or WL Imposed loads (IL): Classrooms 4.0, offices 3.5, Assembly areas 4.0.KN/m2.

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(Examples & Exercises )

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PLASTIC THEORY 1. Introduction: Structural design must satisfy two basic criteria: a) Strength, structure or member to carry load. b) Stiffness, structure or member will not deform or distort more than the permissible limit.

Elastic design method (working stress, allowable stress) was used in the past, which uses allowable stress based on the yield stress producing heavier structure than if the plastic method is used, which is based on the ultimate loads.

Two aspects of structural behavior are important: a) Behavior of steel in the plastic region of the stress-strain curve. b) The tendency of compression members to buckle and become unstable.

2. Basis of Plastic Theory: Tests on actual buildings in USA and UK (1930 Showed that steel frames behave differently from the assumptions made in the elastic theory, and the factor of safety of structural elements was very different from what has been assumed.

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In the elastic design method member capacity is based on yield stress. But steel, because of its ductility is able to stand (resist) large deformations beyond the elastic limit without fracture, and therefore steel possesses reserve (additional) strength beyond its yield strength. The method which uses this reserve strength is the plastic method of analysis.

The plastic theory makes the design more rational; since the level of safety is related to the collapse load and not to the apparent failure at one point. To explain the concepts of plastic analysis, consider an I-beam subjected to an increasing bending moment, M.

Idealized tensile stress-strain diagram for steel.

When the yield stress reaches the extreme fiber as in (b) the nominal moment strength(Mn) of the beam will be equal to the yield moment(My) given by: Mn= My=Zefy Ze is the elastic section modulus.

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Further increase of BM causes the yield to spread inwards from the lower and upper surfaces of the beam (stage c). further increase of BM, the whole section yields(stage d), and every fiber has a strain equal or greater than: €y = fy/Es The nominal moment at this stage is the plastic moment Mp, and is given by: Mp = Fy∫AydA = FyZp Where Zp = ∫AydA is the plastic section modulus.

Further increase in BM results only in rotation, until strain hardening occurs.

In the fully plastic stage, the stress is uniformly equal to the yield stress, and equilibrium occurs when the, neutral axis,(NA) divides the section into two equal parts.

C=T FyA1 = fyA2 Since A1 = A2 and A= A1+ A2 , A1= A2 = A/2 Plastic moment of resistance Mp=fyA1y1+fyA2y2 =fyA/2(y1+y2) 11

Thus, Mp=fyZp Where Zp=A/2(y1+y2) is the plastic modulus of the section .

Therefore ,the plastic modulus may be defined as the combined statical moment of the Cross-sectional area above and below the equal –area axis .

-Shape Factor: The ratio Mp/My is a property of the cross-sectional shape and is independent of the material properties. It is known as the shape factor, v, given by V=Mp/My=Zp/Ze 3.General Requirements for Plastic Design : The following assumptions are made to simplify computations: a.The material obeys Hooks law till the stress reaches fy . b.The yield stress and modulus of elasticity have the same value in compression and tension . c.The material is homogenous and isotropic in both elastic and plastic states . d.The material is assumed to be sufficiently ductile to permit large rotations of the section to take place . e. Plastic hinge rotations large compared with the elastic deformation so that all the rotations are at the plastic hinges. The segments between the plastic hinges are rigid. f. The magnitude of BM caused by external loads will at the most be equal to the plastic moment capacity of the section. A plastic hinge forms when the moment reaches the capacity of the section. g. The influence of normal and shear forces on plastic moments is not considered.

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h. Plane sections remain plane even after bending and the effect of shear is neglected . i.The equilibrium of forces at the time of collapse is considered for the un-deformed state of the structure . j.No instability occurs in any member up to collapse.

The fundamental concept of plastic analysis method is that sufficient plastic hinges are formed at critical sections of the structure ,which transforms the structure into a collapse mechanism .Extensive testing has adequately verified this concept.

-Plastic Hinge Concept: ******(Diagram) Mp = WuL/4 = fybh2/4 (since Zp=bh2/4) My = fybh2/6 =( fybh2)(2/3) = (2/3)Mp From triangle properties: My/(L/2 – x/2) = Mp/(L/2) My/Mp = ((L – x)/2)/(L/2) = 1 – x/L (L – x)/Mp = LMy , but My = (2/3)Mp So, x = L/3

(for this case)

Generally the number of hinges n= (r+1), where r is the number of redundancies (or indeterminacy ).Exceptions e.g. the partial collapse of a beam in a structure .

4. Plastic Collapse Load : It is the load at which a sufficient number of plastic hinges are formed such that a collapse mechanism is created. ( plastic limit load ).

In plastic design redistribution of BM occurs after the first hinge forms at the point of max BM ,and the final collapse load at which the collapse mechanism forms may be much higher than the load at which the first hinge forms .The BM at the first hinge remains constant at Mp, then second hinge forms and BM constant at Mp ,and so on third 13

and final hinge forms to produce a collapse mechanism. Final load will be 33% higher for a fixed beam having a load at the one third point .****(Diagram) -Examples of plastic-collapse mechanisms : ***** (Diagrams) For uniformly distributed load (UDL), location of plastic hinge is not well defined .

5. Conditions of Plastic Analysis : a)Equilibrium: The BM distribution defined by the assumed plastic hinges must be in static equilibrium with the applied loads and reactions. b) Mechanism: There must be sufficient number of plastic and frictionless hinges to form a mechanism.

The ultimate collapse load is reached when a mechanism is formed. c)Plasticity: The BMs in any section of a structure must be less than the plastic moment of the section, - M≤ Mp. 6. The Principle of Virtual Work : It may be stated as follows: If a system of forces in equilibrium is subjected to a virtual displacement, then the work done by the external forces equals the work done by the internal forces: We = wi ; which is the condition of equilibrium While applying this method to determine the moment at collapse, an arbitrary displacement is assumed at a plastic hinge location (one for which only the internal moments at the plastic hinges contribute to the internal work, W i ).(Rotations at points of 14

simple supports and at points where plastic moments are expected to occur in producing the mechanism).

7. Load Factors: A minimum load factor for a simply supported beam with a central point load: Elastically; M=WL/4 = fbcZe Plastically;Mp =WcL/4 = fyZp = fyvZe If we consider fbc = 0.66fy and a mean value of v = 1.14(I-section), we get: Load factor = Mp/M = Wc/W = (1/0.66)x1.14 =1.727 taken 1.7 approx.

(Refer to Table in section 4.2, page 3).

8. Continuous Beams: ***** 9. Portal Frames: ****** A similar behavior is observed for frames with pinned bases.

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For pinned-base frames: Mp = 짜L(wL2/8)(1 + k + (1 + k)1/2)

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For fixed-base frames: Mp = 짜L(wL2/8)(1/(1 + 0.5k + (1 + k)1/2) Where 짜L = 1.7 is the global load factor, k = h2/h1

AXIALLY LOADED COMPRESSION MEMBERS 1. Types of Compression Members : Columns,top chords of trusses, bracing members. Others have compression in some of their parts e.g compression flanges of beams rolled or built- up).

Columns are straight vertical members whose lengths are much greater than their thicknesses. Short vertical members in compression are called struts.

2. Modes of Failure: 15

a/flexural buckling (Euler buckling): Members are subjected to flexure, or bending, when they become unstable . b)Local buckling: Occurs when some part or parts of the section of a column are so thin that they buckle locally in compression before the other modes of buckling can occur. c)Flexural torsional buckling: Failure occurs by twisting (torsion)or by a combination of torsional and flexural buckling.

The tendency of a column to buckle is measured by its slenderness ratio which is the ratio of length to the least radius of gyration (l/r). Also by types of end connections, eccentricity of load, imperfection of column material, and residual stresses from manufacture.

Axial loads are concentric loads(exactly centered). Dead loads, and live loads may never be concentric on columns , and it is doubtful that a perfect axially loaded column will be found in practice.

Other situations are imperfections in x-sectional dimensions, residual stresses, holes for bolts and rivets, erection stresses, and transverse loads which can not be taken in a formula.

(If a bridge truss is hit by a truck, bending of tension members will not be serious as tensile loads will tend to straighten those members ,but bending of any compression member is serious as compressive loads will increase bending in those members.

A major cause of residual stresses is the uneven cooling of shapes after hot rolling , (residual compressive/tensile stress at different points of the section).

Welding can also produce severe (great,harmful) residual stresses as in the case of built –up shapes welded together .

Residual stresses may also be caused when cambering is performed by cold bending , and due to cooling after welding.

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3. EULER FORMULA: The stress at which a column buckles decreases as the column length increases .When it reaches a certain length the buckling stress will be elastic. The buckling load, P, is given by the Euler formula as follows: P =π²EI/L2 Since r =(I/A)1/2 or I = Ar²,substituting and dividing both sides by A, we get: P/A = π2E/(L/r)2 = fe Note that P is independent of steel strength when using Euler formula.

4. Effective lengths of Columns: Effective length is defined as the distance between points of zero moment in the column, (points of inflection ). *******(Diagrams attached) Columns in braced frames (side sway prevented).

5. Sections used for Compression Members: Single angle , double angle tee, channel, I- section round, square, rectangular tubing, box sections; from angles or channels, built –up sections from angles and plates.

Circular hollow sections (CHS) are the favoured shapes for compression members especially with welding, (trusses, space frames). ( Firth of Forth bridge in Edinburgh, Scotland, 1899, constructed from welded circular hollow sections.

Square hollow sections (SHS) and rectangular (RHS) are also favoured in different types of structures.

For increasing loads built-up sections are used.

Table (1): Recommended Structural Shapes: Structure/member

Type of x-Section

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Table Attached : ‫الجدول مرفق‬

------------------------------------------------- ------------------------------------------------

Table(2): Attached…………………………..

Effective length Factors in Codes (a)

(b) (c) (d) (e) (f)

------------------------------------------------- -----------------------------------------------End Conditions: (see attached Fig. ) (a( (b) (c) (d) (e) (f)

6-Design of Compression Members: a-

b-

Single Angle Members: In practice loads may not be concentric and large amounts of eccentricities of load applications are present unless applied through bearing plates.

Built-Up Columns : *Two –angle sections are the most common, (used in light trusses). *For long columns, parts of the columns are widely separated (used in bridges, crane booms ,towers). Members may also be assembled from two plates in contact. *******(Diagrams)

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c) Design Procedure: The strength of long columns is a function of the effective slenderness ratio, (KL/r), and for short columns a function of the yield stress (f y) of the steel. Since the radius of gyration (r) depends on the selected section, the design is a repetitive (many trials) process, (unless design tables are available). Design procedure is as follows: i) Axial force is determined by frame analysis; or by statics, for statically determinate structures. The factored load is determined by summing up the specified loads multiplied by the partial safety factors ( ¥ f ). ii) Select a trial section ,eg, single angle or double angle or hollow section, with reference to the slenderness ratio (Table below will help for choice of trial sections): Type of member

Slenderness ratio (L/r)

1.Single angles

100 – 150

2.Circular,square,rectangular hollow sections,(CHS,SHS,RHS) 3.Single channels

90 – 110 90 - 150

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4.Double angles

80 -120

5.Double channels

40 - 80

6.Single I-section

80 - 150

7.Double I-section

30 - 60

iii) Compute KL/r for the section selected, which should by less than the value in the table below: Maximum Slenderness Ratios of Compression Members Type of Member

KL/r

1-Carrying loads from dead loads and superimposed loads.

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2- Carrying loads resulting from wind 250. and seismic loads only, provided that the deformation does not affect the stress in any part of the structure.

3-Acting as a tie in a roof truss or a bracing system, but subject to possible reversal of stress resulting from wind or seismic forces.

350 20

4-Lacing bars in columns

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5- Elements (components)in built-up sections.

50 iv) Using Fig and tables below ,compute fcd and the design strength Pd = Afcd *******(Diagram) Column buckling curves as per IS 800-2006 are similar to the curves given in the British Code BS 5950 (Part1)-2000, based on the Perry-Robertson approach. The design strength is given by: Pd=Ae Fcd Where (Ae) is the effective area fcr is the Euler buckling stress =π²E/(KL/r)² KL/r is the effective slenderness ratio Alfa ( α ) Is the imperfection factor as follows:

Buckling curve a

b

c

d

α

0.34

0.49

0.76

0.21

v) Compare Pd with Pu . When the strength provided does not exceed the strength required by more the a few per cent the design would be acceptable ;otherwise repeat steps (ii) (iii) and (iv).

7. Column Bases Plates: Bases are used to transmit and distribute the column load over the foundation.

The area of the plate is chosen so that the intensity of loading is less than the bearing capacity of the concrete (and soil under concrete) on which it rests .

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The safety of the column and structure depends mainly upon the stability of the foundations (bases).

The design of a base plate assumes uniform distribution of load under the base and the projecting portions are treated as cantilevers.

-Main Types of Bases: *****(Diagrams) For(a ) and (b), and depending on the values of the axial load and moment, compression may be over the whole base, or partially on the bases and tension on the holding - down bolts. Horizontal loads are resisted by shear in the weld between column and base plates, (by friction) and the bond between the base and concrete. ****(Diagrams) Consider an element at A, and the two cantilever strips 1-mm wide; BMs at A = Mx = wa²/2 , My=wb2/2 Assume the moment in the direction of the greater direction is reduced by the co-existant moment at right angle; By introducing Poisons ratio µ : Mx= Wa²/2- 0.3wb2/2 =w/2(a²- 0.3b2) The moment capacity of the plate is: MP= 1.2fy Ze= 1.2fyt²/6

(Z=t²/6 for cantilever strip)

Mx= Mp, solve for, t, apply partial safety factor for the material. ts = (2.5w(a2 – 0.3b2)¥m0/fy)1/2 ≥ tf where: a<b ,Fy yield strength ,tf is flange thickness of compression member, 22

and 짜mo is partial safety factor for material = 1.10.

Design Of Tension Members 1-Introduction: Tension members are subjected to direct axial tensile loads which may cause elongation of the member, and stress it beyond the yield limit (strain hardening).

The design stress (Fy) is not reduced.

Tension members occur in roof trusses (bottom chord, web tie members ),bridges, wind bracing. Some truss web members and members in towers may be subjected to compression for some loading cases (eg. wind loading).

2-Factors Affecting The Strength of Tension Members : The load causing yielding of the gross section is taken as one limit state . The net section through the bolt holes may be subjected to tensile stresses as high as the ultimate stress. Hence , the rupture strength of the net section at the bolt holes at the ends is considered another limit state. Several factors affect the rupture strength of the net section. they are as follows: a) Effect of Bolt Holes: -Bolts are connected to the members through gusset plates. Bolt holes reduce xsection area and hence affect strength, as presented below: -Methods of fabrication of holes , (by punching or drilling); material around the holes is deformed in shear beyond the ultimate strength .

-Net Area of Cross - Section : When more than one hole is present, the failure lines may occur along sections normal to the axis or may include zigzag sections.

All possible failure paths are to be considered to find the correct net area.

-Effect of Bearing Stress: 23

Failure mode in bearing depends on end distance , bolt diameter and thickness of connected material.

(Failure occurs either by splitting of fastener through the end, or excessive deformation in the material near the hole causing elongation of the hole, or a combination may occur).

-Effect of Shear Lag: As the tensile force is transferred by shear ,hence the tensile stress on the section from first bolt hole up to the last bolt hole will not be uniform.

-Geometry factor: (student homewok) -Ductility factor:

“““

-Spacing of fasteners: “”” -Effect of residual stresses: “ “ “ 3) Angles Under Tension : Angles are connected to other members by means of gusset plates through one of the two legs. This results in eccentric loading , causing non-uniform distribution of stress over the x-section ,and also there is shear lag at the end connections. *****(Diagram) 4) Design of a Tension Member: The type of member is chosen based on type of structure and location of the member (e.g double at bottom chord or a rafter of roof trusses angles or pipes for web members of roof trusses).

The design is iterative (repetitive),involving a choice of a trial section and analysis of its capacity as follows: a)A trial section is chosen from steel sections tables . b)The net area required to carry the design load is obtained using appropriate equation.

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c)From net area ,the gross area may by computed by increasing net area by 25% to 40%. The gross area may also be checked from the yield strength of the gross section. d) The number of bolts or welding is calculated. e) If design strength is either small or too large compared to the design force, a new trial section should be chosen and step(d) is repeated until a satisfactory design is obtained. f)The slenderness of the member is also checked.

5. Lug Angles: When a tension member is subjected to heavy load, the number of bolts or length of weld becomes large and bigger gusset plate will be required. Hence, an additional short angle is used to reduce the joint length and shear lag, called lug angle. ***** (drawings of angle and channel connections) 6.Splices: When the available length is less than the required length of a tension member, splices are provided to connect the additional length.

7.Gussets: A gusset plate is a plate provided at the ends of tension members through which the forces are transferred to the main member.

8.Fatigue Effects: Fatigue stresses result from either a large number of stress reversals or cyclic stresses. Fatigue should be considered in the design of bridge tension members.

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BOLTED CONNECTIONS 1.

Introduction:

Any steel structure is an assembly of members, beams, columns, and tension members, which are fastened or connected together. Members are made of angles, I-sections, channels, or plates, which have to be properly connected, so they will act as a single composite unit. Connections of members in a steel framework facilitate (allow) the flow of forces and moments from one member to another and allow the transfer of forces to the foundation level. ***** (show diagrams & photos).

Connection failure before member failure is not desirable and should be avoided as this may lead to the collapse of the whole structure. The design of bolted joints is more complicated than the design of members. Some standard connections types used in the past have shown satisfactory performance, for which simplified analytical procedures are developed. ***** (show examples).

Connections are classified as: a) Rigid, which can develop full moment capacity of members, and rotation will be very small. b) Semi-rigid, develop less than full moment and will allow more rotation movement than rigid joints.

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c) Simple joints, no movement transfer, hence assumed as hinged (pinned).

Rotational movement will be large.

In real life all connections will be semi-rigid, but some are assumed rigid and some as hinged.

2. Revited Connections: For many years revits were the sole (only) means used for steel connections, but now is no longer used in engineering structures; instead bolts and welding are being used.

3. Bolted Connections: a) There are several types of bolts: *Unfinished or black or C grade bolts. *Turned bolts; precision (grade A), semi precision (grade B) bolts. *Ribbed bolts. *High-strength bolts.

(Students HW on types of bolts). b) Bolt holes: Usually drilled and made larger than bolt diameter (clearance 1.0, 2.0, 3.0, for 14.0 mm, 16,0 to 24.0, and more than24,0 mm). c) Spacing and Edge Distance of Holes: Center-to-center in a line is called pitch, for Zigzag bolts, is called staggered pitch. Minimum spacing is 2.5xdiameter of bolt.

Edge distance and gauge distance are specified in international codes of practice.

4. Behaviour of Bolted Joints: a)Types of Joints: *Lap joints (single shear) 27

*Butt joints (double shear) *Eccentric shear connection *Tension connection *Combined shear and tension connection b) Types of Joint Failures: The possible â&#x20AC;&#x2DC;limit state, or failure modes that may control the strength of a bolted connection are: *Shear failure of bolt *Shear failure of plate *Bearing failure of bolt *Bearing failure of plate *Tensile failure of bolt *Tensile failure of plate ( Show figure) 5. Pin Connection: When two structural members are connected by means of a cylindrical pin, when zero moment and free rotation is required, the friction between the pin and the connected members should be minimized. Pins are used in chain link cables in suspension bridges, diagonal bracing in beams and columns, hinged arches, and truss bridge girders. (Tutorial: Design of pins & example).

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WELDED CONNECTIONS Welding is a method of connecting two pieces of metal by heating to a plastic or fluid state (with or without pressure), so that fusion occurs. Structural welding is nearly all electrical, although some gas welding is used (usually acetylene/oxygen mixture).

Quality of electrical welding (arc welding) depends on electrode size and the current that produces sufficient heat to melt the base metal and minimize electrode splatter.

1.Welding Process: The processes of arc welding used in structural steel are as follows: a) Shielded metal arc welding (SMAW). b) Submerged arc welding (SAW). c) Gas-shielded arc welding(GMAW). d) Flux core arc welding(FCAW). e) Electro slag welding(ESW). f) Slud welding(SW).

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2.Choice of Process: ******* (Student HW).

3. Advantages of welding: *** (Student HW).

4. Types and Properties of Welds: a) Groove welds: Have same strength as the members they join. b) Fillet welds: Fail in shear. c) Slot welds: Fail in shear. d) Plug welds: Fail in shear. *******(Insert diagrams here.)

5. Weld Defects: ****** (Student further reading) 6. Welding quality control: ****** (Student further reading).

7. Non-destructive tests on welds: a) Liquid penetration inspection. b) Magnetic particle inspection. c) Radiographic inspection. d) Ultrasonic inspection. ****** (Student further reading).

8. Types of Joints: The types of welded joints depend on: a) Size and shape of members connected at the joint. b) Type of loading. c) Area available for welding at the joint. d) Cost.

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Five basic types of welded joints can be made in four different welding positions, flat, horizontal, vertical, and overhead, as follows: a) Butt joint b) Lap joint c) T-joint d) corner joint e) Edge joint *******(Homework)

9. Effective Area of Welds: The effective areas of groove or a fillet weld is the product of the effective throat(te) multiplied by the effective length of the weld. (t e) depends on the minimum width of the expected failure plain. a) Groove Welds: ******(insert diagram) b) Fillet Weld: The effective throat dimension is the shortest distance from the root to the face of the weld, and should be less than 3 mm and not greater than (0.7a), where (a) is the size of the weld in mm. *****(insert diagrams) 10. Concave and Convex welds: ****** (Student further reading).

11. Design of welds: The following assumptions are made for the analysis of welds: a) Welds connecting the various parts are homogeneous, isotropic, and elastic.

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b) The parts connected by the welds are rigid and their deformations are therefore neglected. c) Only the stresses due to external forces are considered. The effects of residual stresses, stress concentrations, and the shape of the weld are neglected.

11.1Groove Welds: The stresses shall not exceed those allowed for the parent metal: a) For tension or compression normal to the effective area or parallel to the axis of the weld: Tdw = fyLwte/位mw b) For shear on the effective area: Vdw = Lwtefyw/(3x位mw)1/2 Where Tdw is the design strength of the weld in tension, fy: the smaller of the yield stress of and parent metal in Mpa, te:effective throat thickness of the weld in mm, Lw:effective length of the weld in mm, 位mw:partial safety factor, taken as 1.25 for workshop welding and as 1.5 for site welding, Vdw:design strength of weld in shear.

In case of complete penetration of groove welds in butt joints, design calculations are not required as the weld strength is equal or even greater than the strength of the metal connected. In case of incomplete penetration of groove weld in butt joints, the strength of the weld should be checked to be equal or greater than the strength of member or the applied external force.

11.2 Fillet Welds: The actual distribution of stress in a fillet weld is very complex making analysis of weld behavior not possible. In many cases it is possible to use a simplified 32

approach of average stresses in the weld throat. The use of design tables is preferred.

11.3 Intermittent Fillet Welds: Such welds are used in connection of stiffeners to the web of plate girders. The weld required is computed as a continuous fillet weld, then executed as a chain of intermittent fillet welds of total length as computed.

12. Simple Joints: Simple joints as truss members connections, angle seat connections, web angle and end seat connections, and end plate connections.

12.1 Fillet Welds for Truss Members: *****(insert diagram) Taking moments about A, (clockwise +ve); we get ∑ MA = - P1d – P2d/2 + Ty = 0 P1 = Ty/d – P2/2 …………………………………………………………..(1) P2 = RwLw2 (Rw:weld resistance, Lw2:length) ……………(2) ∑ FH = T – P1 – P2 – p3 =0 …………………………………………… (3) Solving (1) & (3) we get P3 = T(1-y/d) – P2/2 ……………………………………………………(4) Eliminating the eccentricity caused by the unsymmetrical weld (balancing the weld): a. Select weld size and electrode, compute the force resisted by the end weld (P2) using equation (2). b. Compute (P1) using equation (1). c. Compute (P3) using equation (4) or P3 = T – P1 – P2 d. Compute the length Lw1 and Lw3 using: 33

Lw1 = P1/Rw and Lw3 = P3/Rw Alternatively, the total length Lw may be calculated and the length of the end weld (d) subtracted, and the remaining length is allocated to P 1 and P2 in inverse proportion to their distances from the center of gravity (CG). ****(Student further reading and tutorial for other simple joints) 13. Moment Resistant Connections: When beams are connected to columns by means of brackets, the welds may be subjected to either twisting or bending moment, in addition to shear forces.

Stiffened seat connections are also subjected to bending moments and shear forces. These connections are termed (known as) moment resistant connections. ****** ( Further reading & tutorial).

14. Tubular Connections: Tubular members and hollow sections are being used in plane and space frames, because they are more efficient in resisting compression, due to the large radius of gyration. Connection can be using automatic oxyacetelene tube cutting machines, or using gusset plates. Welding may be done by complete penetration groove weld, fillet weld, or a combination of both.

15. Application of Adhesives: Until now, use of adhesives is limited due to lack of knowledge about the mechanism, design calculations, and long-term resistance as well as the working of the sticking surfaces. In 1960s, some bridges were built using adhesives connections, with some additional bolts for added safety. Construction was based on experiments and testing. These bridges are still performing well showing the long-term behavior of adhesives. The quality of adhesives has been greatly improved, and more varieties are available.

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BEAMS 1. General: Beams are structural members that support loads applied transverse to their longitudinal axes. A skeleton by Leonardo da Vinci indicates that he tried to describe beam action in a rational way. Galilio was the first to begin investigating the bending problem in a systematic way in 1638, (bending refers to the study of stresses and deformations in a beam bowed by the action of forces). The load transfer by a beam is primarily by bending and shears. But problems may arise when large values of bending moment and shear occur, and lateral buckling of the whole beam, or local buckling of flanges, or longitudinal buckling of the web, or buckling in the depth under point loads or near the supports may take place.

Generally a beam may be subjected to simple, unsymmetrical, or biaxial bending.

Simple bending takes place if the loading passes through the shear center and is parallel to the principal plane. Unsymmetrical bending occurs if the plane of loading does not pass through the shear center; in this case bending is coupled with torsion. If simple bending occurs about both principal planes, without torsion, the beam undergoes biaxial bending, (eg roof purlin).

Design of beams has to satisfy : 35

a)Strength, to resist the bending moments and shear forces, b) Stability, be safe against buckling, c) Adequate bending stiffness, which is often satisfied by strength. Occasionally (sometimes), bending deflections may be the primary design consideration.

2. Types of Beams: a) Main or primary beams/girders.

Beams that span between adjacent columns. b) Secondary beams/joists, Beams that transfer floor loads to main beams.

Beams may also be classified as follows: a) Floor beam: A major beam of a floor system supporting joists in a building; (a transverse beam in a bridge floor). b) Girder: Same as floor beams. c) Girt: Beam spanning between preipheral columns in industrial buildings, used to support wall cladding such as corrugated metal sheeting. d) Joist: Beam supporting floor construction, but not a major beam. e) Lintels: Beam over wall opening for doors and windows. f) Purlin: Roof beam supported by roof truss.

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g) Rafter: Truss roof beam, usually supporting purlins. h) Spandrel: Exterior beam at floor level, carrying part of floor load and the exterior wall. i) Stringers: Members in bridges parallel to traffic to carry deck slab; they are connected by transverse floor beams. *Simple beam: End moment zero. *Continuous beam: Extends over two or more supports. *Fixed end beam: Its ends rigidly connected to other members, so that the moments can be carried across the connection, 3. Beams Sections: There are many forms of beam x-sections used in practice. Selection of a x-section depends upon use and economy. Beam should possess (have) required strength and should not deflect beyond a certain limit as governed by span to depth ratio (slenderness ratio, L/D). If suitable rolled sections are not available, beams may be fabricated (made) from rolled plates, such sections are called plate girders.

Compound sections can also be made from rolled sections (I & channel sections).

Other sections which can be fabricated are castellated and tapered beams.

Composite steel beams and concrete-encased steel beams are also used. ****(include table of beams types showing applications) 4. Behavior of Beams in Bending: The response of a beam subjected to in-plane bending is dependent upon section properties, applied load, type of support, and yield stress, fy.

A laterally restrained and compact beam may fail by: a) flexure, 37

b) Shear, or c) Bearing.

These three failure conditions form the ( collapse limit states ) for such steel beams. Steel beams may become (unserviceable) due to excessive deflection, which is considered as the (limit state of serviceability).

5. Steel Beams in Flexure: A beam, laterally restrained and compact, loaded by increasing amounts, it deflects and its bending curveture varies along its length. Its behavior is elastic during the first stages of loading.

Applying the assumptions of the elastic theory: a) that plane sections remain plane after bending, b) the strain at any point is proportional to its distance from the neutral axis, the strain will be: ******(insert diagram) Idealised elasto-plastic stress-strain curve for steel 5.1 Elastic Behaviour: At point 1; ξ1 ≤ ξy (yield strain) and stress is directly proportional to strain, then Mc = f1I/c (c is extreme distance from NA ).

I/c is the elastic section modulus, Z, M c = f 1Z Point 2; strain ξ2 = ξy and f2 = fy , and My = f yZ Point 3; When loading is increased the outermost fiber strain, ξ max @ mid-point of the beam, reaches ξ3 ≥ ξy and strain is plastic, but stresses are still at yield stress, f y.

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Stresses are redistributed towards the NA and interior fibers gradually reach yield stress, and M3 = fy (bftf + d0tw)(Zc+Zt) + fytwd2/6 plastic

+ elastic

where bf and tf are breadth and thickness of flange of I-section, d 0 is depth plasticized of web, tw web thickness, d is depth of web not plasticized, Zc and Zt lever arms from NA. Point 4; Upon further loading all X-section becomes plastic, it resists any further rotation under a constant moment. At this stage a (plastic hinge) is formed, and BM is Mp. Curvature increases enormously (greatly)after BM at X-section reaches Mp, and Mp = fy â&#x2C6;ŤAydA = fyZp, but Zp = â&#x2C6;ŤydA, plastic section modulus.(â&#x2C6;Ť integral sign).

The formation of the first plastic hinge is the basis of the conventional method of (elastic design) of determinate and indeterminate beams where the moment distribution is done from an elastic analysis. In the case of indeterminate beams, where the redistribution of BM may take place after the formation of the first plastic hinge, failure occurs only after the formation of sufficient number of plastic hinges causing the beam to become a mechanism. Plastic design is based on the ultimate load which causes the formation of a mechanism in an indeterminate beam. ****(insert diagrams) 6. Laterally Supported Beams in Bending: In most situations steel beams must be restrained against lateral buckling by provision of lateral supports. Upper flange used to support concrete and bridge floors can be assumed as laterally restrained from buckling if it is incorporated (inside) these concrete floors. When the flange is long and slender it may buckle unless lateral support is provided. There are many factors that affect amount of stress that will cause buckling. The tension flange tends to resist buckling, but as BM increases buckling will increase, and twisting or tension will occur leading to 39

failure of the beam. Built-up box sections have a great torsion resistance than Isections and channels that are normally used as beams. What is lateral support ?

Examples are a beam fully incased in concrete, or compression flange incorporated in concrete slab. If slab is only resting on the beam flange this situation must be studied carefully to know if friction can provide full lateral support, (fixed or moving loads, vibrating machinery, traffic on bridges).

Lateral support may be provided by connecting beams or other members inserted for that purpose, such as a system of X-bracing for selected bays of the building.

Intermittent welding of metal roof or floor decks to the compression flange will be sufficient. Corrugated metal roofs connected to purlins furnish (provide) only partial lateral support, similarly wood flooring bolted to steel beams. ******* (Tutorial & HW :beam Design).

7. Shear Strength of Steel Beams: Bending will govern the design of most steel beams, shear may control in cases of short beams carrying heavy concentrated loads. ***(insert diagrams) It is noticed from (a) that a great proportion of shear force is carried by the web and the distribution is nearly uniform. Hence, for design purpose we can compute the average shear stress for most sections, such as I, channel, T, as given by: τav = V/twdw (in the elastic range) ……………………………………..(1) V is the shear force, tw, dw are web thickness and depth.

The nominal shear yielding strength of the web as based on the Von Mises yield criteria (no buckling of web): τy = fy/(3)1/2 =0.58fy , taken 0.60fy ……………………………………..(2) Substituting in (1), we get; Vp = 0.6fytwdw (approx..) no buckling …………………………(3) 40

The usual practice to design beams for flexure then check for shear such that, V ≤ Vd where Vd is the design strength, and V is the factored design shear force due to external loads.

Vd = Vn/V¥0 Vn the nominal shear strength of a X-section may be governed by plastic shear resistance or web governed by buckling. Under pure shear: Vn = Vp

where Vp = Avfyw/(3)1/2

(see eqn 2)

8. Maximum Deflection: Deflection and (vibration) is included in the (limit state for serviceability) and is limited by (codes of practice), usually in terms of deflection to span ratio. Check is done using working (service) loads.

Pre-camber of beams, trusses and girders when the deflection due to the combination of (DL+LL) can be excessive (great). For spans greater than (25)m, camber = deflection due to (DL + ½ LL). It is normally more economical to use heavier sections with large moment of inertia (I-section), than to camber sections. ****(include table of loading, BM, and deflections for simply supported beams).

9. Lattice Beams: Also known as (open web joists, OWJ) are used to support floor and roof systems made from cast-in-place or pre-cast concrete. Top and bottom chords are angled, T-sections, or hollow sections. Lateral stability needs careful consideration, and the floor or roof deck must be attached to top chord such that it forms a continuous lateral support. Depending on span, lattice beams (OWJ) may be more economical than rolled sections. ******(Insert diagram) 10. Castellated Beam: It is a beam with a number of regular openings in its web, formed by flame cutting of a single rolled wide beam in a selected pattern, and the rejoining the segments 41

(parts) to form a beam with regular openings in its web. The new section will be 50% deeper, 2.25 times increase in section modulus, so that it can be used for bigger spans than the original section, and services (electrical cables, water pipesâ&#x20AC;Ś.etc) can be placed within the depth of the section. The applied forces are carried by a combination of pure flexure, and vierendeel action (support of cladding and partially the floor system) in the web. ******(include diagram) 11. Purlins: Purlins are beams used on trusses to support the sloping roof system between adjacent trusses; angles, channels, and Z-sections are used for purlins. They are placed on the main rafters at panel points (if feasible), or at suitable points according to the roof sheeting lengths. Purlins are subjected to bending and twisting. They may be designed as simple, continuous, or cantilever beams. They should be erected with flange facing up slope so that the twisting moment do not cause instability. ******(insert diagram) Sag rods are recommended and can be provided midway or at one-third points of purlins so that moments can be reduced (75% of simple support values). *****(Student further reading)..Tutorial: Design procedure for purlins with examples & HW) 12. Biaxial Bending of Beams: Occurs when a beam is subjected to a loading that produces bending about both major and minor axes, as in the case of purlins. The load acts normal to the longitudinal axis but is inclined to the principal axes of the X-section. Here the load passes through the shear center of the X-section. The shear center is the point through which the loads must act if there is no twisting or torsion of the beam.

The location of the shear center can be determined by equating the internal resisting torsional moment derived (obtained) from the shear flow on the Xsection to the external torque. The shear center is always located on an axis of symmetry; thus the shear center is the centroid of a X-section which has two axes 42

of symmetry. When loads act through the shear center, the problem is simple bending in two perpendicular directions. The load can be resolved in Z and Y directions, each producing bending about the other axis.

Consider bending about the Z axis only: Mz/Mdz ≤ 1.0 Mz: factored BM about Z-Z axis, and M dz is the design moment strength about Z-Z axis.

Similarly : My/Mdy ≤ 1.0 When bending occurs about both Z and Y axes, an interaction (alternate effect) type of formula requires that: Mz/Mdz + My/Mdy ≤ 1.0 For plastic and compact sections this equation becomes: ¥m0Mz/Zp,zfy + My¥m0/Zp,yfy ≤ 1.0 ….with ¥m0=1.1 ******(insert diagrams) 13. Unsymmetrical Bending: If the applied loads do not act through the shear center (SC) of the section, the beam will be subjected to flexure and torsion. An example of loading which gives rise to torsion is as follows: ******(insert diagrams) 14. Unsymmetrical sections: ******(insert diagrams) The bending stresses can be determined about the principal axes (ZZ-YY), but points of max stresses, fz,max and fy,max may not occur at the same point, therefore stresses must be calculated at various points and max value be used in the design.

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DESIGN PROGRAM 1. Design of steel columns and column bases.

2. Design of truss members, types of loads, effect of load reversal.

3. Detailed design considerations for members.

4. Use of relevant codes of practice, design charts, and tables.

REFERANCES: 1. Steel Designers Manual, The Steel Construction Institute.

2. International Codes of Practice for the Design of Steel Structures (eg EC3).

3. University Library..(Civil Eng. Section).

4. Structural steelwork: Design to Limit State Theory, MacGineley,T.T. & Ang, T.C. (Latest edition).

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