PROBABILISTIC LIMIT STATE PREDICTION OF WEB … limit states for the structure (Melchers, 1987; ... (Ellingwood et al., 1982). Structural reliability is concerned with the calculation

Academia Journal of Scientific Research 1(10): 156-171, November 2013 DOI: http://dx.doi.org/10.15413/ajsr.2013.0105 ISSN: 2315-7712 ©2013 Academia Publishing

Research Paper

Probabilistic limit state prediction of web shear yielding in steel beams and columns

Accepted 9th October, 2013 ABSTRACT The limit state formulations for web shear yielding in steel beams and columns as in BS5950 (2000), EC3 (2003) and AISC (2005) have been evaluated in order to ascertain their implied safety levels in a probabilistic environment. Results indicate that, safety levels in structural steel beams and columns under limit state of shear yielding is directly proportional to the design strength of steel, shear resistance factor and web shear coefficient. At 275 N/mm2 design strength of steel with target shear resistance factor and web shear coefficient of unity, BS5950 (2000), EC3 (2003) and AISC (2005) produced a safety index value of 5.50, 6.01, 5.90 for steel beams and 2.88, 4.67 and 3.39 for steel columns respectively. This signifies that the safety incorporated in EC3 (2003) formulations exceeds that in BS5950 (2000) and AISC (2005) by 8.5 and 1.8% respectively for the design of steel beams; and 38.4 and 27.4% for steel columns respectively. As such, EC3 (2003) can be relied upon to produce design optimum safety levels in both steel beams and columns. It is suggested also that some adjustments may be required in the other codes when revised to accommodate optimum web shear yielding formulations for designs. Key words: Web shear, steel beams, columns, safety.

INTRODUCTION Structural steel design is carried out by computing the internal forces and moment acting on components of the structure, followed by the selection of an appropriate cross-section for a given grade of steel. Ultimate limit state design in structural steel beams considers flexure as the main failure criteria to which beams should be sized in order to satisfy both flexural and shear demands. On the other hand, columns are sized and checked for axial load capacity based on the effective slenderness of their overall length about the weaker axis and the slenderness of the cross-section.

In addition, interaction between axial load and bending moment is checked with shear seldom considered. The column web can fail due to shearing effect of tensile and compressive forces applied to it at the vicinity of moment resisting connections before bearing and buckling failure of its flanges due to difference in thickness, inadequate design

procedure and the non-provision of stiffeners at the web shear zone due to fabrication cost (Tahir et al., 2006).

The resistance of a structural member as well as, the loads applied to it is a function of several variables, most of which are random (Melchers, 1999). Therefore, the use of probabilistic approach in the design of structures enables the structural safety to be treated in a more rational manner. Thus, a reliability formulation approach will be used.

The study of structural reliability is concerned with the calculation and prediction of the probability of limit state violation for engineered structures at any stage during their lifetime. In particular, the study of structural safety is concerned with the violation of the ultimate or serviceability limit states for the structure (Melchers, 1987; Madsen et al., 1986).

O. S. Abejide* and I. M. Adamu Department of Civil Engineering, Ahmadu Bello University, Zaria. Nigeria. *Corresponding author. E-mail: [email protected].

Academia Journal of Scientific Research; Abejide and Adamu. 157

Structural reliability is therefore, the branch of structural engineering which is concerned with the analysis and probabilistic assessment of design random variables in order to predict whether a specified limit state would be violated or not, and in doing this, uncertainties inherent in structural designs have to be taken into consideration (Doty, 1985).

This presentation involves the probabilistic approach coded in First Order Reliability Method (FORM) (Gollwitzer et al., 1988) and is used to predict web shear yielding in steel beam and column according to the limit states provided by three steel design codes namely; BS5950 (2000), EC3 (2003) and AISC (2005). It was noted that the probabilistic approach coded in FORM (Gollwitzer et al., 1988) to predict the safety of web shear yielding in steel beams and columns using the limit states provided in the design codes has the following limitations: (i) the column shear yielding considered is that which occur due to horizontal shear force that generates around moment resisting connections due to flexural action of the beam under a non-sway mode; (ii) the steel beams and columns sections considered are hot-rolled or compact I (UB) and H (UC) sections respectively and; (iii) the beams are assumed effectively restrained against lateral and torsional buckling.

The essence of this presentation is to investigate and predict the reliability or safety of the equations governing shear yielding of webs in steel beams and columns provided by BS5950 (2000), EC3 (2005) and AISC (2005) so as to ascertain the code with the most reliable provision for the structural design of these steel sections. Therefore, in carrying out the investigation the objectives or procedure employed are as follows: (a) investigate the governing conditions for the design of structural steel elements subjected to shear; (b) compute the probability of failure and safety indices associated with shear failure given by the three codes (BS5950-1:2000, EC-3:2003 and AISC-360: 2005) and (c) establish a target reliability index for steel beams and columns subjected to shear and optimize the design equations provided by these codes, if necessary. SHEAR IN STEEL BEAMS AND COLUMNS WEBS Web shear yielding in steel beams and columns is considered to be one of the criteria for shear failure (AISC, 1999; 2005; BS5950, 2000; EC3, 2003). The limit state of web shear yielding is attained when the beam or column web is stressed to its full shear capacity (MacGinley and Ang, 1990).

Probability denotes a chance that a particular predefined event occurs (Melchers, 1987). As defined by the three steel design codes, a limit state is a condition in which a structure or component becomes unfit for service. Thus, shear yielding is the limit state of inelastic deformation that occurs due to shear after the yield stress is reached.

Structural failure can be considered as the occurrence of one or more types of undesirable structural response including the violation of pre-defined limit states (Madsen et al., 1986). When an engineering structure is loaded in some ways, it will respond in a manner which depends on the magnitude and type of load(s), the strength as well as, stiffness of the structure.

Whether the response is satisfactory depends on the requirements which must be satisfied (Melchers, 1987). It is in this regard that Summer and Yura (1982) conducted an experiment on a hot-rolled or compact I-section beam under concentrated point load with the beams compression flange braced at the point of load application, and they observed an unexpected sideway squeezing and buckling failure in the beams web.

Uncertainties in analysis and design should not be under estimated, it is because of uncertainties that the question of safety and performance of a structure has arisen (Ellingwood, 1978; MacGinley and Ang, 1990; Ditlevsen and Madsen, 2005). Reliability analysis provides a frame work for incorporating the uncertainties inherent in analysis and design (Ellingwood et al., 1982).

Structural reliability is concerned with the calculation and prediction of the probability of limit state violation for engineered structures at any stage during their lifetime (Melchers, 1987). Until recently, reliability has not been routinely quantified in design but considered tactically, and thus, is accounted for by the use of safety factors provided by codes of practice (AISC, 1999; Melchers, 1999; Ditlevsen and Madsen, 2005).

Beams are structural members with the primary function of resisting moment (AISC-360: 2005). Steel beams can have varying cross-sections which can be rolled or welded. Examples of rolled steel beam sections are Universal Beams (UB), channels, and compound sections formed from the combination of rolled sections. These beams may be cantilevered, simply supported, fixed-end and continuous depending on their formulation for use. Also, they can be used to support columns, floors or carry roof sheeting as purlins.

Columns are structural members with the primary function of resisting axial loads (AISC, 2005). Columns cross-sections may vary depending on aesthetics and strength. Typical cross-sections in structural designs are Universal Columns (UC), circular (CHS), square and rectangular hollow sections (RHS). The primary function of columns is to transfer vertical loads from beams to foundations and sometimes, in combinations with bracings do resist lateral loads from wind. Columns are classified based on their end conditions as pinned, fixed, free and a combination of end conditions.

BS5950 (2000) and EC3 (2003) considers cross-sections of steel beams and columns in four classes namely: plastic; compact; semi-compact and slender. These cross-sections are as defined in the codes. However, AISC (2005) classified the cross-sections into three groups as compact, non-

Academia Journal of Scientific Research; Abejide and Adamu. 158 compact and slender. This was based on the width to thickness ratio of the steel cross-sections, connection of flanges to the web and with a limit set according to the category of compactness which depend on the factored ration of elastic modulus and strength of the steel material.

Shear yielding as defined by AISC (2005) is the limit state of inelastic deformation that occurs due to shear after the yield stress is reached. Yielding of the web represents one of the shear limit states, because the web will completely yield long before the flange begins to yield (Arora and Wang, 2006).

Shear force is the unbalanced force that exist perpendicular to a structural element (Beedle, 1958). In steel beams, shear force is considered to be vertical as contrary to steel columns where it is horizontal (Tahir et al., 2006). Tahir et al. (2006) also noted that horizontal shear force can occur at the vicinity of moment resisting connections due to flexural action of the beam. In these two cases, the shear force developed can be assumed to be resisted by the beam or column web in rolled sections (MacGinley and Ang, 1990).

It is usually assumed that the central portion of the web on an I-beam carries the shear force, with the remainder of the section carrying moment (Clarke and Coverman, 1987). But Tahir et al. (2006) has suggested that the column web of a steel section can fail due to shearing effect of tensile and compressive forces applied to it at the vicinity of moment resisting connections before bearing and buckling failure of its flanges due to differences in thickness, inadequate design procedure and the non provision of stiffeners at the web shear zone due to fabrication cost. Limit state equations governing shear in steel beams and columns

Let, = characteristic dead (permanent) load, =

characteristic imposed (live) load and w = design load. Then, the load combinations as recommended in the various codes are as given: Load combinations BS5950-1:2000:

(1)

EC–3:2003:

(2)

AISC-360:2005 (LRFD):

(3)

Reactions in steel beams

Let, M = Moment at support; = Maximum vertical shear

force; L= Beam span; = Maximum span moment and w =

Design load. Therefore:

(4)

(5)

(6) Shear force in steel columns around moment resisting connections

When M= moment at support; Z= lever arm of beam; =

maximum horizontal shear force, then:

(7)

And:

(8)

Where, D = Overall depth of beam cross-section; and T = Thickness of beams flange. BS 5950:1:2000:

(9)

Where, = shear capacity of member; = design

strength of steel; = shear area.

For rolled I, H and channel sections with load parallel to web:

(10)

Where, t = thickness of web; D = depth of section

For the section to be structurally safe:

Beam: (11)


Where, = Maximum applied vertical shear force.

Column: (12)

Where, = Maximum Applied horizontal shear force.

EC-3:2003:

(13)

Where, = Design plastic shear resistance; =

Shear area; = Partial factor for resistance of cross-

section whatever the class; = Design strength of steel.

For rolled I and H sections with load parallel to web EC-3:2003 specifies that:

ʃ (14)

Where, A = Overall cross-sectional area; b = Overall

breadth; h = Overall depth; = Depth of web; r = root

radius; = flange thickness; = Web thickness; ʃ = Factor

for shear area. For a conservative design:

0.1 (15)

For the section to be structurally safe:

Beam = (16)

Where, = Design Plastic shear resistance; =

Maximum applied vertical shear force.

Where, Column: (17)

= Maximum applied horizontal shear force.

AISC – 360:2005:

Design shear strength = (18)

Where, = Shear resistance factor; = Nominal shear

strength.

(19)

Where, = Design strength of steel; = Web area; =

Web shear coefficient.

(20)

Where, = Thickness of web; d = Overall depth of cross-

section. For webs of rolled I-shaped members that satisfies the condition in equation:

(21)

(22)

For the section to be structurally safe, then:

Beam = (23)

Column = (24)

Where, = Maximum applied vertical shear force; =

Maximum applied horizontal shear force.

Observations and arguments on the limit state equations for shear yielding in steel beams and columns provided by BS 5950 (2000), EC3 (2003) and AISC (2005) The following observations are noted in the three steel design codes: (1) The shear area may be stressed to yield stress at 1/√3, that is, 0.57735 ≈ 0.6 or 60% of the yield stress in tension (fy) that is, (0.6fy) for all three codes. Thus, 0.6fy will be considered deterministic and unchangeable component in the limit state of shear yielding equations. (2) The shear area equation given by BS5950-1:2000 (Equation 10) and AISC-360:2005 (Equation 20) for rolled I and H sections are the same, but contradicts that of EC-3:2003 (Equation 14). (3) EC3 (2003) shear capacity design equation had been

multiplied by a factor of (where, ). AISC

(2005) multiplied its shear capacity design equation by

and ; while, BS5950 (2000) designed formulation or

equation has no multiplying factor.

(4) A target value of shear resistance factor ( ) and web

shear coefficient ( ) equals unity is required by the three

Academia Journal of Scientific Research; Abejide and Adamu. 160 codes for sections that are at least compact (for BS5950 (2000) and EC3 (2003) or semi-compact (for AISC (2005). Therefore, after due considerations of the observations aforementioned, the following is suggested:

- The shear resistance factor ( ) and web shear

coefficient ( ) can be assumed to be inherent in all the

three codes as formulated for the shear yielding limit state equations. Thus, the effect(s) of shear resistance factor

( ) and web shear coefficient ( ) on the limit state

equations governing shear yielding in steel beams and columns can be considered random and stochastic. These will then be evaluated under the load(s) effect(s) specified by each code in order to ascertain the intrinsic safety levels and from which the optimum is clearly indicated. MATERIALS AND METHODS Reliability is defined as the probability of a performance function g(X) greater than zero, that is, P{g(X) >0} . In other words, reliability is the probability that the random variables Xi=(X1, ……., Xn) are in the safe region that is defined by g(X) >0. The probability of failure is defined as the probability P{g(X) < 0}, or it is the probability that the random variables Xi=(X1, ……., Xn) are in the failure region defined by g(X) < 0.

Assuming that R and S are random variables whose statistical distributions are known very precisely as a result of a very long series of measurement; R is a variable representing the variations in strength between nominally identical structures, whereas, S represents the maximum load effects in successive T-yr periods, then, the probability that the structure will collapse during any reference period of duration T years is given as:

(25) Where, FR is the probability distribution function of R and fs the probability density function of S. Note that R and S are statistically independent and must necessarily have the same dimensions. The reliability of the structure is the probability that it will survive when the load is applied, given by:

(26) In basic reliability problems, consideration is given to the effect of a load S and the resistance R offered by the

structure. Both the load and resistant S and R can be described by a known probability density function (Fs) and (Fr) respectively. S can be obtained from the applied load through a structural analysis making sure that R and S are expressed in the same unit.

Considering only safety of a structural element, it would be said that a structural element has failed if its resistance R, is less than the resulting stress S acting on it. The probability of failure Pf of the structural element can be expressed in any of the following ways: Pf = P(R – S) (27) Where R = strength (resistance) and S = loading in the structure. The failure in this case is defined in this region where R-S is less than zero or R is less than S, that is: Pf = P((R – S) ≤ 0) (28) The performance function g(x) is sometimes called the limit state function. It relates the random variables for the limit-state of interest. The limit state function gives a discretized assessment of the state of a structural element as being either failed or safe. It is obtained from traditional deterministic analysis, but uncertain input parameters are identified and quantified. Interpretation of what is considered to be an acceptable failure probability is made with consideration of the sequences of failure, which is predetermined.

The limit state “g(x) = R - S” is a function of material properties, loads and dimensions. The state of the performance function g(x) of a structure or its components at a given limit state is usually modeled in terms of infinite uncertain basic random variable x= (x1, x2, ……….,xn) with joint distribution function given as:

(30)

And

(31)

Where is the joint probability distribution function

of x.

The region of integration of the function g(x) is stated as: g(x) > 0: represents safety; g(x) = 0: represents attainment of the limit state; g(x) < 0: represents failure. The probability of failure is given by P(g(x)<0) and therefore, the reliability index β can be related to probability of failure by the following equation:

= 1 – Φ (β) (32)

Academia Journal of Scientific Research; Abejide and Adamu. 161 A relationship can be drawn between the probability of failure, Pf, and the reliability index, . It however, holds true only when the safety margin, M, is linear in the basic variables, and these variables are normally distributed. This relationship is stated as:

Pf (33)

1 Pf (34)

Where is the standardized normal distribution function.

P P R S P Mf

R s

R s

0 0

0

2 2

35)

Shear design in steel beams and columns according to BS5950-1:2000

Examining the load combinations in Equations (1), (2) and (3), it would be observed that BS5950 (2000) load combination will produce the critical design load. Since, designs are carried out based on the critical reactions obtained after analysis; it is evident that for a given span and variable design loads, critical reactions would be obtained at maximum design load. As such, sections sized using B55950-1:2000 will satisfy the other two building codes (EC-3:2003 and AISC-360:2005). Hence B55950-1:2000 will be used to size the sections and illustrate the design procedure.

A connection needs to be of moment resisting type to facilitate the development of horizontal shear force on the connected column. Also, the three building codes are required to be subjected to loads based on their respective load combinations in order to examine their shear provisions. As such, the following condition will be appropriate, for example, considering a fixed ended steel

beam of 6 m span carrying a characteristic dead load ( )

of 30 kN/m and characteristic imposed load ( ) of 15

kN/m with effective restraint against lateral and torsional buckling is considered as an example as follows:

Design load

= 1.4x30 + 1.6x15 = 66 kN/m

The moments and shear forces on the beam are obtained as:

Span moment: = = 99 kNm

Support moment: M= =198 kNm

Shear force: = 198 kN

The beam is designed by assuming a S275 steel grade and a

trial steel section. Thus, = 275 N/mm2 with a trial

section:

Try 356 × 171 × 67 kg/m UB. The section properties of the trial section are given as: t = 9.1 mm; T= 15.7 mm; r = 10.2 mm; B = 173.2 mm; d= 312.2 mm; D= 364 mm; b/t = 5.52; d/t= 34.3. Hence:

= tD = 3312.4 mm2

Where ( = shear area; t = web thickness; D= Overall

depth of beam). Sectional classification:

= = 1.0

b/T= 5.52 < 9: plastic section d/t= 34.3 < 80: plastic section Beam web is within limits of shear yielding (that is, shear yielding governs shear failure) as:

= 0.6 × 275 × 3312.4 × 10-3 = 546.5 kN

Beam is ok in shear

Where, = Shear capacity of section; = Design strength

of steel; = Shear area of section.

Then, = 0.36 < 0.6: shear force will not reduce moment

capacity. Therefore, the chosen section, that is; 356 × 171 × 67 kg/m UB is adequate in shear, hence, adopted as beam section. The column can be designed as shown: M = 198 kNm Where, M = support moment.

Academia Journal of Scientific Research; Abejide and Adamu. 162 = D-T = 364 – 15.7 = 348.3 mm Where, Z = lever arm of beam; D = Depth of beam; T = Thickness of beams flange.

Try: 254 × 254 × 167 kg/m UC The section properties of the trial column section as given as: D = 289.1 mm; B = 264.15 mm; t = 19.2 mm; T = 31.7 mm; r =12.7 mm; d =200.3 mm.

T

b = 4.17

t

d= 19.4

Sectional classification:

T

b = 4.17 < 9: plastic section

t

d= 10.4 < 80: Plastic section

Shear yielding will govern shear failure Av = tD = 19.2 x 289.1 = 5550.7mm2 Where, Av = Shear area of column; t = Thickness of column web; D = Depth of column section. Ph = 0.6Py Av = 0.6 × 275 × 5550.7 × 10-3 = 915.9 kN Ph = 915.9 kN Ph > Fh column is Ok in shear. Where, Ph = Column shear capacity; Py = Design strength of steel.

0.59 < 0.6 Shear force will not reduce moment

capacity Section 254 × 254 × 167 kg/m UC is adequate in shear, hence, adopted as column section. The design loads for beam, column support moment and shear area for both beam and column according to the three building codes (BS 5950-1:2000, EC-3: 2003, AISC-

360:2005) were calculated using the relevant Equations (1 to 8) and are tabulated in Tables 1 and 2 based on the sections sized as earlier mentioned. Beam section: 356 × 171 × 67 kg/m UB Probability estimates Derivation of Limit State Equations The derivation of the limit state functions is for the beams and columns are as shown: Beams For the beam to be structurally safe against shear failure due to yielding of web, Equation (19) must be satisfied. The corresponding limit state is represented by Equations 36, 37, 38, and 39 given as: G(x) = V res – V applied (36) Where, V res = Ultimate shear resistance; V applied = Applied shear force. BS 5950-1: 2000:

37)

Where, φV = Shear resistance factor; = Design strength of

steel; Av = Shear area of web; =Web shear coefficient;

= Characteristic dead (permanent) load; = characteristic

imposed (live) load; L = Beam span.

EC-3:2003:

(38)

Where, v = Shear resistance factor; = Design strength of

steel; Av = Shear area; =Web shear coefficient; =

Characteristic dead (permanent) load; = characteristic

imposed (live) load; L = Beam span.

AISC-360:2005:

(39)

Where, φV = Shear resistance factor; = Design strength of


Table 1. Design load, shear force and shear area in steel beam according to building codes.

Code Design load, w (kN/m) Shear force (kN) Shear area (mm2)

BS5950-1:2000 66 198 3312.4

EC-3:2003 63 189 3574.7

AISC-360:2005 60 180 3312.4

Column section: 254 × 254 × 167 kg/m UC.

Table 2. Support moment, shear force and shear area in steel column according to building codes.

Code Support moment (M) kNm Shear force ( ) kN Shear area ( ) mm2

BS5950-1:2000 198 568.5 5550.7

EC-3:2003 189 542.6 5844.5

AISC-360:2005 180 516.8 5550.7

steel; Av = Shear area; =Web shear coefficient; =

Characteristic dead (permanent) load; = characteristic

imposed (live) load; L = Beam span. Columns For the column web to be structurally safe against shear failure due to yielding Equation (20) must be satisfied. The corresponding limit state is represented by Equations 40, 41, 42 and 43 given as: G(x) = Vres – V applied (40) Where, V res = Ultimate shear resistance; V applied = Applied shear force; BS 5950-1: 2000:

G(x) (41)

Where; φV = Shear resistance factor; Py = Design strength of

steel; Av = Shear area; Cv = Web shear coefficient; =

Characteristic dead (permanent) load; = Characteristic

imposed (live) load; L = Beam span; Z = Lever arm of beam. EC-3:2003:

G(x) (42)

Where, φV = Shear resistance factor; Fy = Design strength of



imposed (live) load; L = Beam span; Z = Lever arm of beam. AISC-360:2005:

G(x) (43)

Where, φV = Shear resistance factor; Fy = Design strength of



imposed (live) load; L = Beam span; Z = Lever arm of beam.

Computation of safety index ( )

First Order Reliability Method (FORM) coded in FORM5 (Gollwitzer et al., 1988) was employed in the computation, making use of the stochastic model parameters tabulated in Table 3, design parameters in Table 3 and the relevant limit state functions according to the three building codes (BS5950-1: 2000, EC-3:2003 and AISC-360: 2005). The shear resistance factor and web shear coefficient were varied under various design strength of steel and constant load.

Programs were developed in FORTRAN module for shear yielding of beam and column web according to the three design codes investigated. The results are given in Figures 2 to 19. RESULTS AND DISCUSSION

The results are presented in Figures 2 to 19; where, β =

safety index, v = shear resistance factor and Cv = web

shear coefficient. With regards to the safety or values


Table 3. Stochastic model parameters in steel beams and columns.

S/No. Variable Distribution Covariance Mean (EX) Standard deviation (SX)

1 Fy = Steel design strength Log-normal 0.05 275 13.75

2 v = Shear resistance factor Normal 0.10 1.0 0.10

3 Cv = Web shear coefficient Normal 0.10 1.0 0.10

Figure 1. Shear force in steel column around moment resisting connection (EC-3: ENV 1993-1-1: 1992).

Figure 2. Effect of shear resistance factor (φv) and web shear coefficient (Cv) on safety of steel beam against web shear yielding at Py = 275 N/mm2 (BS5950-1:2000).


Figure 3. Effect of shear resistance factor (φv) and web shear coefficient (Cv) on safety of steel beam against web shear yielding at Py = 355N/mm2 (BS5950-1:2000).

Figure 4. Effect of shear resistance factor (φv) and web shear coefficient (Cv) on safety of steel beam against web shear yielding at Py =460 N/mm2 (BS5950-1:2000).

plotted as aforementioned which is obtained from the design formulations for steel beams and columns webs, it can be said that the web thickness is safest against shear failure due to yielding at the highest possible values of the variables (that is, design strength of steel, shear resistance factor and web shear coefficient).

It can be observed that at constant design strength of

steel, the maximum safety index value (maximum value)

was obtained at the maximum allowable value of shear

resistance factor ( v ) and web shear coefficient (Cv) for

both steel beams and columns irrespective of the code. The safety indices values reduce as the shear resistance factor and web shear coefficient decreases simultaneously. It was also observed that, safety of beams and columns webs increases with increases in design strength of steel irrespective of shear resistance factor and web shear coefficient.

Therefore, it is clear that, the safety of both steel beams


Figure 5. Effect of shear resistance factor (φv) and web shear coefficient (Cv) on safety of steel beam against web shear yielding at Fy = 275 N/mm2 (EC-3:2003).




Figure 8. Effect of shear resistance factor (φv) and web shear coefficient (Cv) on safety of steel beam against web shear yielding at Fy = 275 N/mm2 (AISC-360:2005).

Figure 9. Effect of shear resistance factor (φv) and web shear coefficient (Cv) on safety of steel beam against web shear yielding at Fy =355 N/mm2 (AISC-360:2005).

Figure 10. Effect of shear resistance factor (φv) and web shear coefficient (Cv) on safety of steel beam against web shear yielding at Fy = 460 N/mm2 (AISC-360:2005).


Figure 11. Effect of shear resistance factor (φv) and web shear coefficient (Cv) on safety of steel column against web shear yielding at Py = 275 N/mm2 (BS5950-1:2000).




Figure 14. Effect of shear resistance factor (φv) and web shear coefficient (Cv) on safety of steel column against web shear yielding at Fy = 275 N/mm2 (EC-3:2003).

Figure 15. Effect of shear resistance factor (φv) and web shear coefficient (Cv) on safety of steel column against web shear yielding at Fy

= 355 N/mm2 (EC-3:2003).

Figure 16. Effect of shear resistance factor (φv) and web shear coefficient on safety of steel column against web shear yielding at Fy = 460 N/mm2 (EC-3:2003).


Figure 17. Effect of shear resistance factor (φv) and web shear coefficient (Cv) on safety of steel column against web shear yielding at Fy =275 N/mm2 (AISC-360:2005).

Figure 18. Effect of shear resistance factor (φv) and web shear coefficient (Cv) on safety of steel column against web shear yielding at Fy = 355 N/mm2 (AISC-360:2005).

Figure 19. Effect of shear resistance factor (φv) and Web shear coefficient (Cv) on safety of steel column against web shear yielding at Fy = 460 N/mm2 (AISC-360:2005).

Academia Journal of Scientific Research; Abejide and Adamu. 171 and columns webs against shear yielding failure is directly proportional to the design strength of steel, shear resistance factor and web shear coefficient.

However, the best results of web safety for both steel beams and columns were obtained from the safety indices values of EC-3: 2003 code. This implies that EC-3: 2003 code which stipulates shear resistance (capacity) of webs to

be equal to ( , = = 1.0, = 1.0,

= 1.0) and shear area ( ) to be equal to

( ≥ ʃ , ʃ = 1.0) gives the

safest provision for shear yielding of compact (rolled) I and H sections of beams and columns respectively. Conclusion This study presents the inherent safety of web shear yielding in compact or rolled steel beams and columns according to three building codes, namely, BS5950-1: 2000, EC-3: 2003, and AISC-360: 2005, under a probabilistic setting so as to ascertain the code that gives the optimum safety provision for shear yielding of the web.

From the results obtained, it can be concluded that the code that gives the optimum safety provision for shear yielding of web in steel beams and columns is the code that gives the highest safety value when the required design parameters are considered. This code is EC-3: 2003.

Examining the safety indices values in Figures 2 to 19, it would be observed that the values for a given steel design strength, allowable shear resistance factor and web shear coefficient will be at its optimum provision only in the code that provides the highest shear resistance or capacity and the least design load. Unfortunately, none of these building codes satisfied the two conditions as earlier mentioned. Simultaneously, EC-3: 2003 provides the highest shear area and hence, the highest shear resistance (capacity) but AISC-360: 2005 provides the least design load for a given characteristics dead and imposed load. The following suggestions are therefore made for improvement in these codes:

(i) Shear resistance (capacity) = , φv =1.0,

Cv=1.0 for steel beam and column.

(ii) Shear area for

hot rolled I and H sections. (iii) Stiffeners be included in the code provisions for design in columns web panel zone connected to steel beams by means of moment resisting connections so as to reduce the effect of horizontal shear force that may be developed. (iv) Safety indices value of at least 2.5 should be achieved in design formulations for adequate safety.

Where, = Shear resistance factor; ʃ = Factor for shear

area; = Safety factor for steel; = Design strength of

steel; = Web shear coefficient; = Shear area; A =

Cross-sectional area of steel section (Gross); b = Overall

breadth of section; h = Overall depth of section; = Depth

of web; r = root radius; = Flange thickness; = web

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Cite this article as: Abejide OS, Adamu IM (2013). Probabilistic limit state prediction of web shear yielding in steel beams and columns. Acad. J. Sci. Res. 1(10):156-171. Submit your manuscript at http://www.academiapublishing.org/journals/ajsr

PROBABILISTIC LIMIT STATE PREDICTION OF WEB … limit states for the structure (Melchers, 1987; ... (Ellingwood et al., 1982). Structural reliability is concerned with the calculation

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