Epistemics in Science, Engineering and Technology, Vol.3, No.1, 2013, 215-223 WebsJournals
UcheEngineering and Ahmed Epistemics in Science, and Technology
5111-6005, © WebsJournal Incin Science, Engineering and Technology, Vol.3, No.1, 2013, 215-223 Epistemics 5111-6005, © WebsJournal Inc
Reliability Design Format for Steel Plate Girder to BS 5950 (2000) 1
Uche, O.A.U and 2Ahmed A Civil Engineering Department, Faculty of Engineering, Bayero University, Kano. Email: [email protected]
1,2
Abstract The paper presents reliability assessment of deterministic design of steel plate girder considering both ultimate and serviceability limit state in accordance with BS 5950 (2000). The reliability analysis was carried out using First Order Reliability Method (FORM). Design variables such as strength of the material (Py), width of the flange, flange thickness, web thickness as well as the span of the girder were considered random and stochastic. It was shown among the findings that, when the span (L) of the plate girder was kept constant with increase in the magnitude of live-dead load ratio, the safety indices decreases, as deflection criterion was considered. Also, the design of the plate girder in accordance with BS 5950 considering shear and deflection is safe for almost all the range of variables considered. On varying the depth of the plate girder, safety indices increase with increase in depth and decrease with increase in live-dead load ratio, when shear criterion was considered. The deflection is the most critical mode of failure on varying the span and load ratio, with safety index less than the recommended value by JCSS 2000 for structural members with moderate to large consequences of failure of 3.3 to 4.4. Therefore, BS 5950 design result seems unsafe with respect to bending (under high live-dead load ratio) and satisfactory with respect to shear and deflection. It is then, recommended that the design of steel plate girder base on BS 5950 be reviewed to incorporate reliability analysis. Keywords Reliability design, steel plate girder, safety index, deterministic design
1.
Introduction and Concept
The aim of structural design is to produce a safe, aesthetical and economical structure that fulfills its required purpose (Clarke and Converman, 1987). Theoretical knowledge of structural analysis must be combined with knowledge of design principles and theory and the constraints given in the standard code of practice to give a safe design. Although a thorough knowledge of properties of materials, methods of fabrication and erection are essential for the experienced designer. Steel structures in a building consist of a skeletal framework which carries the entire load to which the building is subjected. The steel members are used to carry lateral loads when acting as beams and girders, and axial loads when acting as stanchions. Steel on its own exhibits a number of useful characteristics, such as, structural behaviour up to initial yield point which is nearly elastic, deformation which are directly proportional to the applied load and long-term deformation at normal temperature is not a problem (Ang and McGinley, 2004). The important properties of steel are strength, ductility, impact resistance and weldability. The plate girder is typically I – beams made up from separate structural steel plates, which are welded or bolted or riveted together to form the vertical web and horizontal flanges of the beam. Though sometimes they may be formed in a Z – shaped rather than I – shape. Plate girders are used to carry larger loads over longer spans than are possible with universal or compound beams. Thus, it is use in the buildings for long span floor girder, for crane girder and bridges (Abubakar and Pius, 2007). There are two types that can be selected depending on the cost; one of which has a web so slender that the transverse stiffeners must be added and some time longitudinal stiffeners are added as well. The other types of plate girder have thicker webs and therefore, it is not stiffened. However, in the design of plate girder, as in other design processes, variation and uncertainties in the operation condition causes variation in its performance and durability. The design of any structure must ensure safety at worst loading condition and the deformation of the members during normal working condition must be accepted from the durability, performance and appearance of the structure (McGinley, 1986; Morris and Plum, 1987; Ray, 1998, Uche and Afolayan 2008, ). Thus, the existence of physical uncertainties in the design of plate girder required a reliability assessment to be taken in to account, in other to satisfy the structural integrity. A pre- requisite to building safety is the assurance of the structural integrity to a high level of reliability throughout their lives; such integrity may be threatened by uncertainties that were observed in the design, construction and maintenance stages. The work presented is concerned with the safety associated with the design criteria of grade 43 simply supported steel plate girder considering both ultimate and serviceability limit states.
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1.2 Justification of the research The use of probabilistic approach in the assessment of plate girder performance would provide structural safety to be treated in a more rational manner for the randomly applied load to plate girder. This will provide adequate counteraction of the uncertainties related to the design of a plate girder and increase the level of confidence. In the field of structural engineering it has become necessary to review the way of designing plate girder and have a convincing reliability based design format (Dhillon, 2005). 2.
Materials and Method
2.1 Steel plate girder design The design of plate girder in Ultimate and serviceability limit state covers violation against strength, fatigue and stability against overturning and sway when the ultimate limit states are exceeded. Whereas Serviceability limit state covers deflection, vibration, durability of steel plate girder system. According to BS 5950(2000), a manual design was carried out using plastic section analysis, this is in turn subjected to reliability analysis. 2.2 First–order reliability method (FORM) The probabilistic design of any structure is connected with the probability that the structure will realize its assigned function. Probability denotes the chance that a particular defined event occurs. The First Order Reliability Method (FORM) involves the calculation of the probability of failure or limit state violation for structural element. This aspect is sometimes ignored and relative frequency data assumed to be purely “objective” information (Melchers 1990). It is noted that a subjective probability estimate reflect the degree of insurance about the phenomenon under consideration. The manipulation of subjective estimate and use of compactible data to amend and improve the estimate may be achieved through Baye’s theorem. Consequently, subjective probabilities assessments are sometimes known as “Bayesian probabilities” (Benjamin, 1976; Lindley, 1972). The reconciliation of the two interpretations of the probability is controversial (Fishburn, 1974; Hasofer, 1984). If R is the strength capacity and S is the loading effect of the structural system which are random variables, the main objective of reliability analysis of any system or component is to ensure that R is never exceeded by S, in practice, R and S are usually functions of basic variables, in order to investigate the effect of the variables on the performance of the variables on the performance of a structural system, a limit state equation in term of the basic design variables is required. The state function used in FORM5 is given by g(X). It is defined such that g(xi) = g(x1, x2, ……, xn) = R – S
(1)
where xi for I = 1, 2, ……, n, represent the basic design variables. The limit state of the system can then be express as g(xi) = 0 (2) Graphically, g(X) > 0 Corresponds to a favourable (safe, intact, acceptable) state while g(X) = 0 denote the failure boundary and g(X) < 0 defines the failure (unacceptable) domain. This is show in Figure.1. The introduction of a set of uncorrelated reduced variates where,
i
i xi , i 1, 2 ,....... n xi
( 3)
shows the limit state equation becomes
g xi 1 xi , x 2 2 x 2 ,..........., xn n xn 0
(4)
where and are the means and standard deviations of the design variables. Also the distance D, from a point X’1, X’2, …………., X’n on the failure surface g(X’i) = 0 to the origin of Xi space is given as; 2
2
2
D ( 1 2 ........... n )
216
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The solution to equation (4) and (5) can be transformed into gradient vector where the gradient vector matrix and its transpose are introduced as; G GD (6) G X ( G G ) 1 / 2 D (G t G ) 1 / 2 This implies that:
G X ( G t G )1 / 2
D
(7 )
The minimum distance from the origin describing the variable space to the line representing the failure surface equals beta, β and therefore equation (7) becomes, t
G X t ( G tG )1 / 2
(8 ) ,
,
,
Where G is the gradient vector at the most probable failure point ( 1 , 2 , , n ) and value of β is the measure of the safety of any given design under uncertainties in the decision variables. In scalar form equation (8) becomes
'
i
i
g ' i
i
g ' i
(9)
where the derivatives are performed at ( 1, , 2, , , n, ) . Also introducing the concept of reduced variates, equation (9) can be truncated at first order linear term and simplified to,
g
(10)
g
2.3 Derivation of safe design parameters In other to resist the loads, the resistance properties must be carefully chosen. This selection may not only revolve about the derivation of the ultimate strength equation of the steel plate girder section but also aid in deriving the limit state expression of the various failure modes considered in the loading of reinforced steel plate girder. Hence the limits state equation for the idealized strength and load analysis are derived from the respective failure modes, under moment capacity, shear capacity, and deflection failure modes. i) Bending moment capacity The moment capacity of the steel plate girder
The moment resistance of the steel section
=
( + )
(11)
=
SPy
(12)
where S = the plastic modulus of section and Py = the design strength. The limit state equation for the performance function for bending is given , , ,
,
=
( + )
−
(13)
ii) Shear capacity =
(14)
In which the shear resistance of steel section is giving by: Pv = 0.6Py Av
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where Pv = Shear capacity and Py = design strength, Av = tD. The limit state equation for the performance function for shear is given as , , ,
,
= 0.6
−
(16)
iii) Deflection The deflection at mid – span =
(17)
=
(18)
Where w = un-factored imposed load, l = effective span Allowable deflection
a
The limit state equation for the performance function for deflection is given as g(w, l, E, I) =
−
(19)
The following equations were used to obtained the stochastic model for the basic variables in different limit state COV = N=
( ) ( )
E(X)
S(X) = COV x E(X)
(20) (21) (22)
Where: COV = Coefficient of variation of the basic variables, S(X) = standard deviation of the basic variables, E(X) = mean of basic variables, = bias factor of the basic variables, N = nominal value of the basic variables obtained from the deterministic analysis of the plate girder. With the various performance function or limit state equation derived for various mode of failures and taking into consideration the variability in the design variables, they are rated using the First-Order Reliability Analysis (FORA). Fig. 2 shows a flow chart of the numerical algorithm for the computation of β using FORM5 (Gollwitzer, et al; 1986) computer package, the implied safety levels or safety indices (β) associated with BS 5950 design criteria for different limit states for steel plate girder are computed using the example in Section 2.4. 2.4 Example of a simply supported steel plate girder A grade 43 simply supported steel plate girder design to transmit a uniformly distributed load, point load and dynamic loading, was design in accordance with BS 5950 (2000). The plate girder have a clear span of 20m and was opted for as an economical section that satisfies both the ultimate and serviceability limit states of BS 5950 (2000). 2.5 Result of reliability assessment Reliability analyses of simply supported steel plate girder design was achieved by the use of FORM 5 by estimating the reliability levels at varying value of span, L; design strength, Py; load ratio, flange width, web thickness, and depth. Safety indices were obtained from the programs, and plot of the safety indices versus the varied design variables as shown, considering moment failure criterion, shear failure criterion and deflection failure criterion. 3. Presentation of Results and Discussions The reliability levels implicit in the various design criteria are presented in figures 3 to 9. The results show that the BS5950 (2000) design criteria for steel plate girder has beta values ranging from -0.49 to 7.54 with an average of about 2.77 (i.e., Pf = 2.8 x10-3) for bending; 2.3 to 9.59 with average of 6.98( Pf = 2.8.x 10-10) in shear and 4.77 to 9.42 with average of 8.23(Pf = 1.0 x 10-15) in deflection. The results showed that the bending criterion is more critical whereas the shear and deflection criteria are considered conservatively adequate in
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comparison with JCSS, 2000 recommendation of Safety index (β) values of 3.3 to 4.4 for structure with a moderate to large consequences of failure. It is also observed that for all design criteria, there is a general trend of beta, (β) decreasing with increasing live-dead load (L/D) ratio. However, (β) increases with increase in design strength (Py) and depth of plate girder in shear as depicted in Figures 4 and 7. Figure 3 also revealed that the plate girder design considering bending criterion is safe at lower range of live-dead (L/D) ratios but unsafe at higher ratios of 0.8 to 1.2 in all design strength considered. Also shown in Figure 3 is that the safety of plate girder is enhanced as flange width is increased, this also improves the resistance against bending. The reliability assessment of steel plate girder also shows that, the safety index, β increase with increase in flange thickness and it decrease with increase in livedead load ratio as figure 5. Considering shear and deflection criteria, Figures 6, 7, 8 and 9 confirm conservativeness of the safety index β, as depth and span of plate girders varies with load ratio L/D., while Figure 10 shows the most likely failure point. This general trend where reliability index beta (β), decreases with increase in the load ratio is an indication of the fact that reliability arising from the design to BS 5950(2000) is sensitive to design variables such as the load and geometry and hence the uniformity in reliability level which is an object of the code specification is highly questioned. All the design criteria exhibit some level of inconsistency though the level of inconsistency varies from one criterion to criterion.
9 8 7
SAFETY INDEX
6 5 4 3 2 1 0 -1 -2 0.2
0.4
0.6
o.8
1
LIVE-DEAD LOAD RATIO 245
255
265
275
Figure 3 Variation of safety indices with live-dead load ratio and constant design strength of the material (Py) in bending
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Safety Index
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8 7 6 5 4 3 2 1 0 -1 -2 245
255
265
275
Design Strenght (Py) N/mm2 0.2
0.4
0.6
0.8
1
1.2
Figure 4 Variation of safety indices with the design strength of the material (Py) and constant live-dead load ratio in bending 10 SAFETY INDEX
8 6 4 2 0 -2 -4 0.2
0.4
0.6
0.8
1
1.2
LIVE-DEAD LOAD RATIO 20
25
30
35
40
Figure 5 Variation of safety indices with live-dead load ratio and constant flange thickness in bending 12 SAFETY INDEX
10 8 6 4 2 0 0.2 1000
0.4 1200
0.6
0.8
LIVE-DEAD LOAD RATIO 1400 1600
1 1800
1.2 2000
Figure 6 Variation of safety indices with live-dead ratio at constant depth of flat girder in shear
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12
SAFETY INDEX
10 8 6 4 2 0 1000
1200
1400
1600
1800
2000
DEPTH (mm) 0.2
0.4
0.6
0.8
1
1.2
SAFETY INDEX
Figure 7 Variation of safety indices with depth at constant live-dead ratio of flat girder in shear
10 9 8 7 6 5 4 3 2 1 0 0.2
0.4
0.6
0.8
1
1.2
LIVE-DEAD RATIO 5
10
15
20
25
Figure 8 Variation of safety indices with live-dead load ratio and constant span in deflection
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10 9 8 7 6 5 4 3 2 1 0 5 0.2
10 0.4
15 SPAN (m) 0.6 0.8
20 1
25 1.2
Figure 9 Variation of safety indices with span and constant live-dead load ratio in deflection
Figure 10
Most likely failure point
4. Conclusion Reliability assessment of simply supported steel plate girder considering ultimate and serviceability limit states was investigated using FORM5. The results of the investigation showed that the BS 5950 (2000) design procedure of the plate girder is fairly consistent. When the flange width, flange thickness, web thickness and depth of the plate girder are kept constant, as the magnitude of live-dead load ratio increased the safety of the design section decreased. Therefore, BS 5950 design results seems conservative with respect to shear and deflection and unsafe with respect to bending at higher load ratios. 5. References Abubakar, I. and E. Pius. 2007. Reliability analysis of simply supported steel beam, Australian J. Basic and Applied Sciences,1(1) pp 20-29. Ang, T.C. and T. J. Maginley. 2004. Structural steelwork, design to limit state theory, Elservier Butterwork Heinemann, Oxford. Benjamin, J.R. and C.A. Cornell. 1970. Probability, statistics and decision for civil engineers, McGraw-Hill, New York, pp 105-115. BS 5950, 2000. The structural use of steelwork in building, Part 1, Her Majesty Stationary Office, London. Dhillon, B.S. 2005. Reliability, quality and safety for engineers, CRC press, London. Gollwitzer, S.; Abdo, T. and R. Rackwitz. 1988. First Order Reliability Method.
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(FORM), Manual, RCP, GMBH, Munich, West Germany. Fishburn, A. M. and N. C. Lind. 1974. Exact and Invariant Second Moment Code Format” J. Engineering, Mechanical Division. JCSS, 2000. Probabilistic Model Code, Joint Committee on Structural Safety, 2000/01 IABSE, Publication, London. Melchers, R.E. 1990. Structural reliability analysis and prediction, John Wiley and Sons, New York. McGinley, T. J. 1986. Structural steelwork calculation and detailing, Billing and Sons Limited, London Uche, O.A.U and J. O. Afolayan. 2008. Reliability-based hating of the BS8110 design criteria for reinforced concrete columns. Journal of Engineering and Technology,3(1), pp1-11.
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UcheEngineering and Ahmed Epistemics in Science, and Technology
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Reliability Design Format for Steel Plate Girder to BS 5950 (2000) 1
Uche, O.A.U and 2Ahmed A Civil Engineering Department, Faculty of Engineering, Bayero University, Kano. Email: [email protected]
1,2
Abstract The paper presents reliability assessment of deterministic design of steel plate girder considering both ultimate and serviceability limit state in accordance with BS 5950 (2000). The reliability analysis was carried out using First Order Reliability Method (FORM). Design variables such as strength of the material (Py), width of the flange, flange thickness, web thickness as well as the span of the girder were considered random and stochastic. It was shown among the findings that, when the span (L) of the plate girder was kept constant with increase in the magnitude of live-dead load ratio, the safety indices decreases, as deflection criterion was considered. Also, the design of the plate girder in accordance with BS 5950 considering shear and deflection is safe for almost all the range of variables considered. On varying the depth of the plate girder, safety indices increase with increase in depth and decrease with increase in live-dead load ratio, when shear criterion was considered. The deflection is the most critical mode of failure on varying the span and load ratio, with safety index less than the recommended value by JCSS 2000 for structural members with moderate to large consequences of failure of 3.3 to 4.4. Therefore, BS 5950 design result seems unsafe with respect to bending (under high live-dead load ratio) and satisfactory with respect to shear and deflection. It is then, recommended that the design of steel plate girder base on BS 5950 be reviewed to incorporate reliability analysis. Keywords Reliability design, steel plate girder, safety index, deterministic design
1.
Introduction and Concept
The aim of structural design is to produce a safe, aesthetical and economical structure that fulfills its required purpose (Clarke and Converman, 1987). Theoretical knowledge of structural analysis must be combined with knowledge of design principles and theory and the constraints given in the standard code of practice to give a safe design. Although a thorough knowledge of properties of materials, methods of fabrication and erection are essential for the experienced designer. Steel structures in a building consist of a skeletal framework which carries the entire load to which the building is subjected. The steel members are used to carry lateral loads when acting as beams and girders, and axial loads when acting as stanchions. Steel on its own exhibits a number of useful characteristics, such as, structural behaviour up to initial yield point which is nearly elastic, deformation which are directly proportional to the applied load and long-term deformation at normal temperature is not a problem (Ang and McGinley, 2004). The important properties of steel are strength, ductility, impact resistance and weldability. The plate girder is typically I – beams made up from separate structural steel plates, which are welded or bolted or riveted together to form the vertical web and horizontal flanges of the beam. Though sometimes they may be formed in a Z – shaped rather than I – shape. Plate girders are used to carry larger loads over longer spans than are possible with universal or compound beams. Thus, it is use in the buildings for long span floor girder, for crane girder and bridges (Abubakar and Pius, 2007). There are two types that can be selected depending on the cost; one of which has a web so slender that the transverse stiffeners must be added and some time longitudinal stiffeners are added as well. The other types of plate girder have thicker webs and therefore, it is not stiffened. However, in the design of plate girder, as in other design processes, variation and uncertainties in the operation condition causes variation in its performance and durability. The design of any structure must ensure safety at worst loading condition and the deformation of the members during normal working condition must be accepted from the durability, performance and appearance of the structure (McGinley, 1986; Morris and Plum, 1987; Ray, 1998, Uche and Afolayan 2008, ). Thus, the existence of physical uncertainties in the design of plate girder required a reliability assessment to be taken in to account, in other to satisfy the structural integrity. A pre- requisite to building safety is the assurance of the structural integrity to a high level of reliability throughout their lives; such integrity may be threatened by uncertainties that were observed in the design, construction and maintenance stages. The work presented is concerned with the safety associated with the design criteria of grade 43 simply supported steel plate girder considering both ultimate and serviceability limit states.
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Epistemics in Science, Engineering and Technology, Vol.3, No.1, 2013, 215-223
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1.2 Justification of the research The use of probabilistic approach in the assessment of plate girder performance would provide structural safety to be treated in a more rational manner for the randomly applied load to plate girder. This will provide adequate counteraction of the uncertainties related to the design of a plate girder and increase the level of confidence. In the field of structural engineering it has become necessary to review the way of designing plate girder and have a convincing reliability based design format (Dhillon, 2005). 2.
Materials and Method
2.1 Steel plate girder design The design of plate girder in Ultimate and serviceability limit state covers violation against strength, fatigue and stability against overturning and sway when the ultimate limit states are exceeded. Whereas Serviceability limit state covers deflection, vibration, durability of steel plate girder system. According to BS 5950(2000), a manual design was carried out using plastic section analysis, this is in turn subjected to reliability analysis. 2.2 First–order reliability method (FORM) The probabilistic design of any structure is connected with the probability that the structure will realize its assigned function. Probability denotes the chance that a particular defined event occurs. The First Order Reliability Method (FORM) involves the calculation of the probability of failure or limit state violation for structural element. This aspect is sometimes ignored and relative frequency data assumed to be purely “objective” information (Melchers 1990). It is noted that a subjective probability estimate reflect the degree of insurance about the phenomenon under consideration. The manipulation of subjective estimate and use of compactible data to amend and improve the estimate may be achieved through Baye’s theorem. Consequently, subjective probabilities assessments are sometimes known as “Bayesian probabilities” (Benjamin, 1976; Lindley, 1972). The reconciliation of the two interpretations of the probability is controversial (Fishburn, 1974; Hasofer, 1984). If R is the strength capacity and S is the loading effect of the structural system which are random variables, the main objective of reliability analysis of any system or component is to ensure that R is never exceeded by S, in practice, R and S are usually functions of basic variables, in order to investigate the effect of the variables on the performance of the variables on the performance of a structural system, a limit state equation in term of the basic design variables is required. The state function used in FORM5 is given by g(X). It is defined such that g(xi) = g(x1, x2, ……, xn) = R – S
(1)
where xi for I = 1, 2, ……, n, represent the basic design variables. The limit state of the system can then be express as g(xi) = 0 (2) Graphically, g(X) > 0 Corresponds to a favourable (safe, intact, acceptable) state while g(X) = 0 denote the failure boundary and g(X) < 0 defines the failure (unacceptable) domain. This is show in Figure.1. The introduction of a set of uncorrelated reduced variates where,
i
i xi , i 1, 2 ,....... n xi
( 3)
shows the limit state equation becomes
g xi 1 xi , x 2 2 x 2 ,..........., xn n xn 0
(4)
where and are the means and standard deviations of the design variables. Also the distance D, from a point X’1, X’2, …………., X’n on the failure surface g(X’i) = 0 to the origin of Xi space is given as; 2
2
2
D ( 1 2 ........... n )
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The solution to equation (4) and (5) can be transformed into gradient vector where the gradient vector matrix and its transpose are introduced as; G GD (6) G X ( G G ) 1 / 2 D (G t G ) 1 / 2 This implies that:
G X ( G t G )1 / 2
D
(7 )
The minimum distance from the origin describing the variable space to the line representing the failure surface equals beta, β and therefore equation (7) becomes, t
G X t ( G tG )1 / 2
(8 ) ,
,
,
Where G is the gradient vector at the most probable failure point ( 1 , 2 , , n ) and value of β is the measure of the safety of any given design under uncertainties in the decision variables. In scalar form equation (8) becomes
'
i
i
g ' i
i
g ' i
(9)
where the derivatives are performed at ( 1, , 2, , , n, ) . Also introducing the concept of reduced variates, equation (9) can be truncated at first order linear term and simplified to,
g
(10)
g
2.3 Derivation of safe design parameters In other to resist the loads, the resistance properties must be carefully chosen. This selection may not only revolve about the derivation of the ultimate strength equation of the steel plate girder section but also aid in deriving the limit state expression of the various failure modes considered in the loading of reinforced steel plate girder. Hence the limits state equation for the idealized strength and load analysis are derived from the respective failure modes, under moment capacity, shear capacity, and deflection failure modes. i) Bending moment capacity The moment capacity of the steel plate girder
The moment resistance of the steel section
=
( + )
(11)
=
SPy
(12)
where S = the plastic modulus of section and Py = the design strength. The limit state equation for the performance function for bending is given , , ,
,
=
( + )
−
(13)
ii) Shear capacity =
(14)
In which the shear resistance of steel section is giving by: Pv = 0.6Py Av
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where Pv = Shear capacity and Py = design strength, Av = tD. The limit state equation for the performance function for shear is given as , , ,
,
= 0.6
−
(16)
iii) Deflection The deflection at mid – span =
(17)
=
(18)
Where w = un-factored imposed load, l = effective span Allowable deflection
a
The limit state equation for the performance function for deflection is given as g(w, l, E, I) =
−
(19)
The following equations were used to obtained the stochastic model for the basic variables in different limit state COV = N=
( ) ( )
E(X)
S(X) = COV x E(X)
(20) (21) (22)
Where: COV = Coefficient of variation of the basic variables, S(X) = standard deviation of the basic variables, E(X) = mean of basic variables, = bias factor of the basic variables, N = nominal value of the basic variables obtained from the deterministic analysis of the plate girder. With the various performance function or limit state equation derived for various mode of failures and taking into consideration the variability in the design variables, they are rated using the First-Order Reliability Analysis (FORA). Fig. 2 shows a flow chart of the numerical algorithm for the computation of β using FORM5 (Gollwitzer, et al; 1986) computer package, the implied safety levels or safety indices (β) associated with BS 5950 design criteria for different limit states for steel plate girder are computed using the example in Section 2.4. 2.4 Example of a simply supported steel plate girder A grade 43 simply supported steel plate girder design to transmit a uniformly distributed load, point load and dynamic loading, was design in accordance with BS 5950 (2000). The plate girder have a clear span of 20m and was opted for as an economical section that satisfies both the ultimate and serviceability limit states of BS 5950 (2000). 2.5 Result of reliability assessment Reliability analyses of simply supported steel plate girder design was achieved by the use of FORM 5 by estimating the reliability levels at varying value of span, L; design strength, Py; load ratio, flange width, web thickness, and depth. Safety indices were obtained from the programs, and plot of the safety indices versus the varied design variables as shown, considering moment failure criterion, shear failure criterion and deflection failure criterion. 3. Presentation of Results and Discussions The reliability levels implicit in the various design criteria are presented in figures 3 to 9. The results show that the BS5950 (2000) design criteria for steel plate girder has beta values ranging from -0.49 to 7.54 with an average of about 2.77 (i.e., Pf = 2.8 x10-3) for bending; 2.3 to 9.59 with average of 6.98( Pf = 2.8.x 10-10) in shear and 4.77 to 9.42 with average of 8.23(Pf = 1.0 x 10-15) in deflection. The results showed that the bending criterion is more critical whereas the shear and deflection criteria are considered conservatively adequate in
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comparison with JCSS, 2000 recommendation of Safety index (β) values of 3.3 to 4.4 for structure with a moderate to large consequences of failure. It is also observed that for all design criteria, there is a general trend of beta, (β) decreasing with increasing live-dead load (L/D) ratio. However, (β) increases with increase in design strength (Py) and depth of plate girder in shear as depicted in Figures 4 and 7. Figure 3 also revealed that the plate girder design considering bending criterion is safe at lower range of live-dead (L/D) ratios but unsafe at higher ratios of 0.8 to 1.2 in all design strength considered. Also shown in Figure 3 is that the safety of plate girder is enhanced as flange width is increased, this also improves the resistance against bending. The reliability assessment of steel plate girder also shows that, the safety index, β increase with increase in flange thickness and it decrease with increase in livedead load ratio as figure 5. Considering shear and deflection criteria, Figures 6, 7, 8 and 9 confirm conservativeness of the safety index β, as depth and span of plate girders varies with load ratio L/D., while Figure 10 shows the most likely failure point. This general trend where reliability index beta (β), decreases with increase in the load ratio is an indication of the fact that reliability arising from the design to BS 5950(2000) is sensitive to design variables such as the load and geometry and hence the uniformity in reliability level which is an object of the code specification is highly questioned. All the design criteria exhibit some level of inconsistency though the level of inconsistency varies from one criterion to criterion.
9 8 7
SAFETY INDEX
6 5 4 3 2 1 0 -1 -2 0.2
0.4
0.6
o.8
1
LIVE-DEAD LOAD RATIO 245
255
265
275
Figure 3 Variation of safety indices with live-dead load ratio and constant design strength of the material (Py) in bending
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8 7 6 5 4 3 2 1 0 -1 -2 245
255
265
275
Design Strenght (Py) N/mm2 0.2
0.4
0.6
0.8
1
1.2
Figure 4 Variation of safety indices with the design strength of the material (Py) and constant live-dead load ratio in bending 10 SAFETY INDEX
8 6 4 2 0 -2 -4 0.2
0.4
0.6
0.8
1
1.2
LIVE-DEAD LOAD RATIO 20
25
30
35
40
Figure 5 Variation of safety indices with live-dead load ratio and constant flange thickness in bending 12 SAFETY INDEX
10 8 6 4 2 0 0.2 1000
0.4 1200
0.6
0.8
LIVE-DEAD LOAD RATIO 1400 1600
1 1800
1.2 2000
Figure 6 Variation of safety indices with live-dead ratio at constant depth of flat girder in shear
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SAFETY INDEX
10 8 6 4 2 0 1000
1200
1400
1600
1800
2000
DEPTH (mm) 0.2
0.4
0.6
0.8
1
1.2
SAFETY INDEX
Figure 7 Variation of safety indices with depth at constant live-dead ratio of flat girder in shear
10 9 8 7 6 5 4 3 2 1 0 0.2
0.4
0.6
0.8
1
1.2
LIVE-DEAD RATIO 5
10
15
20
25
Figure 8 Variation of safety indices with live-dead load ratio and constant span in deflection
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10 9 8 7 6 5 4 3 2 1 0 5 0.2
10 0.4
15 SPAN (m) 0.6 0.8
20 1
25 1.2
Figure 9 Variation of safety indices with span and constant live-dead load ratio in deflection
Figure 10
Most likely failure point
4. Conclusion Reliability assessment of simply supported steel plate girder considering ultimate and serviceability limit states was investigated using FORM5. The results of the investigation showed that the BS 5950 (2000) design procedure of the plate girder is fairly consistent. When the flange width, flange thickness, web thickness and depth of the plate girder are kept constant, as the magnitude of live-dead load ratio increased the safety of the design section decreased. Therefore, BS 5950 design results seems conservative with respect to shear and deflection and unsafe with respect to bending at higher load ratios. 5. References Abubakar, I. and E. Pius. 2007. Reliability analysis of simply supported steel beam, Australian J. Basic and Applied Sciences,1(1) pp 20-29. Ang, T.C. and T. J. Maginley. 2004. Structural steelwork, design to limit state theory, Elservier Butterwork Heinemann, Oxford. Benjamin, J.R. and C.A. Cornell. 1970. Probability, statistics and decision for civil engineers, McGraw-Hill, New York, pp 105-115. BS 5950, 2000. The structural use of steelwork in building, Part 1, Her Majesty Stationary Office, London. Dhillon, B.S. 2005. Reliability, quality and safety for engineers, CRC press, London. Gollwitzer, S.; Abdo, T. and R. Rackwitz. 1988. First Order Reliability Method.
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(FORM), Manual, RCP, GMBH, Munich, West Germany. Fishburn, A. M. and N. C. Lind. 1974. Exact and Invariant Second Moment Code Format” J. Engineering, Mechanical Division. JCSS, 2000. Probabilistic Model Code, Joint Committee on Structural Safety, 2000/01 IABSE, Publication, London. Melchers, R.E. 1990. Structural reliability analysis and prediction, John Wiley and Sons, New York. McGinley, T. J. 1986. Structural steelwork calculation and detailing, Billing and Sons Limited, London Uche, O.A.U and J. O. Afolayan. 2008. Reliability-based hating of the BS8110 design criteria for reinforced concrete columns. Journal of Engineering and Technology,3(1), pp1-11.
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Munck Cranes Inc. Assumes no responsibility for, but not limited to, any events or damages which occur as a result of the use of any data contained in this brochure. It should be noted that these figures are supplied to give a general understanding of the specifications relating to any particular capacity of top running, double girder. Crane Girder design is an Excel Spreadsheet Template for the design of simply supported crane girders. The procedure is based on the Steelwork Design Guide to BS 5950-1:2000, Volume 2, Worked Examples, SCI Publication P326, The Steel Construction Institute, 2003.