Latour - Paper SISMICA 2010

SSMICA 2010 8 CONGRESSO DE SISMOLOGIA E ENGENHARIA SSMICA 1

SEISMIC DESIGN OF EXPOSED COLUMN BASE PLATE CONNECTIONS: MONTE CARLO ANALYSIS

LATOUR MASSIMO

Ph.D. Student

University of Salerno

Fisciano (SA) - Italy

RIZZANO GIANVITTORIO

Associate Professor

University of Salerno

Fisciano (SA) - Italy

SUMMARY

The seismic design of moment resisting steel frames, in the usual design practice, is carried out adopting full

strength joints. In this way, the dissipation of the seismic input energy is obtained relying on the plastic hinges

developed at beam ends and at the base of columns rather than on the plastic engagement of the elements

composing the connections. The design of full strength joints can be led in the framework of Eurocode 8

providing to the connections an adequate overstrength. In this work, with reference to base plates connections,

starting from a Monte Carlo simulation accounting for the influence of random material variability of the column

and of the connecting elements, the reliability of the approach provided by Eurocode 8 is analyzed. On the basis

of the results obtained from the statistical analyses a new criteria accounting for both the random variability of

the steel and the overstrength of the connecting members due to the strain hardening is proposed.

1. INTRODUCTION

It is well known that the knowledge of the actual response of a structure subjected to seismic loads can be very

complex because of the large number of sources of uncertainty involved in the design process. In fact, external

loads, environmental factors, material properties and geometry of structures all usually give rise to important

effects on the performance of buildings under severe ground motions. Besides, further uncertainties related to the

lack of understanding of the true structural behaviour and to the simplifications usually assumed in the

mechanical modelling can result in unsatisfactory predictions of the structural performance if these sources of

variability are not properly taken into account.

A first effect, which is mainly due to the randomness related to the definition of the dynamic properties of the

seismic action, to the materials mechanical properties and to the quality of the workmanship is usually called

aleatory uncertainty. A second effect that is related to the deviations of physical models of the structural

elements from reality and to the approximations of the analysis procedure (static or dynamic, linear or non-

linear) represents the so-called epistemic uncertainty. Therefore, it is obvious that, in general, the complete

knowledge of a structural system requires not only the implementation of refined models regarding the

prediction of the structural response from a deterministic point of view but also the application of probabilistic

and statistical concepts to account for the uncertain nature of structures.

Starting from the 70s different techniques for the stochastic analysis of structural systems have been developed.

Essentially three types of methods can be identified: perturbation methods, reliability methods and simulation

methods. Perturbation methods involve the first and second order Taylor series expansion of the terms in the

governing equation around the mean values of the random variables. The variation of the response is then

obtained by solving a set of deterministic equations. In this method no distribution information are required and

the random variables are characterized by the first and second order statistical moments. Reliability methods are

aimed at evaluating the failure probabilities of a structure defining the failure criteria in terms of limit-state

functions defining the surfaces separating the safe and failure sets. Finally, in Monte Carlo simulation method a

deterministic analysis is carried out for a series of parameters generated in agreement with their probability

distributions and then the statistics of the response quantities, such as the mean values, the variance and the

exceedance probabilities are evaluated on the results of the generated sample. This kind of analysis, often

retained a force-brute method, requires usually a great computational effort even though gives indisputable

advantages, as for instance the flexibility, in fact it can be applied generally to all types of engineering problems.

For these reasons, it is advisable, before applying a Monte Carlo analysis, to adequately evaluate the

2 SSMICA 2010 8 CONGRESSO DE SISMOLOGIA E ENGENHARIA SSMICA

cost/effectiveness computational effort needs because, in some cases, i.e. when a large number of solutions are

needed to obtain accurate results and/or many random variables are considered, the procedure can result too

expensive.

With reference to the seismic design of moment resisting steel frames, the modelling of joints is an aspect of

considerable importance, in fact, an affordable modelling of the elements constituting the connections in terms of

strength, stiffness and ductility supply is required, due to the relevance of such parameters in the prediction of

the overall structural response [1-5]. In particular, the prediction of the bending resistance of joints is of great

importance because it affects the location of dissipative zones. Such prediction can be carried out by means of

the so-called component method, which is currently codified in Eurocode 3 [6] with reference both to beam-to-

column joints and to base plate joints.

Furthermore, last version of Eurocode 3 allows, under severe seismic events, to locate the dissipative zones, i.e.

the plastic hinges, either at the connecting elements or at the beam ends. In the former case, within the

framework of the capacity design, the dissipation of seismic input energy is entrusted mainly to beam-to-column

joints and to column base joints. In the latter case joints have to be designed to provide the full strength, in order

to allow the formation of plastic zones at the beams ends and at the base of columns. To this scope, the joints

located in the dissipative zones have to posses an adequate over-strength with respect to the connecting elements,

to guarantee that, at the Ultimate Limit State, the full development of the cyclic yielding of the dissipative parts

is obtained. This last approach is usually considered the best way to provide moment resisting frames with

adequate behaviour in terms of ductility and energy dissipation supply. In order to reach this goal, the approach

suggested by the Eurocode 8 is to design the connecting parts for all the joints typologies, made exception for

full penetration welded connections whose resistance is assumed to be sufficient, requiring that the following

relationship is satisfied:

fyovd RR 1.1 (1)

where dR is the resistance of the connection defined in Eurocode 3, fyR is the plastic resistance of the

connected dissipative member based on the design yield stress of the material, ov is the overstrength factor depending on the ratio between the actual and nominal value of the steel yield strength. In particular, ov can be taken equal to 1.00 if the actual strength of the dissipative elements is determined on the base of specific

experimental tests, such condition is usually verified when the assessment of an existing building is considered

or when steels are taken from specific stocks. In the other situations, even though the overstrength factor should

be specified by the national annex, Eurocode 8 recommends the value ov=1.25. Many authors have pointed out how, the unexpected failure of the connecting elements during the seismic events

of Kobe (1995) and Northridge (1994) has represented a limitation to the energy dissipation capacity of steel

frames, not allowing the full development of beam plastic rotation supply [7]. In particular, in some cases the

reasons of this unsatisfactory behavior have been ascribed to the deficiency of the design criteria which in many

cases were not able to provide the adequate over-strength to the joints, i.e. the criteria were not sufficient to

allow the full development of the beam plastic hinges. In fact, a number of steel buildings, particularly low rise

moment resistant frame systems, developed failure at the column base plate connections and was found by [7]

that the rotational stiffness and strength of the base plates affected the damage of these structures which arised

not only in the column bases but also in other regions of the lateral loading resistant frames.

For these reasons, the main objective of the paper is to analyze the criterion suggested by EC8 to design full

strength column base joints and to propose a recalibration of the parameters involved in the design of base plate

connections in order to provide full strength joints. Thus, basing on a probabilistic approach devoted to compare

the column ultimate resistance and the base joints ultimate strength accounting for random material variability

effects, the following steps have been performed:

1. design of 10 full strength base joints connecting different shapes, 6 unstiffened and 4 stiffened, applying the predicting model for the joint flexural resistance given in EC3;

2. implementation of a Monte Carlo simulation accounting for the random variability of yield strength of the steel constituting base plate, flange and webs of the connected member, of the ultimate stress of the

bolts and of the characteristic value of the cubic compressive strength of the concrete;

3. analysis of the accuracy of the approach proposed by EC8 to design full strength steel joints; 4. proposal of a new design criterion able to properly account for the above said effects, covering the

whole range of steel shapes considered in the Monte Carlo analyses.


2. DESIGN OF BASE JOINT

The connection typology considered in this work is an exposed column base joint with only one bolt row outside

the column flange (Fig.1). It is useful to note that the method proposed by EC3 for the prediction of the plastic

resistance of column-base joints is mainly devoted to Joint Typology which are normally adopted in the

experimental programs. In previous works of the authors [8] the reliability of the Eurocode 3 approach has been

analyzed by evaluating the degree of accuracy in terms of prediction of bending stiffness and plastic resistance.

These analyses have pointed out that the model codified in EC3 is sufficiently accurate in terms of strength

prediction, conversely regarding the prediction of the initial stiffness, even though the approach gives reasonable

results for practical scopes, a lower degree of accuracy has to be expected [4,9]. In any case, the purpose of the

paper is to compare the joint flexural resistance with respect to that of the connected member, so that the

attention is completely focused on the strength. To this scope two deterministic models have been applied. In

particular, the plastic resistance of the joint has been obtained by applying the component method as codified in

last version of Eurocode and the ultimate strength of the column has been derived from the formulation given by

Mazzolani and Piluso [10].

d

bpl

b

Base plate

zz

Stiffeners

hn

0,8s

1,5dm

d

b

hpl

pl

c

bcf

b

Base plate

zzt

hpl

hn

0,8s

3d 1,25mm c

2db

2,4

db

2,4

db

b

Unstiffened column base

Stiffened column base

0,8s

zt,l zc,r

N

e

z

L*=

2000 m

m

b

c

t c

Figure 1. Scheme of the joint typologies considered

With reference to exposed column base plates, different components are individuated as source of stiffness and

strength in EC3 model, namely concrete in compression, base plate in bending, anchor bolts in tension and

column flange and web in compression. This last component is only involved in the definition of strength. In

Eurocode approach the prediction of the base plate moment rotation curve is obtained starting from the definition

of three non overlapping T-stubs, two under the column flanges and one under the connected member web. In

particular, EC3 proposes to consider only the T-stubs under the column flanges when the column base plate has

to bear both to axial load and to bending moments. Basing on translational and rotational equilibria the resistance

of the joint is given by the Eurocode 3 in cases of low and high eccentricity. In particular, considering the

notation given in Fig.2, the following relations are given:

Low eccentricity

e < zc,r

1;

1min

,

,

,

,

,ez

zF

ez

zFM

lc

Rdcr

rc

Rdcl

Rdj (2)

High eccentricity

e > zc,r

1;

1min

,

,

,

,

,ez

zF

ez

zFM

lt

Rdcr

rc

Rdtl

Rdj (3)


where Fcl,Rd and Ftl,Rd are the plastic resistances of the equivalent T-stub in compression/tension under the left column flange, Fcr,Rd, is the plastic resistance of the equivalent T-stub in compression under the right column flange, zcl, zcr, ztl are the lever arms of the springs modelling the equivalent T-stubs in compression/tension reported in Fig.3 and z is the total lever arm, defined in low eccentricity as sum of zcl and zcr and in high eccentricity as sum of ztl and zcr.

In order to evaluate the accuracy of the Eurocode approach to design full strength column base joints, in the first

step of the work the design of 10 full strength joints fastening different column shapes varying in a wide range of

the geometrical parameters, i.e. from HE 120 A to HE 320 B, has been performed. In particular, aiming to

evaluate the influence both of the axial load and of the anchor bolts class on the flexural resistance of the

connection, for each column shape 3 different values of the ratio between the axial load and the squash load and

2 different classes of anchor bolts have been considered. The connections designed can be divided into two

groups: the first one constituted by 6 joints without stiffeners, fastening HE 120 A, HE 160 A, HE 200 A, HE

100 B, HE 120 B and HE 160 B profiles, and the second one constituted by 4 joints with stiffening plates,

connecting HE 300 A, HE 400 A, HE 240 B, HE 320 B columns. Steel grade S235 has been assumed both for

the profiles and for the base plates, while two anchor classes have been considered, i.e. 8.8 and 10.9.

zc,r

N

Kc,l Kc,r

e < zc

zc,l

e

Fc,rFc,l

z

zt,l zc,r

Kt,l Kc,r

e > zc

N

e

Ft,l Fc,r

z

Fig. 2: EC3 mechanical model for column base joints

Preliminary, two criteria have been considered for designing the joints. The first one is the criterion

recommended by Eurocode 8, with ov=1.25, providing the following design formulation of the joint resistance:

cplcplovECj MMM ,,8, 375.11.1 (4)

Regarding Eq.4, proposed by Eurocode 8, it is clear that the two coefficients multiplying the plastic resistance of

the connected member are introduced to account for the two effects above said: the random material variability

and the beam or column overstrength. If reference is made to the coefficient related to the strain-hardening, the

value 1.1, prescribed by Eurocode 8, is not able to cover all the range of real cases both for beams (0) as already demonstrated in [10] and for columns (usually 0.10.2). As an example, in Fig. 3 the overstrength parameter s for HEA and HEB shapes is represented considering different values of the member span-to-depth

ratio.

HE A COLUMNS

10

12

141618202530

f u /f y

1,1

1,2

1,3

1,4

1,5

1,6

0 200 400 600 800 1000

COLUMN DEPTH (mm)

CO

LU

MN

OV

ER

ST

RE

NG

TH

L*=2000 mm

Analyzed casesL*/hc RATIO

HE B COLUMNS

1012141618202530

f u /f y

1,1

1,2

1,3

1,4

1,5

1,6

0 200 400 600 800 1000

COLUMN DEPTH (mm)

CO

LU

MN

OV

ER

ST

RE

NG

TH

L*=2000 mm

Analyzed casesL*/hc RATIO

Figure 3. Overstrength factor s for HEA and HEB shapes


Starting from the above considerations, the second joint design approach is an hybrid approach accounting for

the effect due to the random variability of steel yield stress using the value of ov suggested by EC8 and for the effect due to the strain hardening of the base material by means of the formulation proposed by Mazzolani and

Piluso [10].

In this formulation, the influence of the strain hardening, has been modeled by properly calibrating the main

parameters involved in the definition of the overstrength factor due to the strain hardening, i.e. the width-to-

thickness ratios of the member plates and the bending moment gradient. To this scope the following expression

has been derived:

pMsM max (5)

where Mmax is the beam/column maximum moment accounting for the strain-hardening effect, Mp is the

beam/column plastic moment accounting for the moment-shear interaction effects, is the ratio between the axial load and the squash load (


Table 1 Geometrical properties of unstiffened joints with 8.8 anchor bolts

Eurocode 8 Design Criterion Hybrid Design Approach

Shape tpl

(mm)

bpl (mm)

hpl (mm)

nb Mj

(kNm)

(%)

tpl (mm)

bpl (mm)

hpl (mm)

nb Mj

(kNm)

(%)

HE 120 A 0,1 23 162 342 2 39,7 2,9% 24 162 342 2 43,0 2,9%

HE 120 A 0,2 21 162 342 2 36,8 4,7% 22 162 342 2 39,8 5,3%

HE 120 A 0,3 18 162 342 2 31,9 3,6% 19 162 342 2 34,4 2,4%

HE 160 A 0,1 25 234 380 3 82,0 3,6% 25 234 380 3 82,0 0,1%

HE 160 A 0,2 27 162 380 2 74,0 1,8% 27 162 380 2 74,0 0,6%

HE 160 A 0,3 23 162 380 2 63,8 0,3% 24 162 380 2 67,7 4,2%

HE 200 A 0,1 28 306 418 4 139,5 0,5% 28 306 418 4 139,5 1,4%

HE 200 A 0,2 28 234 418 3 133,9 5,1% 27 234 418 3 126,1 3,1%

HE 200 A 0,3 26 200 418 2 116,0 4,1% 25 200 418 2 109,9 3,0%

HE 100 B 0,1 23 162 328 2 36,3 7,7% 25 162 328 2 42,4 6,9%

HE 100 B 0,2 20 162 328 2 30,9 1,3% 23 162 328 2 39,0 7,5%

HE 100 B 0,3 18 162 328 2 28,8 8,0% 20 162 328 2 33,6 2,0%

HE 120 B 0,1 27 162 348 2 56,0 5,0% 29 162 348 2 64,0 2,5%

HE 120 B 0,2 24 162 348 2 49,5 3,0% 27 162 348 2 60,4 5,9%

HE 120 B 0,3 21 162 348 2 44,3 5,4% 24 162 348 2 53,9 4,7%

HE 160 B 0,1 27 306 388 4 117,4 2,7% 29 306 388 4 134,0 0,9%

HE 160 B 0,2 27 234 388 3 109,2 5,5% 29 234 388 3 123,2 1,7%

HE 160 B 0,3 27 162 388 2 90,8 0,1% 26 234 388 3 112,0 2,7%

Table 2 Geometrical properties of stiffened joints with 8.8 anchor bolts

Eurocode 8 Design Criterion Hybrid Design Approach

Shape tpl

(mm)

bpl (mm)

hpl (mm)

nb Mj

(kNm)

(%)

tpl (mm)

bpl (mm)

hpl (mm)

nb Mj

(kNm)

(%)

HE 300 A 0,1 21 512 713 5 457,6 2,4% 21 512 713 5 457,6 4,2%

HE 300 A 0,2 19 512 713 5 417,7 2,0% 19 512 713 5 417,7 7,9%

HE 300 A 0,3 17 512 713 4 385,4 7,6% 16 512 713 4 353,9 6,3%

HE 400 A 0,1 25 656 813 7 858,9 3,8% 29 656 813 7 927,1 0,0%

HE 400 A 0,2 23 656 813 7 796,9 3,3% 25 656 813 7 848,2 2,6%

HE 400 A 0,3 27 512 813 5 687,8 1,9% 29 512 813 5 732,9 1,5%

HE 240 B 0,1 21 452 663 5 366,9 7,8% 21 596 663 7 427,5 8,7%

HE 240 B 0,2 19 452 663 4 333,1 8,3% 20 452 663 5 363,1 1,9%

HE 240 B 0,3 16 452 663 4 280,2 4,1% 18 452 663 4 331,1 4,1%

HE 320 B 0,1 24 656 743 7 701,3 1,0% 31 656 743 7 820,3 1,3%

HE 320 B 0,2 21 656 743 7 634,4 0,6% 26 656 743 7 743,9 1,9%

HE 320 B 0,3 26 512 743 5 561,4 1,8% 27 656 743 7 667,3 3,1%

In particular, S235 steel grade has been considered, in other works, by means of a regression analysis on the

results of 550 experimental tests, it has been recognized that the log-normal distribution provides good

agreement with the experimental data. Therefore, in the present paper it is assumed that ln(fy) is normally

distributed with a mean value E[ln(fy)] depending on the plate thickness t and given by the following relationship

[13-16]:

ln 0,007 5,7664yE f t (9)

and with a standard deviation equal to 0.07. In Eq. (9) the units to be adopted are N and mm. The ultimate tensile

stress of the bolts, according to available experimental data is assumed to be normally distributed with a ratio

between the tensile mean resistance ant the nominal one equal to 1.20 for bolt class 8.8 and to 1.07 for bolt class

10.9 and with a coefficients of variation equal to 0.07 and to 0.02, respectively. Regarding the concrete

constituting the base footing, class C20/25 has been considered with a mean value of the cylindrical compressive

strength equal to 28 MPa and a coefficient of variation equal to 0.2 [17]. The main statistical parameters of the

distribution characterizing the mechanical properties of the materials constituting the base joints are summarized

in Tab.3.

For each mechanical property, starting from the distributions previously described the generation of the random

values has been performed by means of the Box and Muller approach [18]. Each combination of the generated

random values provides an element of the investigated sample.


Table 3 Statistical parameters for the considered random variables

Random variable Statistical

Distribution

Mean Value

[MPa]

Coefficient of

Variation

Standard

Deviation

S235 Steel lognormal Eq.(9) # 0.07

C20/25 Concrete normal 28 0.2 #

Anchor Bolts (8.8 Class) normal 960 0.07 #

Anchor Bolts (10.9 Class) normal 1070 0.02 #

The resistance of the joint (Mj) has been evaluated by means of the component approach codified in a software

specifically developed by the authors to this scope. The evaluation of the overstrength factor due to the strain-

hardening s, of the non-dimensional axial load , of the column plastic moment Mc allows to calculate the statistical sample of the ratio Ov=Mj /[(s-)Mc] between the joint plastic moment and the column maximum resistance.

MONTE CARLO ANALYSIS

RANDOM GENERATION OF

THE MATERIALS

MECHANICAL PROPERTIES

(BOX AND MULLER)

EVALUATION OF THE BASE

PLATE FLEXURAL

RESISTANCE (EUROCODE 3)

EVALUATION OF THE COLUMN

FLEXURAL RESISTANCE

STATISTICAL ANALYSIS OF THE

RATIO BETWEEN THE RESISTANCE

OF COLUMN BASE JOINT AND OF

FASTENED COLUMN

DESIGN OF THE BASE

PLATE CONNECTIONS

EVALUATION OF THE

ACCURACY OF THE DESIGN

APPROACHES

Fig. 4: Scheme of the Monte Carlo Simulation

Aiming to establish the size of the sample to be investigated, a preliminary test has been performed generating

10000 combinations of the mechanical properties of column, base plate, anchors and concrete, considering two

connections: the first one fastening an HE 160 B column shape with =0.3 and the second one connecting an HE 320 B column with =0.1 The minimum size of the sample has been selected analysing its influence on the mean value and on the standard deviation of the ratio Mj/[(s-)Mc]. From this preliminary analysis, as shown in Fig. 5, it has been observed that a sample size equal to 6000 elements leads to an accurate and stable evaluation

of these statistical parameters characterising the distribution of the overstrength ratio.


Mean Value of Ov

1,260

1,265

1,270

1,275

1,280

1,285

1,290

1,295

1,300

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Sample

Standard Deviation of Ov

0,095

0,100

0,105

0,110

0,115

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Sample

Mean Value of M j,R

950

955

960

965

970

975

980

985

990

995

1000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Sample

Standard Deviation of M j,R

50,000

52,000

54,000

56,000

58,000

60,000

62,000

64,000

66,000

68,000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Sample

Figure 5. Mean value and Standard Deviation for increasing values of the sample size (joint HE 320 B)

4. ACCURACY OF THE DESIGN APPROACH

In order to verify the reliability of the approach provided by Eurocode 8 and of the hybrid design approach, the

Monte Carlo simulation has been performed for each designed joint. The statistical analysis has been applied to

the overstrength factor defined as:

c

j

vMs

MO

(10)

In Figs. 6 and 7, with reference to stiffned and unstiffned joints, to EC8 and Hybrid design approach and to 8.8

and 10.9 bolt class, the mean value and the 5% fractile of the overstrength ratio are represented.

The results of the probabilistic analysis lead to the following considerations:

in case of the analyzed unstiffened joints, Eurocode 8 design criterion leads to the full-strength condition only in few cases. Conversely, the hybrid design approach, which properly accounts for the

influence of the steel strain hardening by means of the coefficient s, provides better results even though

the full strength condition is still not reached in all considered cases. This last result points out that for

these cases the overstrength factor accounting for the material random variability suggested by EC8

equal to 1.25 is inadequate to guarantee the full strength if reference is made to the 5% percentile;

in case of stiffened joints, Eurocode 8 design criterion provides again unsatisfactory results, while the hybrid criterion provides the full strength condition in all analyzed cases. The analysis of these results

shows that the overstrength factor related to the strain hardening recommended by Eurocode 8 is

underestimated, because in most cases (s-) is greater than 1.1. On the other hand, the overstrength factor related to the variability of the steel yielding equal to 1.25 in the examined cases is sufficient to

cover the effects connected to the random material variability;

the influence of the bolt class is low, in fact the results in terms of the overstrength ratio in case of anchor bolts of class 8.8 and 10.9 are sensibly similar.

From the above considerations derives that, even though the hybrid approach is more accurate with respect to

Eurocode 8 approach, both approaches are not able to guarantee the full strength in all considered cases, so that a

more accurate design approach appear auspicable.


EC8 approach Hybrid approach

Un

stif

fned

Jo

int

Sti

ffn

ed J

oin

t

Figure 6. Mean value and 5% fractile of the overstrength ratio in case of 8.8 Anchor bolts

EC8 approach Hybrid approach

Un

stif

fned

Jo

int

Sti

ffn

ed J

oin

t

Figure 7. Mean value and 5% fractile of the ratio in case of 10.9 Anchor bolts

4. PROPOSAL OF AN ALTERNATIVE APPROACH

Aiming to propose an improvement of the Eurocode 8 design criterion for full strength joints, a proper


calibration of the coefficient ov has been performed. To this scope, starting from Eq.5 the ratio between the joint and the member plastic resistance, has been preliminary analyzed considering the following relationship:

cplj

ovMs

M

,

(11)

Assuming that the base plate fails according to mechanism type-1, the joint plastic resistance is given by the

following equation:

u

c

IT

j

e

z

zFM

1

, (12)

where, FT,I is the ultimate resistance of T-stub representing the base plate in bending, z is the lever arm, zc is the

distance between the column axis and the centre of compression and eu is the ultimate eccentricity given by the

ratio between the column maximum moment accounting for strain hardening effects and the design axial load.

By means of Eqs. (11) and (12), and by means of simple analytical developments, the following expression has

been derived for the coefficient ov:

csdcyccfcf

pycpp

ovzNfhtbs

f

m

mhtb

,

,

2

(13)

where bp is the base plate width, hc is the column depth, m is the distance between the bolt axis and the plastic

hinge located in correspondence of the column flange welds, fy,p and fy,c are the yield stresses of the steel

composing the base plate and the column flanges respectively and tp is the plate thickness. The coefficient ov of Eq.12 assumes a value equal to 1 when the variability of the material properties is neglected. Conversely, the

random material variability affects the following terms of Eq.13: fy,p and fy,c , whose mean values can be related

to the thickness of the base plate and of the column flange (Eq.9), the overstrength s and the non-dimensional

axial load . The correlations between the main parameters governing the coefficient ov are given in the following correlation matrix:

%5,

%5,

1632.0479.0348.0

632.01131.0034.0

479.0131.01195.0

348.0034.0195.01

ov

plcf

ovplcf

tt

s

K

tts

(14)

From such matrix it is evident that exists a positive relationship between the parameters s, and the 5% percentile of the overstrength factor. In fact, these parameters are directly influenced by the yield strength of

column web and flanges. This can be well understood considering Eqs.6-7 and taking into account that is defined as the ratio between the axial and the squash load, which is defined as the product of the cross area

section and of the steel yield strength. In addition, considering again Eqs.6-7, it is recognized that affects, in turn, the overstrength due to the strain hardening (s) by influencing the definition of the effective web depth

(dw,e). This behaviour is confirmed by the negative correlation between and s (Eq.13). Moreover an inverse proportion, i.e. a negative correlation, between the overstrength factor and the ratio between the column flange

and plate thickness can be observed. This last result can be justified observing that the mean value of the steel

yield strength depends on the thickness of the considered plate (Eq.9). So that, the design of a full strength

column base joint with a thick column flange and a thin base plate and vice-versa should require the adoption of

a minor or major overstrength factor due to the random material variability.

Hence, aiming to design full strength joints it is required that the 5% fractile of the overstrength factor given in


Eq.3.3 is greater than 1. To this scope, starting from a multiple regression analysis of the statistical data (5%

fractile) the following relationship has been derived:

p

cfov

t

ts 234.0474.0294.0953.0

* (15)

which represents a straight line parallel to that provided by the multiple regression analysis (characterized by a

correlation coefficient equal to R2=0.964) and satisfying the condition *ov >ov. In Fig.8 the comparison between

the value of ov required to obtain full strength joints and that calculated from Eq 15 is reported. It is interesting to note that the results provided by Eq.15 assuming =0 are in agreement with that found in a previous work of the same authors [8] dealing with the design of full strength beam-to-column joints. On the base of the above

considerations, the full strength design of base joints can be lead according to the following design criterion:

cplovj MsM ,** (16)

Figure 8. Comparison between required and predicted overstrength

5. CONCLUSIONS

In this work, with reference to exposed column base joints, the reliability of the design criteria proposed by

Eurocode 8 for designing full strength steel joints has been verified by means of a Monte Carlo simulation

accounting for the random variability effects of steel, concrete and anchor bolts. The results of the performed

analises have pointed out that the design criterion provided by EC8 is not able to guarantee the design of full

strength joints for all the analyzed cases, due to the approximations related both to the coefficient covering the

effect due to the material strain hardening and to the coefficient accounting for the random material variability.

As a consequence a new design criterion, able to properly account for these two effects, has been proposed and

calibrated on the data provided by a wide numerical simulation.

REFERENCES

[1] Aviram A., Stojadinovic B., Der Kiureghian A. (2008). Reliability of exposed column base plate connection

in special moment-resisting frames, 14th

World Conference on Earthquake Engineering, October 12-17,

2008, Beijing, China.

[2] Jaspart J., Wald F., Weynand K., Gresnigt N., (2008). Steel column base classification, HERON Vol.53, No

.

ov Required Vs ov [Eq 15]

1,10

1,15

1,20

1,25

1,30

1,35

1,40

1,45

1,10 1,15 1,20 1,25 1,30 1,35 1,40 1,45

ov [Eq 15]

ov R

eq

uir

ed

SAFE

UNSAFE


[3] Wald F., Sokol Z., Steenhuis M., Jaspart J. (2008). Component Method for Steel column bases, HERON

Vol.53.

[4] Stojadinovic B., Spacone E., Goel S.C., Kwon M. (1998). Influence of Semi-Rigid Column Base Models on

the Response of Steel MRF Buildings, Proceedings of the Sixth U.S. National Conference on Earthquake

Enginneering, Seattle, WA, July.

[5] Idris A., Umar A.I., (2007). Reliability Analysis of Steel Column Base Plates, Journal of Applied Sciences

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[6] CEN (2005b). EN 1993-1-8 Eurocode 3: Design of Steel Structures. Part 1-8: Design of Joints, European

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[7] Bertero V.V., Anderson J.C., Krawinkler, H. (1994). Performance of Steel Building Structures during the

Northridge Earthquake, Report No. UCB/EERC-94/09.

[8] Piluso V., Rizzano G. (2007). Random Variability effects on full-strength end-plate beam-to-column joints,

Journal of Constructional Steel Research (63), pp.658-666.

[9] Latour M., Rizzano G., Sorrentino V. (2009). Theoretical and Experimental Behaviour of Column Base

Joints in Steel Structures, Proceedings of CTA 2009, Padova.

[10] Mazzolani F., Piluso V. (1996). Theory and Design of Seismic Resistant Steel Frames, E & FN Spon.

[11] CEN (2005a). Design of Structures for Earthquake Resistance. Part 1: General Rules, Seismic Actions and

Rules for Buildings, European Committee for Standardization.

[12] CEN (2005c). EN 1992-1-1 Eurocode 2: Design of concrete structures. Part 1-1: General rules and rules

for buildings, European Committee for Standardization.

[13] Totoli G. (2002). Affidabilit dei criteri di progetto dei nodi flangiati: Simulazione di Monte Carlo. Thesis

for civil engineering degree tutors: V.Piluso and G.Rizzano. University of Salerno. [14] Galambos T.V., Mayasandra K., Ravindra M., (1978). Properties of Steel for use in LFRD. ASCE annual

convention & Exposition. Chicago.

[15] Nakashima M., Morino S., Koba S., (1991). Statistical Evaluation of Strength of Steel Beam Columns.

Journal of Structural Engineering, Vol.117, No.11.

[16] Melcher J., Kala Z., Holicky M., Fajkus M., Rozlivka L., (2004). Design Characteristica of Structural Steels

Based on Statistical Analysis of Metallurgical Products. Journal of Constructional Steel Research. No 60.

pp.795-808.

[17] Popovics S. (1998). Strength and Related Properties of Concrete, John Wiley & Sons.

[18] Rubinstein RY. (1981). Simulation and the Monte Carlo Method, John Wiley & Sons.

Latour - Paper SISMICA 2010

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