A parametric analysis for flexible base plate in steel-concrete connection

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POLITECNICO DI MILANO

Scuola di Ingegneria Civile, Ambientale e Territoriale

Laurea Magistrale in Ingegneria Civile

A parametric analysis for flexible base plate in steel-

concrete connection

Relatore: Prof. Giovanni Muciaccia

Correlatore: Angelo Marchisella

Tesi di laurea di:

Rao Muhammad Zakwan

matricula : 884911

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3.4.6 Step 6 ( Analysis ) : ... 41 Chapter 4 ... 43 4 Numerical Analysis ... 43 4.1 Analysis scheme ... 43 4.2 Cases ... 45 4.2.1 Case 1 ... 45 4.2.2 Case 2 ... 47 4.2.3 Case 3 ... 49 4.2.4 Case 4 ... 51 4.2.5 Case 5 ... 52 4.2.6 Case 6 ... 53 4.2.7 Case 7 ... 54 4.2.8 Case 8 ... 56 4.2.9 Case 9 ... 67 Chapter 5 ... 70 5 Discussion... 70

Case 1 ; Effect of varying Lx ... 70

Case 2 ; Effect of varying Ly ... 71

Case 3 ; Effect of varying thickness ... 71

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Case 5,6 ; Effect of changing applied moment and tensile axial load on plate behaviour ... 72

Case 7 ; Effect of varying anchor stiffness ... 72

Case 8 ; Effect of Biaxial moment ... 73

Case 9 ; Effect of number of anchors ... 74

Chapter 6 ... 75

6 Conclusion ... 75

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Table of Figures

Figure 1-1 Components of steel-concrete connection ... 11

Figure 2-1 Column Base assembly and the selection of Components ... 16

Figure 2-2 T-Stub model for equivalent rigid base plate ... 17

Figure 2-3 concrete block geometrical dimensions ... 20

Figure 2-4 Flange of Flexible T-Stub ... 21

Figure 2-5 The T-stub, anchors in tension and base plate in bending assumption of acting forces and deformations of T-stub in tension ... 23

Figure 2-6 Failure modes of T-stub ... 26

Figure 3-1 General problem ... 29

Figure 3-2 Tensile and compression area ... 29

Figure 3-3 Area in compression ... 30

Figure 3-4 Flow chart to Find significant Depth "L" ... 33

Figure 3-5 Recursive meshing ... 37

Figure 3-6 IPE profile node detailing ... 38

Figure 4-1 Top view of base plate Connection ... 43

Figure 4-2 Section of base plate connection ... 44

Figure 4-3 Changing parameters, Lx and m ... 45

Figure 4-4 Spring reactions for case 1 ... 46

Figure 4-5 Spring reactions for case 2 ... 48

Figure 4-6 thickness "t" of base plate ... 49

Figure 4-7 General layout of base plate ... 56

Figure 4-8 spring reactions for case 8.1 ... 57

Figure 4-9 Spring reactions for case 8.2 ... 58

Figure 4-12 Compresson reactions for case 8.5 ... 60

Figure 4-13 Spring reactions (compression) for case 8.6 ... 61

Figure 4-14 Compression reactions for case 8.7 ... 62

Figure 4-15 compression reaction for case 8.8 ... 63

Figure 4-16 Compression reactions for case 8.9 ... 63

Figure 4-17 Compression reactions for cases 8.1-8.6 ... 65

Figure 4-18 Compression reactions for case 8.7-8.9 ... 66

Figure 4-19 Anchors Positions ... 68

Figure 4-20 Spring reactions (compression) for case 9 ... 69

Figure 5-1 (a) constant moment arm Ly (b) varying Lx ... 70

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Table of Tables

Table 3-1 Influence factor ... 35

Table 3-2 Bench mark Data ... 36

Table 3-3 Average stresses ... 39

Table 3-4 Force to be inserted for each node ... 39

Figure 3-5 area in compression ... 41

Figure 3-6 Max and min Tensile anchor force for Benchmark ... 42

Figure 3-7 deformations in the base plate for Benchmark ... 42

Figure 3-8 Von Mises stresses for base plate ... 42

Table 4-1 Analysis Scheme ... 44

Table 4-2 case 3(a) thickness variation ... 49

Table 4-3 results for Case 3(b) ... 50

Table 4-4 Results for case 3(c) ... 50

Table 4-5 Results of Case 4 ... 52

Table 4-6 Case 5 results ... 53

Table 4-7 Results for Varying Moment ... 53

Table 4-8 Results for case 7 ... 55

Table 4-9 Loading combinations moments and axial force ... 57

Table 4-10 case 8.1 Uniaxial moment only ... 57

Table 4-11 Case 8.2 uniaxial moment with axial compression ... 58

Table 4-12 Case 8.3 uniaxial moment with axial tension ... 59

Table 4-13 Case 8.4 biaxial moment only ... 59

Table 4-14 case 8.5 biaxial moment ... 60

Table 4-15 Case 8.6 biaxial moment with axial compression ... 61

Table 4-16 Case 8.7 Biaxial moment with axial compression ... 62

Table 4-17 case 8.8 biaxial moment with axial tension ... 62

Table 4-18 case 8.9 biaxial moment with axial tension ... 63

Table 4-19 Case 8 results for rigid base plate assumption ... 64

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Table of graphs

Graph 4-1 Comparison for Flexible and rigid base plate ... 46

Graph 4-2 effects of variable Ly ... 47

Graph 4-3 Compression resultant for case 1 and 2 ... 48

Graph 4-4 effect of varying thickness ... 50

Graph 4-5 Lx/t Vs Na,max ... 50

Graph 4-6 Ly/t Vs Na,max ... 51

Graph 4-7 RBP Vs FBP for constant Ly/t ... 52

Graph 4-8 Variation of Applied load ... 53

Graph 4-9 FBP and RBP for T =12,16 Vs Varying moment ... 54

Graph 4-10 Effect of anchor stiffness ... 55

Graph 4-11 Anchor elongation ... 55

Graph 4-12 Angle vs loading combinations ... 64

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Notations

FBP

Flexible base plate approach

RBP

Rigid base plate approach

Na,max

Maximum tensile anchor force

Na,min

Minimum tensile anchor force

Ө

Angle between applied moment and X-axis

a

length of base plate

al

effective length of foundation

ar

edge distance

arig

width of rigid plate

b

width of base plate

bl

effective width of foundation

c

equivalent width of footprint

d

diameter of the bolt

k

stiffness coefficient

mpl

bending resistance of base plate

As

net area of the bolt

fck

characteristic value of concrete compressive cylinder strength

f j

concrete bearing strength

fy

yield stress of steel

k

stiffness coefficient

kj

concentration factor

n

distance from the bolt axes to plate edge

m

distance from the bolt axes to the weld edge

t

thickness of the base plate

x

position

A

area

E

Young's modulus, Young's modulus of steel

Ec

Young's modulus of concrete

B

bolt force

F

force

L

length of the anchor bolt

Q

prying force

I’

second moment of area per unit length

M’

bending moment per unit width

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α

characteristic factor

βj

joint coefficient

δ

deformation

γM0

partial safety factor for steel

γc

partial safety factor for concrete

ν

Poisson's ratio

Subscripts :

d

design

eq

equivalent

el

elastic

fl

flexible

r

rigid

p

flexible plate

R

resistance

Sd

acting

W

web

b

effective length of bolt

eff

effective

em

embedment

h

bolt head

j

joint

ini

initial

p

plate

t

tension

T

T-stub

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ABSTRACT – ENG

Keywords

: connection, rigid base plate, elastic base plate

The present thesis addresses the problem of base plate in steel-to-concrete connections. The existing design method assumes the hypothesis of force distribution based on rigid behavior, namely the section under bending remains plane. However, until now there are no provisions on the required level of stiffness to guarantee. In this work, it is proofed that the rigid base plate calculation method without checking the required stiffness may lead to underestimation of the anchor tensile forces, resulting in the selection of smaller anchor bolts consequently resulting in the unsafe design of the base plate. A parametric analysis considering a single benchmark (rectangular plate with regular anchor’s layout) is carried out comparing the results of FEM modeling using plates elements and rigid base calculation method. The relevant parameters are: (i) plate’s thickness; (ii) plates’s and attached profile’s dimensions; (iii) biaxial moment condition with different relative angle between applied moments; (iv) different conditions for the applied axial force.

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ABSTRACT – ITA

Keywords

: connessione, piastra di base rigida, piastra di base elastica

Il presente lavoro di tesi affronta il problema della piastra di base nelle connessioni acciaio-calcestruzzo. Attualmente i metodi di calcolo delle connessioni sono basati sulla ipotesi di piastra rigida, che assume la planarità della sezione sotto le azioni di momento flettente. Ne consegue una distribuzione lineare delle forze negli ancoraggi che è facilmente desumibile dalla soluzione dell’asse neutro con le comuni formule per la presso-flessione deviata per sezioni omogenizzate. Tuttavia, non esiste un criterio univoco per la verifica della rigidezza richiesta al fine di soddisfare la ipotesi di piastra rigida e ci si affida quasi totalmente alla pratica del progettista. Nel lavoro presentato, si propone il confronto tra le forze sugli ancoraggi ottenute da modelli ad elementi finiti con quelli ottenute considerando la piastra rigida. E’ provato che le azioni ottenute con il secondo modello sono inferiori nei casi in cui sia notevole l’impegno flessionale della piastra. Nello specifico, è stata condotta una analisi parametrica considerando uno esempio di riferimento, i.e. piastra rettangolare con disposizione simmetrica degli ancoraggi. I parametri variati sono: (i) spessore della piastra; (ii) dimensioni della piastra e dell’impronta del profilo; (ii) diversi angoli relativi tra i momenti applicati nel caso della flessione biassiale; (iv) differenti casi di azione assiale applicata.

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Chapter 1 1 Introduction

Steel construction is a comparatively newer branch in the field of construction, It includes steel industrial and agricultural halls as well as skyscrapers, which are quite common these days, involves connection techniques ranging from cast-iron fittings to rivets, bolts and welding, whereby for steel constructions the bolted base-plate joints are the most obvious option because of their easier installation techniques. In figure 1.1 a, b, c and d, simple steel plate that is used in connection, wood column and base connection, typical concrete base plate connection and Steel embed plate with welded headed studs are shown respectively.

(a) (b)

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Figure 1-1 Components of steel-concrete connection

Initially, concrete and reinforced concrete construction borrowed fastening techniques from other construction trades, either unchanged or slightly altered. Wood lathe placed in the formwork was anchored through pre-driven nails in the concrete and served as an attachment point for the whole range

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of building systems as well as suspended ceilings. Every year millions of fasteners are used in the construction industry worldwide. The demand for greater flexibility in designing new structures and reinforcing existing concrete structures has also led to increased use of various metallic anchors in practice.

1.1 Design approaches for base plate connection

Among different Steel-concrete connections, base plate connection is of prime importance. The reason is quite simple i.e Base plate connection is used to connect a column with footing. This effectively transfers the load of structures in terms of tensile or compressive forces, moments or shear forces from columns to the foundation. In a base plate connection anchors in tension are of key importance. As these anchor rods have the main purpose of transferring tensile load and shear load, or any of these to the concrete foundation. For designing the anchors in tension there are various theoretical and empirical models to predict anchor failure loads and mechanisms.

To model the response of the base plate two

approaches are found in the literature

1. The base plate is assumed to be rigid

2. The base plate is assumed to be flexible

1.2 Difference between Rigid and Flexible base Plate

The traditional design of moment-resisting column bases involves an elastic analysis based on the assumption that the sections remain

plane. By solving equilibrium equations, the maximum stress

in the concrete foundation (based on linear stress distribution) and the tension in the holding

down assembly may be determined.

The traditional approach for the design of pinned bases results in a base plate thickness of sufficient stiffness to ensure uniform stress under the base plate and therefore the base plate can be modeled as a rigid plate [DeWolf, 1978].

Whilst this procedure has proved satisfactory in service over many years, the approach ignores the flexibility of the base plate in bending (even when it is strengthened by stiffeners), the holding down assemblies and the concrete [DeWolf, Ricker, 1990]. The concept, which was adopted in prEN 1993-1-8: transfers the flexible base plate into an effective rigid plate and allows stress in the concrete foundation equal to the resistance in concentrated compression [Murray, 1983]. A plastic distribution of the internal

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forces is used for calculations at the ultimate limit state. In case of flexible plates, the stresses are concentrated around the footprint of the column section under the plate. In EN 1993-1-8, the flexible model is considered as it better corresponds to reality. The need for rotational ductility in simple column bases implies the use of unstiffened and not too thick plates (especially for configurations with four anchor bolts).

Various researchers (Shelson, 1957; Hawkins, 1968b, 1968a; DeWolf, 1978) experimentally investigated the resistance of the “base plate and concrete block in compression” component. Factors influencing this resistance are the concrete strength, the plate area, the plate thickness, the grout, the location of the plate on the concrete foundation, the size of the concrete foundation and reinforcement.

In this study parametric analysis of base plate connection is performed considering both approaches and the main parameter responsible for this difference is searched. The difference between both approaches is analyzed for different parameters. It is important to mention that tests in this study were conducted under specific boundary and load conditions. Therefore, this analysis involves many uncertainties thus different simplification assumptions are adopted. This method can only be sufficient for some of the possible geometric and loading conditions because of many uncertainties associated with it. It can be quite conservative in some cases and non-conservative in other.

1.1 Research motivation

The main objective of this thesis is to present a parametric analysis of base plate connection. Maximum tensile anchor force must be calculated and compared while taking into account the flexible base plate approach and the rigid base plate approach. In the first part, a parameter is searched for which results of both assumptions coincide. Then the parametric analysis is performed. Following steps are followed for in this analysis.

I) Tensile anchor force is calculated using a rigid base plate approach.

II) Modeling of the base plate connection is done in SAP2000, and results for the flexible base plate assumption are obtained.

III) In SAP2000, an initial parametric analysis is performed to search the parameter, that will minimize the difference between results for both approaches.

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1.2 Research Limitations

The research presented in this thesis is focused on only on the behavior of cast-in-place headed anchors under monotonic tensile load and bending moment, no external shear force is applied, so the current work is subjected to some limitations.

• The thesis focuses on the behavior of anchors under static loads only, the effect of cyclic and fatigue loading of anchors were not considered.

• Edge distance of anchors in the base plate is kept constant in all simulations. • Joint springs are used to depict the behavior of anchor bolts.

1.3 Methodology

The following methodology is adopted for this research,

As a rigid base plate approach can be directly applied in the current case, so this approach is used to find the maximum tensile force on anchors and also compression resultant of the base plate. While

considering the flexible base plate approach, FE analysis is done by using SAP2000 for the base plate connection, and then the comparison is made for the value of tensile force in anchors.

A general modeling procedure for the flexible base plate and a rigid base plate assumption of the benchmark is provided in chapter 3. A similar scheme is followed in all cases. After that parametric analysis is performed.

1.4 Thesis outline

A brief description of the chapters’ contents is given below,

Chapter (1; introduction) Brief introduction of the research conducted and the significance thereof. Insight into the research objectives, limitations, and methodology is also provided.

Chapter (2; state of the art) General description of anchorage systems to concrete structures including their load transfer mechanisms, failure modes, and failure loads. A brief review of previous studies on the research subject is also presented.

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Chapter (3; methodology) Detailed procedure adopted to find the required results for both assumptions is presented.

Chapter (4; Numerical analysis )Test results obtained from the parametric analysis are presented, in this chapter,

Chapter (5; Discussion) Detailed discussion on results obtained by parametric analysis. Chapter (6; conclusion) research conclusions.

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Chapter

2 2 State of the Art

2.1 Basis for the evaluation of the design resistance

According to the component method [Wald, 1995], the evaluation of the design resistance of the

connection, in the case of axial compression/tension and bending moment, requires the

identification, the characterization and the assembly of the constitutive joint components.The

active components in simple column base joints are illustrated in Figure 2.1, namely the

“concrete in compression and bending of base plate” and the “anchor bolts in tension and base

plate in bending” are contributing to the transfer of the axial force, and moments through the

joint, respectively under compression and tension action.

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2.2 Design Approaches

When considering the “base plate and concrete block in compression” component, Usually the

grout layer located between the base plate and the concrete is considered, as it influences the

resistance of the component.

Concerning modelling, Stockwell (1975) introduced the concept of replacing a flexible plate with a non-uniform stress distribution by an equivalent rigid plate with a non-uniform stress distribution. Steenhuis and Bijlaard (1999) and Murray (1983) verified this simple practical method with experiments and suggested improvements. EN 1993-1-8 (CEN, 2005c) has adopted this method in a form suitable for standardization.

Figure 2-2 T-Stub model for equivalent rigid base plate

This method is described below. The resistance is so determined through an equivalent rigid plate concept. Figure 2.2 shows how an equivalent rigid plate can replace a flexible plate when the base plate connection is loaded by bending moment or axial tension[25]. The symbol A is the area of top surface of

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the concrete block, Ap is the area of the plate, Aeq is the area of the equivalent rigid plate and c is the

equivalent width of footprint. The resistance is determined by two parameters: • The bearing strength of the concrete

• The dimensions of the equivalent rigid plate.

2.2.1 Effective width and bearing area

By limiting the deformations of the base plate to the elastic range a uniform stress under the base plate may be assumed. It also ensures that the yield strength of the base plate is not exceeded. The effective bearing area of a flexible base plate is based on an effective width c. The equivalent width c of the T-stub, see Fig. 2.2, can be determined by assuming that:

No plastic deformations will occur in the flange of the T-stub. Therefore, the resistance per unit length of the T-stub flange is taken as the elastic resistance:

𝑀

′

₌

1

6 𝑡

2

𝑓

𝑦𝑑

(2.1)

It is assumed that the T-stub is loaded by a uniform stress distribution. The bending moment per unit length in the base plate acting as a cantilever with span c is [DeWolf, Sarisley, 1980],

𝑀

′

₌

1

2 𝑓

𝑗

𝑐

2

(2.2)

where fj is concrete bearing strength. When these moments are equal, the bending moment resistance of

the base plate is reached and the formula for evaluating c can be obtained from

1

2 𝑓

𝑗

𝑐

2

=

1

6 𝑡

2

𝑓

𝑦

As

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(2.3)

The bearing strength of the concrete underneath the plate is dependent on the size of the concrete block. The edge effect is considered by the following definition of the concentration factor [EN 1993-1-8 (CEN, 2005c)].

(2.4)

Where the geometrical edge conditions, see figure 2.3

(2.5)

(2.6)

If there is no edge effect, it means that the geometrical position of the column base is sufficiently

far away from the edges of the concrete and the value for kj according to [1] is 5. This

concentration factor is used for evaluation of the design value for the bearing strength as follows,

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1

Figure 2-3 concrete block geometrical dimensions

where γc is a partial safety factor for concrete. A reduction factor βj is used for taking into account

that the resistance under the plate might be smaller due to the quality of the grout layer. The

value βj = 2/3 may be used if the grout characteristic strength is more than 0,2 times the

characteristic strength of the concrete foundation fc,g ≥ 0,2 fc and the thickness of the grout is

smaller than 0,2 times the minimum base plate width tg ≤ 0,2 min (a ; b). These conditions are

usually fulfilled. If not, the grout should be checked separately.

The width of the T-stub is now,

(2.8)

The resistance FRd of the T-stub, see Fig. 2.3, should be higher than the loading FSd

(2.9)

The model for the elastic stiffness behavior of the T-stub component "concrete in compression

and plate in bending" is based on a similar interaction between the concrete and the base plate

as assumed for the resistance.

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2.3 Effect of Stiffness coefficient of concrete

František Antonín Emanuel Wald and Zdeněk Sokol [2008] derived a relation for stiffness

coefficient, K

c

of concrete, The elastic stiffness is influenced by the following factors: the flexibility

of the plate, the Young's modulus of the concrete and the size of the concrete block. In their

modelling they considered the stiffness behaviour of a rigid rectangular plate supported by an

elastic half-space. Then the flexible base plate is replaced by a rigid base plate. Then a for practical

use assumptions are made on the deformation of concrete block.

A flexible plate can be expressed in terms of an equivalent rigid plate based on the same

deformations. For this purpose, half of a T-stub flange in compression is modelled as shown in

Fig. 2.4.

Figure 2-4 Flange of Flexible T-Stub

For practical joints Ec ≈ 30 000 N/mm² and Es ≈ 210 000 N/mm², the Cfl is found to be [František

Antonín Emanuel Wald and Zdeněk Sokol,2008].

𝐶

𝑓𝑙

= 1.98𝑡

(2.9)

And

𝐶

𝑟

= 1.25𝑡

(2.9)

The equivalent width is then ,

𝑎

𝑒𝑞,𝑒𝑙

= 𝑡

𝑤

+ 2.5𝑡

(2.10)

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𝑎

_{𝑒𝑞,𝑒𝑙}

=

𝐶

𝑟

2 + 2.5𝑡

𝑎

_{𝑒𝑞,𝑒𝑙}

= 3.125𝑡

(2.12)

The influence of the finite block size compared to the infinite half-space can be neglected in

practical cases.

The quality of the concrete surface and the grout layer influences the stiffness of

this component, as demonstrated in [7] and [8]. Comparison with tests leads to the conclusion

that stiffness reductions are observed from 1,0 till 1,55. Sokol and Wald [2008] proposed a

reduction of the design value of the modulus of elasticity of the upper layer of concrete of

thickness of 30 mm based on tests without a grout layer, with poor grout quality and with high

grout quality respectively. The model proposed here takes the quality of the surface into account

with a stiffness reduction factor equal to 1,5. In conclusion, the formula to calculate the stiffness

coefficient kc of concrete in compression is given in Eq. (2.13)

(2.13)

Where,

aeq,el

equivalent width of the T-stub, a

eq,el

= tw + 2,5t ;

L

length of the T-stub;

t

flange thickness of the T-stub, the base plate thickness;

tw

web thickness of the T-stub, the column web or flange thickness.

It is mentioned that aeq,R is also a sufficiently good approximation for the width of the equivalent

rigid plate. This has a practical advantage for application by designers.

2.4 Bolts in tension

When the column base is loaded by the bending moment, the anchor bolts in the tensile zone are activated to transfer the applied force. This results in elongation of the anchor bolts and bending of the

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base plate [1]. The failure of the tensile zone could be caused by yielding of the plate, failure of the anchor bolts, or a combination of both phenomena.

Figure 2-5 The T-stub, anchors in tension and base plate in bending assumption of acting forces and deformations of T-stub in tension

Model of

the

deformation curve of the T-stub of the base plate is based on similar assumptions which are used for modelling of the T-stub of beam-to-column joints, see [5]. Two cases should be considered for the column bases. In the case the bolts are flexible, and the plate is stiff, the plate is separated from the concrete foundation. In the other case, the edge of the plate is in contact with the concrete resulting in prying off the T-stub and the bolts are loaded by additional prying force Q, which is balanced by the contact force at the edge of the T-stub, see Figure 2.5. Stiffness coefficient of the components is derived for both cases. When there is no contact of the T-stub and the concrete foundation, the deformation of the bolts is given by

(2.14) where F is the tensile force in the bolt, and deformation in T-stub can be calculated by eq 2.15.

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(2.15) The bolt stiffness coefficient for the component method is defined according to [1]

(2.16)

and the stiffness coefficient of the T-stub is

(2.17)

The stiffness coefficient T-stub without contact between the plate and the concrete foundation is,

(2.18)

When there is contact between the plate and the concrete, beam theory is used to derive the model of the T-stub. František Antonín Emanuel Wald and Zdeněk Sokol derived the formula for Prying force while taking in to account the beam theory [6]. This prying force induces extra tensile force in anchors. The formula for prying force is,

𝑄 =

𝐹

2 3(𝑚

2

_{𝑛𝐴 − 2𝐿}

𝑏

𝐼)

2𝑛

2

_{𝐴(3𝑚 + 𝑛) + 3𝐿}

𝑏

𝐼

(2.19)

2.4.1 Stiffness coefficients of plates and bolts

The component method adopted in Eurocode 3 [1] allows the prediction of the base plate stiffness. The boundary is given by

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When the above condition is satisfied, contact will occur and prying forces will develop. However, it is assumed the components are independent in this case, see [7], and the stiffness coefficients of the components are

(2.21)

(2.22) When there is no prying, the condition (2.20) changes to

(2.23)

(2.24)

(2.25)

The stiffness coefficient of the base plate in bending (2.21) or (2.24) and bolts in tension (2.22) or (2.25) should be composed into the stiffness coefficient of the T-stub

(2.26)

2.5 Design resistance for Different failure modes

In the Eurocode 3 [1], three collapse mechanisms of the T-stub are derived. These collapse modes can be used for T-stubs in contact with the concrete foundation. The design resistance corresponding to the collapse modes is the following[25]:

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(2.27)

Mode 1 - plastic mechanism of the plate, see Figure 2.6(b),

(2.28)

Mode 2 - mixed failure of the bolts and the plate, see Figure 2.6(c),

(2.29)

Figure 2-6 Failure modes of T-stub

The design resistance FRd of the T-stub is derived as the smallest value obtained from the

expressions (2.27) to (2.29)

(2.30)

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2.6 Stiffness of anchor Rods

Tsavdaridis, KD, Shaheen, M, Baniotopoulos, C et al (2016), presented in their paper “Analytical Approach of anchor Rod Stiffness and Steel Base-Plate calculation under Tension”, that while modeling column base plate connection under tension and biaxial moment, to avoid complexity due to the number and properties of anchors, we can model anchors as spring at the desired point. The stiffness of this spring is calculated by equation 2.31. 𝐾 = 𝐸𝑠. 𝐴𝑟 0.8𝑛𝑑_𝑟+ 𝐿_𝑟 (2.31) Where, 𝑛 = 𝐸_𝐸𝑠 𝑐

Ec and Es are young’s Modulus for concrete and steel,

Ar and Lr are area and lengths of steel rod

The proposed mathematical equation proves that headed anchor bolts can be represented by mass-spring models taking into account the stiffness of the rod and the concrete. The stiffness obtained by the proposed analytical formula with corresponding FE results is satisfactory. The stiffness calculated by the proposed equation can be used as an input to the available structural software (such as SAP2000) using a spring coefficient to simplify the modelling of the base plate connections.

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Chapter 3 3 Methodology

This chapter discusses the procedure of solving the base plate connection under consideration for the flexible base plate and the rigid base plate. For a rigid base plate approach, hand calculations were performed for each model, while analysis for flexible base plate approach is performed on SAP2000.

3.1 Rigid base Plate Approach

While implementing the assumption of a rigid base plate, prime purpose is to find the compression resultant of the plate, tensile force in anchors and area in compression.

In the rigid base plate approach first step is to locate the neutral axis, which will be further used to, • Find out the anchors if they are in compression or tension.

• Calculate the magnitude and location of the resultant tension and compression forces

Simple statics can be used to develop equations to find the location of the neutral axis of the base plate. as in the current case the applied moment is acting about the x-axis, and only tensile axial force is applied. But it is needed to mention that this application, however, is statically indeterminate as the location of Neutral axis and resultant forces are unknown.

3.1.1 Hypothesis

Assuming the base plate is rigid, a linear-elastic stress/strain relationship can be assumed. Which means compatibility equations can be used to provide the additional equations necessary to solve for the location of neutral axis.

3.1.2 Procedure

Once the neutral axis has been located, the compressive stress in the concrete beneath the base plate can be calculated, and subsequently; the magnitude of the resultant tension and compression forces can be

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calculated. Once the resultant forces are known ,the tensile forces acting on each anchor can be calculated then. Based on the following data for the problem, general equations were derived,

Figure 3-1 General problem

Figure 3-2 Tensile and compression area

Starting with comaptibily equations,

𝑃𝐿

𝐴𝐸

=

σ

_𝑐

. 𝐿

𝐸

_𝑐

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30

Using eq 3.1, the equation for tensile force can be rewritten as,

𝑃𝐿

𝐴𝐸

=

𝑇₁₃. 𝐿 𝐴₁₃. 𝐸_𝑠

(3.2)

Equating tensile and compressive part of equations, 𝑇13. 𝐿 𝐴₁₃. 𝐸_𝑠 =

σ

𝑐

. 𝐿

𝐸

𝑐 (3.3) Here

T13 = tension force in anchor 1 and 3

A13 = tensile stress area of anchors 1 and 3

σc = compression stress in the concrete beneath the base plate

L = thickness of the base plate

Figure 3-3 Area in compression

Shaded area in figure 3.3 represents the part of base plate in compression, so Neutral axis is assumed at distance “x” from edge of plate. Based upon data provided in the problem exploiting compatibility equations following relation can be obtained,

𝑇₁₃. 𝐿 𝐴₁₃. 𝐸_𝑠 [(𝐿𝑦− 35)] =

σ

𝑐

. 𝐿

𝐸

𝑐 𝑥 (3.4)

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31 Rearranging above equation to get T13 in terms of variables,

𝑇₁₃=

_𝐸

σ

𝑐

. 𝑥

. ((𝐿𝑦− 35) − 𝑥) . (𝐴13. 𝐸𝑠)

(3.5)

By satisfying static equilibrium ,

∑F

z

= 0 ;

↑+ve

0.5. σ

𝑐

. 𝑥.

𝐿

𝑥

+ P - 𝑇

13

= 0

σ

𝑐 = 𝑇₁₃− 𝑃 0.5. 𝑥. 𝐿_𝑥 (3.6)

∑M

Y

= 0 ,

C-W +ve

Following equation is obtained for

σ

_𝑐

,

σ

𝑐

=

𝑃 (

𝐿

_{2 − 𝑥) + 𝑀}

𝑦 _𝑥

−

𝑇13

[(

𝐿𝑦− 35

)

− 𝑥

]

0.335𝑥2_{. 𝐿} 𝑥

(3.7)

By solving equation (3.3) and (3.6) the equation for compressive stress is obtained ,

σ

_𝑐 = − 𝑃.

𝐸

𝑐

. 𝑥

[(

0.5. 𝑥2_{. 𝐿}

𝑥. 𝐸𝑐

)

−

((

𝐿𝑦− 35

)

− 𝑥

)

. 𝐴13. 𝐸𝑠

]

(3.8)

Solving equation (3.3) and (3.7),

σ

𝑐 =

[

𝑃

(

𝐿_{2 − 𝑥}𝑦

)

+ 𝑀_𝑥

] [

𝐸_𝑐. 𝑥

]

0.335𝑥3_{. 𝐿} 𝑥. 𝐸𝑐+

((

𝐿𝑦− 35

)

− 𝑥

)

2 . 𝐴₁₃. 𝐸_𝑠 (3.9)

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32

By hit and trial, such value of “x” is to be found for which equations (3.8) and (3.9) give approximately equal results for compressive stress.

When the value of “x” is found, the area in compression, maximum tensile force and maximum compression force can be calculated by using already derived equations which are eq (3.3) and (3.6).

3.2 Concrete Modelling

Concrete base, on which steel plate is mounted through anchors, plays an important in determining the bearing design resistance of a base plate connection. In SAP2000 model concrete behavior is replicated by introducing winkler bed, in which springs attached to the bottom of areas (area springs) activated for compressive load only.

3.2.1 Stiffness of winkler bed

The stiffness that is required to be assigned to each area spring is needed to be found. So initially by using the basic equation of mechanics of materials.

𝐸 =𝑆𝑡𝑟𝑒𝑠𝑠 𝑆𝑡𝑟𝑎𝑖𝑛 𝐸 =𝑃 𝛿∗ 𝐿 𝐴_𝑐 𝐸 =𝐾𝐿 𝐴_𝑐

Rearranging the above equation, 𝐾 =𝐸𝐴𝑐

𝐿

(3.10)

The equation (3.10) will be used to find the stiffness of winkler bed equivalent to concrete base. Here,

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33 E = Modulus of Elasticity of concrete

Ac = Area in compression of Plate

L = Vertical component under the surface of applied stress, where applied stress becomes negligible or 20% see[9] of the applied stress ( Bousinessq’s theory)

K = Stiffness of equivalent spring

In equation (3.10) only modulus of elasticity of concrete is known, Ac and L are still unknown, Flow chart to find significant depth is shown in figure 3.4.

3.3 Boussinesq’s method

After compressive stress and area in compression are known, Bousinesq’s theory can be applied to find the significant depth in concrete, up to which the value of stress remains only than 20% of the total compressive stress.

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34

Figure 3-5 Point load, r and z for bousinessq’s approach

Fig 3.5 is representing a general compressive load Q, r represents the horizontal distance from point of application of compression resultant, and z represents the vertical component at which stress is calculated.

At any point with coordinates R and Z, following equation was developed by boussinesq,

σ

_𝑧

=

3𝑄

2𝜋

.

𝑍

3

𝑅

5

σ

𝑧 = 3𝑄 2

𝜋

. 𝑍3_𝑍2 𝑅5𝑍2 (3.10) (3.11) Rearranging gives,

σ

𝑧 = 3𝑄 2

𝜋

𝑍2 . 𝑍5 𝑅5 (3.12) Here R5 _{= ( r}2_{+ Z}2₎5/2

σ

_𝑍 = 𝑄 𝑍2 . 3 2

𝜋

.

[

1 1 +

(

_𝑍𝑟

)

2

]

5 2 (3.13)

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35 Ip is the influence factor and is equal to,

I

p

= .

3 2𝜋

. [

1 1+(_𝑍𝑟)2

]

5 2 (3.14)

Putting value of Ip in equation 3.13 ,

σ

_𝑧 = 𝑄 𝑍2 . 𝐼𝑝

(3.15)

Table 3-1 Influence factor

In eq 3.15,

Ip = Influence factor depend on (r/Z) = F(r/Z) values can be found from table 3.1

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36

Z = Vertical dimension for point A at load

Q = Point load

By putting r = 0, the significant depth “z” can be found exactly below the compression resultant

Q here. The values of I

p

for different values of “z” are represented in table 3.1.

3.4 SAP modeling

To analyze the benchmark problem with a flexible approach, the benchmark problem is modeled in SAP2000 as shown in the following steps. Data for the benchmark model is given in table 3.2.

Table 3-2 Bench mark Data

Loadings Values

Axial 5 KN

Bending moment (Mx ) 1.5KN-m

Base Plate

Thickness 12 mm

Element type Thin plate

Steel Grade S275

Anchors

Class M12

Effective depth (heff) 70 mm

Footprint profile IPE 100

Ix 1710000

Area 1030 mm2

Concrete

Class C25/30

3.4.1 Step 1 ( Grid generation )

A suitable grid formation is required that will be based on the size of the base plate, the position of anchors and type of IPE profile footprint. A pattern for the grid was drawn for benchmark problem, that will allow to implement and change the grid in accordance with different geometrical parameters.

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37

3.4.2 Step 2 (Material and element definition )

After suitable grid creation next step adopted is the definition of materials, in the case under consideration a concrete of class C25/30 and steel base plate of standard S275 are used. An area element( thin plate element) was defined and materials which were used are recalled from default SAP2000 library

3.4.3 Step 3 (Meshing)

After assigning area element to the grid, suitable recursive meshing is done, which is unique for each dimension of the model but few points are kept constant as the requirement of the model under consideration, i.e at least 2 elements were considered for flange of footprint. As shown in figure 3.5.

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38

3.4.4 Step 4 ( Load modeling)

To take into account the applied forces and/or moments, a load case is defined named ”Applied load” and non-linear static is chosen as an analysis type. In the current case, there is an axial load (tensile) and moment about X-axis in CW direction. As shown in figure 3.8 which is exactly depicting the meshing of IPE 100 foot profile after meshing,

Figure 3-6 IPE profile node detailing

During the meshing of IPE 100 footprint, a total of 57 nodes and 36 area elements were created. So for tensile load, its application is quite simple by dividing the magnitude of applied force into 57 parts and assigning it to each nodal element included in the footprint.

3.4.4.1 Effect of Moment

The effect of the applied moment is transferred to each node of the IPE100 profile footprint. Equation 3.16 is used to calculate stress due to Mx for each node with distance Y from the neutral axis of the IPE 100 footprint.

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39

σ

_𝑐

=

𝑀

𝑥

. 𝑌

𝐼

_𝑥

(3.16)

As Y is changing for each group of nodes, thus there are different stresses for a different group of nodes. Figure 3-8 shows the detailing of nodes of section footprint. A1 and A2 are half of the areas of web and flange respectively, which are used to find nodal forces from average stresses which are reported in table 3.3. Values of A1 and A2 are 160.54 mm2 and 157.025mm2.

Table 3-3 Average stresses

Below N.A

N/mm

2

_{Above N.A}

_N/mm

2

ơ1 6.476 ơ1’ -6.47

ơ2 19.42 ơ2’ -19.42

ơ3 32.38 ơ3’ -32.38

ơ4 40.10 ơ4’ -40.10

Table 3-4 Force to be inserted for each node

Nodes

Force at each node

Values (N)

28-30 F1 65.35 25-27 F1+F2 261.41 22-24 F2+F3 522.82 17-19 F3+F4 775.84 15,16,20,21 F4 449.08 8-14 F4+F5 926.16 1-7 F5 477.07 28-30 F1' -65.35 31-33 F1'+F2' -261.41 34-36 F2'+F3' -522.82 39-41 F3'+F4' -775.84 37,38,42,43 F4' -449.08 44-50 F4'+F5' -926.16 51-57 F5 -477.07

After calculating average stresses, nodal force for each set of nodes is calculated w.r.t A1 and A2 individually and are named F1, F2, F3, F4, F5 for nodes above neutral axis and F1’, F2’, F3’, F4’, F5’ for nodes below neutral axis. Nodal forces are then superimposed for each node as shown in table 3.4. These values are directly provided as input on each nodal set in SAP2000.

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40

3.4.5 Step 5 (modeling of anchors and concrete)

Springs were introduced at an offset of 35 mm from each corner of the base plate to depict the behavior of anchors. Then area springs were applied for winkler bed under the steel plate with compression only mode.

Spring stiffness for anchors = 116.5 KN/mm2 Winkler bed spring stiffness = 192.30 N/mm/mm2

The following procedure is adopted to calculate stiffness for area springs, from equation 3.8 and 3.9, compressive stress can be found but x is unknown here, so iterations are made to find the value of “x” for which these equations provide same results.

σ

𝑐 = − 𝑃.

𝐸

_𝑐

. 𝑥

[(

0.5. 𝑥2_{. 𝐿} 𝑥. 𝐸𝑐

)

−

((

𝐿𝑦− 35

)

− 𝑥

)

. 𝐴13. 𝐸𝑠

]

(3.8)

σ

𝑐 =

[

𝑃

(

𝐿_{2 − 𝑥}𝑦

)

+ 𝑀_𝑥

] [

𝐸_𝑐. 𝑥

]

0.335𝑥3_{. 𝐿} 𝑥. 𝐸𝑐+

((

𝐿𝑦− 35

)

− 𝑥

)

2 . 𝐴₁₃. 𝐸_𝑠 (3.9) Here,

P = Applied tensile force Mx = moment about X-axis

LX = Length of the plate along X-axis Ly = Length of the plate along Y-Axis

X = Distance of Neutral Axis from the edge of the plate σ_𝑐 = Total Compressive stress

For x = 81.52 mm, equation 3.8 and 3.9 produce same results, i.e σ_𝑐 = 0.547 𝑁/𝑚𝑚2

By putting the value of compressive stress in equation 3.3 and 3.6, compression resultant and tensile force in each anchor is calculated also the area in compression can be found as shown in figure 3.8.

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41

Figure 3-5 area in compression

T = Max tensile force = 5288 N Q = Compressive resultant = 5577N Ac = Area in compression = 20380mm2

Compression resultant is then inserted in boussinesq’s equation to find required significant depth “Z” that is found by changing value of “Z” in and keeping r = 0 to reach the exact point where stress remains of 20% of the applied stress.

σ𝑧 =_𝑍𝑄2 . 𝐼𝑝 σ_𝑧 = 0.10949 𝑁

𝑚𝑚2 (20% 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑠𝑡𝑟𝑒𝑠𝑠)

For Z = L = 156mm

Now putting this value in eq 3.10, 𝐾 =𝐸𝐴𝑐 𝐿 Where, E = 30000 N/mm2, Ac = 20380mm2 , L = 156 mm,

thus the spring stiffness per unit area is found to be 192.30 N/mm/mm2

3.4.6 Step 6 ( Analysis ) :

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42

Figure 3-6 Max and min Tensile anchor force for Benchmark

Base plate deformation for the benchmark is shown in figure 3.7.

Figure 3-7 deformations in the base plate for Benchmark

For Von Mises stresses in the base plate are shown in figure 3.12.

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43

Chapter 4 4 Numerical Analysis

Chapter 4 focuses on the scheme of parametric analysis and results are reported. Results for flexible base plate approach are obtained from the analysis of base plate connection on SAP2000, while for the rigid base plate are manually calculated.

4.1 Analysis scheme

Plane and section view of the base plate connection under consideration are labeled and shown in figure 4.1 and 4.2. Analysis scheme is shown in table 4.1.

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44

Figure 4-2 Section of base plate connection

Table 4-1 Analysis Scheme

Case #

Varying parameter

Effects

1 Length of Base plate along x-axis “Lx” (consequently, change in “m” )

Na,max for FBP and RBP 2 Length of base plate along Y-axis ”Ly”

(Consequently, change in m’)

Na,max for FBP and RBP

3 The thickness of base plate “t” Na,max, on case 1 and case 2

4 Ly, with constant Ly/t ratio Na,max, comparison between

RBP and FBP

5 Axial Load “F” Na,max, comparison for FBP and

RBP

6 Bending moment “M” Na,max, Comparison for FBP and

RBP

7 Anchor stiffness “K” Na,max, FBP

8 Angle between applied moment and X-axis, “Ө” and loading conditions

Na,max, compression resultant, tension resultant

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45 Following notation is used in all cases,

Na,max = Na,min = RBP = FBP = Ө =

Maximum tensile anchor force Minimum tensile anchor force Rigid base plate approach Flexible base plate approach

Angle between applied moment and X-axis

4.2 Cases

4.2.1 Case 1

In Case 1, Na,max for the rigid base plate and flexible base plate approach is analyzed, by changing the length of the base plate along the x-axis. The difference in results for both approaches is presented in table 4.2. In the rigid base plate approach, no change in tensile force is expected due to constant moment arm. But in case of flexible base plate assumption, an increase in tensile anchor force is expected with increasing Lx. By changing “Lx”, “m” also changes which is an important factor that influences the flexible behavior of the base plate.

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46

Table 4-2 Case 1 (results)

Case 1

Lx (mm)

Na,max (N)

Compression resultant (N)

Flexible base plate

Rigid base plate

a 230 6572 5288 8202 b 250 6903 5288 9057 c 270 7229 5288 9856 d 350 8432 5288 13665 e 450 9788 5288 18086 f 550 10817 5288 21038

Graph 4-1 Comparison for Flexible and rigid base plate

(a) (b)

Figure 4-4 Spring reactions for case 1

0 2000 4000 6000 8000 10000 12000 0 100 200 300 400 500 600 Max Tensi le force (N ) Lx (mm)

Case 1

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47

Graph 4.1 shows an initial difference between result for both approaches and this difference keeps on increasing, possible reasons for this difference are elaborated in chapter 5 discussion. Figure 4.4 (a) and (b) is representing the spring reactions of base plate for Lx =250 and Lx = 450mm respectively.

4.2.2 Case 2

Case 2 focuses on the variation of length of the base along Y-axis, consequently m’ changes with Ly. All other parameters are constant. Ly and m’ are shown in figure 4.4. table 4.3 represents results for maximum tensile anchor force for flexible base plate assumption and the reigid base plate assumption, and compression resultant for flexible base plate assumption.

Table 4-3 Case 2(results)

Case 2 Ly (mm) Na_max (N) Compression resultant (N) Flexible base plate Rigid base plate

a 230 7457 5752 10264 b 250 6903 5288 9057 c 270 6446 4910 8070 d 350 5989 3897 7367 e 450 6851 3209 9202 f 550 8051 2804 12433

Graph 4-2 effects of variable Ly

0 2000 4000 6000 8000 10000 0 100 200 300 400 500 600 Max Tensi le F orce (N ) Ly (mm)

Case 2

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48

Graph 4-3 Compression resultant for case 1 and 2

Compression resultant for case 1 and 2 is shown in graph 4.3, it can be seen that for increasing Lx the compression resultant increases, while in case of increasing Ly compression resultant first decreases and then starts increasing.

(a) (b)

Figure 4-5 Spring reactions for case 2

Figure 4.5 (a) and (b) are representing spring reactions for Ly = 250 and 450 mm. 0 4000 8000 12000 16000 20000 24000 0 100 200 300 400 500 600 Max Tensi le F orce (N ) Lx, Ly (mm)

Compression resultant

Varying Lx Varying Ly

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49

4.2.3 Case 3

In this case minimization of the initial difference between rigid and flexible base plate approaches is done, which was quite prominent in case 1 and 2.

Case 3 is subdivided into three parts, in 3(a) thickness “t” of the base plate is chosen as a varying parameter, to check if results of two approaches coincide for any thickness. While in case 3b and 3c, case 1 and 2 are remodeled with the updated thickness of the base plate.

Figure 4-6 thickness "t" of base plate

Results for case 3(a) are reported in table 4.3.

Table 4-2 case 3(a) thickness variation

Case 3(a) t(mm) Na,max (N)

Flexible base plate Rigid base plate

a 10 7870 5288

b 12 6903 5288

c 14 6019 5288

d 16 5288 5288

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50

Graph 4-4 effect of varying thickness

In case 3(a) for a thickness 16mm, FE analysis gives the same result for Na,max calculated from rigid base plate assumption. This result is further verified for case 1 and 2.

Table 4-3 results for Case 3(b)

Case 3(b) Lx/t m (mm) Na,max (N) FBP RBP a 15.6 62.5 5287 5288 b 21.8 112.5 6520 5288 c 28.1 162.5 7601 5288 d 34.3 212.5 8404 5288 Graph 4-5 Lx/t Vs Na,max

Table 4-4 Results for case 3(c)

0 2000 4000 6000 8000 10000 0 5 10 15 20 Max Tensi le force (N )

thickness of base plate "t" (mm)

Case 3(a)

Flexible BP approach Rigid BP Approach

0 2000 4000 6000 8000 10000 0 5 10 15 20 25 30 35 40 Max Tensi le force (N ) Lx/t

Case 3 (b)

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51

Case 3(c) Ly/t m’(mm) Na,max (N)

FBP RBP

a 15.6 40 5287 5288

b 21.8 90 4484 3897

c 28.1 140 5135 3209

d 34.3 190 6280 2804

Case 1 and 2 are analyzed again with the new value of thickness, and results are presented in table 4.6 and 4.7

Graph 4-6 Ly/t Vs Na,max

Ratios Lx/t and Ly/t are introduced In case 3b and 3c. It’s evident from graphs that although results of both approaches coincide for the thickness of 16mm, with an increase in the Length/thickness ratio, results for both approaches show a significant difference.

4.2.4 Case 4

In case 4, Ly/t ratio is constant, and case 2 is remodeled with this assumption. Results are presented in table 4-5 and comparison is shown in graph 4.6

0 2000 4000 6000 8000 0 5 10 15 20 25 30 35 40 Max Tensi le force (N ) Ly/t

Case 3 (c)

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52

Table 4-5 Results of Case 4

Case 4 Ly/t Ly Na,max (N)

FBP RBP

A 15.6 250 5290 5288

B 15.6 350 3807 3749

C 15.6 450 3209 3146

D 15.6 550 2804 2783

Graph 4-7 RBP Vs FBP for constant Ly/t

Results obtained for case 4 confirmed that the even for higher Ly, if Ly/t is kept constant, results for both flexible and rigid base plate assumption are very close.

4.2.5 Case 5

In case 5, applied tensile force is enlarged from 5KN to 15KN, tensile anchor force was analyzed for this increment. Results are shown in table 4.6

0 1000 2000 3000 4000 5000 6000 0 100 200 300 400 500 600 Max tensil e force in anc hor(N ) Ly (mm)

Case 4

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53

Table 4-6 Case 5 results

Case Applied Force (N) Na,max (N)

FBP For t = 12mm FBP For t = 16mm RBP a 5000 6903 5290 5288 b 7000 7471.63 5779 5777 c 9000 8109.28 6334 6278 d 11000 8814.62 6912 6793 e 13000 9743.52 7603 7321 f 15000 10499.8 8283 7866

Graph 4-8 Variation of Applied load

4.2.6 Case 6

In this particular study, the variation of external moments been considered. External moments are changed within a range of 1.5 KNm to 4 KNm .

Table 4-7 Results for Varying Moment Case Applied Moment

(N-mm) Na,max (N) FBP For t = 12mm FBP For t = 16mm RBP a 1500000 6903 5290 5280 b 2000000 8819 6698 6655 c 2500000 10690 8111 8028 d 3000000 12568 9526 9408 e 3500000 14449 10944 10784 f 4000000 16334 12363 12163 0 2000 4000 6000 8000 10000 12000 0 2000 4000 6000 8000 10000 12000 14000 16000 Max Tensi le F orce

Applied tensile force

Case 5

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54

Results of varying moments on the base plate of different thickness in terms of Na,max are described in table 4.7 and graph 4.8.

Graph 4-9 FBP and RBP for T =12,16 Vs Varying moment

4.2.7 Case 7

To investigate the effects of the anchor bolt size (i.e stiffness) on connection behavior under bending and tensile load, finite element analysis is performed. A 4-bolt connection was analyzed for different stiffness of anchor bolts. Weak anchor bolts (<1.0 Ko) are excluded case because those cases are undesirable especially for the column bases in high wind or seismic region, which often have to be designed for uplift column forces. Additionally, excessive anchor bolt elongation in a column-base plate connection makes it more difficult to be repaired once the connection is damaged. Results for case 7 are shown in table 4.8. The stiffness Ko introduced for each model is defined as n times the initial stiffness K of bolt.

K = 116.5 KN/mm Ko = n*K 0 4000 8000 12000 16000 20000 0 1000000 2000000 3000000 4000000 5000000 Max Tensi le F orce (N )

Moment about X axis(N.mm)

Case 6

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55

Table 4-8 Results for case 7

Case 7 n Na,max(N)

a 1 6903

b 2 7928

c 3 8617

d 4 9047

Graph 4-10 Effect of anchor stiffness

Graph 4-11 Anchor elongation

0 2000 4000 6000 8000 10000 0 1 2 3 4 5 Max Tensi le force (N ) Anchor Stiffness K°

Case 7

Flexible BP approach 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 1 2 3 4 5 An ch or deform ation (mm) Anchor Stiffness K°

Case 7

Flexible BP approach

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56

Variation for maximum tensile anchor force is reported in graph 4.10, and with increase in anchor stiffness Na, max increases. Similarly graph 4.11 presents the elongation in anchor. With higher stiffness deformation in anchor decreases.

4.2.8 Case 8

In case 8, the behavior of the flexible base plate is analyzed under the effect of the 9 different combinations of the uniaxial moment, biaxial moment and axial loadings. The geometry of the base plate and loading footprint is the same for all subcases. Loading conditions for subcases are shown in the 4.9. Here,

M = 1.5 KN.mm N = 5 KN

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57

Table 4-9 Loading combinations moments and axial force

Combination

M

X

M

Y

Angle = (My/Mx)

N

8.1 1 0 0 0 8.2 1 0 0 -1 8.3 1 0 0 1 8.4 1 0.4142 22.5 0 8.5 1 1 45 0 8.6 1 0.4142 22.5 -1 8.7 1 1 45 -1 8.8 1 0.4142 22.5 1 8.9 1 1 45 1 4.2.8.1 Case 8.1

Case 8.1 includes the only uniaxial moment of 1.5 KN.m about the x-axis. While moment about the y-axis and axial force are set as zero for this case. Maximum tensile anhor force (Na,max) , compression resultant (C,total) and peak value of compression are reported in table 4.10.

Table 4-10 case 8.1 Uniaxial moment only

Mx (KN.m) My(KN.m) Angle (My/Mx) N(KN) Na,max (N) C,total (N) C,max (N)

1.5 0 0 0 5704 11466 346

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58

For a uniaxial bending case, spring reactions are shown in the figure. Spring reactions can be seen on near the corners of the base plate. In further tests, the effect of corner reactions will be discussed.

4.2.8.2 Case 8.2

In case 8.2, an additional axial compression force of 5KN is introduced in the system, while restricting the bending to uniaxial case only. The effect on Na,max and C,total and peak compressive reaction is reported in table 4.11.

Table 4-11 Case 8.2 uniaxial moment with axial compression

1.5 0 0 -5 4647 14353 489

Figure 4-9 Spring reactions for case 8.2

Spring reactions for case 8.2 are shown in the figure 4.6. Peak compression reaction along with the compression resultant is increased as compared to case 8.1.

4.2.8.3 Case 8.3

In combination 8.3, axial tensile force is introduced. Magnitude and direction of bending moment are not changed. Maximum tensile anchor force is increased, compression resultant is decreased as expected, but in the meantime compression on corners of the base plate is increased in comparison with case 8.1.

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Table 4-12 Case 8.3 uniaxial moment with axial tension

1.5 0 0 5 6903 9057 432

Figure 4-10 Spring reactions for case 8.3

For case 8.3 corner effects are much more visible as compared to the cases 8.1 and 8.2. In the figure 4.10 it can be seen that peak compression reactions are located in corners of the base plate.

4.2.8.4 Case 8.4

Only the biaxial moment is introduced in case 8.4, such that winkle is 22.5, Na,max, C,total and C,max are reported in table 4.13.

Table 4-13 Case 8.4 biaxial moment only

1.5 0.621 22.5 0 7047 12627 641

In this case of a biaxial moment when angle is 22.5, a higher concentration of spring reactions was induced in one corner of the base plate while in the meantime corner effect in the opposite corner of base plate becomes significant.

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Figure 4-11 Spring reactions for case 8.4 4.2.8.5 Case 8.5

In combination 8.5, the behavior of the base plate under biaxial bending moment is analyzed, such that moment about x-axis and y-axis is equal. No external axial force in applied in this case. Results are reported in table 4.14.

Table 4-14 case 8.5 biaxial moment

1.5 1.5 45 0 9501 16268 902

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The figure 4.12 shows that spring reactions are concentrated at one corner of the base plate and corner effect is significant in the opposite corner. Peak compression lies in the compression zone of the base plate, though significant corner effect is also induced.

4.2.8.6 Case 8.6

In case 8.6, biaxial moment condition is similar to case 8.4, additionally, an axial compressive force of 5 KN is introduced. results for maximum tensile anchor force, compression resultant and peak compression reaction are reported in the table 4.15.

Table 4-15 Case 8.6 biaxial moment with axial compression

1.5 0.621 22.5 -5 5892 15161 803

Overall compression resultant is increased, but the effect of prying force in the corner which was high in case 8.4 is decreased in this case. Spring reactions for case 8.6 are shown in figure 4.13.

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4.2.8.7 Case 8.7

Case 8.7 is a further modification of case 8.5, where moment about x-axis and y-axis is equal, but an additional compression force of 5 KN is applied on the loaded footprint. Results for Na,max, C,total, C,max are reported in table 4.16.

Table 4-16 Case 8.7 Biaxial moment with axial compression

1.5 1.5 45 -5 8298 18681 1056

Figure 4-14 Compression reactions for case 8.7

The figure 4.14 represents the spring reactions in the base plate for loading condition under consideration. Both compression resultant and corner reaction of the base plate is increased with the application of axial compression force.

4.2.8.8 Case 8.8

In this case, a tenisle axial force is applied along with the biaxial moment. Winkle is taken as 22.5° in this case. Results for Na,max, C,total, C,max are reported in table 4.17.

Table 4-17 case 8.8 biaxial moment with axial tension

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Figure 4-15 compression reaction for case 8.8

Effect of applied tensile axial load is visible in figure 4.15 as there is a decrease in overall compression resultant, but an increase in corner reaction is observed.

4.2.8.9 Case 8.9

Loading combination 8.9 is implemented to analyze the effect of equal biaxial moments, along with a tensile axial force. Results are reported in table 4.18.

Table 4-18 case 8.9 biaxial moment with axial tension

1.5 1.5 45 5 10087 14330 719

A parametric analysis for flexible base plate in steel-concrete connection

POLITECNICO DI MILANO

Scuola di Ingegneria Civile, Ambientale e Territoriale

Laurea Magistrale in Ingegneria Civile

A parametric analysis for flexible base plate in steel-

concrete connection

Relatore: Prof. Giovanni Muciaccia

Correlatore: Angelo Marchisella

Tesi di laurea di:

Rao Muhammad Zakwan

matricula : 884911

Table of Contents

Table of Figures

Table of Tables

Table of graphs

Notations

FBP

Flexible base plate approach

RBP

Rigid base plate approach

Na,max

Maximum tensile anchor force

Na,min

Minimum tensile anchor force

Ө

Angle between applied moment and X-axis

a

length of base plate

al

effective length of foundation

ar

edge distance

arig

width of rigid plate

b

width of base plate

bl

effective width of foundation

c

equivalent width of footprint

d

diameter of the bolt

k

stiffness coefficient

mpl

bending resistance of base plate

As

net area of the bolt

fck

characteristic value of concrete compressive cylinder strength

f j

concrete bearing strength

fy

yield stress of steel

k

stiffness coefficient

kj

concentration factor

n

distance from the bolt axes to plate edge

m

distance from the bolt axes to the weld edge

t

thickness of the base plate

x

position

A

area

E

Young's modulus, Young's modulus of steel

Ec

Young's modulus of concrete

B

bolt force

F

force

L

length of the anchor bolt

Q

prying force