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There is not much guidance available in literature for the design of unstiffened base plates subject to uplift.

The literature presented here outlines the available guidelines for the design of base plates and of anchor bolts. Two models presented here for the design of base plates for hollow sections, which are the IWIMM Model (named here after its authors) and Packer--Birkemoe Model, were firstly derived for bolted connections between hollow sections. [37] and [36] suggest their suitability also for the design of base plates. These models include also guidelines for determining the required number of anchor bolts. Such guidelines are incorporated in the literature review for the design of the steel base plates as their application is only suitable for the particular base plate model they refer to and as they do not account for the interaction between the anchor bolts and the concrete foundation, which is dealt with in the literature review on anchor bolts.

5.2. BASE PLATE DESIGN -- LITERATURE REVIEW

The models presented here differ for their assumptions regarding the failure modes investigated. It is interesting to note that the design guidelines currently available deal with a limited number of base plate layouts.

For each model outlined here, the column sections and the number of bolts considered by the model are specified after the model name.

5.2.1. Murray Model

(H--shaped sections with 2 bolts)

In [32] Murray presents a design procedure for base plates of lightly loaded H--shaped columns with only two anchor bolts subject to uplift. He also notes that to his knowledge no studies have been published on the design of lightly loaded column base plate subjected to uplift loading prior to his [32]. His design model is based on yield line analysis and the yield line pattern assumed is shown in Fig. 18.

The expressions of the internal and external work can be written as follows:

N^*_t = design tension axial load sgand b′ = as defined in Fig. 18

Equating the external and internal work the expression of Ômpcan be written as follows:

Ômp=N^*t

2 sg

b_fc b′bfc

4b′²+ 2b²_fc (32)

The value of b′ which maximises the required plate plastic capacity is obtained differentiating equation (32) for b′ and is equal to:

b′ =b_fc

2 (33)

The presence of the flanges requires b′ to remain always less or equal to dc∕2 and therefore the value of b′ which maximises the plate plastic capacity varies depending upon the column cross--sectional geometry as follows:

b′ =b_fc

The minimum plate thicknesses required under a certain axial load N^*t are obtained substituting equations (34) and (35) into equation (32) as shown below:

t_i≥ N^*_ts_g2

Murray carried out a finite element study to investigate the adequacy of the proposed model. He also validated the reliability of equations (36) and (37) using limited

experimental results, which consisted of 4 base plate specimens with dimensions ranging from 8” x 6” (203.2 x 152.4 mm) to 12” x 8” (304.8 x 203.2 mm) and thicknesses varying from 0.364 in. (9.246 mm) to 0.377 in. (9.576 mm).

This method is included in the design model recommended by the current AISC(US) Manual [5].

bfc∕2

Figure 18 Murray Model Assumed Yield Line Patterns (Ref. [32])

5.2.2. Tensile Cantilever Model (Generic Model)

Tensile Cantilever Method, as it is referred here, assumes that the tension in the anchor bolts spreads out to act over an effective width of plate (be) which is assumed to act as a cantilever in bending ignoring any stiffening action of the column flanges.

dh

Figure 19 Tensile Cantilever Model (Ref. [26]) It can be applied to generic base plate layouts.

Nevertheless it provides conservative designs as it ignores the two way action of the base plates.

Reference [47] suggests a 45 degree angle of dispersion as shown in Fig. 19. This is based on considerations of elastic plate theory as described in reference [13].

The design moment and the design moment capacity are then calculated as:

n_b= number of anchor bolts

bt= distance from face of web to anchor bolt location dh= diameter of the bolt hole

be= 2b_t+ dh

The axial capacity of the base plate can then be determined equating the design moment and the section moment capacity as follows:

N^*_t ≤0.9f_yibet²_i 4

n_b

b_t (40)

or equivalently the minimum base plate thickness t_i under a certain loading condition is calculated as:

t_i= 4N^*t bt

0.9f_yiben_b



⁽⁴¹⁾

5.2.3. IWIMM Model

(CHS with varying number of bolts) The IWIMM Model has been named here after the initials of the authors of the model. [27] The model was firstly derived for the design of CHS bolted connections. [37] and [36] suggest its use also for the design of base plates of CHS columns.

The base plate layout considered by this model is shown in Fig. 20.

The plate thickness is calculated based on the design axial tension load N^*tas follows:

t_i≥ 2N^*_t Ôf_yiπ f₃



⁽⁴²⁾

where:

Ô = 0.9

d₀= outside diameter of a CHS tc= thickness of column section f₃= 12k1



^k3+ k



²₃− 4k1



k1= ln



^rr²₃



k₃= k1+ 2 r₂=d0

2 +a₁ r₃=d0− tc

a1and a22as defined in Fig. 20

[27] recommends to keep the value of a1as small as possible, i.e. between 1.5d_f and 2d_f(where d_f is the nominal diameter of the bolts), while ensuring a minimum of 5 mm clearance between the nut face and the weld around the CHS.

N^*t

a1

do

ti ti

a2

N^*t

Figure 20 Bolted CHS Flange--plate Connection (Ref. [36])

[27] also recommends to determine the number of required anchor bolts as follows:

n_b≥ N^*t

ÔN_tf

 



^{1 − 1f3+f3 ln1}

^

^rr12

^  



⁽⁴³⁾

where:

Ô = 0.9

N_tf= nominal tensile capacity of the bolt r1=d0

2 +2a1

r2=d0

2 +a1

a1= a2

This procedure does not verify the capacity of the concrete foundation and its interaction with the anchor bolts needs to be checked.

Assumptions adopted by this model are an allowance for prying action equal to 1/3 of the ultimate capacity of the anchor bolt (at ultimate state), a continuous base plate, a symmetric arrangement of the bolts around the column profile and a weld capacity able to develop the full yield strength of the CHS.

[28] notes that adopting the above prying coefficient for the bolted CHS connection in the base plate design is conservative due to the greater flexibility of the concrete foundation when compared to the steel to steel connection. [36]

5.2.4. Packer--Birkemoe Model

(RHS with varying number of bolts) The Packer--Birkemoe Model is here named after the authors of the model. [36] This model deals with base plate for RHS as shown in Fig. 21 and it has been validated only for base plates with thickness varying between 12mm and 26mm.

The model includes prying effects in the design procedure. The prying action decreases while increasing a₂as shown in Fig. 21. The value of a₂should be kept less or equal to 1.25 a₁, as no benefit in the base plate performance would be provided beyond such value. a₁is defined as the distance between the bolt line and the face of the hollow section.

Generally 4--5 bolt diameters are used as spacing of the bolts spbut shorter spacing are also possible.

Based on the design loads the required number of anchor bolts should be calculated assuming that the

prying action absorbs about 20--40% of the anchor bolt capacity. The coefficient δ is then calculated as follows:

δ = 1 −d_h

sp (44)

where:

sp= bolt pitch as defined in Fig. 21

The designer should then select a preliminary plate thickness in the following range:

KN^*_b 1 + δ



^{≤ ti≤ KN}

^

^{*b (45)}

where:

K =4a₃10³

Ôf_yisp (where f_yiis in MPa) a₃= a1− df∕2 + tc

N^*_b= design axial tension load carried by one bolt

=N^*_t n_b

d_f= nominal anchor bolt diameter

The value of α represents the ratio of the bending moment per unit width of plate at the bolt line to the bending moment per unit width at the inner hogging plastic hinge. In the case of a rigid base plate α is equal to 0 while for a flexible base plate with plastic hinges forming at both the bolt line and at the inner face of the column (see Fig. 21) α is equal to 1. From equilibrium, the value α for preliminary base plate layout is calculated as follows:

α =



^KÔNt2_i ^{tf− 1}

 ^

^δ(aa2²+ a^{+ d}1^f∕2+ t^c)



⁽⁴⁶⁾

α should be taken as 0 if its value calculated with equation (46) is negative.

The capacity of the steel base plate is then calculated as follows:

ÔN_t=t²_i(1 + δα)nb

K (47)

where:

ÔN_t= axial tension capacity of the base plate ÔN_tcalculated with equation (47) must be greater than N^*t. The actual tension in one bolt, including prying effects, is determined as follows:

N^*_b≈N^*_t

n_b



^{1 +aa3}4



_{1 + δα}^δα

 

⁽⁴⁸⁾

where:

α =



^KNt2_i ⁿb^*t− 1



^1δ

a₄= min



^{1.25a1, a2+d}2^f



The value of α previously calculated in equation (46) does not have to equal the value of α calculated from equation (48) as the former assumes the bolts to be loaded to their full tensile capacity.

It interesting to note how equation (48) provides an estimate of the prying action present in the base plate.

a1

a3

= = = =

tc

N^*t

= =

N^*t

sp

a2

a4

sp

Figure 21 Packer--Birkemoe Model (Ref. [36]) 5.2.5. Eurocode 3 Model

(H--shaped sections with varying number of bolts)

The Eurocode 3 does not provide a specific design procedure for the design of base plates subject to tension. Nevertheless it provides very useful guidelines for the design of bolted beam--to--column connections (Appendix J.3 of [23]) which can be adapted for the design of base plates considering all anchor bolts as bolts on the tension side of the beam--to--column connection.

The design of the end plate or of the column flange of the beam--to--column connection is carried out in terms of equivalent T--stubs as shown in Fig. 22.

e m 0.8a 2 e m

a

emin

tf

tf

e m 0.8r

emin

r

l

Figure 22 T--stub connection in EC3 (Ref. [23]) EC3 considers that the capacity of a T--stub may be governed by the resistence of either the flange, or the bolts, or the web or the weld between flange and web of T--stub. The failure modes considered are three as shown in Fig. 23. The axial capacity is calculated as follows:

F_t.Rd= min



F_t.Rd1, F_t.Rd2, F_t.Rd3



(49) where:

F_t.Rd1=4M_pl.Rd m

F_t.Rd2=2M_pl.Rd+ nΣBt.Rd

m + n F_t.Rd3= ΣBt.Rd

M_pl.Rd=0.25lt²_ffy

γ_MO n = emin≤ 1.25m

l = equivalent effective length calculated in equations (50), (51), (52) and (53)

ΣB_t.Rd= tensile capacity of bolt group γ_MO= partial safety factor

= 1.10 (boxed value from Table 1 of [23]) F_t.Rd1, F_t.Rd2 and F_t.Rd3 = tensile capacities of the

T--stub based on failure modes 1, 2 and 3 respectively

Mode 1: Complete flange yielding

Mode 2: Bolt failure with flange yielding

Mode 3: Bolt failure Ft

Ft

Ft

Q Q

Q Q

Ft

2 + Q Ft

2 + Q



^Bt∕2



^Bt∕2



^Bt∕2



^Bt∕2

Figure 23 Failure modes of a T--stub flange (Ref. [23])

It is interesting to note that the amount of prying action for a certain base plate layout can be obtained as the ratio F_t.Rd∕ΣBt.Rdas shown in Fig. 24.

1 + 2λ2λ 2λ

1 + 2λ

1 2

Mode 3 Mode 2

Mode 1 1



_B^F_t.Rd

λ = n∕m β = 4M_plRd

m



^{Bt.Rd= l t}

2ff_y∕γMO

m



^Bt.Rd

β

Figure 24 Prying action in T--stub for the three failure modes considered in (Ref.

[23])

The tension zone of the end plate should be considered to act as a series of equivalent T--stubs with a total length equal to the total effective length of the bolt pattern in the tension zone, as shown in Fig. 26.[23] The length to be utilised in the design of the equivalent T--stub is calculated as follows:

for bolts outside the tension flange of the beam l_eff.a= min



0.5bp, 0.5w+2mx+0.625ex,

4mx+1.25ex, 2πmx) (50) for first row of bolts below the tension flange of the beam

l_eff.b= min(αm, 2πm) (51)

for other inner bolts

l_eff.c= minp, 4m + 1.25e, 2πm (52) for other end bolts

l_eff.d=min(0.5p+2m+0.625e, 4m+1.25e, 2πm) (53) where:

α = as defined in Fig. 27

It is interesting to note that the failure modes considered for example by equations (52) and (53) are the same as those considered to evaluate the capacity of an unstiffened flange. The yield line patterns of such failure modes are shown in Fig. 25.

e m

p p

Centreline of web

Centreline of web

Centreline of web (a) Combined bolt group action

(b) Separate bolt patterns

(c) Circles around each bolt

Figure 25 Yield line patterns for unstiffened flange (Ref. [23])

Transformation of extension to equivalent T--stub Equivalent T--stub for extension

Portion between flanges bp

w ex

mx

p p

e m e m

ex mx

l_eff.a

l_eff.b l_eff.c

l_eff.d

bp bp∕2 l_eff.a

l_eff.a

Figure 26 Effective lengths of equivalent T--stub flanges representing an end plate (Ref. [23])

1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2π 65.5 5 4.754.54.45

λ₂

α

λ₁= m1

m1+ e λ2= m2

m1+ e

e m1

m2

λ1

Figure 27 Value of Effective lengths of αto calculate equivalent T--stub flanges (Ref. [23])

5.3. DESIGN OF ANCHOR BOLTS --LITERATURE REVIEW

Available design guidelines regarding the behaviour of anchor bolts in tension distinguish between the behaviour of anchor bolts with an anchor head and of hooked anchor bolts and therefore these will be discussed here separately. For the purpose of this paper an anchor head is defined as a nut, flat washer, plate, or bolt head or other steel component used to transmit anchor loads from the tensile stress component to the concrete by bearing. [2]

5.3.1. Anchor bolts with anchor head

The first detailed guidance on the design of anchor bolts is provided by the American Concrete Institute Committee 349 in 1976 in [3]. These recommendations are produced for the design of nuclear safety related structures. Some of the ACI Committee 349 members, very active in the preparation of [3], publish an article [17] where the guidelines provided in [3] are modified to suit concrete structures in general.

The design criteria at the base of [2] and of [17] is that anchor bolts should be designed to fail in a ductile manner, therefore the anchor bolt should reach yielding prior to the concrete brittle failure. This is achieved by ensuring that the calculated concrete strength exceeds the minimum specified tensile strength of the steel.

[2][17]

Typical brittle failure of an isolated anchor bolt is by pulling out of a concrete cone radiating out at 45 degrees from the bottom of the anchor as shown in Fig. 28. [2]

and [17] recommend to calculate its nominal concrete pull--out capacity based on the tensile strength Ô4 f′



c

(where f′cis in psi) or Ô0.33 f′



c_{(where f′}cis in MPa) acting over an effective area which is the projected area of the concrete failure cone.

In both [3] and [17] it is recommended to use a capacity reduction factor of 0.65 in the calculation of the concrete cone capacity, which can be increased to 0.85 in the case the anchor head is beyond the far face reinforcement.

The value of 0.65 applies to the case of an anchor bolt in plain concrete. This intends to be a simplification of a very complex problem. [3][17]

In the current version of ACI349 [2] the capacity reduction factor is equal to 0.65 unless the embedment is anchored either beyond the far face reinforcement, or in a compression zone or in a tension zone where the concrete tension stress (based on an uncracked section) at the concrete surface is less than the tensile strength of the concrete 0.4 f′



^c subjected to strength load combinations calculated in accordance with current loading codes (i.e. AS1170.0 [8]) in which cases a capacity reduction factor of 0.85 can be used. [2] An embedment is defined in [2] as that steel component embedded in the concrete used to transmit applied loads to the concrete structure. The ACI Committee 349 recognises that there is not sufficient data to define more accurate values for the strength reduction factor. [2]

Experimental results have generally verified the results of this approach. [31]

The value of Ô0.33 f′



^crepresents an average value of the concrete stress on the projected area accounting for the stress distribution which occurs along the failure cone surface varying from zero at the concrete surface to a maximum at the bolt end. [31] In calculating the projected area of the failure cone the area of the anchor head should be disregarded as the failure cone initiates at the outside periphery of the anchor head. [2]

Experimental results have shown that the head of a standard bolt, without a plate or washer, is able to develop the full tensile strength of the bolt provided, as specified in [2], that there is a minimum gross bearing area of at least 2.5 times the tensile stress area of the anchor bolt and provided there is sufficient side cover,

that the thickness of the anchor head is at least 1.0 times the greatest dimension from the outermost bearing edge of the anchor head to the face of the tensile stress component and that the bearing area of the anchor head is approximately evenly distributed around the perimeter of the tensile stress component. [2]

The placing of washers or plates above the bolt head to increase the concrete pull--out capacity should be avoided as it only spreads the failure cone away from the bolt--line which may cause overlapping of cones with adjacent anchors or edge distance problems. [31]

Ld

Ld

45^o

Failure plane

Projected surface

Figure 28 Concrete failure cone (Ref. [26]) If reinforcement in the foundation is extended into the area of the failure cone additional strength would be present in practice since the nominal capacity of the failure cone is based on the strength of unreinforced concrete.

The concrete pull--out capacity of a bolt group is calculated as the average concrete tensile strength Ô0.33 f′



c times the effective tensile area of the bolt group. This effective area is calculated as the sum of the projected areas of each anchor part of the bolt group if these projected areas do not overlap; when overlapping occurs overlapped areas should be considered only once in the calculation of the effective tensile area, thus leading to a smaller concrete pull--out capacity if compared to the sum of the concrete pull--out capacities of each anchor in the bolt group considered in isolation.

[2][17]

= πL²d−

2



^cos−1



2L^s_d

 

^πL2d

360⁰ +s

2



L²_d− s4² Shaded

Area

(a) Two Intersecting Failure Cones Ld

Ld

s

s

= πL²d−

2



^cos−1



2L^s_d

 

^πL2d

360⁰ +s

2



L²_d− s4² Area

Circle -- Sector + Triangle (b) Failure Cone Near an Edge 2s

Ld

Ld

− Ld

+ Ld

=

(Note: the inverse cosine term listed in the equations is in degrees)

Figure 29 Calculation of the projected area of two intersecting failure cones or one failure cone near an edge (Ref. [30]) Simple procedures to calculate the effective tensile areas of bolt groups are provided in [30], i.e. the procedure to calculate two intersecting cones is shown in Fig. 29. [30]

Depending upon the bolt group layout other possible failure modes could take place such as the one shown in Fig. 30 where an entire part of the concrete foundation would pull--out. In such cases the effective tensile area should be calculated selecting the smallest projected area due to the possible concrete failure surfaces as shown in Fig. 30. A similar average tensile strength as in the case of the pull--out cones can be adopted. [2][17]

Tension Force

Figure 30 Potential Failure Mode with limited depth (Ref. [2])

Transverse splitting is another failure mode which can occur between anchor heads of an anchor bolt group when their centre--to--centre spacing is less than the anchor bolt depth and is shown in Fig. 31. This failure mode occurs at a load similar to the one required to cause a pull--out cone failure in uncracked concrete and therefore no additional design checks need to be considered. [2][17]

Tension Force

Transverse splitting

Figure 31 Transverse splitting failure mode (Ref. [2])

It is interesting to note that in the case of shallow anchor bolts the angle at the bolt head formed by the failure cone tends to increase from 90 degrees to 120 degrees.

An anchor bolt is classified as shallow when its length is less than 5in. (127 mm). Nevertheless for design purposes caution should be applied is using angles greater than 90 degrees as cracks might be present at the concrete surface. It is recommended not use angles other than 90 degrees. [2][17]

The previous considerations assume the concrete element to be stress--free and only subjected to the anchor bolts loading. [2] and [17] consider the case when there is a state of biaxial compression and tension in the plane of the concrete. The former loading condition would be beneficial to the anchor bolt’s strength while the latter loading state would lead to a significantly decrease in strength. Nevertheless, it is in the opinion of the ACI 349 Committee that a failure cone angle of 90 degrees can still be utilised as it is assumed that any cracking would be controlled by the main reinforcement designed in accordance with current concrete codes, i.e. AS 3600 [10].

The design procedure proposed by ACI 349 and [17] is also recommended by DeWolf in [21].

[21] notes that the use of cored holes, such as shown in Fig. 32, should not reduce the anchorage capacity based on the failure cone, provided that the cored hole does not extend near the bottom of the bolt. This situation should be avoided if the dimensions shown in Fig. 32 are

[21] notes that the use of cored holes, such as shown in Fig. 32, should not reduce the anchorage capacity based on the failure cone, provided that the cored hole does not extend near the bottom of the bolt. This situation should be avoided if the dimensions shown in Fig. 32 are

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