3. Truss system made of cold-formed C-section members
3.3. Design method
3.3.3. Design of structural members
Note that – although not considered as a basis for verification – in-plane bending moments in the chord members calculated from the measured stains in the case of Test 1 are generally 20% that of the calculated values, and by up to 30% higher in the case of Test 4 and 5; this is assumed to be the result of the changed ridge joint.
Taking into account the uncertainties in measurement and evaluation and the limitations and simplifications of the model the results are acceptable, the global numerical model is valid.
states flexural buckling about both axes, flexural-torsional buckling, torsional buckling, lateral-torsional buckling, interaction of flexural buckling and bending (in the following: FB-M interaction) and interaction of flexural buckling and lateral-torsional buckling (in the following: FB-LTB interaction). From the numerous stability failure modes covered by the standard those not involving interaction are not relevant from the design point-of-view, as they provide lower utilisation (e.g.: flexural buckling) or cover phenomena not observed in the tests and not relevant for the cross-section (e.g.: torsional buckling). As relevant failure modes for compression chord members are the N-M, FB-M and FB-LTB interactions considered. The effect of shear in the members is neglected, N-V interaction is taken into account in the design of joints, as detailed in Chapter 3.3.4. The failure modes covered by the standard but not relevant from the structural behaviour point-of-view are not detailed in the thesis.
Due to the similar supporting conditions the internal actions of compression chord members are essentially the same as that of the single members tested in SimpleC and C arrangement, as presented in Chapter 2.2.5, with the difference that chord members are in biaxial bending, hence lateral-torsional buckling is a phenomena to be accounted for. The design formulae of the application rules for N-M and FB-M interactions are presented in Chapter 2.4.3.
The checking formula of the application rule for FB-LTB interaction is following:
0 . / 1
) (
/ ) (
/ ,, 0
, ,
0 ,
, , ,
0 min
⋅ ≤
∆ + + ⋅
⋅
⋅
∆ + + ⋅
⋅
⋅ yb eff zcom M
Ed z Ed
z z M com y eff yb LT
Ed y Ed
y LT M
eff yb
Ed
W f
M M
W f
M M
A f
N
γ κ
γ χ
κ γ
χ (36)
Formula (36), considering formulae (6) and (19) is the general case of checking slender members in compression and bending as it takes into account both flexural buckling and lateral-torsional buckling with reduction factors, and is formally the same as those. Hence, if the member considered has a low slenderness, or – as in the case of the single C-section members – one of the addends is zero, formula (36) yields the checking for FB-M interaction, thus it is not necessary to carry out checks for this as well. Since the orientation of the chord member is fixed and therefore the bending about the weak axis results compression in the web, this formula also covers strength checking on the compression side.
In formula (36) the factors are as follows:
NEd axial compression from the global analysis,
Ed
My, bending about the strong axis from the global analysis,
c Ed Ed
z N y
M , = ⋅ bending about the weak axis,
yc distance of the centroid and the web of the cross-section, χmin = min(χz, χTF) reduction factor for flexural buckling, in the case of the
truss system χmin =χz,
χLT reduction factor for lateral-torsional buckling,
eff yb z
Ed LT LT
A f
N
⋅
⋅
− ⋅
= χ
κ 1 µ but κLT ≤1.0 (37)
15 . 0 15
.
0 ⋅ ⋅ , −
= lat MLT
LT λ β
µ (38)
Ed
My,
∆ and ∆Mz,Ed bending moments about the strong and weak axis, respectively, due to the shift of the centroid.
The factor βM,LT is to be calculated depending on the shape of the bending moment distribution. The calculation of χLT follows the same pattern as the calculation of the reduction
factors in EC3: it is derived from the pertinent relative slenderness, using buckling curve “a”.
To calculate relative slenderness the following formula is to be used:
cr eff y
M W f LT
=
⋅λ
(39)where Mcr stands for the critical moment of the member calculated based on the gross cross-section. As no closed formula is provided in the design code for the calculation of Mcr, this was derived utilizing the finite strip method.
The calculation of buckling lengths is based on the values provided by the standard, the values used are presented in Table 23.
Table 23: Buckling lengths.
Factor Buckling mode
9 .
=0
νy Flexural buckling about the y-y axis (in-plane flexural buckling) 0
.
=1
νz Flexural buckling about z-z axis (out-of-plane flexural buckling) 9
.
=0
νLT Lateral-torsional buckling about z-z axis
Buckling length is calculated as νi⋅l0 =l, with l0 being the system length.
The utilisations calculated using formula (36) for the upper chord members of the specimens used in the tests for the ULS and ultimate load levels are presented in Table 24 with the failed member highlighted.
It is to be noted, that during the design of the first test specimen the out-of-plane eccentricity was neglected, as its value was considered small; the results presented in the tables are calculated taking it into account.
Table 24: Utilisations of the compression chord members.
Member Load level Test
1 2 3 4
1 1.31 1.39 1.56 1.52
ULS 5 0.97 0.99 1.12 1.09
1 1.18 1.25 1.40 1.37
Ultimate
load 5 1.45 1.50 1.69 1.64
The test results show, that the neglected out-of-plane eccentricity and unfavourable joint arrangement yields an unsafe structure, as in the first test the ultimate load level was less than the ULS level. Changing the ridge joint configuration to enable force transfer in the flanges of the chord members the load bearing capacity is enhanced. The utilisation of the specimen used in Test 5 is over 100% at the ULS level and is 169 % at failure, although the ratio of the ultimate load and the ULS load is 1.5; this indicates that the application rule for this arrangement yields conservative design and can be changed to provide more economic results.
The application rule of EC3 to design the upper chord members of the trusses was changed by reducing the out-of-plane eccentricity to be taken into account in the design by 50% to reflect the favourable ridge joint detailing as follows:
/ 1
) 5
. 0 ( /
) (
/ , , 1
, ,
1 ,
, , ,
1 min
⋅ ≤
∆ +
⋅ + ⋅
⋅
⋅
∆ + + ⋅
⋅
⋅ yb eff zcom M
Ed z Ed
z z
M com y eff yb LT
Ed y Ed
y LT M eff yb
Ed
W f
M M
W f
M M
A f
N
γ κ
γ χ
κ γ
χ (40)
The utilisations for the ULS load level calculated using (40) are presented in Table 25.
Table 25: Utilisation of the chord members (modified design method).
Member Load level Test
1 2 3 4
1 1.15 1.15 1.30 1.26
ULS 5 0.85 0.83 0.94 0.91
1 1.03 1.04 1.17 1.14
Ultimate
load 5 1.27 1.25 1.41 1.37
The results show, that if the ridge joint provides full force transfer, the modified design formula yields utilisations under 100% for the ULS load level, the safety against failure is over 1.4, thus the method can be considered valid.
Design of compression brace members
According to the standard compression brace members are to be checked for N-M interaction, flexural buckling about both axis, torsional buckling, torsional-flexural buckling, FB-M interaction. Stability failure modes involving lateral-torsional buckling can be omitted, since in these members no bending about the strong axis is present. Signs of torsional-flexural behaviour modes were not observed during the testing, due to the supporting conditions, hence the design method of these members can be reduced to checking FB-M and N-M interactions. The checking formulae of the application rules are presented in Chapter 2.4.3.
In the development version of the design method these members were handled as members in centric compression and bending, as it was suspected, that the eccentricity of the load introduction is small.
The utilisations calculated according to (6), (7) and (19) using this approach for the ULS load level and the ultimate load of Test 4 are presented in Table 26, with the member failed in Test 4 highlighted.
Table 26: Utilisation of the compression brace members.
Member Load
level Test
17 18 20 22 24
4 0.62 0.80 0.85 0.77 0.42
ULS 5 0.62 0.80 0.85 0.41 0.40
4 0.77 0.99 1.04 0.94 0.53
Ultimate
load 5 0.91 1.18 1.24 0.59 0.61
The member failed in Test 4 had at failure an utilisation of 94%, which points to unsafe design. However, the highest utilisation among the brace members was 124%; this points to the failure is caused by the inaccurate assembly as presented in Chapter 3.2.2. To avoid premature failure due to such reasons, an additional eccentricity was defined to be taken into account in the cross-section checkings of all – compression and tension – brace members, excluding the doubled columns at support.
According to the modified design method, the eccentricity to be taken into account is / 1
/ 0
, ,
0
⋅ ≤
∆ + +
+ ⋅
⋅ eff y M
Ed z Ed
z add Ed M
y eff
Ed
f W
M M
e N f
A N
γ
γ (41)
where:
) 2 . 0 , 8
max( mm b1
eadd = ⋅ the additional eccentricity, (42)
and b1 being the width of the smaller flange of the section as defined in Figure 4.
The utilisations obtained using the modified formula (41) are summarized in Table 27.
Table 27: Utilisation of the compression brace members (modified design method).
Member Load
level Test
17 18 20 22 24
4 0.62 0.80 1.10 0.99 0.55
ULS 5 0.62 0.80 1.10 0.52 0.53
4 0.77 0.99 1.39 1.25 0.69
Ultimate
load 5 0.91 1.18 1.66 0.75 0.79
The utilisation of the member failed in Test 4 at the ULS load level is 99% if calculated with the modified formula and over 124% for the ultimate level, the highest utilisation is 166%.
Based on this the safety of the modified design method is over 1.66, the method is valid.
Design of tension chord and brace members
According to the application rules tension members are to be checked for the cross-section failure modes plastic failure, ultimate failure and combined tension and bending. As in the tests no such failure modes were obtained, the same formulae are used in the modified design methods those of the application rules. The following formulae are to be used in the checkings:
/ 0 ≤1
⋅ ya M
g Ed
f A
N
γ plastic resistance (43)
( )
(
0)
2,Rd 1 3 / 0.3 net u/ M
n r d u A f
F = + ⋅ ⋅ − ⋅ ⋅ γ ultimate resistance (44)
where:
d0 bolt hole diameter, equal to the bolt
diameter,
Anet net cross-sectional area,
fya average yield stress,
fu ultimate stress,
u = max(2e2,p2) e2, p2 bolt distances, (45) r = [number of bolts in the cross-section/number of bolts in bolt layout],
and
2
,Rd net u/ M
n A f
F ≤ ⋅ γ (46)
The interaction of tension and bending is to be checked using / 1
/
/ , , 1
, 1
, ,
, 0
⋅ ≤
⋅ +
⋅ + eff yten yb M
Ed y M
yb ten z eff
Ed z M
ya g
Ed
f W
M f
W M f
A N
γ γ
γ (47)
and if Weff,y,ten ≥Weff,y,com or Weff,z,ten ≥Weff,z,com, then / 1 /
/ , , 1
, 1
, ,
, 0
⋅ ≤
⋅ +
⋅ +
⋅
− Ψ
M yb com y eff
Ed y M
yb com z eff
Ed z M
ya g
Ed vec
f W
M f
W M f
A N
γ γ
γ (48)
is also to be checked.
In case of tension brace members, similarly to the compression brace members, due to the same reasons the smallest value of Weff,z is used in the checkings. Mz,Ed is the bending about the weak axis as calculated using the numerical model and the additional eccentricity is taken into account as defined in (42).
In case of tension chord members in the checkings for the interaction of axial tension and bending Mz,Ed is to be calculated as the product of the acting axial force and the reduced value of the out-of-plane eccentricity.
The utilisations of the tension members in Test 5 for the governing checkings listed are summarized in Table 28 with the chord members highlighted.
Table 28: Utilisation of the tension members.
Member Load
level Test
5 9 15 16 22 23 31
ULS 5 0.24 0.95 0.78 0.65 0.77 0.40 0.77 Ultimate
load 5 0.36 1.43 1.17 0.98 1.16 0.60 1.16
The highest utilisation among the brace members is present in the diagonal member next to the support, with a value over 142%; the chord member with the highest utilisation is 116% at the middle of the lower chord. As no failures involving tension members were obtained in the tests, these values can be considered as minimum safety of the pertinent design methods.