Joint stiffness and the
elastic critical load factor
The susceptibility of moment-resisting frames to global buckling is profoundly
influenced by the stiffness of joints as calculated by the proposed method in
BS EN 1993-1-8. Richard Henderson of the SCI illustrates the potential effects.
NSC 25
May 20
1 Introduction
It is unfortunate that BS EN 1993-1-1 does not adopt a succinct
label for such an important parameter as αcr , the authors instead
choosing the descriptor “factor by which the design loads would
have to be increased to cause elastic instability in a global mode”.
The BS 5950 label in the title above has much greater utility. The
article on the calculation of joint stiffness in the February edition
of New Steel Construction hinted at the effect on stability of the
stiffness of bolted joints and the present article provides an
illustrative example.
2 Example portal frame
2.1 Rigid joints
The structure used in this simple example is a pinned-foot portal
frame with a horizontal rafter. Sufficient restraints are assumed to
be provided to prevent out of plane and lateral torsional
buckling. The frame has a span L of 30 m and a height h to the
centre-line of the rafter of 15 m. The rafter is subject to a uniform
load of 10 kN/m. In order to achieve a high elastic critical load
factor, stiff UB columns have been adopted, consisting of 914 ×
305 UB 224 rolled sections. The rafter is a 533 × 210 UB 101. Hand
analysis has been carried out for amusement and checked by
stick FE analysis, first assuming the joints are infinitely stiff.
2.2 Frame deflections
For the vertical load case, determining the bending moments by
moment distribution requires the stiffness coefficients for the
members at the joint. Assuming symmetry, these are kc = 3EIc/h
for the column and kb = 2EIb/L for the rafter. The distribution
coefficient for the column is given by kc/(kb + kc). No
redistribution is required and the results are obtained directly as
shown in Table 2.1.
The free bending moment in the rafter is 1125 kNm giving a midspan
moment of 413.8 kNm. The bending resistance of the rafter
cross section given in the Blue Book is 901 kNm. The mid-span
deflection of the rafter is given by the difference between the
simply supported deflection and the upward deflection due to
the end moments:
=
( 5wL4
)
384EI– b
= 0.197m
M0L2
8EIb
The lateral deflection of the frame from a horizontal load at rafter
level can be found using the slope-deflection equations and is
given by:
=
Hh2
EIbIc
( LIc + 2hIb )
Assuming a unit load of H = 100 kN, substituting values gives a
horizontal deflection at rafter level of 0.507 m.
2.3 Elastic critical load factor αcr
Using the formula in para. 5.2.1(4) of BS EN 1993-1-1,
cr =
( H)( h
Ed
)
VEd
H,Ed the αcr value for the frame can be calculated. The global stiffness
of the frame (H/δ) is 100/0.507 = 197 kN/m. Substituting the
remaining values gives αcr = 9.9. According to para. 5.2.1(3) of
EC3, the frame is therefore almost stiff enough for second order
effects to be ignored. Increasing the rafter by one serial size
would achieve this, with αcr = 10.6
2.4 Introducing joint flexibility
According to Para 5.1.2 of BS EN 1993-1-8, for elastic global
analysis, joints should be classified according to their rotational
stiffness. If the joint is semi-rigid, the rotational stiffness Sj
corresponding to the design bending moment should be used in
the analysis. A reasonable idea of the joint stiffness is therefore
required to model the structure. The joint must be classified
according to BS EN 1993-1-8 para. 5.2.2 and the initial rotational
stiffness is denoted Sj,ini. The joint is deemed semi-rigid if:
0.5EIb / L ≤ Sj,ini ≤ 25EIb / L or if Kb / Kc < 0.1
For the purpose of this example, the rotational stiffness of the
beam to column joint has been assumed to have the same value
as that calculated in February’s technical article on the
calculation of joint stiffness. The beam in that case was also a
533 deep UB and the joint stiffness calculated:
Sj,ini = kθB = 100 MNm/radian. Using Table 2.1, Kb / Kc = 0.082 but
25EIb / L = 107 MNm/radian so the joint is semi-rigid.
Joint flexibility increases the lateral deflection of the frame
because, in addition to the rotation of the intersection of the
members due to their curvature, the joints themselves rotate. The
effect of this joint flexibility on the lateral deflection can be
determined by assuming the joints behave as rotational springs
and the members are rigid.
A similar approach to the slope-deflection equations results in
the following formula for lateral deflection due to flexible joints:
=
–Hh2
(kA + kB)
Here H is the shear force in the element in kN and the kθ
26
Table 2.1 Vertical load case: bending moments
Technical
Element I value (m4) Stiffness
(kNm/rad)
Distribution
coefficient
FEM (kNm) Bending
moment
(kNm)
Column 3.76e-3 157920 0.9483 - +711.2
Beam 6.15e-4 8610 0.0517 -750 -711.2
/Portal_frames#In-plane_frame_stability
/Portal_frames
/Portal_frames
/Steel_construction_products#Standard_open_sections
/Steel_section_sizes
/The_Blue_Book
/Portal_frames#Rafter_design_and_stability
/Modelling_and_analysis#Modelling
/Portal_frames#Column_design_and_stability