Contact area
Chapter 10 Second-order Direct and Advanced Analysis of Structures
10.3 Methods of analysis
10.3.1 Types of Analysis
10.3.1.1 Finite element analysis for structural instability
The principle of minimum potential energy can be used to solve a buckling problem.
The vanishing of the first variation of the total potential energy functional implies the satisfaction of the equilibrium condition. The vanishing of the second variation of the energy functional means the structural system is in the state of neutral equilibrium. Figure 10.5 illustrates the concept of three different types of equilibrium.
Stable equilibrium Unstable equilibrium
Neutral equilibrium
Figure 10.5 Concept of different types of equilibrium
=0 for equilibrium (Eqn 10.3)
0
> for stable equilibirum
2
=0 for neutal equilibirum (Eqn 10.4)
0
< for unstable equilibirum
It should be noted that, after minimisation, the solution cannot be obtained directly. Instead, a set of equations governing the instability condition will be otained
To derive the Euler buckling load of a column, the energy functional of the column can be written as (Tension +ve),
dx dx
In the case of a simply supported column, the assumption of a half sine curve as in Equation 10.6 will satisfy the deflected shape of the column.
Thus, Equation 10.6 is put into Equation 10.5 and after differentiation and integration, the exact value for the Euler buckling load is obtained as in previous case.
That is,
The steps to develop a finite element for buckling analysis are as follows:
1. Write down the energy functional for the particular type of member. For example, for a general beam-column element as shown in Figure 6, the energy terms corresponding to bending are expressed as,
2
Figure 10.6 Beam-column
2. Depending on the nodal degree of freedom for an element, write down a polynomial for the deflection of the element. If there are 4 degrees of freedom, a cubic polynomial which has also 4 coefficients is used so that the coefficients can be solved. Thus,
3
For x L, v v2, 2
And, after solving,
nodal degree of freedom. After differentiating the functional with respect to the degree of freedom two by two, we obtain the stiffness matrix as follows,
Note that the coefficients are given by,j
4. The condition for the structural system to become unstable is the vanishing of the determinant of the matrix. That is,
0
cr G
L k
k (Eqn 10.21)
To this, NIDA has been developed to calculate the value of the load factor, cr, as shown in Figure 10.7, for the semi-indefinite condition of the eigenvalue.
Because the cubic Hermite function for lateral deflection represents the exact linear solution for the bending for a beam, i.e.
L M x L M x x
EI v2 1 2
2
d 1
d
(Eqn 10.22)
some researchers do not consider the beam-column element as a finite element which implies that the exact expression for the deflection cannot be obtained but approximated by a series of approximate functions such as cubic polynomial.
Moreover, in the present studies of buckling problems, it can be easily seen that the nonlinear solution, which is the half sine curve for a simply supported strut, is far from cubic and therefore the use of several elements per member is needed to obtain an accurate solution.
Figure 10.7 The option of buckling and vibration in NIDA
Alternatively, cr can be obtained by hand calculation by Equation 10.23, provided that the structure is regular portal frame of shallow roof or building frame.
For a multi-storey building frame, the formula is applied for each storey and the minimum cr is taken as the controlling elastic critical load factor.
h
F F
V N
cr (Eqn 10.23)
where F is the factored dead plus live loads on the floor considered V
F is the notional horizontal force taken typically as 0.5% of N F for V building frames
h is the storey height and
N is the notional horizontal deflection of the upper storey relative to the lower storey due to the notional horizontal force F N
10.3.1.2 First-order linear analysis
The method is a conventional method using effective length in Chapter 6 of HKSC. It assumes a linear relationship between force and displacement. It cannot check buckling or material yielding and therefore the output force and moment must be checked to ensure the member is safe. However, the stress due to second-order effects and buckling and stress distribution after yielding are not considered here.
In the first-order linear analysis, the effects of imperfection on member design (i.e. the P- effect) shall be incorporated by using appropriate buckling formulae.
Curves a0, a, b, c and d represent different values of member imperfections and Table 8.7 of HKSC classifies various types of sections into one of these a0-d curves.
It can be observed that the linear analysis currently used by most engineers in Hong Kong has already considered imperfection indirectly via uses of curves a0 to d.
Software claiming to have the ability to do the second order analysis without codified way of considering imperfection is therefore unacceptable.
The P- sway effect is considered by multiplying the moment from linear analysis by the amplification factor
1
cr
cr
. However, the P- effect still needs to be
considered by assuming the effective length equal to the member length for checking.
This linear analysis method cannot be used when the structural is irregular or
cr is less than 5.
10.3.1.3 Second-order P--only elastic analysis (Second-order indirectly analysis)
This analysis method considers the changes in nodal coordinate and sway such that the P-Δ effect is accounted for. The effect of member bowing (P-δ) is not considered here and should be allowed for separately. Member resistance check for P-δ effect to Clause 8.7 of HKSC is required and this P-Δ-only method of analysis and design is under the same limitations of use as the linear analysis.
10.3.1.4 Second-order direct analysis
(Second-order P-- elastic analysis)
In this method, both the P-Δ and P-δ and imperfections effects are accounted for in the computation of bending moment. Checking the buckling resistance of a structure to Clause 6.8.3 is sufficient and member check to Clause 8.9.2 is not needed.
The direct analysis here allows an accurate determination of structural response under loads via the inclusion of the effects of geometric imperfections and stiffness changes directly in the structural analysis and Equations (6.12) to (6.14) of HKSC for section capacity check in the structural analysis are sufficient for structural resistance design.
This method considers both the P- and P- effects such that effective length method for member buckling strength check is not required. This implies significant saving in time as well as improvement in safety.
When the full second-order or P-- analysis is used, we use the appropriate imperfections in Table 6.1. In this method, one need not consider individual stability check nor effective length at all. Cross section capacity check in Equation 10.24 below is sufficient in checking the stability strength of members as,
) 1 sway induced by loads in the frame
= displacement due to member curvature / bowing due to initial moments can be used by replacing Z by plastic modulus, S
My, Mz = external moments about principal y- and z-axes
= section capacity factor. If >1, member fails in section capacity check.
In software NIDA, different values of are indicated by different colours.
For slender sections, the effective area and moduli should be used in Equation 10.24.
For some members influenced by the beam lateral-torsional buckling, the beam buckling moment Mb should be used in place of Mcz in Equation 10.24 (see Section 6.5).
Values of global initial imperfection are taken as 0.5% of height or span.
Values of member initial imperfection should be taken from Table 6.1 of HKSC reproduced below.
Buckling curves referenced in
Table 8.7 L
e0
to be used in Second-order P-- elastic analysis
a0 1/550
a 1/500
b 1/400
c 1/300
d 1/200
Table 10.1 Values of member initial bow imperfection used in design
The curve selection should follow Table 8.7 of HKSC reproduced below.
Table 10.2 Designation of buckling curves for different section types
For first plastic hinge design, the design capacity is considered to have been reached when of any member reaches 1. The design capacity is taken as the load causing the formation of the first plastic hinge for members with plastic (Class 1) or compact (Class 2) sections or first yield for member of semi-compact or slender section which further required reduction of cross-sectional area and moduli. If the sections are not class 1 or 2, their ductility can be obtained by a nonlinear finite element plastic analysis and used in a second-order direct analysis. Figure 10.8 shows the option in NIDA of second-order P-- elastic analysis.
Figure 10.8 The option of second-order P-- elastic analysis (note the deactivated “Enable Plastic Advanced Analysis” icon)
10.3.1.5 Second-order analysis allowing for beam buckling
For beams, especially open section beams, under moment about major principal axis, it will have a tendency to buckle laterally as shown in Figure 10.9 below. A reduction in moment resistance is needed for this type of unrestrained beams. However, the HKSC does not adopt a similar P-delta analysis as for columns in beam buckling check. In fact, the HKSC also does not allow for the P-delta analysis for local buckling check as well. This practice is found not only in HKSC, but also in other codes.
The reason why normally a P-delta type of analysis for frames with beam buckling is not considered is that, unlike column buckling, the effective length does not rely on the sway sensitivity of the frame and therefore one need not worry too much about the accuracy of effective length which can be directly determined from the boundary conditions.
In HKSC, the following equation is used for second-order direct analysis allowing for beam buckling check.
)] 1 in which Mb is the beam buckling moment determined from Equations (8.20) to (8.24) of HKSC as an additional checking equation (see Trahair and Chan, 2005).
Figure 10.9 Lateral-torsional buckling of beam (Courtesy of Professor N.S. Trahair of Sydney University)
10.3.1.6 Advanced analysis and Plastic Analysis
For advanced analysis or second-order plastic analysis, one or two members yield with = 1 do not necessarily indicate structural failure if the structure does not collapse. In Eurocode 3 (2005), plastic analysis can only be used the members are of sufficient rotational capacity to enable redistribution of bending moment. Under Section 5.6(2), this requirement is assumed when plastic (Class 1) section is used and the shear is not larger than 10% of the shear resistance otherwise web stiffeners should be added within a distance h/2 from the plastic hinge location where h is the depth of the cross section.
Plastic strength reserve of steel material is significant as minimum elongation at breakage of 15% is imposed for qualified steel. Elastic design can be considered as an over-conservative in some cases, especially for highly redundant structures.
According to the limit state design, the ultimate design load of a structure should be smaller than the actual load resistance or computed collapse load of the structure which can allow for plastic yielding in some members. A safe and yet economical design should allow no yielding under working load in order to prevent accumulation of strain energy and no collapse at ultimate load using the ultimate load factors.
For collapse load analysis, a plastic hinge will then be inserted into the member end when Equation 10.24 is satisfied and the analysis continues until a plastic collapse mechanism is formed (see Figure 10.1). The members possessing the plastic hinge must have sufficient rotational capacity which can be insured by plastic (Class 1) and doubly symmetric cross section and all members in the whole frame must be compact (Class 2) or plastic (Class 1). The location behind plastic hinges must be adequately restrained against lateral buckling after formation of plastic hinges. Figure 10.10 shows the option in NIDA of Second-order P-- plastic analysis using “plastic hinge” method.
Figure 10.10 Second-order P-- plastic analysis using “plastic hinge” method (note the activated “Enable Plastic Advanced analysis” icon)