Biswatosh PurkayasthaBentley Technical Support Group
Stability of a Structure
The AISC 360-05 Chapter-C specifies that the stability shall be provided for the structure as a whole and for each of its elements.
That means the stability needs to be maintained for the individual members, connections, joints and other building elements as well as the structural system as a whole.
The code recommends using any method that ensures the stability of the structure as a whole and for individual building elements and meets with all the following requirements are permitted.
1. Flexural, shear and axial member deformations and all other deformations that contribute to displacements of the structure.
2. Second-order effects (both P-∆ and P- Ϩ effects)
3. Initial geometric imperfections
4. Stiffness reduction due to inelasticity
5. Uncertainty in stiffness and strength
AISC suggests adopting the Direct Analysis Method (DAM) which satisfies all the aforementioned conditions.
Stability Design Approaches :
From stability consideration of a structure, AISC chapter C suggests the three approaches for determining the required flexural and axial strength of a member in the structure.
(1) Effective Length Factor method (ELM ) (C.2.2a ) : Unless the First Order to Second Order drift ratio is not greater than 1.1, this method demands the determination of actual “K” value of compression members. It is a conventional method which has been adopted by engineers for designing steel columns for a long time. Determination of the Effective Length factor “K” of a member is the cornerstone of this method. The K value accounts for the contribution of boundary conditions to the axial load carrying capacity of a steel column. Since the ELM approach is based on several assumptions on geometry, boundary condition and material properties of columns, sometimes this approach may be very conservative and inappropriate for the design of compression members.
(2) First Order Analysis per C2.2b : This method suggests to perform the first-order elastic analysis using nominal geometry and nominal stiffness. Although the method is derived from the DAM, it is only applicable when the sidesway amplification factor B2 <1.5
Detailed explanation is covered in chapter C2.2 of AISC 360-05.
Following are the few limitations of this method
(a) Structure supports gravity loads primarily through nominally vertical columns, walls or frames.
(b) Second-order effects must be limited.
(c) Inelastic effect must not be significant.
(3) Direct Analysis Method (DAM) :
The Appendix-7 of the AISC 360-05 introduced the DAM which is a new method addressing all the necessary stability requirements suggested by the code. Performing the rigorous Direct Analysis is an advanced approach of stability analysis which considers both geometric and material non-linearity and is far more accurate when compared with the other approximate methods.
Three basic parameters addressed by the DAM.
(3.1) Consideration of the P-∆ and P- Ϩ effect
To address the geometric non-linearity, this method strictly demands the consideration of P-∆ and P- Ϩ effect in a member and the overall structure .
The AISC chapter C2.1 specifies using the Second Order analysis to address those effects.
The AISC 360-05 code states that any second order method that includes the P-∆ and P- Ϩ effect may be used, but the following two methods are mostly used.
(a) Moment Magnification factor method per C-1b
This is a second order analysis done by magnifying the moments determined in the first order elastic analysis. This is an approximate method which is also popularly known as B1 - B2 method as the code specifies the equations eqn- C2-2 and C2-3 to determine the amplification factors for a member’s internal deformation (B1) and for the drift (B2) respectively and use them to calculate the second order flexural and axial strength of the member by eqn- C2-1a and C2-1b .
(b) Direct, Rigorous Second order analysis
Due to the iterative process involved in determining the actual value of forces and displacements on account of the second order effect, it is mostly performed by the computer programs.
(3.2) Geometric Imperfection
Any column used in real life situation never follows the ideal column straightness. Presence of crookedness, initial deformities or out of plumbness are very much feasible.
To account for these pragmatic considerations, AISC came up with the concept of notional load.
Notional Load is a pseudo lateral load to imitate the initial crookedness and out of plumbness of a member . The magnitude of Notional Load at each level is Ni = 0.002Yi , where Yi is the gravity load acting on the ith level. The 0.002 factor is equivalent to the allowable tolerance for initial out of plumbness of each story ( 1/500 times of story height) .
(3.3) Stiffness reduction due to the material Non-Linearity
Stiffness of the members needs to be reduced to account for the inelastic effects due to residual stress and the uncertainty in strength and stiffness. Inelastic effect which is caused by residual stress include stresses due to temperature, as some elements of the hot rolled cross-section cools faster than others and also due to the effects of straightening that must be done to meet ASTM A6 tolerances. Areas with residual stress yields prior to the overall yielding of the section, causing some elements to soften in-elastically prior to reaching their design strength. The loss of stiffness due to residual stresses also increases the frame and member deformations. And this effect is addressed in the DAM by the reduction of Axial Stiffness (EA ) and Flexural Stiffness (EI).
The reduced Axial Stiffness is EA* = 0.8 EA
The reduced Flexural Stiffness is EI* = 0.8 EI τ_{b}
The calculation of τ_{b} which is dependent on the level of axial stress is elaborated in chapter 7.3.3 of the AISC 360-05 .
However, τ_{b} can be assumed 1.0 if the additional notional load of 0.001 times of gravity load is applied.
Other advantages of DAM :
(a) Plain, direct and simple approach.
(b) Eliminates the ambiguity and the intricacy involved in calculation of effective buckling length factor as required by ELM. An engineer needs to assume K=1 in the DAM.
(c) Can be used for all types of steel structures like Braced frame , moment frame and combined frame system.
(d) Convenient and safe design approach with stability consideration.
(e) Performs accurate and exact analysis considering both the geometric and material non-linearity.
The scope of the DAM in STAAD :
STAAD.Pro performs the direct analysis based on the rigorous second order analysis method. However, the moment magnification factor approach is not implemented in STAAD.
STAAD forms the (K+Kg) matrix which accounts for the Geometric non-linearity, is the combination of the Global stiffness matrix and the Global Geometric Stiffness matrix. For the material non linearity, program reduces the axial and flexural stiffness in accordance to the code guidelines.
The DAM in STAAD, which addresses the overall geometric and material non-linearity effect, performs the P-Delta analysis in the background -rigorous second order analysis.
This is an iterative process as the Flexural stiffness coefficient τ_{b} is dependent on the axial force developed in the member.
Procedures to define and perform DAM in STAAD :
Defining DAM parameters .
Go to COMMANDS -> LOADINGS -> DEFINITIONS -> DIRECT ANALYSIS.
Now in the dialogue box,
(1) FLEX PARAMETER: This parameter represents τ_{b} involved in reduced flexural stiffness calculation (0.8* τ_{b} *EI). The default value of τ_{b} is 1.0 but user can define the initial approximate τ_{b} value (Refer to chapter 7-3-(3)).
(2) FYLD PARAMETER: Its default value is 36 ksi but user can input the desired yield strength value.
(3) AXIAL PARAMETER : As the code does not specify any variable reduction coefficient for reduced axial stiffness , the constant 0.8 is taken for reduced axial stiffness determination ( 0.8*EA)
(4) Notional Load Factor: Although the default factor is 0.002, but the program allows user to specify different notional load factor values.
Please note that as the Flexural stiffness coefficient τ_{b} is dependent on the magnitude of the axial stress developed on the member, it also implies that the material starts showing inelasticity as the member force increases
If the defined notional load factor is greater than 0.00299, the program sets the iteration limit to 1 and does not perform any further iterations.
Running the Direct Analysis :
Before running the direct analysis, the load combinations are to be created where the notional loads against the corresponding gravity loads need to be added for every frame levels.
Being a non-linear analysis, the combination should be done by the REPEAT LOAD option. Once the notional load is called in the combination case, Staad automatically applies the lateral loads at each level as the factored proportion of the gravity load that is being applied on that level.
Once the modeling is completely done, go to COMMANDS ->PERFORM DIRECT ANALYSIS.
A dialogue dox will appear which allows the user to select the Design Method.
If the ASD is selected then as per chapter 7.3 of the AISC 360-05, STAAD automatically multiplies the loads internally by 1.6 and the results are subsequently divided by 1.6 to obtain the design forces. The user must ensure of defining the correct ASD load combination.
Additionally, user can specify the tolerance of Tau and displacement values.
Program runs iterations in each step, changing the member characteristics until the maximum change in any τ_{b}_{ }is less than the specified Tau tolerance. If the maximum change in any τ_{b}_{ }is less than 100 times the τ tolerance and the maximum change in any displacement degree of freedom is less than the specified displacement tolerance; then the solution has converged for this case.
The beauty of the Direct Analysis feature in Staad is that, program performs the iterations for both the geometric and material nonlinearity to capture the real effect. Effect of the actual geometric non-linearity is determined by the P-Delta analysis and the material inelasticity by iteration of τ_{b }value.
Validation Staad DAM with Benchmark problem, Case-2 (page-16.1-435) in the AISC 360-05
Input data
B=d = 5 inch , L = 500 inch , P =4 Kips , H = 2 Kip , E = 29732 Ksi , Py = 50 ksi
STAAD.Pro result
After performing the Direct Analysis in STAAD, the maximum moment and maximum displacement reported are 1290.392 Kip-in and 72.588 inch receptively .
Hand Calculation result
REFERENCES
(1) AISC 360-05, Specification for Structural Steel Buildings
(2) AISC Stability Analysis handouts
(3) Structural Steel Design by Jack .C Mormac
(4) Steel Design by William .T. Segui(5) Benchmark Problems