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ANALYSIS OF STEEL STRUCTURES IN STAAD.Pro rozarker Wed, Jun 24 2009 7:45 PM
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Ravi Ozarker, P.Eng., Applications Engineer, Bentley Systems Inc. Thanks to Ray Cutis, Senior Advisory Software Developer for all the help that he has offered me to write this article. article. Thanks to Dr. Bulent Alemdar, FEA Specialist and and Santanu Das, VP, Integrated Engineering Group for reviewing this article in detail and providing their valuable feedback.
1.0 INTRODUCTION: Features Features such as non-linear as non-linear analysis of a st structure ructur e seems to be a grey area to me sometimes. There are terms that will be thrown out at me such as; second order order analysis, plastic plastic analysis analysis etc. I have found that the th e meaning of the word "non-linear" can vary depending upon what the engineer may want to do. This blog is for engineers who are trying to explore structural analysis methods implemented in structural analysis software softwar e such as STAAD.Pro V8i. The goal of this article is to offer offer a basic overview of features such as Linear L inear static, P-Delta, small P-Delta, P-D elta, stress st iffening iffening,, and geometric non-li non-linearity nearity and how t hey are implemented in STAAD.Pro. Note that the t he non-linear and pushover analysis features in STAAD.Pro V8i are a part of the "Advanced "Advanced Analysis License" which is an add-on.
2.0 LINEAR LI NEAR STATIC AND P-DELTA (P-Δ) ANALYSIS: The following following figure figure illustrates a st eel frame frame with some gravity and an d lateral loading. loading. This frame could be a new or existing existing structure. stru cture. There are several analysis options options available to the engineer to analyze this frame depending on what is the final goal. For example, if a new frame is to be designed to come come up with the member sizes, engineers may use p-delta (P-Δ) analysis with effects of small p-delta (p-δ) included etc.
Figure 1: Frame subjected to lateral and gra vity loads
Figure 2 shows a plot of levels of structural analysis and behavior they produce (i.e. applied load (H) vs. displacement (Delta) graph). This graph illustrates the response of the structure depending on what structural analysis method is used. Most engineers are used to the first-order (linear) elastic analysis method. This type of analysis used to be good enough to come up with the force distribution in a particular structure. Once the force distribution is obtained, engineers can obtain stresses in the members and compare them with the allowable stress which used to be 36 ksi. This is all good if you were using the AISCASD codes. American Institute of Steel Construction (AISC) 13 th Edition 2005 Code introduced many new concepts to analyze steel structures. The code specifically addresses how to consider nonlinear effects (P-Delta) in analysis, and provide several guidelines for this purpose. In the past, engineers were more concerned with the stresses not exceeding a particular code defined value, displacements not exceeding a particular code defined value and same applied to slenderness, torsion etc. Today, stability and performance of a structure have become equally important. Most civil engineering structures behave in a linear fashion under service loads. Exceptions are slender structures such as arches and tall buildings, and structures subject to early localized yielding or cracking. Consider the structure shown in Figure 1. Note that this structure is subjected to lateral and vertical forces. The point loads shown in this example may be dead load applied to the structure. The lateral loads may be wind loads applied to the structure. If the lateral loads are applied while the dead loads are acting, the structure would displace laterally. The lateral displacement and the vertical forces would exert an extra moment on the columns which is not taken into account in the linear static analysis. The analysis that would
take this moment into account is known as the P-Delta (P-Δ) analysis or the Big P-Delta analysis. This analysis is performed by first applying the loads laterally to create the displaced shape of the structure and then applying the vertical loads. Once the vertical loads have been applied, the additional moment is converted to lateral forces and is added to the existing lateral forces to obtain a set of updated lateral forces. The updated lateral forces are applied to the structure to obtain an updated displacement. The moment generated by the difference of the previous displacement and the updated displacement and the vertical loads is used to calculate a new moment which is again added to the lateral loads. Engineers are aware of this concept but part of the reason why this is only being discussed recently in the codes is because of the easy availability of computational power and structural analysis software packages like STAAD.Pro. Availability of computational power and structural analysis products does not mean that engineers do not have to worry about the analysis part anymore. I would say that the engineers have to have a thorough understanding of the analysis procedures and how they are implemented in their structural analysis product of choice. It is critical to understand what the results mean in the analysis product; i.e. has the structure fallen down?, Is there global buckling? Is the structure unstable? etc. After all these analysis related issues have been sorted out, the engineer could then concentrate on the design issues.
Figure 2: Load vs. displacement graphs depending on analysis method being used
Figure 2.1 shows the P-Delta (P-Δ) Analysis graph after 35 iterations. Note that the displacement is still about 53 in. From this graph, it is clear that the structure is expected to be stable if it is subjected to lateral loads because the displacement value almost remains unchanged as the number of P-Delta (P-Δ) iterations is increased. In other words, the frame is reached to an equilibrium state so that further iterations are not needed.
Figure 2.1: P-Delta (P-Δ) analysis in STAAD.Pro
3.0 SMALL P-DELTA (p-δ) ANALYSIS: When a column member is subjected to compressive loads, it can have localized deflections throughout it length. The local deflection of the column (known as small delta (p-δ)) and the gravity load together can result to an additional moment which must be taken into account to perform the P-Delta (P-Δ) analysis. The displaced shape of a structure is illustrated in Figure 3 based on which analysis procedure is used (i.e. P-Δ OR P-Δ with effects of p-δ included). Note that the red lines show local deflections of the column members if the P-Delta analysis takes the effects of small P-Delta into account (i.e P-Δ with effects of p-δ included).
Figure 3: Difference between a "regular P-Delta analysis (P-Δ)" and a "P-Delta analysis that takes effects of small P-Delta into account (P-Δ with effects of p-δ included)"
Note that in Figure 3, the left hand side columns have less localized deflections than the columns at the right. The lateral load applied to the building will induce a tensile loading in the columns at the left and try to straighten the columns. This effect is known as the stress stiffening effect. The lateral load applied to the building will induce additional compressive loads in the columns at the right and try to bend the columns more. This effect is also known as the stress stiffening effect but in this case the columns at the right have reduced stiffness. The P-Delta (P-Δ) analysis capability in STAAD.Pro V8i has been enhanced with the option of including the above-mentioned stress stiffening effect of the Kg matrix into the member/plate stiffness. This implementation will also report any global bucking in the structure. The AISC 360-05 Appendix 7 describes a method of analysis, called Direct Analysis, which accounts for the second-order effects resulting from deformation in the structure due to applied loading, imperfections and reduced bending stiffness of members due to the presence of axial load. In STAAD.Pro, this feature is implemented as a non-linear iterative analysis as the stiffness of the members is dependent upon the forces generated by the load. The analysis will iterate, in each step changing the member characteristics until the maximum change in any Tau-b is less than the tau_tolerance. Note that the member stiffness will be changed depending on the load applied.
3.0 NON-LINEAR ANALYSIS: P-Delta (P-Δ) analysis discussed above is a type of non-linear analysis but this section talks about what a true
non-linear analysis is all about. In linear elastic analysis, the material is assumed to be unyielding and its properties invariable, and the equations of equilibrium are formulated on the geometry of the unloaded structure. We assume that the subsequent deflections will be small and will have insignificant effect on the stability and mode of response of the structure. Nonlinear analysis offers several options for addressing problems resulting from the above assumptions. There are two basic sources of nonlinearity: 1. Geometric Non-Linearity: Similar to the P-Delta (P-Δ) analysis feature discussed above but could apply to a structure with any geometry. P-Delta (P-Δ) analysis is not a pure non-linear analysis. 2. Material Non-linearity: Similar to the direct analysis feature in which the member properties are modified based on the load they experience. The direct analysis is not a true non-linear analysis. A true non-linear analysis would consider plastic deformation and inelastic interaction of axial forces, bending shear, and torsion. Let us consider the following example to explain the implementation of "geometric non-linear analysis" feature in STAAD.Pro:
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Figure 4: Three-hinged arch
The system shown above in Figure 4 is a shallow arch consisting of two axial force members. A linear static analysis on the structure using the applied loads will lead the engineer to believe that the structure is stable with the applied loading. A method that only considers material non-linearity may also lead to the same conclusion. The geometric non-linear analysis will illustrate if the shallow arch will snap-through to become a suspension system. Figure 5 shows a load vs. displacement at center node graph that was developed for this STAAD.Pro model. As the load approaches 16 kips, the shallow arch becomes a suspension system. The following video shows the same phenomenon.
Figure 5: Load vs. Displacement at center node
This example in literature is known as "Williams' Toggle" example, and it is one of the very fundamental example found almost in most structural analysis book as shown in Figure 5.1. It is worthwhile to mention that typical Newton Raphson algorithm fails to trace all post-buckling behavior, so Arclength method (or minimum residual method) is required to capture it. Figure 5 shows that the displacements jumps from one to another while snapping through.
Figure 5.1: Williams's Toggle example
Video 1: Non-Linear Analysis of a shallow arch system
A shallow roof system is illustrated in Figure 7. This structure is designed as per the AISC-2005 13th edition code and the loadings from IBC 2006/ASCE 7-05 code. We know that code based loadings are minimum requirements and it is upto the engineer to ensure that the stability and performance criteria are satisfied based on the importance of the structure and customer's needs. Normally, Civil Engineering structures similar to the one shown in Figure 7 will not experience large deformations and when service loads are applied, these structures will behave in a linear fashion. In some instances, engineers are required to study the performance of the structure in the event of an unusually high loading and predict the governing failure mode. This will require the engineer to perform a geometric non-linear or a critical load analysis on the structure. In this case, the model is expected to yield the "true" behavior of a structure closely.
Figure 7: Hanger Structure In this structure, suppose the shallow arch system has to be analyzed for snap through for extremely high loading the engineer could easily utilize the geometric non-linear capability in STAAD.Pro to see if that failure mode is even possible. Figure 8 shows the displacement diagram of arch snap through and the failing members red. It is clear from this analysis that the columns and roof members have to yield before such a failiure mode could occur. Figure 9 shows a similar structure but in this case, the bottom chord members expereinced excessive local deflections during an event of excessive roof loading.
Figure 8: Displacement Diagram of arch snap through. Failing members shown in red.
c Figure 9: Displacement Diagram. Bottom cord members deform first due to excessive roof loading.
4.0 ADDITIONAL COMMENTS: Let us re-visit Figure 2. Figure 2 shows two yellow curves that represent inelastic analysis of structures. The
pushover analysis feature in STAAD.Pro would produce the same curve as the first-order inelastic analysis. The second-order inelastic analysis part is covered in STAAD.Pro without the effects of plastic hinge formation taken into account. In the first-order inelastic analysis option, equations of equilibrium are written in terms of the geometry of the undeformed shape. Inelastic regions can develop gradually. Development of inelastic regions or "plastic hinge" development is not handled by STAAD.Pro V8i's new "Geometric Non-Linear Analysis" feature. The pushover analysis is the only feature that currently handles plastic hinge formation in STAAD.Pro. In the second-order inelastic analysis option, equations of equilibrium are written in terms of the geometry of the deformed shape. It has the potential for accommodating all of the geometric, elastic, and material factors that influence the response of a structure. Again, only the geometric non-linearity is implemented in STAAD.Pro. The equations of equilibrium in STAAD.Pro V8i's new "Geometric Non-Linear Analysis" feature are written in terms of the deformed shape.
References: (1) Matrix Structural Analysis, Second edition, William McGuire (2) Structural Plasticity, CIV E 705 Course Notes, Dr. Don E. Grierson, University of Waterloo
