The static elastic buckling behaviour of axially-loaded columns are tested for different boundary conditions.

Assumptions: 1) the column is pin-ended and perfectly straight without curvature, 2) isotropic, homogeneous, prismatic cross-section, 3) the axial load is applied along the centroidal axis of the column and 4) linearly-elastic range

With these assumptions, the static stability criterion as an eigenvalue problem is formulated and the Euler buckling load is derived as P_e=.

USFOS is dictated as the numerical software for simulation and analysis. It allows nonlinear incremental analysis for elastic and inelastic behaviour using the elastic-plastic hinge method (sudden insertion of plastic hinges) and the refined elasto-plastic hinge method (progressive yielding & hinge formation) respectively.

Useful for both member level and structure level analysis, it basically follows the standard FEM and incorporates the plastic hinge approach of the two surface plasticity model using the modified tangent stiffness relationship. Plastic hinges are allowed to form at either of the two nodes or midspan of an element.

Load increment follows the updated Lagrangian formulation with displacement and load controls which are limited by 1) plastic hinge formations, 2) user-defined total load parameter, 3) excessive elemental nodal forces at the plastic hinges and 4) the nonpositive definiteness of the structural stiffness matrix. The deformation and flow rule are governed by the current stiffness parameter for either hardening or softening systems.

Numerical simulations are at best approximations. Its accuracy is limited by 1) the number of elements and nodes, 2) the FEM model used and 3) the user-defined inputs. A compromise is reached between the accuracy, the model complexity and computational resources. To ensure stability, accuracy and limit error growth, intrinsic and user-defined limits to the maximum load increment and displacement are included.

USFOS is a nonlinear finite element software that utilizes the incremental displacement stiffness method for numerical solution. Each structure is discretized into user-defined elements with local forces and displacements. Due to the nonlinear nature of analysis, the method adopted is to linearize each incremental segment for both forces and displacements. In addition, the analysis is split into the elastic component before yielding, inelastic component and post-collapse component. Elasto-plastic analysis whereby a gradual yielding across a section is performed before a plastic hinge is formed by the connection of opposite yielded zones.

Using the direct stiffness method, the element force-displacement relationship is

(1.1)

Linearization is performed through the elemental incremental form as

(1.2)

Transformations between the local and global elemental displacements and forces are

(1.3)

(1.4)

The elemental incremental force-displacement relationship in global coordinates is

(1.5)

The structural incremental force-displacement relationship in global coordinates is

(1.6)

where is the elastic tangent stiffness matrix.

Once an element has reached elastic boundary surface, it yields and when the yielding propagates throughout the section, rotational stiffness is neglected. In effect, this is similar to inserting a hinge at that completely yielded section. This is the plastic hinge which accounts for the plastic reduction component of the tangent stiffness matrix as well as its geometric deformation component which alters the force matrix. Hence, the element force-displacement relationship under plastic hinge presence in local coordinates is

(1.7)

where is the modified tangent stiffness matrix and is the equilibrium force-correction vector accounting for the change in moment capacity of the element.

The modified element force-displacement relationship in global coordinates is

(1.8)

and the structural modified force-displacement relationship in global coordinates is

(1.9)

where is the modified plastic tangent stiffness matrix and is the structural equilibrium force-correction matrix due to plastic hinge formation.

Post-collapse analysis starts once the structural stability limit is reached due to the formation of collapse mechanism from the plastic hinges as well as tangent stiffness determinant reaches zero. This analysis allows the engineer to simulate the failure process of the user-defined structure and provides further insight into the structural behaviour.

USFOS outputs all user-defined data, finite element data and structural responses in terms of incremental loads, displacements, plastic hinges, buckling, stability and more. With the control node(s), the user has defined the structural criteria for analysis. The results are in incremental form of load step, load factor, control displacement step and critical stages. The absolute loads and displacements have to be accumulated, tabulated and plotted visually as post-processing. This is provided in sections 1.3, 3.b, 3.c, 3.d and 4. The Appendix of this report contains the input files.

The major variation of the USFOS simulation from the theoretical assumptions is that the former allows inelastic behaviour, while the latter is strictly in the linearly-elastic range. By suppressing the formation of plastic hinges and considering only the elastic range of the simulation at the member level, a more balanced comparison may be carried out, provided all other theoretical assumptions are not violated.

The elements used are 2-node linear elements with maximum of six degrees of freedom at each node, thus more elements allow h-convergence and better accuracy towards the theoretical continuous solution.

Due to the inability of the linear shape function of a 2-node element to deform and buckle out of the initial alignment, modeling of the column by a single element is omitted. Numerically, when plastic hinge formation is suppressed, the 1-element column would not buckle out of plane and the results are not accurate, thus only 2 elements are used.

Referring to Figure 1, it shows that the axial load-lateral displacement curves for columns 1(a)-(e). The columns are modeled with 2 elements each with axial load P along the centroidal axis and the control node is at the midspan in the horizontal x-axis direction. The theoretical Euler buckling load is =0.520 MN. The load-displacement curves show that the critical elastic buckling loads agrees well with the theoretical under the various boundary conditions. The buckled deformed models of the fixed-ended and pin-ended columns are shown in Figures 1a and 1d below. The control node is placed at midspan to capture the maximum side sway.

Figure 1a: Fixed-Fixed Column

Figure 1d: Pin-ended Column

With the simulation running in nonlinear elastic mode and suppressing formation of the plastic hinges, elastic buckling of a column can be achieved. The solution is however not realistically correct as the assumptions are too stringent and buckling can only occur in the plastic range, not in the linearly elastic range. Hence the elastic simulation results are non-conservative and safety factors must be used. With this understanding, any practical simulation for column members must incorporate elastic and plastic behaviour into the buckling analysis and theoretical answers might not be realistic depending on the model, assumptions and loading applications.

Difficulties are encountered during analysis. USFOS as a nonlinear FEM program is focused on solution technique but the input parameters governing the analysis have to tried and entered with judgement. Trial-and-error might suffice for simple problems, but is not feasible for complex systems. Trial-and-error is also time-consuming and inherently dangerous.

Another way is to base the analysis on theoretical predictions. This would be satisfactory for simple problems such as the axial-loaded columns above. However, the closed-form theoretical models, assumptions and constraints have to be satisfied before the analysis can be based on. Simplifications and appropriate approximations of actual systems to fit the theoretical models must be performed carefully.

Unlike most other branches of engineering, civil engineering has extensive construction codes that serve as useful guides both for design-analysis as well as construction. The pseudo-theoretical approach based on experimentally-verified findings help narrow the scope of uncertainty for linear complex analysis. However, nonlinear analysis is emerging and issues of design and construction quality continue to cast doubt on usefulness.

The model simulated using USFOS as the numerical algorithm are at best approximations with inherent deficiencies. The input parameters for a specific problems are difficult to judge accurately. Often iterations using pure computations are performed, but attention must be on the stability, convergence and consistency of the numerical methods. The modified Newton-Raphson method for constant stiffness load increment still inherits the deficiencies of convergence difficulties, the need of good initial points and computational inefficiency. Although nonlinear analysis is more accurate, the problem must warrant the need for additional complexity, difficulties and computational resources.

Frames exhibit predominantly flexural-shear deformations. Being unstiffened laterally, unbraced frames exhibit larger lateral deformation under lateral loads than braced frames. The predominantly rigid joints in unbraced frames rotate more under lateral load than braced ones. Buckling of the beam-column members of a vertically-loaded frame can also occur as shown in section 1 above.

In addition, due to the non-uniform spread of induced strains and stresses amongst the frame members and within each member, some critical locations would reach the inelastic range first before others. Therefore due to the spread of plasticity, as the inelastic zones across these highly-stressed sections meet, plastic hinges are formed when the sections are fully yielded. With a plastic hinge, no rotational strength is present and the plastic section is able to rotate freely with only sustained tangential axial loading. The number of plastic hinges formed before collapse is an indication of the structural redundancy. The more hinges formed, the lower the degree of redundancy as a result of the configuration of loading, geometry and material.

Moreover, these frames exhibit post-buckling strength as the formation of plastic hinges allows membrane action to take place. Just like tension field action of steel girders, the axial membrane action (since no bending or torsional stiffness) along the yielded critical frame sections have a stiffening or hardening effect on the overall frame structure.

In addition, the tangent stiffness of the overall structure has three components of 1) elastic stiffness , 2) geometric stiffness and 3) plastic reduction stiffness .

In elastic-plastic or elasto-plastic analysis, increases, reaches maximum when the initial yield surface extends to the bounding surface. Then the force state is allowed to move on the surface.

is affected by the structural geometry, thus along predominantly axially-loaded sections as well as sections between hinges and plastic hinges formed, membrane action is increased due to axial forces. The axial loaded sections are more efficiently stressed such that can be increased. However, the overall deformed structural geometry in relation to the loading conditions can offset any of the gains through membrane action, especially when there are more plastic hinges that deform the structure adversely to load resistance.

In addition, , which decreases , would be the deciding factor of structural stability. As more plastic hinges form, more sections are effectively released in bending and torsional restraints. The overall effect is a softening effect on the frame and would decrease however high the contribution to from membrane action. The limiting stability is reached when the determinant of is zero, below which the frame is unstable. This concept of global instability is captured in the software USFOS by a formulation based on the Current Stiffness Parameter in combination with a Determinant Criterion. Post-collapse analysis is performed by load decrement (unloading).

The unbraced frames studied vary in 1) type of analysis, 2) load level and combination, 3) step size of load increment and 4) equilibrium iterations between each updated Lagrangian of the tangent stiffness. In addition, the output data for control nodal displacement is in incremental form with each load step, thus to get the relative nodal displacement, the accumulated sum of the output displacement for each load step.

A pin-supported, unbraced portal frame with relatively flat roof with slope angle a of 25° is analysed in this section. This unbraced frame is simple and minimally redundant.

The nonlinear elastic analysis is carried out with the use of the tangent stiffness incremental approach. Plastic hinge formation is suppressed. Loading is performed in one load case of gravity and lateral loads. The load-lateral displacement curve is shown in Figure 3b1. As expected, the elastic deformation is large with much high load factor attained due to suppression of plastic hinges, no plastic stiffness reduction and the geometry of the elastic frame remains almost the same. The first element to buckle is element 38 at load level 6.9 and current stiffness parameter of 0.164, shown by the change in the slope of the curve. Further buckling of elements ensures the increase in lateral displacement with lower load increment until the limit of stability is reached at load level 13.34. This occurs when the current stiffness parameter reaches zero and the analysis goes into the post-collapse range with unloading. This is shown by the change in slope of curve from positive to negative at the limit of elastic stability. The buckled deformed model is shown in Figure 3.b(1) on the next page.

Since only elastic-plastic analysis is conducted in this report, the limit load for first hinge is calculated using section 3.b.3 under proportional load increment.

The nonlinear elastic analysis is carried out with the use of the tangent stiffness incremental approach. Allowing plastic hinge formation, true elastic-plastic analysis is carried out. The gravity loading is applied to load level 1.0 and remains thereafter, while the lateral push over load is subsequently applied. This load combination is realistic as it approximates actual service conditions whereby a structure under construction is often braced and subjected to predominantly gravity load. A structure in service is released from temporary bracing and subjected to service loads as well as environmental excitations of wind, rain and the ground. The load-lateral displacement curve is shown in Figure 3b2.

The gravity load is applied to load level 1.0 and remains constant. Referring to Figure 3b2, the lateral displacement of the control node is relatively small and the slope is almost linear, thus the portal frame is still within the elastic range.

For convenience, the horizontal load level is simply added to gravity load level of 1.0. However, once the lateral loads are applied incrementally, the lateral displacement increases rapidly due to changes of geometry as the frame side sways. At horizontal load level of 0.47, the first plastic hinges are formed at global nodes 3 and 7 of

Figure 3.b(1): Elastic

Figure 3.b(2): 2-step & proportional

elements 2 and 7 at the structural connection between the column and the roof. As expected, these are the highest stressed sections of the portal frame due to the abrupt change in geometry affecting both moment redistribution and stress transfer.

The portal frame continues to deform under the combined loads with plastic hinges formed at the critical sections. As more elements buckle and plastic hinges formed along the midspan of the frame rafters, the current stiffness parameter reduces to zero at the limit of plastic stability. At combined load level of 1.47, this is also the point where the first hinges are formed, therefore the portal frame is a minimally redundant structure. The deformed model at ultimate load factor is shown in Figure 3.b(2).

The nonlinear elastic analysis is carried out with the use of the tangent stiffness incremental approach. The analysis in this section is the same as section 3.b.1, except that plastic hinge formation is allowed. Proportional loading is not realistic as it does not approximate actual service conditions. Similar to 2-step loading, the load increment decreases rapidly after load level 1.0 and the limit of plastic stability as well as the formation of the first hinges are reached together, but at lower load level of 1.338. This is also the load level under elastic analysis for limit load at first hinge. The deformed model at ultimate load factor is shown in Figure 3.b(2).

A fixed-ended, unbraced six-story frame with beam span of 8m is analysed in this section. The rotational tilt at the supports is neglected since it is within tolerable limits of design codes. Unlike the portal frame of section 3.b, the six-story frame has connected beams and columns connected perpendicularly for maximal reinforcement in terms of geometry. However, as the frame is tall, it is likely to deform in shear mode with more axial deformation and stress concentration at the lower stories. With equally-spaced floors and higher combined loadings, lateral displacement at the top story is likely to be larger. The lateral loads also create larger bending moments at the base due to the height of the frame. With lateral deformation, the gravity load also induces additional bending moments at the base, thus enhancing the deformation and stress concentration effects. The frame is also symmetry, hence by the Principle of Transmissibility, the analysis results are invariant to the location levels of the lateral loads.

Similar to section 3.b.1, plastic hinges are suppressed. The load-lateral displacement curve is shown in Figure 3c1 which is similar in shape to Figure 3b1. The load increment is large at first, but decreases once the first element 25 at the bottom story buckles with significant increases in lateral displacement. The load level at first buckling is 1.702. Further buckling at the lower floors and then the upper floors weaken the frame with further increase in lateral displacement as the load increment decreases. The limit of elastic stability is reached when the current stiffness parameter reaches zero. However, under the simulation, the buckling and instability of members without plastic hinges still allows load level with negligible stiffness parameter. This is a numerical error due to the iterative nature of the incremental analysis. The peak load level can be interpreted at 2.475 before the cyclic buckling occurs.

The buckled deformed model is shown in Figure 3.c(1) below.

Figure 3.c(1): Elastic

The nonlinear elastic analysis is carried out with the use of the tangent stiffness incremental approach. This analysis follows that of section 3.b.2.

Under gravity load only, the frame has already reached the limit of plastic stability as shown in Figure 3c2 at gravity load level 0.698. Interestingly, the first plastic hinges are located at the midspan of the top story at load level 0.489. Further plastic hinges are formed in succession from top towards the bottom floors. The material and member sections used can account for the analysis results with respect to geometry. The decrease in stiffness however is due to plastic reduction from hinges formed and flexural is further decreased.

The deformed models for 2-step and proportional elastic-plastic analysis are shown in Figures 3.c(2) and 3.c(3) below respectively.

Figure 3.c(2): 2-step

Figure 3.c(3): Proportional

The load-lateral displacement curve is shown in Figure 3c3. It has almost the same shape and magnitude as Figure 3c2. With proportional loading, the frame reaches the limit of plastic stability at combined load level of 0.691 slightly lower than that of 2-step load (gravity only). The inclusion of lateral load accounts for the difference. The first plastic hinges are also at the roof floor at load level 0.473 and the plastic hinge formation propagates downwards towards the bottom floors.

A pin-supported, unbraced two-story frame with horizontal load of 0.75% of the gravity load is analysed in this section. The frame is non-symmetric in-plane geometrically as well as sections used. The transmissibility of forces are altered in unsymmetric frames, hence the locations of lateral loads influence the structural response. If the lateral loads are located at the column of shorter span, the local stiffness is higher, with consequent lesser lateral deflection or higher peak load factor and vice versa. Using simple methods like the cantilever method, the middle column is higher-stressed, the section used is larger and braced at each floor on each side by respective beams. The pin-supported columns reduce the inherent redundancy of the frame such fewer plastic hinges are needed to form collapse mechanism and to reach the limit of plastic stability where the current stiffness parameter is zero.

The nonlinear elastic analysis is carried out with the use of the tangent stiffness incremental approach. Following the analysis of sections 3.b.1 and 3.c.1, plastic hinge formation is suppressed. The curve is shown in Figure 3d1.

The nonlinear elastic analysis is carried out with the use of the tangent stiffness incremental approach with plastic hinge formation allowed. The load-lateral displacement curve is shown in Figure 3d2.

With load increment to load level 1.0, plastic hinges are formed along the middle column connections to the beams where the stress concentration is the highest. The first hinges form at load level 0.736. Due to the lack of geometrical symmetry, the frame deform nonuniformly with larger transverse deformation of the longer-span side which is also the structurally weaker side with respect to stiffness.

Once the lateral load starts to apply, the weakened frame under the gravity load deforms further and more plastic hinges form causing further drop in stiffness.

When second plastic hinge form at the middle column on the lower floor, the stiffness parameter drops rapidly to zero and the limit of plastic stability is reached at load level 1.023. The combined loading then continues into the post-collapse range with unloading.

The nonlinear elastic analysis is carried out with the use of the tangent stiffness incremental approach. Proportional load increment is performed, allowing plastic hinges. As shown in Figure 3d3, sharp load increment is allowed before first hinge form at the middle column with combined load level 0.713, slightly lower than that of 2-step elastic-plastic analysis. The formation of more hinges follow that of section 3.d.2 until the limit of plastic stability is reached at combined load level of 1.009.

The deformed models at ultimate load factors are shown in Figures 3.d(1), 3.d(2) and 3.d(3) for elastic, 2-step and proportional elastic-plastic analysis respectively below.

Figure 3.d(1): Elastic

Figure 3.d(2): 2-step

Figure 3.d(3): Proportional

The results for load factors at buckling and collapse under elastic analysis as well as load factors and plastic hinge formations under elastic-plastic analysis are summarized in Table 1 below.

Table 1: Load factors for Elastic and Inelastic Analysis

l buckle is the load factor at first buckling. l cr is the ultimate load factor under elastic analysis. l first and l u are the load factors at first plastic hinge and ultimate load respectively. The number of plastic hinges formed at point of collapse is also shown at the limit of plastic stability.

It shows that the elastic results for load factors are higher than the elastic-plastic results because of the plastic hinge suppression such that the tangent stiffness used is without modification and reduction. The more plastic hinges formed, the lower the load factors. The number of plastic hinges able to form is an indication of the inherent structural redundancy. Portal frame is least redundant with least plastic hinges, while the six-story frame is most redundant with most hinges. However the more plastic hinges formed, the greater the stiffness reduction in and larger changes in geometry accounting for and geometric stiffness component . Hence, the more hinges, the lower the load factor achieved.

Tables 2, 3 and 4 below show the analysis results for elastic buckling and plastic hinge formation sequences and load factors. Note that in elastic analysis, the buckled elements are listed according to the load factor. In elastic-plastic analysis, the hinges formed at combined gravity and horizontal load factors are shown with notation 2/2 representing a hinge at element 2 at end 2.

Table 2: Buckling and Plastic hinge locations and formation sequence: Portal Frame

Figures 3b1, 3b2 and 3b3 show the load factor-lateral displacement curves of the portal frame.

Table 3: Buckling and Plastic hinge locations and formation sequence: 6-Story Frame

Figures 3c1, 3c2 and 3c3 show the load factor-lateral displacement curves of the symmetric six-story frame. The base tilt is within tolerance and thus neglected.

Table 4: Buckling and Plastic hinge locations and formation sequence: 2-Story Frame

Figures 3d1, 3d2 and 3d3 show the load factor-lateral displacement curves of the non-symmetric two-story frame.

The numerical simulations of an axially-loaded column are compared to the theoretical Euler buckling load at the member level with one and two elements.

Nonlinear elastic and elastic-plastic analyses are carried out for unbraced portal, six-story and two-story frames under combinations of gravity and horizontal loads, step sizes and equilibrium iterations. Elasto-plastic analysis is not conducted where the plasticity is allowed to spread through a section. Elastic analysis provides better lateral displacement prediction, while the elastic-plastic analysis allows peak load prediction, locations of highly-stressed regions as well as the sequence of plastic hinge formation. The proportional load increment produces lower peak load, but with the same general structural response. Although it is not a realistic form of loading, it provides a quick one-step lower bound estimate of the actual loading conditions approximated by the 2-step method. The numerical technique used in the software has been verified satisfactory, but the iterative trial-and-error process of step-size and equilibrium iterations need improvement, especially the need for a pre-processor for a general gauge of recommended values. This is particularly useful for large, complex, interconnected systems at different stages of construction or service where it is virtually impossible for good user-defined input parameters.