Structural Analysis

In general, structural analysis is the process of evaluating the behavior of the structure under the design actions. For the purpose of performance verification, an index is specified for each performance requirement. Typical indexes are section force,

displacement, and strain. Performance is then verified by comparing the value of the index under the appropriate action with the limit value; the value of the index under this action is treated as the demand, while the limit value is considered the capacity of the structure. Linear structural analysis is often sufficient to achieve this end. It is also possible to verify performance by evaluating the load-carrying capacity of the structure and compare it with the design actions, but when this procedure is used, nonlinear structural analysis is required [6].Most structural analysis in these days is based on the finite element method.

In the linear analysis, when the applied load is small, a structure’s deformation is generally small and linearly proportional to the magnitude of the load. Linear analysis, also often called small-displacement analysis, is appropriate for this type of structural behavior. Since the principle of superposition is valid in linear analysis and thus the following equation holds true, and results for various load combinations and influence-line loads are very easily obtained [6]:

∑^𝑁_𝑖=1𝐹_𝑖 = 𝐾(∑^𝑁_𝑖=1𝛼_𝑖𝑈_𝑖) (5.2) where, 𝛼_𝑖= i-th node of load factor, N＝the number of loads to be combined, 𝐹_𝑖= i-th node of load vector to be combined, K= stiffness matrix, 𝑈_𝑖= i-th node of displacement vector due to 𝐹_𝑖．Since, in finite element analysis, the components 𝐹_𝑖 and 𝑈_𝑖 are the load and the displacement at a node, are called the nodal load vector and the nodal displacement vector, respectively.

When strain and displacement become large, linear analysis tends to produce significant errors because their effects are ignored. In such situations, the deformed configuration of a structure is distinctly different from its original configuration. Since a structure’s stiffness depends on its configuration, the structural response changes as deformation progresses; this change in configuration must be taken into consideration in analysis. The relationship between load and deformation is nonlinear. The class of analysis required is called the geometrically nonlinear analysis or finite displacement analysis. The nonlinear analysis, an automatic incremental-iterative solution procedure was performed until they reached to the termination limit.

The weighted residual method and the finite element method lead to the following general discretized equilibrium equation [6]:

𝐹 = 𝐾(𝑈) (5.3)

Where, F is the external force vector, K is the internal force vector, and U is the displacement vector. Usually proportional loading is applied so that Eq.5.3 can be rewritten as

𝛼𝐹₀ = 𝐾(𝑈) (5.4) Where，𝐹₀ is the base load vector and α is the load parameter.

In some kinds of nonlinear analysis, only the displacement caused by a specific load is of interest. However, in many other cases, the load F versus displacement U relationship that satisfies Eq.5.2 is evaluated; and the load F versus displacement U curve is plotted. This curve is called the equilibrium path. Today, most nonlinear structural analysis is conducted using the finite element method. Therefore, Eq.5.2 takes the form of a set of nonlinear equations to which exact solutions are rarely available.

Instead, solutions are obtained numerically. A popular method is the Newton-Raphson technique in which the equation is linearized and solved repeatedly until convergence is reached; when the equilibrium path is targeted, certain variables are picked out and given values. The values of the remaining variables are then calculated. Depending on the variables chosen, three groups of approaches are available: (1) load control; (2) displacement control; and (3) arc-length control [6].

In a load-control approach, the value of α is given and, if the Newton-Raphson method is used, the following set of equations is to be solved repeatedly to obtain the solution:

𝛼𝐹₀ − 𝐾(𝑈^(𝑚)) = 𝐾_𝑇(𝑈^(𝑚))∆𝑈^(𝑚) (5.5) 𝑈^(𝑚+1) = 𝑈^(𝑚)+ ∆𝑈^(𝑚) (5.6) Where, 𝐾_𝑇 is the tangent stiffness matrix defined by ∂K/∂U．Eq.5.5 is the linearized equilibrium equation; a set of simultaneous linear equations. Solving this set, the displacement vector is updated using Eq.5.5. This computation is continued until

∆𝑈^(𝑚) becomes sufficiently small that convergence can be considered as achieved. In the above equations, superscript (m) indicates the number of repetitions of this computation. It is noted here that the construction of the tangent stiffness matrix 𝐾_𝑇 requires an incremental constitutive model (an incremental stress-strain relationship).

Besides, in addition to the update of the displacement vector, the stress states need to be updated [6].

There is a limit load that a structure can carry. To evaluate this limit, analysis must continue beyond the limit, exploring the deterioration of the structure. It is difficult to

use load control for this class of analysis. Instead, displacement control may be used since, in many cases, some displacement components continue to increase even during this stage of structural behavior. In displacement control, one of the displacement components is prescribed and the load factor is then unknown. Therefore, the load factor is one of the variables whose values are computed by the analysis. In using this displacement control approach, it is crucial to use a displacement component that continues to increase beyond the limit load. It should be noted also that displacement control results in an asymmetric tangent stiffness matrix⁡𝐾_𝑇 [6].

In this research, a least square algorithm was applied. For the ultimate limit state, the external force as shown in Equation (5.2) was assumed as combination of dead load and live load. In technical committee of JSCE, the design load cases are defined as follow:

F = 1.7F_𝑑+ 𝛼F_𝑙 (5.7) Where, F_𝑑 is the dead load and F_𝑙 is the live load, α is the live load factor.