Pinned-fixed beams

These beams present a rectangular section of 75 x 150 mm² and a span length of 1.50 m.

They are supported in two points, a pinned and a fixed one. While one of those elements

restricts only the vertical displacement, the other restricts both the vertical and the horizontal one along a line. Vertical spring elements were introduced along the support line, in this latter case, to study the related restraint to this degree of freedom. Such springs would only work for vertical displacements and when in compression. In order to avoid high local stresses, a steel plate was placed between the structure and both structural supports. A uniform mesh of quadrilateral elements was considered. Reinforcing steel bar elements were considered to be embedded in concrete.

In order to correctly simulate the laboratory test, some considerations were taken. A displacement control numerical test was adopted. Three load cases were considered, one representing the real supports with spring elements placed at fixed support, other representing real supports with a vertical displacement restraint at fixed support and other representing the applied displacement. The load is applied at middle span of a steel profile which will load the beam in two points, respectively, at 1/3 and at 2/3 of the span length. An identical increment was considered for the applied displacement in each point load, equal to 1.00 * 10^-4 m (downward). A Newton-Raphson nonlinear search algorithm was used.

Considered parameters are given in Table 5.3. The middle span displacement, the applied load and the pinned support reaction were monitored during the analysis. The bending moment at fixed support was computed through static equilibrium equations [116, 117, 119, 121, 122].

When performing several analysis of the same numerical model, as in model identification or within a probabilistic analysis, the issue of computational cost becomes very important. In order to overcome it, the developed numerical model was simplified. Therefore, both finite element and load step numbers were minimized. Accordingly, three mesh types (with 3983, 1333 and 427 elements) and two different load steps (210 steps of factor 0.50 and 60 steps of factor 2) were studied. Figure 5.9a presents a finite element mesh with 1333 elements.

In a further analysis, the performance of each model was evaluated. In order to assure identical computational conditions, the same computer was used. The computational time and the related error were determined for each analysis. The applied load error was computed through equation (5.2),

(

)

^{[ ]}

i i i i

F F F F

∆ = − (5.2)

in which F₁ⁱ is the applied load for a specific step i of evaluated numerical model and F₀ⁱ is the applied load for the same step i of the reference numerical model. The reference model is the one that presents the most refined mesh and higher number of load steps. Then, the

computed. Finally, the applied load error (θ_F) is computed by dividing this value per two. The pinned support reaction error was computed through equation (5.3),

(

)

^{[ ]}

i i i i

R R R R

∆ = − (5.3)

in which R₁ⁱ is the obtained reaction value for a specific step i of evaluated numerical model and R₀ⁱ is the obtained reaction value for the same step i of the reference numerical model.

The reference model is the same as the one used to determine the applied load error. Then, the maximum and minimum ∆_R-values are determined and the sum of their absolute values is computed. The reaction error (θR) is then computed by dividing this value per two.

Table 5.5 gives the obtained results. It is possible to conclude that the most suitable model, for a further analysis, is the number 3. In fact, by comparing with other models, it is possible to conclude that this model presents a lower error (≈6%) and, at same time, by comparing with reference model (number 0) it reduces the computational cost in almost 90%.

Table 5.5. Simplification results.

Numerical model Finite element

number Step number Computational time [s]

Applied load error - θF [%]

Reaction error - θR [%]

0 3983 210 966.26 - -

1 3983 60 756.41 2.14 3.80

2 1333 210 211.38 4.80 4.58

3 1333 60 141.01 6.00 5.82

4 427 210 79.15 9.16 9.32

5 427 60 41.56 9.23 12.57

A first step calibration procedure was developed, taking into consideration the chosen numerical model, to determine the most suitable spring stiffness value (k = 149.13 kN/m).

Such procedure consisted in identifying the value that optimizes the distance between numerical and experimental data. In order to develop this analysis, an evolutionary strategies optimization algorithm in its plus version [29] was used. During this procedure the other parameter values were fixed. It is verified that the fixed support is not working as a full clamp since the beginning of loading, as the obtained stiffness value is low. This parameter intends to represent the concrete accommodation in fixed support during the initial phase.

The vertical displacement restraint is only assured in a more advanced phase of the test, in which the spring elements are replaced by pinned supports. During the same optimization procedure it was identified this instant, which occurs for load step 30. Accordingly, the load

cases were divided into 30 steps with a factor of 2 in the presence of spring supports, and 30 steps with a factor of 2 with pinned supports. During this last phase the fixed support works as a full clamp, totally restraining the beam rotation.

The collapse mechanism is characterized by two plastic hinges, the first at fixed support and the second beside the point load, which is close to the pinned support. A bending failure mode, with concrete crushing and yielding of longitudinal steel reinforcement, is obtained.

The numerical behavior of analyzed reinforced concrete beams was similar to the one obtained in experimental tests. Obtained results validate the numerical model. Figure 5.9b presents the deformation, crack pattern and horizontal strain of tested beam.

a) b)

Figure 5.9. Numerical model: a) finite element mesh; b) failure mechanism.