Nonlinear Finite Element Analysis of Beam Structures
5.3 History Concepts
⎢
⎣
𝐾11 0 0 1
⎤
⎥
⎦
⎧
⎪⎨
⎪⎩
Δ𝑉1 Δ𝑉ˇ2
⎫
⎪⎬
⎪⎭
=
⎧
⎪⎨
⎪⎩
𝑃1 − 𝑅1 − 𝐾12Δ𝑉ˇ2
Δ𝑉ˇ2
⎫
⎪⎬
⎪⎭
(5.71) Note that the partitioned form as indicated in Eq.(5.70) is made here only to simplify the presentation. The boundary conditions are imposed according to Eq.(5.71) in the original rows and columns without any physical rearrangement of equations.
5.3 History Concepts
5.3.1 Load History
The current level of a loading is determined by the product of a reference value and a scaling factor ℎ𝑙(𝜆). For distributed element loading, the actual relationships are in terms of intensities according to Eq.(5.46), while for discrete nodal loading Eq.(5.61) applies. The load scaling factor, or load history, is given as a function of a parameter 𝜆. The same load history may be assigned to several loadings. 𝜆 is a common parameter to all histories (not only those for loads), and it may be interpreted as a ‘neutral time’-measure that works like a ‘driving wheel’ for the nonlinear solution process. Once a new value of 𝜆 is established, all quantities necessary for carrying out the next solution will then be defined. Section5.4 sums up the solution strategy that is adopted in this work.
Several options for defining loading histories are covered in [6]. In the computer program that is developed as a part of this work (Chapter 11) however, only load histories described by discrete curve points in (𝜆, ℎ𝑙)-coordinates are included. The rules for interpreting the 𝐶0-continuous history function ℎ𝑙(𝜆) are there as follows:
∙ When a given 𝜆 falls between two curve points, the corresponding ℎ𝑙 is deter-mined by linear interpolation.
∙ When𝜆falls either below or above the range of curve points, the corresponding ℎ𝑙 is taken equal to the value at the nearest curve point.
Some typical loading histories are depicted in Fig. 5.2.
5.3.2 Displacement History
The current level of a set of prescribed nodal displacements is determined according to Eq.(5.68), i.e. as the product of a reference value 𝑣ˇ𝑟𝑒𝑓 and a scaling factorℎ𝑑(𝜆𝑠).
𝜆 ℎ𝑙
(failure)
Figure 5.2: Load Histories
The same displacement scaling factor, or displacement history, may be assigned to several sets of prescribed displacements. Displacement histories are defined in exactly the same way as the histories of loading in the preceding subsection.
5.3.3 System History
In this context a system is understood to consist of an assembly of elements with corresponding boundary conditions (constrained DOFs). By identifying each system with a separate number 𝑆𝑖, i.e. the system number, the construction sequence, or system history, of the structural problem in question may then be characterized by a sequence of system numbers related to the parameter 𝜆. In the computer program that is presented in Chapter 11, the system history data are given in terms of a sequence of discrete (𝜆, 𝑆)-values (points). The rules for defining the stepwise constant history function 𝑆(𝜆) from these points are as follows:
∙ When a given 𝜆 falls between two points, the corresponding system number is the one that pertain to the larger 𝜆-value.
∙ When𝜆falls either below or above the range of points, the corresponding system number is taken equal to the one at the nearest point.
The system history may be represented as in Fig.5.3. Of convenience, the sequence of system numbers is here taken in consecutive order from 1 to 𝑛𝑠𝑦; the latter being the number of systems in the structural problem in question.
5.3.4 Time History
The time history relates the actual time 𝑡 to the parameter 𝜆. In this work the function 𝑡(𝜆) will be 𝐶0-continuous in pertinent intervals of 𝜆. However, backward shifts of time are allowed. This option may be of interest in conjunction with the start of a new construction sequence that is materialized in parallel with a previous one (e.g. a cantilevered bridge span that is built from both sides simultaneously). In
𝜆 𝑆
𝑛𝑠𝑦
𝑖+1 𝑖
2 1
Figure 5.3: System History
𝜆 𝑡
Figure 5.4: Time History
the computer program in Chapter11, the time history data are given as a sequence of discrete curve points in (𝜆, 𝑡)-coordinates. The rules for defining the history function 𝑡(𝜆) from these points are as follows:
∙ When a given 𝜆falls between two curve points where the time is unchanged or increased, the corresponding actual time is determined by linear interpolation.
∙ When a given 𝜆 falls between two curve points with a decrease in time, the corresponding actual time is taken as the lower time.
∙ When𝜆falls either below or above the range of curve points, the corresponding time is taken equal to the value at the nearest curve point.
A time history (e.g. for the cantilevered bridge span) may look as depicted in Fig.5.4.
𝑡 𝑇𝑠
Δ𝑡𝑦𝑟 𝑇𝑚𝑎𝑥
𝑇𝑚𝑖𝑛
Figure 5.5: Mean Seasonal Temperature Variation
5.3.5 Mean Seasonal Temperature Variation
Similar to [17], the mean seasonal temperature𝑇𝑠 will be related to the actual time 𝑡 through the simplified, periodic expression
𝑇𝑠(𝑡) = 𝑇𝑚𝑎𝑥 + 𝑇𝑚𝑖𝑛
2 − 𝑇𝑚𝑎𝑥 − 𝑇𝑚𝑖𝑛
2 cos
(︃ 2𝜋 Δ𝑡𝑦𝑟 𝑡
)︃
(5.72) where (𝑇𝑚𝑎𝑥, 𝑇𝑚𝑖𝑛) are the mean values of maximum and minimum seasonal temper-ature, respectively, and Δ𝑡𝑦𝑟 is the duration of a year. Normally, a day (24 hours) is used as the unit of time, as well as the averaging period for the temperature recordings that constitute the statistical basis for (𝑇𝑚𝑎𝑥, 𝑇𝑚𝑖𝑛). The graphical representation of Eq.(5.72) is shown in Fig. 5.5. It is seen that 𝑡= 0 yields 𝑇𝑠=𝑇𝑚𝑖𝑛. Thus, the time may be counted from January 1 in a certain year (northern hemisphere).
Since time dependent effects in concrete often also depend on the corresponding temperature level in a mean sense, an expression like Eq.(5.72) provides a simple and reasonably accurate basis for including such effects with a minimum of data-input.
Time and temperature dependent effects are covered in Chapter9. Note also that the mean seasonal temperature variation implicitly becomes a function of the parameter 𝜆 through the time history concept; thus 𝑇𝑠 =𝑇𝑠(𝑡(𝜆)).
5.3.6 Temperature Deviation History
The current absolute temperature 𝑇 may be obtained by superimposing the mean seasonal component 𝑇𝑠 and a deviation from the mean Δ𝑇, i.e.
𝑇 = 𝑇𝑠 + Δ𝑇 (5.73)
where𝑇𝑠is covered in the preceding subsection. Analogous to the treatment of loads and prescribed displacements, Δ𝑇 will be determined by the product of a reference value Δ𝑇𝑟𝑒𝑓 and a scaling factor ℎ𝑇(𝜆). Thus
Δ𝑇 = ℎ𝑇(𝜆) Δ𝑇𝑟𝑒𝑓 (5.74)
The reference value is specified at the element level through
Δ𝑇𝑟𝑒𝑓 = Δ𝑇1 + 𝑥 𝑔𝑇𝑥 + 𝑦 𝑔𝑇𝑦 + 𝑧 𝑔𝑇𝑧 (5.75) where Δ𝑇1 is the value at node 1, and (𝑔𝑇𝑥, 𝑔𝑇𝑦, 𝑔𝑇𝑧) are the constant temperature gradients in the local (𝑥, 𝑦, 𝑧)-directions, respectively. The scaling factor, or temper-ature deviation history, is related to the parameter 𝜆 according to the same rules as described for the load history function in Subsection 5.3.1. The same temperature deviation history may be assigned to several reference values of temperature.
The split in temperature according to Eq.(5.73) allows for investigating extreme thermal effects at a certain instant of time by including the corresponding tempera-ture deviations in the pertinent interval of𝜆, while long-time effects are continuously taken care of through the mean temperature component. Optionally, all thermal ef-fects may be evaluated through the temperature deviation history concept by letting 𝑇𝑚𝑎𝑥 =𝑇𝑚𝑖𝑛 = 0.