DYNAMIC WIND LOAD ANALYSIS PER RUSSIAN DESIGN CODE
DYNAMIC WIND LOAD ANALYSIS PER RUSSIAN DESIGN CODE
VERIFICATION REPORT
1 General
The purpose of the program is generation of wind pressure forces on buildings and structures together with vibration analysis under the turbulent wind pressure fluctuations.
According to Russian code for loads and actions SNiP 2.01.07-85* [1] (hereafter referred as the Code) the wind load comprises mean wind pressure [pic] and quasi-static wind fluctuation component [pic] with fluctuation coefficient [pic], dependent on terrain roughness and the height above ground. If the structure is susceptible to wind-induced resonance, then the additional dynamic amplification factor (DAF) [pic] should be introduced. The product [pic] will take into account the wind fluctuation together with resonance vibrations of the structure.
1.1 Theoretical background
The wind-induced vibration of the structures is treated as a stationary random process described in terms of the spectral theory of random vibrations [2], [3]. The use of spectral theory in design practice of the USSR was long ago communicated in [4]. Comprehensive theoretical and practical description of wind load analysis according to Russian design code is found in the Design Guide [5].
The buffeting vibration, caused by the wind pressure fluctuation, is predicted from the dynamic equilibrium equations. The random wind loading is represented by A. G. Davenport’s power spectral density [6]. Normalized power spectral function reads as
[pic], [pic] (1)
where f is current frequency, Hz,
[pic] m — longitudinal turbulence length scale,
[pic] — basic wind velocity 10 m above ground.
The solution is obtained by modal expansion of structural equations of the oscillatory system
[pic] (2)
Corresponding eigenproblem
[pic] (3)
yields generalized modal displacements [pic] with [pic] and a set of uncoupled equations
[pic] (4)
[pic], [pic]
where,
[pic] — mass, attached to the j-th degree of freedom (DOF),
[pic] — i-th natural frequency,
[pic] — j-th component of the i-th mode shape vector,
[pic] — quasi-static wind pressure force, applied to the j-th DOF.
Dynamic response to the random wind loading can be obtained by means of generally accepted spectral analysis technique [3]. The response of a structure to the random excitation is assumed harmonic, in the form of modal expansion.
The format of solution sought for follows from the equations (2) to (4). In terms of amplitude values of fluctuating variables, with no allowance for resonance effects, it can be formulated as
[pic] (5)
The resonance effect is determined by the RMS response to the wind fluctuation described by spectral density (1). This results in dynamic amplification factor (DAF) [pic],
[pic] (6)
where [pic] — non-dimensional natural vibration period,
[pic], [pic] — logarithmic vibration decrement.
To obtain the meaningful engineering solution, the RMS quantity should be multiplied by the probability based peak wind pressure factor. It is taken [pic].
The correlation coefficient [pic] of the wind pressures on different points of wind-exposed surface is introduced in a similar way.
More sophisticated dynamic wind loading method [7] is proposed, but little is known about its practical use in design practice.
1.2 Main assumptions and limitations
Some wind engineering problems cannot be solved by standard codified procedures. More sophisticated design methods including non-linear and aeroelastic stability analysis, wind tunnel tests and field measurements should be applied in such cases. Limiting assumptions and limitation are present in the program for this reason.
• Dynamic action of the wind load is represented by the peak quasi-static values of internal forces in the structure.
• The program automatically includes necessary number of natural vibration modes and performes the SRSS summation of modal contributions if the number of modes is greater than one.
• For all vibration modes the same DAF is applied, as calculated for the first vibration mode.
• Wind pressure correlation coefficient for the first vibration mode is applied. For higher modes, the coefficient is taken unity.
• Dynamic wind action can be determined only on vertical or near-vertical surfaces and cannot be applied to roofs.
• Design code does not contain recommendations about the vibrations of vertical surfaces of flexible shells, such as cylindrical shells and cooling towers.
• The program is not applicable to aerodynamic and aeroelastic stability problems.
1.3 Computation algorithm
The basic wind pressure [pic] is determined in the Code on the basis of measured wind velocity at the height of 10 meters with the 10-minutes duration averaging and 50-year return period and relates to the wind velocity by the eq (11.3) of the Code.
Wind pressure and wind force coefficients. The mean wind pressure [pic] is introduced with the multiplier [pic] determined by the Table 11.2, and pulsating pressure with the multiplier [pic], determined by the Table 11.3 of the Code. Here [pic] is effective height, defined in the Code, 11.1.5. Aerodynamic wind pressure coefficients for wind-exposed surfaces, or wind force coefficients for entire structures, [pic], depend on the shape of a structure, and are defined in the Annex D of the Code for typical structural shapes.
If the structure is sufficiently rigid, then the quasi-static application of wind fluctuation pressure is acceptable. Otherwise, the resonant action of fluctuating wind should be taken into account. The dynamic rigidity of the structure is defined by its natural frequencies. The greater is the natural frequency, the less susceptible to wind-induced vibrations is the structure. Limit natural frequencies are presented in the Table 11.5 of the Code. They depend on the basic wind pressure in a region and the logarithmic decrement of a structure. Dynamic calculation of the wind load must include all vibration modes with frequencies less than limit value.
Static wind forces. The static wind force acting on the k-th DOF on the wind-exposed surface of the structure is determined as
[pic]. (7)
Here [pic]— wind pressure area, related to the k-th DOF of the wind-exposed surface,
[pic] — aerodynamic coefficient calculated in accordance with the Annex D of the Code, applicable to both mean and pulsating pressure components.
X- and Z-components of modal dynamic wind forces at a joint are considered as different DOFs of the oscillatory system.
The obtained mean value is summed with the dynamic components of the wind loads. If the structure is not susceptible to wind–induced vibration, then the [pic] is considered as dynamic load component, and calculation is completed with this.
Dynamic forces. If one or more of natural frequencies are below the code-defined limiting value [pic] then the dynamic calculation shall be performed. Dynamic amplification factor [pic] is determined from the graph in Fig. 11.1 as a function of the non-dimensional natural vibration period, obtained by the eq (11.8) of the Code:
[pic] (8)
where
[pic] — basic wind pressure, Pa,
[pic] — pressure coefficient at a reference height,
[pic], H — reference height of the structure,
[pic] — partial load reliability coefficient,
[pic] — fundamental natural frequency.
For all vibration modes, the same DAF is applied, as calculated for the lowest natural frequency.
The modal components of dynamic wind forces of the k-th DOF, corresponding to the i-th natural mode, are
[pic]
This equation is rewritten in the form of
[pic]
where
[pic]
is modal participation factor, independent of joint coordinates,
[pic] — is the wind pressure correlation coefficient for the first vibration mode; for higher modes it is equal to unity. The coefficient [pic] is obtained according to Table 11.6 of the Code.
1.4 Notation
In subsequent parts of the Report, the following notation is used.
i — vibration mode number,
j — number of DOF,
[pic] — wind contribution area, m2,
[pic] — reference height of the joint according to design code, 11.1.4,
[pic] —wind pressure vertical variation coefficient according to Table 11.2 of the Code,
[pic] — wind pressure pulsation coefficient according to Table 11.4 of the Code,
[pic] — dynamic amplification factor, determined from the graph in Fig. 11.1 of the Code,
[pic] — mass, related to the j-th DOF, kg,
[pic] — component of the modal j-th DOF force for mode i, kN,
2. Full-scale building
TYPE OF THE PROBLEM: wind load generation on the full-scale building
REFERENCE: control hand calculations.
1. Main parameters of the building structure
The structure is situated in the III wind loading region, characteristic wind pressure is [pic] kPa. Terrain category is A. Wind is directed along the x-axis, the width of the wind-exposed side of the building is [pic] m. Building plan dimensions are [pic] m, height is H = 63.0 m.
Natural frequencies of the structure (Hz): [pic]. Logarithmic decrement is [pic]. The structural model of the building comprises 96 joints with the attached masses which represent design loading. Total mass of the structure is 2177370 kg.
According to the Table 11.5 of the Code, the limit natural frequency for buildings in the III wind region is [pic] Hz. Three lowest vibration modes shall be taken into account to assess the resonance effect of wind-induced vibrations.
[pic]
Fig. 2.1
To assess the possible interaction between the translational and rotational vibrations, the distribution of masses is assumed asymmetrical in the YZ plane. Mode shapes 1 and 3 are interconnected due to mass asymmetry. The mode 2 is pure translational mode in the YX plane (Fig. 2.3).
[pic]
Fig. 2.2 Along-wind vibration mode
[pic]
Fig. 2.3 Cross-wind vibration mode
[pic]
Fig. 2.4 Rotational vibration mode
2.2 Dynamic wind load analysis
[pic]
Fig. 2.5. Wind contribution areas
The main assumption of the analysis is that the building behaves as a rigid in the horizontal plane. The wind drag force, produced by the wind velocity pressure, is distributed over the joints of the windward façade of the building. The wind contribution areas in Fig. 5 are related to the windward joints of the building. But the aerodynamic wind force coefficients relate the wind velocity pressure to the wind drag force as a whole, including both windward and leeward façades.
Intermediate and final results of dynamic wind load generation are presented in the tables 1 and 2 below. The final results are the static wind forces [pic] in the Table 1, and the dynamic modal wind force components [pic] in the Table 2 (i = 1, … 3, j = 1, … 96). Since the number of modes is more than one, the modal effects (section forces, stresses or displacements) must be summed by use of the SRSS procedure.
It should be remembered that index values j here are the joint numbers, not the DOFs.
Table 2.1
[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |1 |12.60 |0.00 |18.00 |1.20000 |13.27430 |0.70400 |0.00000 |0.00000 |0.00000 | |2 |0.00 |0.00 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |3 |18.90 |4.20 |18.00 |1.20000 |13.28760 |0.70400 |0.15513 |-0.00468 |0.43985 | |4 |0.00 |4.20 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |5 |18.90 |8.40 |18.00 |1.20000 |13.28760 |0.70400 |0.39008 |-0.00599 |1.03105 | |6 |0.00 |8.40 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |7 |18.90 |12.60 |18.00 |1.20000 |13.28760 |0.70400 |0.67723 |-0.00571 |1.68034 | |8 |0.00 |12.60 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |9 |18.90 |16.80 |18.00 |1.20000 |13.28760 |0.70400 |1.00485 |-0.00571 |2.36228 | |10 |0.00 |16.80 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |11 |18.90 |21.00 |21.00 |1.26250 |13.97970 |0.68650 |1.41092 |-0.00566 |3.15829 | |12 |0.00 |21.00 |21.00 |1.26250 |0.00000 |0.68650 |0.00000 |0.00000 |0.00000 | |13 |18.90 |25.20 |25.20 |1.31500 |14.56100 |0.67180 |1.85878 |-0.00606 |3.98091 | |14 |0.00 |25.20 |25.20 |1.31500 |0.00000 |0.67180 |0.00000 |0.00000 |0.00000 | |15 |18.90 |29.40 |29.40 |1.36750 |15.14230 |0.65710 |2.35359 |-0.00597 |4.83690 | |16 |0.00 |29.40 |29.40 |1.36750 |0.00000 |0.65710 |0.00000 |0.00000 |0.00000 | |17 |18.90 |33.60 |33.60 |1.42000 |15.72370 |0.64240 |2.87857 |-0.00596 |5.69164 | |18 |0.00 |33.60 |33.60 |1.42000 |0.00000 |0.64240 |0.00000 |0.00000 |0.00000 | |19 |18.90 |37.80 |37.80 |1.47250 |16.30500 |0.62770 |3.44587 |-0.00532 |6.55754 | |20 |0.00 |37.80 |37.80 |1.47250 |0.00000 |0.62770 |0.00000 |0.00000 |0.00000 | |21 |18.90 |42.00 |42.00 |1.52000 |16.83100 |0.61600 |4.03525 |-0.00508 |7.40079 | |22 |0.00 |42.00 |42.00 |1.52000 |0.00000 |0.61600 |0.00000 |0.00000 |0.00000 | |23 |18.90 |46.20 |63.00 |1.72250 |19.07320 |0.57700 |4.89140 |-0.00462 |8.63792 | |24 |0.00 |46.20 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |25 |18.90 |50.40 |63.00 |1.72250 |19.07320 |0.57700 |5.49285 |-0.00418 |9.34566 | |26 |0.00 |50.40 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |27 |18.90 |54.60 |63.00 |1.72250 |19.07320 |0.57700 |6.10825 |-0.00418 |9.99696 | |28 |0.00 |54.60 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |29 |18.90 |58.80 |63.00 |1.72250 |19.07320 |0.57700 |6.69933 |-0.00253 |10.54990 | |30 |0.00 |58.80 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |31 |6.30 |63.00 |63.00 |1.72250 |6.35775 |0.57700 |2.41809 |-0.00015 |3.66842 | |32 |0.00 |63.00 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |33 |12.60 |0.00 |18.00 |1.20000 |13.27430 |0.70400 |0.00000 |0.00000 |0.00000 | |34 |0.00 |0.00 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |35 |37.80 |4.20 |18.00 |1.20000 |26.57520 |0.70400 |0.33279 |-0.00019 |0.02825 | |36 |0.00 |4.20 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |37 |37.80 |8.40 |18.00 |1.20000 |26.57520 |0.70400 |0.83064 |-0.00037 |0.07147 | |38 |0.00 |8.40 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |39 |37.80 |12.60 |18.00 |1.20000 |26.57520 |0.70400 |1.43725 |-0.00019 |0.12460 | |40 |0.00 |12.60 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |41 |37.80 |16.80 |18.00 |1.20000 |26.57520 |0.70400 |2.12442 |0.00019 |0.18597 | |42 |0.00 |16.80 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |43 |37.80 |21.00 |21.00 |1.26250 |27.95930 |0.68650 |2.97609 |0.00038 |0.26277 | |44 |0.00 |21.00 |21.00 |1.26250 |0.00000 |0.68650 |0.00000 |0.00000 |0.00000 | |45 |37.80 |25.20 |25.20 |1.31500 |29.12200 |0.67180 |3.91081 |0.00039 |0.34805 | |46 |0.00 |25.20 |25.20 |1.31500 |0.00000 |0.67180 |0.00000 |0.00000 |0.00000 | |47 |37.80 |29.40 |29.40 |1.36750 |30.28460 |0.65710 |4.94348 |0.00000 |0.44298 | |48 |0.00 |29.40 |29.40 |1.36750 |0.00000 |0.65710 |0.00000 |0.00000 |0.00000 | |49 |37.80 |33.60 |33.60 |1.42000 |31.44730 |0.64240 |6.03279 |-0.00040 |0.54444 | |50 |0.00 |33.60 |33.60 |1.42000 |0.00000 |0.64240 |0.00000 |0.00000 |0.00000 | |51 |37.80 |37.80 |37.80 |1.47250 |32.61000 |0.62770 |7.21251 |-0.00020 |0.65543 | |52 |0.00 |37.80 |37.80 |1.47250 |0.00000 |0.62770 |0.00000 |0.00000 |0.00000 | |53 |37.80 |42.00 |42.00 |1.52000 |33.66190 |0.61600 |8.42950 |0.00021 |0.77116 | |54 |0.00 |42.00 |42.00 |1.52000 |0.00000 |0.61600 |0.00000 |0.00000 |0.00000 | |55 |37.80 |46.20 |63.00 |1.72250 |38.14650 |0.57700 |10.20640 |0.00066 |0.93985 | |56 |0.00 |46.20 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |57 |37.80 |50.40 |63.00 |1.72250 |38.14650 |0.57700 |11.44130 |0.00022 |1.05959 | |58 |0.00 |50.40 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |59 |37.80 |54.60 |63.00 |1.72250 |38.14650 |0.57700 |12.70790 |-0.00044 |1.18307 | |60 |0.00 |54.60 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |61 |37.80 |58.80 |63.00 |1.72250 |38.14650 |0.57700 |13.91720 |-0.00022 |1.30148 | |62 |0.00 |58.80 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |63 |25.20 |63.00 |63.00 |1.72250 |25.43100 |0.57700 |10.03450 |0.00029 |0.94161 | |64 |0.00 |63.00 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |65 |12.60 |0.00 |18.00 |1.20000 |13.27430 |0.70400 |0.00000 |0.00000 |0.00000 | |66 |0.00 |0.00 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |67 |18.90 |4.20 |18.00 |1.20000 |13.28760 |0.70400 |0.17673 |0.00440 |-0.41160 | |68 |0.00 |4.20 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |69 |18.90 |8.40 |18.00 |1.20000 |13.28760 |0.70400 |0.44049 |0.00561 |-0.95977 | |70 |0.00 |8.40 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |71 |18.90 |12.60 |18.00 |1.20000 |13.28760 |0.70400 |0.75936 |0.00552 |-1.55584 | |72 |0.00 |12.60 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |73 |18.90 |16.80 |18.00 |1.20000 |13.28760 |0.70400 |1.12037 |0.00580 |-2.17641 | |74 |0.00 |16.80 |18.00 |1.20000 |0.00000 |0.70400 |0.00000 |0.00000 |0.00000 | |75 |18.90 |21.00 |21.00 |1.26250 |13.97970 |0.68650 |1.56538 |0.00605 |-2.89581 | |76 |0.00 |21.00 |21.00 |1.26250 |0.00000 |0.68650 |0.00000 |0.00000 |0.00000 | |77 |18.90 |25.20 |25.20 |1.31500 |14.56100 |0.67180 |2.05355 |0.00636 |-3.63316 | |78 |0.00 |25.20 |25.20 |1.31500 |0.00000 |0.67180 |0.00000 |0.00000 |0.00000 | |79 |18.90 |29.40 |29.40 |1.36750 |15.14230 |0.65710 |2.59032 |0.00597 |-4.39423 | |80 |0.00 |29.40 |29.40 |1.36750 |0.00000 |0.65710 |0.00000 |0.00000 |0.00000 | |81 |18.90 |33.60 |33.60 |1.42000 |15.72370 |0.64240 |3.15730 |0.00556 |-5.14741 | |82 |0.00 |33.60 |33.60 |1.42000 |0.00000 |0.64240 |0.00000 |0.00000 |0.00000 | |83 |18.90 |37.80 |37.80 |1.47250 |16.30500 |0.62770 |3.76707 |0.00512 |-5.90252 | |84 |0.00 |37.80 |37.80 |1.47250 |0.00000 |0.62770 |0.00000 |0.00000 |0.00000 | |85 |18.90 |42.00 |42.00 |1.52000 |16.83100 |0.61600 |4.39792 |0.00529 |-6.62984 | |86 |0.00 |42.00 |42.00 |1.52000 |0.00000 |0.61600 |0.00000 |0.00000 |0.00000 | |87 |18.90 |46.20 |63.00 |1.72250 |19.07320 |0.57700 |5.31486 |0.00517 |-7.69873 | |88 |0.00 |46.20 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |89 |18.90 |50.40 |63.00 |1.72250 |19.07320 |0.57700 |5.95114 |0.00440 |-8.28663 | |90 |0.00 |50.40 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |91 |18.90 |54.60 |63.00 |1.72250 |19.07320 |0.57700 |6.59842 |0.00374 |-8.81488 | |92 |0.00 |54.60 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |93 |18.90 |58.80 |63.00 |1.72250 |19.07320 |0.57700 |7.21623 |0.00231 |-9.24948 | |94 |0.00 |58.80 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |95 |6.30 |63.00 |63.00 |1.72250 |6.35775 |0.57700 |2.59761 |0.00029 |-3.19809 | |96 |0.00 |63.00 |63.00 |1.72250 |0.00000 |0.57700 |0.00000 |0.00000 |0.00000 | |∑ |1134.00 | | | |965.9131 | |188.06452 |0.00012 |17.24477 | |NOTE. In the above table 2.1, j values are referred to the X-direction, associated to j-th joint.
Table 2.2
[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |1 |0.00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 | |2 |0.00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 | |3 |16921.50 |1.7143E-01 |5.4165E-03 |-1.5082E-09 |1.9109E-07 |4.0020E-02 |-1.5899E-02 | |4 |16921.50 |1.7143E-01 |-5.4165E-03 |1.4780E-09 |1.9109E-07 |4.0020E-02 |1.5908E-02 | |5 |16921.50 |4.3105E-01 |1.1345E-02 |-1.9305E-09 |4.1261E-07 |9.3812E-02 |-3.6803E-02 | |6 |16921.50 |4.3105E-01 |-1.1345E-02 |1.9003E-09 |4.1267E-07 |9.3812E-02 |3.6803E-02 | |7 |16921.50 |7.4836E-01 |1.8446E-02 |-1.8400E-09 |6.4384E-07 |1.5289E-01 |-5.9537E-02 | |8 |16921.50 |7.4836E-01 |-1.8446E-02 |1.8400E-09 |6.4387E-07 |1.5289E-01 |5.9528E-02 | |9 |16921.50 |1.1104E+00 |2.5326E-02 |-1.8400E-09 |8.7716E-07 |2.1494E-01 |-8.3292E-02 | |10 |16921.50 |1.1104E+00 |-2.5326E-02 |1.8701E-09 |8.7713E-07 |2.1494E-01 |8.3292E-02 | |11 |16921.50 |1.5197E+00 |3.3231E-02 |-1.7797E-09 |1.1158E-06 |2.8010E-01 |-1.0812E-01 | |12 |16921.50 |1.5197E+00 |-3.3158E-02 |1.8400E-09 |1.1157E-06 |2.8010E-01 |1.0813E-01 | |13 |16921.50 |1.9642E+00 |4.0697E-02 |-1.8701E-09 |1.3507E-06 |3.4638E-01 |-1.3327E-01 | |14 |16921.50 |1.9642E+00 |-4.0697E-02 |1.9003E-09 |1.3507E-06 |3.4638E-01 |1.3327E-01 | |15 |16921.50 |2.4451E+00 |4.8822E-02 |-1.8098E-09 |1.5846E-06 |4.1375E-01 |-1.5872E-01 | |16 |16921.50 |2.4451E+00 |-4.8895E-02 |1.7495E-09 |1.5846E-06 |4.1375E-01 |1.5871E-01 | |17 |16921.50 |2.9459E+00 |5.5995E-02 |-1.7797E-09 |1.8088E-06 |4.7960E-01 |-1.8346E-01 | |18 |16921.50 |2.9459E+00 |-5.6069E-02 |1.6892E-09 |1.8088E-06 |4.7960E-01 |1.8345E-01 | |19 |16921.50 |3.4803E+00 |6.4559E-02 |-1.5685E-09 |2.0320E-06 |5.4534E-01 |-2.0802E-01 | |20 |16921.50 |3.4803E+00 |-6.4486E-02 |1.5685E-09 |2.0320E-06 |5.4534E-01 |2.0803E-01 | |21 |16921.50 |4.0232E+00 |7.1220E-02 |-1.4780E-09 |2.2390E-06 |6.0756E-01 |-2.3111E-01 | |22 |16921.50 |4.0232E+00 |-7.1074E-02 |1.5987E-09 |2.2391E-06 |6.0756E-01 |2.3113E-01 | |23 |16921.50 |4.5944E+00 |7.9199E-02 |-1.2669E-09 |2.4396E-06 |6.6805E-01 |-2.5343E-01 | |24 |16921.50 |4.5944E+00 |-7.9272E-02 |1.4177E-09 |2.4396E-06 |6.6805E-01 |2.5341E-01 | |25 |16921.50 |5.1593E+00 |8.5054E-02 |-1.1462E-09 |2.6179E-06 |7.2278E-01 |-2.7340E-01 | |26 |16921.50 |5.1593E+00 |-8.5201E-02 |1.1161E-09 |2.6178E-06 |7.2278E-01 |2.7338E-01 | |27 |16921.50 |5.7374E+00 |9.2008E-02 |-1.1462E-09 |2.7809E-06 |7.7315E-01 |-2.9158E-01 | |28 |16921.50 |5.7374E+00 |-9.1935E-02 |9.0491E-10 |2.7808E-06 |7.7315E-01 |2.9160E-01 | |29 |16921.50 |6.2926E+00 |9.7132E-02 |-6.9376E-10 |2.9145E-06 |8.1591E-01 |-3.0679E-01 | |30 |16921.50 |6.2926E+00 |-9.7059E-02 |5.7311E-10 |2.9145E-06 |8.1591E-01 |3.0681E-01 | |31 |16921.50 |6.8138E+00 |1.0218E-01 |-1.2065E-10 |3.0163E-06 |8.5113E-01 |-3.1871E-01 | |32 |16921.50 |6.8138E+00 |-1.0226E-01 |3.0164E-10 |3.0163E-06 |8.5113E-01 |3.1869E-01 | |33 |0.00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 | |34 |0.00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 | |35 |36289.50 |3.9432E-01 |1.0046E-02 |-6.4688E-11 |3.9984E-07 |2.7562E-03 |-3.2546E-02 | |36 |36289.50 |3.9432E-01 |-1.0046E-02 |-6.4688E-11 |3.9990E-07 |2.7562E-03 |3.2546E-02 | |37 |36289.50 |9.8424E-01 |2.3546E-02 |-1.2938E-10 |8.7336E-07 |6.9727E-03 |-7.6627E-02 | |38 |36289.50 |9.8424E-01 |-2.3546E-02 |-1.2938E-10 |8.7342E-07 |6.9727E-03 |7.6627E-02 | |39 |36289.50 |1.7030E+00 |3.8459E-02 |-6.4688E-11 |1.3695E-06 |1.2157E-02 |-1.2509E-01 | |40 |36289.50 |1.7030E+00 |-3.8616E-02 |-6.4688E-11 |1.3696E-06 |1.2157E-02 |1.2507E-01 | |41 |36289.50 |2.5173E+00 |5.4314E-02 |6.4688E-11 |1.8699E-06 |1.8144E-02 |-1.7576E-01 | |42 |36289.50 |2.5173E+00 |-5.4314E-02 |6.4688E-11 |1.8699E-06 |1.8144E-02 |1.7574E-01 | |43 |36289.50 |3.4373E+00 |7.0953E-02 |1.2938E-10 |2.3821E-06 |2.4989E-02 |-2.2882E-01 | |44 |36289.50 |3.4373E+00 |-7.0796E-02 |1.2938E-10 |2.3820E-06 |2.4989E-02 |2.2884E-01 | |45 |36289.50 |4.4314E+00 |8.7749E-02 |1.2938E-10 |2.8861E-06 |3.2473E-02 |-2.8258E-01 | |46 |36289.50 |4.4314E+00 |-8.7749E-02 |6.4688E-11 |2.8861E-06 |3.2473E-02 |2.8260E-01 | |47 |36289.50 |5.5070E+00 |1.0486E-01 |0.0000E+00 |3.3883E-06 |4.0632E-02 |-3.3712E-01 | |48 |36289.50 |5.5070E+00 |-1.0502E-01 |-6.4688E-11 |3.3882E-06 |4.0632E-02 |3.3708E-01 | |49 |36289.50 |6.6201E+00 |1.2181E-01 |-1.2938E-10 |3.8695E-06 |4.9193E-02 |-3.9013E-01 | |50 |36289.50 |6.6201E+00 |-1.2181E-01 |-1.2938E-10 |3.8695E-06 |4.9193E-02 |3.9009E-01 | |51 |36289.50 |7.8113E+00 |1.3877E-01 |-6.4688E-11 |4.3490E-06 |5.8447E-02 |-4.4282E-01 | |52 |36289.50 |7.8113E+00 |-1.3861E-01 |-6.4688E-11 |4.3491E-06 |5.8447E-02 |4.4286E-01 | |53 |36289.50 |9.0120E+00 |1.5462E-01 |6.4688E-11 |4.7937E-06 |6.7884E-02 |-4.9242E-01 | |54 |36289.50 |9.0120E+00 |-1.5446E-01 |6.4688E-11 |4.7937E-06 |6.7884E-02 |4.9245E-01 | |55 |36289.50 |1.0280E+01 |1.7000E-01 |1.9407E-10 |5.2249E-06 |7.7941E-02 |-5.4042E-01 | |56 |36289.50 |1.0280E+01 |-1.7016E-01 |1.2938E-10 |5.2249E-06 |7.7941E-02 |5.4039E-01 | |57 |36289.50 |1.1523E+01 |1.8398E-01 |6.4688E-11 |5.6083E-06 |8.7871E-02 |-5.8348E-01 | |58 |36289.50 |1.1523E+01 |-1.8429E-01 |6.4688E-11 |5.6083E-06 |8.7871E-02 |5.8345E-01 | |59 |36289.50 |1.2799E+01 |1.9716E-01 |-1.2938E-10 |5.9591E-06 |9.8111E-02 |-6.2271E-01 | |60 |36289.50 |1.2799E+01 |-1.9700E-01 |-1.2938E-10 |5.9590E-06 |9.8111E-02 |6.2275E-01 | |61 |36289.50 |1.4017E+01 |2.0799E-01 |-6.4688E-11 |6.2475E-06 |1.0793E-01 |-6.5562E-01 | |62 |36289.50 |1.4017E+01 |-2.0768E-01 |-1.2938E-10 |6.2475E-06 |1.0793E-01 |6.5566E-01 | |63 |36289.50 |1.5160E+01 |2.1663E-01 |1.2938E-10 |6.4677E-06 |1.1713E-01 |-6.8249E-01 | |64 |36289.50 |1.5160E+01 |-2.1678E-01 |6.4688E-11 |6.4678E-06 |1.1713E-01 |6.8245E-01 | |65 |0.00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 | |66 |0.00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 |0.0000E+00 | |67 |19368.00 |2.2352E-01 |4.9430E-03 |1.6227E-09 |2.1871E-07 |-4.2864E-02 |-1.8208E-02 | |68 |19368.00 |2.2352E-01 |-4.9430E-03 |-1.7953E-09 |2.1875E-07 |-4.2864E-02 |1.8217E-02 | |69 |19368.00 |5.5713E-01 |1.2902E-02 |2.0715E-09 |4.7230E-07 |-9.9952E-02 |-4.2134E-02 | |70 |19368.00 |5.5713E-01 |-1.2986E-02 |-2.2786E-09 |4.7233E-07 |-9.9952E-02 |4.2134E-02 | |71 |19368.00 |9.6044E-01 |2.0861E-02 |2.0370E-09 |7.3693E-07 |-1.6203E-01 |-6.8154E-02 | |72 |19368.00 |9.6044E-01 |-2.0945E-02 |-2.1751E-09 |7.3696E-07 |-1.6203E-01 |6.8145E-02 | |73 |19368.00 |1.4170E+00 |2.9909E-02 |2.1405E-09 |1.0040E-06 |-2.2666E-01 |-9.5344E-02 | |74 |19368.00 |1.4170E+00 |-2.9909E-02 |-2.1060E-09 |1.0040E-06 |-2.2666E-01 |9.5344E-02 | |75 |19368.00 |1.9299E+00 |3.8706E-02 |2.1751E-09 |1.2771E-06 |-2.9395E-01 |-1.2376E-01 | |76 |19368.00 |1.9299E+00 |-3.8706E-02 |-1.9679E-09 |1.2771E-06 |-2.9395E-01 |1.2377E-01 | |77 |19368.00 |2.4838E+00 |4.8173E-02 |2.2441E-09 |1.5460E-06 |-3.6182E-01 |-1.5255E-01 | |78 |19368.00 |2.4838E+00 |-4.8173E-02 |-2.0715E-09 |1.5460E-06 |-3.6182E-01 |1.5255E-01 | |79 |19368.00 |3.0801E+00 |5.7221E-02 |2.0715E-09 |1.8137E-06 |-4.3023E-01 |-1.8169E-01 | |80 |19368.00 |3.0801E+00 |-5.7221E-02 |-2.0370E-09 |1.8137E-06 |-4.3023E-01 |1.8167E-01 | |81 |19368.00 |3.6983E+00 |6.6940E-02 |1.8989E-09 |2.0703E-06 |-4.9645E-01 |-2.1000E-01 | |82 |19368.00 |3.6983E+00 |-6.7023E-02 |-2.0715E-09 |2.0703E-06 |-4.9645E-01 |2.0998E-01 | |83 |19368.00 |4.3548E+00 |7.5317E-02 |1.7262E-09 |2.3258E-06 |-5.6184E-01 |-2.3811E-01 | |84 |19368.00 |4.3548E+00 |-7.5234E-02 |-1.8989E-09 |2.3259E-06 |-5.6184E-01 |2.3813E-01 | |85 |19368.00 |5.0188E+00 |8.4701E-02 |1.7608E-09 |2.5628E-06 |-6.2296E-01 |-2.6453E-01 | |86 |19368.00 |5.0188E+00 |-8.4617E-02 |-1.7953E-09 |2.5628E-06 |-6.2296E-01 |2.6455E-01 | |87 |19368.00 |5.7139E+00 |9.1906E-02 |1.6227E-09 |2.7924E-06 |-6.8149E-01 |-2.9009E-01 | |88 |19368.00 |5.7139E+00 |-9.1989E-02 |-1.4846E-09 |2.7924E-06 |-6.8149E-01 |2.9008E-01 | |89 |19368.00 |6.3980E+00 |1.0003E-01 |1.3810E-09 |2.9964E-06 |-7.3354E-01 |-3.1296E-01 | |90 |19368.00 |6.3980E+00 |-1.0020E-01 |-1.2429E-09 |2.9964E-06 |-7.3354E-01 |3.1293E-01 | |91 |19368.00 |7.0938E+00 |1.0598E-01 |1.1738E-09 |3.1830E-06 |-7.8030E-01 |-3.3378E-01 | |92 |19368.00 |7.0938E+00 |-1.0590E-01 |-1.1738E-09 |3.1830E-06 |-7.8030E-01 |3.3380E-01 | |93 |19368.00 |7.7580E+00 |1.1168E-01 |7.2502E-10 |3.3359E-06 |-8.1877E-01 |-3.5119E-01 | |94 |19368.00 |7.7580E+00 |-1.1151E-01 |-7.5954E-10 |3.3359E-06 |-8.1877E-01 |3.5121E-01 | |95 |19368.00 |8.3779E+00 |1.1461E-01 |2.7620E-10 |3.4524E-06 |-8.4929E-01 |-3.6485E-01 | |96 |19368.00 |8.3779E+00 |-1.1469E-01 |-2.7620E-10 |3.4525E-06 |-8.4929E-01 |3.6484E-01 | |∑ |2177370.00 |4.2540E+02 |-1.6850E-04 |-3.0200E-10 |2.2300E-04 |1.2918E+00 |-3.4000E-05 | |NOTE In the tables 2.1 and 2.2 the quantities j have different meaning, than in the text; They are the joint numbers, not DOFs. To distinguish the different DOFs for each joint, additional notation jX and jZ, relating to x- and z-direction is used.
3. Main loading parameters and computation formulas
Mean wind force is
[pic]
[pic] — aerodynamic force coefficient, calculated in accordance with the Annex D of the Code. For rectangular building, [pic] depends on the ratio [pic] and according to plot in the Fig.D.19 equals to [pic]. End effect coefficient [pic] from the plot in Fig.D.23, when the height/width ratio is [pic].
Dynamic amplification factor for wind resonance effects, [pic] is determined from the graph as a function of the argument by the eq (11.8) of the Code:
[pic]
where [pic] — characteristic wind pressure, Pa,
[pic], H — height of the structure,
[pic] — partial load reliability coefficient.
Modal dynamic wind forces on the k-th DOF, corresponding to the i-th natural mode, are
[pic]
where
[pic]
is modal factor, independent of joint coordinates,
[pic] — is the wind pressure correlation coefficient for the first vibration mode; for higher modes it is equal to unity. The coefficient [pic]is obtained according to Table 11.6 of the Code.
In Table 2.1, the quantities
[pic]
are contributions of each joint to the modal factors [pic].
[pic]
[pic]
[pic]
[pic], j values here are referred to the joint numbers.
3 Full-scale lattice steel tower
TYPE OF THE PROBLEM: wind load generation on the full-scale lattice steel structure.
REFERENCE: control hand calculations for selected DOFs of the system.
3.1 Main parameters of the structure
Height of the tower H = 45.0 m. Wind is directed along the x-axis. Terrain category is A. The structure is situated in the III wind loading region, characteristic wind pressure is [pic] kPa. The structural model of the building comprises 64 joints with the attached masses. It is assumed that that asymmetric behavior of the structure can be neglected and only pure translational vibration in the XY plane is considered. Wind-exposed envelope region of the tower, divided into joint-related contributory areas with x-directed notional mass forces, is shown in the figure below.
[pic]
Fig. 3.1. Wind contributory areas and notional mass forces
Natural frequencies of the structure (Hz) are [pic]. Logarithmic decrement for steel structure is [pic]. According to the Table 11.5 of the SP 2011, the limit natural frequency for steel structures, in the III wind region is [pic] Hz. Only the first vibration mode should be taken into account to assess the resonance effect of wind-induced vibrations.
[pic]
Fig. 3.2. Fundamental mode shape. Natural frequency 2.299 Hz
Wind drag forces, applied to the wind exposed joints, are determined by the exposure factors, which reduce the gross area of the wind exposed envelope of the lattice structure (Fig. 3.1) to the effective wind contributory area, including main structural members, gusset plates, service elements (such as ladders, platforms, railings) and special equipment. Exposure factors are supplied by the user and should be implicitly multiplied by aerodynamic wind force coefficients for spatial lattice structures defined in the Annex D.1.14 of the design code.
3.2 Dynamic wind load analysis
Intermediate and final results of dynamic wind load generation are presented in the Table 3.1 below. The final results are static wind forces [pic] and dynamic modal wind force [pic], kN. The joint numbers coincide with the DOF numbers, j = 1, … 64.
Table 3.1
[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |1 |0.000 |0.1781 |0.00 |0.7500 |0.000 |0.8500 |0.000 |0.0000 |0.0000 |0.000 | |2 |0.000 |0.1781 |0.00 |0.7500 |0.000 |0.8500 |0.000 |0.0000 |0.0000 |0.000 | |3 |12.604 |0.1781 |0.00 |0.7500 |1.072 |0.8500 |0.000 |0.0000 |0.0000 |0.000 | |4 |12.604 |0.1781 |0.00 |0.7500 |1.072 |0.8500 |0.000 |0.0000 |0.0000 |0.000 | |5 |0.000 |0.3161 |7.50 |0.8750 |0.000 |0.8050 |505.167 |0.0000 |0.0133 |0.009 | |6 |0.000 |0.3161 |7.50 |0.8750 |0.000 |0.8050 |505.167 |0.0000 |0.0133 |0.009 | |7 |7.708 |0.3161 |7.50 |0.8750 |0.810 |0.8050 |505.167 |0.0069 |0.0133 |0.009 | |8 |7.708 |0.3161 |7.50 |0.8750 |0.810 |0.8050 |505.167 |0.0069 |0.0133 |0.009 | |9 |0.000 |0.3161 |7.50 |0.8750 |0.000 |0.8050 |505.167 |0.0000 |0.0135 |0.009 | |10 |0.000 |0.3161 |7.50 |0.8750 |0.000 |0.8050 |505.167 |0.0000 |0.0133 |0.009 | |11 |30.833 |0.3161 |7.50 |0.8750 |3.241 |0.8050 |505.167 |0.0280 |0.0135 |0.009 | |12 |0.000 |0.3161 |7.50 |0.8750 |0.000 |0.8050 |505.167 |0.0000 |0.0133 |0.009 | |13 |0.000 |0.3976 |15.00 |1.1250 |0.000 |0.7250 |759.280 |0.0000 |0.1148 |0.120 | |14 |0.000 |0.3976 |15.00 |1.1250 |0.000 |0.7250 |759.280 |0.0000 |0.1148 |0.120 | |15 |15.391 |0.3976 |15.00 |1.1250 |2.616 |0.7250 |759.280 |0.1730 |0.1148 |0.120 | |16 |15.391 |0.3976 |15.00 |1.1250 |2.616 |0.7250 |759.280 |0.1730 |0.1148 |0.120 | |17 |0.000 |0.3776 |18.75 |1.2188 |0.000 |0.6988 |256.051 |0.0000 |0.1702 |0.060 | |18 |0.000 |0.3776 |18.75 |1.2188 |0.000 |0.6988 |256.051 |0.0000 |0.1702 |0.060 | |19 |3.073 |0.3776 |18.75 |1.2188 |0.537 |0.6988 |256.051 |0.0508 |0.1702 |0.060 | |20 |3.073 |0.3776 |18.75 |1.2188 |0.537 |0.6988 |256.051 |0.0508 |0.1702 |0.060 | |21 |0.000 |0.3776 |18.75 |1.2188 |0.000 |0.6988 |256.051 |0.0000 |0.1701 |0.060 | |22 |0.000 |0.3776 |18.75 |1.2188 |0.000 |0.6988 |256.051 |0.0000 |0.1698 |0.060 | |23 |12.292 |0.3776 |18.75 |1.2188 |2.150 |0.6988 |256.051 |0.2029 |0.1701 |0.060 | |24 |0.000 |0.3776 |18.75 |1.2188 |0.000 |0.6988 |256.051 |0.0000 |0.1698 |0.060 | |25 |0.000 |0.3820 |22.50 |1.2813 |0.000 |0.6813 |504.148 |0.0000 |0.2599 |0.181 | |26 |0.000 |0.3820 |22.50 |1.2813 |0.000 |0.6813 |504.148 |0.0000 |0.2599 |0.181 | |27 |8.438 |0.3820 |22.50 |1.2813 |1.569 |0.6813 |504.148 |0.2207 |0.2599 |0.181 | |28 |8.438 |0.3820 |22.50 |1.2813 |1.569 |0.6813 |504.148 |0.2207 |0.2599 |0.181 | |29 |0.000 |0.3796 |26.25 |1.3281 |0.000 |0.6681 |251.055 |0.0000 |0.3529 |0.122 | |30 |0.000 |0.3796 |26.25 |1.3281 |0.000 |0.6681 |251.055 |0.0000 |0.3529 |0.122 | |31 |2.552 |0.3796 |26.25 |1.3281 |0.489 |0.6681 |251.055 |0.0916 |0.3529 |0.122 | |32 |2.552 |0.3796 |26.25 |1.3281 |0.489 |0.6681 |251.055 |0.0916 |0.3529 |0.122 | |33 |0.000 |0.3796 |26.25 |1.3281 |0.000 |0.6681 |251.055 |0.0000 |0.3529 |0.122 | |34 |0.000 |0.3796 |26.25 |1.3281 |0.000 |0.6681 |251.055 |0.0000 |0.3525 |0.122 | |35 |10.208 |0.3796 |26.25 |1.3281 |1.956 |0.6681 |251.055 |0.3663 |0.3529 |0.122 | |36 |0.000 |0.3796 |26.25 |1.3281 |0.000 |0.6681 |251.055 |0.0000 |0.3525 |0.122 | |37 |0.000 |0.2993 |30.00 |1.3750 |0.000 |0.6550 |500.171 |0.0000 |0.4726 |0.327 | |38 |0.000 |0.2993 |30.00 |1.3750 |0.000 |0.6550 |500.171 |0.0000 |0.4726 |0.327 | |39 |6.875 |0.2993 |30.00 |1.3750 |1.075 |0.6550 |500.171 |0.2644 |0.4726 |0.327 | |40 |6.875 |0.2993 |30.00 |1.3750 |1.075 |0.6550 |500.171 |0.2644 |0.4726 |0.327 | |41 |0.000 |0.4029 |33.75 |1.4219 |0.000 |0.6419 |249.320 |0.0000 |0.5943 |0.205 | |42 |0.000 |0.4029 |33.75 |1.4219 |0.000 |0.6419 |249.320 |0.0000 |0.5943 |0.205 | |43 |2.031 |0.4029 |33.75 |1.4219 |0.442 |0.6419 |249.320 |0.1340 |0.5943 |0.205 | |44 |2.031 |0.4029 |33.75 |1.4219 |0.442 |0.6419 |249.320 |0.1340 |0.5943 |0.205 | |45 |0.000 |0.4029 |33.75 |1.4219 |0.000 |0.6419 |249.320 |0.0000 |0.5945 |0.205 | |46 |0.000 |0.4029 |33.75 |1.4219 |0.000 |0.6419 |249.320 |0.0000 |0.5938 |0.205 | |47 |8.125 |0.4029 |33.75 |1.4219 |1.769 |0.6419 |249.320 |0.5361 |0.5945 |0.205 | |48 |0.000 |0.4029 |33.75 |1.4219 |0.000 |0.6419 |249.320 |0.0000 |0.5938 |0.205 | |49 |0.000 |0.4259 |37.50 |1.4688 |0.000 |0.6288 |498.131 |0.0000 |0.7313 |0.503 | |50 |0.000 |0.4259 |37.50 |1.4688 |0.000 |0.6288 |498.131 |0.0000 |0.7313 |0.503 | |51 |5.313 |0.4259 |37.50 |1.4688 |1.263 |0.6288 |498.131 |0.4613 |0.7313 |0.503 | |52 |5.313 |0.4259 |37.50 |1.4688 |1.263 |0.6288 |498.131 |0.4613 |0.7313 |0.503 | |53 |0.000 |0.4594 |41.25 |1.5125 |0.000 |0.6175 |298.063 |0.0000 |0.8675 |0.357 | |54 |0.000 |0.4594 |41.25 |1.5125 |0.000 |0.6175 |298.063 |0.0000 |0.8675 |0.357 | |55 |1.510 |0.4594 |41.25 |1.5125 |0.399 |0.6175 |298.063 |0.1696 |0.8675 |0.357 | |56 |1.510 |0.4594 |41.25 |1.5125 |0.399 |0.6175 |298.063 |0.1696 |0.8675 |0.357 | |57 |0.000 |0.4594 |41.25 |1.5125 |0.000 |0.6175 |298.063 |0.0000 |0.9347 |0.385 | |58 |0.000 |0.4594 |41.25 |1.5125 |0.000 |0.6175 |298.063 |0.0000 |0.8667 |0.357 | |59 |6.042 |0.4594 |41.25 |1.5125 |1.595 |0.6175 |298.063 |0.7314 |0.9347 |0.385 | |60 |0.000 |0.4594 |41.25 |1.5125 |0.000 |0.6175 |298.063 |0.0000 |0.8667 |0.357 | |61 |0.000 |0.5555 |45.00 |1.5500 |0.000 |0.6100 |349.151 |0.0000 |1.0000 |0.483 | |62 |0.000 |0.5555 |45.00 |1.5500 |0.000 |0.6100 |349.151 |0.0000 |1.0000 |0.483 | |63 |2.005 |0.5555 |45.00 |1.5500 |0.656 |0.6100 |349.151 |0.3179 |1.0000 |0.483 | |64 |2.005 |0.5555 |45.00 |1.5500 |0.656 |0.6100 |349.151 |0.3179 |1.0000 |0.483 | |∑ |202.500 | | | |32.567 | |22920.767 |5.6446 | |12.544 | |
3.3 Main loading parameters and computation formulas
Mean wind forces acting on the wind exposed joints, are computed as
[pic]
where [pic] — mean wind pressure,
[pic] — wind contributory area, related to the j-joint.
Product [pic] is effective area, catching the wind pressure. The multiplier [pic] is supplied by the user as an exposure factor. It implies the wind force multiplier.
Dynamic amplification factor [pic] is determined from the graph, Fig. 11.1, as a function of the argument obtained by the eq (11.8) of the design code:
[pic]
taking [pic] for the steel structure.
Here [pic] — characteristic wind pressure, Pa,
[pic] m — reference height,
[pic] — partial load reliability coefficient,
[pic] — fundamental frequency.
X-components of modal dynamic wind forces on the k-th joint, corresponding to the i-th natural mode:
[pic]
Here only the fundamental mode i = 1 is considered.
This equation is rewritten in the form of
[pic]
where
[pic]
is modal factor, independent of joint coordinates,
[pic] — is the wind pressure correlation coefficient for the first vibration mode. The coefficient is obtained from the Table 11.6 of design code.
In Table 1, the quantities
[pic]
are contributions of each joint to the modal factor. Resulting modal factor is
[pic]
4. Reinforced concrete chimney
TYPE OF THE PROBLEM: along-wind vibration of cylindrical chimney.
REFERENCE: control hand calculation.
4.1 Main parameters of the structure
In this verification example, the dynamic calculation of the full-scale reinforced concrete chimney wind load is presented. Height of the chimney H = 415.0 m. Terrain category is A. The structure is situated in the II wind loading region, characteristic wind pressure is [pic] kPa. The structural model of the building comprises 10 joints with the attached masses connected by beam members with variable cross sections. The model of a structure with the attached mass forces is shown in the picture below. The outside diameters of the chimney are introduced through the exposure command lines.
[pic]
Fig. 4.1 Structural model; notional mass forces, expressed by weight units
Natural vibration frequencies of the chimney, are [pic], Hz. Limit frequency, according to the Table 11.5 of the Code is [pic] Hz, when [pic] kPa and [pic] for RC structure. Hence two lowest vibration modes should be taken into account.
[pic]
Fig. 4.2 Natural frequency [pic] Hz
[pic]
Fig. 4.3 Natural frequency [pic] Hz
2. Dynamic wind load analysis
The chimney is considered as stick-type cylindrical structure with zero horizontal dimensions of the structural model. The outside diameters of the chimney is defined as the set of exposure parameters. The roughness of the surface is defined as the CONFIGURATION parameter of load application command.
The summary of calculation is in the tables below. Wind contribution areas are determined as products [pic]. Wind pressure correlation coefficient [pic] is determined for an equivalent rectangular wind pressure area [pic], where [pic] m. According to the Table 11.6 of the Code, [pic]
Table 4.1 Calculation of static forces
[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |1 |0.0 |27.5 |1163.25 |0.750 |6.6066E+07 |0.6644 |248.7570 | |2 |55.0 |50.0 |1785.00 |1.650 |8.2702E+07 |0.6722 |593.9590 | |3 |100.0 |45.0 |1363.50 |2.000 |7.7280E+07 |0.6699 |548.0470 | |4 |145.0 |45.0 |1161.00 |2.225 |6.9405E+07 |0.6662 |516.2540 | |5 |190.0 |45.0 |958.50 |2.410 |5.9634E+07 |0.6607 |457.8910 | |6 |235.0 |45.0 |837.00 |2.590 |5.3985E+07 |0.6571 |427.3410 | |7 |280.0 |45.0 |715.50 |2.710 |4.7205E+07 |0.6521 |379.3030 | |8 |325.0 |45.0 |594.00 |2.750 |3.9477E+07 |0.6451 |316.1430 | |9 |370.0 |45.0 |472.50 |2.750 |3.1402E+07 |0.6359 |247.8650 | |10 |415.0 |22.5 |236.25 |2.750 |3.1402E+07 |0.6359 |123.9330 | |∑ | |415 |9286.5 | | | |3859.493 | |
Table 4.2 Calculation of dynamic forces
[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |1 |0.0 |0.8500 |0.0000E+00 |0.0000 |0.0000 |0.0000 |0.0000 | |2 |55.0 |0.5900 |9.7160E+06 |1.2192 |-9.0307 |12.0796 |56.6958 | |3 |100.0 |0.5400 |7.1100E+06 |3.6945 |-25.0488 |31.7173 |136.2680 | |4 |145.0 |0.5130 |5.3080E+06 |7.9052 |-47.0114 |56.6175 |213.3550 | |5 |190.0 |0.4940 |3.8905E+06 |13.4109 |-65.4685 |82.4253 |254.9750 | |6 |235.0 |0.4760 |2.7995E+06 |21.4263 |-77.1286 |105.3740 |240.3610 | |7 |280.0 |0.4640 |1.9465E+06 |30.2367 |-66.1342 |119.5010 |165.6250 | |8 |325.0 |0.4600 |1.4040E+06 |38.0219 |-27.1525 |131.1730 |59.3586 | |9 |370.0 |0.4600 |1.1725E+06 |42.8693 |33.0276 |157.5330 |-76.9069 | |10 |415.0 |0.4600 |5.5600E+05 |28.8713 |57.0090 |100.6200 |-125.8990 | |∑ | | |3.3903E+07 |187.6553 |-226.9381 |797.0407 |923.8325 | |
Notation in the tables:
[pic] — height of the interval, m,
[pic] — outside diameter, m,
Re — Reynolds number, [pic] according to D.1.11 of the Code.
[pic] — aerodynamic coefficient according to plot in Fig. D.17 of the Code, as a function of Re and relative roughness of the stack (assumed zero, CONFIGURATION = 0 in command line).
[pic] — end effect multiplier, D.1.15 of the Code.
See also notations in the sec. 2.4.
Static forces. The mean wind pressure [pic] is determined by the eq (11.5) of the Code. The static wind force acting on the node k on the wind-exposed surface of the structural model is
[pic].
Here [pic]— wind pressure contribution area, related to the k-th node,
[pic] — the aerodynamic coefficient calculated in accordance with the Annex D of the Code, applicable to both mean and pulsating pressure components.
Dynamic forces. Dynamic wind forces are the pulsating components of the wind load, multiplied by the dynamic amplification factor (DAF). Dynamic wind pressure is determined by the eq (11.7) of the Code:
[pic]
Here ξ is DAF, obtained from the digitized function shown in the plot of the Fig.11.1 of the Code. The function depends on the logarithmic decrement δ of a structure. For buildings and concrete structures δ = 0.30. The argument of the function is:
[pic]
where [pic] Pa is characteristic wind pressure,
[pic] m,
[pic] — load reliability coefficient,
[pic] Hz — the lowest natural frequency.
Resulting DAF is [pic].
Dynamic resonant response of the structure is obtained as the expansion into the natural vibration modes, defined by the eigenvectors [pic], where i are the natural vibration modes and j are the global numbers of the degrees of freedom. In this example, the vibrations only in x-direction are considered and the node numbers coincide with the global degrees of freedom.
Amplitude values of dynamic wind forces for the k-th degree of freedom, corresponding to the i-th natural mode, read as [2]:
[pic]. j, k = 1, …, N
Here N = 10 is total number of DOFs equal to the number of nodes, because only vibration in the x-direction is considered,
[pic] — the mass, related to the j–th node,
[pic]— dynamic amplification factor (DAF),
[pic] the wind pressure correlation coefficient for the first vibration mode; for higher modes it is equal to unity.
Above equation for dynamic modal forces is rewritten in the form of
[pic]. k = 1, …, N
where
[pic]
is the modal factor, not depending on the degrees of freedom.
5 Sign Board
TYPE OF THE PROBLEM: wind load generation on the plane structure
REFERENCE: control hand calculations for selected DOFs of the system.
1. Main parameters of the structure
Layout of the structure conforms with the picture D.2 in the Annex D.1.1 of design code, Fig. 5.1 below. Load generation method can be applied if the board is elevated above the earth surface not less than [pic]. It is assumed by default that the wind is directed along the x-axis and the plane structure is in the YZ plane. The compliance of the STAAD.Pro model with geometric requirements of the code is checked programmatically.
The resultant of the wind pressure shall be placed at a distance 0.25b from the centre of the board. The reference height for wind pressure is [pic].
[pic]
Fig. 5.1 Standard layout of the sign board.
Main dimensions of the structure are [pic] m, [pic] m, [pic] m.
The structural model of the sign board and the wind pressure contribution areas, related to the joints, 3 to 10, is shown in the picture below. The masses of the structure are distributed over the joints 3, 6, 7 and 10. Natural frequencies of the structure (Hz) are[pic]. Logarithmic decrement for the steel structure is [pic].
The building is situated in the III wind loading region, characteristic wind pressure is [pic] kPa. Category B of terrain conditions is assumed. According to the Table 11.5 of design code, the limit natural frequency for structures with [pic] in the III wind region is [pic] Hz. Hence, two lowest vibration modes shall be taken into account to assess the resonance effect of wind-induced vibrations.
These two mode shapes describe the interaction between the translational and rotational vibrations of the structure.
[pic]
Fig. 5.2 Structural model of the sign board
[pic]
Fig. 5.3 Translational mode shape.
[pic]
Fig. 5.4 Rotational mode shape.
2. Dynamic wind load analysis
Aerodynamic wind force coefficient for the sign board is taken [pic] where [pic] is the end effect multiplier, determined by the Annex D.1.15. In order to introduce the required wind pressure eccentricity, the wind contribution areas [pic] were changed by the multipliers [pic] where [pic] are joint coordinates, referenced to the centre of the wind-exposed surface.
Table 5.1
[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] | |1 |0.00 |-0.61896 |0.00000 |0.00000 |0.00000 |0.00000 |0.00000 |0.00000 |0.00000 | |2 |0.00 |0.61896 |0.00000 |0.00000 |0.00000 |0.00000 |0.00000 |0.00000 |0.00000 | |3 |0.75 |-0.99033 |0.00224 |0.09281 |0.20757 |0.00022 |0.00053 |0.20266 |-0.26733 | |4 |3.25 |-0.61896 |0.80385 |0.09073 |0.11125 |0.03627 |0.04876 |0.00000 |0.00000 | |5 |3.25 |0.61896 |3.41532 |0.09073 |-0.11125 |0.15409 |-0.20716 |0.00000 |0.00000 | |6 |0.75 |0.99033 |0.46052 |0.09281 |-0.20757 |0.04472 |-0.10966 |0.20266 |0.26733 | |7 |0.75 |-0.99033 |0.00224 |1.00000 |1.00000 |0.00255 |0.00280 |2.18358 |-1.28791 | |8 |3.25 |-0.61896 |0.80385 |0.99661 |0.63748 |0.43423 |0.30454 |0.00000 |0.00000 | |9 |3.25 |0.61896 |3.41532 |0.99661 |-0.63748 |1.84493 |-1.29392 |0.00000 |0.00000 | |10 |0.75 |0.99033 |0.46052 |1.00000 |-1.00000 |0.52520 |-0.57585 |2.18358 |1.28791 | |∑ |16.00 | |9.36386 | | |3.04221 |-1.82996 |4.77248 |0.00 | |
Notation in the table 5.1:
j — number of joint, corresponds to DOF number.
[pic] — wind contribution area, m2,
[pic] —wind pressure vertical variation coefficient according to Table 11.2 of design code,
[pic] — mean wind force, kN
[pic] — wind pressure pulsation coefficient according to Table 11.4 of design code,
[pic] — joint contribution to modal participation factor, kN,
[pic] — mass, related to the j-th joint, kg,
[pic] — mode shape vector,
[pic] — dynamic modal force at the j-th joint, kN,
here i = 1,2, are the mode numbers.
References
1. SP 20 13330.2011. Loads and Actions. Actualized edition. SNiP 2.01.07-85*. (In Russian)
2. E.Simiu, R.H.Scanlan. Wind Effects on Structures: an Introduction to Wind Engineering. Wiley, 1996.
3. D.E.Newland. An introduction to random vibrations, spectral and wavelet analysis. Longman, 1993, 477 p.
4. M.F.Barstein. Theoretical Basis for the Method Adopted in USSR for the Dynamic Design of Tall Slender Structures for Wind Effects. Wind Effects on Buildings and Structures. Vol. 2. Univ. of Toronto Press, 1968.
5. Kucherenko CNIISK USSR. Wind Actions on Buildings and Structures. Calculations Manual. Moscow, 1978. (In Russian).
6. A.G.Davenport. The Spectrum of Horizontal Gustiness Near the Ground in High Wind. J. Royal Meteorol. Soc. Vol. 87, 1961. 194 – 211.
7. CNIISK. Recommendations on elaborated dynamic calculation of buildings and structures under the action of pulsating component of wind load. Moscow, 1999 (In Russian).
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