Effect of mass parameter on vibration displacement of beam member under air blast loading
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摘要: 为研究质量参数对空爆荷载梁构件振动位移的影响，以等效单自由度法为基本方法，理论推导了柔性及刚性2类梁构件含质量参数在弹性、塑性各阶段的振动位移解，选择矩形截面、圆截面为梁构件典型截面，设计并计算了质量参数在1.00~1.20范围内的13种典型工况。结果表明：对于空爆荷载柔性及刚性2类梁构件，增大其截面面积后，仅考虑质量参数计算振动位移时误差较大，应按质量参数及产生的附加刚度参数耦合效应分析；对于矩形截面梁构件，采用质量参数与刚度参数耦合计算时，其最大弹性位移、最大弹塑性位移和残余变形降低幅度分别约为仅采用质量参数计算结果的4.75、3.28和2.96倍；对于圆截面梁构件，则分别为3.57、2.56和2.32倍；该结论在柔性梁构件及刚性梁构件均适用且无明显差异。Abstract: By exploring the effect of mass parameter on the vibration displacement of beam members under air blast loading, an effective method was proposed to reduce the vibration displacement of beam members by increasing mass. An equivalent single degree of freedom (SDOF) system was used to analyze vibration displacement for beam members. The displacement formulas with mass parameter for flexible and rigid beam members in each stage under air blast loading were respectively established. These stages included elastic forced vibration, elastic free vibration, plastic forced vibration, plastic free vibration, and rebound vibration. Rectangular and circular sections were selected as typical cross sections of beam members, and 13 typical calculation cases with mass parameters ranging from 1.00 to 1.20 were designed. The vibration displacement-time history curves, maximum elastic displacement, maximum elastic-plastic displacement, and residual deformation were calculated and analyzed. Taking the data with a mass parameter value of 1.0 as the reference value, the displacement reduction rate of other calculation cases relative to the reference value could be obtained. The difference between the types of beam members for displacement reduction rate was further analyzed. The results are as follows. For flexible and rigid beam members subjected to air blast loading, increasing the cross-sectional area and considering only the mass parameter will result in a smaller reduction in vibration displacement. Therefore, the displacement should be analyzed according to the coupling effect of the mass parameter and additional stiffness parameter. For beam members with rectangular cross-sections, the reduction ranges of maximum elastic displacement, maximum elastic-plastic displacement, and residual deformation calculated from the coupled effect of mass parameter and stiffness parameter are about 4.75, 3.28, and 2.96 times that of mass parameter alone. For beams with circular cross-section, the data are 3.75, 2.56, and 2.32 times. These conclusions apply to both flexible beam members and rigid beam members, and there is no significant difference.
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Key words:
- air blast loading /
- mass parameter /
- beam member /
- dynamic response /
- stiffness /
- residual deformation
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图 1 工况C1～C5两类梁构件的弹塑性振动位移时程曲线
Figure 1. Displacement-time curves of elastoplastic vibration for beam members of cases C1–C5
图 2 工况C1与C6～C9两类梁构件的弹塑性振动位移时程曲线
Figure 2. Displacement-time curves of elastoplastic vibration for beam members of cases C1 and C6–C9
图 3 工况C1与C10～C13两类梁构件的弹塑性振动位移时程曲线
Figure 3. Displacement-time curves of elastoplastic vibration for beam members of cases C1 and C10–C13
图 4 本文中相对基准工况的位移降低率曲线
Figure 4. Displacement reduction ratios of the calculation cases relative to the reference case
表 1 质量参数典型工况
Table 1. Typical calculation cases for mass parameter
工况（矩形、圆） αm αk 工况（矩形） αm αk 工况（圆） αm αk C1 1.00 1.00 C6 1.05 1.16 C10 1.05 1.10 C2 1.05 1.00 C7 1.10 1.33 C11 1.10 1.21 C3 1.10 1.00 C8 1.15 1.52 C12 1.15 1.32 C4 1.15 1.00 C9 1.20 1.73 C13 1.20 1.44 C5 1.20 1.00 
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表 2
$\beta=2 $ 时位移降低率以及两类梁构件差异性结果Table 2. Displacement reduction ratios and difference between flexible beam members with
$\beta=2 $ and rigid beam members工况 柔性梁构件 刚性梁构件 Δ2T/% Δ2m/% Δ2r/% ΔT/% Δm/% Δr/% ΔT/% Δm/% Δr/% C2 2.16 2.51 2.85 1.09 2.77 4.45 1.07 −0.26 −1.60 C3 4.18 4.84 5.49 2.15 5.32 8.49 2.03 −0.48 −3.00 C4 6.08 7.01 7.93 3.19 7.69 12.18 2.89 −0.68 −4.25 C5 7.87 9.04 10.20 4.21 9.89 15.56 3.66 −0.85 −5.36 C6 9.78 9.11 8.44 11.80 8.59 5.38 −2.02 0.52 3.06 C7 18.24 17.01 15.78 21.85 16.05 10.24 −3.61 0.96 5.54 C8 25.60 23.91 22.22 30.46 22.57 14.68 −4.86 1.34 7.54 C9 32.04 29.97 27.89 37.87 28.32 18.76 −5.83 1.65 9.13 C10 7.30 6.96 6.62 8.33 6.70 5.07 −1.03 0.26 1.55 C11 13.77 13.15 12.52 15.66 12.66 9.66 −1.89 0.48 2.86 C12 19.54 18.67 17.80 22.13 17.99 13.85 −2.59 0.68 3.95 C13 24.70 23.62 22.55 27.88 22.78 17.68 −3.18 0.84 4.87
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表 3
$\beta=5 $ 时位移降低率以及两类梁构件差异性结果Table 3. Displacement reduction ratios and difference between flexible beam members with
$\beta=5 $ and rigid beam members工况 柔性梁构件 刚性梁构件 Δ5T/% Δ5m/% Δ5r/% ΔT/% Δm/% Δr/% ΔT/% Δm/% Δr/% C2 1.87 2.71 2.91 0.62 3.25 3.91 1.25 −0.54 −1.00 C3 3.64 5.21 5.60 1.23 6.25 7.50 2.41 −1.04 −1.90 C4 5.31 7.53 8.09 1.83 9.02 10.82 3.48 −1.49 −2.73 C5 6.89 9.70 10.40 2.41 11.58 13.88 4.48 −1.88 −3.48 C6 10.34 8.72 8.31 12.58 7.68 6.46 −2.24 1.04 1.85 C7 19.25 16.29 15.55 23.14 14.43 12.25 −3.89 1.86 3.30 C8 26.99 22.91 21.89 32.08 20.40 17.48 −5.09 2.51 4.41 C9 33.75 28.73 27.47 39.69 25.71 22.21 −5.94 3.02 5.26 C10 7.58 6.77 6.56 8.75 6.24 5.61 −1.17 0.53 0.95 C11 14.29 12.78 12.40 16.38 11.82 10.67 −2.09 0.96 1.73 C12 20.26 18.16 17.63 23.10 16.83 15.26 −2.84 1.33 2.37 C13 25.60 22.99 22.34 29.03 21.36 19.44 −3.43 1.63 2.90
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