表格及公式演示

表格及公式测试

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表 1. 续表2-24

截面简图截面积（A）图示轴线至边缘距离（y1x）对于图示轴线的惯性矩、截面抵抗矩及回转半径（I、W、及i）
$\frac{h (a + b)}{2}$ $s = \frac{b}{2}$ $\begin{array}{l} I = \frac{h (b^{4} - a^{4})}{48 (b - a)} \\ W = \frac{h (b^{4} - a^{4})}{24 b (b - a)} \\ i = 0.204 \sqrt{b^{2} + a^{2}} \end{array}$
$\begin{array}{l} \frac{π (D^{2} - d^{2})}{4} = 0.785 (D^{2} - d^{2}) = π (R^{2} - r^{2}) \\ (D = 2 R d = 2 r) \end{array}$ $y = \frac{D}{2}$ $\begin{array}{l} I = \frac{π (D^{4} - d^{4})}{64} = 0.0491 (D^{4} - d^{4}) = \frac{π}{4} (R^{4} - r^{4}) \\ W = 0.0982 \frac{D^{4} - d^{4}}{D} = x \frac{R^{4} - r^{4}}{4 R} \\ i = \frac{\sqrt{D^{2} + d^{2}}}{4} \end{array}$
$\begin{array}{l} π D t \\ π (D - t) t \end{array}$ $y = \frac{D}{2}$ $\begin{array}{l} I \approx \frac{π D^{2}}{8} t \\ W \approx 0.7854 D^{2} t \\ i \approx 0.354 D \end{array}$

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图 1.
$= 0.2287 \frac{1}{m}$

对于钢筋混凝土结构,

G=0.425E。

3.分离双室箱形截面

现用示例来叙述其计算过程。

〔例3-31〕试计算例3-28所示截面的扇性特性(图3-47)。

解:

取箱梁顶板中心。作为参考旋转中心,列表计算扇性坐标ω(表3- 104) ,为此先计算左边一侧单室的ψ值,即 :

图 2.
$\begin{array}{l} ψ = \frac{2 F}{\oint \frac{d s}{δ}} = \frac{2 \times 3 \times 5}{\frac{5}{0.3} + \frac{5}{0.4} + 2 \times \frac{3}{0.4}} \\ = 0.6792 \end{array}$

表 2. 扁性坐标计算表3-104

点号板号 $ρ$ $\int_{0}^{s} ρ d s$ $\int_{0}^{s} \frac{d s}{δ}$ $ψ \int_{0}^{s} \frac{d s}{δ}$ $\bar{ω} = \int_{0}^{s} (ρ - \frac{φ}{δ}) d s$
1 0-1 0 0 0* 0 0
2 1-2 0 0 16.6667 11.3200 -11.3200
3 2-3 7.5 22.5 24.1667 21.3478 06524
4 3-4 3 37.5 36.6667 24.9040 12.5960
5 4-1 2.6 30.0 44.1667 30 0
6 2-5 - - - - -11.3200

板O-1段不在闭口箱上,故该项积分为O。

将表中值绘于图3-47b )上。

绘χ坐标图 (图3-47C)。按式 (3 -33 )计算惯性矩 Iy

板5-5' $I_{y (1)} = \frac{10 \times 10}{2} \times \frac{2}{3} \times 10 \times 0.3 \times 2 = 200 m^{4}$

板2-3 $I_{y (2)} = 7.5 \times 7.5 \times 3 \times 0.4 \times 2 = 135 m^{4}$

2'-3'

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截面积（A）	图示轴线至边缘距离（y1x）	对于图示轴线的惯性矩、截面抵抗矩及回转半径（I、W、及i）
$\frac{h (a + b)}{2}$	$s = \frac{b}{2}$	$\begin{array}{l} I = \frac{h (b^{4} - a^{4})}{48 (b - a)} \\ W = \frac{h (b^{4} - a^{4})}{24 b (b - a)} \\ i = 0.204 \sqrt{b^{2} + a^{2}} \end{array}$
$\begin{array}{l} \frac{π (D^{2} - d^{2})}{4} = 0.785 (D^{2} - d^{2}) = π (R^{2} - r^{2}) \\ (D = 2 R d = 2 r) \end{array}$	$y = \frac{D}{2}$	$\begin{array}{l} I = \frac{π (D^{4} - d^{4})}{64} = 0.0491 (D^{4} - d^{4}) = \frac{π}{4} (R^{4} - r^{4}) \\ W = 0.0982 \frac{D^{4} - d^{4}}{D} = x \frac{R^{4} - r^{4}}{4 R} \\ i = \frac{\sqrt{D^{2} + d^{2}}}{4} \end{array}$
$\begin{array}{l} π D t \\ π (D - t) t \end{array}$	$y = \frac{D}{2}$	$\begin{array}{l} I \approx \frac{π D^{2}}{8} t \\ W \approx 0.7854 D^{2} t \\ i \approx 0.354 D \end{array}$

点号	板号	$ρ$	$\int_{0}^{s} ρ d s$	$\int_{0}^{s} \frac{d s}{δ}$	$ψ \int_{0}^{s} \frac{d s}{δ}$	$\bar{ω} = \int_{0}^{s} (ρ - \frac{φ}{δ}) d s$
1	0-1	0	0	0*	0	0
2	1-2	0	0	16.6667	11.3200	-11.3200
3	2-3	7.5	22.5	24.1667	21.3478	06524
4	3-4	3	37.5	36.6667	24.9040	12.5960
5	4-1	2.6	30.0	44.1667	30	0
6	2-5	-	-	-	-	-11.3200