列表计算两铰拱的内力

列表计算两铰拱的内力

郭昊宁 1533104 1153310415

本文将从一道实例题运用Excel 表格列表计算三铰拱的内力。 例题1：两铰拱及其所受荷载如图所示拱的轴线为抛物线方程

y =

4f

⋅x (l -x )l

试计算反力并绘制内力图。

题目分析：该题为超静定问题，可利用力法求解水平力Fh ，再利用求三铰拱内力时的公式

0-y ⎤⎧M 0⎫ ⎧M ⎫⎡1

⎪⎪⎢⎥⎪0⎪⎨F Q ⎬=⎢0cos ϕ-sin ϕ⎥⎨F Q ⎬ ⎪F ⎪⎢0-sin ϕ-cos ϕ⎥⎪F ⎪⎩N ⎭⎣⎦⎩H ⎭即可求出剪力、轴力、弯矩，再利用Excel 即可绘制相应的剪力图、轴力图和弯矩图。 具体步骤：

1. 画出对应简支梁弯矩图 2. 解力法方程 3. 解得Fh

4. 利用公式求剪力、轴力、弯矩

5. 利用坐标变换公式在两铰拱上绘制弯矩图。

由于第一部分存在积分电脑难以完成，所以用力法计算水平推力Fh 部分人工手写完成。 再利用手写得出的水平推力编程让电脑自动运算即可得出相应剪力、轴力、弯矩。

其他荷载同样可以采用本方法思路。

注：坐标变换公式为 X=x0-f*sinθ Y=y0+f*conθ

X0,y0为拱轴线坐标，f 为剪力、轴力、弯矩，θ为拱上点的切线与x 轴夹角。 X ，Y 为所得的坐标值

现将表格数据公布如下：

x

[***********]141516

y=x*(16-x)/16斜率K 剪力Fq0弯矩M0夹角

00.93751.752.4375

33.43753.753.9375

43.93753.753.4375

32.43751.750.9375

010.8750.750.6250.50.3750.250.125

0-0.13-0.25-0.38-0.5-0.63-0.75-0.88-17.57.57.57.57.5-2.5-2.5-2.5-2.5-2.5-2.5-2.5-2.5-2.5-2.5-2.5-2.507.51522.53027.52522.52017.51512.5107.552.500.78540.718830.64350.55860.463650.358770.244980.12435

0-0.1244-0.245-0.3588-0.4636-0.5586-0.6435-0.7188-0.7854

正弦0.707110.65850.60.530.447210.351120.242540.12403

0-0.124-0.2425-0.3511-0.4472-0.53-0.6-0.6585-0.7071余弦Fh=2850/512=5.5弯矩M 剪力Fq 轴力Fn 0.7071075.5664062501.367257253-9.2393444650.7525775.56640625-2.2814941411.978821045-9.127932176

0.85.56640625-5.2587890632.66015625-8.9531250.8479985.56640625-8.9318847663.409797868-8.6952951090.8944275.56640625-13.300781254.218831379-8.3328470720.9363295.56640625-8.365478516-4.295318664-4.3341799820.9701435.56640625-4.125976563-3.775408069-4.7938682140.9922785.56640625-0.582275391-3.171122414-5.213334938

15.566406252.265625-2.5-5.56640625

0.9922785.566406254.417724609-1.79026697-5.8335086110.9701435.566406255.874023438-1.075304431-6.0065463390.9363295.566406256.634521484-0.386327224-6.089797190.8944275.566406256.699218750.253304576-6.0967790950.8479985.566406256.0681152340.830193652-6.045300409

0.85.566406254.7412109381.33984375-5.9531250.7525775.566406252.7185058591.784062428-5.8354091370.7071075.5664062502.168276653-5.703810559

表1

剪力x 剪力y 轴力x 轴力y 弯矩x 弯矩y

-0.1933593750.1933593752.177734375-2.[1**********]0.7393874451.235342923.003595133-1.3523230091.150237440.7658000681.680781252.1756253.790625-0.63752.3155273441.3292968752.6385621493.0158005624.53616573-0.0203651693.4733889461.6800776873.622656253.75468755.24218750.5156254.5948290211.8103419595.3016374142.6331335625.5072773972.0847602745.2937315612.6542158386.1831341913.0174632356.3875612752.1997549026.100069633.3497214787.0786658653.3081730777.2155448722.2131410267.0072222373.879722101

83.582.1445312584.2265625

8.9555889423.5822115388.7588141032.0080128219.054795134.375861049.9478400743.5413602949.514399511.[1**********].142465994.[1**********].972870293.3651541110.287243151.[1**********].23295364.[1**********].022656253.045312511.091145831.[1**********].299598173.[1**********].088000352.[1**********].931999060.[1**********].321609462.[1**********].160781251.96437512.8093750.162514.284472662.[1**********].234962671.[1**********].71911873-0.[1**********].179014861.[1**********].306640630.[1**********].65559896-1.[1**********]0

表2

表1为运用Fh 计算剪力、轴力、弯矩的的过程

表2为将计算出的剪力、轴力、弯矩画在拱上的坐标变换

以下为Excel 绘制的剪力图、轴力图、弯矩图：

图1

图2 图3