Rok akademicki 2015/16

POLITECHNIKA POZNAŃSKA WYDZIAŁ BUDOWNICTWA I INŻYNIERI ŚRODOWISKA

Budownictwo niestacjonarne II stopnia semestr I

PROJEKT NR 2 Z MECHNIKI KONSTRUKCJI

OBLICZENIE NIEWYZNACZALNEGO ŁUKU PARABOLICZONEGO METODĄ SIŁ

Opracował :

Damian Jany

$$\frac{f}{L} = \frac{\frac{13}{4}}{13} = \frac{1}{4} \geq \frac{1}{5}\mathrm{,\ wiec\ luk\ jest\ wyniosly}$$

$$\frac{h}{L} = \frac{1}{12} \leq \frac{1}{10}\mathrm{wiec\ w\ obliczeniach\ uwzgledniamy\ tylko\ wplyw\ M}$$

Równanie łuku parabolicznego:

$$y = \frac{4f}{L^{2}} x \left( L - x \right) = \frac{4 \frac{13}{4}}{13^{2}} x \left( 13 - x \right) = \frac{1}{13} x \left( 13 - x \right)$$

Kąt nachylenia stycznej do krzywej w danym punkcie:

$$tg\Phi = y^{'} = \frac{\text{dy}}{\text{dx}} = \frac{4f}{L^{2}} \left( L - 2x \right) = \frac{4 \frac{13}{4}}{13^{2}} \left( 13 - 2x \right) = \frac{1}{13} \left( 13 - 2x \right) = 1 - \frac{2x}{13}$$

$$\Phi = arctg\left\lbrack 1 - \frac{2x}{13} \right\rbrack$$

SSN=2

Układ równań kanonicznych:

$$\left\{ \begin{matrix} \delta_{11} X_{1} + \delta_{12} X_{2} + \delta_{1P} = 0 \\ \delta_{21} X_{1} + \delta_{22} X_{2} + \delta_{2P} = 0 \\ \end{matrix} \right.\ $$

Układ podstawowy:

Stan X₁ = 1

M₁ + 1(2,3077−y) = 0

M₁ = y − 2, 3077

$$M_{1} = \frac{1}{13} x \left( 13 - x \right) - 2,3077 = - \frac{1}{13}x^{2} + x - 2,3077$$

Stan X₂ = 1

M₂ + 1(3−x) = 0

M₂ = −(3−x)

M₂ = x − 3

Stan P

$$M_{P} = \left\{ \begin{matrix} 30\ \mathrm{dla\ x \in < 0;1 >} \\ 0\ \mathrm{dla\ x \in < 1;3 >} \\ - 6 \frac{\left( x - 3 \right)^{2}}{2}\mathrm{dla\ x \in < 3};8\mathrm{>} \\ - 6 5 \left( x - 5,5 \right)\mathrm{dla\ x \in < 8};13\mathrm{>} \\ \end{matrix} \right.\ $$

Korzystam z metody Simpsona numerycznego całkowania:

$$\int_{a}^{b}{f\left( x \right)dx =}\frac{x}{3} \left( f_{0} + 4 f_{1} + 2 f_{2} + 4 f_{3} + \ldots + 2 f_{n - 2} + 4 f_{n - 1} + f_{n} \right)$$

x = 0, 5m

$$\delta_{11} = \sum\int_{x}^{}\frac{M_{1} M_{1}}{EI cos\Phi}dx = \frac{101,824}{\text{EI}}$$

$$\delta_{12} = \sum\int_{x}^{}\frac{M_{1} M_{2}}{EI cos\Phi}dx = \frac{- 66,825}{\text{EI}}$$

$$\delta_{22} = \sum\int_{x}^{}\frac{M_{2} M_{2}}{EI cos\Phi}dx = \frac{2495,786}{\text{EI}}$$

$$\delta_{1P} = \sum\int_{x}^{}\frac{M_{1} M_{P}}{EI cos\Phi}dx = \frac{3179,621}{\text{EI}}$$

$$\delta_{2P} = \sum\int_{x}^{}\frac{M_{2} M_{P}}{EI cos\Phi}dx = \frac{- 47773,437}{\text{EI}}$$

Otrzymane wartości podstawiam do równania kanonicznego:

$$\left\{ \begin{matrix} \frac{101,824}{\text{EI}} X_{1} + \frac{- 66,825}{\text{EI}} X_{2} + \frac{3179,621}{\text{EI}}0 \\ \frac{- 66,825}{\text{EI}} X_{1} + \frac{2495,786}{\text{EI}} X_{2} + \frac{- 47773,437}{\text{EI}} = 0 \\ \end{matrix} \right.\ $$

X₁ = −18, 998 kN

X₂ = 18, 633 kN

Kontrola kinematyczna:

$\overset{\overline{}}{M} = - 1 + \frac{1}{3}*x$

OSTATECZNY WYKRES

x	y	y' = tgφ	φ	1/cosφ	M1	M2	MP	d11	d22	d12	d1p	d2p	M	M	kontr. kin.
0	0,000	1,000	0,7854	1,414	-2,308	-3,000	30,000	7,531	12,728	9,791	-97,907	-127,279	17,943	-1,000	-25,375575
0,5	0,481	0,923	0,7454	1,361	-1,827	-2,500	30,000	4,542	8,506	6,216	-74,588	-102,068	18,126	-0,833	-20,556456
1	0,923	0,846	0,7023	1,310	-1,385	-2,000	30,000	2,511	5,240	3,628	-54,413	-78,597	19,039	-0,667	-16,627102
1	0,923	0,846	0,7023	1,310	-1,385	-2,000	0,000	2,511	5,240	3,628	0,000	0,000	-10,961	-0,667	9,57195377
1,5	1,327	0,769	0,6557	1,262	-0,981	-1,500	0,000	1,214	2,839	1,856	0,000	0,000	-9,317	-0,500	5,87702286
2	1,692	0,692	0,6055	1,216	-0,615	-1,000	0,000	0,461	1,216	0,748	0,000	0,000	-6,942	-0,333	2,81431676
2,5	2,019	0,615	0,5517	1,174	-0,288	-0,500	0,000	0,098	0,294	0,169	0,000	0,000	-3,836	-0,167	0,75073435
3	2,308	0,538	0,4939	1,136	0,000	0,000	0,000	0,000	0,000	0,000	0,000	0,000	0,000	0,000	0,000
3,5	2,558	0,462	0,4324	1,101	0,250	0,500	-0,750	0,069	0,275	0,138	-0,207	-0,413	3,817	0,167	0,70064019
4	2,769	0,385	0,3672	1,071	0,462	1,000	-3,000	0,228	1,071	0,494	-1,483	-3,214	6,865	0,333	2,45158777
4,5	2,942	0,308	0,2985	1,046	0,635	1,500	-6,750	0,421	2,354	0,996	-4,482	-10,593	9,143	0,500	4,78293468
5	3,077	0,231	0,2268	1,026	0,769	2,000	-12,000	0,607	4,105	1,579	-9,473	-24,631	10,652	0,667	7,28788762
5,5	3,173	0,154	0,1526	1,012	0,865	2,500	-18,750	0,758	6,324	2,189	-16,417	-47,426	11,392	0,833	9,60469086
6	3,231	0,077	0,0768	1,003	0,923	3,000	-27,000	0,855	9,027	2,777	-24,997	-81,239	11,362	1,000	11,3955987
6,5	3,250	0,000	0,0000	1,000	0,942	3,500	-36,750	0,888	12,250	3,298	-34,630	-128,625	10,563	1,167	12,3236892
7	3,231	-0,077	-0,0768	1,003	0,923	4,000	-48,000	0,855	16,047	3,703	-44,439	-192,567	8,995	1,333	12,0287556
7,5	3,173	-0,154	-0,1526	1,012	0,865	4,500	-60,750	0,758	20,488	3,940	-53,191	-276,591	6,658	1,500	10,1037804
8	3,077	-0,231	-0,2268	1,026	0,769	5,000	-75,000	0,607	25,657	3,947	-59,209	-384,856	3,551	1,667	6,07347193
8,5	2,942	-0,308	-0,2985	1,046	0,635	5,500	-90,000	0,421	31,650	3,652	-59,758	-517,902	0,425	1,833	0,81464114
9	2,769	-0,385	-0,3672	1,071	0,462	6,000	-105,000	0,228	38,571	2,967	-51,922	-674,991	-1,971	2,000	-4,2227878
9,5	2,558	-0,462	-0,4324	1,101	0,250	6,500	-120,000	0,069	46,533	1,790	-33,041	-859,069	-3,635	2,167	-8,6749734
10	2,308	-0,538	-0,4939	1,136	0,000	7,000	-135,000	0,000	55,652	0,000	0,000	-1073,289	-4,569	2,333	-12,109019
10,5	2,019	-0,615	-0,5517	1,174	-0,288	7,500	-150,000	0,098	66,048	-2,540	50,806	-1320,952	-4,773	2,500	-14,009518
11	1,692	-0,692	-0,6055	1,216	-0,615	8,000	-165,000	0,461	77,841	-5,988	123,497	-1605,464	-4,245	2,667	-13,768333
11,5	1,327	-0,769	-0,6557	1,262	-0,981	8,500	-180,000	1,214	91,153	-10,518	222,727	-1930,297	-2,987	2,833	-10,677158
12	0,923	-0,846	-0,7023	1,310	-1,385	9,000	-195,000	2,511	106,106	-16,324	353,687	-2298,967	-0,998	3,000	-3,9223096
12,5	0,481	-0,923	-0,7454	1,361	-1,827	9,500	-210,000	4,542	122,822	-23,620	522,118	-2715,012	1,721	3,167	7,41876935
13	0,000	-1,000	-0,7854	1,414	-2,308	10,000	-225,000	7,531	141,421	-32,636	734,303	-3181,981	5,172	3,333	24,379774
								101,824	2495,786	-66,825	3179,621	-47773,437			1,15E-12

	T	N
0	-0,258301	0,25830102
0,5	0,80542637	1,32161562
1	1,95237221	2,46718977
1,5	3,18549205	3,69775484
2	4,50587933	5,01414853
2,5	5, 91198021	6,41453388
3	7,3987148	7,8935298
3,5	6,23271991	10,6985226
4	4,97094284	13,1969679
4,5	3,61986415	15,3252145
5	2,19116063	17,0202467
5,5	0,70189464	18,2249183
6	-0,8260032	18,893748
6,5	-2,3670393	18,9982535
7	-3,894131	18,5306608
7,5	-5,3809239	17,5050677
8	-6,804005	15,9557442
8,5	-5,27726	13,932989
9	-3,7894006	14,348961
9,5	-2,3594365	14,7525897
10	-1,0012809	15,0758311
10,5	0,27610226	15,3271649
11	1,46810458	15,5157822
11,5	2,57365163	15,6508543
12	3,59433261	15,7410332
12,5	4,53359777	15,7941577
13	5,39608338	15,8171201

Va	18,633
Ha	-18,998