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<title>chapter03_16</title>
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<h1 id="Chapter-3-:-Parallel-Forces-In-A-Plane">Chapter 3 : Parallel Forces In A Plane<a class="anchor-link" href="#Chapter-3-:-Parallel-Forces-In-A-Plane">¶</a></h1>
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<h2 id="Example-3.3-1,Page-No:64">Example 3.3-1,Page No:64<a class="anchor-link" href="#Example-3.3-1,Page-No:64">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">math</span>
<span class="c">#Initilization of variables</span>
<span class="n">W</span><span class="o">=</span><span class="mi">1000</span> <span class="c">#N</span>
<span class="n">Lab</span><span class="o">=</span><span class="mi">1</span> <span class="c">#m</span>
<span class="n">Lac</span><span class="o">=</span><span class="mf">0.6</span> <span class="c">#m</span>
<span class="n">theta</span><span class="o">=</span><span class="mi">60</span> <span class="c">#degree #angle made by the beam with the horizontal</span>
<span class="c">#Calculations</span>
<span class="n">Q</span><span class="o">=</span><span class="p">(</span><span class="n">W</span><span class="o">*</span><span class="n">Lac</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="o">*</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">180</span><span class="p">)))</span><span class="o">/</span><span class="p">(</span><span class="n">Lab</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="o">*</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">180</span><span class="p">)))</span> <span class="c">#N # from eq'n 2</span>
<span class="n">P</span><span class="o">=</span><span class="n">W</span><span class="o">-</span><span class="n">Q</span> <span class="c">#N # from eq'n 1</span>
<span class="c">#Results</span>
<span class="nb">print</span><span class="s">"The load taken by man P is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">P</span><span class="p">),</span><span class="s">"N"</span>
<span class="nb">print</span><span class="s">"The load taken by man Q is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Q</span><span class="p">),</span><span class="s">"N"</span>
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<pre>The load taken by man P is 400.0 N
The load taken by man Q is 600.0 N
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<h2 id="Example-3.3-2,Page-No:64">Example 3.3-2,Page No:64<a class="anchor-link" href="#Example-3.3-2,Page-No:64">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">math</span>
<span class="c">#Initilization of variables</span>
<span class="n">F</span><span class="o">=</span><span class="mi">1000</span> <span class="c">#N</span>
<span class="n">Lab</span><span class="o">=</span><span class="mi">1</span> <span class="c">#m</span>
<span class="n">Lbc</span><span class="o">=</span><span class="mf">0.25</span> <span class="c">#m</span>
<span class="n">Lac</span><span class="o">=</span><span class="mf">1.25</span> <span class="c">#m</span>
<span class="c">#Calculations</span>
<span class="n">Rb</span><span class="o">=</span><span class="p">(</span><span class="n">F</span><span class="o">*</span><span class="n">Lac</span><span class="p">)</span><span class="o">/</span><span class="n">Lab</span> <span class="c">#N # from eq'n 2</span>
<span class="n">Ra</span><span class="o">=</span><span class="n">Rb</span><span class="o">-</span><span class="n">F</span> <span class="c">#N # fom eq'n 1</span>
<span class="c">#Results</span>
<span class="nb">print</span><span class="s">"The reaction (downwards)at support A is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Ra</span><span class="p">),</span><span class="s">"N"</span>
<span class="nb">print</span><span class="s">"The reaction (upwards)at support B is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Rb</span><span class="p">),</span><span class="s">"N"</span>
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<pre>The reaction (downwards)at support A is 250.0 N
The reaction (upwards)at support B is 1250.0 N
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<h2 id="Example-3.3-3,Page-No:65">Example 3.3-3,Page No:65<a class="anchor-link" href="#Example-3.3-3,Page-No:65">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">math</span>
<span class="c">#Inilitization of variables</span>
<span class="n">Lab</span><span class="o">=</span><span class="mi">12</span> <span class="c">#m</span>
<span class="n">Mc</span><span class="o">=</span><span class="mi">40</span> <span class="c">#kN-m </span>
<span class="n">Md</span><span class="o">=</span><span class="mi">10</span> <span class="c">#kN-m</span>
<span class="n">Me</span><span class="o">=</span><span class="mi">20</span> <span class="c">#kN-m</span>
<span class="n">Fe</span><span class="o">=</span><span class="mi">20</span> <span class="c">#kN #force acting at point E</span>
<span class="c">#Calculations</span>
<span class="n">Xa</span><span class="o">=-</span><span class="p">(</span><span class="n">Fe</span><span class="p">)</span> <span class="c">#kN #take sum Fx=0</span>
<span class="n">a</span><span class="o">=</span><span class="n">Me</span><span class="o">+</span><span class="n">Md</span><span class="o">-</span><span class="n">Mc</span> <span class="c">#N #take moment at A</span>
<span class="n">Rb</span><span class="o">=</span><span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">Lab</span><span class="p">)</span><span class="o">**-</span><span class="mi">1</span>
<span class="n">Ya</span><span class="o">=-</span><span class="n">Rb</span> <span class="c">#N #take sum Fy=0</span>
<span class="c">#Results</span>
<span class="nb">print</span><span class="s">"The vertical reaction (upwards) at A is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Ya</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span><span class="s">"kN"</span>
<span class="nb">print</span><span class="s">"The horizontal reaction (towards A) is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Xa</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="s">"kN"</span>
<span class="nb">print</span><span class="s">"The reaction (downwards) at B is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Rb</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span><span class="s">"kN"</span>
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<pre>The vertical reaction (upwards) at A is 0.833 kN
The horizontal reaction (towards A) is -20.0 kN
The reaction (downwards) at B is -0.833 kN
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<h2 id="Example-3.3-5,Page-No:66">Example 3.3-5,Page No:66<a class="anchor-link" href="#Example-3.3-5,Page-No:66">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">math</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="c">#Initilization of variables</span>
<span class="n">W</span><span class="o">=</span><span class="mi">1000</span> <span class="c">#N</span>
<span class="n">Lad</span><span class="o">=</span><span class="mf">7.5</span> <span class="c">#m</span>
<span class="n">Lae</span><span class="o">=</span><span class="mf">1.5</span> <span class="c">#m</span>
<span class="n">La1</span><span class="o">=</span><span class="mf">3.75</span> <span class="c">#m #distance of 1st 1000N load from pt A</span>
<span class="n">La2</span><span class="o">=</span><span class="mi">5</span> <span class="c">#m #distance of 2nd 1000N load from pt A</span>
<span class="n">La3</span><span class="o">=</span><span class="mi">6</span> <span class="c">#m # distance of 3rd 1000N load from pt A</span>
<span class="c"># Calculations (part1)</span>
<span class="c">#using matrix to solve the given eqn's 1 & 2</span>
<span class="n">A</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span> <span class="p">,</span><span class="o">-</span><span class="mf">2.5</span><span class="p">],[</span><span class="mf">3.5</span> <span class="p">,</span><span class="o">-</span><span class="mi">5</span><span class="p">]])</span>
<span class="n">B</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1000</span><span class="p">,</span><span class="mi">7250</span><span class="p">])</span>
<span class="n">C</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">)</span>
<span class="c">#Resuts</span>
<span class="nb">print</span><span class="s">"The reaction at F i.e Rf is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">C</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span><span class="s">"N"</span>
<span class="nb">print</span><span class="s">"The reaction at D i.e Rd is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">C</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span><span class="s">"N"</span>
<span class="c">#Calculations (part 2)</span>
<span class="c">#Consider combined F.B.D of beams AB,BC &CD. Take moment at A</span>
<span class="n">Re</span><span class="o">=</span><span class="p">((</span><span class="n">W</span><span class="o">*</span><span class="n">La1</span><span class="p">)</span><span class="o">+</span><span class="p">(</span><span class="n">W</span><span class="o">*</span><span class="n">La2</span><span class="p">)</span><span class="o">+</span><span class="p">(</span><span class="n">W</span><span class="o">*</span><span class="n">La3</span><span class="p">)</span><span class="o">+</span><span class="p">(</span><span class="n">C</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">Lad</span><span class="p">)</span><span class="o">-</span><span class="p">(</span><span class="n">C</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">La3</span><span class="p">))</span><span class="o">/</span><span class="n">Lae</span> <span class="c">#N</span>
<span class="n">Ra</span><span class="o">=</span><span class="n">C</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">Re</span><span class="o">-</span><span class="n">C</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">W</span><span class="p">)</span> <span class="c">#N #Taking sum of forces in Y direction</span>
<span class="c">#Results</span>
<span class="nb">print</span><span class="s">"The reaction at pt E i.e Re is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Re</span><span class="p">),</span><span class="s">"N"</span>
<span class="nb">print</span><span class="s">"The reaction at pt A i.e Ra is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Ra</span><span class="p">),</span><span class="s">"N"</span> <span class="c">#acting vertically downwards</span>
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<pre>The reaction at F i.e Rf is 3500.0 N
The reaction at D i.e Rd is 1000.0 N
The reaction at pt E i.e Re is 833.0 N
The reaction at pt A i.e Ra is -333.0 N
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<h2 id="Example-3.3-7,Page-No:69">Example 3.3-7,Page No:69<a class="anchor-link" href="#Example-3.3-7,Page-No:69">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">math</span>
<span class="c">#Initilization of variables</span>
<span class="n">Ws</span><span class="o">=</span><span class="mi">2</span> <span class="c">#kN #weight of scooter</span>
<span class="n">Wd</span><span class="o">=</span><span class="mf">0.5</span> <span class="c">#kN #weight of driver</span>
<span class="n">Lab</span><span class="o">=</span><span class="mi">1</span> <span class="c">#m</span>
<span class="n">Led</span><span class="o">=</span><span class="mf">0.8</span> <span class="c">#m</span>
<span class="n">Leg</span><span class="o">=</span><span class="mf">0.1</span> <span class="c">#m</span>
<span class="c">#Calculations</span>
<span class="n">Rc</span><span class="o">=</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">Leg</span><span class="p">)</span><span class="o">+</span><span class="p">(</span><span class="n">Wd</span><span class="o">*</span><span class="n">Led</span><span class="p">))</span><span class="o">/</span><span class="n">Lab</span> <span class="c">#kN #take moment at E</span>
<span class="n">Ra</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="n">Wd</span><span class="o">-</span><span class="n">Rc</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="c">#kN # as Ra=Rb,(Ra+Rb=2*Ra)</span>
<span class="n">Rb</span><span class="o">=</span><span class="n">Ra</span> <span class="c"># kN</span>
<span class="c">#Results</span>
<span class="nb">print</span><span class="s">"The reaction at wheel A is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Ra</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="s">"kN"</span>
<span class="nb">print</span><span class="s">"The reaction at wheel B is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Rb</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="s">"kN"</span>
<span class="nb">print</span><span class="s">"The reaction at wheel C is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Rc</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="s">"kN"</span>
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<pre>The reaction at wheel A is 0.95 kN
The reaction at wheel B is 0.95 kN
The reaction at wheel C is 0.6 kN
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<h2 id="Example-3.3-8,Page-No:69">Example 3.3-8,Page No:69<a class="anchor-link" href="#Example-3.3-8,Page-No:69">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">math</span>
<span class="c">#Initilization of variables</span>
<span class="n">W1</span><span class="o">=</span><span class="mi">15</span> <span class="c">#N #up</span>
<span class="n">W2</span><span class="o">=</span><span class="mi">60</span> <span class="c">#N #down</span>
<span class="n">W3</span><span class="o">=</span><span class="mi">10</span> <span class="c">#N #up</span>
<span class="n">W4</span><span class="o">=</span><span class="mi">25</span> <span class="c">#N #down</span>
<span class="n">Lab</span><span class="o">=</span><span class="mf">1.2</span> <span class="c">#m</span>
<span class="n">Lac</span><span class="o">=</span><span class="mf">0.4</span> <span class="c">#m</span>
<span class="n">Lcd</span><span class="o">=</span><span class="mf">0.3</span> <span class="c">#m</span>
<span class="n">Ldb</span><span class="o">=</span><span class="mf">0.5</span> <span class="c">#m</span>
<span class="n">Lad</span><span class="o">=</span><span class="mf">0.7</span> <span class="c">#m</span>
<span class="n">Leb</span><span class="o">=</span><span class="mf">0.417</span> <span class="c">#m #Leb=Lab-x</span>
<span class="c">#Calculations</span>
<span class="c">#(a) A single force</span>
<span class="n">Ry</span><span class="o">=</span><span class="n">W1</span><span class="o">-</span><span class="n">W2</span><span class="o">+</span><span class="n">W3</span><span class="o">-</span><span class="n">W4</span> <span class="c">#N #take sum Fy=0</span>
<span class="n">x</span><span class="o">=</span><span class="p">((</span><span class="o">-</span><span class="n">W2</span><span class="o">*</span><span class="n">Lac</span><span class="p">)</span><span class="o">+</span><span class="p">(</span><span class="n">W3</span><span class="o">*</span><span class="n">Lad</span><span class="p">)</span><span class="o">-</span><span class="p">(</span><span class="n">W4</span><span class="o">*</span><span class="n">Lab</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">Ry</span><span class="p">)</span> <span class="c">#m</span>
<span class="c">#(b) Single force moment at A</span>
<span class="n">Ma</span><span class="o">=</span><span class="p">(</span><span class="n">Ry</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="c">#N-m</span>
<span class="c"># Single force moment at B</span>
<span class="n">Mb</span><span class="o">=</span><span class="n">W2</span><span class="o">*</span><span class="n">Leb</span> <span class="c">#N-m</span>
<span class="c">#Results</span>
<span class="nb">print</span><span class="s">"The reaction for single force (a) is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Ry</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="s">"N"</span>
<span class="nb">print</span><span class="s">"The distance of Ry from A is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span><span class="s">"m"</span>
<span class="nb">print</span><span class="s">"The moment at A is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Ma</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="s">"N-m"</span>
<span class="nb">print</span><span class="s">"The moment at B is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Mb</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="s">"N-m"</span>
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<pre>The reaction for single force (a) is -60.0 N
The distance of Ry from A is 0.783 m
The moment at A is -47.0 N-m
The moment at B is 25.02 N-m
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<h2 id="Example-3.3-9,Page-No:71">Example 3.3-9,Page No:71<a class="anchor-link" href="#Example-3.3-9,Page-No:71">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">math</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="c">#Initilization of variables</span>
<span class="n">Ra</span><span class="o">=</span><span class="mi">5000</span> <span class="c">#N</span>
<span class="n">Ma</span><span class="o">=</span><span class="mi">10000</span> <span class="c">#Nm</span>
<span class="n">alpha</span><span class="o">=</span><span class="mi">60</span> <span class="c">#degree #angle made by T1 with the pole</span>
<span class="n">beta</span><span class="o">=</span><span class="mi">45</span> <span class="c">#degree #angle made by T2 with the pole</span>
<span class="n">theta</span><span class="o">=</span><span class="mi">30</span> <span class="c">#degree #angle made by T3 with the pole</span>
<span class="n">Lab</span><span class="o">=</span><span class="mi">6</span> <span class="c">#m</span>
<span class="n">Lac</span><span class="o">=</span><span class="mf">1.5</span> <span class="c">#m</span>
<span class="n">Lcb</span><span class="o">=</span><span class="mf">4.5</span> <span class="c">#m</span>
<span class="c">#Calculations</span>
<span class="n">T3</span><span class="o">=</span><span class="n">Ma</span><span class="o">/</span><span class="p">(</span><span class="mf">4.5</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="o">*</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">180</span><span class="p">)))</span> <span class="c">#N #take moment at B</span>
<span class="c"># Now we use matrix to solve eqn's 1 & 2 simultaneously,</span>
<span class="n">A</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="o">-</span><span class="mf">0.707</span><span class="p">,</span> <span class="mf">0.866</span><span class="p">],[</span><span class="mf">0.707</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">]])</span>
<span class="n">B</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">2222.2</span><span class="p">,</span><span class="mf">8848.8</span><span class="p">])</span>
<span class="n">C</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">)</span>
<span class="c">#Results</span>
<span class="nb">print</span><span class="s">"Tension in wire 1 i.e T1 is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">C</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span><span class="mi">1</span><span class="p">),</span><span class="s">"N"</span> <span class="c">#answer may vary due to decimal variance</span>
<span class="nb">print</span><span class="s">"Tension in wire 2 i.e T2 is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">C</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="mi">1</span><span class="p">),</span><span class="s">"N"</span>
<span class="nb">print</span><span class="s">"Tension in wire 3 i.e T3 is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">T3</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="s">"N"</span>
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<pre>Tension in wire 1 i.e T1 is 8104.7 N
Tension in wire 2 i.e T2 is 6784.2 N
Tension in wire 3 i.e T3 is 4444.4 N
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<h2 id="Example-3.3-10,Page-No:76">Example 3.3-10,Page No:76<a class="anchor-link" href="#Example-3.3-10,Page-No:76">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">math</span>
<span class="c">#Initilization of variables</span>
<span class="n">w</span><span class="o">=</span><span class="mi">2000</span> <span class="c">#N/m</span>
<span class="n">Lab</span><span class="o">=</span><span class="mi">3</span> <span class="c">#m</span>
<span class="c">#Calculations</span>
<span class="n">W</span><span class="o">=</span><span class="n">w</span><span class="o">*</span><span class="n">Lab</span><span class="o">/</span><span class="mi">2</span> <span class="c">#N# Area under the curve</span>
<span class="n">Lac</span><span class="o">=</span><span class="p">(</span><span class="mf">0.6666</span><span class="p">)</span><span class="o">*</span><span class="n">Lab</span> <span class="c">#m#centroid of the triangular load system</span>
<span class="n">Rb</span><span class="o">=</span><span class="p">(</span><span class="n">W</span><span class="o">*</span><span class="n">Lac</span><span class="p">)</span><span class="o">/</span><span class="n">Lab</span> <span class="c">#N #sum of moment at A</span>
<span class="n">Ra</span><span class="o">=</span><span class="n">W</span><span class="o">-</span><span class="n">Rb</span> <span class="c">#N</span>
<span class="c">#Results</span>
<span class="nb">print</span><span class="s">"The resultant of the distibuted load lies at "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Lac</span><span class="p">),</span><span class="s">"m"</span>
<span class="nb">print</span><span class="s">"The reaction at support A is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Ra</span><span class="p">),</span><span class="s">"N"</span>
<span class="nb">print</span><span class="s">"The reaction at support B is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Rb</span><span class="p">),</span><span class="s">"N"</span>
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<pre>The resultant of the distibuted load lies at 2.0 m
The reaction at support A is 1000.0 N
The reaction at support B is 2000.0 N
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<h2 id="Example-3.3-12,Page-No:78">Example 3.3-12,Page No:78<a class="anchor-link" href="#Example-3.3-12,Page-No:78">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">math</span>
<span class="c"># Initilization of variables</span>
<span class="n">w1</span><span class="o">=</span><span class="mf">1.5</span> <span class="c">#kN/m # intensity of varying load at the starting point of the beam</span>
<span class="n">w2</span><span class="o">=</span><span class="mf">4.5</span> <span class="c">#kN/m # intensity of varying load at the end of the beam</span>
<span class="n">l</span><span class="o">=</span><span class="mi">6</span> <span class="c">#m # ength of the beam</span>
<span class="c"># Calculations</span>
<span class="c"># The varying load distribution is divided into a rectangle and a right angled triangle</span>
<span class="n">W1</span><span class="o">=</span><span class="n">w1</span><span class="o">*</span><span class="n">l</span> <span class="c">#kN # where W1 is the area of the load diagram(rectangle ABED)</span>
<span class="n">x1</span><span class="o">=</span><span class="n">l</span><span class="o">/</span><span class="mi">2</span> <span class="c">#m # centroid of the rectangular load system</span>
<span class="n">W2</span><span class="o">=</span><span class="p">(</span><span class="n">w2</span><span class="o">-</span><span class="n">w1</span><span class="p">)</span><span class="o">*</span><span class="n">l</span><span class="o">/</span><span class="mi">2</span> <span class="c">#kN # where W1 is the area of the load diagram(triangle DCE)</span>
<span class="n">x2</span><span class="o">=</span><span class="mi">2</span><span class="o">*</span><span class="n">l</span><span class="o">/</span><span class="mi">3</span> <span class="c">#m # centroid of the triangular load system</span>
<span class="n">W</span><span class="o">=</span><span class="n">W1</span><span class="o">+</span><span class="n">W2</span> <span class="c">#kN # W is the resultant</span>
<span class="n">x</span><span class="o">=</span><span class="p">((</span><span class="n">W1</span><span class="o">*</span><span class="n">x1</span><span class="p">)</span><span class="o">+</span><span class="p">(</span><span class="n">W2</span><span class="o">*</span><span class="n">x2</span><span class="p">))</span><span class="o">/</span><span class="n">W</span> <span class="c">#m # where x is the distance where the resultant lies</span>
<span class="c">#Results</span>
<span class="nb">print</span><span class="s">"The resultant of the distributed load system is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">W</span><span class="p">),</span><span class="s">"kN"</span>
<span class="nb">print</span><span class="s">"The line of action of the resulting load is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="s">"m"</span>
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<pre>The resultant of the distributed load system is 18.0 kN
The line of action of the resulting load is 3.5 m
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<h2 id="Example-3.3-13,Page-No:78">Example 3.3-13,Page No:78<a class="anchor-link" href="#Example-3.3-13,Page-No:78">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">math</span>
<span class="c"># Initiization of variables</span>
<span class="n">W1</span><span class="o">=</span><span class="mi">10</span> <span class="c">#kN #point load acting at D</span>
<span class="n">W2</span><span class="o">=</span><span class="mi">20</span> <span class="c">#kN # point load acting at C at an angle of 30 degree</span>
<span class="n">W3</span><span class="o">=</span><span class="mi">5</span> <span class="c">#kN/m # intensity of udl acting on span EB of 4m</span>
<span class="n">W4</span><span class="o">=</span><span class="mi">10</span> <span class="c">#kN/m # intensity of varying load acting on span BC of 3m</span>
<span class="n">M</span><span class="o">=</span><span class="mi">25</span> <span class="c">#kN-m # moment acting at E</span>
<span class="n">theta</span><span class="o">=</span><span class="mi">30</span> <span class="c">#degree # angle made by 20 kN load with the beam</span>
<span class="n">Lad</span><span class="o">=</span><span class="mi">2</span> <span class="c">#m</span>
<span class="n">Leb</span><span class="o">=</span><span class="mi">4</span> <span class="c">#m</span>
<span class="n">Laf</span><span class="o">=</span><span class="mi">6</span> <span class="c">#m #distance between the resultant of W3 & point A</span>
<span class="n">Lac</span><span class="o">=</span><span class="mi">11</span> <span class="c">#m</span>
<span class="n">Lag</span><span class="o">=</span><span class="mi">9</span> <span class="c">#m #distance between the resultant of W4 and point A</span>
<span class="n">Lbc</span><span class="o">=</span><span class="mi">3</span> <span class="c">#m</span>
<span class="n">Lab</span><span class="o">=</span><span class="mi">8</span> <span class="c">#m</span>
<span class="c"># Calculations</span>
<span class="n">Xa</span><span class="o">=</span><span class="mi">20</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="o">*</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">180</span><span class="p">))</span> <span class="c">#kN # sum Fx=0</span>
<span class="n">Rb</span><span class="o">=</span><span class="p">((</span><span class="n">W1</span><span class="o">*</span><span class="n">Lad</span><span class="p">)</span><span class="o">+</span><span class="p">(</span><span class="o">-</span><span class="n">M</span><span class="p">)</span><span class="o">+</span><span class="p">(</span><span class="n">W3</span><span class="o">*</span><span class="n">Leb</span><span class="o">*</span><span class="n">Laf</span><span class="p">)</span><span class="o">+</span><span class="p">(</span><span class="n">W2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="o">*</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">180</span><span class="p">))</span><span class="o">*</span><span class="n">Lac</span><span class="p">)</span><span class="o">+</span><span class="p">((</span><span class="n">W4</span><span class="o">*</span><span class="n">Lbc</span><span class="o">*</span><span class="n">Lag</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="n">Lab</span> <span class="c">#kN # taking moment at A</span>
<span class="n">Ya</span><span class="o">=</span><span class="n">W1</span><span class="o">+</span><span class="p">(</span><span class="n">W2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="o">*</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">180</span><span class="p">)))</span><span class="o">+</span><span class="p">(</span><span class="n">W3</span><span class="o">*</span><span class="n">Leb</span><span class="p">)</span><span class="o">+</span><span class="p">(</span><span class="n">W4</span><span class="o">*</span><span class="n">Lbc</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">-</span><span class="n">Rb</span> <span class="c">#kN # sum Fy=0</span>
<span class="n">Ra</span><span class="o">=</span><span class="p">(</span><span class="n">Xa</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">Ya</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mf">0.5</span> <span class="c">#kN # resultant at A</span>
<span class="c">#Results</span>
<span class="nb">print</span><span class="s">"The horizontal reaction at A i.e Xa is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Xa</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="s">"kN"</span>
<span class="nb">print</span><span class="s">"The vertical reaction at A i.e Ya is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Ya</span><span class="p">),</span><span class="s">"kN"</span>
<span class="nb">print</span><span class="s">"The reaction at A i.e Ra is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Ra</span><span class="p">),</span><span class="s">"kN"</span>
<span class="nb">print</span><span class="s">"The reaction at B i.e Rb is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">Rb</span><span class="p">),</span><span class="s">"kN"</span>
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<pre>The horizontal reaction at A i.e Xa is 17.32 kN
The vertical reaction at A i.e Ya is 10.0 kN
The reaction at A i.e Ra is 20.0 kN
The reaction at B i.e Rb is 45.0 kN
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<h2 id="Example-3.3-14,Page-No:79">Example 3.3-14,Page No:79<a class="anchor-link" href="#Example-3.3-14,Page-No:79">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">math</span>
<span class="c"># Initilization of variables</span>
<span class="n">h</span><span class="o">=</span><span class="mi">4</span> <span class="c">#m #height of the dam wall</span>
<span class="n">rho_w</span><span class="o">=</span><span class="mi">1000</span> <span class="c"># kg/m^3 # density of water</span>
<span class="n">rho_c</span><span class="o">=</span><span class="mi">2400</span> <span class="c"># kg/m^3 # density of concrete</span>
<span class="n">g</span><span class="o">=</span><span class="mf">9.81</span> <span class="c"># m/s^2</span>
<span class="c"># Calculations</span>
<span class="n">P</span><span class="o">=</span><span class="p">(</span><span class="n">rho_w</span><span class="o">*</span><span class="n">g</span><span class="o">*</span><span class="n">h</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="c"># The resultant force due to water pressure per unit length of the dam</span>
<span class="n">x</span><span class="o">=</span><span class="p">(</span><span class="mf">0.6666</span><span class="p">)</span><span class="o">*</span><span class="n">h</span> <span class="c">#m # distance at which the resutant of the triangular load acts </span>
<span class="n">b</span><span class="o">=</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">P</span><span class="o">*</span><span class="n">h</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">h</span><span class="o">*</span><span class="n">rho_c</span><span class="o">*</span><span class="n">g</span><span class="p">))</span><span class="o">**</span><span class="mf">0.5</span> <span class="c"># m # eq'n required to find the minimum width of the dam</span>
<span class="c"># Results</span>
<span class="nb">print</span><span class="s">"The minimum width which is to be provided to the dam to prevent overturning about point B is "</span><span class="p">,</span><span class="nb">round</span><span class="p">(</span><span class="n">b</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span><span class="s">"m"</span>
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<pre>The minimum width which is to be provided to the dam to prevent overturning about point B is 1.491 m
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