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Week 5 - Bracing Stiffness Requirements.slides.html
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Week 5 - Bracing Stiffness Requirements.slides.html
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<h1 id="Required-Stiffness-of-Slab-Braces">Required Stiffness of Slab Braces<a class="anchor-link" href="#Required-Stiffness-of-Slab-Braces">¶</a></h1>
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<h2 id="Brace-Loading">Brace Loading<a class="anchor-link" href="#Brace-Loading">¶</a></h2>
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<source src="https://github.com/damontallen/Construction-Lectures-Fall-2014/raw/master/Lecture%20Audio/Week%205/Brace%20Stiffness%20Requirements%20(optional)/Brace%20Loading.mp3"
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<p><a href="http://i.imgur.com/SBLJJNE.png" id="P-delta Effect">
<img title="P-delta Effect" alt = "P-delta Effect"
src="http://i.imgur.com/SBLJJNE.png"
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onmouseout="this.src='http://i.imgur.com/SBLJJNE.png'"
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<p>Slab braces need to be designed with the capacity to resist horizontal loading, and by code, this is at least 2% of the vertical load. This 2% is a resultant of the <a href="http://en.wikipedia.org/wiki/Couple_(mechanics)#Forces_and_couples">moment couple</a> that is generated by a shore with an out of plumbness of 1:500, or 1/4" in 10 ft, multiplied by a factor of safety of 5.</p>
$$P_{hor} \times \text{ Horizontal Moment Arm} = P \times \Delta$$<p><em>(Hover over the image to see the motion that causes $\Delta$.)</em></p>
$$P_{hor} \times 10' \times \frac{12"}{1'} = P \times \frac{1}{4}"$$$$P_{hor} \times 120" = P \times 0.25"$$$$P_{hor} = P \times \frac{0.25"}{120"} \approx P \times \frac{1}{500}$$$$P_{hor} = P \times 0.004 $$<p>and when the factor of safety of 5 is applied,</p>
$$P_{hor} = P \times 0.004 \times 5 = P \times 0.02 $$
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<h2 id="$P$---$\Delta$-Effect">$P$ - $\Delta$ Effect<a class="anchor-link" href="#$P$---$\Delta$-Effect">¶</a></h2>
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<p>However, to resist this horizontal load, the brace must stretch a little more than its initial length.</p>
$$\text{The change in length of the brace} = \text{Tension in the brace}\times \text{Stiffness of the brace}$$<p>Where:</p>
$$\text{Stiffness of the brace} = \text{Lumber stiffness}+\text{Connection stiffness}$$$$\Delta_{br} = T_{br} \left(\frac{L_{br}}{A E} + k_{connection}\right)$$<p>where,</p>
<p>$k_{connection}$ is the connection stiffness.</p>
<p>If this stretching is greater than the expected stretching of $ \frac{h}{500} \cos(\theta)$ then the moment will be greater than the design moment.</p>
<p>This greater moment in turn must be resisted be the brace, which in turn results in more stretching, etc. This is known as the <a href="http://en.wikipedia.org/wiki/P-Delta_Effect">P-Delta Effect</a>.</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">arcsin</span>
<span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span><span class="n">sin</span>
<span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">rad2deg</span><span class="p">,</span> <span class="n">deg2rad</span>
<span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">ceil</span>
<span class="n">nails</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">nail_str</span><span class="o">=</span><span class="mi">141</span><span class="o">*</span><span class="mf">1.25</span>
<span class="n">A</span> <span class="o">=</span> <span class="mf">1.5</span><span class="o">*</span><span class="mf">0.75</span> <span class="c">#brace cross section area</span>
<span class="n">E</span> <span class="o">=</span><span class="mf">1.6E6</span> <span class="c">#Species Young's modulus</span>
<span class="n">h</span> <span class="o">=</span> <span class="mi">96</span> <span class="c">#Height of the shoring</span>
<span class="n">G</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="c"># EWP vs 0.5 for DFL specific gravity</span>
<span class="n">d</span> <span class="o">=</span> <span class="mf">0.177</span> <span class="c">#16d common nail diameter</span>
<span class="n">deg_ang</span> <span class="o">=</span> <span class="mi">45</span> <span class="c">#angle of brace in degrees</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">deg2rad</span><span class="p">(</span><span class="n">deg_ang</span><span class="p">)</span>
<span class="n">L</span> <span class="o">=</span> <span class="n">h</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">K</span><span class="p">(</span><span class="n">nails</span><span class="p">):</span>
<span class="sd">"""Determines the nail, connection, board, and brace stiffnesses """</span>
<span class="n">k_2</span> <span class="o">=</span> <span class="n">A</span><span class="o">*</span><span class="n">E</span><span class="o">/</span><span class="n">L</span> <span class="c">#board stiffness</span>
<span class="n">k_nail</span> <span class="o">=</span> <span class="mi">80000</span><span class="o">*</span><span class="n">G</span><span class="o">**</span><span class="mf">1.5</span><span class="o">*</span><span class="n">d</span><span class="o">**</span><span class="mf">0.8</span> <span class="c">#nail stiffness</span>
<span class="n">k_1</span> <span class="o">=</span> <span class="n">nails</span><span class="o">*</span><span class="n">k_nail</span> <span class="c">#connection stiffness</span>
<span class="n">k_eq</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">((</span><span class="mi">2</span><span class="o">/</span><span class="n">k_1</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="o">/</span><span class="n">k_2</span><span class="p">)</span> <span class="c">#stiffness provided</span>
<span class="k">return</span> <span class="p">(</span><span class="n">k_nail</span><span class="p">,</span><span class="n">k_1</span><span class="p">,</span><span class="n">k_2</span><span class="p">,</span><span class="n">k_eq</span><span class="p">)</span>
<span class="n">k_nail</span><span class="p">,</span><span class="n">k_1</span><span class="p">,</span><span class="n">k_2</span><span class="p">,</span><span class="n">k_eq</span> <span class="o">=</span> <span class="n">K</span><span class="p">(</span><span class="n">nails</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">delta</span><span class="p">(</span><span class="n">angle</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">old_d</span><span class="p">):</span>
<span class="n">M</span> <span class="o">=</span> <span class="n">old_d</span> <span class="o">*</span> <span class="n">P</span>
<span class="n">P_hor</span> <span class="o">=</span> <span class="n">M</span><span class="o">/</span><span class="n">h</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">P_hor</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
<span class="n">d_br</span> <span class="o">=</span> <span class="n">T</span><span class="o">/</span><span class="n">k_eq</span>
<span class="k">return</span> <span class="n">d_br</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">P_delta</span><span class="p">(</span><span class="n">P</span><span class="p">):</span>
<span class="n">old_d</span> <span class="o">=</span> <span class="n">h</span><span class="o">/</span><span class="mi">500</span>
<span class="n">angle</span> <span class="o">=</span> <span class="n">theta</span>
<span class="n">Delta</span> <span class="o">=</span> <span class="p">[</span><span class="n">old_d</span><span class="p">]</span>
<span class="n">steps</span> <span class="o">=</span> <span class="mi">10</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">steps</span><span class="p">):</span>
<span class="n">d</span> <span class="o">=</span> <span class="n">delta</span><span class="p">(</span><span class="n">angle</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">old_d</span><span class="p">)</span>
<span class="n">Delta</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="n">old_d</span> <span class="o">=</span> <span class="n">d</span>
<span class="n">i</span> <span class="o">=</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">steps</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
<span class="k">return</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">Delta</span><span class="p">)</span>
<span class="n">P</span> <span class="o">=</span> <span class="mi">200000</span>
<span class="p">(</span><span class="n">i1</span><span class="p">,</span><span class="n">Delta1</span><span class="p">)</span> <span class="o">=</span> <span class="n">P_delta</span><span class="p">(</span><span class="n">P</span><span class="p">)</span>
<span class="n">P</span> <span class="o">=</span> <span class="mi">290000</span>
<span class="p">(</span><span class="n">i2</span><span class="p">,</span><span class="n">Delta2</span><span class="p">)</span> <span class="o">=</span> <span class="n">P_delta</span><span class="p">(</span><span class="n">P</span><span class="p">)</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">import</span> <span class="nn">matplotlib</span>
<span class="c"># plot = matplotlib.pyplot.plot</span>
<span class="n">plt</span> <span class="o">=</span> <span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span>
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<div class=" highlight hl-ipython3"><pre><span class="k">def</span> <span class="nf">plot</span><span class="p">():</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax1</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">l_1</span><span class="p">,</span> <span class="o">=</span> <span class="n">ax1</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">i1</span><span class="p">,</span><span class="n">Delta1</span><span class="p">,</span><span class="s">'o-g'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"Stable"</span><span class="p">)</span>
<span class="n">l_2</span><span class="p">,</span> <span class="o">=</span> <span class="n">ax1</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">i2</span><span class="p">,</span><span class="n">Delta2</span><span class="p">,</span><span class="s">'x-r'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"Unstable"</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s">"Itteration"</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s">"Delta Increments (in)"</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">'P-$\Delta$ Effect in Slab Bracing'</span><span class="p">)</span>
<span class="n">lines</span><span class="o">=</span><span class="p">[</span><span class="n">l_1</span><span class="p">,</span><span class="n">l_2</span><span class="p">]</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">lines</span><span class="p">,</span> <span class="p">[</span><span class="n">l</span><span class="o">.</span><span class="n">get_label</span><span class="p">()</span> <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="n">lines</span><span class="p">],</span><span class="s">'upper left'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="k">True</span><span class="p">)</span>
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<h2 id="Stability-Example">Stability Example<a class="anchor-link" href="#Stability-Example">¶</a></h2>
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<source src="https://github.com/damontallen/Construction-Lectures-Fall-2014/raw/master/Lecture%20Audio/Week%205/Brace%20Stiffness%20Requirements%20(optional)/Stability%20Example.mp3"
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<p>Looking at two braces, one with adequate stiffness, and another that isn't quite stiff enough. We can start them out with the initial displacement of L/500. When we load them us with 2% of the vertical load, we can calculate the resulting stretch. This can then be use to recalculate the horizontal force and the resulting displacement.</p>
<p>We can repeat this process, until it becomes clear, that either the brace will stabilize, or it be unstable.</p>
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<h2 id="How-to-Check-Stiffness">How to Check Stiffness<a class="anchor-link" href="#How-to-Check-Stiffness">¶</a></h2>
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<source src="https://github.com/damontallen/Construction-Lectures-Fall-2014/raw/master/Lecture%20Audio/Week%205/Brace%20Stiffness%20Requirements%20(optional)/How%20to%20Check%20Stiffness.mp3"
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<p>Generally, with the factors of safety applied to the horizontal load requirements, stiffness is not going to be an issue. However, if you have an unusual situation, where you have a thick slab, small braces, or no concrete columns intersect the slab form, you can do a quick check to see if you should add more braces.</p>
<p>Calculate the value of the equation below, and if it is less than zero you will need to add more braces to resist any $P-\Delta$ effects.</p>
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$$\frac{ Z' \cos(\theta)}{801 \Sigma P (n) G^{1.5} d^{0.8}} + L_{br}\left(\frac{ 1}{n A E'}-\frac{ \sin(2\theta)\cos(\theta)}{10 \Sigma P}\right) \le 0 \text{ when brace stiffness is inadequate}$$
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<p>Where:</p>
<p>$\Sigma P$ is the portion of the vertical load that the brace need to resist (lbs).</p>
<p>$n$ is the number of tension braces used to resist the horizontal load in one direction.</p>
<p>$Z'$ is the adjusted strength for the type of nails used to fasten the ends of each brace (lbs).</p>
<p>$d$ is the diameter of the nails used to fasten the braces (in) (Table L4 of the Timber NDS).</p>
<p>$G$ is the specific gravity of the brace lumber (Found in the material proerties tables of the Timber Supplement <em>[also Table 12.3.3A in the NDS]</em>)</p>
<p>$L_{br}$ is the length of a brace (in).</p>
<p>$A$ is the cross sectional area of a brace (in$^2$).</p>
<p>$E'$ is the modified Young's modulus of the species and grade of the brace (psi).</p>
<p>$\theta$ is the angle of the brace.</p>
<p><em>(This is adapted from the information about the "<a href="http://www.fierabolzano.it/bauschau2005/congress/EN_1995-1-1_%20%28E%29.pdf">Eurocode 5 (1995) nail-stiffness formulation, Section 7, Serviceability limit-states</a>" found in <a href="http://www.saice.org.za/downloads/journal/vol48-4-2006/vol48_n4_c.pdf">Journal of the South African Institution of Civil Engineering • Volume 48 Number 4 December 2006: Effective stiffness of nailed multiple-ply timber members</a>.)</em></p>
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<h2 id="Viable-Alternative">Viable Alternative<a class="anchor-link" href="#Viable-Alternative">¶</a></h2>
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<p>If you are still unsure of yourself, or do not think that your liability insurance will cover you, then contact an engineer to check your design.</p>
<p>Knowing when you are out of your depth, is a good thing to be able to know.</p>
<p><em>(Click on the blue right arrow to see the references used in this lecture.)</em></p>
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<h2 id="References:">References:<a class="anchor-link" href="#References:">¶</a></h2>
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<ul>
<li><a href="http://www.fierabolzano.it/bauschau2005/congress/EN_1995-1-1_%20%28E%29.pdf">EN 1995-1-1:2004 (E) Eurocode 5 Design of timber structures Section 7, Table 7.1</a></li>
<li><a href="http://www.saice.org.za/downloads/journal/vol48-4-2006/vol48_n4_c.pdf">Journal of the South African Institution of Civil Engineering • Volume 48 Number 4 December 2006: Effective stiffness of nailed multiple-ply timber members</a></li>
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<p><a href="https://ufl.instructure.com/courses/">Class website</a> <em>(Use this link to if you are taking the course on e-learning.)</em></p>
<p><a href="http://damontallen.github.io/Construction-Lectures-Fall-2014/">Github.io version of course website</a> <em>(Do not use this link if you are taking this course in Summer A or B.)</em></p>
<p><a href="http://ipython.org/">IPython.org</a> <em>(IPython is the opensource software used in the development of much of this course.)</em></p>
<p><a href="http://nbviewer.ipython.org/github/damontallen/Construction-Lectures-Fall-2014/blob/master/Resources.ipynb">Complete Software List</a></p>
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