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Week 3 - Timber Beam Lecture.slides.html
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Week 3 - Timber Beam Lecture.slides.html
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<h1 id="Timber-Beams">Timber Beams<a class="anchor-link" href="#Timber-Beams">¶</a></h1>
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<p><iframe width="420" height="315" src="//www.youtube.com/embed/U56QJoelQVc" frameborder="0" allowfullscreen></iframe></p>
<p><em>(Be aware this lecture may take some time to load due to embedded windows onto external resources. Since the equations in this lecture are the last to load, they will not look correct until loading has finished. Depending on your internet connection speed this may take up to a minute.)</em></p>
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<h2 id="Timber-Beam-Failure-Modes">Timber Beam Failure Modes<a class="anchor-link" href="#Timber-Beam-Failure-Modes">¶</a></h2>
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<p>Here you can see an animation of the three failure modes for a timber beam that we will be checking this semester, shear, bending, and deflection.</p>
<p><em>Scroll down to see all three beam failure modes.</em></p>
<p><center><strong>Shear Failure Mode</strong></center></p>
<p><center>This is a strength failure mode which is typically seen in shorter beams.</center>
<a href="http://imgur.com/wYc4xpn"><img src="http://i.imgur.com/wYc4xpn.gif" alt = "Shear Failure Mode" title="Shear Failure Mode" /></a></p>
<p><center><strong>Bending Failure Mode</strong></center></p>
<p><center>This is a strength failure mode which is seen in average length beams.</center>
<a href="http://imgur.com/JYCBH71"><img src="http://i.imgur.com/JYCBH71.gif" alt = "Bending Failure Mode" title="Bending Failure Mode" /></a></p>
<p><center><strong>Excessive Deflection Failure Mode</strong></center></p>
<p><center>This is a serviceability failure mode which is typically seen in longer beams.</center>
<a href="http://imgur.com/dnxzNRS"><img src="http://i.imgur.com/dnxzNRS.gif" alt = "Deflection Failure Mode" title="Deflection Failure Mode" /></a></p>
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<p>Don't know and want to know why the shear stress failure presents as a horizontal crack, read <a href="http://fp.optics.arizona.edu/optomech/references/OPTI_222/OPTI_222_W10.pdf">this</a>.</p>
<p><em>(Click the blue down arrow to hear a brief description of the failure mode we won't be looking at.)</em></p>
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<h3 id="Wind-Uplift">Wind Uplift<a class="anchor-link" href="#Wind-Uplift">¶</a></h3>
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<p>When a timber beam is loaded in a way such that the compression side of the beam in bending is on the opposite side from the loading, i.e. wind uplift, then we need to check beam stability. In a steel beam this is know as lateral torsional buckling. In timber we check this with the $C_L$ adjustment factor.</p>
<p>Because we are always loading the beams, in this course, on the same side as the compression, we will not be using $C_L$. If you want to learn about it I suggest reading through the beam examples in the "Structural Wood Design Solved Example Problems" of your wood design package.</p>
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<h2 id="Timber-Beam-Analysis">Timber Beam Analysis<a class="anchor-link" href="#Timber-Beam-Analysis">¶</a></h2>
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<p>The process of analyzing a timber beam goes like this.</p>
<ol>
<li>Determine the section properties, in Table 1B of the Supplement, based on the cross sectional size of the beam.</li>
<li>Find out the material properties of the beam based on the species, grade, and size of the beam by looking in either Table 4A, 4B or 4D.</li>
<li>Apply the adjustment factors to $F_b$, $F_v$, and $E$ to find the allowable stresses.</li>
<li>Calculate the actual stresses for bending, and shear, and find the actual deflections of the beam.</li>
<li>Make comparisons between the actual and allowable values.</li>
<li>Write a summary of your results and conclude if the beam is adequate.</li>
</ol>
<p>In hand solution we will typically go back and forth between steps 3 and 4 looking at the the allowable and actual values for bending, shear, and deflection rather than finding all of the allowable vales then all of the actual values. In the design process we can stop as soon as we find a failure mode rather than doing additional work.</p>
<p>You should be aware that in this course I will want you to solve all the values regardless if you find a failure mode early on.</p>
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<p>Let's work through an example to see the whole process.</p>
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<p><strong>Typical Problem Statement:</strong></p>
<p>A 2x10 Douglas Fir-South Grade No.2 beam is used as a floor joist. The joists span are 10'-6",
and they are spaced 16" on center. Wet service conditions prevail. The deflection
limits are L/360 for live load and L/240 for total load. <em>(This comes from Chapter 16 of the
<a href="http://www2.iccsafe.org/states/florida_codes/">Florida Building Code</a> (FBC).)</em>
The floor is subjected to a uniform dead load of 15 psf and live load of 75 psf.</p>
<p><a href="http://i.imgur.com/AM7Die1.png"><img src="http://i.imgur.com/AM7Die1.png" title="Loaded Beam" alt = "Loaded Beam" style = "max-width:300px; max-height:300px; border:1px solid blue; float:left; margin-right:3px;"/></a></p>
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<source src="https://github.com/damontallen/Construction-Lectures-Fall-2014/raw/master/Lecture%20Audio/Week%203/Timber%20Beam/Typical%20Problem%20Statement.mp3"
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<p>Values from problem statement:</p>
<ul>
<li><p>$\text{Span: }$</p>
<ul>
<li>$L = 10\text{' }6\text{" }= 10.5 ft$</li>
</ul>
</li>
<li><p>$\text{Grade: No.2}$</p>
</li>
<li><p>$\text{Size: 2x10}$</p>
</li>
<li><p>$\text{Spacing: }$</p>
<ul>
<li>$s = 16\text{" }= 1.333 ft \text{ (on center)}$</li>
</ul>
</li>
<li><p>$\text{Live Load: }$</p>
<ul>
<li>$LL = 75 psf$</li>
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</li>
<li><p>$\text{Dead Load: }$</p>
<ul>
<li>$DL = 15 psf$</li>
</ul>
</li>
<li><p>$\text{Deflection Limits: }$</p>
<ul>
<li>$\frac{L}{360} \text{ for Live Load}$</li>
<li>$\frac{L}{240} \text{ for Dead Load}$</li>
</ul>
</li>
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<p>Note about this course:</p>
<p>To be clear, for simplicity reasons, I generally will give you a deflection limit to use so you will not have to look it up. I often will give you a deflection limit that is more conservative than you you will find in the table but in practice you should look at the construction specifications for the limits you need to use.</p>
<p><em>(Click on the blue down arrow to see the Florida Building Code Deflection Limits table and notes.)</em></p>
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<p><a href="http://i.imgur.com/HDSPyHh.png"><img src="http://i.imgur.com/HDSPyHh.png" title="FBC: Table 1604.3 Deflection Limits" style = "max-width:300px; max-height:300px; border:1px solid blue; float:left; margin-right:3px;"/></a></p>
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<p>Table 1604.3 Notes:</p>
<p>a. For structural roofing and siding made of formed metal sheets, the total load deflection shall not exceed l/60. For secondary roof structural members supporting formed metal roofing, the live load deflection shall not exceed l/150. For secondary wall members supporting formed metal siding, the design wind load deflection shall not exceed l/90. For roofs, this exception only applies when the metal sheets have no roof covering.</p>
<p>b. Interior partitions not exceeding 6 feet in height and flexible, folding and portable partitions are not governed by the provisions of this section. The deflection criterion for interior partitions is based on the horizontal load defined in Section 1607.13.</p>
<p>c. See Section 2403 for glass supports.</p>
<p>d. For wood structural members having a moisture content of less than 16 percent at time of installation and used under dry conditions, the deflection resulting from L + 0.5 D is permitted to be substituted for the deflection resulting from L + D.</p>
<p>e. The above deflections do not ensure against ponding. Roofs that do not have sufficient slope or camber to assure adequate drainage shall be investigated for ponding. See Section 1611 for rain and ponding requirements and Section 1503.4 for roof drainage requirements.</p>
<p>f. The wind load is permitted to be taken as 0.42 times the "component and cladding" loads for the purpose of determining deflection limits herein.</p>
<p>g. For steel structural members, the dead load shall be taken as zero.</p>
<p>h. For aluminum structural members or aluminum panels used in skylights and sloped glazing framing, roofs or walls of sunroom additions or patio covers, not supporting edge of glass or aluminum sandwich panels, the total load deflection shall not exceed l/60. For continuous aluminum structural members supporting edge of glass, the total load deflection shall not exceed l/175 for each glass lite or l/60 for the entire length of the member, whichever is more stringent. For aluminum sandwich panels used in roofs or walls of sunroom additions or patio covers, the total load deflection shall not exceed l/120.</p>
<p>i. For cantilever members, l shall be taken as twice the length of the cantilever.</p>
<p>j. Screen surfaces shall be permitted to include a maximum of 25% solid flexible finishes.</p>
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<h2 id="Looking-up-Additional-Beam-Properties">Looking up Additional Beam Properties<a class="anchor-link" href="#Looking-up-Additional-Beam-Properties">¶</a></h2>
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<p>Because I did not indicate otherwise in the problem statement we will assume that this is a floor subject to construction loading. This means that my load duration factor, $C_D = 1.25$. Note that I will indicate what load duration factor to use by saying "wind loading" $(C_D= 1.6)$, "design for multiple reuses" or "assume long term load duration" $(C_D = 1.0)$.</p>
<p>From Table 1B in the <a href="http://www.awc.org/pdf/AWC_NDS-2015-Supplement_web-viewonly_1411.pdf">Supplement</a> we find:</p>
<ul>
<li>$\text{Thickness} = b = 1 \frac{1}{2}" = 1.5 \text{inches} $</li>
<li>$\text{Width} = d = 9 \frac{1}{4}" = 9.25 \text{inches} $</li>
<li>$\text{Section Area}=A=13.88 in^2$</li>
</ul>
<p>Because the problem statement did not have any indication that the beam was in the weak orientation, or "on the flat", we will assume that it is in the strong orientation. This means that the bending is occurring about the X-X axis.</p>
<ul>
<li>$\text{Section Modulus} = S_{xx} = 21.39 in^3 $</li>
<li>$\text{Moment of Inertia} = I_{xx} = 98.93 in^4 $</li>
</ul>
<p>In equations I will drop the xx (or yy) subscript, for convenience, making these values $S$ and $I$. You should also note that if you know the values for $b$ and $d$ you can easily calculate the values for $A$, $S$ and $I$ because all sawn lumber timber sections are rectangular.</p>
$$A = b \times d$$$$S = \frac{1}{6}\times b \times d^2$$$$I = \frac{1}{12}\times b \times d^3$$<p>(Of course if the beam is on the flat the values for $b$ and $d$ switch in these equations.)</p>
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<h3 id="Beam-Geometric-Properties:">Beam Geometric Properties:<a class="anchor-link" href="#Beam-Geometric-Properties:">¶</a></h3>
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<p>In summary the beam geometric values are:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">beam</span><span class="o">.</span><span class="n">geometry_latex</span>
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$$b = 1.50" \text{, }d = 9.25" \text{, }A = 13.88 in^2 \text{, }S = 21.39 in^3 \text{, }I = 98.93 in^4 \text{, }$$ $$L = 10.50' = 126.00"$$
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<h2 id="Loads">Loads<a class="anchor-link" href="#Loads">¶</a></h2>
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<p>We know from the problem statement that:</p>
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<li>$\text{Live Load }=LL=75 psf$</li>
<li>$\text{Dead Load }=DL=15 psf$</li>
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<p>But they are not in the form we can use them in our beam equations. Beams are analyzed using line loads not pressure. We need to use the tributary width to convert these loads to line loads.</p>
<p>The tributary width is half the distance from one beam to the next on both sides. To make sense of this let's make the variable for the spacing from one beam to the next to be lower case $s$.</p>
<p>$W_{trib} = \frac{1}{2} \times s + \frac{1}{2} \times s = s$</p>
<p>In our case the spacing is $16"$, so $W_{trib} = 16"$.</p>
<p>The variable for line loads is usually $w$, and we find $w$ by multiplying the pressure by the tributary width.</p>
<p><em>(Click the blue down arrow to see the line load calculations.)</em></p>
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<p>For the Live Load:</p>
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<li>$w_{live} = LL \times W_{trib} $</li>
<li>$w_{live} = 75 psf \times 16" \times \frac{1 ft}{12 in} \text{ (for the unit conversion)}$</li>
<li>$w_{live} = 100 \frac{lb}{ft} \text{ or } 100 plf$</li>
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<p>and for the Dead Load:</p>
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<li>$w_{dead} = DL \times W_{trib} $</li>
<li>$w_{dead} = 15 psf \times 16" \times \frac{1 ft}{12 in} $</li>
<li>$w_{dead} = 20 plf$</li>
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<p>The total line load $w$ is just the super position of the two (just add them):</p>
$$w = w_{dead} + w_{live}$$$$w = 20 plf + 100 plf$$$$w = 120 plf$$
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<h2 id="Unmodified-Stresses-from-Tables-4A,-4B,-or-4D">Unmodified Stresses from Tables 4A, 4B, or 4D<a class="anchor-link" href="#Unmodified-Stresses-from-Tables-4A,-4B,-or-4D">¶</a></h2>
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<p>We know that the species of lumber of the beam is Douglas Fir-South and the grade is No.2. We also know that the beam is $2"$ thick so what table do you think we should look in?</p>
<ul>
<li>Table 4A (page 32) Reference Design Values for Visually Graded Dimension Lumber (2" - 4" thick)</li>
<li>Table 4B (page 39) Reference Design Values for Visually Graded Southern Pine Dimension Lumber (2" - 4" thick)</li>
<li>Table 4D (page 46) Reference Design Values for Visually Graded Timbers (5" x 5" and larger)</li>
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<p><em>(Click the light blue down arrow to find out what table we need to use.)</em></p>
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<p>Well the beam is $2"$ thick, so we don't use Table 4D; it is for members that are $5" \times 5"$ or larger.</p>
<p>The species is not Southern Pine, so we don't use Table 4B.</p>
<p>That means we need to look up the unmodified allowable stresses in Table 4A of the <a href="http://www.awc.org/pdf/AWC_NDS-2015-Supplement_web-viewonly_1411.pdf">Supplement</a>.</p>
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<h3 id="Table-4A-Values">Table 4A Values<a class="anchor-link" href="#Table-4A-Values">¶</a></h3>
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<p>On page 34 of the in the Supplement we find that,</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">beam</span><span class="o">.</span><span class="n">unmodified_stresses_latex</span>
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$$\text{The unmodified stresses for a No.2 grade, Douglas Fir-South, member are:}$$
$${}$$
$$F_b = 850 psi\text{ - for bending}$$
$$F_t = 525 psi\text{ - for axial tension}$$
$$F_v = 180 psi\text{ - for shear}$$
$$F_{c \perp} = 520 psi\text{ - for crushing (compresion perpendicular to grain)}$$
$$F_c = 1350 psi\text{ - for axial compression (compresion parallel to grain)}$$
$$E = 1.2e+06 psi\text{ - the unadjusted Young's modulus}$$
$$E_{min} = 440000 psi\text{ - unadjusted Young's modulus for column loading analysis}$$
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<h2 id="Loading-Analysis">Loading Analysis<a class="anchor-link" href="#Loading-Analysis">¶</a></h2>
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<p>For timber beams under a uniform line loading we generally check check bending first since it is the most likely failure mode. We will then check shear and finally deflection.</p>
<p>In this course we are generally only concerned with one load combination; that is the construction total load. We always will use it for bending and shear and most of the time we will use it for deflection as well. Since this is or first example it is a more complete example, e.g. we will be checking deflections for the Live Load <em>and</em> the Total Load.</p>
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<h3 id="Bending-Analysis:">Bending Analysis:<a class="anchor-link" href="#Bending-Analysis:">¶</a></h3>
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<p>The total uniform line load of our beam $w=120plf$. For a beam with a uniform line load, the value of the bending moment can be found with the following equation.</p>
$$\text{Moment} = \frac{\text{Line Load} \times (\text{Beam Span (or Length)})^2}{8}$$$$M = \frac{w L^2}{8}$$$$M = \frac{120 plf \times (10.5 ft)^2}{8}$$<p>so,</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">beam</span><span class="o">.</span><span class="n">M_latex</span> <span class="c">#in foot pounds</span>
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$$M = 1654 \text{ ft-lb} = 19840 \text{ in-lb}$$
$${}$$
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<h4 id="Actual-Bending-Stress">Actual Bending Stress<a class="anchor-link" href="#Actual-Bending-Stress">¶</a></h4>
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<p>The actual stresses are denoted using lower case $f$ instead of the upper case $F$ for the allowable stresses. The allowable bending stress is $F_b'$ so our actual bending stress is $f_b$. You may recall from your previous classes that the bending moment,</p>
<p>$\text{Moment} = \text{Maximum Bending Stress} \times \text{Section Modulus}$</p>
<p>or,</p>
<p>$M = f_b \times S$</p>
<p>so,</p>
<p>$f_b = \frac{M}{S}$</p>
<p>where $f_b$ is the stress at the extreme fibers of the top and bottom surfaces.</p>
<p>You may have also seen the relationship written as $\sigma = \frac{M \times c}{I}$ but $S = \frac{I}{c}$.</p>
<p><em>(Click the blue down arrow to see our beam's actual bending stress.)</em></p>
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<div class=" highlight hl-ipython3"><pre><span class="n">beam</span><span class="o">.</span><span class="n">f_b_latex</span>
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$$f_b = \frac{M}{S}$$
$${}$$
$$f_b = \frac{1654 ft\text{-}lb}{21.39 in^3}$$
$${}$$
$f_b = 927.7 psi$
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<p><em>(Note that this is equation 3.3-1 on page 15 in the 2015 Timber NDS.)</em></p>