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Bernoulli-Euler beam theory

Peter Mackenzie-Helnwein

June 27, 2019
+class="cmr-12">June 28, 2019

Contents

@@ -44,7 +44,7 @@

 8.2 Single span beam with a single concentrated force
 8.3 Single span beam with a concentrated force and distributed +href="#x1-120008.3" id="QQ2-1-17">Single span beam with a concentrated force and distributed load using the stiffness method

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8.2
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PIC +

Figure 2: Deformed shape for different locations, α = a∕ℓ, for EI = 1, +ℓ = 1, P = 1
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Figure 3: Rotation for different locations, α = a∕ℓ, for EI = 1, ℓ = 1, +P = 1
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Figure 4: Moment diagrams for different locations, α = a∕ℓ, for EI = 1, +ℓ = 1, P = 1
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+ id="x1-11016r57">
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PIC +

Figure 5: Shear diagrams for different locations, α = a∕ℓ, for EI = 1, ℓ = 1, +P = 1
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8.3 Single span beam with a concentrated force and distributed load using +the stiffness method

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8.3 Single span beam with a concentrated force and distributed load using -the stiffness method

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diff --git a/docs/help/theory.idv b/docs/help/theory.idv index 75ec446..bdd0790 100644 Binary files a/docs/help/theory.idv and b/docs/help/theory.idv differ diff --git a/docs/help/theory.lg b/docs/help/theory.lg index 2936d62..aa510b2 100644 --- a/docs/help/theory.lg +++ b/docs/help/theory.lg @@ -175,13 +175,13 @@ File: beam.png --- needs --- theory.idv[41] ==> theory39x.png --- --- needs --- theory.idv[42] ==> theory40x.png --- --- needs --- theory.idv[43] ==> theory41x.png --- ---- empty picture --- theory.idv[43] --- +File: forceDeflection.png --- needs --- theory.idv[44] ==> theory42x.png --- ---- empty picture --- theory.idv[44] --- +File: forceSlope.png --- needs --- theory.idv[45] ==> theory43x.png --- ---- empty picture --- theory.idv[45] --- +File: forceMoment.png --- needs --- theory.idv[46] ==> theory44x.png --- ---- empty picture --- theory.idv[46] --- +File: forceShear.png --- needs --- theory.idv[47] ==> theory45x.png --- --- empty picture --- theory.idv[47] --- --- needs --- theory.idv[48] ==> theory46x.png --- @@ -202,21 +202,12 @@ File: beam.png --- empty picture --- theory.idv[55] --- --- needs --- theory.idv[56] ==> theory54x.png --- --- empty picture --- theory.idv[56] --- ---- needs --- theory.idv[57] ==> theory55x.png --- ---- empty picture --- theory.idv[57] --- ---- needs --- theory.idv[58] ==> theory56x.png --- ---- empty picture --- theory.idv[58] --- ---- needs --- theory.idv[59] ==> theory57x.png --- ---- empty picture --- theory.idv[59] --- ---- needs --- theory.idv[60] ==> theory58x.png --- ---- empty picture --- theory.idv[60] --- ---- needs --- theory.idv[61] ==> theory59x.png --- ---- empty picture --- theory.idv[61] --- --- characters --- Font("cmr","10","10","109") Font("cmr","17","17","100") Font("cmr","12","12","100") Font("cmr","8","8","100") +Font("cmr","6","6","100") Font("cmmi","8","8","100") Font("cmsy","8","8","100") Font("cmti","10","10","109") diff --git a/docs/help/theory.log b/docs/help/theory.log index 97c703e..24f060b 100644 --- a/docs/help/theory.log +++ b/docs/help/theory.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017) (preloaded format=latex 2017.5.23) 27 JUN 2019 23:03 +This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017) (preloaded format=latex 2017.5.23) 28 JUN 2019 00:10 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -556,39 +556,78 @@ l. 395 Writing theory.idv[40] (theory39x.png) [23] l. 412 Writing theory.idv[41] (theory40x.png) l. 428 Writing theory.idv[42] (theory41x.png) -l. 431 Writing theory.idv[43] (theory42x.png) - 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5481 strings out of 492996 - 63062 string characters out of 6132730 - 256704 words of memory out of 5000000 - 8938 multiletter control sequences out of 15000+600000 + 5499 strings out of 492996 + 63319 string characters out of 6132730 + 256745 words of memory out of 5000000 + 8952 multiletter control sequences out of 15000+600000 11558 words of font info for 44 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 - 30i,10n,32p,874b,506s stack positions out of 5000i,500n,10000p,200000b,80000s + 30i,11n,32p,874b,506s stack positions out of 5000i,500n,10000p,200000b,80000s -Output written on theory.dvi (31 pages, 68352 bytes). +Output written on theory.dvi (39 pages, 71172 bytes). diff --git a/docs/help/theory.pdf b/docs/help/theory.pdf index d34f1ad..1d70fe2 100644 Binary files a/docs/help/theory.pdf and b/docs/help/theory.pdf differ diff --git a/docs/help/theory.tex b/docs/help/theory.tex index fe1636c..39b1a63 100644 --- a/docs/help/theory.tex +++ b/docs/help/theory.tex @@ -426,32 +426,72 @@ \subsection{Single span beam with a single concentrated force} \label{B9} \end{equation} \begin{equation} + v(x) = \left\{ \begin{array}{ll} + \frac{(1-\alpha)Px(x^2 - \alpha(2-\alpha)\ell^2)}{6\,EI} & xa + \end{array} \right. \label{B10} \end{equation} + +\begin{figure}[htbp] + \begin{center} + \includegraphics[]{forceDeflection.png} + \caption{Deformed shape for different locations, $\alpha=a/\ell$, for $EI=1$, $\ell=1$, $P=1$} + \label{default1} + \end{center} +\end{figure} + \begin{equation} + \theta(x) = \left\{ \begin{array}{ll} + \frac{(1-\alpha)P( 3 x^2 - \alpha(2-\alpha)\ell^2)}{6\,EI} & xa + \end{array} \right. \label{B11} \end{equation} + +\begin{figure}[htbp] + \begin{center} + \includegraphics[]{forceSlope.png} + \caption{Rotation for different locations, $\alpha=a/\ell$, for $EI=1$, $\ell=1$, $P=1$} + \label{default2} + \end{center} +\end{figure} + \begin{equation} + M(x) = \left\{ \begin{array}{ll} + (1-\alpha) P x & xa + \end{array} \right. \label{B12} \end{equation} + +\begin{figure}[htbp] + \begin{center} + \includegraphics[]{forceMoment.png} + \caption{Moment diagrams for different locations, $\alpha=a/\ell$, for $EI=1$, $\ell=1$, $P=1$} + \label{default3} + \end{center} +\end{figure} + \begin{equation} + V(x) = \left\{ \begin{array}{ll} + (1-\alpha)P & xa + \end{array} \right. \label{B13} \end{equation} -\begin{equation} - \label{B14} -\end{equation} -\begin{equation} - \label{B15} -\end{equation} -\begin{equation} - \label{B16} -\end{equation} -\begin{equation} - \label{B17} -\end{equation} -\begin{equation} - \label{B18} -\end{equation} + +\begin{figure}[htbp] + \begin{center} + \includegraphics[]{forceShear.png} + \caption{Shear diagrams for different locations, $\alpha=a/\ell$, for $EI=1$, $\ell=1$, $P=1$} + \label{default4} + \end{center} +\end{figure} \subsection{Single span beam with a concentrated force and distributed load using the stiffness method} diff --git a/docs/help/theory.toc b/docs/help/theory.toc deleted file mode 100644 index 9cc5c79..0000000 --- a/docs/help/theory.toc +++ /dev/null @@ -1,11 +0,0 @@ -\contentsline {section}{\numberline {1}Introduction}{1} -\contentsline {section}{\numberline {2}Kinematics}{2} -\contentsline {section}{\numberline {3}Constitutive relations}{2} -\contentsline {section}{\numberline {4}Stress resultants}{3} -\contentsline {section}{\numberline {5}Equilibrium}{3} -\contentsline {section}{\numberline {6}Governing equation}{4} -\contentsline {section}{\numberline {7}Finding moment, shear force, and slope from the displacement function}{4} -\contentsline {section}{\numberline {8}Examples}{5} -\contentsline {subsection}{\numberline {8.1}Single span beam with constant distributed force}{5} -\contentsline {subsection}{\numberline {8.2}Single span beam with a single concentrated force}{6} -\contentsline {subsection}{\numberline {8.3}Single span beam with a concentrated force and distributed load using the stiffness method}{8} diff --git a/docs/help/theory.xref b/docs/help/theory.xref index 7fc8691..1aed13b 100644 --- a/docs/help/theory.xref +++ b/docs/help/theory.xref @@ -12,7 +12,7 @@ \:CrossWord{)QQQ2-1-10}{1}{3}% \:CrossWord{)QQQ2-1-11}{1}{3}% \:CrossWord{)QQQ2-1-12}{1}{3}% -\:CrossWord{)QQQ2-1-13}{1}{3}% +\:CrossWord{)QQQ2-1-17}{1}{3}% \:CrossWord{)Qx1-20001}{1}{3}% \:CrossWord{)Qx1-20011}{1}{4}% \:CrossWord{1cAp0}{x1-20011}{5}% @@ -67,22 +67,25 @@ \:CrossWord{)Qx1-11008r52}{1}{23}% \:CrossWord{)Qx1-11009r53}{1}{23}% \:CrossWord{)Qx1-11010r54}{1}{24}% -\:CrossWord{)Qx1-11011r55}{1}{24}% -\:CrossWord{)Qx1-11012r56}{1}{25}% -\:CrossWord{)Qx1-11013r57}{1}{25}% -\:CrossWord{)Qx1-11014r58}{1}{25}% -\:CrossWord{)Qx1-11015r59}{1}{26}% -\:CrossWord{)Qx1-11016r60}{1}{26}% -\:CrossWord{)Qx1-11017r61}{1}{26}% -\:CrossWord{)Qx1-11018r62}{1}{27}% -\:CrossWord{)Qx1-120008.3}{1}{27}% -\:CrossWord{)Qx1-12001r63}{1}{27}% -\:CrossWord{)Qx1-12002r64}{1}{28}% -\:CrossWord{)Qx1-12003r65}{1}{28}% -\:CrossWord{)Qx1-12004r66}{1}{28}% -\:CrossWord{)Qx1-12005r67}{1}{29}% -\:CrossWord{)Qx1-12006r68}{1}{29}% -\:CrossWord{)Qx1-12007r69}{1}{30}% -\:CrossWord{)Qx1-12008r70}{1}{30}% -\:CrossWord{)Qx1-12009r71}{1}{30}% -\:CrossWord{)Qx1-12010r72}{1}{31}% +\:CrossWord{)Qx1-110112}{1}{25}% +\:CrossWord{2cAp0}{x1-110112}{26}% +\:CrossWord{)Qx1-11012r55}{1}{27}% +\:CrossWord{)Qx1-110133}{1}{28}% +\:CrossWord{3cAp0}{x1-110133}{29}% +\:CrossWord{)Qx1-11014r56}{1}{30}% +\:CrossWord{)Qx1-110154}{1}{31}% +\:CrossWord{4cAp0}{x1-110154}{32}% +\:CrossWord{)Qx1-11016r57}{1}{33}% +\:CrossWord{)Qx1-110175}{1}{34}% +\:CrossWord{5cAp0}{x1-110175}{35}% +\:CrossWord{)Qx1-120008.3}{1}{36}% +\:CrossWord{)Qx1-12001r58}{1}{36}% +\:CrossWord{)Qx1-12002r59}{1}{36}% +\:CrossWord{)Qx1-12003r60}{1}{36}% +\:CrossWord{)Qx1-12004r61}{1}{37}% +\:CrossWord{)Qx1-12005r62}{1}{37}% +\:CrossWord{)Qx1-12006r63}{1}{38}% +\:CrossWord{)Qx1-12007r64}{1}{38}% +\:CrossWord{)Qx1-12008r65}{1}{38}% +\:CrossWord{)Qx1-12009r66}{1}{39}% +\:CrossWord{)Qx1-12010r67}{1}{39}% diff --git a/docs/help/theory41x.png b/docs/help/theory41x.png new file mode 100644 index 0000000..8fbbfa4 Binary files /dev/null and b/docs/help/theory41x.png differ diff --git a/docs/help/theory42x.png b/docs/help/theory42x.png new file mode 100644 index 0000000..28637c5 Binary files /dev/null and b/docs/help/theory42x.png differ diff --git a/docs/help/theory43x.png b/docs/help/theory43x.png new file mode 100644 index 0000000..7ca29cc Binary files /dev/null and b/docs/help/theory43x.png differ diff --git a/docs/help/theory44x.png b/docs/help/theory44x.png new file mode 100644 index 0000000..faeec77 Binary files /dev/null and b/docs/help/theory44x.png differ diff --git a/resources.qrc b/resources.qrc index 95b20a5..d0b984c 100644 --- a/resources.qrc +++ b/resources.qrc @@ -4,8 +4,6 @@ images/I-Beam.png images/Rectangle.png docs/help/theory.html - 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