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Bernoulli-Euler beam theory

Peter Mackenzie-Helnwein

January 16, 2019
+class="cmr-12">January 17, 2019

1 Introduction

-

This tool employs the Bernoulli-Euler beam theory. This theory, also +

+ + + +
+

+

PIC +

Figure 1: Deformation of the Bernoulli-Euler beam. Definition of coordinate +axes and components of displacement.
+
+ +
+

This tool employs the Bernoulli-Euler beam theory. This theory, also known as shear rigid beam theory, is based on the kinematic assumption that

-

Any plane cross section perpendicular to the undeformed beam’s +

Any plane cross section perpendicular to the undeformed beam’s axis remain plane and perpendicular to the axis throughout the deformation.

-

This allows us to reduce the three-dimensional problem to a single unknown +

This allows us to reduce the three-dimensional problem to a single unknown function, v(x), known as deflection of the beam. -

2 Kinematics

-

Navier’s assumption leads to +

Navier’s assumption leads to

-
′ -u(x,y) = - yv (x) v(x,y) = v(x ) +src=
(1)
-

+

+This displacement field induces an axial strain of

+
ε(x,y) = ∂u(x,y)-= - yv ′′(x ) +src=
(2)
-

-

+

+Equation (2) states that a fiber parallel to the beam axis stretches in the bottom +portion of the beam (y < 0) and contracts if the fiber is located above the beam +axis (y > 0). The beam axis itself does not stretch. +

3 Constitutive relations

+

For a slender beam, we can ignore stress components acting perpendicular to the +beam’s axis. Thus, the constitutive relations can be simplified as the 1D-version +of Hooke’s law: +

- σ(x,y) = Eε(x,y)
(3)
-

+

+where E is the modulus of elasticity. +

The imposed state of deformation induces normal stress proportional to the +strain field (2) as

+
σ(x,y) = - Eyv′′(x) +src=
(4)
-

-

+

+This relation states that (i) the stress varies linearly with the distance from the +beam’s axis, vanishing at the axis, and (ii) the stress is proportional to the +curvature of the beam. +

4 Stress resultants

-

The beam sees two stress resultants: the internal moment, +

The beam sees two stress resultants: the internal moment,

4 (5)
-

+

and the transverse shear force,

+
∫ -V (x ) = - τ (x,y)dA - A xy +V (x ) = - τxy(x,y)dA + A
(6)
-

+

Substituting (4) into (5) yields @@ -124,27 +162,27 @@

4 ∫ ∫ - 2 ′′ ′′ 2 ′′ -M (x ) = A Ey v (x)dA = Ev (x) A y dA = EIv (x) +M (x ) = Ey2v ′′(x)dA = Ev ′′(x) y2dA = EIv ′′(x) + A A (7) -

+

where

-
∫ -I = y2dA - A + 2 +I = A y dA
(8)
-

+ +

is the area moment of inertia or, short, moment of inertia. -

Note that the modulus of elasticity,

Note that the modulus of elasticity, E, characterizes the material, the moment of inertia, I, characterized the shape of the cross section, and the second @@ -153,11 +191,11 @@

4 ′′(x), characterizes the deformation (curvature) of the beam. -

+

5 Equilibrium

-

Equilibrium is formulated in terms of shear forces,

Equilibrium is formulated in terms of shear forces, V (x), and internal moments, 5 (9) -

+

where w(x) is the distributed lateral load per length. 5 x) is defined positive if pointing against the (upward) positive y-axis. - -

Moment equilibrium around the out-of-plane axis on the same element +

Moment equilibrium around the out-of-plane axis on the same element yields

+
′ M (x) = V (x) .
(10)
-

+

A system for which equations (9) and (10) are sufficient to determine the @@ -204,7 +242,7 @@

5 7) and respective boundary conditions. -

Equations (

Equations (9) and (10) may be combined into one equation as

@@ -216,17 +254,16 @@

5 (11) -

+

Equation (11) replaces both equilibrium equations (9) and (10). - -

+

6 Governing equation

-

The governing equation is obtained by assuming the displacement function, +

The governing equation is obtained by assuming the displacement function, v(x), as the primary unknown and expressing 6 11) using (7) to obtain +

(EI (x)v′′(x ))′′ + w (x ) = 0 +src=
(12)
-

+

This equation is known as the governing equation of the Bernoulli-Euler beam. -

If the beam possesses a constant cross section and is made of one material, +

If the beam possesses a constant cross section and is made of one material, then EI(x) = 6

′′′′ -EIv (x)+ w(x) = 0 +src=
(13) -

+

Equation (13) is what is implemented in this program. - -

+

7 Finding moment, shear force, and slope from the displacement function

-

Solving (

Solving (13) and applying suitable boundary conditions yields the displacement function, v(7 (14) -

+

It is positive if the cross section rotates counter-clockwise during deformation. -

The moment follows from (

The moment follows from (7) as

7
′′ ′ -M (x) = EI (x )v (x) = EI (x)θ(x) . +src=
(15)
- -

+

The transverse shear force follows from (11) as

+
′ ( ′′ )′ V (x ) = M (x) = EI (x)v (x)
(16)
-

+

or, for constant EI, simplifies to

@@ -324,7 +361,7 @@

7 (17) -

+

diff --git a/docs/help/theory.idv b/docs/help/theory.idv index 95e7e74..f2cc1ff 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 b909aba..3b525df 100644 --- a/docs/help/theory.lg +++ b/docs/help/theory.lg @@ -132,6 +132,7 @@ Css: table.multline, table.multline-star {width:100%;} Css: td.gather {text-align:center; } Css: table.gather {width:100%;} Css: div.gather-star {text-align:center;} +File: beam.png --- needs --- theory.idv[2] ==> theory0x.png --- --- needs --- theory.idv[3] ==> theory1x.png --- --- needs --- theory.idv[4] ==> theory2x.png --- diff --git a/docs/help/theory.log b/docs/help/theory.log index ad143f0..72c1cdf 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) 16 JAN 2019 16:05 +This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017) (preloaded format=latex 2017.5.23) 17 JAN 2019 14:35 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -480,40 +480,51 @@ LaTeX Font Info: Try loading font information for U+msb on input line 20. 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Definition of coordinate axes and components of displacement.} + \end{center} + \label{Fig:1} +\end{figure} This tool employs the Bernoulli-Euler beam theory. This theory, also known as \emph{shear rigid beam theory}, is based on the kinematic assumption that \begin{quote} Any plane cross section perpendicular to the undeformed beam's axis remain plane and perpendicular to the axis throughout the deformation. @@ -34,20 +41,27 @@ \section{Kinematics} v(x,y) = v(x) \label{Eq:1} \end{equation} +This displacement field induces an axial strain of \begin{equation} - \varepsilon(x,y) = \frac{\partial u(x,y)}{\partial x} = -y\, v''(x) + \varepsilon(x,y) = \frac{\partial u(x,y)}{\partial x} = -y\, v''(x) ~. \label{Eq:2} \end{equation} +Equation~\eqref{Eq:2} states that a fiber parallel to the beam axis stretches in the bottom portion of the beam ($y<0$) and contracts if the fiber is located above the beam axis ($y>0$). The beam axis itself does not stretch. \section{Constitutive relations} +For a slender beam, we can ignore stress components acting perpendicular to the beam's axis. Thus, the constitutive relations can be simplified as the 1D-version of Hooke's law: \begin{equation} \sigma(x,y) = E\,\varepsilon(x,y) \label{Eq:3} \end{equation} +where $E$ is the modulus of elasticity. + +The imposed state of deformation induces normal stress proportional to the strain field~\eqref{Eq:2} as \begin{equation} - \sigma(x,y) = -E y\,v''(x) + \sigma(x,y) = -E y\,v''(x) ~. \label{Eq:4} \end{equation} +This relation states that (i)~the stress varies linearly with the distance from the beam's axis, vanishing at the axis, and (ii)~the stress is proportional to the curvature of the beam. \section{Stress resultants} The beam sees two stress resultants: the internal moment, diff --git a/docs/help/theory.xref b/docs/help/theory.xref index bf1eb5f..130c9d2 100644 --- a/docs/help/theory.xref +++ b/docs/help/theory.xref @@ -2,26 +2,28 @@ \:CrossWord{TITLE+}{Bernoulli-Euler beam theory}{1}% \:CrossWord{)M1x0}{;}{3}% \:CrossWord{)Qx1-10001}{1}{3}% -\:CrossWord{)Qx1-20002}{1}{3}% -\:CrossWord{)Qx1-2001r1}{1}{3}% -\:CrossWord{)Qx1-2002r2}{1}{4}% -\:CrossWord{)Qx1-30003}{1}{4}% -\:CrossWord{)Qx1-3001r3}{1}{4}% -\:CrossWord{)Qx1-3002r4}{1}{5}% -\:CrossWord{)Qx1-40004}{1}{5}% -\:CrossWord{)Qx1-4001r5}{1}{5}% -\:CrossWord{)Qx1-4002r6}{1}{6}% -\:CrossWord{)Qx1-4003r7}{1}{6}% -\:CrossWord{)Qx1-4004r8}{1}{6}% -\:CrossWord{)Qx1-50005}{1}{7}% -\:CrossWord{)Qx1-5001r9}{1}{7}% -\:CrossWord{)Qx1-5002r10}{1}{8}% -\:CrossWord{)Qx1-5003r11}{1}{8}% -\:CrossWord{)Qx1-60006}{1}{9}% -\:CrossWord{)Qx1-6001r12}{1}{9}% -\:CrossWord{)Qx1-6002r13}{1}{9}% -\:CrossWord{)Qx1-70007}{1}{10}% -\:CrossWord{)Qx1-7001r14}{1}{10}% -\:CrossWord{)Qx1-7002r15}{1}{10}% -\:CrossWord{)Qx1-7003r16}{1}{11}% -\:CrossWord{)Qx1-7004r17}{1}{11}% +\:CrossWord{)Qx1-10011}{1}{4}% +\:CrossWord{1cAp0}{x1-10011}{5}% +\:CrossWord{)Qx1-20002}{1}{6}% +\:CrossWord{)Qx1-2001r1}{1}{6}% +\:CrossWord{)Qx1-2002r2}{1}{6}% +\:CrossWord{)Qx1-30003}{1}{7}% +\:CrossWord{)Qx1-3001r3}{1}{7}% +\:CrossWord{)Qx1-3002r4}{1}{8}% +\:CrossWord{)Qx1-40004}{1}{8}% +\:CrossWord{)Qx1-4001r5}{1}{8}% +\:CrossWord{)Qx1-4002r6}{1}{8}% +\:CrossWord{)Qx1-4003r7}{1}{9}% +\:CrossWord{)Qx1-4004r8}{1}{9}% +\:CrossWord{)Qx1-50005}{1}{10}% +\:CrossWord{)Qx1-5001r9}{1}{10}% +\:CrossWord{)Qx1-5002r10}{1}{10}% +\:CrossWord{)Qx1-5003r11}{1}{11}% +\:CrossWord{)Qx1-60006}{1}{11}% +\:CrossWord{)Qx1-6001r12}{1}{12}% +\:CrossWord{)Qx1-6002r13}{1}{12}% +\:CrossWord{)Qx1-70007}{1}{12}% +\:CrossWord{)Qx1-7001r14}{1}{13}% +\:CrossWord{)Qx1-7002r15}{1}{13}% +\:CrossWord{)Qx1-7003r16}{1}{13}% +\:CrossWord{)Qx1-7004r17}{1}{14}% diff --git a/docs/help/theory11x.png b/docs/help/theory11x.png index b60e251..d0c31a5 100644 Binary files a/docs/help/theory11x.png and b/docs/help/theory11x.png differ diff --git a/docs/help/theory15x.png b/docs/help/theory15x.png index 5d648a5..0e01030 100644 Binary files a/docs/help/theory15x.png and b/docs/help/theory15x.png differ diff --git a/docs/help/theory1x.png b/docs/help/theory1x.png index 41fdfdc..5fc1d79 100644 Binary files a/docs/help/theory1x.png and b/docs/help/theory1x.png differ diff --git a/docs/help/theory3x.png b/docs/help/theory3x.png index 03f66e4..8a0b36a 100644 Binary files a/docs/help/theory3x.png and b/docs/help/theory3x.png differ diff --git a/docs/help/theory4x.png b/docs/help/theory4x.png index da83d6c..37b20b5 100644 Binary files a/docs/help/theory4x.png and b/docs/help/theory4x.png differ diff --git a/docs/help/theory5x.png b/docs/help/theory5x.png index 0da7351..cca5de2 100644 Binary files a/docs/help/theory5x.png and b/docs/help/theory5x.png differ diff --git a/docs/help/theory6x.png b/docs/help/theory6x.png index 2985139..4fd7bda 100644 Binary files a/docs/help/theory6x.png and b/docs/help/theory6x.png differ diff --git a/docs/help/theory7x.png b/docs/help/theory7x.png index 3f4fa87..065d975 100644 Binary files a/docs/help/theory7x.png and b/docs/help/theory7x.png differ diff --git a/helpwindow.ui b/helpwindow.ui index acbbd8d..ae3492a 100644 --- a/helpwindow.ui +++ b/helpwindow.ui @@ -42,14 +42,18 @@ <html><head><meta name="qrichtext" content="1" /><title>Bernoulli-Euler beam theory</title><style type="text/css"> p, li { white-space: pre-wrap; } </style></head><body style=" font-family:'.SF NS Text'; font-size:13pt; font-weight:400; font-style:normal;"> -<h2 style=" margin-top:16px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><span style=" font-size:x-large; font-weight:600;">Bernoulli-Euler beam theory</span></h2> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Peter Mackenzie-Helnwein</p> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><br /></p> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">December 20, 2018 </p> -<h3 style=" margin-top:14px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><span style=" font-size:large; font-weight:600;">1 </span><a name="x1-10001"></a><span style=" font-size:large; font-weight:600;">I</span><span style=" font-size:large; font-weight:600;">ntroduction</span></h3> -<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">This tool employs the Bernoulli-Euler beam theory. This theory, also known as shear rigid beam theory, is based on the kinematic assumption that </p> -<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Any plane cross section perpendicular to the undeformed beam’s axis remain plane and perpendicular to the axis throughout the deformation.</p> -<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">This allows us to reduce the three-dimensional problem to a single unknown function, v(x), known as deflection of the beam. </p> +<h2 align="center" style=" margin-top:16px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><span style=" font-size:x-large; font-weight:600;">Bernoulli-Euler beam theory</span></h2> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Peter Mackenzie-Helnwein</p> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><br /></p> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">January 17, 2019 </p> +<h3 style=" margin-top:14px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><span style=" font-size:large; font-weight:600;">1 </span><a name="x1-10001"></a><span style=" font-size:large; font-weight:600;">I</span><span style=" font-size:large; font-weight:600;">ntroduction</span> </h3> +<hr /> +<p align="center" style=" margin-top:12px; margin-bottom:12px; margin-left:52px; margin-right:52px; -qt-block-indent:0; text-indent:0px;"><img src="beam.png" /><a name="x1-10011"></a> <br /></p> +<p style=" margin-top:0px; margin-bottom:0px; margin-left:100px; margin-right:68px; -qt-block-indent:0; text-indent:0px;"><span style=" font-weight:600;">Figure 1: </span>Deformation of the Bernoulli-Euler beam. Definition of coordinate axes and components of displacement. </p> +<hr /> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">This tool employs the Bernoulli-Euler beam theory. This theory, also known as <span style=" font-style:italic;">shear rigid beam theory</span>, is based on the kinematic assumption that </p> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:16px; margin-right:16px; -qt-block-indent:0; text-indent:0px;">Any plane cross section perpendicular to the undeformed beam’s axis remain plane and perpendicular to the axis throughout the deformation.</p> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">This allows us to reduce the three-dimensional problem to a single unknown function, <span style=" font-style:italic;">v</span>(<span style=" font-style:italic;">x</span>), known as <span style=" font-style:italic;">deflection </span>of the beam. </p> <h3 style=" margin-top:14px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><span style=" font-size:large; font-weight:600;">2 </span><a name="x1-20002"></a><span style=" font-size:large; font-weight:600;">K</span><span style=" font-size:large; font-weight:600;">inematics</span></h3> <p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Navier’s assumption leads to </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> @@ -57,28 +61,32 @@ p, li { white-space: pre-wrap; } <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory0x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-2001r1"></a>(1)</p></td></tr></table> -<p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><br /></p> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-2001r1"></a>(1)</p></td></tr></table> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">This displacement field induces an axial strain of </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory1x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-2002r2"></a>(2)</p></td></tr></table> -<h3 style=" margin-top:14px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><span style=" font-size:large; font-weight:600;">3 </span><a name="x1-30003"></a><span style=" font-size:large; font-weight:600;">C</span><span style=" font-size:large; font-weight:600;">onstitutive relations</span> </h3> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-2002r2"></a>(2)</p></td></tr></table> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Equation (<a href="#x1-2002r2"><span style=" text-decoration: underline; color:#0000ff;">2</span></a>) states that a fiber parallel to the beam axis stretches in the bottom portion of the beam (<span style=" font-style:italic;">y &lt; </span>0) and contracts if the fiber is located above the beam axis (<span style=" font-style:italic;">y &gt; </span>0). The beam axis itself does not stretch. </p> +<h3 style=" margin-top:14px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><span style=" font-size:large; font-weight:600;">3 </span><a name="x1-30003"></a><span style=" font-size:large; font-weight:600;">C</span><span style=" font-size:large; font-weight:600;">onstitutive relations</span></h3> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">For a slender beam, we can ignore stress components acting perpendicular to the beam’s axis. Thus, the constitutive relations can be simplified as the 1D-version of Hooke’s law: </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory2x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-3001r3"></a>(3)</p></td></tr></table> -<p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><br /></p> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-3001r3"></a>(3)</p></td></tr></table> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">where <span style=" font-style:italic;">E </span>is the modulus of elasticity. </p> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">The imposed state of deformation induces normal stress proportional to the strain field (<a href="#x1-2002r2"><span style=" text-decoration: underline; color:#0000ff;">2</span></a>) as </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory3x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-3002r4"></a>(4)</p></td></tr></table> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-3002r4"></a>(4)</p></td></tr></table> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">This relation states that (i) the stress varies linearly with the distance from the beam’s axis, vanishing at the axis, and (ii) the stress is proportional to the curvature of the beam. </p> <h3 style=" margin-top:14px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><span style=" font-size:large; font-weight:600;">4 </span><a name="x1-40004"></a><span style=" font-size:large; font-weight:600;">S</span><span style=" font-size:large; font-weight:600;">tress resultants</span></h3> <p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">The beam sees two stress resultants: the internal moment, </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> @@ -86,80 +94,80 @@ p, li { white-space: pre-wrap; } <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory4x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-4001r5"></a>(5)</p></td></tr></table> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-4001r5"></a>(5)</p></td></tr></table> <p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">and the transverse shear force, </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory5x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-4002r6"></a>(6)</p></td></tr></table> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-4002r6"></a>(6)</p></td></tr></table> <p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Substituting (<a href="#x1-3002r4"><span style=" text-decoration: underline; color:#0000ff;">4</span></a>) into (<a href="#x1-4001r5"><span style=" text-decoration: underline; color:#0000ff;">5</span></a>) yields </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory6x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-4003r7"></a>(7)</p></td></tr></table> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-4003r7"></a>(7)</p></td></tr></table> <p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">where </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory7x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-4004r8"></a>(8)</p></td></tr></table> -<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">is the area moment of inertia or, short, moment of inertia. </p> -<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Note that the modulus of elasticity, E, characterizes the material, the moment of inertia, I, characterized the shape of the cross section, and the second derivative of the deflection, v′′(x), characterizes the deformation (curvature) of the beam. </p> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-4004r8"></a>(8)</p></td></tr></table> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">is the <span style=" font-style:italic;">area moment of inertia </span>or, short, <span style=" font-style:italic;">moment of inertia</span>. </p> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Note that the modulus of elasticity, <span style=" font-style:italic;">E</span>, characterizes the material, the moment of inertia, <span style=" font-style:italic;">I</span>, characterized the shape of the cross section, and the second derivative of the deflection, <span style=" font-style:italic;">v</span>′′(<span style=" font-style:italic;">x</span>), characterizes the deformation (curvature) of the beam. </p> <h3 style=" margin-top:14px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><span style=" font-size:large; font-weight:600;">5 </span><a name="x1-50005"></a><span style=" font-size:large; font-weight:600;">E</span><span style=" font-size:large; font-weight:600;">quilibrium</span></h3> -<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Equilibrium is formulated in terms of shear forces, V (x), and internal moments, M(x). Equilibrium of forces on an beam element of infinitesimal length, formulated in the y-direction, yields </p> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Equilibrium is formulated in terms of shear forces, <span style=" font-style:italic;">V </span>(<span style=" font-style:italic;">x</span>), and internal moments, <span style=" font-style:italic;">M</span>(<span style=" font-style:italic;">x</span>). Equilibrium of forces on an beam element of infinitesimal length, formulated in the <span style=" font-style:italic;">y</span>-direction, yields </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory8x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-5001r9"></a>(9)</p></td></tr></table> -<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">where w(x) is the distributed lateral load per length. w(x) is defined positive if pointing against the (upward) positive y-axis. </p> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-5001r9"></a>(9)</p></td></tr></table> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">where <span style=" font-style:italic;">w</span>(<span style=" font-style:italic;">x</span>) is the distributed lateral load per length. <span style=" font-style:italic;">w</span>(<span style=" font-style:italic;">x</span>) is defined positive if pointing against the (upward) positive <span style=" font-style:italic;">y</span>-axis. </p> <p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Moment equilibrium around the out-of-plane axis on the same element yields </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory9x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-5002r10"></a>(10)</p></td></tr></table> -<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">A system for which equations (<a href="#x1-5001r9"><span style=" text-decoration: underline; color:#0000ff;">9</span></a>) and (<a href="#x1-5002r10"><span style=" text-decoration: underline; color:#0000ff;">10</span></a>) are sufficient to determine the internal moment and shear functions is called statically determinate. Otherwise, the system is called statically indeterminate. Solving these equations for the latter requires consideration of the kinematic relation (<a href="#x1-4003r7"><span style=" text-decoration: underline; color:#0000ff;">7</span></a>) and respective boundary conditions. </p> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-5002r10"></a>(10)</p></td></tr></table> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">A system for which equations (<a href="#x1-5001r9"><span style=" text-decoration: underline; color:#0000ff;">9</span></a>) and (<a href="#x1-5002r10"><span style=" text-decoration: underline; color:#0000ff;">10</span></a>) are sufficient to determine the internal moment and shear functions is called <span style=" font-style:italic;">statically determinate</span>. Otherwise, the system is called <span style=" font-style:italic;">statically indeterminate</span>. Solving these equations for the latter requires consideration of the kinematic relation (<a href="#x1-4003r7"><span style=" text-decoration: underline; color:#0000ff;">7</span></a>) and respective boundary conditions. </p> <p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Equations (<a href="#x1-5001r9"><span style=" text-decoration: underline; color:#0000ff;">9</span></a>) and (<a href="#x1-5002r10"><span style=" text-decoration: underline; color:#0000ff;">10</span></a>) may be combined into one equation as </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory10x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-5003r11"></a>(11)</p></td></tr></table> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-5003r11"></a>(11)</p></td></tr></table> <p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Equation (<a href="#x1-5003r11"><span style=" text-decoration: underline; color:#0000ff;">11</span></a>) replaces both equilibrium equations (<a href="#x1-5001r9"><span style=" text-decoration: underline; color:#0000ff;">9</span></a>) and (<a href="#x1-5002r10"><span style=" text-decoration: underline; color:#0000ff;">10</span></a>). </p> <h3 style=" margin-top:14px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><span style=" font-size:large; font-weight:600;">6 </span><a name="x1-60006"></a><span style=" font-size:large; font-weight:600;">G</span><span style=" font-size:large; font-weight:600;">overning equation</span></h3> -<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">The governing equation is obtained by assuming the displacement function, v(x), as the primary unknown and expressing M(x) in (<a href="#x1-5003r11"><span style=" text-decoration: underline; color:#0000ff;">11</span></a>) using (<a href="#x1-4003r7"><span style=" text-decoration: underline; color:#0000ff;">7</span></a>) to obtain </p> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">The governing equation is obtained by assuming the displacement function, <span style=" font-style:italic;">v</span>(<span style=" font-style:italic;">x</span>), as the primary unknown and expressing <span style=" font-style:italic;">M</span>(<span style=" font-style:italic;">x</span>) in (<a href="#x1-5003r11"><span style=" text-decoration: underline; color:#0000ff;">11</span></a>) using (<a href="#x1-4003r7"><span style=" text-decoration: underline; color:#0000ff;">7</span></a>) to obtain </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory11x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-6001r12"></a>(12)</p></td></tr></table> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-6001r12"></a>(12)</p></td></tr></table> <p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">This equation is known as the governing equation of the Bernoulli-Euler beam. </p> -<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">If the beam possesses a constant cross section and is made of one material, then EI(x) = EI = const. and (<a href="#x1-6001r12"><span style=" text-decoration: underline; color:#0000ff;">12</span></a>) simplifies to </p> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">If the beam possesses a constant cross section and is made of one material, then <span style=" font-style:italic;">EI</span>(<span style=" font-style:italic;">x</span>) = <span style=" font-style:italic;">EI </span>= <span style=" font-style:italic;">const. </span>and (<a href="#x1-6001r12"><span style=" text-decoration: underline; color:#0000ff;">12</span></a>) simplifies to </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory12x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-6002r13"></a>(13)</p></td></tr></table> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-6002r13"></a>(13)</p></td></tr></table> <p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Equation (<a href="#x1-6002r13"><span style=" text-decoration: underline; color:#0000ff;">13</span></a>) is what is implemented in this program. </p> -<h3 style=" margin-top:14px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><span style=" font-size:large; font-weight:600;">7 </span><a name="x1-70007"></a><span style=" font-size:large; font-weight:600;">F</span><span style=" font-size:large; font-weight:600;">inding moment, shear force, and slope from the displacement function</span></h3> -<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Solving (<a href="#x1-6002r13"><span style=" text-decoration: underline; color:#0000ff;">13</span></a>) and applying suitable boundary conditions yields the displacement function, v(x), for the beam. The slope, θ(x), is obtained through differentiation as </p> +<h3 style=" margin-top:14px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><span style=" font-size:large; font-weight:600;">7 </span><a name="x1-70007"></a><span style=" font-size:large; font-weight:600;">F</span><span style=" font-size:large; font-weight:600;">inding moment, shear force, and slope from the displacement function</span> </h3> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">Solving (<a href="#x1-6002r13"><span style=" text-decoration: underline; color:#0000ff;">13</span></a>) and applying suitable boundary conditions yields the displacement function, <span style=" font-style:italic;">v</span>(<span style=" font-style:italic;">x</span>), for the beam. The slope, <span style=" font-style:italic;">θ</span>(<span style=" font-style:italic;">x</span>), is obtained through differentiation as </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory13x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-7001r14"></a>(14)</p></td></tr></table> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-7001r14"></a>(14)</p></td></tr></table> <p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">It is positive if the cross section rotates counter-clockwise during deformation. </p> <p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">The moment follows from (<a href="#x1-4003r7"><span style=" text-decoration: underline; color:#0000ff;">7</span></a>) as </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> @@ -167,21 +175,21 @@ p, li { white-space: pre-wrap; } <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory14x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-7002r15"></a>(15)</p></td></tr></table> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-7002r15"></a>(15)</p></td></tr></table> <p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">The transverse shear force follows from (<a href="#x1-5003r11"><span style=" text-decoration: underline; color:#0000ff;">11</span></a>) as </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory15x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-7003r16"></a>(16)</p></td></tr></table> -<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">or, for constant EI, simplifies to </p> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-7003r16"></a>(16)</p></td></tr></table> +<p style=" margin-top:12px; margin-bottom:12px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;">or, for constant <span style=" font-style:italic;">EI</span>, simplifies to </p> <table border="0" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px;" cellspacing="2" cellpadding="0"> <tr> <td> <p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><img src="theory16x.png" /></p></td> <td> -<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-7004r17"></a>(17)</p></td></tr></table> +<p align="center" style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><a name="x1-7004r17"></a>(17)</p></td></tr></table> <p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px;"><br /></p></body></html> diff --git a/mainwindow.ui b/mainwindow.ui index c5f3e01..bb49441 100644 --- a/mainwindow.ui +++ b/mainwindow.ui @@ -15,173 +15,122 @@ - - + + + + + 0 + 0 + + - ft + w (k/ft) - - + + + + + 0 + 1 + + - Beam properties + Right support - + - + - I = in^4 + free - - Qt::AlignCenter + + + + + + pin + + + true - + - E = ... ksi + fixed - - Qt::AlignCenter + + + + + + slider - - - - - 0 - 0 - - - - 1 - + + - 100.000000000000000 + 100 - - 1.000000000000000 + + Qt::Vertical - - - - Material + + + + 0.000000000000000 - - false + + 100.000000000000000 - - - - - - Softwood - - - - - Hardwood - - - - - Steel - - - - - Concrete 4 ksi - - - - - Concrete 6 ksi - - - - - - - - - Dimensions + + + + + 0 + 0 + - - - - - Length - - - - - - - - 0 - 0 - - - - 1.000000000000000 - - - 1000.000000000000000 - - - 10.000000000000000 - - - - - - - ft - - - - - - - - - w (k/ft) + ft - - + + - - 2 - 1 + + 1 + 0 - - + + 100 - - - - - - P (kips) + + Qt::Horizontal + + + 0 + 1 + + Left support @@ -220,6 +169,32 @@ + + + + + 3 + 1 + + + + + + + + + + + + 0 + 0 + + + + P (kips) + + + @@ -233,63 +208,19 @@ - - - - Right support - - - - - - free - - - - - - - pin - - - true - - - - - - - fixed - - - - - - - slider - - - - - - + + + 0 + 0 + + xP - - - - 100 - - - Qt::Vertical - - - @@ -300,522 +231,31 @@ - - - - 0.000000000000000 - - - 100.000000000000000 - - - - - - - - 0 - 1 - - - - true - - - color='#ff0000', background='#00ff00' - - - - Cross section - - - - 12 - - - - - - 0 - 1 - - - - 0 - - - - - - - B - - - - - - - in - - - - - - - - 0 - 0 - - - - 2.000000000000000 - - - - - - - Qt::Vertical - - - - 20 - 40 - - - - - - - - - 0 - 0 - - - - 8.000000000000000 - - - - - - - H - - - - - - - in - - - - - - - - 0 - 0 - - - - - - - :/images/Rectangle.png - - - false - - - Qt::AlignCenter - - - - - - - - - - - - 0 - 0 - - - - 0.500000000000000 - - - - - - - tw - - - - - - - in - - - - - - - in - - - - - - - B - - - - - - - tf - - - - - - - - 0 - 0 - - - - 1.000000000000000 - - - - - - - H - - - - - - - in - - - - - - - - 0 - 0 - - - - 6.000000000000000 - - - - - - - Qt::Vertical - - - - 20 - 40 - - - - - - - - in - - - - - - - - 0 - 0 - - - - 4.000000000000000 - - - - - - - Qt::Vertical - - - - 20 - 40 - - - - - - - - - 0 - 0 - - - - - - - :/images/I-Beam.png - - - false - - - Qt::AlignCenter - - - - - - - - - - - in - - - - - - - Qt::Vertical - - - - 20 - 40 - - - - - - - - - 0 - 0 - - - - 0.500000000000000 - - - - - - - in - - - - - - - - 0 - 0 - - - - 0.500000000000000 - - - - - - - H - - - - - - - tf - - - - - - - t_bottom - - - - - - - in - - - - - - - in - - - - - - - - 0 - 0 - - - - 4.000000000000000 - - - - - - - t_top - - - - - - - - 0 - 0 - - - - 0.750000000000000 - - - - - - - in - - - - - - - - 0 - 0 - - - - 6.000000000000000 - - - - - - - B - - - - - - - - 0 - 0 - - - - - - - :/images/BoxChannel.png - - - false - - - Qt::AlignCenter - - - - - - - - - - - - Rectangular - - - - - I-beam - - - - - Channel/Box - - - - - - - - - + + - 1 + 0 0 + + 1 + - 100 + 100.000000000000000 - - Qt::Horizontal + + 1.000000000000000 - + 2 - 2 + 5 @@ -920,6 +360,632 @@ + + + + QFrame::NoFrame + + + QFrame::Raised + + + 0 + + + + -1 + + + 0 + + + 0 + + + 0 + + + 0 + + + + + Dimensions + + + + + + + 0 + 0 + + + + 1.000000000000000 + + + 1000.000000000000000 + + + 10.000000000000000 + + + + + + + ft + + + + + + + Length + + + + + + + + + + Material + + + false + + + + + + + Softwood + + + + + Hardwood + + + + + Steel + + + + + Concrete 4 ksi + + + + + Concrete 6 ksi + + + + + + + + + + + + 0 + 1 + + + + true + + + color='#ff0000', background='#00ff00' + + + + Cross section + + + + 12 + + + + + + 0 + 1 + + + + 0 + + + + + + + B + + + + + + + in + + + + + + + + 0 + 0 + + + + 2.000000000000000 + + + + + + + Qt::Vertical + + + + 20 + 40 + + + + + + + + + 0 + 0 + + + + 8.000000000000000 + + + + + + + H + + + + + + + in + + + + + + + + 0 + 0 + + + + + + + :/images/Rectangle.png + + + false + + + Qt::AlignCenter + + + + + + + + + + + + 0 + 0 + + + + 0.500000000000000 + + + + + + + + 0 + 0 + + + + 6.000000000000000 + + + + + + + tf + + + + + + + in + + + + + + + Qt::Vertical + + + + 20 + 40 + + + + + + + + + 0 + 0 + + + + 1.000000000000000 + + + + + + + in + + + + + + + + 0 + 0 + + + + 4.000000000000000 + + + + + + + in + + + + + + + in + + + + + + + H + + + + + + + + 0 + 0 + + + + + + + :/images/I-Beam.png + + + false + + + Qt::AlignCenter + + + + + + + tw + + + + + + + B + + + + + + + + + + + in + + + + + + + B + + + + + + + in + + + + + + + Qt::Vertical + + + + 20 + 40 + + + + + + + + in + + + + + + + H + + + + + + + + 0 + 0 + + + + + + + :/images/BoxChannel.png + + + false + + + Qt::AlignCenter + + + + + + + in + + + + + + + + 0 + 0 + + + + 4.000000000000000 + + + + + + + tf + + + + + + + + 0 + 0 + + + + 0.750000000000000 + + + + + + + t_top + + + + + + + + 0 + 0 + + + + 6.000000000000000 + + + + + + + t_bottom + + + + + + + + 0 + 0 + + + + 0.500000000000000 + + + + + + + + 0 + 0 + + + + 0.500000000000000 + + + + + + + in + + + + + + + Qt::Vertical + + + + 20 + 40 + + + + + + + + + + + + + Rectangular + + + + + I-beam + + + + + Channel/Box + + + + + + + + + + + Beam properties + + + + + + I = in^4 + + + Qt::AlignCenter + + + + + + + E = ... ksi + + + Qt::AlignCenter + + + + + + + + + diff --git a/resources.qrc b/resources.qrc index 48f0a74..3768ff1 100644 --- a/resources.qrc +++ b/resources.qrc @@ -27,5 +27,6 @@ docs/help/theory16x.png images/W-Logo_Purple_RGB.png docs/about.html + docs/help/beam.png