From 9a614e7f4fb5fef3e5fa7e9351e1647089fb0482 Mon Sep 17 00:00:00 2001 From: Peter Mackenzie-Helnwein Date: Thu, 27 Jun 2019 20:15:47 -0700 Subject: [PATCH] moving to the next release New target: support for force method and stiffness method fundamentals --- SimpleBeam.pro.user.4.9-pre1 | 327 ++++++++++++++++++++ docs/about.html | 9 +- docs/help/theory.4ct | 4 + docs/help/theory.4tc | 4 + docs/help/theory.aux | 47 ++- docs/help/theory.dvi | Bin 31500 -> 55112 bytes docs/help/theory.html | 571 +++++++++++++++++++++++++++-------- docs/help/theory.idv | Bin 4053 -> 12505 bytes docs/help/theory.lg | 55 ++++ docs/help/theory.log | 143 ++++++--- docs/help/theory.pdf | Bin 134773 -> 143250 bytes docs/help/theory.tex | 202 ++++++++++++- docs/help/theory.xref | 47 ++- docs/help/theory15x.png | Bin 1399 -> 1399 bytes docs/help/theory17x.png | Bin 0 -> 868 bytes docs/help/theory18x.png | Bin 0 -> 1056 bytes docs/help/theory19x.png | Bin 0 -> 1299 bytes docs/help/theory20x.png | Bin 0 -> 1740 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Deploy Configuration + + ProjectExplorer.DefaultDeployConfiguration + + 1 + + + false + false + 1000 + + true + + false + false + false + false + true + 0.01 + 10 + true + 1 + 25 + + 1 + true + false + true + valgrind + + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + + 2 + + SimpleBeam + + Qt4ProjectManager.Qt4RunConfiguration:/Users/pmackenz/Development/Educational/SimpleBeam/SimpleBeam.pro + SimpleBeam.pro + + 3768 + false + true + false + true + false + false + true + + /Users/pmackenz/Development/Educational/build-SimpleBeam-Desktop_Qt_5_12_1_clang_64bit-Release/SimpleBeam.app/Contents/MacOS + + 1 + + + + ProjectExplorer.Project.TargetCount + 1 + + + ProjectExplorer.Project.Updater.FileVersion + 20 + + + Version + 20 + + diff --git a/docs/about.html b/docs/about.html index 88b9d36..1731275 100644 --- a/docs/about.html +++ b/docs/about.html @@ -1,5 +1,5 @@

Simple Beam Educational Tool

-

(version 1.0, released Winter 2019)

+

(version 1.1, released June 2019)

by Peter Mackenzie-Helnwein
pmackenz@uw.edu
@@ -14,3 +14,10 @@

(version 1.0, released Winter 2019)

  • a constant distributed load

    • on deflection, rotation, transverse shear and the bending moment in a single span beam.

    + +

    Version history

    +
      +
    1. version 1.0, released March 2019: single span beam under concentrated force and distributed force.
      2. +
    2. version 1.1, released June 2019: adding functionality for teaching the force method (method of superposition).
      3. +
    + diff --git a/docs/help/theory.4ct b/docs/help/theory.4ct index 3241f03..5bd936c 100644 --- a/docs/help/theory.4ct +++ b/docs/help/theory.4ct @@ -7,4 +7,8 @@ \doTocEntry\tocsection{5}{\csname a:TocLink\endcsname{1}{x1-50005}{QQ2-1-6}{Equilibrium}}{10}\relax \doTocEntry\tocsection{6}{\csname a:TocLink\endcsname{1}{x1-60006}{QQ2-1-7}{Governing equation}}{11}\relax \doTocEntry\tocsection{7}{\csname a:TocLink\endcsname{1}{x1-70007}{QQ2-1-8}{Finding moment, shear force, and slope from the displacement function}}{12}\relax +\doTocEntry\tocsection{8}{\csname a:TocLink\endcsname{1}{x1-80008}{QQ2-1-9}{Examples}}{14}\relax +\doTocEntry\tocsubsection{8.1}{\csname a:TocLink\endcsname{1}{x1-90008.1}{QQ2-1-10}{Single span beam with constant distributed force}}{14}\relax +\doTocEntry\tocsubsection{8.2}{\csname a:TocLink\endcsname{1}{x1-100008.2}{QQ2-1-11}{Single span beam with a single concentrated force}}{20}\relax +\doTocEntry\tocsubsection{8.3}{\csname 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    @@ -21,11 +18,11 @@

    Bernoulli-Euler beam theory

    Peter Mackenzie-Helnwein

    January 17, 2019
    +class="cmr-12">June 27, 2019

    1 Introduction

    -

    1 -

    -

    +

    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 +

    +

    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 + deformation.

    +

    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 ) -
    		(1)
    -

    +" class="math-display" >(1) +

    This displacement field induces an axial strain of -

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

    +" class="math-display" >(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 +

    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)
    -

    +" class="math-display" >(3) +

    where E is the modulus of elasticity. -

    The imposed state of deformation induces normal stress proportional to the +

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

    -
    σ (x, y) = - Eyv ′′(x) . -
    		(4)
    -

    +" class="math-display" >(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,
    ∫ -M (x) = - yσ(x,y)dA - A -
    		(5)
    -

    + +M (x) = - A yσ(x,y)dA +" class="math-display" >(5) +

    and the transverse shear force, -

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

    +" class="math-display" >(6) + +

    Substituting (4) into (5) yields -

    ∫ ∫ -M (x ) = Ey2v ′′(x)dA = Ev ′′(x) y2dA = EIv ′′(x) +src=
    		(7)
    -

    +" class="math-display" >(7) +

    where -

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

    +I = y2dA + A +" class="math-display" >(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 @@ -191,11 +173,11 @@

    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 x). Equilibrium of forces on an beam element of infinitesimal length, formulated in the y-direction, yields -

    V ′(x) = - w(x) -
    		(9)
    -

    +" class="math-display" >(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)
    -

    +src="theory9x.png" alt="M ′(x) = V (x) . +" class="math-display" >(10) +

    A system for which equations (9) and (10) are sufficient to determine the @@ -241,11 +219,11 @@

    5 statically indeterminate. Solving these equations for the latter requires consideration of the kinematic relation (7) and respective + boundary conditions. -

    Equations (

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

    @@ -253,17 +231,16 @@

    5

    		(11)
    -

    +" class="math-display" >(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 -
    		(12)
    -

    +src="theory11x.png" alt="( ) + EI (x)v′′(x )′′ + w (x ) = 0 +" class="math-display" >(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) = EI = const. and (12) simplifies to -

    EIv′′′′(x)+ w(x) = 0 -
    		(13)
    -

    +src="theory12x.png" alt=" ′′′′ +EIv (x)+ w(x) = 0 +" class="math-display" >(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 θ(x), is obtained through differentiation as -

    θ(x) = v′(x) . -
    		(14)
    -

    +" class="math-display" >(14) +

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

    The moment follows from (

    The moment follows from (7) as -

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

    +" class="math-display" >(15) +

    The transverse shear force follows from (11) as -

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

    +src="theory15x.png" alt="V (x ) = M ′(x) = (EI (x)v′′(x))′ +" class="math-display" >(16) +

    or, for constant EI, simplifies to -

    V(x) = EIv ′′′(x) . -
    		(17)
    -

    -

    + +" class="math-display" >(17) +

    +

    +

    8 Examples

    +

    +

    8.1 Single span beam with constant distributed force

    +

    Both bending stiffness, EI, and distributed load, w(x) = w0, are constant over +the length of the beam. Thus, (13) simplifies to +
    +
    + w0 +v ′′′′(x) = - --- + EI +
    		(18)
    +

    +
    +
    + w0x +v′′′(x ) = -----+ c1 + EI +
    		(19)
    + +

    +
    +
    + w x2 +v′′(x) = - -0---+ c1x+ c2 + 2EI +
    		(20)
    +

    +
    +
    + 3 +θ(x) = v′(x) = - w0x--+ 1c1x2 + c2x+ c3 + 6EI 2 +
    		(21)
    +

    +
    +
    + 4 +v (x ) = - w0x-+ 1c1x3 + 1c2x2 + c3x + c4 + 24EI 6 2 +
    		(22)
    +

    +
    +
    + ′′ w0x2- +M (x) = EIv (x ) = - 2 + EIc1x + EIc2 +
    		(23)
    +

    +
    +
    +V(x) = M ′(x) = EIv ′′′(x ) = - w x + EIc + 0 1 +
    		(24)
    +

    +Pinned on both ends yields the boundary conditions +
    + +
    +( ) ( ) +||{ v(0) ||} ||{ 0 ||} + M (0) = 0 +||( v(ℓ) ||) ||( 0 ||) + M (ℓ) 0 +
    		(25)
    +

    +
    +
    + ( ) +⌊ ⌋ ( ) || 0 || + 0 0 0 1 || c1|| |||| 0 |||| +| 0 EI 0 0 | { c2} { w0 ℓ4 } +|⌈ ℓ3∕6 ℓ2∕2 ℓ 1 |⌉ | c3| = | 24EI- | + EIℓ EI 0 0 |( c |) |||| 2 |||| + 4 |( w0-ℓ- |) + 2 +
    		(26)
    +

    +
    +
    + ( w ℓ ) +( ) |||| --0- |||| +|| c1 || ||| 2EI ||| +{ c2 } { 0 } +| c3 | = | w0ℓ3- | +|( c4 |) |||| - 24EI |||| + ||( ||) + 0 +
    		(27)
    +

    +
    +
    + 4 ( )( 2 ) +v(x) = -w0ℓ- x- 1- x- x--- x-- 1 + 24EI ℓ ℓ ℓ2 ℓ +
    		(28)
    +

    +
    +
    + ′′ w0-ℓ2x-( x) +M (x) = EIv (x) = 2 ℓ 1- ℓ +
    		(29)
    +

    +
    + +
    + w ℓ( x ) +V (x) = M ′(x ) = EIv ′′′(x) =--0- 1 - 2-- + 2 ℓ +
    		(30)
    +

    +Shear vanishes at x = ℓ and, thus, +
    +
    + 2 +max M = M (ℓ∕2) = w0-ℓ + 8 +
    		(31)
    +

    +By symmetry, rotation vanishes at x = ℓ and, thus, +
    +
    + 5w0ℓ4- +max |v| = - min v = - v(ℓ∕2) = 384EI +
    		(32)
    +

    + +

    +

    8.2 Single span beam with a single concentrated force

    +
    +
    + +
    		(33)
    +

    +
    +
    + +
    		(34)
    +

    +
    +
    + + +
    		(35)
    +

    +
    +
    + +
    		(36)
    +

    +
    +
    + +
    		(37)
    +

    +
    + +
    + +
    		(38)
    +

    +
    +
    + +
    		(39)
    +

    +
    +
    + +
    		(40)
    +

    + +
    +
    + +
    		(41)
    +

    +
    +
    + +
    		(42)
    +

    +

    +

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

    +
    + +
    + +
    		(43)
    +

    +
    +
    + +
    		(44)
    +

    +
    +
    + +
    		(45)
    +

    + +
    +
    + +
    		(46)
    +

    +
    +
    + +
    		(47)
    +

    +
    +
    + + +
    		(48)
    +

    +
    +
    + +
    		(49)
    +

    +
    +
    + +
    		(50)
    +

    +
    +
    + +
    		(51)
    +

    +
    +
    + +
    		(52)
    +

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+ ui->textBrowser->setSource(QUrl("qrc:/docs/help/theory.html")); + // // adjust size of application window to the available display // diff --git a/helpwindow.ui b/helpwindow.ui index ae3492a..0dcac23 100644 --- a/helpwindow.ui +++ b/helpwindow.ui @@ -32,164 +32,16 @@ - Bernoulli-Euler beam theory + true <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0//EN" "http://www.w3.org/TR/REC-html40/strict.dtd"> -<html><head><meta name="qrichtext" content="1" /><title>Bernoulli-Euler beam theory</title><style type="text/css"> +<html><head><meta name="qrichtext" content="1" /><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 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"> -<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="theory0x.png" /></p></td> -<td> -<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 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 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 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"> -<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="theory4x.png" /></p></td> -<td> -<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 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 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 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, <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 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 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 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, <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 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 <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 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, <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 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"> -<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="theory14x.png" /></p></td> -<td> -<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 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 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.cpp b/mainwindow.cpp index 7b181e2..0326a01 100644 --- a/mainwindow.cpp +++ b/mainwindow.cpp @@ -13,8 +13,8 @@ #include -#define MAX_FORCE 100. -#define MAX_W 10. +#define MAX_FORCE 50. +#define MAX_W 5. MainWindow::MainWindow(QWidget *parent) : QMainWindow(parent), @@ -73,55 +73,59 @@ void MainWindow::on_loadWslider_valueChanged(int value) { m_w = MAX_W * double(value) / 100. / 12.; ui->loadW->setValue(m_w*12.); - this->updatePlots(); + //this->updatePlots(); } void MainWindow::on_loadW_valueChanged(double arg1) { ui->loadWslider->setValue(int(100.*arg1/MAX_W)); + this->updatePlots(); } void MainWindow::on_loadW_editingFinished() { double arg1 = ui->loadW->value(); ui->loadWslider->setValue(int(100.*arg1/MAX_W)); + this->updatePlots(); } void MainWindow::on_loadP_valueChanged(double arg1) { ui->loadPslider->setValue(int(100.*arg1/MAX_FORCE)); + this->updatePlots(); } void MainWindow::on_loadP_editingFinished() { double arg1 = ui->loadP->value(); ui->loadPslider->setValue(int(100.*arg1/MAX_FORCE)); + this->updatePlots(); } void MainWindow::on_loadPslider_valueChanged(int value) { - m_P = MAX_FORCE * double(value) / 100.; - ui->loadP->setValue(m_P); - this->updatePlots(); + ui->loadP->setValue(MAX_FORCE * double(value) / 100.); + //this->updatePlots(); } void MainWindow::on_loadXP_valueChanged(double arg1) { ui->loadXPslider->setValue(int(100.*arg1*12./m_length)); - //this->updatePlots(); + this->updatePlots(); } void MainWindow::on_loadXP_editingFinished() { double arg1 = ui->loadXP->value(); ui->loadXPslider->setValue(int(100.*arg1*12./m_length)); + this->updatePlots(); } void MainWindow::on_loadXPslider_valueChanged(int value) { m_xP = m_length * double(value)/100.; ui->loadXP->setValue(m_xP/12.); - this->updatePlots(); + //this->updatePlots(); } void MainWindow::on_leftBCfree_clicked() @@ -535,10 +539,15 @@ void MainWindow::doAnalysis(void) { this->computeI(); m_solver->setParameters(m_E, m_I, m_length, leftBC, rightBC); + + m_P = ui->loadP->value(); + m_w = ui->loadW->value(); + m_xP = 12*ui->loadXP->value(); + m_status = m_solver->doAnalysis(m_w, m_P, m_xP); }; -void MainWindow::updateResultPlot(QCustomPlot * plot, QVector &x, QVector &y) +void MainWindow::updateResultPlot(QCustomPlot * plot, QVector &x, QVector &y, double xP) { plot->clearPlottables(); @@ -553,6 +562,25 @@ void MainWindow::updateResultPlot(QCustomPlot * plot, QVector &x, QVecto plot->graph(0)->rescaleAxes(); plot->graph(1)->rescaleAxes(true); + + if (xP > 0.0) + { + QVector XP; + QVector YP; + + double minY = MIN(y); + double maxY = MAX(y); + + XP.append(xP); YP.append(minY); + XP.append(xP); YP.append(maxY); + + plot->addGraph(); + QPen pen(Qt::darkGreen, 1); + pen.setStyle(Qt::DashDotLine); + plot->graph(2)->setPen(pen); + plot->graph(2)->setData(XP, YP); + } + plot->replot(); }; @@ -585,11 +613,13 @@ void MainWindow::updatePlots() d = &ZEROS; } + double xP = ui->loadXP->value(); + this->updateSystemPlot(); - this->updateResultPlot(ui->momentPlot, *X, *M); - this->updateResultPlot(ui->shearPlot, *X, *V); - this->updateResultPlot(ui->slopePlot, *X, *th); - this->updateResultPlot(ui->dispPlot, *X, *d); + this->updateResultPlot(ui->momentPlot, *X, *M, xP); + this->updateResultPlot(ui->shearPlot, *X, *V, xP); + this->updateResultPlot(ui->slopePlot, *X, *th, xP); + this->updateResultPlot(ui->dispPlot, *X, *d, xP); } void MainWindow::on_action_Load_triggered() @@ -833,3 +863,8 @@ void MainWindow::on_actionBackground_triggered() HelpWindow *help = new HelpWindow(); help->show(); } + +void MainWindow::on_action_Quit_triggered() +{ + this->close(); +} diff --git a/mainwindow.h b/mainwindow.h index c61ffb9..7e3e5fa 100644 --- a/mainwindow.h +++ b/mainwindow.h @@ -6,6 +6,11 @@ #include "./qcp/qcustomplot.h" #include "definitions.h" +#include + +#define MAX(vec) *std::max_element(vec.constBegin(), vec.constEnd()) +#define MIN(vec) *std::min_element(vec.constBegin(), vec.constEnd()) + class Solver; namespace Ui { @@ -23,7 +28,7 @@ class MainWindow : public QMainWindow void updateSystemPlot(void); void doAnalysis(void); - void updateResultPlot(QCustomPlot * plot, QVector &x, QVector &y); + void updateResultPlot(QCustomPlot * plot, QVector &x, QVector &y, double xP = -1.); void computeI(void); private slots: @@ -74,6 +79,8 @@ private slots: void on_action_About_triggered(); void on_actionBackground_triggered(); + void on_action_Quit_triggered(); + private: Ui::MainWindow *ui; diff --git a/mainwindow.ui b/mainwindow.ui index 5a4c44b..01c375e 100644 --- a/mainwindow.ui +++ b/mainwindow.ui @@ -6,7 +6,7 @@ 0 0 - 779 + 782 726 @@ -76,6 +76,9 @@ + + -100 + 100 @@ -87,11 +90,14 @@ - 0.000000000000000 + -100.000000000000000 100.000000000000000 + + 0.020000000000000 + @@ -999,7 +1005,7 @@ 0 0 - 779 + 782 22 @@ -1062,6 +1068,9 @@ &Quit + + Ctrl+Q + @@ -1075,6 +1084,9 @@ &About + + Ctrl+/, Ctrl+B + diff --git a/resources.qrc b/resources.qrc index 3768ff1..811d750 100644 --- a/resources.qrc +++ b/resources.qrc @@ -28,5 +28,20 @@ images/W-Logo_Purple_RGB.png docs/about.html docs/help/beam.png + docs/help/theory17x.png + docs/help/theory18x.png + docs/help/theory19x.png + docs/help/theory20x.png + docs/help/theory21x.png + docs/help/theory22x.png + docs/help/theory23x.png + docs/help/theory24x.png + docs/help/theory25x.png + docs/help/theory26x.png + docs/help/theory27x.png + docs/help/theory28x.png + docs/help/theory29x.png + docs/help/theory30x.png + docs/help/theory31x.png