From 42f603e311a0710e002bd21a833977ea8f14f77e Mon Sep 17 00:00:00 2001 From: Peter Mackenzie-Helnwein Date: Thu, 27 Jun 2019 23:08:08 -0700 Subject: [PATCH] adding examples to the documentation --- docs/help/theory.4ct | 25 +- docs/help/theory.4tc | 25 +- docs/help/theory.aux | 114 ++++---- docs/help/theory.dvi | Bin 55112 -> 68352 bytes docs/help/theory.html | 567 +++++++++++++++++++++++++--------------- docs/help/theory.idv | Bin 12505 -> 20205 bytes docs/help/theory.lg | 25 +- docs/help/theory.log | 139 +++++----- docs/help/theory.pdf | Bin 143250 -> 166444 bytes docs/help/theory.tex | 143 +++++++++- docs/help/theory.toc | 11 + docs/help/theory.xref | 150 ++++++----- docs/help/theory32x.png | Bin 0 -> 7709 bytes docs/help/theory33x.png | Bin 0 -> 8146 bytes docs/help/theory34x.png | Bin 0 -> 3353 bytes docs/help/theory35x.png | Bin 0 -> 1328 bytes docs/help/theory36x.png | Bin 0 -> 1865 bytes docs/help/theory37x.png | Bin 0 -> 2287 bytes docs/help/theory38x.png | Bin 0 -> 12112 bytes docs/help/theory39x.png | Bin 0 -> 2944 bytes docs/help/theory40x.png | Bin 0 -> 3025 bytes resources.qrc | 10 + 22 files changed, 781 insertions(+), 428 deletions(-) create mode 100644 docs/help/theory.toc create mode 100644 docs/help/theory32x.png create mode 100644 docs/help/theory33x.png create mode 100644 docs/help/theory34x.png create mode 100644 docs/help/theory35x.png create mode 100644 docs/help/theory36x.png create mode 100644 docs/help/theory37x.png create mode 100644 docs/help/theory38x.png create mode 100644 docs/help/theory39x.png create mode 100644 docs/help/theory40x.png diff --git a/docs/help/theory.4ct b/docs/help/theory.4ct index 5bd936c..1a99c38 100644 --- a/docs/help/theory.4ct +++ b/docs/help/theory.4ct @@ -1,14 +1,15 @@ \expandafter\ifx\csname doTocEntry\endcsname\relax \expandafter\endinput\fi -\doTocEntry\tocsection{1}{\csname a:TocLink\endcsname{1}{x1-10001}{QQ2-1-1}{Introduction}}{3}\relax -\doTocEntry\toclof{1}{\csname a:TocLink\endcsname{1}{x1-10011}{}{\ignorespaces Deformation of the Bernoulli-Euler beam. 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NcSAzX%%B(W?_VnD%EBernoulli-Euler beam theory
June 27, 2019
+

Contents

+ +

1 Introduction

-


Introduction +


+ id="x1-20011">
-

-

+

PIC

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

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 + id="x1-30002">Kinematics +

Navier’s assumption leads to
+ id="x1-3001r1">
u(x,y) = - yv′(x)   v(x,y) = v(x )
(1)
-

+

This displacement field induces an axial strain of
+ id="x1-3002r2">
ε(x,y) = ∂u(x,y)-= - yv′′(x) .
            ∂x
(2)
-

+

Equation (2) states that a fiber parallel to the beam axis stretches in the bottom +href="#x1-3002r2">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 + id="x1-40003">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:
+ id="x1-4001r3">
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 +

The imposed state of deformation induces normal stress proportional to the strain field (2) as +href="#x1-3002r2">2) as
+ id="x1-4002r4">
σ (x, y) = - Eyv ′′(x) .
(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, + id="x1-50004">Stress resultants +

The beam sees two stress resultants: the internal moment,
+ id="x1-5001r5">
          ∫
 
 M (x) = -  A yσ(x,y)dA
(5)
-

+

and the transverse shear force,
+ id="x1-5002r6">
         ∫
@@ -136,35 +164,35 @@ <h3 class=4
(6)
-

+

Substituting (4) into (5) yields +href="#x1-4002r4">4) into (5) yields
+ id="x1-5003r7">
        ∫     2 ′′           ′′   ∫   2         ′′
 M  (x ) =   Ey  v (x)dA = Ev  (x)   y dA = EIv  (x)
          A                       A
(7)
-

+

where
+ id="x1-5004r8">
    ∫
 I =    y2dA
      A
(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 @@ -174,10 +202,10 @@

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

+

5 Equilibrium

-

Equilibrium is formulated in terms of shear forces, Equilibrium

+

Equilibrium is formulated in terms of shear forces, V (x), and internal moments, 5 y-direction, yields
+ id="x1-6001r9">
V ′(x) = - w(x)
(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
+ id="x1-6002r10">
M ′(x) = V (x) .
(10)
-

+

A system for which equations (9) and (10) are sufficient to determine the +href="#x1-6001r9">9) and (10) 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 (7) and respective +href="#x1-5003r7">7) and respective boundary conditions. -

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

Equations (9) and (10) may be combined into one equation as
+ id="x1-6003r11">
  ′′       ′
 M  (x) = V (x) = - w (x )
(11)
-

+

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

+href="#x1-6003r11">11) replaces both equilibrium equations (9) and (10). +

6 Governing equation

-

The governing equation is obtained by assuming the displacement function, + id="x1-70006">Governing equation +

The governing equation is obtained by assuming the displacement function, v(x), as the primary unknown and expressing M(x) in (11) using (7) to +href="#x1-6003r11">11) using (7) to obtain
+ id="x1-7001r12">
(          )
  EI (x)v′′(x )′′ + w (x ) = 0
(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 +href="#x1-7001r12">12) simplifies to
+ id="x1-7002r13">
   ′′′′
 EIv  (x)+  w(x) = 0
(13)
-

+

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

+href="#x1-7002r13">13) is what is implemented in this program. +

7 Finding moment, shear force, and slope from the displacement + id="x1-80007">Finding moment, shear force, and slope from the displacement function

-

Solving (13) and applying suitable boundary conditions yields the displacement +

Solving (13) and applying suitable boundary conditions yields the displacement function, v(x), for the beam. The slope, 7 + id="x1-8001r14">

θ(x) = v′(x) .
(14) -

+

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

The moment follows from (7) as +

The moment follows from (7) as
+ id="x1-8002r15">
M (x) = EI (x )v′′(x) = EI (x)θ′(x) .
(15)
-

+

The transverse shear force follows from (11) as +href="#x1-6003r11">11) as
+ id="x1-8003r16">
V (x ) = M ′(x) = (EI (x)v′′(x))′
(16)
-

+

or, for constant EI, simplifies to
+ id="x1-8004r17">
V(x) = EIv ′′′(x) .
(17)
-

-

-

8 Examples

+

+

8 Examples

+

8.1 Single span beam with constant distributed force

-

Both bending stiffness, 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 +href="#x1-7002r13">13) simplifies to
+ id="x1-10001r18">
           w0
 v ′′′′(x) = - ---
            EI
(18)
-

+

+ id="x1-10002r19">
          w0x
@@ -366,59 +394,59 @@ <h4 class=8.1
(19)
-

+

+ id="x1-10003r20">
          w x2
 v′′(x) = - -0---+ c1x+ c2
           2EI
(20)
-

+

+ id="x1-10004r21">
                   3
 θ(x) = v′(x) = - w0x--+ 1c1x2 + c2x+ c3
                 6EI    2
(21)
-

+

+ id="x1-10005r22">
             4
 v (x ) = - w0x-+  1c1x3 + 1c2x2 + c3x + c4
          24EI    6       2
(22)
-

+

+ id="x1-10006r23">
            ′′       w0x2-
 M (x) = EIv  (x ) = -  2  + EIc1x  + EIc2
(23)
-

+

+ id="x1-10007r24">
V(x) = M ′(x) = EIv ′′′(x ) = - w x + EIc
                              0        1
(24)
-

+

Pinned on both ends yields the boundary conditions
+ id="x1-10008r25">
8.1
(25)
-

+

+ id="x1-10009r26">
                               (        )
@@ -443,10 +471,10 @@ <h4 class=8.1
(26)
-

+

+ id="x1-10010r27">
          (    w ℓ   )
@@ -458,29 +486,29 @@ <h4 class=8.1
(27)
-

+

+ id="x1-10011r28">
           4  (      )(  2        )
 v(x) = -w0ℓ- x- 1- x-   x--- x-- 1
        24EI  ℓ      ℓ   ℓ2   ℓ
(28)
-

+

+ id="x1-10012r29">
           ′′     w0-ℓ2x-(    x)
 M (x) = EIv (x) =   2  ℓ  1-  ℓ
(29)
-

+

+ id="x1-10013r30">
8.1
(30)
-

+

Shear vanishes at x = and, thus,
+ id="x1-10014r31">
                       2
 max  M =  M (ℓ∕2) = w0-ℓ
                      8
(31)
-

+

By symmetry, rotation vanishes at x = and, thus,
+ id="x1-10015r32">
                              5w0ℓ4-
 max  |v| = - min v = - v(ℓ∕2) = 384EI
(32)
-

+

-

+

8.2 Single span beam with a single concentrated force

+ id="x1-110008.2">Single span beam with a single concentrated force +

+
+                  v′1′′′(x)  =  0                           (33)
+                   v′′′(x)  =  c                           (34)
+                    1′′        1
+                   v1(x)  =  c1x + c2                    (35)
+                    ′         1-  2
+            θ1(x ) = v1(x) =   2c1x + c2x + c3            (36)
+                              1-  3   1-  2
+                   v1(x)  =   6c1x +  2c2x + c3x + c4    (37)
+        M  (x) = EIv ′′(x) =  EIc  x + EIc                (38)
+          1′         1′′′           1       2
+V1(x) = M 1(x) = EIv 1 (x) = EIc1                        (39)
+
+
+
+                    ′′′′
+                   v2 (x) =   0                           (40)
+                   v′2′′(x) =   d1                          (41)
+                     ′′
+                    v2(x) =   d1x + d2                    (42)
+            θ2(x) = v′(x) =   1d1x2 + d2x + d3            (43)
+                     2        2
+                    v (x) =   1d x3 + 1-d x2 + d x + d    (44)
+                     2        6  1    2  2     3     4
+        M2 (x) = EIv ′′2(x) =   EId1x + EId2                (45)
+          ′         ′′′
+V2(x) = M 2(x) = EIv 2 (x) =  EId1                        (46)
+
+
Boundary conditions +
+
+(   v(0) = v (0) )    ( 0 )
+||{          1     ||}    ||{   ||}
+   M (0) = M1 (0)  =    0
+||(   v(ℓ) = v2(ℓ) ||)    ||( 0 ||)
+   M (ℓ) = M2 (ℓ)       0
+
(47)
+

+Continuity conditions:
+ id="x1-11004r48">

-
(33)
-

+src="theory35x.png" alt="v1(a) = v2(a) and θ1(a) = θ2(a) +" class="math-display" >(48) +

+Equilibrium of forces for the interval [a - ϵ,a + ϵ]:
+ id="x1-11005r49">

-
(34)
-

+src="theory36x.png" alt="liϵ→m0 [V(a - ϵ)- P - V (a+ ϵ)] = 0 ⇒ V1(a)- V2(a) = P +" class="math-display" >(49) +

+Moment equilibrium for the interval [a - ϵ,a + ϵ]:
+ id="x1-11006r50">
- 
-
(35)
-

+src="theory37x.png" alt="lϵ→im0 [M (a- ϵ) + ϵV(a - ϵ)- M (a + ϵ)+ ϵV(a + ϵ)] = 0 ⇒ M1 (a) = M2 (a) +" class="math-display" >(50) +

+ id="x1-11007r51"> +

-
(36)
-

+src="theory38x.png" alt="⌊ 0 0 0 1 0 0 0 0 ⌋ ( ) ( ) +| | || c1 || || 0 || +| 0 EI 0 0 0 0 0 0 | ||| ||| ||| ||| +|| || |||| c2 |||| |||| 0 |||| +|| 0 0 0 0 ℓ3∕6 ℓ2∕2 ℓ 1 || |||| |||| |||| |||| +|| || ||| c3 ||| ||| 0 ||| +|| 0 0 0 0 EI ℓ EI 0 0 || ||{ c4 ||} ||{ 0 ||} +|| || = +| a3∕6 a2∕2 a 1 - a3∕6 - a2∕2 - a - 1 | ||| d1 ||| ||| 0 ||| +|| || |||| |||| |||| |||| +|| a2∕2 a 1 0 - a2∕2 - a - 1 0 || |||| d2 |||| |||| 0 |||| +|| || ||| d ||| ||| 0 ||| +|⌈ EIa EI 0 0 - EIa - EI 0 0 |⌉ |||| 3 |||| |||| |||| + ( d4 ) ( P ) + EI 0 0 0 - EI 0 0 0 +" class="math-display" >(51) +

+Using α = a∕ℓ, the integration constants are obtained as
+ id="x1-11008r52">

-
(37)
-

+src="theory39x.png" alt=" { ( ) } + (1---α)P- α--α2 --3α-+-2-Pℓ2- +{c1, c2, c3, c4} = EI , 0, - 6EI , 0 +" class="math-display" >(52) +

+and
+ id="x1-11009r53">

-
(38)
-

+src="theory40x.png" alt=" { ( 2) 2 3 3} +{d1, d2, d3, d4} = --αP-, αP-ℓ, - α--2+-α---Pℓ-, α--Pℓ- + 2EI EI 6EI 6EI +" class="math-display" >(53) +

+ id="x1-11010r54">

-
(39)
-

+src="theory41x.png" alt=" +" class="math-display" >(54) +

+ id="x1-11011r55">

-
(40)
-

+src="theory42x.png" alt=" +" class="math-display" >(55) +

+ id="x1-11012r56">

-
(41)
-

+src="theory43x.png" alt=" +" class="math-display" >(56) +

+ id="x1-11013r57">

-
(42)
-

-

+src="theory44x.png" alt=" +" class="math-display" >(57) +

+
+
+
+
(58)
+

+
+
+
+
(59)
+

+
+
+
+
(60)
+

+
+ +
+
+
(61)
+

+
+
+
+
(62)
+

+

8.3 Single span beam with a concentrated force and distributed load using + id="x1-120008.3">Single span beam with a concentrated force and distributed load using the stiffness method

- + id="x1-12001r63">

-
(43)
-

+src="theory50x.png" alt=" +" class="math-display" >(63) +

+ id="x1-12002r64">

-
(44)
-

+src="theory51x.png" alt=" +" class="math-display" >(64) +

+ id="x1-12003r65">

-
(45)
-

- +src="theory52x.png" alt=" +" class="math-display" >(65) +

+ id="x1-12004r66"> +

-
(46)
-

+src="theory53x.png" alt=" +" class="math-display" >(66) +

+ id="x1-12005r67">

-
(47)
-

+src="theory54x.png" alt=" +" class="math-display" >(67) +

+ id="x1-12006r68">

+src=
(68)
+

-" class="math-display" >(48) -

+ id="x1-12007r69">

-
(49)
-

+src="theory56x.png" alt=" +" class="math-display" >(69) +

+ id="x1-12008r70">

-
(50)
-

+src="theory57x.png" alt=" +" class="math-display" >(70) +

+ id="x1-12009r71">

-
(51)
-

+src="theory58x.png" alt=" + +" class="math-display" >(71) +

+ id="x1-12010r72">

-
(52)
-

+src="theory59x.png" alt=" +" class="math-display" >(72) +

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[25] -l. 341 Writing theory.idv[49] (theory48x.png) -l. 344 Writing theory.idv[50] (theory49x.png) -l. 347 Writing theory.idv[51] (theory50x.png) +l. 443 Writing theory.idv[47] (theory46x.png) +l. 446 Writing theory.idv[48] (theory47x.png) +l. 449 Writing theory.idv[49] (theory48x.png) [26] -l. 350 Writing theory.idv[52] (theory51x.png) - [27] (./theory.aux) ) +l. 452 Writing theory.idv[50] (theory49x.png) +l. 458 Writing theory.idv[51] (theory50x.png) + [27] +l. 461 Writing theory.idv[52] (theory51x.png) +l. 464 Writing theory.idv[53] (theory52x.png) +l. 467 Writing theory.idv[54] (theory53x.png) + [28] +l. 470 Writing theory.idv[55] (theory54x.png) +l. 473 Writing theory.idv[56] (theory55x.png) + [29] +l. 476 Writing theory.idv[57] (theory56x.png) +l. 479 Writing theory.idv[58] (theory57x.png) +l. 482 Writing theory.idv[59] (theory58x.png) + [30] +l. 485 Writing theory.idv[60] (theory59x.png) + [31] (./theory.aux) ) Here is how much of TeX's memory you used: - 5427 strings out of 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