Appendix ===== Presentation ------------ We consider the gantry opposite, subject to various loads. +-----------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |We consider the gantry opposite, subject to various loads. | + .. image:: images/10000000000001D2000001672BDC41EDF32CABE4.png + + | :width: 3.5626in | Degree 1 hyperstaticity. Hyperstatic unknown: X: moment in C. | + :height: 2.5654in + + | | Vertical loading distributed :math:`p` over :math:`{C}_{1}C`. Two forces :math:`{F}_{1}`, :math:`{F}_{2}` and a couple in :math:`{C}_{1}`. | + + + | | | +-----------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ Degree 1 hyperstaticity. Hyperstatic unknown: :math:`X` Charges applied: * moment in :math:`C`, * vertical loading distributed :math:`p` over :math:`{C}_{1}{C}_{2}`, * force :math:`\mathrm{F1}`, :math:`\mathrm{F2}` applied in :math:`{C}_{1}`, * :math:`\Gamma` torque applied in :math:`{C}_{1}` .. image:: images/100000000000012D000000BA3A22FD6020CAD4B5.png :width: 1.6972in :height: 1.1543in .. _RefImage_100000000000012D000000BA3A22FD6020CAD4B5.png: :math:`\mathrm{tan}(\alpha )=\frac{\mathrm{2a}}{l}=0.4(\Rightarrow {(\mathrm{cos}(\alpha ))}^{-1}=\sqrt{1.16}=1.077033)` :math:`\mathrm{tan}(\beta )=\frac{l}{2(a+h)}=\frac{1}{1.2}` :math:`b=\frac{l}{\mathrm{2cos}(\alpha )};\mathrm{sin}(\alpha )=\frac{a}{b}` Isostatic stresses under real load distributed :math:`p` on :math:`{C}_{1}C` ------------------------------------------------------------------------- Isostatic support reactions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ :math:`{H}_{A}+{H}_{B}=0` :math:`{V}_{A}+{V}_{B}=\frac{\mathrm{pl}}{\mathrm{2cos}(\alpha )}` :math:`{\mathrm{lV}}_{B}=\frac{{\mathrm{pl}}^{2}}{\mathrm{8cos}(\alpha )}` Part :math:`\mathrm{CB}` is articulated and loaded only at its ends :math:`(\begin{array}{}{H}_{B}\\ {V}_{B}\end{array})\wedge \mathrm{BC}=0\iff {H}_{B}=-{V}_{B}\mathrm{tan}(\beta )` Hence the isostatic reactions :math:`{H}_{A}=\frac{\mathrm{pl}}{\mathrm{8cos}(\alpha )}\mathrm{tan}(\beta )`; :math:`{V}_{A}=\frac{3\mathrm{pl}}{\mathrm{8cos}(\alpha )}`; :math:`{H}_{B}=\frac{-\mathrm{pl}}{\mathrm{8cos}(\alpha )}\mathrm{tan}(\beta )`; :math:`{V}_{B}=\frac{\mathrm{pl}}{\mathrm{8cos}(\alpha )}` **Note:** :math:`\frac{l\mathrm{tan}(\beta )}{8\mathrm{cos}(\alpha )}=\frac{\mathrm{bl}}{8(a+h)}` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Solicitations ~~~~~~~~~~~~~~~ **Beam** :math:`{\mathrm{AC}}_{1}` +-----------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |:math:`\begin{array}{}{N}_{\mathrm{iso}}=\frac{-3\mathrm{pl}}{\mathrm{8cos}(\alpha )}\\ {V}_{\mathrm{iso}}=\frac{\mathrm{pl}}{\mathrm{8cos}(\alpha )}\mathrm{tan}(\beta )\\ {M}_{\mathrm{iso}}=\frac{-\mathrm{pl}}{\mathrm{8cos}(\alpha )}\mathrm{y.tan}(\beta )\end{array}`| + .. image:: images/100000000000007B00000080327158A9E207E182.png + + | :width: 1.2811in | | + :height: 1.2535in + + | | | + + + | | | +-----------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ **Beam** :math:`{C}_{2}B` +-----------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |:math:`\begin{array}{}{N}_{\mathrm{iso}}=\frac{-\mathrm{pl}}{\mathrm{8cos}(\alpha )}\\ {V}_{\mathrm{iso}}=\frac{-\mathrm{pl}}{\mathrm{8cos}(\alpha )}\mathrm{tan}(\beta )\\ {M}_{\mathrm{iso}}=\frac{-\mathrm{pl}}{\mathrm{8cos}(\alpha )}\mathrm{y.tan}(\beta )\end{array}`| + .. image:: images/1000000000000070000000809E493CF23E17DE52.png + + | :width: 1.1665in | | + :height: 1.1819in + + | | | + + + | | | +-----------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ **Beam** :math:`{C}_{1}C` +-----------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |:math:`\begin{array}{cc}{N}_{\mathrm{iso}}& \text{=}-{H}_{A}\mathrm{cos}(\alpha )-{V}_{A}\mathrm{sin}(\alpha )+\frac{\mathrm{px}}{\mathrm{cos}(\alpha )}\mathrm{sin}(\alpha )\\ & \text{=}-\frac{\mathrm{pl}}{8}(\mathrm{tan}(\beta )+3\mathrm{tan}(\alpha )-8\mathrm{tan}(\alpha )\frac{x}{l})\end{array}` | + .. image:: images/1000000000000106000000F97237DECA6E26F809.png + + | :width: 2.4028in | :math:`\begin{array}{cc}{V}_{\mathrm{iso}}& \text{=}{H}_{A}\mathrm{sin}(\alpha )-{V}_{A}\mathrm{cos}(\alpha )+\frac{\mathrm{px}}{\mathrm{cos}(\alpha )}\mathrm{cos}(\alpha )\\ & \text{=}\frac{\mathrm{pl}}{8}(\mathrm{tan}(\beta )\mathrm{tan}(\alpha )-3+8\frac{x}{l})\end{array}` | + :height: 2.1717in + + | | :math:`\begin{array}{cc}{M}_{\mathrm{iso}}& \text{=}-\frac{{\mathrm{px}}^{2}}{\mathrm{2cos}(\alpha )}+{V}_{A}x-{H}_{A}y\\ & \text{=}\frac{p}{\mathrm{cos}(\alpha )}(\frac{-{x}^{2}}{2}+\frac{3\mathrm{lx}}{8}-\frac{\mathrm{ly}\mathrm{tan}(\beta )}{8})\end{array}` | + + + | | with :math:`{M}_{\mathrm{iso}}=0` :math:`C` | + + + | | | +-----------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ :math:`{M}_{\mathrm{iso}}=\frac{-\mathrm{pl}}{8(a+h)}(2{s}^{2}(\frac{a+h}{b})-s(\mathrm{2a}+\mathrm{3h})+\mathrm{bh})` with :math:`s=\frac{x}{\mathrm{cos}(\alpha )}\in [\mathrm{0,}b]` **Beam** :math:`{\mathrm{CC}}_{2}` +----------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |:math:`\begin{array}{cc}{N}_{\mathrm{iso}}& \text{=}{H}_{B}\mathrm{cos}(\alpha )-{V}_{B}\mathrm{sin}(\alpha )\\ & \text{=}-\frac{\mathrm{pl}}{8}(\mathrm{tan}(\beta )+\mathrm{tan}(\alpha ))\end{array}` | + .. image:: images/100000000000013B000000C629026E018D1EA36B.png + + | :width: 2.889in | :math:`\begin{array}{cc}{V}_{\mathrm{iso}}& \text{=}{H}_{B}\mathrm{sin}(\alpha )+{V}_{B}\mathrm{cos}(\alpha )\\ & \text{=}-\frac{\mathrm{pl}}{8}(\mathrm{tan}(\beta )\mathrm{tan}(\alpha )-1)\end{array}` :math:`\begin{array}{cc}{M}_{\mathrm{iso}}& \text{=}{H}_{B}y-{V}_{B}(l-x)\\ & \text{=}-\frac{\mathrm{pl}}{\mathrm{8cos}(\alpha )}(\mathrm{y.tan}(\beta )-(l-x))\end{array}` with :math:`{M}_{\mathrm{iso}}=0` in :math:`C` | + :height: 1.8118in + + | | | + + + | | | +----------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ Diagrams ~~~~~~~~~~ :math:`(b=\frac{l}{\mathrm{2cos}(\alpha )})` .. image:: images/1000000000000308000000FF05AA8138C4D73811.png :width: 6.4575in :height: 2.1138in .. _RefImage_1000000000000308000000FF05AA8138C4D73811.png: Concentrated force stresses :math:`{F}_{1}` (downward) ----------------------------------------------------------------- Supportive reactions ~~~~~~~~~~~~~~~~~~ +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ |:math:`{H}_{A}+{H}_{B}=0;` :math:`{V}_{A}+{V}_{B}={F}_{1};` :math:`(\begin{array}{}{H}_{A}\\ {V}_{A}\end{array})\wedge \mathrm{AC}=0=(\begin{array}{}{H}_{B}\\ {V}_{C}\end{array})\wedge \mathrm{BC};`| | + + .. image:: images/100000000000012D000000E9E1660AAA6525BEA3.png + | | :width: 2.2854in | + + :height: 1.6874in + | | | + + + | | | +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ From where: :math:`{H}_{A}=\frac{1}{2}{F}_{1}\mathrm{tan}(\beta )`; :math:`{V}_{A}=\frac{1}{2}{F}_{1}`; :math:`{H}_{B}=\frac{-1}{2}{F}_{1}\mathrm{tan}(\beta )`; :math:`{V}_{B}=\frac{1}{2}{F}_{1}` Solicitations ~~~~~~~~~~~~~~~ .. csv-table:: "**Beam** :math:`{\mathrm{AC}}_{1}` **:**", ":math:`{N}_{\mathrm{iso}}=\frac{-1}{2}{F}_{1}` :math:`{V}_{\mathrm{iso}}=\frac{1}{2}{F}_{1}\mathrm{tan}(\beta )` :math:`{M}_{\mathrm{iso}}=\frac{-1}{2}{F}_{1}y\mathrm{tan}(\beta )`" "**Beam** :math:`{C}_{2}B` **:**", ":math:`{N}_{\mathrm{iso}}=\frac{-1}{2}{F}_{1}` :math:`{V}_{\mathrm{iso}}=\frac{-1}{2}{F}_{1}\mathrm{tan}(\beta )` :math:`{M}_{\mathrm{iso}}=\frac{-1}{2}{F}_{1}y\mathrm{tan}(\beta )`" "**Beam** :math:`{C}_{1}C` **:**", ":math:`{N}_{\mathrm{iso}}=\frac{-1}{2}{F}_{1}(\mathrm{tan}(\beta )\mathrm{cos}(\alpha )+\mathrm{sin}(\alpha ))` :math:`{V}_{\mathrm{iso}}=\frac{1}{2}{F}_{1}(\mathrm{tan}(\beta )\mathrm{sin}(\alpha )-\mathrm{cos}(\alpha ))` :math:`{M}_{\mathrm{iso}}=\frac{-1}{2}{F}_{1}(y\mathrm{tan}(\beta )-x)`" "**Beam** :math:`{\mathrm{CC}}_{2}` **:**", ":math:`{N}_{\mathrm{iso}}=\frac{-1}{2}{F}_{1}(\mathrm{tan}(\beta )\mathrm{cos}(\alpha )+\mathrm{sin}(\alpha ))` :math:`{V}_{\mathrm{iso}}=\frac{1}{2}{F}_{1}(\mathrm{tan}(\beta )\mathrm{sin}(\alpha )-\mathrm{cos}(\alpha ))` :math:`{M}_{\mathrm{iso}}=\frac{-1}{2}{F}_{1}(y\mathrm{tan}(\beta )-(l-x))`" Diagrams (:math:`{F}_{1}` down) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. image:: images/10000000000002A8000000DEB33AFAF001209DD4.png :width: 5.7528in :height: 1.7854in .. _RefImage_10000000000002A8000000DEB33AFAF001209DD4.png: Solicitations under concentrated force :math:`{F}_{2}` (to the left) -------------------------------------------------------- Supportive reactions ~~~~~~~~~~~~~~~~~~ +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ |:math:`\begin{array}{}{H}_{A}+{H}_{B}={F}_{2};\\ {V}_{A}+{V}_{B}=0;\\ {\mathrm{lV}}_{B}+{\mathrm{hF}}_{2}=0;\end{array}` | | + + .. image:: images/1000000000000132000000C2C7933BFEBA320EC3.png + | :math:`(\begin{array}{}{H}_{B}\\ {V}_{B}\end{array})\mathrm{ans}\mathrm{BC}=0` | :width: 2.4335in | + + :height: 1.5728in + | | | + + + | | | +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ From where: :math:`{H}_{A}={F}_{1}(1-\frac{h}{l}\mathrm{tan}(\beta ))`; :math:`{V}_{A}={F}_{2}\frac{h}{l}`; :math:`{H}_{B}={F}_{1}\frac{h}{l}\mathrm{tan}(\beta )`; :math:`{V}_{B}=-\mathrm{F2}\frac{h}{l}`; **Note:** :math:`\frac{h}{l}\mathrm{tan}(\beta )=\frac{h}{2(a+h)}` :math:`(1-\frac{h}{l}\mathrm{tan}(\beta ))=\frac{\mathrm{2a}+h}{2(a+h)}` :math:`\mathrm{tan}(\beta )\mathrm{sin}(\alpha )–\mathrm{cos}(\alpha )=\frac{-\mathrm{hl}}{\mathrm{2b}(a+h)}`, :math:`\mathrm{tan}(\beta )\mathrm{cos}(\alpha )–\mathrm{sin}(\alpha )=\frac{{l}^{2}–4({a}^{2}+\mathrm{ah})}{\mathrm{4b}(a+h)}` Solicitations ~~~~~~~~~~~~~~~ .. csv-table:: "**Beam** :math:`{\mathrm{AC}}_{1}`:", ":math:`{N}_{\mathrm{iso}}=-{F}_{2}\frac{h}{l}` :math:`{V}_{\mathrm{iso}}={F}_{2}(1-\frac{h}{l}\mathrm{tan}(\beta ))` :math:`{M}_{\mathrm{iso}}=-{F}_{2}y(1-\frac{h}{l}\mathrm{tan}(\beta ))`" "**Beam** :math:`{C}_{2}B`:", ":math:`{N}_{\mathrm{iso}}={F}_{2}\frac{h}{l}` :math:`{V}_{\mathrm{iso}}={F}_{2}\frac{h}{l}\mathrm{tan}(\beta )` :math:`{M}_{\mathrm{iso}}=-{F}_{2}y\frac{h}{l}y\mathrm{tan}(\beta )`" "**Beam** :math:`{C}_{1}C`:", ":math:`{N}_{\mathrm{iso}}={F}_{2}((1-\frac{h}{l}\mathrm{tan}(\beta ))\mathrm{cos}(\alpha )–\frac{h}{l}\mathrm{cos}(\alpha ))` :math:`{V}_{\mathrm{iso}}={F}_{2}((1–\frac{h}{l}\mathrm{tan}(\beta ))–\frac{h}{l}\mathrm{cos}(\alpha ))` :math:`{M}_{\mathrm{iso}}={F}_{2}(\frac{h}{l}x-(1–\frac{h}{l}\mathrm{tan}(\beta ))y)`" "**Beam** :math:`C{C}_{2}` **:**", ":math:`{N}_{\mathrm{iso}}={F}_{2}\frac{h}{l}(\mathrm{tan}(\beta )\mathrm{cos}(\alpha )+\mathrm{sin}(\alpha ))` :math:`{V}_{\mathrm{iso}}={F}_{2}\frac{h}{l}(\mathrm{tan}(\beta )\mathrm{sin}(\alpha )-\mathrm{cos}(\alpha ))` :math:`{M}_{\mathrm{iso}}={F}_{2}\frac{h}{l}(y\mathrm{tan}(\beta )-(l-x))`" Diagrams ~~~~~~~~~~ .. image:: images/100000000000029C000000EDCA1A24F3881A8B90.png :width: 6.098in :height: 2.0827in .. _RefImage_100000000000029C000000EDCA1A24F3881A8B90.png: Stress under concentrated torque :math:`\Gamma` (positive) -------------------------------------------------------------- Supportive reactions ~~~~~~~~~~~~~~~~~~ +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ |:math:`\begin{array}{}{H}_{A}+{H}_{B}=0;\\ {V}_{A}+{V}_{B}=0;\\ {\mathrm{lV}}_{B}+\Gamma =0;\end{array}` | | + + .. image:: images/100000000000012D000000BA53EF896D4CDEBDCA.png + | :math:`(\begin{array}{}{H}_{B}\\ {V}_{B}\end{array})\wedge \mathrm{BC}=0;` | :width: 2.4492in | + + :height: 1.4827in + | | | + + + | | | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ Where: :math:`{H}_{A}=-\Gamma \mathrm{tan}\frac{\beta }{l}`, :math:`{V}_{A}=\frac{\Gamma }{l}`, :math:`{H}_{B}=\Gamma \mathrm{tan}\frac{\beta }{l}`, :math:`{V}_{A}=\frac{-\Gamma }{l}` **Note:** :math:`\frac{\mathrm{tan}(\beta )}{l}=\frac{1}{2(a+h)}` Solicitations ~~~~~~~~~~~~~~~ .. csv-table:: "**Beam** :math:`A{C}_{1}`:", ":math:`{N}_{\mathrm{iso}}=\frac{-\Gamma }{l}` :math:`{V}_{\mathrm{iso}}=\frac{-\Gamma \mathrm{tan}(\beta )}{l}` :math:`{M}_{\mathrm{iso}}=\frac{\Gamma y\mathrm{tan}(\beta )}{l}`" "**Beam** :math:`{C}_{2}B`:", ":math:`{N}_{\mathrm{iso}}=\frac{-\Gamma }{l}` :math:`{V}_{\mathrm{iso}}=\frac{\Gamma \mathrm{tan}(\beta )}{l}` :math:`{M}_{\mathrm{iso}}=\frac{\Gamma y\mathrm{tan}(\beta )}{l}`" "**Beam** :math:`{C}_{1}C`:", ":math:`{N}_{\mathrm{iso}}=\frac{\Gamma }{l}(\mathrm{tan}(\beta )\mathrm{cos}(\alpha )–\mathrm{sin}(\alpha ))` :math:`{N}_{\mathrm{iso}}=\frac{-\Gamma }{l}(\mathrm{tan}(\beta )\mathrm{sin}(\alpha )+\mathrm{cos}(\alpha ))` :math:`{N}_{\mathrm{iso}}=\frac{\Gamma }{l}(x+y\mathrm{tan}(\beta )–l)`" "**Beam** :math:`C{C}_{2}` **:**", ":math:`{N}_{\mathrm{iso}}=\frac{\Gamma }{l}(\mathrm{tan}(\beta )\mathrm{cos}(\alpha )+\mathrm{sin}(\alpha ))` :math:`{N}_{\mathrm{iso}}=\frac{\Gamma }{l}(\mathrm{tan}(\beta )\mathrm{sin}(\alpha )-\mathrm{cos}(\alpha ))` :math:`{N}_{\mathrm{iso}}=\frac{\Gamma }{l}(y\mathrm{tan}(\beta )–(l-x))`" Diagrams (:math:`\Gamma` positive) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. image:: images/1000000000000343000001154B3444319539B639.png :width: 5.9835in :height: 1.9827in .. _RefImage_1000000000000343000001154B3444319539B639.png: Solicitations under the :math:`X` hyperstatic moment ---------------------------------------- Supportive reactions ~~~~~~~~~~~~~~~~~~ +---------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ |:math:`{H}_{A}+{H}_{B}=0;` | | + + .. image:: images/1000000000000151000000B7847C7CFB1B3CBF5C.png + | :math:`{V}_{A}+{V}_{B}=0;` :math:`{\mathrm{lV}}_{B}=0;` :math:`{H}_{B}(a+h)–X=0;` | :width: 3.0228in | + + :height: 1.5965in + | | | + + + | | | +---------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ Hence the reactions: :math:`{H}_{a}=\frac{-X}{a+h}`, :math:`{V}_{A}=0`, :math:`{H}_{B}=\frac{X}{a+h}`, :math:`{V}_{B}=0` Solicitations ~~~~~~~~~~~~~~~ .. csv-table:: "**Beam** :math:`{\mathrm{AC}}_{1}`:", ":math:`{N}_{X}=0` :math:`{V}_{X}=\frac{-X}{a+h}` :math:`{M}_{X}=\frac{X}{a+h}y`" "**Beam** :math:`{C}_{2}B`:", ":math:`{N}_{X}=0` :math:`{V}_{X}=\frac{X}{a+h}` :math:`{M}_{X}=\frac{X}{a+h}y`" "**Beam** :math:`{C}_{1}C`:", ":math:`{N}_{X}=\frac{X}{a+h}\mathrm{cos}(\alpha )` :math:`{V}_{X}=\frac{X}{a+h}\mathrm{sin}(\alpha )` :math:`{M}_{X}=\frac{X}{a+h}y=\frac{X}{a+h}(h+x\mathrm{tan}(\alpha ))`" "**Beam** :math:`{\mathrm{CC}}_{2}` **:**", ":math:`{N}_{X}=\frac{X}{a+h}\mathrm{cos}(\alpha )` :math:`{V}_{X}=\frac{X}{a+h}\mathrm{sin}(\alpha )` :math:`{M}_{X}=\frac{X}{a+h}y`" Diagrams ~~~~~~~~~~ .. image:: images/10000000000003010000015D27D5B04219FF9F97.png :width: 6.4035in :height: 2.9043in .. _RefImage_10000000000003010000015D27D5B04219FF9F97.png: Requests under fictional one-off loads in :math:`C` ------------------------------------------------------------ In order to calculate the displacements in :math:`C`, using the Principle of Virtual Works (:math:`\mathrm{cf.}` paragraph :math:`[\S 8]`), it is necessary to establish the load diagrams under the action of two "fictional" forces :math:`f` and :math:`g` applied in :math:`C`. Supportive reactions ~~~~~~~~~~~~~~~~~~ +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ |:math:`{H}_{A}+{H}_{B}=-f;` | | + + .. image:: images/100000000000012D000000E266E8AC0680ABB2F0.png + | :math:`{V}_{A}+{V}_{B}=-g;` :math:`(\begin{array}{}{H}_{A}\\ {V}_{A}\end{array})\wedge \mathrm{AC}=0=(\begin{array}{}{H}_{B}\\ {V}_{B}\end{array})\wedge \mathrm{BC}` | :width: 2.0854in | + + :height: 1.6311in + | | | + + + | | | +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ From where: :math:`{H}_{A}=\frac{-1}{2}(f+g\mathrm{tan}(\beta ))`, :math:`{V}_{A}=\frac{-1}{2}(g+f\mathrm{cot}(\beta ))` :math:`{H}_{B}=\frac{-1}{2}(f-g\mathrm{tan}(\beta ))`, :math:`{V}_{B}=\frac{-1}{2}(g-f\mathrm{cot}(\beta ))` Solicitations ~~~~~~~~~~~~~~~ .. csv-table:: "**Beam** :math:`{\mathrm{AC}}_{1}` **:**", ":math:`n=\frac{1}{2}(g+f\mathrm{cot}(\beta ))` :math:`v=\frac{-1}{2}(f+g\mathrm{tan}(\beta ))` :math:`m=\frac{1}{2}(f+g\mathrm{tan}(\beta ))`" "**Beam** :math:`{C}_{2}B`:", ":math:`n=\frac{1}{2}(g-f\mathrm{cot}(\beta ))` :math:`v=\frac{-1}{2}(f-g\mathrm{tan}(\beta ))` :math:`m=\frac{-1}{2}(f-g\mathrm{tan}(\beta ))y`" "**Beam** :math:`{C}_{1}C`:", ":math:`n=\frac{1}{2}(f+g\mathrm{tan}(\beta ))\mathrm{cos}(\alpha )+\frac{1}{2}(g+f\mathrm{cot}(\beta ))\mathrm{sin}(\alpha )` :math:`v=\frac{-1}{2}(f+g\mathrm{tan}(\beta ))\mathrm{sin}(\alpha )+\frac{1}{2}(g+f\mathrm{cot}(\beta ))\mathrm{cos}(\alpha )` :math:`m=\frac{1}{2}(f+g\mathrm{tan}(\beta ))y-\frac{1}{2}(g+f\mathrm{cot}(\beta ))x`" "**Beam** :math:`{\mathrm{CC}}_{2}` **:**", ":math:`n=\frac{-1}{2}(f-g\mathrm{tan}(\beta ))\mathrm{cos}(\alpha )+\frac{1}{2}(g-f\mathrm{cot}(\beta ))\mathrm{sin}(\alpha )` :math:`v=\frac{-1}{2}(f-g\mathrm{tan}(\beta ))\mathrm{sin}(\alpha )-\frac{1}{2}(g-f\mathrm{cot}(\beta ))\mathrm{cos}(\alpha )` :math:`m=\frac{-1}{2}(f-g\mathrm{tan}(\beta ))y-\frac{1}{2}(g-f\mathrm{cot}(\beta ))(l-x)`" Diagrams ~~~~~~~~~~ Here are the diagrams of solicitations under the action of the two "fictional" forces :math:`f` and :math:`g`. Here we consider: :math:`f\ge 0,g\ge \mathrm{fcot}(\beta )` **.** .. image:: images/100000000000037600000129AFAACC0851F87FFF.png :width: 6.3543in :height: 2.1209in .. _RefImage_100000000000037600000129AFAACC0851F87FFF.png: Determining the :math:`X` hyperstatic moment --------------------------------------------- We place ourselves in elasticity; we only consider the flexural energy, the beams being slender. The natural state is assumed to be virgin (no prestresses or support movements). The complementary potential is then: :math:`F\ast (X)={\int }_{\mathrm{poteaux}}\frac{{({M}_{\mathrm{iso}}+{M}_{1}X)}^{2}}{{\mathrm{EI}}_{1}}+{\int }_{\mathrm{charpentes}}\frac{{({M}_{\mathrm{iso}}+{M}_{1}X)}^{2}}{{\mathrm{EI}}_{2}}` It is stationary at equilibrium, hence: :math:`\delta \mathrm{.}X=\left[{\int }_{\mathrm{pot}}\frac{{M}_{1}^{2}}{{\mathrm{EI}}_{1}}+{\int }_{\mathrm{charp}}\frac{{M}_{1}^{2}}{{\mathrm{EI}}_{2}}\right]\mathrm{.}X=-{\int }_{\mathrm{pot}}\frac{{M}_{1}{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{1}}-{\int }_{\mathrm{charp}}\frac{{M}_{1}{M}_{\mathrm{iso}}}{{\mathrm{Ei}}_{2}}=S` The flexibility coefficient :math:`\delta` is the sum of: :math:`{\int }_{\mathrm{pot}}\frac{{M}_{1}^{2}}{{\mathrm{EI}}_{1}}=\frac{\mathrm{2h}}{{\mathrm{3EI}}_{1}}{(\frac{h}{a+h})}^{2}` :math:`{\int }_{\mathrm{charp}}\frac{{M}_{1}^{2}}{{\mathrm{EI}}_{2}}=\frac{\mathrm{2b}}{{\mathrm{EI}}_{2}}[{(\frac{h}{a+h})}^{2}+\frac{1}{3}{(\frac{a}{a+h})}^{2}+\frac{\mathrm{ah}}{{(a+h)}^{2}}]` either: :math:`E\mathrm{.}\delta =\frac{2}{{(a+h)}^{2}}[\frac{{h}^{3}}{{\mathrm{3I}}_{1}}+\frac{b({\mathrm{3h}}^{2}+{a}^{2}+\mathrm{3ah})}{{\mathrm{3I}}_{2}}]` **Digital application:** In the example under consideration: :math:`{I}_{1}={\mathrm{2I}}_{2}=5.0E-4{m}^{4}`, :math:`h=\mathrm{2a}=8m`, :math:`l=20m`, :math:`b=\frac{l}{2}\sqrt{1.16}` From where: :math:`\gamma =\frac{2}{E{(a+h)}_{1}^{\mathrm{2I}}}\underset{2353.45347{m}^{3}}{\underset{\underbrace{}}{\frac{{h}^{2}}{3}(h+\frac{\mathrm{19b}}{2})}}` We study the various loads one after the other to calculate the second members :math:`S`. Distributed load :math:`p` on :math:`{C}_{1}C` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The second member :math:`S` due to :math:`f` is: :math:`-{\int }_{\mathrm{pot}}\frac{{M}_{1}{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{1}}=\frac{\mathrm{3h}}{{\mathrm{3EI}}_{1}}(\frac{h}{a+h})(\frac{\mathrm{pblh}}{8(a+h)})=\frac{2}{E{(a+h)}^{2}{I}_{1}}\frac{{\mathrm{ph}}^{3}\mathrm{bl}}{24}` :math:`-{\int }_{{\mathrm{CC}}_{2}}\frac{{M}_{1}{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{2}}=\frac{{\mathrm{pb}}^{2}\mathrm{hl}}{8(a+h){\mathrm{EI}}_{2}}(\frac{1}{2}\frac{h}{a+h})+(\frac{a}{6}\frac{a}{a+h})=\frac{2}{E{(a+h)}^{2}{I}_{2}}\frac{{\mathrm{phb}}^{\mathrm{2l}}(\mathrm{3h}+a)}{48}` :math:`\begin{array}{}-{\int }_{{C}_{1}C}\frac{{M}_{1}{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{2}}=\frac{1}{{\mathrm{EI}}_{2}}\frac{\mathrm{pl}}{8{(a+h)}^{2}}{\int }_{0}^{b}\left[{\mathrm{2s}}^{2}\frac{a+h}{b}–s(\mathrm{2a}+\mathrm{3h})+\mathrm{bh}\right]\left[h+s\frac{a}{b}\right]\mathrm{ds}\\ =\frac{1}{E{(a+h)}^{2}{I}_{2}}\frac{{\mathrm{plb}}^{2}}{48}({h}^{2}+\mathrm{2ah}+{a}^{2})\end{array}` From where: :math:`S=\frac{2}{E{(a+h)}^{2}}\frac{\mathrm{plb}}{96}\left[\frac{{\mathrm{4h}}^{3}}{{I}_{1}}+\frac{\mathrm{hb}(\mathrm{3h}+a)}{{I}_{2}}+\frac{b({h}^{2}–\mathrm{2ah}-{a}^{2})}{{I}_{2}}\right]` **Digital application:** :math:`{I}_{1}=2{I}_{2}` **;** :math:`h=\mathrm{2a}` **;** :math:`p=3000{\mathrm{N.m}}^{1}` (down) :math:`S=\frac{2}{E{(a+h)}^{2}{I}_{1}}\underset{43946021.89{\mathrm{N.m}}^{4}}{\underset{\underbrace{}}{\frac{{\mathrm{plbh}}^{2}}{96}\left[\mathrm{4h}+\frac{13}{2}b\right]}}` From where: • the moment in :math:`C`: :math:`X=18672994\mathrm{N.m}` • the reaction in :math:`A`: :math:`{H}_{A}=p\frac{\mathrm{bl}}{8(a+h)}-\frac{X}{a+h}=\frac{\frac{\mathrm{pbl}}{8-X}}{a+h}`, :math:`{H}_{A}=5175.37N` :math:`{V}_{A}=\frac{\mathrm{3pb}}{4}-0`, :math:`{V}_{A}=24233.24N` Point load :math:`{F}_{1}` in :math:`C` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The second member is obtained using: :math:`-{\int }_{\mathrm{pot}}\frac{{M}_{1}{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{1}}=\frac{\mathrm{2h}}{{\mathrm{3EI}}_{1}}(\frac{h}{a+h})(\frac{{F}_{1}\mathrm{lh}}{4(a+h)})=\frac{2}{E{(a+h)}^{2}{I}_{1}}\frac{{F}_{1}{\mathrm{lh}}^{3}}{12}` :math:`-{\int }_{\mathrm{charp}}\frac{{M}_{1}{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{2}}=\frac{\mathrm{2b}}{{\mathrm{EI}}_{2}}\frac{{F}_{1}\mathrm{lh}}{4(a+h)}(\frac{1}{2}(\frac{h}{a+h})+\frac{1}{6}(\frac{a}{a+h}))=\frac{2}{E{(a+h)}^{2}{I}_{2}}\frac{{F}_{1}\mathrm{blh}(\mathrm{3h}+a)}{24}` From where: :math:`S=\frac{2}{E{(a+h)}^{2}}\frac{{F}_{1}\mathrm{lh}}{24}\left[\frac{{\mathrm{2h}}^{2}}{{I}_{1}}+\frac{b(\mathrm{3h}+a)}{{I}_{2}}\right]` **Digital application:** :math:`{I}_{1}=2{I}_{2}` **;** :math:`h=\mathrm{2a}` **;** :math:`{F}_{1}=20000N` (down) :math:`S=\frac{2}{E{(a+h)}^{2}{I}_{1}}\underset{97485127.76{\mathrm{N.m}}^{4}}{\underset{\underbrace{}}{\frac{{F}_{1}{\mathrm{lh}}^{2}}{24}\left[\mathrm{2h}+7b\right]}}` From where: • the moment in :math:`C`: :math:`X=41422.161\mathrm{N.m}` • the reaction in :math:`A`: :math:`{H}_{A}=\frac{1}{4}{F}_{1}\frac{l}{a+h}-\frac{X}{a+h}=\frac{\frac{{F}_{1}l}{4-X}}{a+h}`, :math:`{H}_{A}=4881.4866N` :math:`{V}_{A}=\frac{1}{2}{F}_{1}-0`, :math:`{V}_{A}=10000.0N` Point load :math:`{F}_{2}` in :math:`{C}_{1}` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The second member is obtained using: :math:`-{\int }_{{\mathrm{AC}}_{1}}\frac{{M}_{1}{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{1}}=\frac{h}{{\mathrm{3EI}}_{1}}(\frac{h}{a+h})\frac{{F}_{2}h(\mathrm{2a}+h)}{2(a+h)}=\frac{2}{E{(a+h)}^{2}{I}_{1}}\frac{{F}_{2}{h}^{3}(\mathrm{2a}+h)}{12}` :math:`-{\int }_{{C}_{2}B}\frac{{M}_{1}{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{1}}=\frac{h}{{\mathrm{3EI}}_{1}}(\frac{h}{a+h})\frac{(-{F}_{2}{h}^{2})}{2(a+h)}=\frac{2}{E{(a+h)}_{1}^{\mathrm{2I}}}\frac{{F}_{2}{h}^{3}(\mathrm{2a}+h)}{12}` :math:`-{\int }_{{C}_{1}C}\frac{{M}_{1}{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{2}}=\frac{b}{{\mathrm{EI}}_{2}}\frac{{F}_{2}h(\mathrm{2a}+h)}{2(a+h)}\left[\frac{1}{2}(\frac{h}{a+h})+\frac{1}{6}(\frac{a}{a+h})\right]=\frac{2}{E{(a+h)}^{2}{I}_{2}}\frac{{F}_{2}\mathrm{bh}({\mathrm{3h}}^{2}+\mathrm{7ah}+{\mathrm{2a}}^{2})}{24}` :math:`-{\int }_{{\mathrm{CC}}_{2}}\frac{{M}_{1}{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{2}}=\frac{b}{{\mathrm{EI}}_{2}}\frac{-{F}_{2}{h}^{2}}{2(a+h)}\left[\frac{1}{2}(\frac{h}{a+h})+\frac{1}{6}(\frac{a}{a+h})\right]=\frac{2}{E{(a+h)}^{2}{I}_{2}}\frac{-{F}_{2}{\mathrm{bh}}^{2}(\mathrm{3h}+a)}{24}` :math:`S=\frac{2}{E{(a+h)}^{2}}\frac{{F}_{1}\mathrm{lh}}{24}\left[\frac{{\mathrm{2h}}^{2}}{{I}_{1}}+\frac{b(\mathrm{3h}+a)}{{I}_{2}}\right]` **Digital application:** :math:`{I}_{1}=2{I}_{2}` **;** :math:`h=\mathrm{2a}` ****; ** :math:`{F}_{2}=10000N` (to the left) :math:`S=\frac{2}{E{(a+h)}^{2}{I}_{1}}\underset{19497025.55{\mathrm{N.m}}^{4}}{\underset{\underbrace{}}{\frac{{F}_{2}{h}^{2}a}{12}\left[\mathrm{2h}+7b\right]}}` From where: • the moment in :math:`C`: :math:`X=8284.4321\mathrm{N.m}` • the reaction in :math:`A`: :math:`{H}_{A}={F}_{2}\frac{\mathrm{2a}+h}{2(a+h)}-\frac{X}{a+h}=\frac{{F}_{2}(a+\frac{h}{2})-X}{a+h}`, :math:`{H}_{A}=5976.297N` :math:`{V}_{A}=\frac{{F}_{2}h}{l}`, :math:`{V}_{A}=4000.0N` Punctual couple :math:`\Gamma` in :math:`{C}_{1}` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The second member is obtained using: :math:`-{\int }_{\mathrm{pot}}\frac{{M}_{1}{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{1}}=\frac{-\mathrm{2h}}{{\mathrm{3EI}}_{1}}(\frac{h}{a+h})\frac{\Gamma h}{2(a+h)}=\frac{2}{E{(a+h)}^{2}{I}_{1}}\frac{-\Gamma {h}^{3}}{6}` :math:`-{\int }_{{C}_{1}C}\frac{b}{{\mathrm{EI}}_{2}}=\frac{\Gamma (h+\mathrm{2a})}{2(a+h)}\frac{b}{{\mathrm{EI}}_{2}}\left[\frac{1}{2}(\frac{h}{a+h}+\frac{1}{6}(\frac{a}{a+h}))\right]=\frac{2}{E{(a+h)}^{2}{I}_{2}}\frac{\Gamma (h+\mathrm{2a})(\mathrm{3h}+a)b}{24}` :math:`-{\int }_{{\mathrm{CC}}_{2}}\frac{{M}_{1}{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{2}}=\frac{b}{{\mathrm{EI}}_{2}}\frac{-\Gamma h}{2(a+h)}\left[\frac{1}{2}(\frac{h}{a+h})+\frac{1}{6}(\frac{a}{a+h})\right]=\frac{2}{E{(a+h)}^{2}{I}_{2}}\frac{-\Gamma \mathrm{hb}(\mathrm{3h}+a)}{24}` From where: :math:`S=\frac{2}{E{(a+h)}^{2}}\frac{-\Gamma }{12}\left[\frac{{\mathrm{2h}}^{3}}{{I}_{1}}+\frac{\mathrm{ab}(\mathrm{3h}+a)}{{I}_{2}}\right]` **Digital application:** :math:`{I}_{1}=2{I}_{2}` **;** :math:`h=\mathrm{2a}` ****; ** :math:`\Gamma =-100000\mathrm{N.m}` (clockwise) :math:`S=\frac{2}{E{(a+h)}^{2}{I}_{1}}\underset{11571281.93{\mathrm{N.m}}^{4}}{\underset{\underbrace{}}{\frac{-\Gamma }{6}\left[{h}^{3}-\mathrm{ab}(\mathrm{3h}+a)\right]}}` From where: • the moment in :math:`C`: :math:`X=4916.7243\mathrm{N.m}` • the reaction in :math:`A`: :math:`{H}_{A}=\frac{-\Gamma }{2(a+h)}-\frac{X}{a+h}=\frac{\frac{-\Gamma }{2}-X}{a+h}`, :math:`{H}_{A}=4576.394N` :math:`{V}_{A}=\frac{\Gamma }{l}`, :math:`{V}_{A}=5000.0N` Recap ~~~~~~~~~~~~~ .. csv-table:: "**CAS**", "**Moment in** :math:`C` ", "**Reactions in** :math:`A(N)` ", "" "", ":math:`(\mathrm{N.m})` "," :math:`{H}_{A}` "," :math:`{V}_{A}`" ":math:`p` out of :math:`{C}_{1}C` ", "18672.994", "5175.37", "24233.240" ":math:`{F}_{1}` in :math:`C` ", "41422.161", "4881.487", "10000.000" ":math:`{F}_{2}` in :math:`{C}_{1}` ", "8284.432", "5976.297", "4000.000" ":math:`\Gamma` in :math:`{C}_{1}` ", "4916.724", "4576.394", "5000.000" "TOTAL ", "73296.311", "22033.31", "43233.24" **Note** Reminder: in post :math:`{\mathrm{AC}}_{1}`: normal effort= :math:`-{V}_{A}`, shear effort= :math:`{H}_{A}`. Calculating the displacement in :math:`C` --------------------------------- We also only consider the elastic flexural energy (slender beams). By applying the Principle of Virtual Works on the structure subject to the fictional forces in paragraph :math:`[\S 6]`, working in the movements sought, we calculate the numbers :math:`w` and :math:`d` depending linearly on :math:`f` and :math:`g`: :math:`f{u}_{c}+g{v}_{c}={\int }_{\mathrm{pot}}\frac{m({M}_{\mathrm{iso}}+{\mathrm{XM}}_{1})}{{\mathrm{EI}}_{1}}+{\int }_{\mathrm{charp}}\frac{m({M}_{\mathrm{iso}}+{\mathrm{XM}}_{1})}{{\mathrm{EI}}_{2}}=w+\mathrm{Xd},\forall (f,g)` Distributed load :math:`p` on :math:`{C}_{1}C` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ :math:`{\int }_{\mathrm{pot}}\frac{m{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{1}}=\frac{\mathrm{2h}}{{\mathrm{3EI}}_{1}}\frac{\mathrm{ghl}}{4(a+h)}\frac{-\mathrm{pbhl}}{8(a+h)}=\frac{2}{E{(a+h)}^{2}{I}_{1}}\frac{-{\mathrm{gpbh}}^{3}{l}^{2}}{96}` :math:`{\int }_{{C}_{1}C}\frac{m{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{2}}=\frac{2}{E{(a+h)}^{2}{I}_{2}}\frac{-{\mathrm{plhb}}^{2}}{384}(\mathrm{2f}(a+h)+\mathrm{gl})(h-a)` :math:`{\int }_{{\mathrm{CC}}_{2}}\frac{{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{2}}=\frac{b}{{\mathrm{3EI}}_{2}}\frac{\mathrm{pbhl}}{8(a+h)}(\frac{\mathrm{fh}}{2}-\frac{\mathrm{glh}}{4(a+h)})=\frac{2}{E{(a+h)}^{2}{I}_{2}}\frac{-{\mathrm{pb}}^{2}{\mathrm{lh}}^{2}(\mathrm{gl}-\mathrm{2f}(a+h))}{192}` From where: :math:`w=\frac{2}{E{(a+h)}^{2}}\frac{-\mathrm{pbhl}}{384}(\frac{{\mathrm{4glh}}^{2}}{{I}_{1}}+\frac{\mathrm{glb}(\mathrm{3h}+a)-\mathrm{efb}{(a+h)}^{2}}{{I}_{2}})` **Digital application:** :math:`{I}_{1}={\mathrm{2I}}_{2}`; :math:`h=\mathrm{2a}`; :math:`p=3000{\mathrm{N.m}}^{-1}` (down) :math:`w=\frac{2}{E{(a+h)}^{2}{I}_{1}}(\mathrm{gl}(\mathrm{2h}+\frac{5}{2}b)-\frac{9}{2}\mathrm{fbh})\underset{-215406.5922{\mathrm{N.m}}^{3}}{\underset{\underbrace{}}{\frac{-{\mathrm{pbh}}^{2}l}{192}}}` Point load :math:`{F}_{1}` in :math:`C` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ :math:`{\int }_{\mathrm{pot}}\frac{m{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{1}}=\frac{\mathrm{2h}}{{\mathrm{3EI}}_{1}}\frac{\mathrm{ghl}}{4(a+h)}\frac{-{F}_{1}\mathrm{hl}}{4(a+h)}=\frac{2}{E{(a+h)}^{2}{I}_{1}}\frac{-{F}_{1}{\mathrm{gh}}^{3}{l}^{2}}{48}` :math:`{\int }_{\mathrm{charp}}\frac{m{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{2}}=\frac{\mathrm{2b}}{{\mathrm{3EI}}_{2}}\frac{\mathrm{ghl}}{4(a+h)}\frac{-{F}_{1}\mathrm{hl}}{4(a+h)}=\frac{2}{E{(a+h)}^{2}{I}_{2}}\frac{-{F}_{1}{\mathrm{gbh}}^{2}{l}^{2}}{48}` Hence (we can see that :math:`w` does not depend on :math:`f` for this load): :math:`w=\frac{2}{E{(a+h)}^{2}}\frac{-{F}_{1}{\mathrm{gh}}^{2}{l}^{2}}{48}(\frac{h}{{I}_{1}}+\frac{b}{{I}_{2}})` **Digital application:** :math:`{I}_{1}={\mathrm{2I}}_{2}`, :math:`h=\mathrm{2a}`, :math:`{F}_{1}=20000N` (down) :math:`w=\frac{\mathrm{2g}}{E{(a+h)}^{2}{I}_{1}}\frac{-{F}_{1}{h}^{2}{l}^{2}}{48}\underset{-3155100365.0{\mathrm{N.m}}^{5}}{\underset{\underbrace{}}{(h+\mathrm{2b})}}` Point load :math:`{F}_{2}` in :math:`{C}_{1}` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ :math:`\begin{array}{}{\int }_{\mathrm{pot}}\frac{m{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{1}}=\frac{h}{{\mathrm{3EI}}_{1}}\frac{{F}_{2}h}{2(a+h)}\left[-(\mathrm{2a}+h)(\frac{\mathrm{fh}}{2}+\frac{\mathrm{ghl}}{4(a+h)})+h(\frac{-\mathrm{fh}}{2}+\frac{\mathrm{ghl}}{4(a+h)})\right]\\ =\frac{2}{E{(a+h)}^{2}{I}_{1}}\frac{-{F}_{2}{h}^{3}}{24}(\mathrm{agl}+\mathrm{2f}{(a+h)}^{2})\end{array}` :math:`{\int }_{\mathrm{charp}}\frac{m{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{2}}=\frac{2}{E{(a+h)}^{2}{I}_{2}}\frac{-{F}_{2}{\mathrm{bh}}^{2}}{24}(\mathrm{agl}+\mathrm{2f}{(a+h)}^{2})` From where: :math:`w=\frac{2}{E{(a+h)}^{2}}\frac{-{F}_{2}{h}^{2}}{24}(\mathrm{agl}+\mathrm{2f}{(a+h)}^{2})(\frac{h}{{I}_{1}}+\frac{b}{{I}_{2}})` **Digital application:** :math:`{I}_{1}={\mathrm{2I}}_{2}`, :math:`h=\mathrm{2a}`, :math:`{F}_{2}=10000N` (to the left) :math:`w=\frac{2}{E{(a+h)}^{2}{I}_{1}}(\mathrm{gl}+\mathrm{9gh})\underset{-3151003.65{\mathrm{N.m}}^{4}}{\underset{\underbrace{}}{\frac{-{F}_{2}{h}^{3}(h+\mathrm{2b})}{48}}}` Punctual couple :math:`\Gamma` in :math:`{C}_{1}` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ :math:`\begin{array}{}{\int }_{\mathrm{pot}}\frac{m{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{1}}=\frac{h}{{\mathrm{3EI}}_{1}}\frac{\Gamma h}{2(a+h)}\left[(\frac{\mathrm{fh}}{2}+\frac{\mathrm{glh}}{4(a+h)})+(\frac{-\mathrm{fh}}{2}+\frac{\mathrm{glh}}{4(a+h)})\right]\\ =\frac{2}{E{(a+h)}^{2}{I}_{1}}\frac{\Gamma {h}^{3}\mathrm{lg}}{24}\end{array}` :math:`\begin{array}{}{\int }_{\mathrm{charp}}\frac{m{M}_{\mathrm{iso}}}{{\mathrm{EI}}_{2}}=\frac{b}{{\mathrm{3EI}}_{2}}\frac{\Gamma h}{2(a+h)}\left[-(\mathrm{2a}+h)(\frac{\mathrm{fh}}{2}+\frac{\mathrm{glh}}{4(a+h)})+h(\frac{-\mathrm{fh}}{2}+\frac{\mathrm{glh}}{4(a+h)})\right]\\ =\frac{-2}{E{(a+h)}^{2}{I}_{2}}\frac{\Gamma \mathrm{bh}}{24}(\mathrm{agl}+\mathrm{2f}{(a+h)}^{2})\end{array}` **Digital application:** :math:`{I}_{1}={\mathrm{2I}}_{2}`; :math:`h=\mathrm{2a}`; :math:`\Gamma =-100000\mathrm{N.m}` :math:`w=\frac{2}{E{(a+h)}^{2}{I}_{1}}\underset{-3151003.65{\mathrm{N.m}}^{4}}{\underset{\underbrace{}}{\frac{\Gamma {h}^{2}}{24}}}(\mathrm{gl}(h-b)-\mathrm{9fhb})` Calculation of :math:`d=\int \frac{m\cdot {M}_{1}}{\mathrm{EI}}` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ :math:`{\int }_{\mathrm{pot}}\frac{{\mathrm{mM}}_{1}}{{\mathrm{EI}}_{1}}=\frac{\mathrm{2h}}{{\mathrm{3EI}}_{1}}\frac{\mathrm{glh}}{4(a+h)}\frac{h}{a+h}=\frac{2}{E{(a+h)}^{2}{I}^{1}}\frac{{\mathrm{glh}}^{3}}{12}` :math:`{\int }_{\mathrm{charp}}\frac{m{M}_{1}}{{\mathrm{EI}}_{2}}=\frac{\mathrm{2b}}{{\mathrm{EI}}_{2}}\left[\frac{1}{2}(\frac{h}{a+h})+\frac{1}{6}(\frac{a}{a+h})\right]\frac{\mathrm{glh}}{4(a+h)}=\frac{2}{E{(a+h)}^{2}{I}_{2}}\frac{\mathrm{glbh}(\mathrm{3h}+a)}{24}` From where (we can see that :math:`d` does not depend on :math:`f`): :math:`d=\frac{2}{E{(a+h)}^{2}}\frac{\mathrm{glh}}{24}(\frac{{\mathrm{2h}}^{2}}{{I}_{1}}+\frac{b(\mathrm{3h}+a)}{{I}_{2}})` **Digital application:** :math:`{I}_{1}=2{I}_{2}`, :math:`h=\mathrm{2a}` :math:`d=\frac{2}{E{(a+h)}^{2}{I}_{1}}g\frac{{\mathrm{lh}}^{2}}{24}\underset{-4874.2564{\mathrm{N.m}}^{4}}{\underset{\underbrace{}}{(\mathrm{2h}+\mathrm{7b})}}` Summary of :math:`{u}_{c}` and :math:`{v}_{c}` trips ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ :math:`{I}_{1}=5.0E-4{m}^{4}` :math:`E=210000\mathrm{MPA}` .. csv-table:: "**CAS**", ":math:`X` "," :math:`X\stackrel{ˉ}{d}` "," :math:`{w}_{v}`" "press on :math:`{C}_{1}C` ", "18672.994", "91016960.3", "—184930109.4" ":math:`{F}_{1}` in :math:`C` ", "41422.161", "201902233.4", "—315100365.0" ":math:`{F}_{2}` in :math:`{C}_{1}` ", "8284.432", "40380445.6", "—63020073.0" ":math:`\Gamma` in :math:`{C}_{1}` ", "4916.724", "23965373.4", "14775091.25" .. csv-table:: "**CAS**", ":math:`{w}_{h}` "," :math:`{u}_{c}(m)` "," :math:`{v}_{c}(m)`" "press on :math:`{C}_{1}C` ", "83519999.94", "0.0110476", "—0.012422374" ":math:`{F}_{1}` in :math:`C` ", "0.00", "0.00", "—0.01497330" ":math:`{F}_{2}` in :math:`{C}_{1}` ", "—226872262.8", "—0.03000956", "—0.00299466", "—0.00299466" ":math:`\Gamma` in :math:`{C}_{1}` ", "206790328.5", "0.0273532", "—0.001215646" **Note:** :math:`d=\frac{2}{E{(a+h)}^{2}{I}_{1}}g\stackrel{ˉ}{d}`, with: :math:`\stackrel{ˉ}{d}=4874.2564{m}^{4}` :math:`w=\frac{2}{E{(a+h)}^{2}{I}_{1}}(g{w}_{v}+f{w}_{h})` see above :math:`{u}_{c}=\frac{2}{E{(a+h)}^{2}{I}_{1}}{w}_{H}`; :math:`{v}_{c}=\frac{2}{E{(a+h)}^{2}{I}_{1}}({w}_{v}+X\stackrel{ˉ}{d})` :math:`\frac{2}{E{(a+h)}^{2}{I}_{1}}=1.32275132E-10{N}^{-1}{m}^{-4}` **Comparison** **Aster- analytical reference (R.)** .. csv-table:: "CAS ", "", "**Moment in** :math:`C(\mathrm{N.m})` ", "**Reaction** :math:`{H}_{A}(N)` ", "**Reaction** :math:`{V}_{A}(N)` ", "**Move** :math:`{u}_{c}(m)` ", "**Move** :math:`{v}_{c}(m)`" ":math:`P` out of :math:`{C}_{1}C` ", "R: *Aster*:", "18672.994 18673.20", "5175.37 5175.36", "24233.24 24233.2", "0.0110476 0.0110472", "—0.012422374 —0.0124233" ":math:`{F}_{1}` in :math:`C` ", "R: *Aster*:", "41422.161 41422.40", "4881.487 4881.47", "10000.00 10000.0", "0.00000 0.0000", "—0.01497330 —0.0" ":math:`{F}_{2}` in :math:`{C}_{1}` ", "R: *Aster*:", "8284.432 8284.34", "5976.297 5976.31", "4000.00 4000.0", "—0.03000956 —0.0300098", "—0.00299466 —0.00299450" ":math:`\Gamma` in :math:`{C}_{1}` ", "R: *Aster*:", "4916.724 4916.62", "4576.394" 4576.38", "5000.00 5000.0", "0.0273532 0.0273536", "—0.001215646 —0.00121583" **Note:** The*Aster* calculation was done by taking very slender elements, so that: :math:`{\mathrm{Sl}}^{2}\ll I`. Thus, the flexural energy is predominant. The values for the *Aster* calculation are from test case :math:`\mathrm{VPCS}` called :math:`\mathrm{SSLL14}`, with the following data: :math:`{I}_{1}=5.0E-4{m}^{4}`; :math:`{I}_{2}=2.5E-4{m}^{4}`; :math:`E=210000\mathrm{MPa}` :math:`h=\mathrm{2a}=8m`; :math:`l=20m`; :math:`b=\frac{l}{2}\sqrt{1.16}` :math:`p=3000\mathrm{N.m}` (down), :math:`{F}_{1}=20000N` (down), :math:`{F}_{2}=10000N` (to the left), :math:`\Gamma =-100000\mathrm{Nm}` (clockwise)