From 468b4a22cd27e431dccbef46dfde1d878e5dd674 Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Thu, 16 Feb 2023 15:07:08 +0000 Subject: [PATCH] notes on strain energy --- ...ot-2023-02-16T14:19:48,580021845+00:00.png | Bin 0 -> 8109 bytes ...ot-2023-02-16T14:22:41,358964129+00:00.png | Bin 0 -> 13529 bytes ...ot-2023-02-16T14:23:06,415583742+00:00.png | Bin 0 -> 11758 bytes ...ot-2023-02-16T14:26:48,809335151+00:00.png | Bin 0 -> 6982 bytes ...ot-2023-02-16T14:40:43,750030500+00:00.png | Bin 0 -> 5022 bytes ...ot-2023-02-16T14:41:32,526245370+00:00.png | Bin 0 -> 13655 bytes .../Strain Energy Methods Notes.pdf | Bin 0 -> 430850 bytes .../lecture_slides/MMME2053 SE L1 Slides.pdf | Bin 0 -> 5662422 bytes .../lecture_slides/MMME2053 SE L2 Slides.pdf | Bin 0 -> 5670665 bytes .../lecture_slides/MMME2053 SE L3 Slides.pdf | Bin 0 -> 5638008 bytes .../2053_mechanics_of_solids/strain_energy.md | 111 ++++++++++++++++++ 11 files changed, 111 insertions(+) create mode 100644 uni/mmme/2053_mechanics_of_solids/images/vimscrot-2023-02-16T14:19:48,580021845+00:00.png create mode 100644 uni/mmme/2053_mechanics_of_solids/images/vimscrot-2023-02-16T14:22:41,358964129+00:00.png create mode 100644 uni/mmme/2053_mechanics_of_solids/images/vimscrot-2023-02-16T14:23:06,415583742+00:00.png create mode 100644 uni/mmme/2053_mechanics_of_solids/images/vimscrot-2023-02-16T14:26:48,809335151+00:00.png create mode 100644 uni/mmme/2053_mechanics_of_solids/images/vimscrot-2023-02-16T14:40:43,750030500+00:00.png create mode 100644 uni/mmme/2053_mechanics_of_solids/images/vimscrot-2023-02-16T14:41:32,526245370+00:00.png create mode 100644 uni/mmme/2053_mechanics_of_solids/lecture_notes/Strain Energy Methods Notes.pdf create mode 100644 uni/mmme/2053_mechanics_of_solids/lecture_slides/MMME2053 SE L1 Slides.pdf create mode 100644 uni/mmme/2053_mechanics_of_solids/lecture_slides/MMME2053 SE L2 Slides.pdf create mode 100644 uni/mmme/2053_mechanics_of_solids/lecture_slides/MMME2053 SE L3 Slides.pdf create mode 100755 uni/mmme/2053_mechanics_of_solids/strain_energy.md diff --git a/uni/mmme/2053_mechanics_of_solids/images/vimscrot-2023-02-16T14:19:48,580021845+00:00.png b/uni/mmme/2053_mechanics_of_solids/images/vimscrot-2023-02-16T14:19:48,580021845+00:00.png new file mode 100644 index 0000000000000000000000000000000000000000..bea434bef47e09358dd0e6208288369f221ba6a5 GIT binary patch literal 8109 zcmbVx2Q*xN*Y44K8C~>IBWfhdB%;meo#?$sFQY_{I*1Vj38D-MB8Wi<(V|X>@+T3| zNf}*4uix>0>#lp>cdc92wHC9SnR9-7?|JsK_kMPYp}qzUPx;7=sE3kv+79f`0&n)taPLmdNM0e4?NA6M}p=Rj9i-yjb^ z~JpkZQ(t@j)h81qkxjUG;&XerO$UN6F(A;KLjRjQ7QL(pFR18gf-z}!bZ7hHK zo* zRAP&p@gDFW%1qYS3gaNhdJCz54H|NaU1qLi6z5Z?LtK&m=fsR=GOAoug0VM_`(Dwb zN|f@rG-|N>I@V>%`;1xZy29XR$-Y13GME+mOrjM{e=xk)1FxbgVS$Ci8Juto5G|CT z4qSjQS!DXvFP3o#YEb2RhfBITV^LHv#H?mP7)A+E;O(8F*g*FXND*+^yO}Js9irgO4~OZT3g!> zJ7|(LGOuHOnu_8~hw8j5t*21(xfk2fey7a7SiYt0J?~91;QXYQPo#wLtM9BN3<}04 zx#dmBN#QY@%m>M>_faUk2>gcw)B7ukzqNVI;f6KA1H)v10Pd>G*JJI6xt@T6CQO`*57 zvG%F!!QP#ci5HhQWS$R)ZBKk*Q*r}Z`x5OhIJ_&aXRABRy&?)bx)>kxI10)fEE>eP z<)ORA-UIuH}=OPTk_Yt}x*FKCNL3lclFM z+PM)inlH=Ux^SNJ{IlDe>7yIEZZfTXdaZkMp^p&ue*LUNazp^bGXI~|nG7O;JLbZK z(*Q;UJPRBpAASa7acf}j*$S)OZ+h`zmH;5jcNc_#SvA9zOs#74pQ${+pVo0vpt-mVp9qSilQ(+Ad1ik7He$Rw78i8-IT1N zc|{QRr0LxA`$-3|_-4Y2%L5Aaq}y8L|2Hmye;|a~8vV7_tXvLjn99Vk+fBxWswA1YQWiJ`k z6!Iuy>g(<0&w1SH+X-xwa<`f!4772NzG}t$+LpJO6GDPHEuU_*71w^`;CVyOrDjID zzfLMFOmh>PO3%viz4Z*~NxQuAvGmKQgd#_{5ghxxv z4?O}(QTY6~F0LCE#w@dQHdf6p6E~^+g(}$EBj9N?>TsX_tepDBVZwyOBb6Gnva0ta z_BFG#0RFoIHBGvnd9*YHN2p`gUl?q~#B_ggK=1U=uQO)0h`;+l>}G`i^8R5*Vh4^_ zo)2P6lTp+duU^(aoZMd;r2AH2J7&ALR4QLrM7*whgFUosW{75jn=M^EO2)1*h{r~< zML_|A%Mg>O*3%z~$9b!d`Wk_S1rjV`>5U87zQzY5gk%b|82m3qWTxBY=1L98g7eDP z9!%L2W%V*GBypB>0rwHi{}xTL_#F8ViI$1 znskj>Jgb|5I@}_*1nEL`ocI&W5v2>*rtG+p<}2^;#r2&()aB*B!4H#p?jMhTKM^7l zW1(7GW5N>+wL^%GJ-)SH<-{SJuR?1(vcIzyA4K{kDLR0VidQz2ta+W>P|- z?GDbm;mw7Lez6Qscss$+y@%THLZvq^`%syo?Xj6^41YElv%x%3Dom}gyI2skZvdZi z1I50J>I2V^UYN+yTI3i~SM1$-qZ%S?6zDqfH$Lb5pO%Vv6NC@8+Tv zf|R^$JDfjr7*>Bd+Rmb}Xm#QMcJ3Gx9!YDNS zra+oW3n#9TOx^iAQJJJwbd{=U^_Oq^D={U0Hyw7%{+5`%m^;~C5Z=ma=yTKMN&}A% ziS%eK2Pq+G+M5)3Gre;d(q{z1Vk zS9DDxmyEjVS$xPqw1D|r9rX$h?-WkUQbdYzozhjLq60oZR%hILaY4!=G0zK@RNPas z1Du2nmH_;6iRv;~xe`&8YB;B>>Qv=YFQ_A6>=ptU{?!nAYE*sSKQu_iP+7Rr^mX-z zEc}`U_Q#H_k7C@<>JW#)E@F7XG5?A#AEl0{^x@@}hH7LvKFsBJOIs|b-FAWp&^4@- z(Pz)oL9^W<%T=n#CIKUDny7keBhag1wHdE>$0YVyCGVpu&Fi4q zGwoi);UQ@ZT=ASw`kcB~8*K#6wiXszKa)MlH;PT(;u4f%4;E-pnWd@xmy_CdjtZXZ z*8-MJS>~U05-J?OlqwJpTmSh0KT{BTr-8LUT#-7bNy_vqrmBTOPmtOD(%Ob1N`?I_ zSpd2fI-%p8BB(>VX{s#&He*H6XTB}+>x-*Pl?Ai~?`xG)tcwL?xJ1 zy$52Gx0UYrTv1e+@Ojz3B8YLu6h-IyZHJph5BI46_t1b%u<`&7-I(Abw=epDdxg)3 z)9*;2t8to#_gt3%aOi((E$zCiIVp#iKIn*f*Kt_kr-V*8(%kq~kihy|oB(K%_o{#R za~+61WUcK#Tt6TLNcXqVmml_|0Ee|Vp-*zG34v#~eeQ0x{S`7bn_59%uLGZgp>yMx zkfpW`;qDpoi=qK`{6qV`Cypu?+o#?ik}c(3r9LBox3qPU#nzXuVlE~1w

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+tags: [ strain_energy ] +uuid: 9cbe6fab-c733-45d0-b8e7-b3d949b3eeb7 +lecture_slides: [ ./lecture_slides/MMME2053 SE L1 Slides.pdf, ./lecture_slides/MMME2053 SE L2 Slides.pdf, ./lecture_slides/MMME2053 SE L3 Slides.pdf ] +lecture_notes: [ ./lecture_notes/Strain Energy Methods Notes.pdf ] +worked_examples: [ ./worked_examples/MMME2053 SE WE Slides.pdf ] +exercise_sheets: [ ./exercise_sheets/Strain Energy Methods Exercise Sheet.pdf, ./exercise_sheets/Strain Energy Methods Exercise Sheet Solutions.pdf ] +--- + +# Strain Energy Definition + +This section refers to the first two slide sets. + +Strain energy in a body is equal to the work done on the body by the applies loads: + +$$U = \int_0^u P\mathrm du$$ + +![](./images/vimscrot-2023-02-16T14:19:48,580021845+00:00.png) + +## Bending + +\begin{equation} +U = \int_0^\Phi M\mathrm d\Phi = \int^L_0\frac{M^2}{2EI}\delta s +\label{eqn:bendingeqn1} +\end{equation} + +![](./images/vimscrot-2023-02-16T14:22:41,358964129+00:00.png) + +If this material represents an element of a larger beam of length $L$ and curvature of radius $R$ + +![](./images/vimscrot-2023-02-16T14:41:32,526245370+00:00.png) + +The strain energy within this element will be: + +$$\delta U = \frac12 M\delta \Phi$$ + +From the elastic beam bending equation we know: + +$$\frac MI = \frac ER$$ + +and as the angle subtended by the element is equal to the change in slope, the expression for the +arc created by the element is: + +$$\delta s = R\delta \Phi$$ + +Eliminating $R$ from the above two equations and rearranging gives + +$$\delta \Phi = \frac{M}{EI}\delta s$$ + +This can be substituted into \label{eqn:bendingeqn1} (left and middle) to get + +$$\delta U = \frac{M^2}{2EI}\delta s$$ + +Integrate over full length of beam to get total strain energy: + +$$U = \int^L_0\frac{M^2}{2EI}\delta s$$ + +## Torsion + +$$U = \int_0^\theta T\mathrm d\theta = \int^L_0 \frac{T^2}{2GJ} \delta s$$ + +Derivations for the equation above is analogous to those for bending and axial loads, and can be +found in the second lecture slide set (p9-p11). + +![](./images/vimscrot-2023-02-16T14:23:06,415583742+00:00.png) + +## Elastic Axial Loading + +\begin{equation} +U = \frac12 Pu = \int^L_0\frac{P^2}{2EA}\delta s +\label{eqn:elasticaxialloading1} +\end{equation} + +![](./images/vimscrot-2023-02-16T14:26:48,809335151+00:00.png) + +If this material represents an element, of length $\delta s$, of a larger beam of length $L$, and +the change in length of this element due to the applied load $P$ is $\delta u$ ten the strain energy +within the element is + +$$\delta U = \frac12 P\delta u$$ + + +![](./images/vimscrot-2023-02-16T14:40:43,750030500+00:00.png) + +> Note that there are transverse strains/displacements due to Poisson's effects but there are no +> transverse stresses/loads, thus there is no work done in the transverse direction. + +Axial strain is: + +$$\epsilon = \frac{\delta u}{\delta s}$$ + +Equating this to Hooke's law yields: + +$$\delta u = \frac{P}{EA}\delta s$$ + +which can be substituted into equation \ref{eqn:elasticaxialloading1} (left and middle) to get + +$$\delta U = \frac{P^2}{2EA}\delta s$$ + +Integrate over full length of beam to get total strain energy: + +$$U = \int^L_0\frac{P^2}{2EA}\delta s$$ + +# Castigliano's Theorem + +This section refers to the third slide set. + +

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