[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-300760-105":59,"doc-detail-300760-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","fundamental-mechanics-of-materials-equations-sheet","Fundamental Mechanics of Materials Equations - Sheet","","Fundamental Mechanics of Materials Equations compiles core relationships used in strength-of-materials analysis, covering axial deformation, torsion in circular shafts, beam flexure and elastic curve relations, shear-force and bending-moment diagram construction, and plane stress and plane strain transformations. It also provides generalized Hooke’s law links between stresses and strains, principal stresses/strains and maximum shear formulations, failure theories such as Mises equivalent stress, column buckling using Euler load and stress, pressure-vessel stress equations, and geometry properties of plane figures.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/fundamental-mechanics-of-materials-equations-sheet/300760/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/fundamental-mechanics-of-materials-equations-sheet/300760.png","ImageObject",300,407,{"name":92,"@type":93},"Kurz","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-26","2026-09-19",true,{"@type":102,"interactionType":103,"userInteractionCount":44},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What are the key equations for axial deformation and torsion in the sheet?","Question",{"text":112,"@type":113},"The sheet lists relations for average normal stress and strain in axial members and includes torsion equations for maximum torsion shear stress in a circular shaft, polar moment of inertia J, and angle of twist.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are shear-force and bending-moment diagrams constructed?",{"text":117,"@type":113},"It provides six rules for constructing shear-force and bending-moment diagrams, linking relationships among V, M, and loads across beam segments.",{"name":119,"@type":110,"acceptedAnswer":120},"Which failure and buckling formulas are included?",{"text":121,"@type":113},"It includes failure theories such as Mises equivalent stress for plane stress and Euler buckling load and Euler buckling stress for columns, along with the radius of gyration definition.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},300760,1790461763,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":44,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":24,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":144},2336478945635,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Fundamental Mechanics of Materials Equations  \nBasic definitions  \nAverage normal stress in an axial 􀀂avg 􀀂 FA  \nAverage direct shear stress  \n􀀃avg 􀀂 VAV~~  ~~Average bearing stress  \n􀀂b 􀀂 FAb  \nAverage normal strain in an axial 􀀃avg 􀀂 􀀄L~~ ~~ 􀟝􀯧􀯥􀯔􀯡􀯦􀯩􀯘􀯥􀯦􀯘 =  \nmember  \nmember  \n􀟂􀝀 􀟂􀝓  \n~~ ~~ 􀝋􀝎 ~~ ~~ 􀝋􀝎  \n􀝀 􀝓  \n􀟪 􀝏􀝅􀝃􀝉􀜽  \n􀟝 􀝁􀝌􀝏􀝅􀝈􀝋􀝊  \n􀟬 􀝐􀜽􀝑  \n􀟛 􀝃􀜽􀝉􀝉􀜽  \n􀟥 􀝊􀝑  \n􀟜 􀟂 􀝀􀝁􀝈􀝐􀜽  \n􀟙 􀜽􀝈􀝌ℎ􀜽  \n􀟮 􀝌ℎ􀝅  \n􀟱 􀝋􀝉􀝁􀝃􀜽  \n􀟠 􀝐ℎ􀝁􀝐􀜽  \n􀟂􀝐  \n􀝐  \n􀟛 = 􀜿ℎ􀜽􀝊􀝃􀝁 􀝅􀝊 􀜽􀝊􀝃􀝈􀝁 􀝂􀝎􀝋􀝉 90°  \nAverage normal strain caused by temperature change 􀀃T 􀀂 􀀅􀀄T  \nHooke’s Law (one-dimensional)  \n􀀂 􀀂 E􀀃 and 􀀃 􀀂 G􀀆 Poisson’s ratio  \n􀀅 􀀂 􀀆 ~~ 􀀃~~􀀃latng~~ ~~  \nRelationship between E, G, and ν G 􀀂  E   \n2(1 􀀇 􀀇) Definition of allowable stress  \n􀀂allow 􀀂 􀀂~~failu~~FSre~~ ~~ or 􀀃allow 􀀂 􀀃~~failu~~FSre Factor of safety  \nFS 􀀂 􀀂~~􀀂~~ac~~ilu~~tu~~ ~~ or FS 􀀂 􀀃~~􀀃~~ac~~ilu~~tu  \nAxial deformation  \nDeformation in axial members  \n􀀄  \n􀀂  \nFL  \nAE  \nor  \n􀀄 􀀂 ∑i   \nFi Li   \nAi Ei  \nForce-temperature-deformation relationship  \n􀀄 􀀂 FLAE 􀀇 􀀅􀀄TL  \nTorsion  \nMaximum torsion shear stress in a circular shaft 􀀃max 􀀂 TcJ  \nwhere the polar moment of inertia J is defined as  \nJ 􀀂 􀀈2[R4 􀀆 r4 ] 􀀂 􀀈32 [D4 􀀆 d4 ] Angle of twist in a circular shaft  \n􀀉 􀀂 TJLG  \nor  \n􀀉 􀀂 ∑i   \n Ti Li   \nJi Gi  \nPower transmission in a shaft P 􀀂 T􀀊  \n􀝃􀝁􀜽􀝎􀝏  \n􀝎2􀜶1 = 􀝎1􀜶2  \n􀝎1 􀟱1 = 􀝎2 􀟱 2  \n􀝓􀜽􀝐􀝐􀝏 = 􀜰􀝉/􀝏  \nℎ􀝌 = 6600 􀝅􀝊 ∙ 􀝈􀜾/􀝏  \nSix rules for constructing shear-force and bending-moment diagrams  \nRule 1:  \nRule 2:  \nRule 3:  \nRule 4:  \nRule 5:  \nRule 6:  \n􀀄V  \n􀀄V  \ndV  \ndx  \n􀀄M  \ndM  \ndx  \n􀀄M  \n􀀂 P0  \n􀀂 V2 􀀆 V1 􀀂 ∫xx12 w(x) dx  \n􀀂 w(x)  \n􀀂 M2 􀀆 M1 􀀂 ∫xx12 V dx 􀀂 V  \n􀀂􀀆M0  \nFlexure  \nFlexure formula  \n􀀂x 􀀂􀀆 MIy or 􀀂max 􀀂 MIc 􀀂 MS~~ ~~ where Unsymmetric bending of arbitrary cross sections  \n⎡ I z 􀀆 I y⎤ ⎡􀀆I y 􀀇 I z⎤  \n􀀂x 􀀂  ~~ ~~IzyIz~~ ~~􀀆~~ ~~yIz~~ ~~  My 􀀇 ~~ ~~Iyz~~ ~~􀀆~~ ~~Iy2yzzz~~ ~~ Mz Unsymmetric bending of symmetric cross sections  \n􀀂x 􀀂 MyI~~ ~~yz 􀀆 MzI~~ ~~z~~ ~~y~~ ~~ tan 􀀋 􀀂 MMyzIIzy Horizontal shear stress associated with bending  \n􀀃H  \n􀀂  \nVQ  \nIt  \n􀝓ℎ􀝁􀝎􀝁 􀜳 = ∑􀝕􀴤􀯜􀜣 􀯜  \nShear flow formula  \nVQq 􀀂 I  \nS 􀀂 Ic  \n􀜿􀝋􀝉􀝌􀝋􀝏􀝅􀝐􀝁 􀜾􀝁􀜽􀝉􀝏  \n􀝊 =  \n􀜧􀮻  \n􀜧􀮺  \n􀟪􀮺 =  \n􀟪􀮻 =  \n−􀜯􀝕  \n􀜫􀯍  \n−􀝊􀜯􀝕  \n􀜫􀯍  \nShear flow, fastener spacing, and fastener shear relationship  \nqs 􀀈 nf Vf 􀀂 nf 􀀃f Af For circular cross sections,  \n􀜸􀯕􀯘􀯔􀯠 􀜳  􀝊􀜸􀯙􀯔􀯦􀯧􀯘􀯡􀯘􀯥  \nor 􀝍 = =  \n􀜫 􀝏  \nQ 􀀂 112 d 3 (solid sections)  \nQ 􀀂 23 [R3 􀀆 r 3 ] 􀀂 112 [D3 􀀆 d 3 ] (hollow sections)  \nBeam deflections  \nElastic curve relations between w, V, M, θ, and v for constant EI  \nDeflection 􀀂 v  \nSlope 􀀂 Moment  \nShear V  \ndv  \ndx  \nM  \n􀀂 􀀌  \n􀀂 EI  \nd 2 v  \ndx 2  \ndM d 3 v  \n􀀂 ~~ ~~ 􀀂 EI ~~ ~~dx dx 3  \nLoad w 􀀂 ddVx 􀀂 EI dd4xv4  \nFundamental Mechanics of Materials Equations  \nPlane stress transformations  \nNormal and shear stresses on an arbitrary plane  \n􀀂n 􀀂 􀀂x cos2 􀀃 􀀃 􀀂y sin2 􀀃 􀀃 2􀀄xy sin 􀀃 cos 􀀃  \n􀀄nt 􀀂 􀀄(􀀂x 􀀄 􀀂y )sin 􀀃cos 􀀃 􀀃 􀀄xy ( cos2 􀀃 􀀄 sin2 􀀃) or  \n􀟪􀯡 = 􀟪􀯫~~ ~~+2~~ ~~􀟪􀯬 + 􀟪􀯫~~ ~~−2~~ ~~􀟪􀯬 cos 2􀟠 + 􀟬􀯫􀯬 sin 2􀟠  \n􀟪􀯫 + 􀟪􀯬 􀟪􀯫 − 􀟪􀯬  \n􀟪􀯧 = ~~ ~~ − ~~ ~~ cos 2􀟠 − 􀟬 sin 2􀟠  \n2 2 􀯫􀯬  \n􀟪􀯫 − 􀟪􀯬  \n􀟬􀯡􀯧 = − sin 2􀟠 + 􀟬 cos 2􀟠  \n2 􀯫􀯬  \nPrincipal stress magnitudes  \n􀀂p1,p2 􀀂 􀀂x~~ ~~􀀃2~~ ~~􀀂y~~ ~~ 􀀅  ~~ ~~~~ ~~   Orientation of principal planes  \n􀀄xy tan 2􀀃 􀀂 ~~ ~~  \np (􀀂x 􀀄 􀀂y )2  \nMaximum in-plane shear stress magnitude  \n􀀄max 􀀂 􀀅  ~~ ~~~~ ~~  or 􀀄max 􀀂 􀀂p1~~ ~~~~ ~~􀀂p2~~ ~~  \n􀀂avg 􀀂 􀀂x~~ ~~􀀃2~~ ~~􀀂y~~ ~~  \nAbsolute maximum shear stress magnitude  \n􀀄 􀀂 􀀂max 􀀄 􀀂min  abs max 2  \nNormal, stress invariance  \n􀀂x 􀀃 􀀂y 􀀂 􀀂n 􀀃 􀀂t 􀀂 􀀂p1 􀀃 􀀂p2  \nPlane strain transformations  \nNormal and shear strain in arbitrary directions  \n􀀆n 􀀂 􀀆x cos2 􀀃 􀀃 􀀆y sin2 􀀃 􀀃 􀀅xy sin 􀀃 cos 􀀃  \n􀀅nt 􀀂 􀀄2(􀀆x 􀀄 􀀆y )sin 􀀃cos 􀀃 􀀃 􀀅xy (cos2 􀀃 􀀄 sin2 􀀃) or  \n􀟝􀯡 = 􀟝􀯫~~ ~~+2~~ ~~􀟝􀯬 + 􀟝􀯫~~ ~~−2~~ ~~􀟝􀯬 cos 2􀟠 + 􀟛􀯬 sin 2􀟠  \n􀟝􀯧 = 􀟝􀯫~~ ~~+2~~ ~~􀟝􀯬 − 􀟝􀯫~~ ~~−2~~ ~~􀟝􀯬 cos 2􀟠 − 􀟛􀯬 sin 2􀟠  \n􀟛 = − 􀟝􀯫~~ ~~−2~~ ~~􀟝􀯬 sin 2􀟠 + 􀟛􀯬 cos 2􀟠  \nPrincipal strain magnitudes  \n􀀆p1,p2 􀀂  \n􀀆x 􀀃 􀀆y ~~ ~~ 􀀅  \n2  \n~~ ~~􀀆x~~ ~~~~ ~~􀀆y~~ ~~ 2 􀀃 ~~ ~~􀀅xy2~~ ~~ ⎟ 2  \nOrientation of principal strainstan 2􀀃 􀀂  􀀅xy   \np 􀀆x 􀀄 􀀆y Maximum in-plane shear strain  \n~~ ~~􀀅~~m~~2ax 􀀂 􀀅  ~~ ~~􀀆x~~ ~","cbCaiptRldGnHxzH","https://ap.wps.com/l/cbCaiptRldGnHxzH","pdf",1308933,"English","# Fundamental Mechanics of Materials Equations\n## Stress, strain, and material laws\n## Axial deformation, torsion, and beam flexure\n## Shear-force and bending-moment diagrams\n## Plane stress/strain transformations and invariants\n## Generalized Hooke’s law\n## Pressure vessels and failure theories\n## Column buckling and plane-figure properties","[{\"question\":\"What are the key equations for axial deformation and torsion in the sheet?\",\"answer\":\"The sheet lists relations for average normal stress and strain in axial members and includes torsion equations for maximum torsion shear stress in a circular shaft, polar moment of inertia J, and angle of twist.\"},{\"question\":\"How are shear-force and bending-moment diagrams constructed?\",\"answer\":\"It provides six rules for constructing shear-force and bending-moment diagrams, linking relationships among V, M, and loads across beam segments.\"},{\"question\":\"Which failure and buckling formulas are included?\",\"answer\":\"It includes failure theories such as Mises equivalent stress for plane stress and Euler buckling load and Euler buckling stress for columns, along with the radius of gyration definition.\"}]","Fundamental Mechanics of Materials Equations - Sheet | PDF",1789777138,13]