[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-671660-105":59,"doc-detail-671660-en":129},{"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":122,"head_meta":124,"extra_data":126,"updated_unix":128},105,"en","mechanics-of-materials-equation-sheet-chapters-1-14-formulas-and-key-relations","Mechanics of Materials Equation Sheet - Chapters 1-14 - formulas and key relations","","Mechanics of Materials Equation Sheet compiles core governing relations for stress, strain, elasticity, and structural response. It covers simple stress and strain, elastic properties (E, G, ν), axial deformation, torsion, bending moment–shear–deflection links, transverse shear, thin-walled pressure vessels, failure criteria (distortion energy and brittle limits), and plane stress transformations using Mohr’s circle. Additional sections include beam deflections, strain energy, generalized Hooke’s law, and column buckling with critical load and stress expressions.",{"@graph":69,"@context":121},[70,84,104],{"@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/mechanics-of-materials-equation-sheet-chapters-1-14-formulas-and-key-relations/671660/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":98,"encodingFormat":97,"isAccessibleForFree":99,"interactionStatistic":100},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/mechanics-of-materials-equation-sheet-chapters-1-14-formulas-and-key-relations/671660.png","ImageObject",300,407,{"name":92,"@type":93},"Alexandar","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-02",true,{"@type":101,"interactionType":102,"userInteractionCount":8},"InteractionCounter",{"@type":103},"ViewAction",{"@type":105,"mainEntity":106},"FAQPage",[107,113,117],{"name":108,"@type":109,"acceptedAnswer":110},"What equations are provided for simple stress and strain in Chapters 1–2?","Question",{"text":111,"@type":112},"The sheet includes stress measures based on force and area and average shear relations based on shear force and area, along with related strain expressions such as average elongation or deformation forms.","Answer",{"name":114,"@type":109,"acceptedAnswer":115},"How does the sheet handle plane stress transformations and principal stresses?",{"text":116,"@type":112},"It provides stress transformation equations for σx′, σy′, and τx′y′, then uses Mohr’s circle parameters (σavg and radius R) to locate principal planes and compute maximum shear and in-plane shear relationships.",{"name":118,"@type":109,"acceptedAnswer":119},"What failure criteria are summarized for combined stresses?",{"text":120,"@type":112},"It lists distortion energy failure conditions for ductile behavior and maximum stress/brittle limits based on principal stresses, including max shear stress statements tied to allowable yield strength.","https://schema.org",{"og:url":83,"og:type":123,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":125,"canonical":83},"index,follow",{"doc_id":127,"site_id":62},671660,1793577832,{"code":4,"msg":5,"data":130},{"doc_id":127,"user_id":131,"nickname":92,"user_avatar":132,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":133,"file_id":134,"file_url":135,"file_type":136,"file_size":137,"view_count":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":14,"language":138,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":139,"faqs":140,"seo_title":141,"seo_description":67,"update_tm":128,"read_time":24},962090768658,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Mechanics of Materials Equation Sheet  \nGeneral  \nsin θa   sin θb   sin θc  \n= =  \na b c  \nc2 = a2 + b2 − 2abcos θc W = mg  \ng = 32.2 ft/sec2 = 9.81 m/sec2 N = kg·m slug =  lb  sec2 , (ft/ sec2 )  \n N   lb   \nPa = psi =  \nm2 , in2  \nChapter 1 , 2: Simple Stress and Strain σ = NA , τavg = VA  \nAcircle = πr2 = π4 d2 δ   L′ − L  π  \n􀀥avg = L = L , γ = 2 − θ Chapter 3: Elastic Properties of Materials  \n􀀥lat σ = E 􀀥 , ν = − ~~ ~~  \n􀀥long τ = G γ , G = 2~~ ~~(1~~ ~~ν)  \nChapter 4: Axial Load EAdudx = N (x)  \nδ = 􀀡0 L 􀀥 dx = 􀀡0 L NA(xE) dx  \nConstant N, A, E : δ = 􀀥 L = NALE Displacement kinematics:  \nSeries : δ1 + δ2 + · · ·  \nParallel : δ1 = δ2 = · · ·δB/A = uB − uA  \n􀀥T = α∆T, δT = α∆TL Chapter 5: Torsion  \nτ = TJρ , τmax = TJc  \nJshaft = π2 c4 , Jtube = π2 (c4o − c4i) P = T ω , W = N·m/s, hp = ft·lb/s GJddφx = T (x), φ = 􀀡0 L TJ(xG) dx  \nConstant T A G : φ = TL, , JG  \nTwist angle kinematics:  \nSeries : φ 1 + φ2 + · · ·  \nParallel : φ 1 = φ2 = · · · Thin-walled tube:  \nτavg = 2tTAm , q = τavg t = 2TAm φ = 4TAL2mG 􀀢 dts  \nChapter 6: Bending  \nShear and Moment Relations: ddVx = w , ∆V = 􀀡 w (x)dx  \ndMdx = M, ∆M = 􀀡 V (x)dx σx = − MzIzy , σmax = MIc  \nIrect = 112 bh3 , Icircle = 14πr4 I = ¯I+ Ad  \nσx = − MzIzy + MyIyz  \nMy Iz   Iz tan α = = tan θ  \nMz Iy Iy Chapter 7: Transverse Shear  \nτ = V~~ ~~QIt , Q = 􀀡A′ y dA′ = y¯′ A′  \nAsemicircle = 12πr2 , y¯ = 43rπ V Q  \nq = τ t = I , Vfastener = q s Chapter 8: Thin-Walled Pressure Vessels  \npr  pr  pr  \nσhoop = t , σlong = 2t , σsphere = 2tr is inside radius.  \nCombined Stresses  \nP   M c  V Q  Tc  \nσ = ± A ± I , τ = ± It ± J Chapter 10: Failure Criteria τabs−max occurs at max radius of  \nMax distortion energy failure criteria:(σ2p1 − σp1σp2 + σ2p2 ) ≤ (σY )2  \nBrittle materials max (|σp1 | , |σp2 |) ≤ σY  \nChapter 9: Plane Stress Transformations  \n1  \nσavg = 2 (σx + σy )  \n(σy , −⌧xy )  \nσp2 = σavg −  R    σp1 = σavg + R  \n2✓p1 σ  \n2✓s  (σx , ⌧xy )  \n⌧   \n⌧inmpalxane = R = s ✓σx~~ ~~−2~~ ~~σy~~ ~~ ◆2 +(⌧xy )2  \nσx′ = σx~~ ~~+2~~ ~~σy + σx~~ ~~−2~~ ~~σy cos 2θ+τxy sin 2θ  \nσy′ = σx~~ ~~+2~~ ~~σy − σx~~ ~~−2~~ ~~σy cos 2θ−τxy sin 2θ  \nτx′ y′ = − σx~~ ~~−2~~ ~~σy sin 2θ + τxy cos 2θ Center of Mohr’s Circle:  \nσx + σy  \n(σ, τ ) = (σavg , 0), where σavg =  \n2  \nRadius of Mohr’s Circle:  \nR = 􀀥 􀀣 σx~~ ~~−2~~ ~~σy~~ ~~ 􀀤 2 + (τxy)2 Plot (σx , τxy) and (σy , −τxy), + τ ↓ . Principal plane with no shear  \n3 Mohr circles for three planes.  \n(3D principal stresses, σ 1 > σ2 > σ3 ) :  \nτ abs =  \nmax  \nσmax − σmin  \n2 ,  \nσ 1 = σmax , σ3 = σmin When σp1 , σp2 same sign,  \nτabsmax = max 􀀣 ~~ ~~|σp21~~ ~~| , |σp22~~ ~~| 􀀤 ,  \noccurs out of plane When σp1 , σp2 different signs,  \nσp1 − σp2  \nτ abs = ~~ ~~ = τ max max 2 in plane ,  \noccurs in plane Max shear stress failure criteria:  \nτ abs ≤ σY max 2  \nWhen σp1 , σp2 same signs, max (|σp1 | , |σp2 |) ≤ σY ,  \nWhen σp1 , σp2 opposite signs,|σp1 − σp2| ≤ σY  \nσp1 , σp2 = σavg ± R  \n τxy  tan 2θp = (σx − σy)/2  \nCheck θ p with σ x′ transformation [eqn. to](eqn. to) determine if θp equals θp2 , or θp1; or use Mohr’s circle. Other angle is ±90◦ rotated. τ = 0 at principal orientations. On Mohr’s circle (σ, τ ) = (σp1 , 0), and (σ, τ ) = (σp2 , 0) .  \nMax in-plane shear  \nτinmaplxane = R = 􀀥 􀀣 σx~~ ~~−2~~ ~~σy~~ ~~ 􀀤 2 + (τxy)2  \ntan 2θs = − tan12θp = − ~~ ~~(σx~~ ~~xσyy)/2 ,θs = θp ± 45◦  \nDirection of τmax-in-plane determined from τx′ y′ transformation eqn. Normal stress at this orientation:  \nσx + σy  \nσx′ = σy′ = σavg = 2  \nAuthor: Dr. Lonny L. Thompson, Clemson University  \nChapter 12: Beam Deflections  \nRelationships between w , V, M, θ , and v Deflection = v  \ndv Slope angle: = θ  \ndx  \nEI d2 v  \nMoment: M = ~~ ~~ = EI ~~ ~~ρ dx2  \nShear: V = dMdx = ddx 􀀣EId2~~ ~~vdx2 􀀤  \nLoad: w = ddVx = ddx22 􀀣EId2~~ ~~vdx2 􀀤 Method of Integration, determine internal moment as a function of position M (x) .  \nd2 v  \nEI ~~ ~~ = M (x)  \ndx2  \nθ (x) = dvdx = 1EI (􀀡 M (x) dx + C1 )  \nv (x) = 1EI (􀀡 (􀀡 M (x) dx) dx+C1 x+Cs) C1 and C2 are determined from two end conditions (d","cbCaict0xsH8c7TH","https://ap.wps.com/l/cbCaict0xsH8c7TH","pdf",508779,"English","# General\n# Chapter 1-2: Simple Stress and Strain\n# Chapter 3: Elastic Properties of Materials\n# Chapter 4: Axial Load\n# Chapter 5: Torsion\n# Chapter 6: Bending\n# Chapter 7: Transverse Shear\n# Chapter 8: Thin-Walled Pressure Vessels\n# Chapter 9: Plane Stress Transformations\n# Chapter 10: Failure Criteria\n# Chapter 12: Beam Deflections\n# Chapter 13: Column Buckling\n# Chapter 14: Strain Energy","[{\"question\":\"What equations are provided for simple stress and strain in Chapters 1–2?\",\"answer\":\"The sheet includes stress measures based on force and area and average shear relations based on shear force and area, along with related strain expressions such as average elongation or deformation forms.\"},{\"question\":\"How does the sheet handle plane stress transformations and principal stresses?\",\"answer\":\"It provides stress transformation equations for σx′, σy′, and τx′y′, then uses Mohr’s circle parameters (σavg and radius R) to locate principal planes and compute maximum shear and in-plane shear relationships.\"},{\"question\":\"What failure criteria are summarized for combined stresses?\",\"answer\":\"It lists distortion energy failure conditions for ductile behavior and maximum stress/brittle limits based on principal stresses, including max shear stress statements tied to allowable yield strength.\"}]","Mechanics of Materials Equation Sheet - Chapters 1-14 - formulas and key relations | PDF"]