{"id":2168,"date":"2019-09-25T16:21:55","date_gmt":"2019-09-25T15:21:55","guid":{"rendered":"https:\/\/www.nickzom.org\/blog\/?p=2168"},"modified":"2024-03-08T12:23:15","modified_gmt":"2024-03-08T11:23:15","slug":"how-to-calculate-and-solve-for-vertical-strain-rock-mechanics","status":"publish","type":"post","link":"https:\/\/www.nickzom.org\/blog\/2019\/09\/25\/how-to-calculate-and-solve-for-vertical-strain-rock-mechanics\/","title":{"rendered":"How to Calculate and Solve for Vertical Strain | Rock Mechanics"},"content":{"rendered":"

To calculate vertical strain, five essential parameters are needed, and these parameters are Vertical Stress (\u03c3v<\/sub>), Poisson’s Ratio (v), Principal Horizontal Stress Component 1 (\u03c3H1<\/sub>), Principal Horizontal Stress Component 2 (\u03c3H2<\/sub>),<\/strong> and Young’s Modulus (E).<\/strong><\/p>\n

The formula for calculating the vertical strain:<\/p>\n

\u03b5v<\/sub>\u00a0=\u00a0(\u03c3v<\/sub>\u00a0– v\u03c3H1<\/sub>\u00a0– v\u03c3H2<\/sub>)<\/sup>\u00a0\/\u00a0E<\/sub><\/p>\n

Where:<\/p>\n

\u03b5v<\/sub>\u00a0= Vertical Strain
\n\u03c3v<\/sub>\u00a0= Vertical Stress
\nv = Poisson’s Ratio
\n\u03c3H1<\/sub>\u00a0= Principal Horizontal Stress Component 1
\n\u03c3H2<\/sub>\u00a0= Principal Horizontal Stress Component 2
\nAnd then, E = Young’s Modulus<\/p>\n

So, let’s solve an example;
\nFind the vertical strain with vertical stress of 44, Poisson’s ratio of 5, principal horizontal stress component 1 of 6, principal horizontal stress component 2 of 4, and then young’s modulus of 12.<\/p>\n

Hence, this implies that;<\/p>\n

\u03c3v<\/sub> = Vertical Stress = 44
\nv = Poisson’s Ratio = 5
\nThen, \u03c3H1<\/sub> = Principal Horizontal Stress Component 1 = 6
\n\u03c3H2<\/sub> = Principal Horizontal Stress Component 2 = 4
\nE = Young’s Modulus = 12<\/p>\n

\u03b5v<\/sub>\u00a0=\u00a0(\u03c3v<\/sub>\u00a0– v\u03c3H1<\/sub>\u00a0– v\u03c3H2<\/sub>)<\/sup>\u00a0\/\u00a0E<\/sub>
\n\u03b5v<\/sub>\u00a0=\u00a0(44 – 5(6) – 5(4))<\/sup>\u00a0\/\u00a012<\/sub>
\nSo, \u03b5v<\/sub>\u00a0=\u00a0(44 – 30 – 20)<\/sup>\u00a0\/\u00a012<\/sub>
\n\u03b5v<\/sub>\u00a0=\u00a0-6<\/sup>\u00a0\/\u00a012<\/sub>
\n\u03b5v<\/sub>\u00a0= -0.5<\/p>\n

Therefore, the\u00a0vertical strain\u00a0<\/strong>is\u00a0-0.5.<\/strong><\/p>\n

Calculating the Vertical Stress when the Vertical Strain, Poisson’s Ratio, Principal Horizontal Stress Component 1, Principal Horizontal Stress Component 2 and Young Modulus are Given<\/strong><\/h3>\n

\u03c3v<\/sub> = (\u03b5v<\/sub> x E) + v\u03c3H1<\/sub> + v\u03c3H2<\/sub><\/p>\n

Where:<\/p>\n

\u03c3v<\/sub>\u00a0= Vertical Stress
\n\u03b5v<\/sub>\u00a0= Vertical Strain
\nv = Poisson’s Ratio
\nAlso, \u03c3H1<\/sub>\u00a0= Principal Horizontal Stress Component 1
\n\u03c3H2<\/sub>\u00a0= Principal Horizontal Stress Component 2
\nE = Young’s Modulus<\/p>\n

So, let’s solve an example;
\nFind the vertical stress with a vertical strain of 52, Poisson’s ratio of 12, principal horizontal stress component 1 of 15, principal horizontal stress component 2 of 10 and then young’s modulus of 13.<\/p>\n

Hence, this implies that;<\/p>\n

\u03b5v<\/sub> = Vertical Strain = 52
\nv = Poisson’s Ratio = 12
\n\u03c3H1<\/sub> = Principal Horizontal Stress Component 1 = 15
\n\u03c3H2<\/sub> = Principal Horizontal Stress Component 2 = 10
\nAnd then, E = Young’s Modulus = 13<\/p>\n

\u03c3v<\/sub> = (\u03b5v<\/sub> x E) + v\u03c3H1<\/sub> + v\u03c3H2<\/sub>
\nThen, \u03c3v<\/sub> = (52 x 13) + 12(15) + 15(10)
\n\u03c3v<\/sub> = 676 + 180 + 150
\n\u03c3v<\/sub> = 1006<\/p>\n

Therefore, the\u00a0vertical stress\u00a0<\/strong>is\u00a01006.<\/strong><\/p>\n

Calculating the Poisson’s Ratio when the Vertical Strain, Vertical Stress, Principal Horizontal Stress Component 1, Principal Horizontal Stress Component 2 and Young Modulus are Given<\/strong><\/h4>\n

v = \u03c3v<\/sub> – (\u03b5v<\/sub> x E)<\/sup> \/ \u03c3H1<\/sub> + \u03c3H2<\/sub><\/sub><\/p>\n

Where:<\/p>\n

v = Poisson’s Ratio
\n\u03b5v<\/sub>\u00a0= Vertical Strain
\n\u03c3v<\/sub>\u00a0= Vertical Stress
\n\u03c3H1<\/sub>\u00a0= Principal Horizontal Stress Component 1
\n\u03c3H2<\/sub>\u00a0= Principal Horizontal Stress Component 2
\nE = Young’s Modulus<\/p>\n

So, let’s solve an example;
\nFind the Poisson’s ratio with a vertical strain of 16, vertical stress of 110, principal horizontal stress component 1 of 7, principal horizontal stress component 2 of 13 and then young’s modulus of 6.<\/p>\n

Hence, this implies that;<\/p>\n

\u03b5v<\/sub> = Vertical Strain = 16
\n\u03c3v<\/sub> = Vertical Stress = 110
\n\u03c3H1<\/sub> = Principal Horizontal Stress Component 1 = 7
\n\u03c3H2<\/sub> = Principal Horizontal Stress Component 2 = 13
\nE = Young’s Modulus = 6<\/p>\n

v = \u03c3v<\/sub> – (\u03b5v<\/sub> x E)<\/sup> \/ \u03c3H1<\/sub> + \u03c3H2<\/sub><\/sub>
\nv = 110 – (16 x 6)<\/sup> \/ 7 + 13<\/sub>
\nHence, v = 110 – 96<\/sup> \/ 20<\/sub>
\nv = 14<\/sup> \/ 20<\/sub>
\nv = 0.7<\/p>\n

Therefore, the P<\/strong>oisson’s ratio\u00a0<\/strong>is\u00a00.7.<\/strong><\/p>\n

Calculating the Principal Horizontal Stress Component 1 when the Vertical Strain, Vertical Stress, Poisson’s Ratio, Principal Horizontal Stress Component 2 and Young Modulus are Given.<\/strong><\/h4>\n

\u03c3H1<\/sub> = -((Ev<\/sub> x E) – \u03c3v<\/sub> + v\u03c3H2<\/sub><\/sup>\u00a0\/ v<\/sub>)<\/p>\n

Where;<\/p>\n

\u03c3H1<\/sub>\u00a0= Principal Horizontal Stress Component 1
\n\u03b5v<\/sub>\u00a0= Vertical Strain
\n\u03c3v<\/sub>\u00a0= Vertical Stress
\nv = Poisson’s Ratio
\n\u03c3H2<\/sub>\u00a0= Principal Horizontal Stress Component 2
\nE = Young’s Modulus<\/p>\n

Then, let’s solve an example;
\nFind the principal horizontal stress component 1 with a vertical strain of 15, Poisson’s ratio of 10, vertical stress of 11, principal horizontal stress component 2 of 8 and young’s modulus of 14.<\/p>\n

Hence, this implies that;<\/p>\n

\u03b5v<\/sub> = Vertical Strain = 15
\n\u03c3v<\/sub> = Vertical Stress = 11
\nv = Poisson’s Ratio = 10
\n\u03c3H2<\/sub> = Principal Horizontal Stress Component 2 = 8
\nE = Young’s Modulus = 14<\/p>\n

\u03c3H1<\/sub> = -((Ev<\/sub> x E) – \u03c3v<\/sub> + v\u03c3H2<\/sub><\/sup>\u00a0\/ v<\/sub>)
\n\u03c3H1<\/sub> = -((15 x 14) – 11 + 10(8)<\/sup> \/ 10<\/sub>)
\nSo, \u03c3H1<\/sub> = -(210 – 11 + 80<\/sup> \/ 10<\/sub>)
\n\u03c3H1<\/sub> = -(199 + 80<\/sup> \/ 10<\/sub>)
\n\u03c3H1<\/sub> = -(279<\/sup> \/ 10<\/sub>)
\nThen, \u03c3H1<\/sub> = -(27.9)
\n\u03c3H1<\/sub> = -27.9<\/p>\n

Therefore, the\u00a0principal horizontal stress component 1 <\/strong>is -27.9.<\/strong><\/p>\n

Read more:\u00a0How to Calculate and Solve for Horizontal Strain | Rock Mechanics<\/a><\/p>\n

Calculating the Principal Horizontal Stress Component 2 when the Vertical Strain, Vertical Stress, Poisson’s Ratio, Principal Horizontal Stress Component 1 and Young Modulus are Given<\/strong><\/h3>\n

\u03c3H2<\/sub> = -((Ev<\/sub> x E) – \u03c3v<\/sub> + v\u03c3H1<\/sub><\/sup>\u00a0\/ v<\/sub>)<\/p>\n

Where;<\/p>\n

\u03c3H2<\/sub>\u00a0= Principal Horizontal Stress Component 2
\n\u03b5v<\/sub>\u00a0= Vertical Strain
\n\u03c3v<\/sub>\u00a0= Vertical Stress
\nv = Poisson’s Ratio
\n\u03c3H1<\/sub>\u00a0= Principal Horizontal Stress Component 1
\nE = Young’s Modulus<\/p>\n

So, let’s solve an example;
\nFind the principal horizontal stress component 2 with a vertical strain of 30, Poisson’s ratio of 18, vertical stress of 17, principal horizontal stress component 1 of 11 and then young’s modulus of 19.<\/p>\n

Hence, this implies that;<\/p>\n

\u03b5v<\/sub> = Vertical Strain = 30
\n\u03c3v<\/sub> = Vertical Stress = 17
\nv = Poisson’s Ratio = 18
\n\u03c3H1<\/sub>\u00a0= Principal Horizontal Stress Component 1 = 11
\nE = Young’s Modulus = 19<\/p>\n

\u03c3H2<\/sub> = -((Ev<\/sub> x E) – \u03c3v<\/sub> + v\u03c3H1<\/sub><\/sup>\u00a0\/ v<\/sub>)
\n\u03c3H2<\/sub> = -((30 x 19) – 17 + 18(11)<\/sup> \/ 18<\/sub>)
\nThen, \u03c3H2<\/sub> = -(570 – 17 + 198<\/sup> \/ 18<\/sub>)
\n\u03c3H2<\/sub> = -(533 + 198<\/sup> \/ 18<\/sub>)
\n\u03c3H2<\/sub> = -(751<\/sup> \/ 18<\/sub>)
\nSo, \u03c3H2<\/sub> = -(41.72)
\n\u03c3H2<\/sub> = -41.72<\/p>\n

Therefore, the\u00a0principal horizontal stress component 2\u00a0<\/strong>is\u00a0-41.72.<\/strong><\/p>\n

Calculating the Young Modulus when the Vertical Strain, Vertical Stress, Poisson’s Ratio, Principal Horizontal Stress Component 1 and Principal Horizontal Stress Component 2 are Given.<\/strong><\/h4>\n

E = (\u03c3v<\/sub> – v\u03c3H1<\/sub> – v\u03c3H2<\/sub>)<\/sup> \/ \u03b5v<\/sub><\/sub><\/p>\n

Where;<\/p>\n

E = Young’s Modulus
\n\u03b5v<\/sub>\u00a0= Vertical Strain
\n\u03c3v<\/sub>\u00a0= Vertical Stress
\nv = Poisson’s Ratio
\n\u03c3H1<\/sub>\u00a0= Principal Horizontal Stress Component 1
\nAnd also, \u03c3H2<\/sub> = Principal Horizontal Stress Component 2<\/p>\n

Then, let’s solve an example;
\nFind the young’s modulus with a vertical strain of 20, vertical stress of 60, Poisson’s ratio of 2, principal horizontal stress component 1 of 7, and then the principal horizontal stress component 2 of 9.<\/p>\n

Hence, this implies that;<\/p>\n

\u03b5v<\/sub> = Vertical Strain = 20
\n\u03c3v<\/sub> = Vertical Stress = 60
\nAnd then, v = Poisson’s Ratio = 2
\n\u03c3H1<\/sub> = Principal Horizontal Stress Component 1 = 7
\n\u03c3H2<\/sub> = Principal Horizontal Stress Component 2 = 9<\/p>\n

E = (\u03c3v<\/sub> – v(\u03c3H1<\/sub>) – v(\u03c3H2<\/sub>))<\/sup> \/ \u03b5v<\/sub><\/sub>
\nE = (60 – 2(7) – 2(9))<\/sup> \/ 20<\/sub>
\nThen, E = (60 – 14 – 18)<\/sup> \/ 20<\/sub>
\nE = (28)<\/sup> \/ 20<\/sub>
\nE = 1.4<\/p>\n

Therefore, the\u00a0young’s modulus\u00a0<\/strong>is\u00a01.4.<\/strong><\/p>\n

How to Calculate Vertical Strain Using the Nickzom Calculator<\/strong><\/h3>\n

Nickzom Calculator \u2013\u00a0The Calculator Encyclopedia<\/strong> is capable of calculating the vertical strain.<\/p>\n

Hence, to get the answer and workings of the vertical strain, use the Nickzom Calculator – The Calculator Encyclopedia. <\/strong>First, you need to obtain the app.<\/p>\n

Moreover, you can get this app via multiple means, such as<\/p>\n

Web<\/strong>\u00a0\u2013\u00a0https:\/\/www.nickzom.org\/calculator-plus<\/a><\/p>\n

To get access to the\u00a0professional\u00a0<\/strong>version via web, you need to\u00a0register<\/strong>\u00a0and\u00a0subscribe\u00a0<\/strong>for\u00a0NGN 1,500\u00a0<\/strong>per\u00a0annum<\/strong>\u00a0to have utter access to all functionalities.
\nYou can also try the\u00a0demo\u00a0<\/strong>version via\u00a0https:\/\/www.nickzom.org\/calculator<\/a><\/p>\n

Android (Paid)<\/strong>\u00a0\u2013\u00a0https:\/\/play.google.com\/store\/apps\/details?id=org.nickzom.nickzomcalculator<\/a>
\nAndroid (Free)<\/strong>\u00a0\u2013\u00a0https:\/\/play.google.com\/store\/apps\/details?id=com.nickzom.nickzomcalculator<\/a>
\nApple (Paid)<\/strong>\u00a0\u2013\u00a0https:\/\/itunes.apple.com\/us\/app\/nickzom-calculator\/id1331162702?mt=8<\/a>
\nOnce, you have obtained the calculator encyclopedia app, proceed to the\u00a0Calculator Map,\u00a0<\/strong>then click on\u00a0Geology<\/strong>\u00a0<\/strong>under\u00a0Add-on.<\/strong><\/p>\n

\"\"<\/p>\n

So, Click on Rock Mechanics<\/strong>\u00a0<\/strong>under\u00a0Geology<\/strong><\/p>\n

\"How<\/p>\n

Then, click on Vertical<\/strong>\u00a0<\/strong>Strain\u00a0<\/strong>under\u00a0Rock Mechanics<\/strong><\/p>\n

\"\"<\/p>\n

So, the screenshot below displays the page or activity to enter your values, to get the answer for the vertical strain according to the respective parameters which are the Vertical Stress (\u03c3v<\/sub>), Poisson’s Ratio (v), Principal Horizontal Stress Component 1 (\u03c3H1<\/sub>), Principal Horizontal Stress Component 2 (\u03c3H2<\/sub>)<\/strong> and Young’s Modulus (E).<\/strong><\/p>\n

\"How<\/p>\n

Then, enter the values appropriately and accordingly for the parameters as required by the Vertical Stress (\u03c3v<\/sub>)<\/strong> is 44<\/strong>, Poisson’s Ratio (v)<\/strong> is 5<\/strong>, Principal Horizontal Stress Component 1 (\u03c3H1<\/sub>)<\/strong> is 6<\/strong>, Principal Horizontal Stress Component 2 (\u03c3H2<\/sub>)<\/strong> is 4 <\/strong>and Young’s Modulus (E)<\/strong> is 12<\/strong>.<\/p>\n

\"\"<\/p>\n

Finally, Click on Calculate<\/p>\n

\"How
\n\"\"<\/p>\n

As\u00a0you can see from the screenshot above,\u00a0Nickzom Calculator<\/strong>\u2013 The Calculator Encyclopedia solves the vertical strain and also presents the formula, workings and steps too.<\/p>\n","protected":false},"excerpt":{"rendered":"To calculate vertical strain, five essential parameters are needed, and these parameters are Vertical Stress (\u03c3v), Poisson’s Ratio…","protected":false},"author":5,"featured_media":13152,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Calculate Vertical Strain","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn the steps, guidelines, and formula on How to accurately Calculate and Solve for Vertical Strain | Rock Mechanics","_yoast_wpseo_opengraph-title":"","_yoast_wpseo_opengraph-description":"","_yoast_wpseo_twitter-title":"","_yoast_wpseo_twitter-description":"","_lmt_disableupdate":"","_lmt_disable":"","_sitemap_exclude":false,"_sitemap_priority":"","_sitemap_frequency":"","_yoast_wpseo_focuskw_text_input":"","csco_display_header_overlay":false,"csco_singular_sidebar":"","csco_page_header_type":"","footnotes":""},"categories":[48],"tags":[658,84,64,87,667,670,671,657,669,668,672],"class_list":["post-2168","post","type-post","status-publish","format-standard","has-post-thumbnail","category-geology","tag-add-on","tag-calculator-encyclopedia","tag-geology","tag-nickzom-calculator","tag-poissons-ratio","tag-principal-horizontal-stress-component-1","tag-principal-horizontal-stress-component-2","tag-rock-mechanics","tag-vertical-strain","tag-vertical-stress","tag-youngs-modulus","cs-entry"],"yoast_head":"\nHow to Calculate and Solve for Vertical Strain | Rock Mechanics<\/title>\n<meta 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