31:[] 2f:{"title":"Time Effect on Elastic Modulus Code Comparison","cells":[{"id":0,"value":"md`# Time Effect on Elastic Modulus Code Comparison`","pinned":false,"hidden":false,"mode":"ojs"},{"id":388,"value":"md` In this notebook elastic modulus development of concrete with time is compared based on different codes. `","pinned":false,"hidden":false,"mode":"ojs"},{"id":392,"value":"md`\nThe elastic modulus with respect to time is plotted for several design codes below. As it can be seen from given chart below LARSA4D Built-in Curve is very close to EN 1992-1-1_2004.\n\nThe same material parameter values are used for each code: elastic modulus at day 28, cement type, and temperature. The adjusted or effective concrete age due to effect of temperature is taken into account while calculating modulus of elasticity.\n\nFor the example chart and table, elastic modulus, cement type, and temperature are taken as following, respectively: 35000 MPa, 32.5 N, 20 °C.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":491,"value":"test_material = ({\n Ecm: 24.85, // GPa, Fc_4\n fcm: 40, // MPa; used by CEB-FIP 90 / fib MC2010 only but not implemented in LARSA 4D\n cement_type: \"32.5 N\", // Choices: \"32.5 N\", \"32.5 R\", \"42.5 N\"\n})","pinned":true,"hidden":false,"mode":"ojs"},{"id":497,"value":"test_environment = ({\n temperature: 20 // Celcius (20C gives no temperature effect in CEB-FIP 78, 90/2010, and EN1992-1-1:2004)\n})","pinned":true,"hidden":false,"mode":"ojs"},{"id":17,"value":"chart({\n xAxis: {\n title: \"Time (days)\",\n min: .25,\n max: 150\n },\n yAxis: {\n title: \"Elastic Modulus (GPa)\",\n min: 5,\n max: 30\n },\n\n legend: {\n color: ['red', 'green', 'orange', 'blue'],\n tickSize: 0\n},\n \nseries: [\n { title: \"Built-In Curve\", points: t => Built_in_curve(test_material, test_environment, t), color: 'red'},\n { title: \"CEB-FIP 78\", points: t => ceb_fip_78 (test_material, test_environment, t), color: 'green' },\n { title: \"CEB-FIP 90/2010\", points: t => ceb_fip_90_and_2010 (test_material, test_environment, t), color: 'orange'}, \n { title: \"EN1992/2004\", points: t => en1992_1_1_2004 (test_material, test_environment, t), color: 'blue' },\n],\n})\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":649,"value":"table([\n { Code: \"LARSA 4D Built-in Curve\",\n \"Day 10\": Built_in_curve(test_material, test_environment, 10),\n \"Day 150\": Built_in_curve(test_material, test_environment, 150)\n },\n { Code: \"CEB-FIP 78\",\n \"Day 10\": ceb_fip_78(test_material, test_environment, 10),\n \"Day 150\": ceb_fip_78(test_material, test_environment, 150)\n },\n { Code: \"CEB-FIP 90/2010\",\n \"Day 10\": ceb_fip_90_and_2010(test_material, test_environment, 10),\n \"Day 150\": ceb_fip_90_and_2010(test_material, test_environment, 150)\n },\n { Code: \"EN1992-1-1:2004\",\n \"Day 10\": en1992_1_1_2004(test_material, test_environment, 10),\n \"Day 150\": en1992_1_1_2004(test_material, test_environment, 150)\n }\n],\n{ columns: {\n \"Day 10\": { formatter: (val, i) => Math.round(val*10)/10 },\n \"Day 150\": { formatter: (val, i) => Math.round(val*10)/10 },\n}}\n)","pinned":true,"hidden":false,"mode":"ojs"},{"id":6,"value":"md`# LARSA 4D Built-in Curve`","pinned":false,"hidden":false,"mode":"ojs"},{"id":36,"value":"tex`E_c(t) = E_{ci}\\cdot(\\frac{t}{4+0.857 \\cdot t})^{0.5}`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":911,"value":"md`for t < 28, after which Ec(t) = Eci`","pinned":false,"hidden":false,"mode":"ojs"},{"id":29,"value":"Built_in_curve = function(material, environment, t) {\n if (t < 0) return 0;\n if (t <= 28)\n return material.Ecm * (t / (4 + 0.857 * t))**0.5;\n return material.Ecm;\n}\n\n","pinned":true,"hidden":false,"mode":"ojs"},{"id":59,"value":"md`# CEB-FIP 78 `","pinned":false,"hidden":false,"mode":"ojs"},{"id":64,"value":"tex `E_c(t) = 1.25 \\cdot 9.5 \\cdot f_{cm}(t)^{1/3}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":70,"value":"tex `f_{cm}(t) = (0.1314 \\cdot ln(t)+0.175) \\cdot f_{ck,\\infty} + 8`","pinned":false,"hidden":false,"mode":"ojs"},{"id":78,"value":"tex` t_{adjusted} = \\frac{(temperature(t)+10) \\cdot t}{30} `","pinned":false,"hidden":false,"mode":"ojs"},{"id":602,"value":"md` E_c(t_o) is calculated by using f_cm(t_o) (strength of concrete at time t_o) and f_cm(t_o) value is corrected according to f_c(t_o)/f_c,infinity(t_o) vs time graph given by Figure e.1. The above f_cm(t) equation is derived according to this graph. f_c,infinity value is determined by equating E_c(28) to E_cm since f_c,infinity is a constant depending on material properties.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":261,"value":"ceb_fip_78 = function(material, environment, t) {\n let fck_infinity_fck = 0.1314*Math.log(28)+0.175;\n let fck_infinity = ((material.Ecm/9.5/1.25)**3-8)/fck_infinity_fck;\n let t_adjusted = ((environment.temperature+10)*t)/30; //e.8\n let f_cm_t = (0.1314*Math.log(t_adjusted)+0.175)*fck_infinity + 8; // from figure e.1 and [2.1]\n return 1.25*9.5*(f_cm_t**(1/3));\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":88,"value":"md`# CEB-FIP 90 / fib MC2010`","pinned":false,"hidden":false,"mode":"ojs"},{"id":90,"value":"tex `E_{cm}(t) = E_{ci} \\cdot Beta_e `","pinned":false,"hidden":false,"mode":"ojs"},{"id":96,"value":"tex `Beta_e =Beta_{cc}^{0.5} `","pinned":false,"hidden":false,"mode":"ojs"},{"id":103,"value":"tex ` Beta_{cc} = e^{s*(1-(\\frac{28}{t_{adjusted}})^{0.5})} `","pinned":false,"hidden":false,"mode":"ojs"},{"id":120,"value":"tex `t_{adjusted} = e^{13.65-(\\frac{4000}{273+temperature(t)})}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":254,"value":"t_adjusted = function(environment, t) {\n // Used by both CEB-FIP 90/2010 and EN1992\n return Math.exp(13.65-4000/(273 + environment.temperature))*t ;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":129,"value":"ceb_fip_90_and_2010 = function(material, environment, t) {\n let t_adj = t_adjusted(environment, t);\n \n let s; // taken from explanation of eq (2.1-54)\n if (material.fcm > 60) \n s = 0.20; // this is not implemented in LARSA 4D\n else if (material.cement_type == \"32.5 N\")\n s = 0.38;\n else if (material.cement_type == \"32.5 R\" || material.cement_type == \"42.5 N\") \n s = 0.25;\n else\n s = 0.20;\n \n let beta_cc = Math.exp(s*(1-(28/t_adj)**(0.5))); // eq(2.1-54)\n \n let beta_e = beta_cc**0.5; // eq(2.1-58)\n \n return beta_e*material.Ecm; // eq(2.1-57)\n } \n \n// eq numbers is taken from CEB-FIP 90 ","pinned":true,"hidden":false,"mode":"ojs"},{"id":151,"value":"md `# EN1992-1-1:2004 `","pinned":false,"hidden":false,"mode":"ojs"},{"id":161,"value":"tex ` E{cm}(t) = E{ci} \\cdot (\\frac{f_{cm}(t)}{f_{ci}})^{0.3} `","pinned":false,"hidden":false,"mode":"ojs"},{"id":165,"value":"tex `f_{cm}(t) = Beta_{cc} \\cdot f{cm}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":172,"value":"tex `Beta_{cc} = e^{s \\cdot (1-(\\frac{28}{t})^{0.5})}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":192,"value":"tex `time_{adjusted} = e^{13.65-(\\frac{4000}{273+temperature(t)})}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":158,"value":"en1992_1_1_2004 = function(material, environment, t) {\n let t_adj = t_adjusted(environment, t);\n\n let s;\n if (material.cement_type == \"32.5 N\")\n s = 0.38;\n else if (material.cement_type == \"32.5 R\" || material.cement_type == \"42.5 N\") \n s = 0.25;\n else\n s = 0.20;\n\n let beta_cc = Math.exp(s*(1-(28/t_adj)**(0.5)));\n \n // f_cm_t normally calculated here as beta_cc multiplies by Fcm,\n // but it is would be divided by Fcm in the next step.\n \n return material.Ecm * beta_cc**0.3;\n}\n\n// According to the explanation given for equation (B10), located at Annex B, temperature adjusted concrete age is replaced by t_T in the equation (3.2).","pinned":true,"hidden":false,"mode":"ojs"},{"id":42,"value":"import {chart} with { data as data }\nfrom \"@joshdata/simple-line-chart\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":645,"value":"import {table} from \"@tmcw/tables\"","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}