```{r, echo = FALSE, results = "hide"} typ <- if(match_exams_call() %in% c("exams2pdf", "exams2nops")) ".pdf" else ".svg" sup <- paste0("steinbrenner_fdmt",typ) include_supplement(sup, recursive = TRUE) ## stress under center of single footing ## equation from Steinbrenner round2 <- function(x, n) { posneg = sign(x) z = abs(x)*10^n z = z + 0.5 z = trunc(z) z = z/10^n z * posneg } b <- sample(seq(0.25, 1, 0.02),1) ## m, 1/2 * width of footing fa <- sample(c(1, 1.5, 2, 3, 5, 10), 1) ## a/b values for using "Steinbrenner" tables a <- fa*b ## m, 1/2 * length of footing a2 <- 2*a ## b2 <- 2*b ## p <- sample(seq(100, 400, 1),1) ## kPa, base pressure including concrete d <- sample(seq(round(b/2,2),b,0.01),1) ## m, thickness of footing t <- sample(seq(d, 2*d,0.01),1) ## m, depth of footing base bs <- round(b/2,2) ## m, width of column V <- a2*b2*d ## m^3, volume of footing Vs <- bs*bs*(t-d) ## m^3 volume of column VB <- round(V+Vs,4) gamma_B <- 26 ## kN/m^3, specific weight of reinforced concrete gamma <- sample(seq(16,18,0.5),1) p_red <- p-VB*(gamma)/a2/b2 fz <- sample(c(0.25, 0.5, 0.75, 1, 1.5, 2, 3, 4, 6), 1) ## z/b values for "Steinbrenner" table z <- fz*b ## m R <- sqrt(a^2+b^2+z^2) i <- (atan(a*b/z/R)+a*b*z/R*(1/(a^2+z^2)+1/(b^2+z^2)))/2/pi i <- round2(i,4) sig_p <- 4*p_red*i sol <- fmt(c(p_red,sig_p),2) ``` Question ======== A single footing with the width $b=`r b2`$ m and the length $a=`r a2`$ m is embedded $t=`r t`$ m deep into a soil with $\gamma=`r gamma`$ kN/m$^3$. At this depth the footing transfers the bearing pressure $p=`r p`$ kN/m$^2$ (including concrete parts) to the ground. The volume of the concrete parts (footing plus column) insinde the ground is $V_b=`r VB`$ m$^3$. The reinforced concrete has the specific weight $\gamma=`r gamma_B`$ kN/m$^3$. For a settlement calculation, the stress $\sigma_p$ induced by the footing under the center of the footing at the depth $z=`r z`$ m should be calculated. \ ![](steinbrenner_fdmt`r typ`) Answerlist ---------- * How large is the reduced bearing pressure in kN/m$^2$? * How large is the stress $\sigma_p$ in kN/m$^2$? Solution ======== Answerlist ---------- * The reduced bearing pressure is $p_{red}=p - \frac{V_b (\gamma_b-\gamma)}{a \cdot b} = `r fmt(p_red,6)`=`r sol[1]`$ kN/m$^2$. * For the stress below the center of the footing $a'=a/2$ and $b'=b/2$ has to be used. With $a'/b'=`r fa`$ and $z/b'=`r fz`$ we get the Steinbrenner value $i=`r fmt(i,4)`$. The stress is $\sigma_p=4 \cdot i \cdot p_{red} = `r sol[2]`$ kN/m$^2$. Meta-information ================ exname: Stress below footing (Steinbrenner) extype: cloze exclozetype: num|num exsolution: `r paste(sol, collapse = "|")` extol: 1 exextra[difficulty,numeric]: 1 exextra[category,character]: Settlement