Numerical Analysis

Assignment $5(80pts)$

Problem Set I: Due $09$:$50$ AM$.$ May $24($Fri$)$

Problem Set II: Due $09$:$50$ AM$,$ May $31($Fri$)$

Problem Set I: Problems $1-10$

Fig. $1$ shows a reinforced concrete beam under a concentrated load.

$P=200kN$ at midspan.

$($a$)$ section

$(1)$ Lowarrig

Fig. $1$ Reinforced concrete beam under a concentric load at midspan

For this beam, the distribution of curvature $\frac{d^{2}y}{d{x}^2}=-(x)$ is given by:

$(x)=\frac{M(x)}{E_e*I_r(x)}$

where,

$)$

$b,h$ and $L=350mm,650mm,$ and $8m,$ respectively:

$I_{}=\frac{b{h}^3}{12}$: and

$E_c=2{10}^{4}M{P}_L.$

Problems $1$ and $2$: Mathematical expressions and plot

$(10$ pts$)$ Find mathematical expressions of $M(x)$ in $kN*m$ and $(x)$ in

$\frac{1}{m}.$

$(10$ pts$)$ Plot $M(x)$ and $(x),$ separately for the whole span.

Problems $3-5$: Hand calculations

$(10$ pts$)$ Using the Trapezoidal rule, evaluate ${}_i={\displaystyle \underset{0}{\overset{x_i}{}}}(x)dx$ for

$i=0,1,2,3,$ and $4$ corresponding to $x_0=0,x_1=1,000mm,$cdots,$x_4=4,000mm.$

Plot the distributions of ${}_i$ along the half span, $0x4m.[]=$mod.

$(10pts)$ Apply the boundary condition that $(4,000mm)=0$ at midspan

$($or equivalently, ${}_4=0)$ and revise the values of ${}_i$ from problem $3.$ Then

plot the distributions of ${}_i$ along the half span, $0x4m.$

$(10$ pts$)$ Using the Trapezoidal rule, evaluate $y_i={\displaystyle \underset{0}{\overset{x_i}{}}}(x)dx$ for

$i=0,1,$cdots,$4$ corresponding to $x_0=0,x_1=1,000mm,$cdots,$x_4=4,000mm.$

Plot the distributions of $y_i$ along the half span, $0x4m,[y]=m.$

Note that boundary condition requires that $y(0)=0($or equivalently.

$\{$:$y_0=0).$ Plot the deflections $y_i$ along the half span, $0x4m.$

Problems $6-9$: Matlab coding

$(10$ pts$)$ Using the Trapezoidal rule, evaluate ${}_i={\displaystyle \underset{0}{\overset{x_i}{}}}(x)dx$ for

$i=0,1,2,$cdots,$80$ corresponding to $x_0=0,x_1=50mm,$cdots,$x_{20}=4,000mm.$

Plot the distributions of ${}_1$ along the half span, $0x4m.[]=$ rod $.$

$(10$ pts$)$ Apply the boundary condition that $(4,000mm)=0$ at midspan

$($or equivalently, ${}_m=0)$ and revise the values of ${}_i$ from problem $6.$

Then plot the distributions of ${}_i$ along the half span, $0x4m.$

$(10$ pts$)$ Estimate error for ${}_w.$

$(10$ pts$)$ Using the Trapezoidal rule, evaluate $y_i={\displaystyle \underset{0}{\overset{x_i}{}}}(x)dx$ for

$i=0,1,$cdots,$80$ corresponding to $x_0=0,x_1=50mm,$cdots,$x_{50}=4,000mm.$ Plot

the distributions of $y_i$ along the half span, $0x4m.[y]=m.$ Note

that $y(0)=0($or equivalently. $y_0=0).$

Problems $10$: Hand calculations

$(10$ pts$)$ Using Gauss Legendre Quadrature, find $(4,000mm).$ Use $n=3$

for a half$-$span, $0x4m.$