Rectangular Cross Section

$b$	: Width of rectangular section
$h$	: Height of rectangular section
$M$	: Bending moment
$x$	: Distance from compression edge to neutral axis
$d$	: Effective depth (distance from compressive edge to tensile re-bar)
$d_c$	: Distance from compressive edge to compressive re-bar
$c$	: Distance from gravity center of the section to tensile re-bar
$c_{c}$	: Distance from gravity center of the section to compressive re-bar
$\sigma_{c}$	: Compressive stress of Concrete
$\sigma_{st}$	: Tensile stress of Tensile re-bar
$\sigma_{sc}$	: Compressive stress of Compressive re-bar
$n$	: Ratio of elastic moduluses of rebar and concrete (=15)
$A_{st}$	: Total section area of tensile re-bars
$A_{sc}$	: Total section area of compressive rebars

Single Reinforcing Rectangular Section under Bending Moment

\begin{equation} x=\cfrac{n A_{st}}{b} \left[-1+\sqrt{1+\cfrac{2 b d}{n A_{st}}} \right] \end{equation} \begin{equation} \sigma_c=\cfrac{2 M}{b x \left(d-\cfrac{x}{3}\right)} \end{equation} \begin{equation} \sigma_{st}=\cfrac{M}{A_{st} \left(d-\cfrac{x}{3}\right)}=n \sigma_c \cfrac{d-x}{x} \end{equation}

Double reinforcing rectangular section under bending moment

\begin{equation} x=-n\cfrac{A_{st}+A_{sc}}{b}+\sqrt{\left[\cfrac{n (A_{st}+A_{sc})}{b}\right]^2+\cfrac{2 n}{b} (d A_{st} + d_c A_{sc})} \end{equation} \begin{equation} \sigma_c=\cfrac{M}{\cfrac{b x}{2} \left(d-\cfrac{x}{3}\right) + n A_{sc} \cfrac{x-d_c}{x}(d-d_c)} \end{equation} \begin{equation} \sigma_{st}=n \sigma_c \cfrac{d-x}{x} \end{equation} \begin{equation} \sigma_{sc}=n \sigma_c \cfrac{x-d_c}{x} \end{equation}

Shearing Stress of Concrete and Required Section Area of Stirrup in Rectangular Cross Section

$S$	: Shearing force
$\tau_c$	: Shearing stress of concrete
$\tau_{ca}$	: Allowable shearing stress of concrete
$\sigma_{sa}$	: Allowable stress of re-bar
$s$	: Installation interval of stirrup
$V_s$	: Shearing force shared by stirrup
$A_v$	: Required section area of stirrup

\begin{equation} \tau_c=\cfrac{S}{b\cdot (d-x/3)} \end{equation} \begin{equation} V_s=S-\cfrac{\tau_{ca}\cdot b\cdot (d-x/3)}{2} \end{equation} \begin{equation} A_v=\cfrac{V_s \cdot s}{\sigma_{sa}\cdot (d-x/3)} \end{equation}

Shearing Stress of Concrete and Normal Stress of Re-bar due to Torsional Moment

(Source: Specifications for Highway Bridges, Part III Concrete Bridges, Japan Road Association)

Shearing stress of concrete due to torsional moment

$M_T$	: Tortional moment
$\tau_t$	: Shearing stress of concrete due to torsional moment
$K_t$	: Coefficient for stress calculation

\begin{equation} \tau_t=\cfrac{M_T}{K_t} \end{equation}

$b$	: Shorter side of cross section without covers of both sides
$h$	: Longer side of cross section without covers of both sides

At the center of longer side

\begin{equation} K_t=\cfrac{b^2 \cdot h}{\eta_1} \end{equation}

At the center of shorter side

\begin{equation} K_t=\cfrac{b^2 \cdot h}{\eta_1 \cdot \eta_2} \end{equation}

$h/b$	1	2	3	5	10	20	$\infty$
$\eta_1$	4.80	4.07	3.74	3.43	3.20	3.10	3.00
$\eta_2$	1.000	0.795	0.753	0.743	0.742	0.742	0.742

Normal stress of re-bars due to torsional moment

$\sigma_{stl}$	: Stress of lateral re-bar (stirrup) due to torsional moment
$\sigma_{sta}$	: Stress of axial re-bar due to torsional moment
$a$	: Instalation interval of lateral re-bar (stirrup)
$A_{wt}$	: Section area of one lateral re-bar with installation interval $a$
$A_{lt}$	: Total section area of axial re-bars for torsional moment
$b_t$	: Width of cross section without covers of both sides
$h_t$	: Height of cross section without covers of both sides

\begin{align} &\sigma_{stl}=\cfrac{M_T \cdot a}{1.6\cdot b_t\cdot h_t\cdot A_{wt}} \\ &\sigma_{sta}=\cfrac{M_T \cdot (b_t+h_t)}{0.8\cdot b_t\cdot h_t\cdot A_{lt}} \end{align}

Double Reinforcing Rectangular Section under Compressive Axial force and Bending Moment

\begin{equation} e=M/N \qquad N \text{: Compressive axial force} \end{equation} \begin{align} &x^3-x^2\cdot 3\left(\cfrac{h}{2}-e\right)+x\cdot\cfrac{6n}{b}[A_{st} (e+c)+A_{sc} (e-c_{c})] \\ &-\cfrac{6n}{b}\left[A_{st} \left(c+\cfrac{h}{2}\right)(e+c)+A_{sc} \left(\cfrac{h}{2}-c_{c}\right)(e-c_{c})\right] =0 \end{align} \begin{equation} \sigma_c=\cfrac{M}{\cfrac{b x}{2}\left(\cfrac{h}{2}-\cfrac{x}{3}\right)+\cfrac{n A_{sc}}{x} c_{c} \left(c_{c}-\cfrac{h}{2}+x\right)+\cfrac{n A_{st}}{x} c \left(c+\cfrac{h}{2}-x\right)} \end{equation} \begin{equation} \sigma_{st}=\cfrac{n \sigma_c}{x} \left(c+\cfrac{h}{2}-x\right) \end{equation} \begin{equation} \sigma_{sc}=\cfrac{n \sigma_c}{x} \left(c_{c}-\cfrac{h}{2}+x\right) \end{equation}

Double Reinforceing Rectangular Section under Tensile Axial force and Bending Moment

\begin{equation} e=M/N \qquad N \text{: Tensile axial force} \end{equation} \begin{align} &x^3-x^2\cdot 3\left(\cfrac{h}{2}+e\right)-x\cdot\cfrac{6n}{b}[A_{st} (e-c)+A_{sc} (e+c_{c})] \\ &+\cfrac{6n}{b}\left[A_{st} \left(c+\cfrac{h}{2}\right)(e-c)+A_{sc} \left(\cfrac{h}{2}-c_{c}\right)(e+c_{c})\right] =0 \end{align} \begin{align} \sigma_c&=\cfrac{M}{\cfrac{b x}{2}\left(\cfrac{h}{2}-\cfrac{x}{3}\right)+\cfrac{n A_{sc}}{x} c_{c} \left(c_{c}-\cfrac{h}{2}+x\right)+\cfrac{n A_{st}}{x} c \left(c+\cfrac{h}{2}-x\right)} \\ &=\cfrac{N}{\cfrac{n A_{st}}{x}\left(c+\cfrac{h}{2}-x\right)-\cfrac{n A_{sc}}{x} \left(c_{c}-\cfrac{h}{2}+x\right)-\cfrac{b x}{2}} \end{align} \begin{equation} \sigma_{st}=\cfrac{n \sigma_c}{x} \left(c+\cfrac{h}{2}-x\right) \end{equation} \begin{equation} \sigma_{sc}=\cfrac{n \sigma_c}{x} \left(c_{c}-\cfrac{h}{2}+x\right) \end{equation}

Concrete and Re-bar Stresses of Circular Cross Section Column

$r$	: Radius of circular cross section
$r_s$	: Location of re-bars in radial direction
$M$	: Bending moment
$N$	: Axial force
$\sigma_c$	: Compressive stress of Concrete
$\sigma_{st}$	: Tensile stress of re-bar
$\sigma_{sc}$	: Compressive stress of re-bar
$n$	: Ratio of elastic moduluses of rebar and concrete (=15)
$A_{s}$	: Total section area of re-bars
$\alpha$	: Location of neutral axis in radian ($0 < \alpha < \pi$)

Full compressive section

\begin{equation} A_i=\pi r^2 + n A_s \qquad I_i=\cfrac{\pi r^4}{4}+\cfrac{n A_s r_s^2}{2} \end{equation} \begin{align} &\text{Concrete stresses} & &\sigma_{c1}=\cfrac{N}{A_i}+\cfrac{M}{I_i}\cdot r & &\sigma_{c2}=\cfrac{N}{A_i}-\cfrac{M}{I_i}\cdot r \\ &\text{Re-bar stresses} & &\sigma_{s1}=\cfrac{n r_s}{r}\cdot\sigma_{c1} & &\sigma_{s2}=\cfrac{n r_s}{r}\cdot\sigma_{c2} \\ \end{align}

Not full compressive section

Using following equation, $\alpha$ can be calculated by Bisection method.

\begin{equation} e=M/N \end{equation} \begin{equation} \cfrac{e}{r}=\cfrac{\cfrac{\sin 4\alpha}{32}+\cfrac{\pi-\alpha}{8}+\cfrac{\sin^3\alpha\cdot\cos\alpha}{3}+\cfrac{n A_{s}}{4 r^2}\left(\cfrac{r_s}{r}\right)^2} {\cfrac{\sin^3\alpha}{3}+\cfrac{\sin\alpha\cdot\cos^2\alpha}{2}+\cfrac{(\pi-\alpha)\cos\alpha}{2}+\cfrac{n A_{s} \cos\alpha}{2 r^2}} \end{equation}

After getting $\alpha$, concrete stress and re-bar stress can be calculated using following equations.

\begin{equation} \sigma_c=\cfrac{M (1+\cos\alpha)}{2 r^3 \left\{\cfrac{\sin 4\alpha}{32}+\cfrac{\pi-\alpha}{8}+\cfrac{\sin^3\alpha\cdot\cos\alpha}{3}\right\}+\cfrac{n A_{s} r_s^2}{2 r}} \end{equation} \begin{align} &\sigma_{st}=n \sigma_c \cfrac{\cfrac{r_s}{r}-\cos\alpha}{1+\cos\alpha} \\ &\sigma_{sc}=n \sigma_c - \cfrac{\sigma_{st}+n \sigma_c}{r+r_s}(r-r_s) \end{align}

Checking of $\alpha$ value

\begin{equation} \cos\alpha=\cfrac{n \cfrac{r_s}{r}-\cfrac{\sigma_{st}}{\sigma_c}}{\cfrac{\sigma_{st}}{\sigma_c}+n} \end{equation}

Shearing stress of concrete and requied section area of shear re-bar will be calculated using a square cross section model with equivalent cross section area to the circular cross section.