# Stress Calculation Formulas for RC members

## 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

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

### Double reinforcing rectangular section under bending moment

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

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

### 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
$$\tau_t=\cfrac{M_T}{K_t}$$
 $b$ : 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

$$K_t=\cfrac{b^2 \cdot h}{\eta_1}$$

At the center of shorter side

$$K_t=\cfrac{b^2 \cdot h}{\eta_1 \cdot \eta_2}$$
$h/b$ 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

$$e=M/N \qquad N \text{: Compressive axial force}$$ \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} $$\sigma_c=\cfrac{M}{\cfrac{b x}{2}\left(\cfrac{h}{2}-\cfrac{x}{3}\right)+\cfrac{n A_{sc}}{x} c_{c} \left(c_{c}-\cfrac{h}{2}+x\right)+\cfrac{n A_{st}}{x} c \left(c+\cfrac{h}{2}-x\right)}$$ $$\sigma_{st}=\cfrac{n \sigma_c}{x} \left(c+\cfrac{h}{2}-x\right)$$ $$\sigma_{sc}=\cfrac{n \sigma_c}{x} \left(c_{c}-\cfrac{h}{2}+x\right)$$

### Double Reinforceing Rectangular Section under Tensile Axial force and Bending Moment

$$e=M/N \qquad N \text{: Tensile axial force}$$ \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} $$\sigma_{st}=\cfrac{n \sigma_c}{x} \left(c+\cfrac{h}{2}-x\right)$$ $$\sigma_{sc}=\cfrac{n \sigma_c}{x} \left(c_{c}-\cfrac{h}{2}+x\right)$$

## 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

$$A_i=\pi r^2 + n A_s \qquad I_i=\cfrac{\pi r^4}{4}+\cfrac{n A_s r_s^2}{2}$$ \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.

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

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

$$\sigma_c=\cfrac{M (1+\cos\alpha)}{2 r^3 \left\{\cfrac{\sin 4\alpha}{32}+\cfrac{\pi-\alpha}{8}+\cfrac{\sin^3\alpha\cdot\cos\alpha}{3}\right\}+\cfrac{n A_{s} r_s^2}{2 r}}$$ \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

$$\cos\alpha=\cfrac{n \cfrac{r_s}{r}-\cfrac{\sigma_{st}}{\sigma_c}}{\cfrac{\sigma_{st}}{\sigma_c}+n}$$

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.