Calculating shear reinforcement, also known as stirrup design, is a critical step in ensuring the structural integrity of concrete members like beams against shear forces. It involves a systematic approach based on design codes and the specific properties of the structural element.
The process of calculating shear reinforcement involves several key steps, starting from determining the applied shear force to designing the appropriate stirrup configuration. The aim is to ensure the concrete section can safely resist the design shear force, often by providing steel stirrups to carry any excess shear not resisted by the concrete itself.
Key Steps in Shear Reinforcement Calculation
Here's a breakdown of the calculation process, incorporating the provided reference information:
1. Determine Design Shear Force ($V_u$)
First, identify the maximum factored shear force that the section needs to resist. This is the design shear force ($V_u$).
- Reference Example: The provided reference states a specific calculation for a shear force: "Shear Force T = 2657 x 7.55. = 100.30 Kn." This value represents the total shear force acting on the section for which reinforcement is being designed.
2. Establish Section Properties
Identify the critical dimensions of the concrete section.
- Effective Depth ($d$): This is the distance from the extreme compression fiber to the centroid of the tensile reinforcement.
- Reference Example: The effective depth of the section is given as "650 mm."
- Width of Section ($b$): The width of the web for beams or the dimension of the column. (While not explicitly given, it's implied by formulas involving
bd).
3. Calculate Nominal Shear Stress ($\tau_v$)
The nominal shear stress is the shear force per unit area of the concrete section.
- Reference Formula: The reference mentions "For Nominal shear stress. τ v= Vu." In typical design codes, this is calculated as:
$ \tau_v = \frac{V_u}{b \cdot d} $
4. Determine Permissible Shear Stress of Concrete ($\tauc$) and Maximum Shear Stress ($\tau{c,max}$)
These values are crucial for assessing whether the concrete alone can resist the shear and if the section size is adequate.
- Permissible Shear Stress ($\tau_c$): This is the shear strength of concrete without shear reinforcement. It depends on the grade of concrete and the percentage of longitudinal tensile steel reinforcement.
- Reference Mention: The reference highlights "Permissible shear stress."
- Percentage of Steel ($p_t$ or $p$): This percentage directly influences $\tau_c$.
- Reference Formula: "Percentage of steel = 100 x Ast. bd."
$ pt = \frac{100 \cdot A{st}}{b \cdot d} $
where $A_{st}$ is the area of longitudinal tensile steel.
- Reference Formula: "Percentage of steel = 100 x Ast. bd."
- Maximum Shear Stress ($\tau_{c,max}$): This is the absolute maximum shear stress a concrete section can safely withstand, even with reinforcement, to prevent crushing of the concrete web. If $\tauv$ exceeds $\tau{c,max}$, the section dimensions must be increased.
5. Check for Shear Reinforcement Requirement
Compare the calculated nominal shear stress ($\tau_v$) with the permissible shear stress of concrete ($\tauc$) and the maximum permissible shear stress ($\tau{c,max}$).
- Case 1: $\tau_v \le \tau_c$
- If $\tau_v$ is less than or equal to $\tau_c$, the concrete section can theoretically resist the shear. However, "For Minimum Shear Reinforcement," minimum shear reinforcement (stirrups) is always required in beams to prevent brittle failure and control cracks.
- Case 2: $\tau_v > \tau_c$ but $\tauv \le \tau{c,max}$
- If $\tau_v$ exceeds $\tauc$ but is within $\tau{c,max}$, shear reinforcement (stirrups) must be designed to resist the excess shear force.
- Case 3: $\tauv > \tau{c,max}$
- If $\tauv$ exceeds $\tau{c,max}$, the section dimensions are insufficient, and the beam or structural member needs to be redesigned with a larger cross-section.
6. Design Shear Reinforcement (Stirrups)
If shear reinforcement is required (Case 1 or Case 2), the stirrups must be designed to carry the shear force not resisted by the concrete.
-
Shear Force to be Resisted by Stirrups ($V_{us}$):
$ V_{us} = V_u - V_c $
where $V_c = \tau_c \cdot b \cdot d$ is the shear resistance of concrete. -
Minimum Shear Reinforcement:
- Reference Formula: "Asv = 0.4. bSv 0.87 x fy." This formula typically represents the minimum area of shear reinforcement required. It is usually presented as:
$ \frac{A_{sv}}{b \cdot S_v} \ge \frac{0.4}{0.87 \cdot fy} $
where $A{sv}$ is the area of the stirrup legs crossing the shear plane, $S_v$ is the spacing of the stirrups along the beam, and $f_y$ is the yield strength of the stirrup steel. From this, the maximum spacing for minimum reinforcement can be derived.
- Reference Formula: "Asv = 0.4. bSv 0.87 x fy." This formula typically represents the minimum area of shear reinforcement required. It is usually presented as:
-
Calculating Spacing of Stirrups ($S_v$):
- Once the type and diameter of stirrups are chosen, $A_{sv}$ (area of the chosen stirrup) can be calculated.
- Reference Context: "Providing 12.0 mm dia 2.00 legged stirrups." This means you select, for instance, 12 mm diameter stirrups with 2 legs (common for beams).
- For 12mm dia 2-legged stirrups, $A_{sv} = 2 \times \frac{\pi}{4} \times (12 \text{ mm})^2$.
- Reference Formula: The reference provides "Sv = Asv x 0.87 x fy." This formula, as given, seems incomplete for calculating spacing directly from $A{sv}$. In common design practices, the required spacing for stirrups is calculated based on the shear force to be carried by the stirrups ($V{us}$):
$ Sv = \frac{A{sv} \cdot 0.87 \cdot fy \cdot d}{V{us}} $
This formula ensures that the chosen stirrups at that spacing provide sufficient resistance.
7. Final Checks and Detailing
After calculating the required spacing, it's crucial to check against maximum permissible spacing limits specified by relevant building codes. These limits ensure adequate crack control and load transfer. Common limits include:
- Maximum spacing not exceeding $0.75d$ for vertical stirrups.
- Maximum spacing not exceeding $450 \text{ mm}$.
- Maximum spacing according to minimum shear reinforcement requirements.
The final selected spacing for the stirrups will be the minimum of the calculated required spacing and the various code-specified maximum spacing limits.
Example Summary Table of Key Parameters (from reference):
| Parameter | Value | Notes |
|---|---|---|
| Design Shear Force ($V_u$) | 100.30 kN | Derived from 2657 x 7.55 as per reference |
| Effective Depth ($d$) | 650 mm | |
| Stirrup Diameter (Chosen) | 12.0 mm | As per reference |
| Number of Stirrup Legs | 2.00 | As per reference |
By following these steps, engineers can accurately calculate and specify the necessary shear reinforcement for concrete structures, ensuring their safety and durability.
