Multi-Span RC Beam Design
Comprehensive Structural Analysis
Global Max Demand
Peak Factored $w_u$
0.0 kN/m
Peak Moment $M_u$
0.0 kN·m
Peak Shear $V_u$
0.0 kN
Overall Beam Check
Section Status
Singly Reinf.
Cross Section
Comprehensive Structural Detailing (Elevation View)
Multi-Span SFD & BMD Visualizer
Span-by-Span Section Design Summary
| Limit State | Location | Demand (kN·m | kN) |
Req. Steel Area (mm² | mm²/mm) |
Provided Detailing | Capacity (kN·m | kN) |
Prov. Steel Area (mm² | mm²/mm) |
|---|
Bar Bending Schedule
| Mark | Span | Location | Shape | Size | Qty | Cut Length (m) |
Total Length (m) |
|---|
7. Bill of Quantities (Material Takeoff)
| Material / Item | Quantity | Unit | Remarks |
|---|
Bar Bending Schedule (BBS) Formulas & Bend Deductions
1. Standard Bend Deductions (Elongation)
* Where $d$ is the nominal diameter of the reinforcement bar.
| Angle of Bend | Bend Deduction | Example (Ø10mm) |
|---|---|---|
| 45° Bend (Slight bend) | 1d | 10 mm |
| 90° Bend (L-bar, ties) | 2d | 20 mm |
| 135° Bend (Seismic hook) | 3d | 30 mm |
| 180° Bend (U-Hook) | 4d | 40 mm |
2. Cutting Length Formulas ($L$)
-
Straight Bar with 90° Hooks (e.g., Bottom Steel):
$$L = \text{Clear Span} + 2(L_d) - 2(2d)$$
-
Closed Stirrup / Tie (with 135° Hooks):
Let $a, b$ = internal dimensions of stirrup. Hook length $\approx 12d$.
$$L = 2(a + b) + 24d$$
-
Top Support Bar (Curtailed):
$$L = 0.3L_1 + \text{Support Width} + 0.3L_2$$
Design Methodology Workflow
graph TD
A[Start: Define Geometry & Boundaries] --> B[Input Material Properties & Loads]
B --> C[Compute Factored Loads wu]
subgraph Structural Analysis
C --> D[Generate Stiffness Matrix & Load Vectors]
D --> E[Solve Nodal Displacements/Rotations]
E --> F[Calculate Bending Moments Mu & Shears Vu]
end
subgraph Flexural Design ULS
F --> G[Extract Peak Positive & Negative Moments]
G --> H{Is Mu > Mmax limit?}
H -- No --> I[Design Singly Reinforced Section]
H -- Yes --> J[Design Doubly Reinforced Section]
I --> K[Calculate As,req]
J --> K
K --> L[Select Rebar & Verify φMn ≥ Mu]
end
subgraph Shear Design ULS
L --> M[Extract Critical Shear Vu at supports]
M --> N[Calculate Concrete Capacity Vc]
N --> O{Is Vu > φVc / 2?}
O -- No --> P[Provide Nominal / Min Stirrups]
O -- Yes --> Q[Calculate Required Vs & Spacing s]
P --> R[Verify φVn ≥ Vu]
Q --> R
end
subgraph Detailing & BBS
R --> S[Check Rebar Spacing & Clearances]
S --> T[Calculate Bar Cut Lengths & Bend Deductions]
T --> U[Generate Bar Bending Schedule]
end
U --> V[End: Output Design Report]
Detailed Design Theories
2.1 Structural Analysis (Matrix Stiffness Method)
Continuous beams are statically indeterminate. The tool utilizes the Direct Stiffness Method to solve for internal forces.
- Stiffness Matrix ($[K]$): For each span with length $L$ and moment of inertia $I$, the rotational stiffness coefficients are defined as $\frac{4EI}{L}$ for the near end and $\frac{2EI}{L}$ for the far end.
- Fixed End Moments (FEM): For a uniformly distributed load $w_u$, the FEM at the supports are calculated as: $$FEM = \frac{w_u \cdot L^2}{12}$$
- Nodal Equilibrium: The global stiffness matrix $[K]$ and the load vector $\{F\}$ are assembled. The nodal rotations $\{\theta\}$ are solved using: $$[K] \{\theta\} = \{F\}$$
- Final Moments & Shears: Span moments are computed by superimposing the FEMs with the moments induced by the calculated nodal rotations.
2.2 Flexural Design (Ultimate Limit State)
Flexural design ensures the section can safely resist the maximum bending moments. The tool primarily utilizes the Equivalent Rectangular Stress Block (Whitney Stress Block) for concrete in compression.
Singly Reinforced Section
When the applied moment $M_u$ is relatively small, only tension reinforcement is required.
- Concrete Compressive Force: $C = 0.85 f'_c \cdot a \cdot b$
- Steel Tensile Force: $T = A_s f_y$
- Equilibrium: $a = \frac{A_s f_y}{0.85 f'_c b}$
- Design Capacity: $$\phi M_n = \phi A_s f_y \left( d - \frac{a}{2} \right)$$
- Required Ratio ($\rho$): $$\rho = \frac{0.85 f'_c}{f_y} \left( 1 - \sqrt{1 - \frac{2 R_n}{0.85 f'_c}} \right)$$ where $R_n = \frac{M_u}{\phi b d^2}$
Doubly Reinforced Section
If required $\rho$ exceeds the maximum allowable limit, steel is added to the compression zone ($A'_s$).
- Max Singly Reinforced Moment: $M_{max}$
- Excess Moment: $\Delta M = M_u - \phi M_{max}$
- Compression Steel Req: $$A'_s = \frac{\Delta M}{\phi f'_s (d - d')}$$
- Total Tension Steel: $A_s = A_{s1} + A'_s$
2.3 Shear Design
Shear failure is sudden and brittle, making shear capacity crucial. $\phi V_n \ge V_u$ where $V_n = V_c + V_s$
- Concrete Shear Capacity ($V_c$) (ACI 318 approach): $$V_c = 0.17 \lambda \sqrt{f'_c} b_w d$$ (For Eurocode 2, $V_{Rd,c}$ is based on the reinforcement ratio and size effect factor).
- Steel Shear Capacity ($V_s$): If $V_u > \frac{\phi V_c}{2}$, stirrups are required. $$V_s = \frac{A_v f_{yt} d}{s}$$ where $A_v$ is the cross-sectional area of the stirrup legs and $s$ is the spacing.
- Maximum Spacing ($s_{max}$): Usually limited to $\frac{d}{2}$ or $600\text{mm}$. If $V_s > 0.33 \sqrt{f'_c} b_w d$, $s_{max}$ is reduced to $\frac{d}{4}$ or $300\text{mm}$.
2.4 Detailing and Deflection Control
- Minimum Thickness: To control deflections, codes prescribe minimum $\frac{L}{h}$ ratios (e.g., $L/16$ for simply supported, $L/18.5$ for continuous spans per ACI).
- Curtailment: Top reinforcement at supports is generally curtailed at $0.3L$ from the face of the support, while bottom reinforcement is continuous across the span with proper anchorage.
- Clear Cover: Required to protect steel from corrosion and fire (typically 40mm for standard internal beams).
Design Equations
1. Flexural Capacity ($\phi M_n$)
2. Shear Capacity ($\phi V_n$)
Standard Design Reference Tables
Typical Live Loads (LL)
| Occupancy / Use | Metric (kPa) | Imp (psf) |
|---|---|---|
| Residential / Hotel Rooms | 2.0 | 40 |
| Offices | 2.5 | 50 |
| Classrooms | 3.0 | 60 |
| Corridors (Above 1st Flr) | 4.0 | 80 |
| Retail / Assembly Areas | 4.8 | 100 |
| Heavy Storage / Library | 6.0+ | 125+ |
Typical Material Strengths
| Material | Metric (MPa) | Imp (psi) |
|---|---|---|
| Concrete (Standard) | 21, 25, 28 | 3000, 4000 |
| Concrete (High Strength) | 35, 40, 50 | 5000, 6000 |
| Steel Rebar (Grade 40) | 280 | 40,000 |
| Steel Rebar (Grade 60) | 420 | 60,000 |
| Steel Rebar (B500B/Gr 75) | 500 | 75,000 |
Minimum Clear Cover ($c_c$)
| Exposure Condition | Metric (mm) | Imp (in) |
|---|---|---|
| Slabs/Walls (Dry) | 20 | 0.75 |
| Beams/Cols (Internal) | 40 | 1.5 |
| Weather Exposed | 40 - 50 | 1.5 - 2.0 |
| Cast Against Earth | 75 | 3.0 |
Min. Thickness ($h$) - Deflection Control
| Support Condition | 1-Way Slab |
|---|---|
| Simply Supported | $L / 20$ |
| One End Cont. | $L / 24$ |
| Both Ends Cont. | $L / 28$ |
| Cantilever | $L / 10$ |
| Support Condition | Beams |
|---|---|
| Simply Supported | $L / 16$ |
| One End Cont. | $L / 18.5$ |
| Both Ends Cont. | $L / 21$ |
| Cantilever | $L / 8$ |
* ACI 318 baseline for $f_y = 420$ MPa (60 ksi). For other yield strengths, multiply by $(0.4 + f_y / 700)$.
Rebar Spacing Limits- Min Spacing: To allow aggregate passing, generally $> 25\text{mm}$ (1"), or $d_b$, or $1.33 \times$ max agg size.
References
- ACI 318: Building Code Requirements for Structural Concrete and Commentary. American Concrete Institute.
- EN 1992-1-1 (Eurocode 2): Design of concrete structures - Part 1-1: General rules and rules for buildings. European Committee for Standardization.
- SS EN 1992-1-1: Singapore National Annex to Eurocode 2. Enterprise Singapore.
- IS 456: Plain and Reinforced Concrete - Code of Practice. Bureau of Indian Standards.
- BS 8110-1: Structural use of concrete. Code of practice for design and construction. British Standards Institution.
- GB 50010: Code for design of concrete structures. Ministry of Housing and Urban-Rural Development of the PRC.
- JSCE: Standard Specifications for Concrete Structures - Design. Japan Society of Civil Engineers.
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