🔧 Shear Design Method - Truss Model

According to Section 6.2.3 of Eurocode2 (EC2, BSEN1992-1-1:2004), shear reinforcement design, truss model with compression struct with inclined angle θ and tension links ties with inclined angle α. Since vertical shear links is the most commonly used arrangement, this article focuses on the case where the shear links is vertical and only the inclination of the concrete compression strut, θ, is considered.

📐 Principle Behind Truss Model

In this principle, shear resistance is developed through the interaction of tensile vertical reinforcement, inclined compression struts, and the bottom tension chord. The shear capacity is then determined by satisfying force equilibrium First Principle Logic.

🔎 Derivation of Shear Resistance V_Rd,s

The shear resistance, V_Rd,s, is provided by the vertical shear links acting in tension. The derivation below illustrates how V_Rd,s is obtained from force equilibrium, in accordance with Eurocode 2. Eq 6.8,BSEN1992-1-1:2004

Shear Resistence from vertical reinforcement

\begin{aligned}V_{Rd,s} &= \frac{A_{sw}}{s}zf_{ywd}cot\theta\end{aligned}

Eq. 6.8, BSEN 1992-1-1:2004

1. Horizontal length between tension ties = z • cotθ

2. Number of A_sw provided in tension ties, n = z • cotθ / s

3. Total Shear Link Area provided in tension ties =n • A_sw=(z • cotθ / s) • A_sw

4. Material Strength of shear links = f_ywd

5. Shear Resistance (V_Rd,s)=Material Strength of Reinforcement • Shear Link Area=(f_ywd) • (z • cotθ / s • A_sw)=(A_sw/s)zf_ywdcotθ

🔎 Derivation of Maximum Shear Resistance , V_Rd,max

The maximum shear resistance, V_Rd,max, is governed by the capacity of the inclined concrete compression strut. The derivation below illustrates how V_Rd,max is obtained from force equilibrium, in accordance with Eurocode 2. Eq 6.9,BSEN1992-1-1:2004

Maximum Shear Resistance

\begin{aligned}V_{Rd,max} &= \alpha_{cw}b_{w}zv_{1}f_{cd}/(cot\theta + tan\theta)\end{aligned}

Eq. 6.9, BSEN 1992-1-1:2004

1. Component of depth of compression strut = z • cosθ

2. Area of compression strut = b_w • z • cosθ

3. Material strength of concrete = α_cw • v₁ • f_ck

4. Force of Compression strut, F_cc =Material strength • Area=α_cw • v₁ • f_ck • b_w • z • cosθ

5. Max. Shear Resistance, V_Rd,max =Vertical Component of Force of Compression Strut=F_cc • sinθ=(α_cw• v₁• f_ck • b_w • z • cosθ) • sinθ=α_cwb_wv₁f_cdz / (cotθ + tanθ) where cosθsinθ = 1 / (cotθ + tanθ)

🔢 Formula of Additional Longitindinal Force, ΔF_td

Last by not least, as mentioned above, the longitudinal reinforcement, acting as the bottom tension chord, needs to carry additional tensile force due to the applied shear force, V_Ed, as indicated below: Cl 6.2.3(7) & Eq 6.18,BSEN1992-1-1:2004

Additional Longitindinal Force

\begin{aligned}ΔF_{td} &= 0.5 V_{Ed}cot\theta\end{aligned}

Eq. 6.18, BSEN 1992-1-1:2004

📝Summary & Key Takeaways

Truss model is the principle taken by eurocode 2 for shear resistance analysis, where the applied shear force, V_Ed, is resisted by inclined compression struts, vertical shear links, and the longitudinal bottom tension chord.
Shear resistance of a reinforced concrete section, V_Rd,s in Eurocode 2, is the design shear capacity provided by shear links, with equation $V_{Rd,s} = \frac{A_{sw}}{s}zf_{ywd}cot\theta$ Eq. 6.8, BSEN 1992-1-1:2004.
Maximum shear resistanceV_Rd,max is govern by capacity of the inclined concrete compression strut, with equation $V_{Rd,max} = \alpha_{cw}b_{w}zv_{1}f_{cd}/(cot\theta + tan\theta)$ Eq. 6.9, BSEN 1992-1-1:2004.
Additional longitindinal force which carried by the bottom tension longitindinal reinforcement is $ΔF_{td} = 0.5 V_{Ed}cot\theta$ Eq. 6.18, BSEN 1992-1-1:2004.

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CivilSimple Team

The CivilSimple Team writes practical engineering guides for the profession and the curious. All articles are reviewed for technical accuracy before publication.