Design of RC Beams for Shear

Uncracked Elastic Beam Behavior

Look at the shear and bending moment diagrams. The acting shear stress distribution on the beam:

The acting stresses distributed across the cross-section.

The shear stress acting on the rectangular beam.

The equation of the shear stress for a rectangular beam is given as:

Note: The maximum 1st moment occurs at the neutral axis (NA).

The ideal shear stress distribution can be described as:

A realistic description of the shear distribution is shown as:

The shear stress acting along the beam can be described with a stress block:

Using Mohr’s circle, the stress block can be manipulated to find the maximum shear and the crack formation.

Inclined Cracking in Reinforced Concrete Beams

Typical Crack Patterns for a deep beam.

Flexural-shear crack – Starts out as a flexural crack and propagates due to shear stress.

Flexural cracks in beams are vertical (perpendicular to the tension face).

For deep beam the cracks are given as:

The shear cracks inclined (diagonal) intercept crack with longitudinal bars plus vertical or inclined reinforcement.

For deep beam the cracks are given as:

The shear cracks fail due two modes:

- Shear-tension failure

– Shear-compression failure

Shear Strength of RC Beams without Web Reinforcement

v_cz – shear in compression zone

v_a – Aggregate Interlock forces

v_d = Dowel action from longitudinal bars

Note: v_cz increases from (V/bd) to (V/by) as crack forms.

Total Resistance = v_cz + v_ay +v_d (when no stirrups are used)

Strength of Concrete in Shear (No Shear Reinforcement)

(1) Tensile Strength of concrete affect inclined cracking load

(2) Longitudinal Reinforcement Ratio, r_w

(3) Shear span to depth ratio, a/d — >(M/(Vd))

(4) Size of Beam Increase Depth – Reduced shear stress at inclined cracking:

(5) Axial Forces – Axial tension -Decreases inclined cracking load – Axial Compression

Increases inclined cracking load (Delays flexural cracking)

Function and Strength of Web Reinforcement

Function:

Web Reinforcement is provided to ensure that the full flexural capacity can be developed. (desired a flexural failure mode – shear failure is brittle)

- Acts as “clamps” to keep shear cracks from widening

• Uncracked Beam: Shear is resisted uncracked concrete.

• Flexural Cracking: Shear is resisted by v_cz, v_ay, v_d

• Flexural Cracking: Shear is resisted by v_cz, v_ay, v_d and v_s

V_s increases as cracks widen until yielding of stirrups then stirrups provide constant resistance.

Designing to Resist Shear

Shear Strength (ACI 318 Sec 11.1)

Shear Strength Provided by Concrete

Lightweight Concrete:
Shear Strength Provided by Shear Reinforcement

Minimum Shear Reinforcement: (11.5.5)

Except

(provides additional 50 psi of shear strength)

Note:

Typical Shear Reinforcement

Stirrup – perpendicular to axis of members – (minimum labor – more material)

Design Procedure for Shear

(1) Calculate V_u

(2) Calculate fV_c Eqn 11-3 or 11-5 (no axial force)

(3) Check

(4)

Also:

(5)

Check:

(6) Solve for required stirrup spacing(strength) Assume # 3, #4, or #5 stirrups

(7) Check minimum steel requirement (eqn 11-13)

(8) Check maximum spacing requirement (ACI 11.5.4)

(9) Use smallest spacing from steps 6,7,8

Note: A practical limit to minimum stirrup spacing is 4 inches.

Location of Maximum Shear for Beam Design

Non-pre-stressed members:

Sections located less than a distance d from face of support may be designed for same shear, V_u, as the computed at a distance d.

When:

1. The support reaction introduces compression into the end regions of the member

2. No concentrated load occurs with in d from face of support .