Section A A: Slab & Beam Elevation

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1 CE 331, Spring 2011 Flexure Strength of Reinforced Concrete s 1 / 5 A typical reinforced concrete floor system is shown in the sketches below. The floor is supported by the beams, which in turn are supported by the columns. If the width of a concrete (= column spacing below) is greater than twice its span (= beam spacing below), then the is said to be a one way (bends in principally in one direction only). The beams run left to right in the Plan View below. Section A A below indicates that the beams are T shaped: the extends down below the soffit (underside) of the. Exterior Span Interior Span Exterior Span span Span Plan View of Floor System A A t bw h Section A A: Slab & Elevation (clear span) column Elevation View Figure 1. Plan View, Elevation View and Cross section of typical cast in place RC floor system. Isometric views of the reinforced concrete floor system are shown on the following page.

2 CE 331, Spring 2011 Flexure Strength of Reinforced Concrete s 2 / 5 Figure 2a. Isometric View of Reinforced Concrete Floor System Figure 2b. Cut away showing beam dimensions

3 CE 331, Spring 2011 Flexure Strength of Reinforced Concrete s 3 / 5 Placement of Reinforcement Concrete is strong in compression but weak in tension and cracks under relatively small tensile stresses. The crack patterns in a three span continuous beam are shown below. Steel reinforcement is placed in the tension zones of reinforced concrete beams, as indicated in the next figure. +'ve M steel -'ve M steel Elevation View The steel reinforcement is covered with a minimum thickness of concrete to protect it from moisture which can lead to corrosion. Reinforced concrete design is governed by the American Concrete Institute (ACI). ACI clear cover requirements for cast in place concrete are shown below. ACI Clear Cover Requirements: clear cover φ stirrup φ bar / 2 stirrup Cross Section Concrete cast against and permanently exposed to earth Concrete exposed to earth or weather #6 bar and larger #5 bar and smaller Concrete not exposed to earth or weather Slabs, walls, joists s and columns Clear Cover, in

4 CE 331, Spring 2011 Flexure Strength of Reinforced Concrete s 4 / 5 Factored Moments due to Dead + Live Loads (M u ) Because concrete frames are highly indeterminate, the moments due to factored dead and live loads are typically calculated with the aid of a computer program. Alternatively, designers often use the American Concrete Institute (ACI) moment coefficients (shown below) which represent the envelope of moments due to dead load plus various live load span load patterns. (See Pg. 144 of the FE reference for the moment coefficients.) (clear column M u ACI Moment Coefficients Stress and Strain in a Reinforced Concrete flexure strength is usually calculated considering the stress and strain distributions across the section. The stress strain relations for steel and concrete are shown below. A reinforced concrete beam must be analyzed differently than a wood or steel beam. Differences include the presence of two different materials, non linear stress strain behavior, and tensile cracking of concrete. Simplifying assumptions exist for the analysis of a reinforced concrete beam an imminent flexure failure, as shown in the table below..

5 CE 331, Spring 2011 Flexure Strength of Reinforced Concrete s 5 / 5 Complication Concrete ruptures under relatively small tensile stresses. The steel stress strain curve is bi linear The concrete stress strain curve is nonlinear Simplifying Assumption The strength of the concrete in tension is neglected. Steel reinforcement carries all of the tensile force. The steel has yielded at failure An equivalent rectangular stress distribution (stress block) is used to approximate the curvilinear stress distribution at failure. C c T s Neutral Axis.003 c f'c 0.85f'c a=β 1 c a/2 C c d ε s f s f s T s Cross- Section Strain Actual Stress Equivalent (Whitney Stress Block) Stress Stress Resultants f'c, psi β 1 <= 4, , , , >=8, φ ε s min for beams = ε s