Design of Concrete Structure I Dr. Ali Tayeh First Semester 2009

DESIGN OF BEAMS FOR MOMENTS Lecture 4 DESIGN OF BEAMS FOR MOMENTS

Design of Beam For Moment The main two objectives of design is to attain the required strength and ductility. Design moment strength (also known as moment resistance) Internal ultimate moment

Flexural Stress The beam is a structural member used to support the internal moments and shears C = T M = C*(jd) = T*(jd)

Stress – strain distribution across beam depth

Flexural Stress The compressive zone is modeled with a equivalent stress block.

Flexural Stress The equivalent rectangular concrete stress distribution has what is known as a b1 coefficient is proportion of average stress distribution covers.

Flexural Stress Requirements for analysis of reinforced concrete beams [1] Stress-Strain Compatibility – Stress at a point in member must correspond to strain at a point. [2] Equilibrium – Internal forces balances with external forces

Flexural Stress Example of rectangular reinforced concrete beam. (1) Setup equilibrium.

Flexural Stress (2) Find flexural capacity.

Flexural Stress (3) Need to confirm et > ey

Strain Limit Method for Analysis and Design If the net tensile strain in the extreme tension fibers, et is small, being equal or less than a compression-controlled strain limit, a brittle mode of failure is expected For the tension-controlled state, the strain limit et = 0.005 corresponds to reinforcement Ratio = 0.63, where is the balanced reinforcement ratio for the balanced strain et =0.002 in the extreme tensile reinforcement.

Mode of failure

Mode of failure Compression Failure The concrete will crush before the steel yields. This is a sudden failure. The beam is known as an over-reinforced beam.

Tension Failure The reinforcement yields before the concrete crushes. The concrete crushes is a secondary compression failure. The beam is known as an under-reinforced beam.

Balanced Failure The concrete crushes and the steel yields simultaneously. The beam is known as an balanced-reinforced beam.

Critical thinking In order to prevent such a state of behavior in flexural members, a strain greater than in the extreme tensile reinforcement has to be required in design. For example, if 60 grade steel (410 Mpa) is used as reinforcement, the yield strain The design has to be based on (termed at the level of the extreme tensile reinforcement layer) sufficiently larger than 0.002 in flexural members to ensure ductile performance.

Notes =balanced neutral axis depth at limit strain =effective depth to the extreme tensile reinforcement layer

Minimum steel requirement

Example 1 A singly reinforced concrete beam has the cross-section shown in Figure below. Determine if the beam is tension- or compression-controlled and if it is satisfied ACI code .

1 OK 2 Brittle behavior as the section is compression-controlled and does not satisfy ACI code

1 OK 2 Transition zone satisfy ACI code requirement is satisfied

Example 2 A singly reinforced concrete beam has the cross-section shown in Figure below. Calculate the nominal moment strength if.

OK Not Satisfy ACI code

OK Satisfy ACI code