1. Chapter 4 - Flexure

2. Lecture Goals Structures Basic Concepts Rectangular Beams

3. Rectangular Beams:- Simply Supported

4. Continuous Beams

5. Flexural Stress The beam is a structural member used to support the internal moments and shears. It would be called a beam-column if a compressive force existed. C = T M = C*(jd) = T*(jd)

6. Flexural Stress The stress in the block is defined as:  M*y) / I S xx = I / (y max ) The equation for S xx modulus for calculating maximum compressive stress. Sign Convention: +M, -M

7. Basic Safety Equation for Flexure: Art. 9.1.1 (SBC 304)  design strength  required Strength design strength=  nominal strength, e.g.  Mn,  Vn, …. required strength=load effect resulting from factored loads, Mu, Vu, …. For Beam design, the design criteria become  Mn  Mu and  Vn  Vu Mu= factored load moment, (Moment computed from combination of factored loads) Vu= factored load shear, (Shear computed from combination of factored loads)  : Strength reduction factor to account for variations in dimensions, and material strengths.

8. Flexural Stress The three main stages of the beam. Stage 1: No external loads acting on the beam. Stage 2: Service loading on the beam. Stage 3: Beam failure.

9. Flexural Stress The moment-curvature diagram show the five stages of the beam. The plot is of the curvature angle, , verse the moment.  = (  / y) = [  / E ] / y = [(My / I) / E] / y  = M / ( E I )

10. Flexural Stress The beam fails first in shear and the second beam fails in bending moment.

11. Flexural Stress There are three types of flexural failure of a structural member. Steel may reach its yield strength before the concrete reaches its maximum. (Under-reinforced section). Steel reaches yield at same time as concrete reaches ultimate strength. (Balanced section). Concrete may fail before the the yield of steel due to the presence of a high percentage of steel in the section. (Over-reinforced section).

12. Flexural Stress Steel may reach its yield strength before the concrete reaches its maximum. (Under-reinforced section).

13. Flexural Stress Steel reaches yield at same time as concrete reaches ultimate strength. (Balanced section).

14. Flexural Stress Concrete may fail before the the yield of steel due to the presence of a high percentage of steel in the section. (Over-reinforced section).