Flexural Analysis and Design of Beams

Chapter Two

Flexural Analysis and Design of Beams

Flexural Design of Beams (and One-Way Slabs):

     The basic assumptions made in flexural design are:

1.     Sections perpendicular to the axis of bending that are plane before bending remains plane after bending.

2.     A perfect bond exists between the reinforcement and the concrete such that the strain in the reinforcement is equal to the strain in the concrete at the same level.

3.     The strains in both the concrete and reinforcement are assumed to be directly proportional to the distance from the neutral axis (ACI 10.2.2).

4.     Concrete is assumed to fail when the compressive strain reaches 0.003 (ACI

10.2.3).

5.     The tensile strength of concrete is neglected (ACI 10.2.5).

6.     The stresses in the concrete and reinforcement can be computed from the strains using stress-strain curves for concrete and steel, respectively.

7.     The compressive stress-strain relationship for concrete may be assumed to be rectangular, trapezoidal, parabolic, or any other shape that results in prediction of strength in substantial agreement with the results of comprehensive tests (ACI 10.2.6). ACI 10.2.7 outlines the use of a rectangular compressive stress distribution which is known as the Whitney rectangular stress block.

Structural Design Requirements:

The design of a structure must satisfy three basic requirements:

 1) Strength to resist safely the stresses induced by the loads in the various structural members.

2) Serviceability to ensure satisfactory performance under service load conditions, which implies providing adequate stiffness to contain deflections, crack widths and vibrations within acceptable limits.

 3) Stability to prevent overturning, sliding or buckling of the structure, or part of it under the action of loads.

There are two other considerations that a sensible designer should keep in mind: Economy and aesthetics.

Design Methods (Philosophies)

Two methods of design have long prevalent.

Working Stress Method: focuses on conditions at service loads.

  Strength Design Method: focusing on conditions at loads greater than the service loads when failure may be imminent. The Strength Design Method is deemed conceptually more realistic to establish structural safety.

The Working-Stress Design Method

     This method is based on the condition that the stresses caused by service loads without load factors are not to exceed the allowable stresses which are taken as a fraction of the ultimate stresses of the materials, fc’ for concrete and fy for steel.

The Ultimate – Strength Design Method

     At the present time, the ultimate-strength design method is the method adopted by most prestigious design codes. In this method, elements are designed so that the internal forces produced by factored loads do not exceed the corresponding reduced strength capacities.

Reduced strength  provided factored loads

Or Design strength ≥ Factored loads

The factored loads are obtained by multiplying the working loads (service loads) by factors usually greater than unity.

Safety Provisions (the strength requirement)

     Safety is required to insure that the structure can sustain all expected loads during its construction stage and its life span with an appropriate factor of safety. There are three main reasons why some sort of safety factor is necessary in structural design:

 • Variability in resistance. *Variability of fc’ and fy, *assumptions are made during design and *differences between the as-built dimensions and those found in structural drawings.

• Variability in loading. Real loads may differ from assumed design loads, or distributed differently.

• Consequences of failure. *Potential loss of life, *cost of clearing the debris and replacement of the structure and its contents and *cost to society.

Safety Provisions (the strength requirement)

     The strength design method, involves a two-way safety measure.

·        The first of which involves using load factors, usually greater than unity to increase the service loads.

·        The second safety measure specified by the ACI Code involves a strength reduction factor multiplied by the nominal strength to obtain design strength. The magnitude of such a reduction factor is usually smaller than unity

Reinforced Concrete Beam Behavior:

      A small transverse load is placed on a concrete beam with tensile reinforcing and that the load is gradually increased in magnitude until the beam fails. As this takes place the beam will go through three distinct stages before collapse occurs.

     These are: (1) the un-cracked concrete stage, (2) the concrete cracked-elastic stresses stage, and (3) the ultimate-strength stage. A relatively long beam is considered for this discussion so that shear will not have a large effect on its behavior.

1-    Un-cracked Concrete Stage.

     At small loads when the tensile stresses are less than the modulus of rupture (the bending tensile stress at which the concrete begins to crack), the entire cross section of the beam resists bending, with compression on one side and tension on the other. Fig. 1 shows the variation of stresses and strains for these small loads.

2-    Concrete Cracked-Elastic Stresses Stage

    As the load is increased after the modulus of rupture of the concrete is exceeded, cracks begin to develop in the bottom of the beam. The moment at which these cracks begin to form (when the tensile stress in the bottom of the beam equals the modulus of rupture) the cracking moment, Mcr.

     As the load is further increased, these cracks quickly spread up to the vicinity of the neutral axis, and then the neutral axis begins to move upward. The cracks occur at those places along the beam where the actual moment is greater than the cracking moment, as shown in Fig. 2(a). Now that the bottom has cracked, another stage is present because the concrete in the cracked zone obviously cannot resist tensile stresses-the steel must do it. This stage will continue as long as the compression stress in the top fibers is less than about and as long as the steel stress is less than its yield stress.

     The stresses and strains for this range are shown in Fig. 2(b). In this stage the compressive stresses vary linearly with the distance from the neutral axis or as a straight line. The straight-line stress-strain variation normally occurs in reinforced concrete beams under normal service-load conditions because at those loads the stresses are generally less than . To compute the concrete and steel stresses in this range, the transformed-area method is used.

     The service or working loads are the loads that are assumed to actually occur when a structure is in use or service. Under these loads, moments develop which are considerably larger than the cracking moments. Obviously the tensile side of the beam will be cracked.

 

Fig. 1: un-cracked concrete section.

Fig. 2: Concrete cracked-elastic stresses stage.

3-    Beam Failure Ultimate-Strength Stage     

 As the load is increased further so that the compressive stresses are greater than  the tensile cracks move further upward, as does the neutral axis, and the concrete compression stresses begin to change appreciably from a straight line. For this initial discussion it is assumed that the reinforcing bars have yielded. The stress variation is much like that shown in Fig. 3.

 

  Fig. 3: Ultimate-strength stage.

     To illustrate the three stages of beam behavior which have been described, a moment-curvature diagram is shown in Fig. 4. For this diagram,  is the angle change of the beam section over a certain length and is computed by the following expression in which  is the strain in a beam fiber at some distance y from the neutral axis of the beam:     

          When the moment is increased beyond the cracking moment, the slope of the curve will decrease a little because the beam is not quite as stiff as it was in the initial stage before the concrete cracked. The diagram will follow almost a straight line from Mcr to the point where the reinforcing is stressed to its yield point. Until the steel yields, a fairly large additional load is required to appreciably increase the beam’s deflection. After the steel yields, the beam has very little additional moment capacity, and only a small additional load is required to substantially increase rotations as well as deflections. The slope of the diagram is now very flat.

Fig. 4: Moment-curvature diagram for reinforced   

concrete beam with tensile reinforcing only.

Cracking Moment

     The area of reinforcing as a percentage of the total cross-sectional area of a beam is quite small (usually 2% or less), and its effect on the beam properties is almost negligible as long as the beam is un-cracked. Therefore an approximate calculation of the bending stresses in such a beam can be obtained based on the gross properties of the beam’s cross section. The stress in the concrete at any point a distance Y from the neutral axis of the cross section can be determined from the following:   

      Section 9.5.2.3 of the ACI Code states that the cracking moment of a section may be determined with ACI Equation 9-9, ( for normal weight concrete with f'c in MPa).The cracking moment is as follows:

       (ACI Equation 9-9)

Ex. 1:

a)  Assuming the concrete is un-cracked, compute the bending stresses in the extreme   fibers of the beam below for a bending moment of 25kN.m. The concrete has an f'c of 25MPa and a modulus of rupture  

(b) Determine the cracking moment of the section.

Sol.:

Bending stresses:

Cracking moment:   

Elastic Stresses -Concrete Cracked

      When the bending moment is sufficiently large to cause the tensile stress in the extreme fibers to be greater than the modulus of rupture, it is assumed that all of the concrete on the tensile side of the beam is cracked and must be neglected in the flexure calculations. The cracking moment of a beam is normally quite small compared to the service load moment. The bottom of the beam cracks while load applied. The cracking of the

beam does not necessarily mean that the beam is going to fail. The reinforcing bars on the tensile side begin to pick up the tension caused by the applied moment.

      On the tensile side of the beam an assumption of perfect bond is made between the reinforcing bars and the concrete. Thus the strain in the concrete and in the steel will be equal at equal distances from the neutral axis. But if the strains in the two materials at a particular point are the same, their stresses cannot be the same since they have different modulus of elasticity. Thus their stresses are in proportion to the ratio of their modulus of elasticity. The ratio of the steel modulus to the concrete modulus is called the modular ratio n:

     If the modular ratio for a particular beam is 10, the stress in the steel will be 10 times the stress in the concrete at the same distance from the neutral axis (as in Fig. 5).

Fig. 5: distribution of stresses.

     The steel bars are replaced with an equivalent area of fictitious concrete (nAs referred to as the transformed area) which supposedly can resist tension. On the tensile side a dashed line is shown because the diagram is discontinuous, the concrete is assumed to be cracked and unable to resist tension. The value shown opposite the steel is  fs/n because it must be multiplied by n to give the steel stress is.

       The steps that to be taken necessary for determining the stresses and resisting moments for reinforced concrete beams are:

1-    Locate the neutral axis, which is assumed to be located a distance x from the compression surface of the beam.

2-    The first moment of the compression area of the beam cross section about the neutral axis must equal the first moment of the tensile area about the neutral axis. The resulting quadratic equation can be solved by completing the squares or by using the quadratic formula.

3-    After the neutral axis is located, the moment of inertia of the transformed section is calculated, and the stresses in the concrete and the steel are computed with the flexure formula.

Ex. 2: Calculate the moment of inertia and all the stresses for the beam shown below having n= 9 and fr= 3.1Mpa and different stages of loading.

a)     small moment M= 35kN.m ,

b)     medium moment (M= cracked moment) ,

c)      large moment M= 95kN.m,

d)     if the allowable stresses are fc = 9.3 MPa and fs =140 MPa find the moment capacity of cracked section. )

a)     For small moment all the section (concrete and steel ) will support the loads the steel area will be equal to (n-1) As then:

Area moment above neutral axis = area moment under neutral axis

To determine cracked moment equal

For M=95 kN.m the section will be cracked and area of concrete under tension under neutral axis will be neglected as shown in figure below:

 

 

d)      

Ex. 3: Calculate the bending stresses in the beam shown in Figure below by using the transformed area method: n = 9 and M = 90kN.m.

Sol.: Taking Moments about Neutral Axis   

Ex.4: Determine the moment of inertia for the section shown below (n=9):

 

    Then y=155.7 mm

 

Tee and L- sections:

The analysis of these sections depend on position of neutral axis

For analysis let y = hf then find moment of area

    Compression area.

 Tension area.

Then     

And

Ex. 5: For the beam shown determine the moment of inertia if :

 mm3

       

 

            

   350

Ex. 6: For the beam shown determine the moment of inertia if n=9:

    by trial  and error    (

Doubly reinforced sections (With compression reinforcement):

For some reason we need to use compression reinforcement as well as to tension reinforcement which will increase the section capacity, reduce deflection at late ages,  and joins the shear or web reinforcement.

To calculate the bending stresses then

 and 

Ex.7: For the beam shown determine the bending stresses if M=200kN.m and  n = 10:

y= 210mm

To calculate the bending stresses then