Analysis of Flexural Strength of Beams

A very brief introduction to the calculation of the ultimate or nominal flexural strength of beams is presented in this module. An analysis of singly and doubly reinforced beams is introduced in this section

LOAD FACTORS

Section 9.2 of the ACI code presents the load factors and combinations that are to be used for reinforced concrete design. The required strength, U, or the load-carrying ability of a particular reinforced concrete member, must at least equal the largest value obtained by substituting into ACI Equations 9-1 to 9-7. The following equations conform to the requirements of the International Building Code (IBC)1 as well as to the values required by ASCE/SEI 7-10.2
For SI, refer to NSCP 2015 code specifications

STRENGTH REDUCTION FACTORS

Strength reduction factors are used to take into account the uncertainties of material strengths, inaccuracies in the design equations, approximations in analysis, possible variations in dimensions of the concrete sections and placement of reinforcement, the importance of members in the structures of which they are part, and so on. The code (9.3) prescribes φ values or strength reduction factors for most situations. Among these values are shown in the right figure.
Also, shown is the variation of φ in net tensile strain (epsilon sub t) and c/dt for Grade 60 reinforcement and for prestressing steel

notations and symbols used:

Symbols and notations used in analysis and design of beams under flexural load is listed as shown in the left figure.

ASSUMPTIONS IN STRENGTH DESIGN IN FLEXURE

Shown in the figure are the assumptions used in analysis of beams under flexural loads
Take note that for NSCP 2010, the factor β sub 1 shall be taken as 0.85 for f'c ≤ 28 MPa, and shall be reduced continuously at a rate of 0.05 for each 7 MPa of strength in excess of 28 MPa, but shall be taken less than 0.65

BALANCED VS OVERREINFORCED VS UNDERREINFORCED

Balanced design refers to design so proportioned that the maximum stresses in concrete (with strain of 0.003) and steel (with strain of fy/Es) are reached simultaneously once the ultimate load is reached, causing them to fail simultaneously.
Underreinforced design is a design in which the steel reinforcement is lesser than what is required for balanced condition. If the ultimate load is approached, the steel will begin to yield although the compression concrete is still understressed. Failure under this condition is ductile and will give warning to the user of the structure to decrease the load
Overreinforced design is a design in which the steel reinforcement is more than what is required for its balanced condition. If the beam is overreinforced, the steel will not yield before failure. As the load is increased, deflections are not noticeable although the compression concrete is highly stressed, and failure occurs suddenly without warning to the user of the structure.

equations used for analysis of rectangular beam reinforced under tension only

Stress and Strain Diagram for Singly Reinforced Beam

BALANCED C AND BALANCED STEEL RATIO

MAXIMUM STEEL REINFORCEMENT

MINIMUM REINFORCEMENT OF FLEXURAL MEMBERS

MINIMUM THICKNESS FOR NON-PRESTRESSED BEAMS

CONCRETE PROTECTION FOR REINFORCEMENT

Steps in Solving the Design of Singly Reinforced Beam under Flexure, Design Strength and Required Steel Area

BEAMS REINFORCED FOR TENSION AND COMPRESSION

Thee steel that is occasionally used on the compression sides of beams is called compression steel, and beams with both tensile and compressive steel are referred to as doubly-reinforced beams.
Compression steel helps the beam withstand stress reversals that might occur during earthquakes. Continuous compression bars are also helpful for positioning stirrups and keeping them in place during concrete placement and vibration.
Doubly reinforced beam is analyzed by dividing the beam into two couples Mn1 and Mn2. Mn1 is couple due to compression concrete and the part of the tension steel As1. Mn2 is the couple due to the compression steel A's and other part of the tension steel area As2.

If the compression steel yields, then A's=As2, otherwise, A's=As2fy/f's, where f's is the stress of compression steel and given by the equation on the left
According to Section 410.4.3.of NSCP, for members with compression reinforcement, the portion (rho sub b) is equalized by compression reinforcement and not be reduced by the 0.75 factor

STEPS IN SOLVING AS, A'S AND MU OF A DOUBLY REINFORCED RECTANGULAR BEAM

SUPPLEMENTARY MATERIALS:

20220905 - CENG132 Module 2a.pdf

20220912 - CENG132 Module 2b.pdf

20220913 - CENG132 Module 2b.2.pdf

RECORDED MEETINGS

wby-fdzq-kzq (2022-09-05 19:01 GMT+8)

seq-jqzb-ztz (2022-09-06 18:08 GMT+8)

COURSEWORK#2

Coursework2.pdf

reflection:

The second module tackled the strength of singly and doubly reinforced concrete beams in accordance to ACI/NSCP code. I have learned that before, reinforced concrete design made use of allowable-stress design (WSD) not until the ultimate strength design was introduced. This is, according to our instructor, the ultimate strength design (also known as LRFD method) uses a more realistic consideration of safety and provides more economical designs. Also, I have grasped that strains in concrete vary in proportion to distances from neutral axis. It would be helpful to use an assumed shape when analyzing stress distribution in a beam, in order to yield a reasonable derivation of beam formulas. Additionally, I learned that space or aesthetic requirements limit beams to such small sizes that compression steel is needed in addition to tensile steel. This is to increase the moment capacity by adding another resisting couple in the beam. I found out that this is very effective in reducing long term deflections due to shrinkage and plastic flow. Although the derivation of formulas is more complex than singly reinforced, I was able to understand and analyzed on how to find the required steel area (for both tension and compression steel) and on how to design of doubly reinforced rectangular beam as well.

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