This page provides engineering theory and calculations for reinforced concrete beams subjected to bending moments. It includes both the traditional elastic analysis method and the modern ultimate load design approach used in structural engineering codes.
The content covers stress distribution, neutral axis calculations, moment of resistance, and reinforcement behaviour in both tension and compression zones.
A reinforced concrete beam is a structural element designed to resist bending loads using a combination of concrete and steel reinforcement. Concrete provides compressive strength, while steel reinforcement carries tensile forces, allowing the beam to safely resist applied loads.
This page includes note covering the theory of reinforced concrete beams. The theory generally used in the past has bee based on elastic method of analysis. In the last twenty years the elastic theory has beean superseded by the ultimate load method in the codes applicable to the construction industry. The notes below consider , in outline, both methods.
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A s = Area of tension reinforcement A' s = Area of Compression reinforcement b = width or effective width of section b w = average width of web d = effective depth of section (compression face to centre of reinforcement ) x= depth to neutral axis of section Ic = Second moment of Area of transformed section C = resultant compressive force in concrete T = resultant tensile force in reinforcement d' = depth to compression reinforcement Ec = Youngs modulus for reinforcement Es = Youngs modulus for concrete nsc = modular ratio = Es /Ec h f = thickness of flange L = effective span of beam M = Design moment at section σc = Stress in concrete σs = Stress in reinforcement Mu = Design Ultimate moment at section Mrc = Moment of Resistance of beam based on concrete strength Mrs = Moment of Resistance of beam based on reinforcement strength |
The basis of this theory is provided on webpage Composite beams
Consider a rectangular beam subject to bending forces such that here are compressive forces on the top surface and tensile force
on the bottom surface. The beam is subject to a bending moment M and is reinforced in its tensile region by a number
or reinforcement bars of total cross section are As. The centre of the reinfocement is at a depth d from the upper surface
and is identified as the effective depth. The neutral axis of the section is at a depth x from the upper surface. The position
of the neutral axis is to be calculated. The concrete is assumed to only contribute to the strengh resisting bending
in the compressive region above the neutral axis i.e all tensile forces are resisted by the steel reinforcement.
The modular ratio nsc = Es /Ec. Multiplying the area of the reinforcement bars by the modular ratio results in the area reinforcement being effectively transformed into an equivalent area of concrete....Taking moments of areas about the neutral axis results in the equation
This when rearranged results in the quadratic
The only relevant solution to this equation is the positive value
The second moment of area Ic of the transformed section is
The maximum stress in the concrete at the extreme fibre is
The strain diagram shown above may be used to calculate the stress in the reinforcement. The strain diagram is linear throughout the depth since the beam section is assumed to remain plane during bending. Therfore
Moment of Resistance of beam
This is the moment withstood by the beam when the stress in the concrete or the steel reaches the maximum allowable value. The value for the concrete and steel is calculated easily from the stress diagram
The moment of resistance of the beam according to elastic theory is the lesser of Mrc or Mrs
The critical /economic section of a beam is when the reinforcement area is adjusted such that Mrs is equal to Mrc
For a beam of given cross section dimensions there is a point when increasing the area of the reinforcement does not provide a resulting increase in the moment of resistance because the concrete becomes the limiting factor. This limitation can be overcome by adding reinforcement in the compression zone of the beam. Consider a beam as shown below
Taking moments of area about the neutral axis
The second moment of area of the transformed section is therefore
The maximum stress in the concrete at the extreme fibre is
Ultimate load theory for concrete beams is based on the calculated moment of resistance (ultimate moment of reistance) being based on
the failure strength of concrete in compression and the yield strength of the reinforcement of the steel in tension both modified by
a margin of safety. However concrete generally fails suddenly while the failure of steel is generally
progressive . The margins are typically 1,5 for concrete and 1,15 for steel. When the characteristic
stength for concrete is based on the cube strength it is necessry that it is multiplied by 2/3 because the stength in bending
is always significantly less than that resulting from the cube strength.
The design strength for concrete and steel in bending is therefore typically
The ultimate load theory analysis is based on the assumptions that plane surfaces remain plane during bending and there is no contribution to the bending strength of the beam from concrete in tension. The strain therefore varies uniformly down the beam section. The stress diagram however is not linear but has a rectangular parabolic shape as shown below.
A reasonably accurate approximation of this curve which is used in strength calculations is shown below
The moment of resistance in this instance is identified as the ultimate moment of resitance Mu and is easily calculated as shown below
The section considered above identifies the ultimate moment of resistance when the neutral axis
is 0,5.d . A lower design moment would have a requirement for a section with a reduced value of x to the
neutral axis. It is easily possible by evaluating moment of resistance of a section resulting from different
values of the ratio of x/d by taking moments about the tensile reinforcement.
Note:The stress distribution, as used in the relevant code and as used for this exercise, is actually slightly different
from that shown above.
A plot of K vs r= x/d is easily plottted
| r | 0.13 | 0.15 | 0.2 | 0.25 | 0.3 | 0.35 | 0.4 | 0.45 | 0.5 |
| K | 0.049569975 | 0.056649375 | 0.07371 | 0.089859375 | 0.1050975 | 0.119424375 | 0.13284 | 0.145344375 | 0.1569375 |
| z | 0.9415 | 0.9325 | 0.91 | 0.8875 | 0.865 | 0.8425 | 0.82 | 0.7975 | 0.775 |
The neutral axis is the location within the beam where stress changes from compression to tension.
Concrete has very low tensile strength and cracks under tension, so steel reinforcement is used to carry tensile forces.
Elastic theory assumes linear stress distribution, while ultimate load theory considers actual failure conditions and nonlinear behaviour.
The moment of resistance is the maximum bending moment a beam can withstand before failure.
It provides a more accurate representation of failure behaviour and includes safety factors for materials and loading conditions.
Useful relevant Links
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