Chapter 2 - Flexural Analysis of RC Beams Flashcards
What is the definition of curvature and give the 3 relationships to it. (State what each term means)
Curvature (ϕ) is defined as the second derivative of deflection.
ϕ = (d^2 v)/(dx^2 ) = ε/y = 1/r
Where:
v = deflection.
ε = strain.
y = distance from the point where ε is measured to the neutral axis.
r = radius of curvature.
Where does concrete cracking occur during flexure and how does this impact the strength of a section?
Concrete cracking occurs below the neutral axis for when the bottom of section is in tension and above the neutral axis when the top of the section is in tension.
Cracked concrete doesn’t offer any structural strength so cracking reduces the second moment of area of the section.
What impact does a reduced second moment of area have on the flexural strength of a section?
A reduced second moment of area reduces the flexural stiffness of a section and makes it more suseptible to larger deflections.
Describe the shape of the moment-cuvature diagram and describe what is happening at each stage.
Moment on the y-axis and curvature on the x-axis.
Stage 1 - Uncracked Section
Steel and concrete are linear elastic and the concrete in the tension zone is uncracked. The gradient of this line is EIu (flexural stiffness of the uncracked section) and continues until Mc (moment of first crack).
Stage 2 - Cracking Section
Concrete cracks in the tension zone and materials are linear elastic. Discontinuity in the graph due to all stress that were in the concrete transferring to the tensile steel. Stage 2 is a horizontal line (curvature increases without an increase in loading).
Stage 3 - Cracked Section
Material are linear and the neutral axis moves up the section to equilibriate the forces. The gradient of this line is EIc (flexural stiffness of the cracked section) and continues until My (moment of tensile steel yielding).
Section 4 - Inelastic Section
Both materials behave inelastically and the concrete in the compression zone reaches εcu2 (ultimate compressive strain of concrete) and MRd (maximum design resistance of the section) is reached. The shape of this stage is a rainbow.
Section 5 - Section Fails
What is the concept of modular ratio?
Modular ratio is a tool that allows the second moment of area of a material to be transformed into that of another material.
What is the use of modular ratio is flexural analysis?
It allows a composite section (steel and concrete) to be analysed as a single, equivalent material.
What is the equation for the cracking moment of an RC section? (State what each term means)
Mc = (fctm*Iu)/(h-xu)
Where:
fctm = maximum concrete tensile stress.
Iu = uncracked second moment of area of the section.
h = height of the section.
xu = neutral axis of the uncracked section.
What are the features of an under-reinforced section?
The steel in the tension zone will yield before the concrete in the compression zone crushes.
The neutral axis will shift up the section (for positive bending moments) as the load increases and cause a longer lever arm at failure.
Failure is gradual and ductile so is a safe mode of failure.
From a side view of the section, flexural cracks are long and few horizontal cracks above the high neutral axis.
What are the features of an over-reinforced section?
The concrete in the compression zone crushes before the steel in the tension zone yields.
The neutral axis will shift down the section (for positive bending moments) as the load increases and cause a shorter lever arm at failure.
Failure is brittle and sudden so is an undesirable mode of failure.
From a side view of the section, flexural cracks are short and a lot horizontal cracks above the low neutral axis.
What is a balanced section?
A section where the concrete in the compression zone will crush at the same time that the steel in the tension zone yields.
What is the analysis procedure for a single reinforced beam to calculate MRd of the section?
- Assume a balanced section failiure mechanism.
- Find the neutral axis depth from the edge fibers of the compression zone by using the equilibrium Fc (force in the compression zone) = Fst (force in the tension steel).
- Calculate the lever arm (z) knowing that z = d - 0.4x.
- Calculate the bending moment resistance of the section of the section (MRd) using MRd = Fcz = Fstz.
What is the analysis procedure for a double reinforced beam to calculate MRd of the section?
- Assume a balanced section failiure mechanism and that the compression and tension steel has yielded.
- Find the neutral axis depth from the edge fibers of the compression zone by using the equilibrium Fc (force in the compression zone) + Fsc (force in the compression steel) = Fst (force in the tension steel).
- Calculate the lever arm between the tension steel and compression zone (z1) knowing that z1 = d - 0.4x.
- Calculatet the lever arm between the tension and compression steel (z2) where z2 = d - d2.
- Calculate the bending moment resistance of the section of the section (MRd) using MRd = Fcz1 + Fscz2.
- Confirm the tension and compression steel have yielded.
What should you do if your assumption that the compression steel and/or the tension steel has yielded is wrong?
Express the strain in the steel as a function of the depth of the neutral axis into the compression zone and substitute this into the force equilibrium.
What are the values of εcu2 (maximum compressive strain in concrete) and εsyd (yield strain of steel)?
εcu2 = 0.0035
εsyd = 0.00217
