Beams Flashcards
What is fcd, the concrete design stress given by?
fcd = acc * fck / Yc
where acc = 0.85
yc = the partial safety factor for concrete
What is epsilon,cu?
The ultimate strain in concrete = 0.0035
What is the value of the partial factor in concrete?
1.5
What is the value of the partial safety factor in steel?
1.15
Describe the stress-strain graph of steel?
Hits a yield point, then yield plateau, the srain hardening where it increases in stress then decreases, then failure.
What does f,yd equal?
f,yd is the design yield strength of steel = f,yk / Ys
What does the moment equal in relation to flexural stiffness and phi?
M = EI*phi (phi is the curvature)
Describe the moment-curvature realtionship of concrete and steel in a graph:
1 - Uncracked, concrete and steel are both linear elastic
2 - Concrete cracks in tension, the tensile in the concrete is transferred to the steel.
3 - Inelastic, both steel and concrete are ineastic, concrete reaches the ultimate strain
4 - Failure
What is the modular ratio, a,e?
Modular ratio = E of steel / E of concrete
What is the 3 step method to find the second moment of area of an uncracked section?
1 - Find the modular ratio
2 - Find x,u using the formula table
3 - Find I,u using the formula table
What is the moment at first crack, Mc equal to?
Mc = (fcttm*Iu) / (h-xu)
If concrete fails in tension, what happens to the neutral axis?
It moves up
Before first crack, what does Fc equal?
Fc = Force steel + force of concrete in tension
If steel (tension steel) yields before concrete crushes, what type of failure is this?
Under-reinforced failure.
If concrete crushes before steel in tension yields, what type of failure is this?
Over-reinforced failure
In under-reinforced failure, what yields first, what happens to the neutral axis, and describe 3 features of the failure?
Steel yields first, neutral axis moves up. The failure is ductile, gradual and predictable,.
What is shear failure, and how does it appear?
Diagonal fractures or slips along a shear plane.
Often at a 45° angle to the applied load.
Brittle materials show jagged, irregular fractures.
Ductile materials may twist or stretch before failure.
In over-reinforced failure, what yields first, what happens to the neutral axis, and describe 3 features of the failure?
Concrete fails first, neutral axis moves down the cross-section. The failure is sudden, brittle and unpredictable.
What is the desired failure mechanism?
Under-reinforced failure.
What is tension failure, and how does it appear?
Brittle materials have straight, clean fractures perpendicular to the load.
Ductile materials show necking and stretching before fracturing, with a fibrous or jagged fracture surface.
Fibrous materials may tear along thread-like structures.
What is compression failure, and how does it appear?
Brittle materials crush or shatter, with cracks radiating from the compression point.
Ductile materials experience plastic deformation, often bulging or buckling.
Compression can lead to crushing zones or indentations.
Columns may exhibit buckling or bending before failure.
What does a balanced section mean?
Steel yields at the same time as concrete crushes, which can be used in design.
In the balanced section, what does the angle phi equal?
Ecu / x = esyd / (d-x)
What does epsilon,syd equal?
0.00217
For a balanced section, what does x equal?
0.617
What is the step by step process of finding the moment of resistance M,Rd of a beam?
1 - Assume a failure mechanism
2 - Find the neutral axis depth from equilibrium, Fc = Fs
Fc = 0.8xbfcd , Fs = As*fyd
3 - Find length of lever arm, z = d -0.4x
4 - M,rd = Fc * z
What is the lever arm, z?
The distance between Fs and Fc
What does M,rd balanced equal?
0.211bd^2 * fck
If x < or > 0.617d, what does this mean?
If x > 0.616d, then over reinforced, if less than it is under reinforced failure.
For a doubly reinforced section, what does F,sc + F,c equal?
F,st (steel in tension)
What is the 3 step process of calculating M,rd from a doubly reinforced section?
1 - Find N.A using Fst = Fsc + Fc
2 - Check whether tension and compression reinforcement have yielded
3 - Calculate Mrd using the lever arm distance
When analysing a doubly reinforced section, and the compression steel has not yielded, what do you do?
1 - Find the stress of the compression steel (which will have an unknown x)
2 - Repeat Fc + Fsc = Fst, using fsc instead of fyk
3 - This will give a new value of x, which you can use to find Mrd.
What is K? And why is it useful?
K is the normalised moment, and is useful in the design process as it helps us identify whether the beam is over or under reinforced. Can also be used to determine the lever arm
What does K equal?
Mrd / fckbd^2
If k < K’ what does this mean?
The beam is under-reinforced?
What is K’?
0.211
When caculating the area of steel reinforcement required, and K < K’, is it a singly or double reinforced section?
Single reinforcement only
When calculating the area of steel required to resist a design moment, and you find K > K’, is there need for double reinforcment?
Yes
What is the 3 step process involved with designing a singly reinforced area of steel to resist a design moment?
1 - Calculate K using formula
2 - Calculate z using formula
3 - Calculate area of steel required using Med = Mrd and Mrd = Fst*z
A simply supported reinforced concrete beam spans 10m and has an effective depth of 700mm and
breadth 300mm. The beam supports a floor that is 8 m wide along the entire length of the beam.
The loading on the floor is q,k = 2.5kN/m^2 and g,k = 5kN / m^2. y,Q = 1.5 and y,K = 1.35.
How would you go about finding the M,Ed, depsite not being told it in the question?
1 - Find the UDL, q,k * y,Q + g,k * y,K = 10.5
w = 10.5 * 8
2 - M,Ed = (wL^2) / 8
3 - Can then go on to find area of steel required as we now know M,Ed….
When designing an area of steel reinforcement for a doubly reinforced section, what is the 6 step process to find the area of tension and compression reinforcement?
1 - Find K
2 - Calculate the moment of resistance of the balanced section
3 - Calculate As,bal
4 - Find the moment of resistance resisted by the compression reinforcement
5 - Find As2
6 - Total of tension reinfocement = As,bal + As,2
7 - Check if the compression steel has yielded.
Name 4 reasons for providing compression reinforcement:
1 - Fabrication ease (when adding shear links
2 - Changes mode of failure from compression to tension
3 - Increases ductility and curvature ductility
4 - Reduces sustained deflections.
In a simply supported beam, what is the max moment with relation to the length?
Max M = 9L^2 / 8
In a fully fixed beam, what is the max moment with relation to the length?
9L&2 / 24
What does analysis of the flanged section depend on?
The location of the neutral axis
What is b,eff of a flange?
The top effective length
What is hf in relation to a flanged section?
The thickness of the flange
What is the web of a flanged section?
The skinny bottom thingy,
If the concrete stress block does not extend into the web of a flanged section, what do we do for calculations involved with analysis and desing?
Analysis and design proceeds as for a rectangular section with width b,eff. Design Proceeds as for a rectangular section with width beff.
How do we work out whether the concrete stress block is in the flange?
1 - Assume yes it is, then:
2 - Find depth of NA for a rectangular beam with width beff.
3 - Find depth of concrete stress block
4 - Check 0.8x < hf
5 - If 0.8x < hf, then proceed as a rectangular beam.
When tackling a flange question, we calculate that 0.8x > hf, what do we do from here?
We equate forces to find the actual value of x in the web.
Calculating F,cf F,st and F,cw
Then in analysis we take moments to find M,Rd
When tackling an analysis question on a flanged section to find M,Rd, and 0.8x>hf, what is the 5 step process to tackle the question (assume concrete stress block is in the flange to begin with) ?
1 - Assume concrete stress block is in the flange, calculate 0.8x (which is greater than hf)
2 - Find the 0.8x, by splitting in to 3 sections, 2 either side of the flange and 1 in the middle of the web.
3 - Find the moment in web and flange
4 - M,Rd = Moment in web + moment in the flange
When designing a flanged beam, what is the step by step process to determine how much steel reinforcment is required?
1 - Find lever arm z using K based of b,eff
2 - Use z to find 0.8x
3 - If 0.8x < hf then proceed as a rectangular beam with width beff
4 - If 0.8x > hf, proceed as follows
5 - Find Moment resisted by flanges
6 - Find moment resisted by web, by subtracting M,cf from the design moment
7 Design steel to balance forces in the web and flange.
What is felxural reinforcment used for?
To resist bending, to ensure failure develops gradually.
What is shear reinforcement used for?
To resist shear, avoiding brittle failure.
What is the appearance of fexural failure?
Bending, with cracks appearing and crush at the top and bottom.
What is the appearance of shear failure?
Diagonal crack extending from the support to the load.
In an uncracked elastic beam, waht would you observe about the principal compressive trajectories near the bottom and near the top of the beam?
At the bottom the trajectories are steep, and near the top they flatten.
What happens when the tensile principal stresses exceed the concrete tensile stress?
The concrete will crack.
When a beam is being tested, what is the appearance of the shear cracks between the 2 loads?
Vertical
When a beam is being tested, what happens to the shear cracks as you move away from the loads to the supports?
They are more diagonal.
What is av / d?
The shear span ratio, av is the shear span.
What is the shear moment equal to?
M = V *av
If shear Moment < M,Rd, what type of failure will occur?
Shear failure
If the shear moment is greater than M,Rd, what type of failure will occur?
It will ensure flexural failure
What does the shear span ratio have to be to ensure ductile failure?
Very large, slender beam (av / d roughly to equal 6 ish)
When the shear span ratio is very low, what type of failure will occur?
Inclined cracks joining the load and support are developed - compressice failure, failure of the compression strut.
When the shear span ratio is short (between 1 and 2.5) describe the failure mode that will occur:
- Inclined cracks are developed, internal forces are redistributed and beams can carry additional load in part by arch action
1: Shear tension failure, tension reinforcement will fail
2 : Shear compression failure, the final failure is caused by the crushing of the compression zone over the crack
If the shear span ratio is slender (between 2.5 and 6), describe the failure that will occur:
- The diagonal crack starts from the last flexural crack and turns gradually into a crack, a flexural-shear crack
-The diagonal crack encounters resistance as it moves into the zone of compression, becoming flatter.
-As the load increases, the tensiomn crack extends gradually.
Describe the failure of a beam will a high shear span ratio (greater than 6):
Thebeam will fail in flexure prior to the formation of inclined cracks.
What is the shear resistance of a beam without shear reinforcement?
Vc
What is the total force that needs to be transferred in a crack in beams without shear reinforcement?
The shear in the compression zone + shear transferred across the crack by the interlock of the aggregate particles + the dowel action of the longitudanal reinforcement.
What is aggregate interlock?
Aggregate interlock occurs because the faces of the crack are not smooth, but rough. Pieces of the aggregate protrufe from the faves of the crack and provide a bearing surface to transfer shear
What is Dowel action?
Dowel action occurs where a reinforcement bar bridges a crack. It refers to the transfer of shear by double curvature or reverse bending of the short length of the rebar at crack.
What is the total force that needs to be transferred in a crack of a beam with shear reinforcment?
The shear carried by the concrete + the shear transferred by tension in the shear reinforcement.
If the shear reistance is greater than the demand resistance, is shear reinforcement needed?
No
What is the main assumption made with the variable inclination strut method?
All shear is resisted by the provision of shear reinforcement with no contribution from the shear capacity of the concrete itself.
Describe the variable strut inclination method:
-Used for deisng of reinforcment concrete beams in shear,
-Concrete acts as the compression member and as the diagonal compression members are angles at theta to the horizontal.
-Bottom chord is the horizontal tension steel and the vertical links are the shear links.`
What angle is the inclination strut between?
22 degrees and 45 degrees.
If V,Rd max (45 degrees) < V,Ed what needs to happen?
The section of the beam needs to change.
What is V Rd,s , and what does it need to be?
It is the strength of the shear reinforcement, and needs to be greater than V,Ed
What is the 5 step process when determining the amount of shear reinforcement required in a shear design question?
Use formula book for all of this btw
1 - Check if shear reinforcement is required: calculate VRd,c and determine if it is less than VEd
2 - Find the inclination of the strut and check VRd,max > VEd
3 - Determine area of shear links required
4 - Check pw > pw,min
5 - Check sL,max > s
Why do we limit compressive stress in concrete?
To avoid longitudunal cracks, micro-cracks and excessive creep.
Why do we limit tensile stress in steel?
To avoid inelastic strain, unacceptable cracking and unacceptable deformation.
What 3 things can cracking impair?
Function, durability and appearance.
Cracking can arise in concrete due to bending, shear or torsion etc, name 2 other ways cracking can occur:
Tnrljgn plastic shrinkage and expansive chemical reactions with the hardened concrete.
What is w max?
A calculated maximum allowed crack width.
What is our aim with cracks?
To limit them, so that they do not impair the proper functioning or durability of the structure or cause its appearance to be unnaceptable
Our aim is to control cracking after the first crack and have strength great than Mcr, how can this be achieved?
-A minimum area of reinforcement should be placed to withstand the cracking moment
-The diameter of the bars should be dependant on the steel stress, and should be limited in order to maintain and control the crack width.
In a crack, what should the tensile steel force equal?
The tensile concrete force
As,min * stress of steel = kc * k * f,ct,eff*Act
For beams, what is the minimum tension reinforcment required?
As,min > 0.26 fctm/fy * b * d
For beams, what is the minimum shear reinforcementIt required?
Asw / sbw > (0.08 root fck) / fyl
As the reinforcement stress increases, what happens to the minimum allowable crack?
decreases`
What is tension stiffening?
A phenomonom which leads to an increase in the stiffness of a concrete section due to the transmission of stresses from the reinforcing bar to the boundary concrete in the tension between two adjacent cracks.
What is creep?
Creep is permanent deflection of concrete under long term loading.
What is creep modelled as?
An effective reduction in the modulus of elasticity
What is shrinkage?
Deflection due to drying or chemical reactions.
What is the step by step process of answering a displacment question?
1 - Calculate uncracked section properties
2 - Calculate cracked section properties
3 - Calculate deflections
4 - Calculate adjusted deflections.
When calculating adjusted deflections, what is the step by step process, and what do you need to know?
1 - Calculate J, which involves Mc and M
2 - Mc = (fctmIu) / (h-xu) , where fctm is 3.5MPa
3 - M = 9L^2 / 8
4 - Displacement = JSc + (1-J)Su
