Lecture :2.3 - Analysis of PSC member Problem no 1

now let us deal with some uh numerical uh based on analysis of a pre-stressed cross section so first numerical print the data carefully

and listen carefully this is the data that is one rectangular quantitative beam of a cross section 30 cm by 20 cm that is 300 by 200

mm wide is processed by means of 15 wires of 5 mm diameter located at 6.5 centimeter from the bottom of the beam and three wires of the

diameter 5 mm at 2.5 centimeter from the top okay so just see here these are the 15 wires 15 wires diameter get 5 mm which are

located at a 65 mm from the bottom and three wires of the same diameter are located at a distance of 25 mm from the

top so this is these are the prestressing wires okay and these wires are assumed to be pressurized at 840 newton per mm square at 40 mph in

stress then calculate what first see calculate the stresses and extreme fibers of a mid span section mid span section piggy nicole

when the beam is supporting its own weight over a span of six meter okay and second if the uniformly distributed live load of a six kilo newton

per meter is imposed on a beam evaluate the maximum working stress in concrete the density of a concrete here is assumed to be a 24 kilo newton per meter cube so friends so this is the problem which

is given to us uh hope so given data the span of the beam is six meter and here we need to analyze it for two uh conditions that when it is

supporting its sulphide only and when it is subjected to the live load condition it is

uh taken to explain very well the conditions of at transfer and at service load

okay solution so let us see the solution of this now first of all we need to find the distance of the prestressing force from a neutral axis

we need to find the resultant of this crystallising force cookie is processed by 15 wire at bottom and three wire at top

so somewhere uh in the cross section there must be position of a resultant prestressing force p so we need to find it simple by the formula that

by which we were finding the y bar that is a 1 by 1 plus a 2 y 2 d dash a n y n by summation of a in the same way we will find it that is why i need to find a distance

from the bottom so how many wires so 15 wires at a distance of 65 mm so 15 into 65

plus three wires at a distance of 275 mm from the bottom because 300k or purchase mmp distance for a topsail for this distance will be a 275

from bottom so 15 wires into its distance from bottom plus three wires into its distance from bottom again

divided by total number of wire will give us location of the resultant of pressurizing force and i found it 100 mm from the bottom as you can see in figure

y that which is 100 mm from the bottom okay now the given cross section is rectangular y bar will be

at the middle of it depth so the neutral axis is lying at the distance of 150 mm and y is found to be 100 mm so can you tell me the eccentricity

eccentricity of the cable simple eccentricity will be 150 minus this hundred that is the 50 mm so at 50 mm our resultant of

pre-stressing force is lying so simply we can convert this problem into a simple problem that uh one force p at a distance of 50 mm from neutral axis okay now the prestressing force how much

will the processing force will be involved here so the stress multiplied by total area of steel this is what done over here that is stressing each wire into the number of

wire into the area of each wire stressing each wire is 840 into area of each wire pi by 4 into 5 square multiplied by how many bars are there 18

wires so i got this much of newton force if motor motors are 3 that is uh sorry three into ten raised to five newton so

it is of course how much it will be it will be i guess three hundred kilo newton okay so this much is the processing force uh which is a precision force of this 3

into 10 raised to 5 newton acting at eccentricity of 50 mm below the neutral axis now this is the case area of cross section b by d we will get

area of cross section moment of inertia since the cross section is a rectangular by bd cube by trial you will get moment of inertia

and section modulus again section is rectangular so distance of its neutral axis from top and bottom will be same so

yt will be equal to yb uh so no need to calculate zt and zp differently but wherever this is the condition where the

yt and yv are not same in a symmetrical uh cross sections you need to calculate zt and zb separately but here zt will be

equal to zb or jokey have bd square by 6 or i by y you can calculate anyhow so here zb and zte 3 into 10 raised to 6

and mq sulphate of beam self rate of beam is nothing but the cross section of being multiplied by density density given here that is a 24

cross section keep my 0.3 into 0.2 so sulfate will be 1.44 kilo newton per meter and moment due to this will be wl square by 8 so six is the span that's why

moment will be of a six point forty eight kilo newton meter live load is given to us that is it is a six kilo newton per meter and moment due to the life load is it

can be written as 6 into 6 square by 8 that is 27 kilo newton meter so these are the moment due to dead load and life load

now the stresses stress at transfer i would like to give a stress at this world that is

transfer stress at transfer what is this condition whenever certain beam is pre-stressed with a prestressing

tendon and if it is placed on its position wherever it may be and if it is not subjected to live load it is

only subjected to or it is only supporting its own weight only then this stage is called as transfer because why because we are only transferring a pre-stress through the tendon to that

beam so this is the stage at transfer and when the same beam is subjected to life load then that that would be a

stress at service okay these are the two conditions now why we need to consider this to just apart from the solution of this problem we need to discuss it

as it is very important at a time of transferring the stress to the concrete or to the beam supported here without any load uh we have discussed that is the moment

induced due to this eccentric processing will be of hogging in nature so job beam will be hog message of hogging hoga so tensile stress would be induced on top fiber and

compressive stress will be induced on bottom fiber we know that concrete is a big intention in resisting tension though we need to take care that the tensile

stressed induced wherever foreign which we need to avoid okay and when this being is subjected to a live load

certain library so it will sag okay now the bottom most fibers are subjective to tensile stresses

and again these stresses should be within permissible limit okay again it should be within permissible limit to avoid a crack of a section a visible crack in a

section okay so that's why that's it what are the permissible limits we will be discussing it in design section okay now we are performing analysis

okay back to the solution so stress at transfer now it is clear what is stress stress at transfer so this is the stresses transfer when

the beam is supporting circuit only so this complex cross section is converted into the simple one that is this is this is the resultant processing force which is

lying at a 50 mm from the neutral axis okay centricity is 50 mm now at this cross section this will be the p pre-stressing force geohemic i'm going to calculate ei then

so i'm considering only mgmg moment due to the date load which is acting like sagging moment is yes sagging moment okay but since the prestressing is

acting at eccentricity e or this this scenario can be represented like this force or a guy plus a century city

which is a moment which will be equal to p into e and moment due to the deadlock to high so our problem becomes

like this or our problem turns into this that this is the cross section subjected to the force speech is a direct stress

when the p is placed at an eccentricity into e moment moment due to dead load okay why because tendons are kept below the

neutral axis over here so this will be the scenario so for this scenario what will be the stresses so direct stress due to pre-stressing will be p by a so just putting the value of

p and cross sectional area a i found direct stress of 5 mpa now bending stress due to the stress will be p e into z t or v into z here z t and z be uh

having a same value so uh again i found it by putting the values again i found it of 5 mba this is the coincidence that the direct stress and the prestressing stress

bears the same value but this will not happen in every problem okay sulfate stress that is the stress due to the moment due to self weight okay

so mg mg give us this annual stress so mg by z again i found it off 2.216 mpa so these are the stresses irrespective of the science of

p into z will be negative at top and positive at bottom why because it are the is at bottom okay

and now the stress will be due to the self weight selfie what you say this is mg by z and mg by okay so plus two point one six minus two point one six

is stresses at this stage of transfer correct so just go for a resultant resultant will be the addition of this that is five minus five will be zero and

plus two point one six two point one six positive iii it means that top fiber by two point one six mb compressive stresses okay and again if we go for

addition of this that is a plus five yet plus hey plus five ten ten minus two point one six here i got again plus seven point eight four it means that bottom fiber pay

seven point eight four magnitude compressive no problem compressive stresses concrete can handle very easily

as concrete is very strong in uh taking compression okay so easy equation this will be the equation f t f

t will be like this and it will be at plus 2.16 and resultant stresses at bottom will be like this just we have written it in the form of

equations okay so remember that at its stage of transfer we found both the top fiber and bottom fiber both

are under compression now stresses at service service methods okay so again this will be the

p j last time this cross section will be subjected to p mg plus mq mq

mq both will having the same direction but due to the pre-stressing sorry okay now this will be uh given like this this problem can be given like this that is the sphere p is

acting at the eccentricity e and due to this eccentricity there will be a moment of p into e which will be opposite opposite to these two mg and mq okay so for this uh

live load stresses have you have a live load stresses i mean just uh have a look over the stress distribution then of course as uh saying previously that there will be a direct stress of 5mp

which will be equal to p by again then pe by zt which will be a bending stresses due to the eccentric pressure

as it is it will be there and plus mg by zt or mg by zb it will be also as it is as it was in previous case

but we need to add one new thing mq by set t and mq by zp which will be a compressive at top fiber tensile at bottom fiber okay and the

magnitude of it is 9 mba and -9 npa at uh bottom five okay so resultant stress okay my addition i found that

this is the resultant stress okay and if i want to write in uh the form of the equation this is like this under this

uh first square bracket these are the stress due to the prestressing and in second square bracket these are the stresses due to

dead load and live load okay so after adding these i found top

uh fiber stresses are subjected to 11.16 mba japanese it was near about two point something two point six seven or two point one six

uh like this right now this time these are increase increased due to what uh due to the addition of life load because sagging moment increased

so don't worry a compressor stresses birthday but we need to check the bottom stresses this time bottom stresses is being changed if you can see here

sorry now it is a negative negative minus 1.16 yes stresses are ahead so bottoms main tensile stress area when live load is there at a time of

surface bottom most fiber are subjected to tensile stresses okay so this is the difference between the

stage of transfer and stage of service