Basic design concepts in Limit State Method

First we will have a brief discussion about the limit state method. A limit state is a state of impending failure. Beyond this state, the structure will not be safe or serviceable.

Safety factorsSafety factors can be explained as follows:A structure should satisfy the condition that The resistance it can offer is always greater than or equal to the effects due to design loads.This condition can be represented by the equation:

1.1

Φ R_n ≥ γS_n

where

R_n = Nominal or characteristic value of resistance R
S_n = Nominal or characteristic value of load effect S
Φ = resistance factor
γ = load factor

Nominal or characteristic value of R is given by the code. For example,if we choose to use Fe 415 steel for a concrete member, we take it’s value of R given by the code. Similarly, if we choose to use M20 concrete, we get the R value from the code.

Characteristic strength of steel is denoted as f_y
and
Characteristic strength of concrete is denoted as f_ck

characteristic load is defined as The load that has a 95 per cent probability of not being exceeded during the life of the structure.

The above definition of ‘characteristic load’ involves some statistical calculations. But in the absence of statistical data about loads, the characteristic values of loads can be taken from the following codes:

Type of Load	Relevant code
Dead Load	IS 875 (Part 1-3)
Live Load	”
Wind Load	”
Seismic Load	IS 1893 (Part 1):2002

It is possible that the value R_n is not calculated accurately due to the uncertainties in material strength, dimensions, theoretical assumptions etc., So, to be on the safer side, we have to take a smaller value than R_n  Thus Φ will be less than unity.

Similarly there will be uncertainties about S_n . There can be overloading during some occasions. So, to be on the safer side, we have to take a larger value than S_n  . Thus γ will be greater than unity.

Partial safety factorsThe code uses the partial safety factor system:
In this system, the safety factor for resistance, ie., Φ   in 1.1, is separated into two ' Partial safety factors '.
• For concrete it is γ_c• For steel it is γ_s   

To get the resistance offered by concrete, γc  is applied to f_ck as (0.67 fck)/γ_c 
To get the resistance offered by  steel, γs  is applied to f_y as f_y/γ_s

In the case of concrete, the quantity 0.67 is coming in due to the following reason:
f_ck  is obtained by the experimental results in the lab. But the actual strength of concrete in a real structure will be less than what is obtained in the lab. This actual strength is taken as 0.67f_ck

We have seen earlier that Φ is less than unity. Now we see that γ_c and γ_s  into which Φ is separated appear in the denominator. But as we will soon see, their values given by the code are greater than unity. Thus by applying them in the denominator, we get the same effect as using a safety factor whose value is less than unity.

Similarly, the safety factor for load, ie., γ in 1.1 is separated into three partial safety factors:
• For Dead Load (DL), it is γ_D• For Live Load (LL), it is γ_L• For Wind Load OR Earthquake Load, it is γ_Q

(Only one of Wind or Earthquake Load is considered at a time. The probability that the earthquake and extreme wind will occur at the same time is taken to be negligible.)

• To get the DL, γ_D is applied to DL as γ_D x DL• To get the LL, γ_L is applied to LL as γ_L x LL• To get the Wind or Earthquake load QL, γ_Q is applied to QL as γ_Q x QL

The above safety factors which are multiplied to the loads are all greater than one. So when multiplied, they will give a greater load. This will result in more safety.
But for γ_D , the value is taken as one in some cases. Examples of such cases are:
• When stability against overturning or sliding is considered.
• When reversal of stresses is considered when dead load is combined with Wind load or Earthquake load.

Values to be used for the Partial safety factors

We have seen the various partial safety factors and the methods to apply them. Now we shall see the actual values of the partial safety factors that we must use while designing a concrete structure. For this we must first know two types of limit states:

• Ultimate limit states and
• Serviceability limit states

• Ultimate limit state is a condition specified by the code. If the structure reaches this stage, the code assumes that the failure of the structure is impending. The structure may start to slide, overturn, collapse or buckle etc.,
• Serviceability limit state is also a condition specified by the code. If the structure reaches this stage, the code assumes that the structure is beginning to cause discomfort to the occupants and/or the structure is beginning to malfunction. This may be in the form of excessive deflection, crack widths, vibration, leakage etc.,

A structure should be designed in such a way that ‘ultimate limit states’ will not be reached. And then, the final design should be checked for ‘serviceability limit states’.

Now we look at the values to be taken for the partial safety factors:

FOR THE MATERIALS CONCRETE AND STEEL,

For ultimate limit states:

• γ_c = 1.5
• γ_s = 1.15

A higher value is given for concrete because it's qualities are more variable than steel

For serviceability limit states:

• γ_c = 1.0
• γ_s = 1.0

Here we notice that, for serviceability limit states, a value of 1.0 is given. This is because if we assume a lower strength for materials by using higher partial safety factors, we will get higher deflections, crack widths etc., We want to know the actual deflections and crack widths so that we can ensure that such a state is not reached. But in the case of ultimate limit states, we want to ensure that the ultimate state is not reached even when a lower material strength is assumed. So we give values 1.5 and 1.15 (note that these values greater than unity are applied in the denominator, thus reducing the final values).

Here we notice that, for serviceability limit states, a value of 1.0 is given. This is because if we assume a lower strength for materials by using higher partial safety factors, we will get higher deflections, crackwidths etc., We want to know the actual deflections and crack widths so that we can ensure that such a state is not reached. But in the case of ultimate limit states, we want to ensure that the ultimate state is not reached even when a lower material strength is assumed. So we give values 1.5 and 1.15 (note that these values greater than unity are applied in the denominator, thus reducing the final values).

FOR LOADS,For ultimate limit states:
The following three different combinations of loads have to be calculated and the maximum value from among the three has to be taken.

Combination 1:
• γ_D = 1.5
• γ_L = 1.5
Giving Ultimate load (UL) = 1.5(DL + LL)

Combination 2:
• γ_D = 1.5
• γ_Q = 1.5
Giving UL = 1.5(DL + QL)

OR

• γ_D= 0.9
• γ_Q = 1.5
Giving UL = 0.9 DL + 1.5 QL
(0.9 is to be used for DL when the stability against overturning or stress reversal is critical)

Combination 3:
• γ_D = 1.2
• γ_L = 1.2
• γ_Q = 1.2
Giving UL = 1.2(DL + LL + QL)
(A lower value of 1.2 is given in this combination because it is assumed that all the loads will not act at the same time at their peak values)

Thus for the ultimate limit state, we see that the safety factors for loads are greater than 1.0, unless it is for special conditions. This is because we want to ensure that the limit state is not reached even at higher loads.

For Serviceability limit states:
The following three different combinations of loads have to be calculated and the maximum value from among the three has to be taken.

Combination 1:
• γ_D = 1.0
• γ_L = 1.0
Giving Serviceability load SL = 1.0(DL + LL)

Combination 2:
• γ_D = 1.0
• γ_Q = 1.0
Giving SL = 1.0(DL + QL)

Combination 3:
• γ_D = 1.0
• γ_L =  0.8
• γ_Q = 0.8
Giving SL = 1.0 DL + 0.8 LL + 0.8 QL
(A lower value of 0.8 is being used because, when LL  and QL are combined, it is improbable that both will reach their peak values at the same time)

All the above values are given in Table 18 of IS 456.

So now we know the exact values that must be used for the various partial safety factors. In the case of loads, we get the ‘Characteristic values’ of the loads from the code, and then apply the appropriate partial safety factors. But for the materials, steel and concrete, the application is not so ‘direct’. Some calculations are involved. Let us see the case of concrete first:

Curve showing the relationship between the stress and strain in concrete:

Fig 21 in section 38.1 of the code shows the relation ship between stress and strain in concrete. Strain is plotted along the X axis, and Stress is plotted along the Y axis. The top most curve is plotted using the values obtained by doing tests in the lab. But we have seen that these values are multiplied by 0.67 to get the values in a real structure. So, the curve corresponding to 0.67f_ck is the characteristic curve. From this curve, the following information can be obtained:

• when the strain increases, the stress also increases parabolically in the initial stage
• This increase continues upto a point when the strain reaches 0.002. The stress has increased to a value of 0.67f_ck• After this point the stress remains constant even if strain is increased.
• The strain can increase only upto an ultimate value of 0.0035. That is., when the strain is 0.0035, the code assumes that concrete has reached the ultimate state (the state of impending failure).

When we design a structure, we need to know the strength of concrete, which is obtained after applying the partial safety factor. So this partial safety factor is applied to the characteristic curve, and thus the bottom most curve is obtained. Thus 0.67f_ck becomes equal to (0.67/1.5)f_ck = 0.447f_ck. From this bottom most curve, we can obtain the design values.

If we know the equation of this design curve, we can find the value of stress developed in concrete for a particular value of strain. The equation consists of two parts. One for the parabolic portion and the other for the straight line portion. These are as follows:

Where fc = stress, and ε = strain

We will see the application of this curve in subsequent chapters.

Curve showing the relationship between the stress and strain in steel:

As in the case of concrete, in the case of steel also, we get the design values from the code. Fig 23 in section 38.1 of the code shows the relation ship between stress and strain in steel. Fig 23A is for cold worked bars (Fe415 and Fe500). 23B is for mild steel (Fe250).

Curve for mild steel:
From this curve, the following information can be obtained:

• When the strain increases, the stress also increases linearly in the initial stage
• This increase continues upto a point when the stress reaches a value of f_y . The strain at this point is f_y/E_s . The behavior of steel upto this point is linear elastic. That is if we remove the load, steel will regain its original length
• This point is called the yield point. And this point is well defined for mild steel.
• Beyond this point, when the strain increases, the stress remains constant.

The design curve is obtained by dividing every stress value by the partial safety factor for steel, 1.15. This is same as multiplying by (1/1.15) = 0.87 .The values thus obtained are plotted to get the lower curve.

Curve for cold worked bars:
From this curve, the following information can be obtained:

• when the strain increases, the stress also increases linearly in the initial stage
• There is no specific yeild point
• The linear elastic behavior continues upto a point where stress is equal to 0.8f_y
• After this point, the material is elastic but not linear. That is, if we remove the load, it will not fully regain it's orginal length. This behaviour continues upto a point where stress is equal to f_y . At this point, if we remove the load, there will be a residual strain of 0.002
• Beyond this point, when the strain increases, the stress remains constant.
• The design curve is obtained by dividing every stress value by the partial safety factor for steel, 1.15. This is same as multiplying by (1/1.15) = 0.87 .This is shown by the lower curve.
• The design curve for Fe 415 and Fe 500 are similar. The value of f_y  has to be changed from 415 to 500.
• We can obtain any values from the straight line portions. But in the non linear portions, it is not easy. The non linear portion is approximated as small linear segments. The code gives the end points of these small linear segments. Any point on these segments, other than the end points should be calculated using linear interpolation. The following table gives the coordinates of the end points:

Table for Fe415:

Strain	Stress (N/mm2)	Calculation
0.000	0.00
0.00144	288.7	(415 x 0.80)/1.15
0.00163	306.7	(415 x 0.85)/1.15
0.00192	324.8	(415 x 0.90)/1.15
0.00241	342.8	(415 x 0.95)/1.15
0.00276	351.8	(415 x 0.975)/1.15
≥ 0.00380	360.9	(415 x 1.00)/1.15

Table for Fe500:

Strain	Stress (N/mm2)	Calculation
0.000	0.00
0.00174	347.8	(500 x 0.80)/1.15
0.00195	369.6	(500 x 0.85)/1.15
0.00226	391.3	(500 x 0.90)/1.15
0.00277	413.0	(500 x 0.95)/1.15
0.00312	423.9	(500 x 0.975)/1.15
≥ 0.00417	434.8	(500 x 1.00)/1.15

We will see more details about the above calculations and the application of these curves in subsequent chapters. In the next chapter we will see some basic details about the analysis of sections by Working stress method.

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