Yield Line Analysis For Slabs, Study notes of Physics

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Module

Yield Line Analysis for

Slabs

Lesson

Basic Principles, Theory and

One-way Slabs

The above discussion clearly indicates the need of adopting the inelastic analysis or collapse limit state analysis for all structures. However, there are sufficient justifications for adopting the inelastic analysis for slabs as evident from the following limitations of the elastic analysis of slabs:

(i) Slab panels are square or rectangular.

(ii) One-way slab panels must be supported along two opposite sides only; the other two edges remain unsupported.

(iii) Two-way slab panels must be supported along two pairs of opposite sides, supports remaining unyielding.

(iv) Applied loads must be uniformly distributed.

(v) Slab panels must not have large opening.

Therefore, for slabs of triangular, circular and other plan forms, for loads other than uniformly distributed, for support conditions other than those specified above and for slabs with large openings (Figs.12.30.1a to d); the collapse limit state analysis has been found to be a powerful and versatile method.

Inelastic or limit analysis is similar to the plastic analysis of continuous steel beams which is based on formation of plastic hinges to form a mechanism of collapse. However, full plastic analysis of reinforced concrete beams and frames is tedious and time consuming. One important advantage of the reinforced concrete slabs over the reinforced concrete beams and frames is that the slabs are mostly under-reinforced. This gives a large rotation capacity of slabs, which may be taken as the presence of sufficient ductility.

12.30.2 Yield Line Theory

The yield line theory, thus, is an ultimate or factored load method of analysis based on bending moment on the verge of collapse. At collapse loads, the slab begins to crack as they are mostly under-reinforced, with the yielding of reinforcement at points of high bending moment. With the propagation of cracks the yield lines are developed gradually. Finally, a mechanism is formed when the slab collapses due to uncontrolled rotation of members. Yield lines, therefore, are lines of maximum yielding moments of the reinforcement of slab. The essence is to find out the locations of the appropriate yield lines.

Yield line analysis, though first proposed by Ingerslev in 1923 (vide, “The strength of rectangular slabs”, by A. Ingerslev, J. of Institution of Structural Engineering, London, Vol. 1, No.1, 1923, pp. 3-14), Johansen is more known for his large extension of the analysis (please refer to (i) Brutlinieteorier, Jul. Gjellerups Forlag, Copenhagen, 1943, by K.W. Johansen, English translation, Cement and Concrete Association London 1962; and (ii) Pladeformler, 2d ed., Polyteknisk Forening, Copenhagen, 1949, by K.W. Johansen, English translation,” Yield Line formulas for slabs, Cement and Concrete Association, London, 1972). Its importance is reflected in the recommendation of the use of this method of analysis of slabs in the Note of cl. 24.4 of IS 456. It is to note that only under-reinforced bending failure is considered in this theory ignoring the effects due to shear, bond and deflection. Effect of in-plane forces developed is also ignored.

As mentioned earlier, reinforced concrete slabs are mostly under- reinforced and they fail in flexure. Figures 12.30.2a and b show such a reinforced concrete simply supported one-way slab subjected to short term loading. The maximum bending moment is along CC, at a distance of L/2 from the supports,

where the deflection and curvature will be the maximum. The curvature φ is

expressed by (Fig. 12.30.2c):

φ = ∈max/ x (12.1)

where ∈max is the compressive strain in concrete in the outermost fibre

and x is the depth of the neutral axis. Figure 12.30.2b shows the plastic curvature

φ p distributed over a short length DE adjacent to the failure section and the

ignored elastic curvatures φ e elsewhere on the slab.

Figure 12.30.3 shows the schematic moment-curvature diagrams of the slab of Fig.12.30.2a. It is evident from Fig.12.30.3 that up to the point D, when the crack first appears anywhere along the line CC of Figs.12.30.2a, the line OD is elastic. Thereafter, the slope of the line changes gradually with the progress of cracks. Accordingly, the stiffness of the slab is reduced. The reinforcement starts yielding at point F, when the ultimate strength is almost reached. The two paths FH for the mild steel reinforcing bars and FI for the deformed bars show marginal increase in moment capacities. Two points H and I are the respective failure points showing larger disproportionate deformations and curvatures when the maximum moment capacity i.e., the resistance of the cross-section Mp is

reached. It may be noted that at these points (H and I), the plastic curvatures φ p

are much larger than the elastic curvature φ e. Neglecting these marginal moment

capacities in the regions beyond F, the moment curvature diagram is idealised as OEG. The extended zone of increasing curvature at nearly constant moment and

considering φ p as much larger than φ e , it is justified to neglect the elastic

curvature φ e totally. This amounts to shifting the point E to E′ in Fig. 12.30.3.

The first crack starting anywhere along CC of the one-way slab of Fig.12.30.2a, after reaching the maximum moment capacity Mp , proceeds forming plastic hinges. In the process, the crack line or yield line CC is formed when the slab collapses forming a mechanism. It is worth mentioning that even at collapse; the two segments AC and BC remain elastic and plane.

It us thus seen that only one yield line is needed to form a mechanism with two real hinges for the collapse of the statically determinate one-way slab of Fig.12.30.2a. In case of statically indeterminate slab, the clamped-clamped slab of Fig.12.30.4a, however, can sustain the loads without collapse even after the formation of one or more yield lines. Figures 12.30.4c to f show the bending moment and deflection diagrams of the slab with increasing loads w 1 < w 2 < w 3 < w 4. It is known that during the elastic range, the negative moment at the clamped ends ( w 1 l^2 /12) is twice of the positive moment at mid-span ( w 1 l^2 /24), as shown in Fig.12.30.4c. Assuming that the slab has equal reinforcement for positive and negative moments, the highly stressed clamped ends start yielding first. Though the support line hinges rotate, the restraining moments continue to act till the mid-span moment becomes equal to the moment at the supports. Accordingly, a third hinge is formed (Fig.12.30.4f). Now, the three hinges form the mechanism and the slab collapses showing large deflection and curvature. The moment diagram just before the collapse is shown in Fig.12.30.4f.

It is clear from the above discussion that such mechanism is possible with the bending moment diagram of Fig.12.30.4f, if the slab is having adequate reinforcement to resist equal moments at the support and mid-span. The elastic bending moment ratio of 1:2 between the mid-span and support could be increased to 1:1 by the redistribution of moment, which depends on the reinforcement provided in the supports and mid-span sections and not on the elastic bending moment diagram shown in Fig.12.30 4c.

12.30.4 Assumptions

The following are the assumptions of the yield line analysis of reinforced concrete slabs.

1. The steel reinforcement is fully yielded along the yield lines at collapse. Rotation following yield is at constant moment.
2. The slab deforms plastically at collapse and is separated into segments by the yield lines. The individual segments of the slab behave elastically.
3. The elastic deformations are neglected and plastic deformations are only considered. The entire deformations, therefore, take place only along the yield lines. The individual segments of the slab remain plane even in the collapse condition.
4. The bending and twisting moments are uniformly distributed along the yield lines. The maximum values of the moments depend on the capacities of the section based on the amount of reinforcement provided in the section.

5. The yield lines are straight lines as they are the lines of intersection between two planes.

12.30.5 Rules for Yield Lines

The first requirement of the yield line analysis is to assume possible yield patterns and locate the axes of rotation.

It has been observed that assuming the possible yield patterns and locating the axes of rotation are simple to establish for statically determinate or indeterminate (simply supported or clamped) one-way slabs. For other cases, however, suitable guidelines are needed for drawing the yield lines and locating the axes of rotation.

It is worth mentioning that other cases of two-way slabs will have sufficient number of real or plastic hinges to form a mechanism while they will be on the verge of collapse. The yield lines will divide the slabs into a number of segments, which will rotate as rigid bodies about the respective axes of rotation. The axes of rotations will be located along the lines of support or over columns, if provided as point supports. The yield line between two adjacent slab segments is a straight line, as the intersection of two-plane surfaces is always a straight line. The yield line should contain the point of intersection, if any, of the two axes of rotation of two adjacent segments as such point of intersection is common to the two planes.

The two terms, positive and negative yield lines, are used in the analysis to designate the yield lines for positive bending moments having tension at the bottom and negative bending moments having tension at the top of the slab, respectively.

The following are the guidelines for predicting the yield lines and axes of rotation:

1. Yield lines between two intersecting planes are straight lines.
2. Positive yield line will be at the mid-span of one-way simply supported slabs.
3. Negative yield lines will occur at the supports in addition to the positive yield lines at the mid-span of one-way continuous slabs.

Thus, the collapse load satisfying the lower bound theorem is always lower than or equal to the true collapse load. On the other hand, the collapse load satisfying the upper bound theorem is always higher than or equal to the true collapse load.

The yield line analysis is an upper bound method in which the predicted failure load of a slab for given moment of resistance (capacity) may be higher than the true value. Thus, the solution of the upper bound method (yield line analysis) may result into unsafe design if the lowest mechanism could not be chosen. However, it has been observed that the prediction of the most probable true mechanism in slab is not difficult. Thus, the solution is safe and adequate in most of the cases. However, it is always desirable to employ a lower bound method, which is totally safe from the design point of view.

12.30.7 Methods of Analysis

After predicting the general yield pattern and locating the axes of rotation, the specific pattern and locations of axes of rotation and the collapse load for the slab can be determined by one of the two methods given below:

(1) Method of segmental equilibrium

(2) Method of virtual work.

The two methods are briefly explained below.

(1) Method of segmental equilibrium

In this method, equilibrium of the individual slab segments causing the collapse forming the required mechanism is considered to arrive at a set of simultaneous equations. The solutions of the simultaneous equations give the values of geometrical parameters for finalising the yield pattern and the relation between the load capacity and resisting moment.

Thus, in segmental equilibrium method, each segment of the slab is studied as a free body (Fig.12.30.5b) which is in equilibrium at incipient failure under the action of the applied loads, moment along the yield lines, and reactions or shear along the support lines. Since, yield moments are principal moments, twisting moments are zero along the yield lines. Further, in most of the cases, shear forces are also zero. Thus, with reference to Fig.12.30.5b, the vector sum of moments along yield lines AO

and OB (segment AOB) is equal to moments of the loads on the segment AOB about the axis of rotation 1-1 just before the collapse.

It should be specially mentioned that equilibrium of a slab segment should not be confused with the general equilibrium equation of the true equilibrium method, which is lower bound. Strip method of slab design, developed by A. Hillerborg, (vide “Equilibrium theory for reinforced concrete slabs” (in Swedish), Betong, vol.4 No. 4, 1956, pp. 171-182) is a true equilibrium method resulting in a lower bound solution of the collapse load, which is safe and preferable too. The governing equilibrium equation for a small slab element of lengths dx and dy is

2 2 2 2 2 2

Mx Mxy My w x x y y

where w is the external load per unit area; Mx and My are the bending moments per unit width in x and y directions, respectively; and Mxy is the twisting moment. However, strip method of analysis is beyond the scope of this course. For more information about strip method, the reader may refer to Chapter 15 of “Design of concrete structures” by A.H. Nilson, Tata- McGraw – Hill Publishing Company Limited, New Delhi.

(2) Method of virtual work

This method is based on the principle of virtual work. After predicting the possible yield pattern and the axes of rotation, the slab, which is in equilibrium with the moments and loads on the structure, is given an infinitesimal increase in load to cause the structure further deflection. The principle of virtual work method is that the external work done by the loads to cause a small virtual deflection should be equal to the internal work done by the yield moments to cause the rotation in accommodating the virtual deflection. The relation between the applied loads and the ultimate resisting moments of the slab is obtained by equating the internal and external works. As the elastic deflections and rotations are small compared to the plastic deformations and rotations, they are neglected in the governing equation. Further, the compatibility of deflection must be maintained. The work equation is written as follows:

∑ w Δ = ∑ M θ l (12.3)

where

w = collapse load,

Δ = vertical deflection through which the collapse load w moves,

Figures12.30.6a and b show one-way continuous slab whose one span of width one metre is considered. The factored moments of resistance are MA and MB at the continuous supports A and B where negative yield lines have formed and MC at C where the positive yield line has formed. All the moments of resistance are for one metre width. Figure 12.30.6c shows the free body diagrams of the two segments of slab, which are in equilibrium. We now apply the method of segmental equilibrium in the free body diagrams of Fig.12.30.6c.

(1) Method of segmental equilibrium

The line CC, where the positive yield line has formed is at a distance of x from AA. The shear V = 0 at C as the bending moment is the maximum (positive) there. Here there are two unknowns x , the locations of the positive yield line and w , the collapse load, which are determined from the two equations of equilibrium

∑ V = 0 and ∑ M = 0.

From the statical analysis, we know that the vertical reactions VA and VB at A and B, respectively are, (assuming MA > MB ):

VA = ( wL /2) + ( MA – MB ) / L (12.4) VB = ( wL /2) – ( MA – MB ) / L (12.5)

Now, ∑ V = 0 gives: VA – wx = 0

Substituting the expression of VA from Eq. 12.4 in Eq. 12.6, we have, wL /2 + ( MA

- MB ) / L – wx = 0, which gives:

w = -2 ( MA – MB ) / L (L -2 x ) (12.7)

∑ M = 0 gives:

VA x – MA – wx^2 /2 – Mc = 0 (12.8)

Substituting the expression of VA from Eq. 12.4 in Eq. 12.8, we have,

{ wL /2 + ( MA – MB ) / L } x – MA – wx^2 /2 – MC = 0 (12.9)

Substituting the expression of w from Eq. 12.7 into Eq. 12.9, we have,

( MB – MA ) x^2 + 2 ( MA + MC ) Lx – ( MA + MC ) L^2 = 0 (12.10)

Equation 12.10 will give the values of x for known values of MA , MB and MC. Equation12.7 will give the value of w after getting the value of x from Eq. 12.10.