Design of a RC slab, Advanced Structural Design

Design of a reinforced concrete slab - Lorenzo Sostegni

P:• Top: 1φ / 30 cm– 12Bottom: 1φ / 30 cm– 12

Q:• Top: 1φ / 30 cm– 12Bottom: 1φ / 25 cm– 12

R:• 27Design of a reinforced concrete slab - Lorenzo Sostegni4. Principal directions and bending momentsTop: 1φ / 30 cm– 12Bottom: 1φ / 30 cm– 12

S:• Direction 1φ / 30 cm– 12Direction 1φ / 30 cm– 12

Figure 4.3: On the left the bottom reinforcement and on the right the top reinforcement

To compute the resisting bending moment associated to this optimal design we consider a section of width 1 m which has only the bottom reinforcement; furthermore, we consider the reinforcement yielded.

Table 4.3: Resisting bending moment [kN m/m]

Now we want to compare the acting and resisting bending moments in these 4 points as the direction changes: in order to do that we need to resume some equations.

2 2M = M cos θ + 2

M sinθ cosθ + M sin θ (4.1)nu xu xyu yu+ + 2 + 2m = m cos θ + m sin θ (4.2)nu xu yu 22 −−− sin θ (4.3)cos θ + m= mm yuxunu+ + +2 2M = M cos θ + M sin θ (4.4)rd rd,x rd,y2 2− − −M = −(M cos θ + M sin θ) (4.5)rd rd,x rd,y+ 2M = M cos (θ − θ ) (4.6)1 1opt 2−M = M cos (θ − θ ) (4.7)2 2opt+ + 2M = M cos (θ − θ ) (4.8)1rd,opt rd,opt 2− −M = −M cos (θ − θ ) (4.9)2rd,opt rd,opt 28Design of a reinforced concrete slab - Lorenzo Sostegni4. Principal directions and bending momentsEquation 4.1 comes directly from the Mohr’s circle and it represents the acting bendingmoment as the angle θ varies. The second couple of relations (4.2, 4.3) allows us tocompute the bending moment following the Wood-Armer method for both bottom andtop reinforcement while the third one (4.4, 4.5) computes the

resisting bending moment associated to the orthogonal reinforcement layout done in chapter 3. The fourth (4.6,4.7) and the fifth couple (4.8, 4.9) are associated to the optimal design done following the principal directions. In particular, the first one computes the resisting bending moment in the case in which we choose as resisting bending moments the principal ones, and therefore it is an ideal condition, due to practical limitations. The second one computes the resisting bending moments with the choice we have done in table 4.3. These sets of equations are graphically represented in the graphs below.

Figure 4.4: Acting bending moment compared to that computed by the Wood-Armer method in point P 29

Design of a reinforced concrete slab - Lorenzo Sostegni

4. Principal directions and bending moments

Figure 4.5: Acting bending moment compared to that computed by the Wood-Armer method in point Q

Figure 4.6: Acting bending moment compared to that computed by the Wood-Armer method in point R 30

Principal directions and bending moments

Figure 4.10: Acting versus resisting bending moments (optimal design) in point R

Figure 4.11: Acting versus resisting bending moments (optimal design) in point S

From these figures it is easy to see how the optimal design is very appealing. In fact, every curve fits very well the acting bending moment. In any case also this design is not perfect due to the limitations imposed by the Eurocode on the minimum amount of steel reinforcement in stretched zones and on the bar spacing. This fact is particularly evident in point R and S, where the resisting bending moment is very high with respect to the acting one.

Figure 4.12: Acting versus resisting bending moments in point P

Figure 4.13: Acting versus resisting bending moments in point Q

4. Principal directions and bending moments

What we can observe from these graphs is that the design choices we made so far are correct, since the acting bending moment is always in between negative and positive bending moments. Anyway, as we have seen with the optimal design, it is possible to design the steel reinforcement of a plate by considerations on the principal directions in order to have a total amount of reinforcement that is drastically less than the one we obtain from the previous design. Even if this consideration seems decisive, we have to consider that principal directions change in every point of the plate and therefore we will need a very complicated reinforcement layout. Moreover from a practical point of view it is very easy to lay.

Design of a reinforced concrete slab - Lorenzo Sostegni

4. Principal directions and bending moments

Orthogonal reinforcement parallel to the slab edges. Another great problem are the Eurocode limitations which impose large limits to the design of a minimum reinforcement layout. In conclusion, the design of a steel reinforcement layout with bars parallel to the edges of the slab remains the best choice.

Design of a reinforced concrete slab - Lorenzo Sostegni

Collapse load

With reference to the reinforcement layout with bars parallel to the bars edges, assess the slab collapse load based on yield line theory and check the slab safety with respect to the design load considering different global and local collapse mechanisms.

The yield line theory by Johansen provides the slab collapse load by considerations on the balance of internal and external work done respectively by the reinforcement and the external load, which is supposed uniformly distributed across the plate. Before computations we need to mind these rules on the yield line pattern:

• Yield lines are generally straight
• Yield lines

terminate at a slab boundary or at the intersection with concurring yield• lines

Yield lines pass trough the intersection of the axes of rotation of adjacent slab plane• elements

Axes of rotation generally lie along lines of supports and pass over columns•

Any symmetry of the slab (geometry and reinforcement layout) should be reflected in• the yield line pattern. However, non symmetric patterns may occur since uniquenessof the collapse mechanism is not ensured.

Since we cannot be sure to analyze all the possible patterns we are going to make asimplification: we will consider the slab symmetric with respect to the horizontal axispassing trough the abscissa η = 0.5. This assumption is correct if we consider the geometryof the slab but may be rough if we consider the reinforcement layout, which is differentbecause of the clamped edge. However, since only one of the four edges is clamped wemay suppose that this hypothesis will not affect so much the final result.

5.1

Global collapse

Since the slab is assumed to be symmetric with respect to the mid horizontal axis, the global yield line patterns analyzed must be symmetric too and they are represented in the next figure. Both the patterns are parameterized with α and β, and the slab is subdivided in a grid which has the spacing of a/10 (b/10) m. The zig zag purple lines indicate yield lines that form on the bottom side of the plate and are associated to positive bending moment, while the dashed purple one forms on the top side of the plate and therefore is associated to negative bending moments.

375. Collapse load

5.1. Global collapse

5.1.1 First global pattern

The internal work is formulated as:

XW = (m θ l + m θ l )i x,i x y,i y,i y x,ii

Where m is the resisting bending moment associated to the i-th yield line in one direction and θ is the angle associated to the rigid body motion of a part of the

On the other hand, the external work is computed as follows:

ZZ Xp w(x, y) dA = p VW = u u ie A i

Where V is the volume of the solid which has the base enclosed by the slab boundary and itwo or more yield lines and the height equal to the maximum vertical displacement δ. The load p is taken outside the sum since we made the hypothesis of uniformly loaded slab.

Applying the Principle of Virtual Works we can derive the collapse load.

W = We iX Xp δ V /δ = δ (m θ /&d