Concrete and Steel Structures

Prestressed concrete beams

A prestressed concrete beam is one in which a compressive force is applied to the concrete before or during service, usually by high-tensile tendons, so that tensile stresses under working loads are reduced or eliminated. The tendon is normally placed with an eccentricity e from the centroid of the section. The stress at any fibre is then the sum of a uniform direct stress and a bending stress, \sigma = \dfrac{P}{A} \pm \dfrac{Pe\,y}{I}, where P is the effective prestressing force. A tendon placed above the centroid compresses the fibre on the tendon side, and the opposite fibre is put in tension. Applied loads then add their own stresses, and the net stress at a fibre is found by superposition. The fibre stress is zero when the prestress-induced compression cancels the stress due to the load. This explains why a cantilever with a hogging moment can end up with zero net stress at the top fibre:

The kern point is the location of a prestressing force that produces no tension at a given extreme fibre. For a rectangular section the kern distance is b/6 or d/6 from the centroid. If the force acts at the kern point, the stress at the opposite fibre is zero under the prestress alone. A force applied at the kern point of a section increases the moment that can be carried before the extreme fibre begins to crack. The extra cracking moment equals the prestress times the combined kern-related distance, and it depends on the force and eccentricity, not on the geometry alone:

For a sagging moment, the prestress force that acts at an eccentricity below the centroid places the soffit in compression. The maximum moment that can be applied without tension at the soffit is found by equating the tensile stress of the applied moment with the compression from the prestress. For a tendon at eccentricity e below the centroid, the moment limit is M = P\left(e + \dfrac{D}{6}\right) for a rectangular section of depth D. The prestress force is the product of the tendon area, number of tendons and effective stress:

Several graphical and analytical methods check whether flexural stresses lie within the allowable limits at transfer and at service. The limit zone method (Magnel–Guyon) and Magnel's graph plot the permissible range of prestressing force and eccentricity, and a design point inside the zone satisfies all stress limits. The Hoyer effect is different: it is the gradual rise of tendon stress over the transmission length in pretensioned members, caused by bond. The load balancing method is a design concept that replaces the prestress by equivalent transverse loads, and it is used to analyse deflections rather than to check stress limits:

• Kern point — the point through which a compressive force can act without producing tension at a chosen fibre. e.g. A force placed exactly at the kern distance from the centroid leaves the far fibre stress-free under prestress alone.
• Transfer stage — the moment when prestress is applied and the self-weight is the only other load, so the stresses at this stage are checked separately from service stresses.

Modular ratio and transformed section

The modular ratio is defined as m = \dfrac{E_s}{E_c}, the Young's modulus of the steel divided by the Young's modulus of the concrete. It is used in the elastic theory of reinforced concrete, where the steel area is converted into an equivalent area of concrete. The modular ratio is defined with Young's moduli, not with shear moduli, and it is always the steel modulus in the numerator:

In the transformed section method, each steel bar of area A_s is replaced by an equivalent concrete area m A_s. For an uncracked rectangular beam, the centroid is found by taking moments of the concrete area and the transformed steel area about a reference line such as the top fibre. The distance from the top fibre to the centroid is \bar{y} = \dfrac{\sum A_i y_i}{\sum A_i}, where the steel area is weighted by m. The distance from the centroid to the steel level is then found by subtraction. This is the basis of the uncracked section analysis:

For a cracked section in which the concrete in the tension zone is neglected, the neutral axis depth x follows from equating the first moment of the compression area about the neutral axis with the transformed steel area times the distance to the steel. For a rectangular section of width b, effective depth d and steel area A_s, the condition is \dfrac{b x^2}{2} = m A_s (d - x). Once x is known, the strain at the steel level follows from plane sections remaining plane, \varepsilon_s = \varepsilon_c \dfrac{d - x}{x}, and the steel stress is \sigma_s = E_s \varepsilon_s:

• Modular ratio — the ratio E_s/E_c used to transform steel into concrete. e.g. With E_s = 2\times10^5 MPa and E_c = 2\times10^4 MPa, m = 10.
• Cracked section — a section in which tension concrete is ignored and only the compression zone and steel carry stresses.

Fillet weld shear capacity

A fillet weld is approximately triangular in cross-section, and its strength is governed by the shear stress on the throat, which is the shortest distance from the root to the weld face. For a 45-degree fillet of leg size s, the effective throat is t = 0.7 s. The design shear strength of a fillet weld in IS 800:2007 is f_{wd} = \dfrac{f_u}{\sqrt{3}\,\gamma_{mw}}, where f_u is the ultimate stress of the weaker part joined. The partial safety factor is \gamma_{mw} = 1.25 for shop welding and 1.5 for site welding. The safe force is the design strength times the throat area, P = f_{wd}\, t\, L:

When the same design force is carried by welds of different leg sizes, the required length is inversely proportional to the throat thickness. Since the throat is proportional to the leg size s, a larger weld needs a shorter length to transfer the same force. This follows directly from P = f_{wd}\, t\, L with f_{wd} constant:

For a circular tube welded to a plate and loaded in torsion, the weld acts as a ring of shear stress along the circumference. If the resisting force per unit length is q = f_{wd}\, t, the torque carried by a weld of mean radius r is T = q\,(2\pi r)\,r = 2\pi r^2 q. The shear stress is taken on the throat area, which is 0.7 times the leg size, so the value of t must be the throat, not the full leg size:

• Throat thickness — the effective shear thickness of a fillet weld, 0.7 \times leg size for equal-leg fillets. e.g. A 6 mm leg gives a throat of about 4.2 mm.
• Shop weld versus site weld — shop welds use \gamma_{mw} = 1.25 and therefore have a higher design strength than site welds with \gamma_{mw} = 1.5.

Design shear strength of RC beams

In IS 456:2000, the nominal shear stress is \tau_v = \dfrac{V_u}{b d}, where V_u is the factored shear force, b the width and d the effective depth. The design shear strength \tau_c of concrete depends on the grade of concrete and the percentage of longitudinal tension steel. When \tau_v exceeds \tau_c, shear reinforcement must be provided. The shear force carried by the concrete is V_c = \tau_c b d, and the remaining shear V_{us} = V_u - V_c must be resisted by stirrups:

For vertical stirrups, the shear resisted is V_{us} = \dfrac{0.87 f_y A_{sv} d}{s_v}, where A_{sv} is the total area of all legs of the stirrup and s_v is the spacing. Solving this for s_v gives the spacing required for a given V_{us}. IS 456 limits the maximum spacing to 0.75 d or 300 mm, whichever is smaller. For a given stirrup spacing, the maximum shear that stirrups can carry is found from the same expression:

Under pure torsion, the factored shear force is zero, and IS 456 uses an equivalent shear stress \tau_{ve} = \dfrac{V_u}{bd} + \dfrac{1.6\,T_u}{b^2 d}. Setting the equivalent shear stress equal to the limiting design shear strength gives the torsional moment capacity of the section. The expression is T_u = \dfrac{\tau_c\, b^2 d}{1.6} for pure torsion:

• Design shear strength \tau_c — the shear stress that concrete alone can carry, read from the code table according to f_{ck} and 100 A_{st}/(bd).
• Equivalent shear stress — the combined stress that accounts for shear and torsion together in IS 456 design.
• Minimum shear reinforcement — provided wherever \tau_v > \tau_c, to control diagonal cracking.

Limiting strains in balanced RC section

In a balanced section, the steel reaches its yield strain at the same instant the concrete reaches its ultimate compressive strain. IS 456:2000 takes the maximum compressive strain in concrete at the extreme fibre as \varepsilon_{cu} = 0.0035. For Fe 415 steel, the yield strain of the bars is taken as \varepsilon_{st} = \dfrac{0.87 f_y}{E_s} + 0.002. Substituting f_y = 415 N/mm^2 and E_s = 2\times10^5 N/mm^2 gives about 0.0038. The balanced tensile steel strain for Fe 415 is therefore approximately 0.0038:

The depth of the neutral axis at the balanced condition is x_{u,max}, and the ratio x_{u,max}/d depends on the grade of steel. The values in IS 456 are 0.53 for Fe 250, 0.48 for Fe 415 and 0.46 for Fe 500. A section with x_u < x_{u,max} is under-reinforced, and failure is ductile because the steel yields first. A section with x_u > x_{u,max} is over-reinforced, and the concrete crushes before the steel yields, giving a brittle compression failure:

In a balanced doubly reinforced section, the neutral axis depth is at x_{u,max}. If the tension steel is reduced, the tensile force falls and the neutral axis moves up, making the section under-reinforced. Increasing the compression steel shares the compressive load, so the concrete compression zone shrinks and again the section becomes under-reinforced. The opposite changes push the section toward over-reinforced behaviour:

• Balanced section — a section in which the steel yield and the concrete crushing strains are reached simultaneously.
• Limiting depth — x_{u,max} = k\,d, with k taken from IS 456 for the grade of steel.
• Compression failure — the mode of failure of an over-reinforced beam, where the concrete crushes before the steel yields.

Bolted connections

When a bolted connection carries a load through a bracket or an eccentric point, the bolt forces are found by analysing the bolt group. The load is first assumed to be shared equally among the bolts, which gives a direct shear on each bolt equal to P/n, where n is the number of bolts. If the line of action of the load is away from the centroid of the bolt group, a torsional moment M = P e acts on the group. This moment produces a torsional shear on each bolt proportional to its distance r from the centroid, F_t = \dfrac{M r}{\sum r^2}:

The torsional shear acts perpendicular to the radius joining the bolt to the centroid. The resultant force on each bolt is the vector sum of the direct and torsional shears. Since the two components may act in the same or in different directions, the resultant must be found by vector addition, not by simple arithmetic addition of magnitudes. For a bolt that lies on the same side as the direct shear, the components can add directly, but the general method is always vector addition:

• Polar moment of the bolt group — \sum r^2, the sum of the squares of the distances of all bolts from the centroid of the group.
• Critical bolt — the bolt carrying the largest resultant force, which is used to design the bolt value required.
• Bolt value — the design strength of a single bolt in shear or bearing, against which the resultant force is compared.

Plastic moment and collapse mechanism

The plastic moment M_p = f_y Z_p is the moment that a steel section can carry when the entire cross-section has yielded. The shape factor is the ratio of the plastic modulus to the elastic modulus, \text{SF} = \dfrac{Z_p}{Z_e}, and it measures the reserve of strength beyond first yield. The shape factor of a rectangular section is 1.5, and that of a solid circular section is about 1.7. For an unsymmetric or built-up section the plastic and elastic moduli must be computed from the section geometry, and the shape factor is then found as their ratio:

For an angle or any section whose area is not symmetric about the neutral axis, the plastic neutral axis lies where the area above it equals the area below it. The plastic modulus is then Z_p = \sum A_i \bar{y}_i, where \bar{y}_i is the distance of each area from the plastic neutral axis. The elastic modulus is computed about the elastic centroidal axis, and the shape factor is the ratio of the two values. A worked example is an angle whose plastic neutral axis lies at the position where half the area is on each side: