Design of Footings

Q1. Design a reinforced-concrete spread footing for a reinforced-concrete wall 4 m long and 300 mm wide. The wall carries a total service load of 1000 kN. The safe bearing capacity of the soil is 150 kN/m². Use M20 concrete and Fe415 steel. [14] (2065 Kartik, Q4a)

A continuous strip footing under a uniformly loaded wall is designed per metre length. For the 2065 wall question, the service line load is 1000/4 = 250 kN/m. A preliminary 10% weight allowance gives B = 1.10 × 250/150 = 1.833 m, so a trial 1.9 m width may be used before the final weight/pressure check. The projection on each side of the 300 mm wall is (B − 0.3)/2.

Transverse bottom reinforcement carries bending of the projections from the wall face. Check one-way shear at the applicable distance from the face and provide longitudinal distribution steel. A uniformly loaded continuous wall normally does not produce the same isolated-column punching mechanism. End regions, discontinuous wall loads and openings require their own consideration.

Q2. Describe the steps for the design of a rectangular RC footing. Why is shear reinforcement not provided in footings? [4] (2065 Shrawan, Q4a)

A footing spreads the column/wall load onto the soil so the bearing pressure stays within the safe bearing capacity (SBC). Shear reinforcement is normally avoided in footings because it is difficult to place and anchor in a relatively shallow, wide member; instead the depth is chosen large enough that the concrete alone resists both one-way and two-way (punching) shear — a deeper footing is cheaper and simpler than adding stirrups.

The foundation must satisfy two different groups of requirements: geotechnical performance, including bearing resistance, settlement, sliding and overturning; and structural resistance, including bending, shear, punching, anchorage and transfer of column/wall forces. A plan area based on SBC alone is not a complete footing design.

1. Plan area: make a preliminary service-load/SBC calculation, choose dimensions, then recalculate actual self-weight, soil cover, eccentricity and pressure. Check the geotechnical report's gross/net convention.
2. Structural loading: derive the net factored upward pressure from the correct load combination. Maintain consistent units and keep the downward column force in the equilibrium model.
3. Depth from shear: check one-way shear at d from the face and two-way (punching) shear on the perimeter at d/2; increase d until τ_v ≤ the permissible value without stirrups.
4. Flexure: M at the column face = p_u·(cantilever projection)²2 → A_st; check development length of the bars beyond the critical section.

═══ 9.2 ═══

For a centred square footing of side B beneath a square column of side c, the projection is a = (B − c)/2. Under uniform net factored pressure q_u, the bending moment per metre strip at the column face is M_u = q_ua²/2, obtained by integrating the upward load times its lever arm. For a rectangular footing, use the appropriate projection and bar-layer depth in each direction.

One-way shear per metre strip at distance d from the face is V_u = q_u(a − d), provided the section lies within the footing. The punching perimeter is at d/2 from the column faces. For a rectangular interior column, b₀ = 2(c₁ + c₂ + 2d), and the net punching force is the column load minus upward pressure inside the perimeter. Edge/corner columns and eccentric moment transfer need modified assessment.

═══ 9.3 ═══

Shear reinforcement is not inherently forbidden in every foundation. In ordinary isolated pads it is usually simpler and more reliable to increase depth until concrete resists one-way and punching shear, because anchoring and placing links in a wide footing is inconvenient. The final choice follows the governing code and constructability, not a rule that concrete is incapable of needing reinforcement.

Q1. Design an isolated footing for a column of size 500 mm × 500 mm carrying a service axial load of 1600 kN. The allowable bearing capacity of the soil is 125 kN/m² at a depth of 1.25 m below ground level. Use Fe500 TMT steel in both the column and footing. Use M25 concrete in both the footing and the column. Prepare construction drawings in plan and elevation. [10+2+2] (2082 Bhadra, Q5a)

Paper data: 500 × 500 mm column, service axial load 1600 kN, allowable soil pressure 125 kN/m² at depth 1.25 m, M25 concrete and Fe500 steel. Study assumptions: the allowable pressure is gross, the column load acts at footing level, groundwater is below the base, soil unit weight is 18 kN/m³, concrete unit weight is 25 kN/m³, and the footing is cast on levelling concrete with 50 mm nominal bottom cover.

1. Choose and verify the base. Adopt 4.10 × 4.10 m, A = 16.81 m², and a trial overall thickness D = 775 mm. Soil cover above the footing is 1.25 − 0.775 = 0.475 m. Conservatively treating that cover as uniform:

q_{service,gross} = 1600/16.81 + 25 × 0.775 + 18 × 0.475
= 95.181 + 19.375 + 8.550 = 123.11 kN/m² < 125

A 3.8 m base selected using only a preliminary 10% weight allowance would not satisfy this same final-weight check. Plan size and thickness must be iterated together.

2. Obtain net structural loading. P_u = 1.5 × 1600 = 2400 kN and q_u,net = 2400/16.81 = 142.772 kN/m². Projection a = (4.10 − 0.50)/2 = 1.80 m. For two layers of 16 mm bottom bars, depths are 717 mm and 701 mm; use the conservative common design depth d = 700 mm.

3. Flexure per metre width. M_u = 142.772 × 1.80²/2 = 231.291 kN·m/m. Section equilibrium gives x_u = 37.559 mm and A_st,required = 777.09 mm²/m. Minimum HYSD steel is 0.0012 × 1000 × 775 = 930 mm²/m, so provide 16 mm bars at 200 mm centres each way, A_st = 1005.31 mm²/m.

4. One-way shear. At d = 0.70 m from the column face, V_u = 142.772(1.80 − 0.70) = 157.049 kN/m. Nominal stress = 157049/(1000 × 700) = 0.224 MPa. At the provided steel percentage, a conservative M25 Table 19 value of 0.29 MPa exceeds this stress. No beneficial thin-slab factor is used.

5. Punching shear. The critical square side is c + d = 1.20 m, so b₀ = 4.80 m. Force outside the perimeter is V_p = 2400 − 142.772 × 1.20² = 2194.408 kN. Hence τ_p = 2194408/(4800 × 700) = 0.653 MPa, below 0.25√25 = 1.25 MPa for the square column.

6. Flexural anchorage. For 16 mm Fe500 deformed bars in M25 tension, L_d = 16 × 435/(4 × 1.4 × 1.6) = 776.79 mm. Approximately 1750 mm is available from the column face toward the edge before nominal end cover, so the straight projection is adequate for this check. Also check the other relevant bar sections and edge termination.

7. Column-footing bearing. The factored bearing stress at the column base is 2400000/(500 × 500) = 9.60 MPa, below the unenhanced 0.45f_ck = 11.25 MPa. Minimum continued column steel or dowels are still required. The ordinary minimum dowel area is 0.5% of the supported column area = 1250 mm², with at least four bars.

An illustrative eight-16 mm starter arrangement provides 1608.50 mm². Its compression development length in M25 is approximately 621.43 mm after the 25% bond-stress increase, which can be accommodated within this footing. The actual starter sizes, column lap and anchorage must match the column's designed bars; the paper does not specify its longitudinal cage.

Study arrangement: 4.10 m square, 775 mm overall thickness, 16 mm bottom bars at no more than 200 mm centres each way. With 50 mm nominal side cover and 16 mm bars, 21 bars each way allow approximately 199 mm centre spacing. The calculations establish the stated gravity checks under the assumptions above, not site settlement, seismic load transfer or a construction approval.

Q2. Design an isolated footing for a column of size 400 mm × 400 mm reinforced with 8 longitudinal bars of 25 mm diameter. The column carries a service axial load of 1200 kN and a service bending moment of 120 kN·m. Take the safe bearing capacity of the soil as 150 kN/m². Use M20 concrete and Fe415 steel. [12] (2081 Bhadra, Q5b)

1. Design basis and load level. The foundation depth and soil unit weight are unspecified; assume 1.5 m and 18 kN/m3. The supplied column cage is eight 25 mm bars and must be carried into the connection design. Treat the quoted 150 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.5 m and groundwater below the base. Use M20 concrete and Fe415 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as service. With the explicitly assumed gravity factor 1.5, service column load is 1200.000 kN and ultimate column load is 1800.000 kN. The ultimate moments used are 180.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 3.45 m, L = 3.45 m, D = 625 mm and A = 11.9025 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 550 mm. Footing weight plus uniformly assumed soil cover contributes 31.375 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 132.194 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 149.728 and q_min = 114.660 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 151.229 kN/m². The maximum including moment gradients is 177.529 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.525 m and a_y = 1.525 m. Face moments per metre width, M = q_u,maxa²/2, are 206.433 and 206.433 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.3147 and 0.3147 MPa, below the respective Table 19 concrete resistances 0.3225 and 0.3225 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 950 by 950 mm and b₀ = 3800 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 1663.516 kN. Direct punching stress is 0.7959 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

Eccentric-load qualification. Uniform average punching stress alone is insufficient when moment is transferred. A conservative elastic perimeter check assigning the whole column moment to shear gives an additional extreme stress 0.2720 MPa. For a critical rectangle of width u and depth v, use I_line,x = uv²/2 + v³/6 and I_line,y = vu²/2 + u³/6, then τ_max = V_p/(b₀d) + M_x(v/2)/(dI_line,x) + M_y(u/2)/(dI_line,y). The sum here is 1.0679 MPa, below the stated limit. This supplementary model is not presented as a replacement for the adopted code's unbalanced-moment transfer provisions; verify those and the column connection in a final design.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 752.19 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1475.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 11.250 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 800.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q3. Design a footing supporting a square column of size 400 mm × 400 mm. The column carries a service load of 1500 kN, and the safe bearing capacity of the soil is 150 kN/m². Use M20 concrete and Fe415 steel. [10] (2081 Baishakh, Q4b)

1. Design basis and load level. Assume 1.5 m foundation depth and soil unit weight 18 kN/m3; no column moment is supplied. Treat the quoted 150 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.5 m and groundwater below the base. Use M20 concrete and Fe415 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as service. With the explicitly assumed gravity factor 1.5, service column load is 1500.000 kN and ultimate column load is 2250.000 kN. The ultimate moments used are 0.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 3.60 m, L = 3.60 m, D = 625 mm and A = 12.9600 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 550 mm. Footing weight plus uniformly assumed soil cover contributes 31.375 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 147.116 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 147.116 and q_min = 147.116 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 173.611 kN/m². The maximum including moment gradients is 173.611 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.600 m and a_y = 1.600 m. Face moments per metre width, M = q_u,maxa²/2, are 222.222 and 222.222 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.3314 and 0.3314 MPa, below the respective Table 19 concrete resistances 0.3320 and 0.3320 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 950 by 950 mm and b₀ = 3800 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 2093.316 kN. Direct punching stress is 1.0016 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 752.19 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1550.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 14.062 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 800.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q4. Design an isolated footing supporting a square column of size 400 mm × 400 mm, reinforced with 8 longitudinal bars of 20 mm diameter and carrying an axial service load of 1000 kN. The footing base is to be 1 m below ground level, where the allowable bearing capacity of the soil is 100 kN/m². Use M15 concrete and Fe415 steel. Also check the transfer of load from the column to the footing. [14] (2080 Bhadra, Q5b)

1. Design basis and load level. Retain M15 as explicitly requested by this historical paper, and use the supplied 1 m founding depth. Assume soil unit weight 18 kN/m3. The column has eight 20 mm longitudinal bars. Treat the quoted 100 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1 m and groundwater below the base. Use M15 concrete and Fe415 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as service. With the explicitly assumed gravity factor 1.5, service column load is 1000.000 kN and ultimate column load is 1500.000 kN. The ultimate moments used are 0.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 3.60 m, L = 3.60 m, D = 525 mm and A = 12.9600 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 450 mm. Footing weight plus uniformly assumed soil cover contributes 21.675 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 98.835 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 98.835 and q_min = 98.835 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 115.741 kN/m². The maximum including moment gradients is 115.741 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.600 m and a_y = 1.600 m. Face moments per metre width, M = q_u,maxa²/2, are 148.148 and 148.148 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.2958 and 0.2958 MPa, below the respective Table 19 concrete resistances 0.3314 and 0.3314 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 850 by 850 mm and b₀ = 3400 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 1416.377 kN. Direct punching stress is 0.9257 MPa; the concrete-only limit k_s0.25√f_ck is 0.9682 MPa.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 902.62 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1550.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 9.375 MPa, compared with the unenhanced 0.45f_ck = 6.750 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 800.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Historical-paper warning: M15 is retained because the original question specifies it. It is not the ordinary minimum grade for new reinforced-concrete design under IS 456:2000. A current project must use the governing minimum grade and be recalculated.

Q5. Design an isolated footing for a column of size 450 mm × 500 mm reinforced with 6 longitudinal bars of 20 mm diameter. The column carries a factored axial load of 1100 kN and a factored uniaxial bending moment of 120 kN·m at its base. Take the foundation depth as 1.5 m and the safe bearing capacity of the soil as 100 kN/m². Use M20 concrete and Fe500 steel. [12] (2080 Baishakh, Q4a)

1. Design basis and load level. The axial load and moment are already factored. Assume they arise from a common 1.5 gravity combination when recovering service actions for the bearing check. The column has six 20 mm bars. Treat the quoted 100 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.5 m and groundwater below the base. Use M20 concrete and Fe500 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as factored. With the explicitly assumed gravity factor 1.5, service column load is 733.333 kN and ultimate column load is 1100.000 kN. The ultimate moments used are 120.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 3.55 m, L = 3.60 m, D = 475 mm and A = 12.7800 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 400 mm. Footing weight plus uniformly assumed soil cover contributes 30.325 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 87.706 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 98.139 and q_min = 77.273 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 86.072 kN/m². The maximum including moment gradients is 101.721 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.550 m and a_y = 1.550 m. Face moments per metre width, M = q_u,maxa²/2, are 122.193 and 122.193 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.2924 and 0.2924 MPa, below the respective Table 19 concrete resistances 0.3606 and 0.3606 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 850 by 900 mm and b₀ = 3500 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 1034.155 kN. Direct punching stress is 0.7387 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

Eccentric-load qualification. Uniform average punching stress alone is insufficient when moment is transferred. A conservative elastic perimeter check assigning the whole column moment to shear gives an additional extreme stress 0.2899 MPa. For a critical rectangle of width u and depth v, use I_line,x = uv²/2 + v³/6 and I_line,y = vu²/2 + u³/6, then τ_max = V_p/(b₀d) + M_x(v/2)/(dI_line,x) + M_y(u/2)/(dI_line,y). The sum here is 1.0285 MPa, below the stated limit. This supplementary model is not presented as a replacement for the adopted code's unbalanced-moment transfer provisions; verify those and the column connection in a final design.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 906.25 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1500.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 4.889 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 1125.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q6. Design a footing for a rectangular column of size 30 cm × 35 cm reinforced with 8 bars of 20 mm diameter. The column is subjected to a factored axial load of 1000 kN and a factored bending moment of 80 kN·m. The allowable bearing capacity of the soil is 140 kN/m² at a depth of 1.6 m. Use M25 concrete and TMT reinforcement for both the column and footing. Sketch all the required reinforcement. [14] (2079 Bhadra, Q4b)

1. Design basis and load level. Assume Fe500 for the unspecified TMT grade, and soil unit weight 18 kN/m3. Recover service actions with the explicitly assumed common factor 1.5. The column has eight 20 mm bars. Treat the quoted 140 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.6 m and groundwater below the base. Use M25 concrete and Fe500 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as factored. With the explicitly assumed gravity factor 1.5, service column load is 666.667 kN and ultimate column load is 1000.000 kN. The ultimate moments used are 80.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 2.70 m, L = 2.75 m, D = 450 mm and A = 7.4250 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 375 mm. Footing weight plus uniformly assumed soil cover contributes 31.950 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 121.737 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 137.409 and q_min = 106.065 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 134.680 kN/m². The maximum including moment gradients is 158.188 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.200 m and a_y = 1.200 m. Face moments per metre width, M = q_u,maxa²/2, are 113.895 and 113.895 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.3480 and 0.3480 MPa, below the respective Table 19 concrete resistances 0.3694 and 0.3694 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 675 by 725 mm and b₀ = 2800 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 934.091 kN. Direct punching stress is 0.8896 MPa; the concrete-only limit k_s0.25√f_ck is 1.2500 MPa.

Eccentric-load qualification. Uniform average punching stress alone is insufficient when moment is transferred. A conservative elastic perimeter check assigning the whole column moment to shear gives an additional extreme stress 0.3210 MPa. For a critical rectangle of width u and depth v, use I_line,x = uv²/2 + v³/6 and I_line,y = vu²/2 + u³/6, then τ_max = V_p/(b₀d) + M_x(v/2)/(dI_line,x) + M_y(u/2)/(dI_line,y). The sum here is 1.2106 MPa, below the stated limit. This supplementary model is not presented as a replacement for the adopted code's unbalanced-moment transfer provisions; verify those and the column connection in a final design.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 776.79 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1150.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 9.524 MPa, compared with the unenhanced 0.45f_ck = 11.250 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 525.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q7. Design an isolated square footing of uniform thickness for a column of size 400 mm × 400 mm subjected to an axial service load of 650 kN. Take the safe bearing capacity of the soil as 170 kN/m². Use M20 concrete and Fe500 steel. Show the reinforcement arrangement in plan and section. [12] (2078 Bhadra, Q4a)

1. Design basis and load level. Use the stated service load. Founding depth is not given; assume 1.5 m and soil unit weight 18 kN/m3. Treat the quoted 170 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.5 m and groundwater below the base. Use M20 concrete and Fe500 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as service. With the explicitly assumed gravity factor 1.5, service column load is 650.000 kN and ultimate column load is 975.000 kN. The ultimate moments used are 0.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 2.35 m, L = 2.35 m, D = 375 mm and A = 5.5225 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 300 mm. Footing weight plus uniformly assumed soil cover contributes 29.625 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 147.325 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 147.325 and q_min = 147.325 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 176.550 kN/m². The maximum including moment gradients is 176.550 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 0.975 m and a_y = 0.975 m. Face moments per metre width, M = q_u,maxa²/2, are 83.917 and 83.917 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.3972 and 0.3972 MPa, below the respective Table 19 concrete resistances 0.4008 and 0.4008 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 700 by 700 mm and b₀ = 2800 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 888.490 kN. Direct punching stress is 1.0577 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 906.25 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 925.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 6.094 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 800.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q8. Design an isolated footing to carry a factored axial load of 1600 kN from a column of size 400 mm × 400 mm. The column has longitudinal bars of 20 mm diameter. Take the bearing capacity of the soil as 180 kN/m². For the footing, adopt M20 concrete and Fe415 high-strength deformed bars. Check the footing for shear. [10] (2076 Ashwin, Q2b)

1. Design basis and load level. Do not factor 1600 kN again. Assume a 1.5 gravity factor to infer service load, a 1.5 m founding depth and soil unit weight 18 kN/m3. Treat the quoted 180 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.5 m and groundwater below the base. Use M20 concrete and Fe415 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as factored. With the explicitly assumed gravity factor 1.5, service column load is 1066.667 kN and ultimate column load is 1600.000 kN. The ultimate moments used are 0.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 2.70 m, L = 2.70 m, D = 550 mm and A = 7.2900 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 475 mm. Footing weight plus uniformly assumed soil cover contributes 30.850 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 177.169 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 177.169 and q_min = 177.169 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 219.479 kN/m². The maximum including moment gradients is 219.479 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.150 m and a_y = 1.150 m. Face moments per metre width, M = q_u,maxa²/2, are 145.130 and 145.130 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.3119 and 0.3119 MPa, below the respective Table 19 concrete resistances 0.3293 and 0.3293 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 875 by 875 mm and b₀ = 3500 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 1431.962 kN. Direct punching stress is 0.8613 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 752.19 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1100.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 10.000 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 800.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q9. Design a reinforced-concrete footing to carry a column load of 1250 kN from a square column of size 400 mm × 400 mm with longitudinal bars of 20 mm diameter. The bearing capacity of the soil is 140 kN/m². Take the foundation depth as 1.8 m and the unit weight of earth as 18 kN/m³. Use M20 concrete and Fe415 steel. Sketch the reinforcement arrangement in plan and section. [13] (2076 Chaitra, Q4b)

1. Design basis and load level. Treat the unqualified 1250 kN column load as service load for this study. The foundation depth and soil unit weight are supplied. Use the actual number and arrangement of column bars when detailing the connection. Treat the quoted 140 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.8 m and groundwater below the base. Use M20 concrete and Fe415 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as service. With the explicitly assumed gravity factor 1.5, service column load is 1250.000 kN and ultimate column load is 1875.000 kN. The ultimate moments used are 0.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 3.50 m, L = 3.50 m, D = 550 mm and A = 12.2500 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 475 mm. Footing weight plus uniformly assumed soil cover contributes 36.250 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 138.291 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 138.291 and q_min = 138.291 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 153.061 kN/m². The maximum including moment gradients is 153.061 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.550 m and a_y = 1.550 m. Face moments per metre width, M = q_u,maxa²/2, are 183.865 and 183.865 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.3464 and 0.3464 MPa, below the respective Table 19 concrete resistances 0.3592 and 0.3592 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 875 by 875 mm and b₀ = 3500 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 1757.812 kN. Direct punching stress is 1.0573 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 752.19 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1500.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 11.719 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 800.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q10. Design an isolated reinforced-concrete footing to carry an axial load of 1500 kN. The supported column has a cross-section of 350 mm × 350 mm and contains 8 longitudinal bars of 20 mm diameter. Take the bearing capacity of the soil as 175 kN/m². Use M20 concrete and Fe415 steel. [10] (2075 Chaitra, Q5a)

1. Design basis and load level. Treat the unqualified axial load as service load. Assume founding depth 1.5 m and soil unit weight 18 kN/m3. The supplied column contains eight 20 mm bars. Treat the quoted 175 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.5 m and groundwater below the base. Use M20 concrete and Fe415 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as service. With the explicitly assumed gravity factor 1.5, service column load is 1500.000 kN and ultimate column load is 2250.000 kN. The ultimate moments used are 0.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 3.25 m, L = 3.25 m, D = 650 mm and A = 10.5625 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 575 mm. Footing weight plus uniformly assumed soil cover contributes 31.550 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 173.562 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 173.562 and q_min = 173.562 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 213.018 kN/m². The maximum including moment gradients is 213.018 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.450 m and a_y = 1.450 m. Face moments per metre width, M = q_u,maxa²/2, are 223.935 and 223.935 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.3242 and 0.3242 MPa, below the respective Table 19 concrete resistances 0.3246 and 0.3246 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 925 by 925 mm and b₀ = 3700 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 2067.737 kN. Direct punching stress is 0.9719 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 752.19 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1400.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 18.367 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 612.50 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q11. Design an isolated footing for a square column of size 450 mm × 450 mm, reinforced with 8 bars of 20 mm diameter and carrying a service load of 1600 kN. Take the bearing capacity of the soil as 250 kN/m² and the foundation depth as 1.5 m. Adopt M20 concrete and Fe500 steel. Also check the development length of the reinforcement and the bearing stress in the concrete. [14] (2073 Shrawan, Q4a)

1. Design basis and load level. Use the supplied 1.5 m founding depth and assume soil unit weight 18 kN/m3. The column contains eight 20 mm bars. Development length and concrete bearing must both be checked. Treat the quoted 250 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.5 m and groundwater below the base. Use M20 concrete and Fe500 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as service. With the explicitly assumed gravity factor 1.5, service column load is 1600.000 kN and ultimate column load is 2400.000 kN. The ultimate moments used are 0.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 2.75 m, L = 2.75 m, D = 675 mm and A = 7.5625 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 600 mm. Footing weight plus uniformly assumed soil cover contributes 31.725 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 243.295 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 243.295 and q_min = 243.295 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 317.355 kN/m². The maximum including moment gradients is 317.355 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.150 m and a_y = 1.150 m. Face moments per metre width, M = q_u,maxa²/2, are 209.851 and 209.851 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.2909 and 0.2909 MPa, below the respective Table 19 concrete resistances 0.2940 and 0.2940 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 1050 by 1050 mm and b₀ = 4200 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 2050.116 kN. Direct punching stress is 0.8135 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 906.25 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1100.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 11.852 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 1012.50 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q12. Design an isolated footing supporting a square column of size 400 mm × 400 mm that carries a service load of 1200 kN. The allowable soil pressure is 150 kN/m², and the unit weight of the soil above the footing base is 18 kN/m³. Use M20 concrete and Fe415 steel. [10] (2074 Chaitra, Q4a)

1. Design basis and load level. The soil unit weight is supplied; assume 1.5 m foundation depth because it is not specified. Treat the quoted 150 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.5 m and groundwater below the base. Use M20 concrete and Fe415 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as service. With the explicitly assumed gravity factor 1.5, service column load is 1200.000 kN and ultimate column load is 1800.000 kN. The ultimate moments used are 0.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 3.20 m, L = 3.20 m, D = 575 mm and A = 10.2400 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 500 mm. Footing weight plus uniformly assumed soil cover contributes 31.025 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 148.213 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 148.213 and q_min = 148.213 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 175.781 kN/m². The maximum including moment gradients is 175.781 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.400 m and a_y = 1.400 m. Face moments per metre width, M = q_u,maxa²/2, are 172.266 and 172.266 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.3164 and 0.3164 MPa, below the respective Table 19 concrete resistances 0.3208 and 0.3208 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 900 by 900 mm and b₀ = 3600 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 1657.617 kN. Direct punching stress is 0.9209 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 752.19 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1350.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 11.250 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 800.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q13. Design a footing for a square column of size 350 mm × 350 mm reinforced with eight 16 mm bars, subjected to a factored axial load of 1100 kN and moment of 60 kN·m. Allowable bearing capacity is 150 kN/m² at a depth of 1.5 m. Use M20 concrete and Fe500 steel for the footing, and M30 concrete and Fe500 steel for the column. Assume the moment is reversible and sketch the reinforcement in plan and sections. [14] (2074 Ashwin, Q4a)

1. Design basis and load level. Use M20 in the footing and M30 in the supported column. The moment is reversible, so check both signs; the symmetrical pressure envelope below covers them. Assume a common 1.5 gravity factor for service-load recovery. The column has eight 16 mm bars. Treat the quoted 150 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.5 m and groundwater below the base. Use M20 concrete and Fe500 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as factored. With the explicitly assumed gravity factor 1.5, service column load is 733.333 kN and ultimate column load is 1100.000 kN. The ultimate moments used are 60.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 2.65 m, L = 2.65 m, D = 475 mm and A = 7.0225 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 400 mm. Footing weight plus uniformly assumed soil cover contributes 30.325 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 134.751 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 147.648 and q_min = 121.855 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 156.639 kN/m². The maximum including moment gradients is 175.984 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.150 m and a_y = 1.150 m. Face moments per metre width, M = q_u,maxa²/2, are 116.370 and 116.370 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.3300 and 0.3300 MPa, below the respective Table 19 concrete resistances 0.3606 and 0.3606 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 750 by 750 mm and b₀ = 3000 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 1011.890 kN. Direct punching stress is 0.8432 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

Eccentric-load qualification. Uniform average punching stress alone is insufficient when moment is transferred. A conservative elastic perimeter check assigning the whole column moment to shear gives an additional extreme stress 0.2000 MPa. For a critical rectangle of width u and depth v, use I_line,x = uv²/2 + v³/6 and I_line,y = vu²/2 + u³/6, then τ_max = V_p/(b₀d) + M_x(v/2)/(dI_line,x) + M_y(u/2)/(dI_line,y). The sum here is 1.0432 MPa, below the stated limit. This supplementary model is not presented as a replacement for the adopted code's unbalanced-moment transfer provisions; verify those and the column connection in a final design.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 906.25 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1100.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 8.980 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 612.50 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q14. Design an isolated footing to carry a column load of 1300 kN and a bending moment of 100 kN·m about each of the column's two axes. The column has a cross-section of 500 mm × 500 mm and longitudinal bars of 25 mm diameter. The bearing capacity of the soil is 220 kN/m². Take the foundation depth as 1.70 m and the unit weight of the soil as 18.5 kN/m³. Use M25 concrete and Fe415 steel. [14] (2071 Chaitra, Q3b)

1. Design basis and load level. Treat the unqualified axial load and both 100 kN-m moments as service actions. Evaluate the combined corner pressure from both moments, not two unrelated uniaxial maximum-pressure checks. Treat the quoted 220 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18.5 kN/m³, foundation depth 1.7 m and groundwater below the base. Use M25 concrete and Fe415 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as service. With the explicitly assumed gravity factor 1.5, service column load is 1300.000 kN and ultimate column load is 1950.000 kN. The ultimate moments used are 150.000 and 150.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 3.05 m, L = 3.05 m, D = 675 mm and A = 9.3025 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 600 mm. Footing weight plus uniformly assumed soil cover contributes 35.837 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 175.585 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 217.879 and q_min = 133.291 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 209.621 kN/m². The maximum including moment gradients is 273.063 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.275 m and a_y = 1.275 m. Face moments per metre width, M = q_u,maxa²/2, are 221.949 and 221.949 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.3072 and 0.3072 MPa, below the respective Table 19 concrete resistances 0.3085 and 0.3085 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 1100 by 1100 mm and b₀ = 4400 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 1696.359 kN. Direct punching stress is 0.6426 MPa; the concrete-only limit k_s0.25√f_ck is 1.2500 MPa.

Eccentric-load qualification. Uniform average punching stress alone is insufficient when moment is transferred. A conservative elastic perimeter check assigning the whole column moment to shear gives an additional extreme stress 0.3099 MPa. For a critical rectangle of width u and depth v, use I_line,x = uv²/2 + v³/6 and I_line,y = vu²/2 + u³/6, then τ_max = V_p/(b₀d) + M_x(v/2)/(dI_line,x) + M_y(u/2)/(dI_line,y). The sum here is 0.9525 MPa, below the stated limit. This supplementary model is not presented as a replacement for the adopted code's unbalanced-moment transfer provisions; verify those and the column connection in a final design.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 644.73 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1225.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 7.800 MPa, compared with the unenhanced 0.45f_ck = 11.250 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 1250.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q15. Design the isolated footing for a column of size 350 mm × 500 mm subjected to a design axial load of 2000 kN and a design bending moment of 80 kN·m. Take the allowable bearing capacity of the soil as 175 kN/m². [14] (2069 Chaitra, Q4a)

1. Design basis and load level. Interpret design actions as factored. Assume a common gravity factor 1.5 for the bearing check, M25/Fe415, foundation depth 1.5 m and soil unit weight 18 kN/m3 because these are omitted. Treat the quoted 175 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.5 m and groundwater below the base. Use M25 concrete and Fe415 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as factored. With the explicitly assumed gravity factor 1.5, service column load is 1333.333 kN and ultimate column load is 2000.000 kN. The ultimate moments used are 80.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 3.10 m, L = 3.25 m, D = 650 mm and A = 10.0750 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 575 mm. Footing weight plus uniformly assumed soil cover contributes 31.550 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 163.891 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 173.664 and q_min = 154.118 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 198.511 kN/m². The maximum including moment gradients is 213.170 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.375 m and a_y = 1.375 m. Face moments per metre width, M = q_u,maxa²/2, are 201.513 and 201.513 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.2966 and 0.2966 MPa, below the respective Table 19 concrete resistances 0.3074 and 0.3074 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 925 by 1075 mm and b₀ = 4000 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 1802.605 kN. Direct punching stress is 0.7837 MPa; the concrete-only limit k_s0.25√f_ck is 1.2500 MPa.

Eccentric-load qualification. Uniform average punching stress alone is insufficient when moment is transferred. A conservative elastic perimeter check assigning the whole column moment to shear gives an additional extreme stress 0.1008 MPa. For a critical rectangle of width u and depth v, use I_line,x = uv²/2 + v³/6 and I_line,y = vu²/2 + u³/6, then τ_max = V_p/(b₀d) + M_x(v/2)/(dI_line,x) + M_y(u/2)/(dI_line,y). The sum here is 0.8846 MPa, below the stated limit. This supplementary model is not presented as a replacement for the adopted code's unbalanced-moment transfer provisions; verify those and the column connection in a final design.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 644.73 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1325.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 11.429 MPa, compared with the unenhanced 0.45f_ck = 11.250 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 875.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q16. Design an isolated footing for a column of size 450 mm × 450 mm with 8 bars of 20 mm diameter as longitudinal reinforcement, carrying a service axial load of 1000 kN and a service moment of 110 kN·m. Take the safe bearing capacity of soil as 140 kN/m². Use M20 concrete and Fe415 steel. [12] (2082 Baishakh, Q5b)

1. Design basis and load level. The loads are explicitly service actions. Assume foundation depth 1.5 m and soil unit weight 18 kN/m3. The column contains eight 20 mm bars. Treat the quoted 140 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.5 m and groundwater below the base. Use M20 concrete and Fe415 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as service. With the explicitly assumed gravity factor 1.5, service column load is 1000.000 kN and ultimate column load is 1500.000 kN. The ultimate moments used are 165.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 3.35 m, L = 3.35 m, D = 550 mm and A = 11.2225 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 475 mm. Footing weight plus uniformly assumed soil cover contributes 30.850 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 119.957 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 137.512 and q_min = 102.401 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 133.660 kN/m². The maximum including moment gradients is 159.993 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.450 m and a_y = 1.450 m. Face moments per metre width, M = q_u,maxa²/2, are 168.193 and 168.193 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.3284 and 0.3284 MPa, below the respective Table 19 concrete resistances 0.3382 and 0.3382 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 925 by 925 mm and b₀ = 3700 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 1385.637 kN. Direct punching stress is 0.7884 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

Eccentric-load qualification. Uniform average punching stress alone is insufficient when moment is transferred. A conservative elastic perimeter check assigning the whole column moment to shear gives an additional extreme stress 0.3045 MPa. For a critical rectangle of width u and depth v, use I_line,x = uv²/2 + v³/6 and I_line,y = vu²/2 + u³/6, then τ_max = V_p/(b₀d) + M_x(v/2)/(dI_line,x) + M_y(u/2)/(dI_line,y). The sum here is 1.0929 MPa, below the stated limit. This supplementary model is not presented as a replacement for the adopted code's unbalanced-moment transfer provisions; verify those and the column connection in a final design.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 752.19 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1400.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 7.407 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 1012.50 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q17. Design an isolated rectangular footing for a column of size 300 mm × 400 mm reinforced with eight 20 mm bars and made of M25 concrete. The column carries a factored axial load of 1200 kN and a factored moment of 120 kN·m. Sketch the footing reinforcement in plan and sections and check concrete bearing stress and development length. Use M20 concrete for the footing and Fe415 steel. Assume soil bearing capacity 200 kN/m² at 1.25 m below ground level. [14] (2072 Chaitra, Q4b)

1. Design basis and load level. Use M20 footing concrete and the supplied M25 column. Recover service actions using the explicit 1.5 gravity assumption. Use the given 1.25 m founding depth and assume soil unit weight 18 kN/m3. The column cage has eight 20 mm bars. Treat the quoted 200 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.25 m and groundwater below the base. Use M20 concrete and Fe415 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as factored. With the explicitly assumed gravity factor 1.5, service column load is 800.000 kN and ultimate column load is 1200.000 kN. The ultimate moments used are 120.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 2.35 m, L = 2.45 m, D = 525 mm and A = 5.7575 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 450 mm. Footing weight plus uniformly assumed soil cover contributes 26.175 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 165.124 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 199.153 and q_min = 131.096 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 208.424 kN/m². The maximum including moment gradients is 259.466 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.025 m and a_y = 1.025 m. Face moments per metre width, M = q_u,maxa²/2, are 136.301 and 136.301 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.3315 and 0.3315 MPa, below the respective Table 19 concrete resistances 0.3387 and 0.3387 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 750 by 850 mm and b₀ = 3200 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 1067.130 kN. Direct punching stress is 0.7411 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

Eccentric-load qualification. Uniform average punching stress alone is insufficient when moment is transferred. A conservative elastic perimeter check assigning the whole column moment to shear gives an additional extreme stress 0.3036 MPa. For a critical rectangle of width u and depth v, use I_line,x = uv²/2 + v³/6 and I_line,y = vu²/2 + u³/6, then τ_max = V_p/(b₀d) + M_x(v/2)/(dI_line,x) + M_y(u/2)/(dI_line,y). The sum here is 1.0447 MPa, below the stated limit. This supplementary model is not presented as a replacement for the adopted code's unbalanced-moment transfer provisions; verify those and the column connection in a final design.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 752.19 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 975.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 10.000 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 600.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q18. Design an isolated square footing of uniform thickness for a 400 mm × 400 mm column subjected to axial load 600 kN and moment 50 kN·m at service state. Take allowable soil bearing pressure 150 kN/m², M20 concrete and Fe415 steel. [10] (2067 Ashadh, Q2b)

1. Design basis and load level. The paper specifies service actions and a square uniform-depth footing. Assume founding depth 1.5 m and soil unit weight 18 kN/m3. Treat the quoted 150 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 18 kN/m³, foundation depth 1.5 m and groundwater below the base. Use M20 concrete and Fe415 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as service. With the explicitly assumed gravity factor 1.5, service column load is 600.000 kN and ultimate column load is 900.000 kN. The ultimate moments used are 75.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 2.50 m, L = 2.50 m, D = 425 mm and A = 6.2500 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 350 mm. Footing weight plus uniformly assumed soil cover contributes 29.975 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 125.975 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 145.175 and q_min = 106.775 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 144.000 kN/m². The maximum including moment gradients is 172.800 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.050 m and a_y = 1.050 m. Face moments per metre width, M = q_u,maxa²/2, are 95.256 and 95.256 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.3456 and 0.3456 MPa, below the respective Table 19 concrete resistances 0.3779 and 0.3779 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 750 by 750 mm and b₀ = 3000 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 819.000 kN. Direct punching stress is 0.7800 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

Eccentric-load qualification. Uniform average punching stress alone is insufficient when moment is transferred. A conservative elastic perimeter check assigning the whole column moment to shear gives an additional extreme stress 0.2857 MPa. For a critical rectangle of width u and depth v, use I_line,x = uv²/2 + v³/6 and I_line,y = vu²/2 + u³/6, then τ_max = V_p/(b₀d) + M_x(v/2)/(dI_line,x) + M_y(u/2)/(dI_line,y). The sum here is 1.0657 MPa, below the stated limit. This supplementary model is not presented as a replacement for the adopted code's unbalanced-moment transfer provisions; verify those and the column connection in a final design.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 752.19 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1000.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 5.625 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 800.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q1. Design a rectangular footing to carry an axial column load of 1150 kN and a bending moment of 250 kN·m from a square column of size 600 mm × 600 mm. The column has longitudinal reinforcement of 20 mm diameter. The bearing capacity of the soil is 200 kN/m². Take the foundation depth as 1.5 m and the unit weight of earth as 17 kN/m³. Use M20 concrete and Fe415 steel. [16] (2072 Kartik, Q1b)

1. Design basis and load level. This is a single-column footing under axial load and moment, not a two-column combined footing. Treat the unqualified actions as service loads. A square is used as a special rectangular trial base; other aspect ratios can be proportioned and rechecked. Treat the quoted 200 kN/m² bearing pressure as an allowable gross service pressure. Use concrete unit weight 25 kN/m³, soil unit weight 17 kN/m³, foundation depth 1.5 m and groundwater below the base. Use M20 concrete and Fe415 steel for the footing. These ground/water and gross-pressure conventions must be confirmed for a real site.

The supplied axial load is treated as service. With the explicitly assumed gravity factor 1.5, service column load is 1150.000 kN and ultimate column load is 1725.000 kN. The ultimate moments used are 375.000 and 0.000 kN·m. Do not apply another factor to an already factored action.

2. Adopt a trial base and recompute its weight. Use B = 3.15 m, L = 3.15 m, D = 650 mm and A = 9.9225 m². With 50 mm nominal cover and two layers of 16 mm bars, use the conservative common effective depth d = 575 mm. Footing weight plus uniformly assumed soil cover contributes 30.700 kN/m².

3. Service bearing check. q_avg = P_service/A + footing/soil weight = 146.598 kN/m². For full contact, add/subtract 6M_x/(BL²) and 6M_y/(LB²) at the corners. The extreme pressures are q_max = 194.589 and q_min = 98.607 kN/m². The maximum is below the allowable pressure and the minimum is nonnegative, so the assumed full-contact pressure model is admissible.

4. Net structural loading. The mean net ultimate upward pressure is P_u/A = 173.847 kN/m². The maximum including moment gradients is 245.834 kN/m². To obtain a conservative strip design, apply this maximum uniformly to each projection. This overestimates the bending/shear of a nonuniform pressure distribution; it is a stated upper-bound strip calculation, not a claim that the actual pressure is uniform.

The projections from the column faces are a_x = 1.275 m and a_y = 1.275 m. Face moments per metre width, M = q_u,maxa²/2, are 199.817 and 199.817 kN·m/m.

5. One-way shear. At d from the column face, V = q_u,max(a − d) per metre strip. The nominal stresses are 0.2993 and 0.2993 MPa, below the respective Table 19 concrete resistances 0.2999 and 0.2999 MPa for the provided reinforcement. No thin-slab enhancement is used.

6. Punching shear. At d/2 from the faces, the critical rectangle is 1175 by 1175 mm and b₀ = 4700 mm. Subtract the upward reaction inside that perimeter from the column force: V_p = 1484.982 kN. Direct punching stress is 0.5495 MPa; the concrete-only limit k_s0.25√f_ck is 1.1180 MPa.

Eccentric-load qualification. Uniform average punching stress alone is insufficient when moment is transferred. A conservative elastic perimeter check assigning the whole column moment to shear gives an additional extreme stress 0.3543 MPa. For a critical rectangle of width u and depth v, use I_line,x = uv²/2 + v³/6 and I_line,y = vu²/2 + u³/6, then τ_max = V_p/(b₀d) + M_x(v/2)/(dI_line,x) + M_y(u/2)/(dI_line,y). The sum here is 0.9038 MPa, below the stated limit. This supplementary model is not presented as a replacement for the adopted code's unbalanced-moment transfer provisions; verify those and the column connection in a final design.

7. Anchorage. For the 16 mm deformed bottom bars, L_d = 16(0.87f_y)/(4τ_bd) = 752.19 mm. The smaller straight projection beyond the column face, allowing for edge cover, is 1225.00 mm, so it can develop the full design stress under the assumed bond conditions. Actual bar termination and other critical sections must also be checked.

8. Column-to-footing transfer. Average axial bearing stress at the column base is 4.792 MPa, compared with the unenhanced 0.45f_ck = 9.000 MPa for the footing. Any bearing-area enhancement requires the effective supporting area to be demonstrated and is capped by the code; the supported column's grade must also be considered. Continue the designed column bars or provide adequate dowels, with at least 0.5% of the column area = 1800.00 mm² and at least four bars under the ordinary rule. Check actual compression/tension development and the stated column cage. A nonzero column moment cannot be transferred by an average compression check alone.

Conclusion and drawing. The trial schedule passes the listed bearing, flexural and conservative shear calculations under the stated assumptions. Draw its plan and matching section with bar directions, top of footing, column position, cover and starter development. For a rectangular base, comply with the required short-direction central-band distribution instead of automatically using a uniform transverse mesh. Settlement, groundwater, seismic combinations and the complete connection detail remain project-specific requirements.

Q1. Describe the design procedure for a mat foundation. [4] (2081 Bhadra, Q5a)

A mat or raft foundation is a common reinforced-concrete base supporting several columns or walls. It may be appropriate where individual footings overlap, ground bearing resistance is low or a common base helps manage differential settlement.

Start with geotechnical data, groundwater level, allowable bearing pressure, total/differential settlement limits and the complete column/wall load combinations. Fix the mat's extent and trial thickness, calculate its weight and locate the resultant of vertical loads and overturning moments. Check bearing, uplift, sliding and overturning as applicable.

A conventional rigid-base method is appropriate only when the relative rigidity and column/load distribution justify it. A flexible mat may need plate or finite-element analysis with a justified soil model. The subgrade modulus is not a universal soil constant; its value and scale dependence need geotechnical input. Prevent tensile soil reactions in the model unless a real hold-down system is provided.

Obtain moment and shear envelopes in both directions, design top and bottom reinforcement, and check punching at interior, edge and corner columns, openings and concentrated wall loads. Revise thickness or use suitable local thickening if required. Finally check cracking, settlement compatibility, construction joints, temperature/shrinkage reinforcement, waterproofing and anchorage. Draw both reinforcement mats and all local strengthening zones.

═══ 9.5 ═══

Q2. Describe the design procedure for a mat foundation. [5] (2075 Ashwin, Q4b)

A mat or raft foundation is a common reinforced-concrete base supporting several columns or walls. It may be appropriate where individual footings overlap, ground bearing resistance is low or a common base helps manage differential settlement.

Start with geotechnical data, groundwater level, allowable bearing pressure, total/differential settlement limits and the complete column/wall load combinations. Fix the mat's extent and trial thickness, calculate its weight and locate the resultant of vertical loads and overturning moments. Check bearing, uplift, sliding and overturning as applicable.

A conventional rigid-base method is appropriate only when the relative rigidity and column/load distribution justify it. A flexible mat may need plate or finite-element analysis with a justified soil model. The subgrade modulus is not a universal soil constant; its value and scale dependence need geotechnical input. Prevent tensile soil reactions in the model unless a real hold-down system is provided.

Obtain moment and shear envelopes in both directions, design top and bottom reinforcement, and check punching at interior, edge and corner columns, openings and concentrated wall loads. Revise thickness or use suitable local thickening if required. Finally check cracking, settlement compatibility, construction joints, temperature/shrinkage reinforcement, waterproofing and anchorage. Draw both reinforcement mats and all local strengthening zones.

═══ 9.5 ═══

Q3. Explain in detail all the design steps of RCC mat foundation design. [6] (2069 Chaitra, Q3a)

A mat or raft foundation is a common reinforced-concrete base supporting several columns or walls. It may be appropriate where individual footings overlap, ground bearing resistance is low or a common base helps manage differential settlement.

Start with geotechnical data, groundwater level, allowable bearing pressure, total/differential settlement limits and the complete column/wall load combinations. Fix the mat's extent and trial thickness, calculate its weight and locate the resultant of vertical loads and overturning moments. Check bearing, uplift, sliding and overturning as applicable.

A conventional rigid-base method is appropriate only when the relative rigidity and column/load distribution justify it. A flexible mat may need plate or finite-element analysis with a justified soil model. The subgrade modulus is not a universal soil constant; its value and scale dependence need geotechnical input. Prevent tensile soil reactions in the model unless a real hold-down system is provided.

Obtain moment and shear envelopes in both directions, design top and bottom reinforcement, and check punching at interior, edge and corner columns, openings and concentrated wall loads. Revise thickness or use suitable local thickening if required. Finally check cracking, settlement compatibility, construction joints, temperature/shrinkage reinforcement, waterproofing and anchorage. Draw both reinforcement mats and all local strengthening zones.

═══ 9.5 ═══

Q1. Write down the procedure for the design of shear reinforcement. Also explain how isolated footings are designed under punching shear. [8] (2067 Ashadh, Q3a)

1. Obtain factored shear at the relevant section. The section at distance d from a support face is permitted only when the applicable loading and support conditions justify it; a concentrated load close to the support needs particular care.
2. Calculate d, τ_v, the effective tension steel percentage and the concrete shear strength from Table 19.
3. Compare τ_v with Table 20's τ_c,max. If it exceeds this limit, revise the section/materials rather than continuing a stirrup calculation.
4. Calculate V_us = max(0, V_u − τ_cbd). Deduct only a permitted, properly located and anchored bent-up-bar contribution.
5. Select stirrup diameter and number of legs; calculate spacing and round down to a practical value.
6. Check minimum reinforcement, maximum spacing, anchorage and the longitudinal bar arrangement. Show the length of each close-spaced support zone and the central zone on the drawing.

═══ 5.3 ═══

Punching (two-way) shear is the tendency of the column to punch through the slab along a pyramid whose critical perimeter is at d2 from the column face (perimeter b₀ = 2(c₁ + c₂ + 2d) for a rectangular column). The nominal punching stress

τ_v = V_ub₀·d ≤ k_s·0.25√f_ck,   k_s = (0.5 + β_c) ≤ 1

Here β_c is the shorter-to-longer dimension ratio of the column. For a centred interior column under uniform net pressure, V_u equals pressure times the area outside the critical perimeter. With nonuniform pressure, integrate the actual pressure outside it. If the stress exceeds the concrete-only limit, increase depth or use an appropriately designed alternative. Either one-way shear, punching, anchorage, flexure or practical detailing can govern the final thickness.