Recommendations for Estimating the Resistance of Soil Below the Maximum Scour Level in the Design of Well Foundations of Bridges

The Introduction of IRC 45 outlines the general features of design for road bridges, referencing the Standard Specifications and Code of Practice for Road Bridges, Section 1. Key formulas include load combinations such as 1.1D + 1.4(Wc + Ep + Wor S) (Equation 2) and 1.1D + 1.6L (Equation 3), where D is dead load, Wc is wheel load, Ep is impact factor, Wor is other relevant loads, and S is a factor. Additionally, the Ultimate Resistance Method is supported by a table relating the ratio D/B (depth to breadth) to a factor Q, which varies with D/B as follows:

Here, + denotes the angle of internal friction of soil. These provide foundational parameters for bridge design per IRC 45.

Sources: Clause 1.4, Clause 1.1D +- B -- 1.4 (Wc + Ep + Wor S) ... (2), Clause 1.1D + 1.6L ... (3), TABLE: II. ULTIMATE RESISTANCE METHOD (Vide Annexure 2)

The scope and application of IRC 45 cover the Standard Specifications and Code of Practice for Road Bridges, Section 1 - General Features of Design. Key formulas include load combinations such as 1.1D ± B - 1.4(Wc + Ep + Wor S) (Clause 1.4) and 1.1D + 1.6L (Clause 1.4), where D is dead load, B is braking force, Wc is centrifugal force, Ep is earthquake force, Wor is wind force, and S is a factor. For wells with square or rectangular bases, a constant Q is given in Table I, and for circular bases, a shape factor of 0.6 is applied.

The Ultimate Resistance Method uses values of Q based on the ratio D/B (depth to width) as shown in the table below:

This Q factor relates to the angle of internal friction of soil and is essential for foundation design.

Sources: Clause 1.4, Table II Ultimate Resistance Method

Key formulas and specifications for Basic Assumptions and Soil Properties in IRC 45 include:

• The base resisting moment (Mb) for well foundations is given by Mb = Q × W × B × tan φ, where:

  • W = total vertical load (with load factors),
  • B = width (rectangular) or diameter (circular) of the well base,
  • φ = angle of internal friction of soil,
  • Q = shape and D/B ratio dependent constant (see table below).

• For circular bases, multiply Mb by 0.6 to account for spherical rupture surface.

• Table of Q values for square/rectangular wells based on D/B ratio:

• Intermediate Q values can be linearly interpolated.

These assumptions are based on observed behavior of well foundation models and soil mechanics principles as per IRC 45 clauses 1.2 and 5.5.

Sources: Clause 1.2, Clause 5.5, Table II Ultimate Resistance Method

The Method of Calculation in IRC 45 involves calculating base pressures using elastic theory with subgrade moduli, as outlined in Clause 4. The procedure for calculating soil resistance is detailed in Clause 3. Key steps include:

• Use elastic theory to determine base pressures considering the subgrade modulus of soil.
• Calculate soil resistance based on soil properties and loading conditions.

Unfortunately, the exact formulas and tables are not provided in the retrieved context. Typically, these calculations involve:

• Base pressure, p = load / contact area, adjusted by subgrade modulus.
• Soil resistance derived from modulus of subgrade reaction (k) and deformation.

For precise formulas and tables, refer to the full text of IRC 45 Clauses 3 and 4.

Sources: Clause 3, Clause 4

For base pressure calculations in well foundations as per IRC 45:1972, the key formulas and tables are:

• Base Resisting Moment (Mb) for rectangular bases is given by: [ Mb = Q \times W \times B \times \tan \phi ] where:

  • W = total vertical load (with load factors)
  • B = width of the base (or diameter for circular wells)
  • ( \phi ) = angle of internal friction of soil
  • Q = shape and D/B ratio dependent constant (see Table below)

• For circular bases, multiply Mb by 0.6 to account for spherical rupture surface.

• Table for Q values (for square/rectangular wells):

• Intermediate Q values can be linearly interpolated.

• Side resisting moments involve passive and active earth pressure coefficients (Kp and Ka) calculated using Coulomb's theory with wall friction angle = 3/4 ( \phi ), limited to 22°.

These are per Clause 5.5 and related notes in IRC 45:1972.

Sources: Clause 5.5, Table 2 in Clause 5.5

The Side Resisting Moment (Ms) for well foundations per IRC 45 is calculated about the point of rotation located at 0.2D above the base. The key formula is:

Ms = 0.1 × γ × D² × (Kp - KA) × L

where:

• γ = unit weight of soil
• D = depth of embedment
• Kp = passive earth pressure coefficient
• KA = active earth pressure coefficient
• L = width of the well (or unit width for per unit width calculation)

This formula accounts for the ultimate soil pressure distribution on the front and back faces of the well (Clause 5.2).

Additional related moments include:

• Base Resisting Moment (Mb) = Q × W × B × tan φ, where Q depends on D/B ratio and shape (Table 2 in Clause 5.5).
• Resisting moment due to friction on front and back faces (Mr) is also considered (Clause 5.3).

The total resisting moment must satisfy: 0.7 (Mb + Ms + Mr) ≥ Applied moment (Clause 6).

Table 2: Q values for rectangular wells (Clause 5.5):

For circular bases, multiply Q by 0.6.

This ensures stability against overturning by soil pressures on well sides.

Sources: Clause 5.2, Clause 5.3, Clause 5.5, Clause 6, Table 2

Key formulas and specifications for frictional resistance and moments in well foundations per IRC 45:1972 are as follows:

1. Base Resisting Moment (Mb):

• For rectangular bases, Mb = Q × W × B × tan φ, where B is width, W is total vertical load per unit width, φ is soil friction angle, and Q is a shape and D/B ratio factor.
• For circular bases, multiply Mb by 0.6 to account for spherical rupture surface.
• Table for Q values (for square/rectangular wells):

2. Side Resisting Moment (Ms):

• Ms = 0.1 × γ × D × (Kp - Ka) × L, where γ is soil unit weight, D is depth, Kp and Ka are passive and active earth pressure coefficients, and L is width of well.

3. Resisting Moment due to Friction (Mr):

• Calculated from frictional forces on front and back faces, considering wall friction angle (δ). Total friction force per unit width is proportional to sin δ times soil pressure.

4. Total Resistance Check:

• Factor of safety ensured by 0.7 × (Mb + Ms + Mr) ≥ Applied moment M.

5. Additional Loads:

• Applied moments include live load (L), water current (Wc), earth pressure (Ep), wind (W), seismic force (S), and moments due to tilt and shift.

These are per Clauses 5.2, 5.3, 5.5, 1.25, and 10.2 of IRC 45:1972.

Sources: Clause 5.2, Clause 5.3, Clause 5.5, Clause 1.25, Clause 10.2

The Total Resisting Moment of Soil (Mt) for well foundations per IRC 45 is the sum of Base Resisting Moment (Mb), Side Resisting Moment (Ms), and Moment due to friction (Mr), expressed as:

Mt = Mb + Ms + Mr (Clause 5.4)

• For a rectangular base, Mb = Q × W × B × tan φ, where W is the total vertical load, B is the base width, φ is the soil friction angle, and Q is a shape and D/B ratio dependent constant (Clause 5.5).

• For circular bases, multiply Mb by 0.6 to account for spherical rupture surface (Clause 5.5).

• Values of Q for rectangular wells are given in Table 2:

• Side resisting moment Ms and friction moment Mr depend on soil parameters (y, Kp, Ka, φ), well dimensions, and are calculated using formulas in Clause 5.2 and Clause 1.1.

• Factor of safety is ensured by verifying 0.7 × (Mb + Ms + Mr) ≥ total applied moment M (Clause 5.5).

This provides a comprehensive method to evaluate soil resisting moments for well foundations.

Sources: Clause 5.4, Clause 5.5, Table 2, Clause 5.2, Clause 1.1

The key factor of safety and load combination specifications per IRC 45 are as follows:

• Factor of Safety (FoS): The total resisting moment of soil (Mt) reduced by a strength factor (0.7) should not be less than the total applied moment (M) about the point of rotation, i.e., 0.7(Mb + M3 + Mr) ≥ M, where 0.7 accounts for soil strength reduction factors (Clause 1.5).

• Load Factors:

  • Dead load (D): 1.1
  • Water current force (Wc): 1.4 (due to 20% velocity estimation error)
  • Buoyancy (B): 1.0
  • Wind or seismic forces (W or S): 1.4 when no live load, else 1.25
  • Earth pressure (Ep): 1.4

• Load Combinations:

Where L = live load including braking, etc. (Clause 1.4).

These factors ensure safety against variations in loads and soil strength, with a total strength reduction factor of 0.7 applied to soil resistance moments (Clause 1.5).

Sources: Clause 1.4, Clause 1.5, Table 5.5

The Elastic Theory Method in IRC 45 (Annexure 1) is used to calculate soil pressures at the base and sides of well foundations under design loads using subgrade moduli. It provides the elastic soil pressure distribution but requires verification with the Ultimate Resistance Method for factor of safety (Clause 1.4). The key aspect is that elastic theory calculates base pressures considering soil stiffness, while ultimate resistance checks soil failure capacity. The provided table relates the ratio D/B (depth to width) with a factor Q used in ultimate resistance calculations (Annexure 2), where Q increases with D/B, indicating higher soil resistance with depth. For example:

This table is essential for ultimate resistance checks complementing the elastic theory method.

Sources: Clause 1.4, Annexure 1, Annexure 2

The Ultimate Resistance Method in IRC 45 (1972) calculates the base resisting moment (Mb) for well foundations considering soil pressure and geometry. For a rectangular base, Mb is derived by integrating soil forces about a rotation point, involving parameters like vertical load W, base width B, and soil friction angle φ. For circular bases, a factor of 0.6 is applied to account for spherical rupture surfaces. The simplified formula is:

Mb = Q × W × B × tan φ

where:

• B = width (rectangular) or diameter (circular) of the well,
• W = total vertical load including load factors,
• φ = angle of internal friction of soil,
• Q = shape and D/B ratio dependent constant from the table below.

For circular bases, multiply Q by 0.6.

Intermediate Q values can be linearly interpolated. This method accounts for soil resistance distribution and geometry effects as per IRC 45 clauses 5.5 and Annexure 2.

Sources: Clause 5.5, Annexure 2, Table II Ultimate Resistance Method

The key formulas and specifications for Conditions of Stability of well foundations per IRC 45 are as follows:

• Load Combination for Stability Check: [1.1D + B + 1.25(L + W_c + E_p + W_{or} S)] where D = dead load, B = buoyancy, L = live load (including braking), W_c = water current force, E_p = earth pressure, S = seismic force (Clause 1.25).

• Factor of Safety Criterion: The reduced total resistance moment (0.7(M_b + M_s + M_r)) must be not less than the total applied moment (M) about the rotation point (Clause 1.25).

• Base Resisting Moment ((M_b)) for Rectangular Wells: [M_b = Q W B \tan \phi] where (B) = base width, (W) = total vertical load with load factors, (Q) = shape and D/B ratio dependent constant (Table below), (\phi) = soil friction angle (Clause 5.5).

For circular bases, multiply (M_b) by 0.6.

• Side Resisting Moment ((M_s)) for Rectangular Wells: [M_s = 0.180 \gamma (K_p - K_a) L B D^2 \sin \delta] where (\gamma) = soil unit weight, (K_p, K_a) = passive and active earth pressure coefficients, (L) = width perpendicular to force, (\delta) = angle of friction between soil and well sides (Clause 5.2).

• Total Resisting Moment: [M_t = M_b + M_s + M_r] where (M_r) = moment due to other resistances (Clause 5.4).

• Stability Conditions:

  1. Maximum soil side reaction must not exceed maximum passive pressure at any depth.
  2. Maximum base soil pressure (\sigma_1) must not exceed allowable soil pressure.
  3. Minimum base soil pressure (\sigma_2) must be (\geq 0) (no tension) (Clause 4).

• Additional Notes:

  • Moments due to tilt and shift of wells must be included (Clause 1.25).
  • Horizontal soil reactions and frictional forces are considered with coefficients (\mu) and soil parameters (K_H, K_O) (Elastic Theory, Annexure 1).

These formulas and tables provide the framework to check the stability of well foundations under combined loads and moments as per IRC 45:1972.

Sources: Clause 1.25, Clause 5.5, Clause 5.2, Clause 5.4, Clause 4

The design procedure in IRC 45 involves a systematic stepwise approach for soil resistance and structural adequacy checks. Key steps include:

• Step 3, 4, and 6: Initial checks for soil resistance and structural parameters.
• Step 7: If any conditions in Steps 3, 4, or 6 fail, the well foundation must be redesigned accordingly.
• Step 8: Repeat the entire design process considering load combinations separately for wind and seismic cases.

This iterative process ensures safety and serviceability under different loading scenarios. Although the exact formulas and tables are not provided in the retrieved context, the emphasis is on verifying soil resistance and structural adequacy under various load combinations, followed by redesign if criteria are not met.

Sources: Clause None: Step 7, Clause None: Step 8

Annexure 1 of IRC 45 provides the Elastic Theory method for calculating base pressures on well foundations assuming the soil is elastic and homogeneous (Hooke's Law). Key assumptions include linear increase of soil reaction with lateral deflection (p = K_H z), and the well acting as a rigid body under horizontal force H and moment M_o at scour level.

Key parameters:

• A: base area
• B: base width parallel to H
• D: well depth below scour
• K_O, K_H: vertical and horizontal subgrade reaction coefficients
• m = K_H / K_O
• M = M_o + H.D (total moment at base)

Base pressure equations involve moments of inertia and soil reactions, with stability conditions ensuring maximum soil pressure does not exceed allowable and no tension occurs at base.

Important formulas include:

• Total horizontal soil reaction on sides: P = m K_O L D^2 / 3

• Moment of soil reaction about base: M_p = m K_O L D^3 / 12

• Moment equilibrium: M = M_B + M_p + p' P L D

• Base pressure extremes: σ_1 = (W - p' P)/A + K_O B/2 σ_2 = (W - p' P)/A - K_O B/2

• Stability condition: m K_e > γ (K_p - K_a)

Where symbols are as defined in the annexure.

The method assumes rotation about the base and uses soil friction coefficients and pressure coefficients from Coulomb's theory.

The provided table in Annexure 2 (Ultimate Resistance Method) gives Q values for different D/B ratios but is separate from Elastic Theory.

This summary is based on the detailed equations and explanations in Annexure 1 of IRC 45-1972.

Sources: Annexure 1: Elastic Theory Calculations, Equations (1) to (9), Clause 1.25, Table in Annexure 2

Annexure 2 of IRC 45 (1972) provides the Ultimate Soil Resistance method for well foundations, focusing on failure modes and soil resistance calculation. Key points include:

• The ultimate soil resistance depends on the depth-to-width ratio (D/B) of the well base, with values of Q given as:

• The design load factor formula is: 1.1D + B + 1.25 (L + Wc + Ep + S) (Equation 5), where:

  • L = live load including braking
  • Wc = water current force
  • Ep = earth pressure
  • S = seismic force

• Stability conditions require that maximum soil side reaction does not exceed passive pressure and base soil pressure remains within allowable limits (no tension).

• Failure modes consider soil around the base and sides, with the ultimate resistance computed using factors of safety.

This method complements the elastic theory (Annexure 1) by addressing ultimate failure rather than just stress distribution.

For detailed equations and stability criteria, refer to the full Annexure 2 text in IRC 45:1972.

Sources: TABLE: II. ULTIMATE RESISTANCE METHOD (Vide Annexure 2), Clause 1.25