GATE Civil Engineering (CE) Syllabus

Give the fixed-end moments for a fixed beam of span $L$ carrying a central point load $W$.

$$M_{AB} = -\frac{WL}{8}, \qquad M_{BA} = +\frac{WL}{8}$$

Give the fixed-end moments for a fixed beam of span $L$ under a uniformly distributed load $w$ per unit length.

$$M_{AB} = -\frac{wL^{2}}{12}, \qquad M_{BA} = +\frac{wL^{2}}{12}$$

In limit state design of RCC (IS 456:2000), what are the two principal limit states checked?

Limit State of Collapse (strength: flexure, shear, torsion, compression) and Limit State of Serviceability (deflection and cracking). Design ensures safety against collapse and satisfactory performance in service.

State the partial safety factors for loads and materials used in IS 456 limit state design.

Load factor for the design load combination (DL + LL) is $1.5$. Partial safety factor for material strength: concrete $\gamma_{mc} = 1.5$ and steel $\gamma_{ms} = 1.15$. Thus design strengths are $f_{cd}=0.446f_{ck}$ (peak) and $f_{yd}=0.87f_y$.

For limit state flexure (IS 456), what is the depth of the rectangular stress block and the limiting depth of neutral axis for Fe415 steel?

The equivalent concrete stress block depth is $0.42x_u$ from the top for the lever arm, with peak stress $0.446f_{ck}$. Limiting neutral axis depth ratio $\frac{x_{u,max}}{d}$: $0.53$ for Fe250, $0.48$ for Fe415, and $0.46$ for Fe500.

Write the limiting moment of resistance of a singly reinforced rectangular RCC section (Fe415).

$$M_{u,lim} = 0.36\,f_{ck}\,b\,x_{u,max}\left(d - 0.42x_{u,max}\right)$$ For Fe415 this gives $M_{u,lim} = 0.138\,f_{ck}\,b\,d^{2}$.

Compare under-reinforced, balanced, and over-reinforced RCC sections.

Under-reinforced ($x_u < x_{u,max}$): steel yields before concrete crushes, giving ductile failure with warning — preferred design. Balanced ($x_u = x_{u,max}$): steel reaches yield strain exactly as concrete reaches $0.0035$. Over-reinforced ($x_u > x_{u,max}$): concrete crushes before steel yields, giving sudden brittle failure — not permitted by IS 456.

What is the principle of prestressed concrete and why is it used?

Prestressing introduces internal compressive stresses in concrete (via tensioned high-strength tendons) before service loads are applied, so that tensile stresses from loading are neutralised. This allows concrete to resist tension, eliminates cracking, and permits longer spans and slender sections.

Differentiate pre-tensioning from post-tensioning in prestressed concrete.

Pre-tensioning: tendons are tensioned against external abutments before casting concrete; after concrete hardens, tendons are released and prestress is transferred by bond. Post-tensioning: concrete is cast around ducts, and tendons are tensioned and anchored after hardening, transferring prestress through end anchorages.

List the major losses of prestress in pre-tensioned and post-tensioned members.

Pre-tensioning losses: elastic shortening of concrete, creep of concrete, shrinkage of concrete, and relaxation of steel. Post-tensioning adds: friction (wobble and curvature) and anchorage slip; elastic shortening loss is usually negligible if tendons are stressed sequentially/simultaneously.

In steel structures, distinguish between the behaviour of tension members and compression members.

Tension members fail by yielding of the gross section or rupture of the net section (no buckling); strength is straightforward. Compression members can fail by buckling (flexural, torsional, or flexural-torsional) before reaching yield; their strength depends on slenderness ratio and is reduced for long, slender members.

Write the design tensile strength governing equations for a steel tension member (IS 800).

The least of: (a) gross-section yielding $T_{dg} = \frac{f_y A_g}{\gamma_{m0}}$; (b) net-section rupture $T_{dn} = \frac{0.9 f_u A_n}{\gamma_{m1}}$; and (c) block shear. Here $\gamma_{m0}=1.10$, $\gamma_{m1}=1.25$.

Define slenderness ratio for a compression member and its significance.

Slenderness ratio $\lambda = \frac{KL}{r}$, where $KL$ is the effective length and $r=\sqrt{I/A}$ is the radius of gyration. A higher slenderness ratio means greater tendency to buckle and lower compressive (buckling) strength. IS 800 limits it (e.g., $\leq 180$ for members carrying loads).

Compare bolted and welded connections in steel structures.

Bolted connections: faster site assembly, easily dismantled/inspected, but reduce net section (holes) and may loosen under vibration. Welded connections: full member section is effective, neater appearance and lighter joints, but require skilled labour, are sensitive to fatigue and residual stresses, and are hard to inspect.

Write the strength of a fillet weld and define throat thickness.

Design strength of a fillet weld $= \frac{f_u}{\sqrt{3}\,\gamma_{mw}}\times (\text{throat thickness} \times \text{effective length})$. Effective throat thickness $t = 0.7s$ for a $90^\circ$ weld, where $s$ is the leg/weld size; $\gamma_{mw}=1.25$ (shop) or $1.5$ (site).

State the three-phase (block) diagram quantities used in soil and the basic phase relation.

Soil is a three-phase system: solids, water, and air, with total volume $V = V_s + V_w + V_a$ and void volume $V_v = V_w + V_a$. Fundamental relation: $$Se = wG$$ where $S$ = degree of saturation, $e$ = void ratio, $w$ = water content, $G$ = specific gravity of solids.

Define void ratio and porosity, and give the relation between them.

Void ratio $e = \frac{V_v}{V_s}$ (volume of voids to volume of solids). Porosity $n = \frac{V_v}{V}$ (volume of voids to total volume). Relation: $$n = \frac{e}{1+e}, \qquad e = \frac{n}{1-n}.$$

Write the expressions for bulk (total), dry, and saturated unit weights of soil.

$$\gamma_{bulk} = \frac{(G + Se)\gamma_w}{1+e}, \quad \gamma_{dry} = \frac{G\gamma_w}{1+e}, \quad \gamma_{sat} = \frac{(G+e)\gamma_w}{1+e}.$$ Submerged unit weight $\gamma' = \gamma_{sat} - \gamma_w$.

Define the Atterberg limits and the plasticity index.

Atterberg limits mark water-content boundaries between consistency states of fine soil: Liquid Limit ($w_L$, liquid-plastic boundary), Plastic Limit ($w_P$, plastic-semisolid boundary), and Shrinkage Limit ($w_s$). Plasticity Index $I_p = w_L - w_P$ is the range of water content over which soil is plastic.

State the equation of the A-line used in the plasticity chart for soil classification.

$$I_p = 0.73\,(w_L - 20)$$ Soils plotting above the A-line are clays (C); below are silts (M) or organic soils (O). It is used in the Unified Soil Classification System / IS classification.

In the Unified Soil Classification System, what do the symbols W, P, M, and C denote?

For coarse soils: W = well graded, P = poorly graded. For fine soils/secondary: M = silt, C = clay. Prefixes G = gravel and S = sand; e.g., GW = well-graded gravel, CH = high-plasticity clay, ML = low-plasticity silt.

Define the coefficient of uniformity $C_u$ and coefficient of curvature $C_c$ for gradation.

$$C_u = \frac{D_{60}}{D_{10}}, \qquad C_c = \frac{(D_{30})^{2}}{D_{10}\,D_{60}}$$ Well-graded sand requires $C_u > 6$ and $1 \leq C_c \leq 3$; for gravel $C_u > 4$. $D_{10}$ is the effective size.

State Darcy's law for laminar flow of water through soil.

$$q = k\,i\,A \quad\text{or}\quad v = k\,i$$ where $q$ = discharge, $k$ = coefficient of permeability, $i = \frac{h}{L}$ = hydraulic gradient, $A$ = cross-sectional area, and $v$ = discharge (Darcy) velocity. Valid for laminar flow (Reynolds number low).

Define effective stress and write Terzaghi's effective stress equation.

Effective stress is the stress carried by the soil skeleton (grain-to-grain contact) that governs strength and deformation. $$\sigma' = \sigma - u$$ where $\sigma$ = total stress, $u$ = pore water pressure, $\sigma'$ = effective stress.

What is a flow net and what two families of lines compose it?

A flow net is a graphical grid representing two-dimensional steady seepage, consisting of flow lines (paths of water particles) and equipotential lines (lines of equal total head). The two families intersect at right angles forming approximately square fields.

Write the formula for seepage discharge through a flow net.

$$q = k\,H\,\frac{N_f}{N_d}$$ where $k$ = permeability, $H$ = total head loss, $N_f$ = number of flow channels, and $N_d$ = number of equipotential drops.

Define critical hydraulic gradient and the condition for quick sand (boiling).

Critical hydraulic gradient is the upward gradient at which effective stress becomes zero: $$i_c = \frac{\gamma'}{\gamma_w} = \frac{G-1}{1+e}.$$ When the seepage gradient reaches $i_c$ (typically $\approx 1$), the soil loses all shear strength and behaves as a quick (boiling) condition.

State the assumptions of Terzaghi's one-dimensional consolidation theory.

Key assumptions: soil is homogeneous and fully saturated; soil grains and water are incompressible; flow and compression are one-dimensional (vertical); Darcy's law is valid; coefficient of permeability $k$ and compressibility are constant; and there is a unique linear relationship between effective stress and void ratio.

Write the governing differential equation of Terzaghi's 1-D consolidation and define $c_v$.

$$\frac{\partial u}{\partial t} = c_v \frac{\partial^{2} u}{\partial z^{2}}, \qquad c_v = \frac{k}{m_v \gamma_w}$$ where $u$ = excess pore pressure, $c_v$ = coefficient of consolidation, $k$ = permeability, and $m_v$ = coefficient of volume compressibility.

Write the formula for primary consolidation settlement of a normally consolidated clay.

$$S_c = \frac{C_c\,H}{1+e_0}\,\log_{10}\!\left(\frac{\sigma'_0 + \Delta\sigma'}{\sigma'_0}\right)$$ where $C_c$ = compression index, $H$ = layer thickness, $e_0$ = initial void ratio, $\sigma'_0$ = initial effective stress, and $\Delta\sigma'$ = stress increment.

Define the degree of consolidation and give the time factor relations for $U \leq 60\%$.

Degree of consolidation $U$ is the ratio of settlement at time $t$ to ultimate settlement. Time factor $T_v = \frac{c_v t}{d^{2}}$ ($d$ = drainage path). For $U \leq 60\%$: $$T_v = \frac{\pi}{4}U^{2};$$ for $U > 60\%$: $T_v = 1.781 - 0.933\log_{10}(100 - U\%)$.

State the Mohr-Coulomb failure criterion for soil shear strength.

$$\tau_f = c + \sigma_n \tan\phi$$ where $\tau_f$ = shear strength at failure, $c$ = cohesion, $\sigma_n$ = normal stress on the failure plane, and $\phi$ = angle of internal friction. In effective stress terms: $\tau_f = c' + \sigma'\tan\phi'$.

For the Mohr-Coulomb criterion, what is the inclination of the failure plane to the major principal plane?

The failure plane is inclined at an angle $\theta = 45^\circ + \frac{\phi}{2}$ to the major principal plane (i.e., to the plane on which $\sigma_1$ acts).

Compare cohesive (clay) and cohesionless (sand) soils in terms of shear strength parameters.

Cohesionless soils (sand, gravel): shear strength comes only from friction, $c \approx 0$, so $\tau_f = \sigma_n\tan\phi$. Cohesive soils (clay): strength comes mainly from cohesion; for saturated clay under undrained (UU) conditions $\phi_u \approx 0$ and $\tau_f = c_u$ (undrained cohesion).

Name the common laboratory tests used to measure shear strength of soil.

Direct shear (shear box) test, triaxial compression test (UU, CU, CD types), unconfined compression test (for saturated clay, giving $q_u = 2c_u$), and the laboratory/field vane shear test (for soft clays).

Write the relationship between coefficient of permeability $k$, coefficient of consolidation $c_v$, and compressibility.

$$k = c_v\,m_v\,\gamma_w$$ where $m_v = \frac{a_v}{1+e_0}$ is the coefficient of volume compressibility and $a_v = -\frac{de}{d\sigma'}$ is the coefficient of compressibility.