GATE Civil Engineering Structural Engineering Syllabus

For a simply supported beam of span $L$ with a central point load $W$, give the maximum bending moment and maximum shear force.

Maximum bending moment $M_{max} = \frac{WL}{4}$ at midspan; maximum shear force $V_{max} = \frac{W}{2}$ at the supports.

For a simply supported beam of span $L$ carrying a uniformly distributed load $w$ per unit length, give the maximum bending moment and end reactions.

$M_{max} = \frac{wL^2}{8}$ at midspan; each support reaction $= \frac{wL}{2}$.

For a cantilever of length $L$ with a point load $W$ at the free end, give the maximum bending moment and where it occurs.

$M_{max} = WL$ (hogging) at the fixed support; shear force is constant at $W$ along the length.

Define normal (direct) stress and engineering normal strain.

Normal stress $\sigma = \frac{P}{A}$ (force per unit area). Normal strain $\varepsilon = \frac{\Delta L}{L}$ (change in length per original length).

State Hooke's law and define Young's modulus.

Within the elastic limit, stress is proportional to strain: $\sigma = E\varepsilon$, where $E$ (Young's modulus) is the constant of proportionality.

Define Poisson's ratio.

$\nu = -\frac{\text{lateral strain}}{\text{longitudinal strain}}$, the ratio of transverse contraction strain to axial extension strain under uniaxial loading.

Write the relationships connecting elastic constants $E$, $G$, $K$, and $\nu$.

$E = 2G(1+\nu)$ and $E = 3K(1-2\nu)$, where $G$ is shear modulus and $K$ is bulk modulus.

Give the formula for axial elongation of a prismatic bar under load $P$.

$\delta = \frac{PL}{AE}$, where $L$ is length, $A$ cross-sectional area, $E$ Young's modulus.

State the simple (Euler–Bernoulli) bending equation.

$\frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R}$, where $M$ = bending moment, $I$ = second moment of area, $\sigma$ = bending stress at distance $y$ from neutral axis, $E$ = Young's modulus, $R$ = radius of curvature.

List the key assumptions of simple bending theory.

Material is homogeneous, isotropic and obeys Hooke's law; plane sections remain plane after bending; the beam is initially straight with constant cross-section; the radius of curvature is large; stresses stay within the elastic limit; and the neutral axis passes through the centroid.

Where does the bending stress reach its maximum across a beam cross-section, and where is it zero?

Bending stress is maximum at the extreme fibres (farthest $y$ from the neutral axis) and zero at the neutral axis. It varies linearly with $y$.

Define section modulus $Z$ and write the maximum bending stress in terms of it.

$Z = \frac{I}{y_{max}}$. Maximum bending stress $\sigma_{max} = \frac{M}{Z}$.

Give the section modulus of a rectangular cross-section of width $b$ and depth $d$.

$Z = \frac{bd^2}{6}$ (since $I = \frac{bd^3}{12}$ and $y_{max} = \frac{d}{2}$).

Write the formula for transverse shear stress distribution in a beam (Jourawski / shear formula).

$\tau = \frac{VQ}{Ib}$, where $V$ = shear force, $Q$ = first moment of area above (or below) the level considered, $I$ = second moment of area, $b$ = section width at that level.

For a rectangular beam section, state the maximum shear stress in terms of average shear stress, and where it occurs.

$\tau_{max} = \frac{3}{2}\tau_{avg} = \frac{3V}{2bd}$, occurring at the neutral axis. Shear stress is zero at the top and bottom fibres.

What is the ratio of maximum to average shear stress for a solid circular section?

$\tau_{max} = \frac{4}{3}\tau_{avg}$, occurring at the neutral axis.

Define the shear centre of a cross-section.

The shear centre is the point through which a transverse load must act to produce bending without twisting (no torsion). For sections with two axes of symmetry it coincides with the centroid.

Where does the shear centre lie for a channel (C) section and for sections made of intersecting rectangles (e.g., angle, T)?

For a channel section the shear centre lies outside the section on the side opposite the web (on the axis of symmetry). For sections formed of intersecting thin rectangles (angle, T, cruciform), the shear centre lies at the point of intersection of the legs.

State the torsion equation for a circular shaft (uniform torsion).

$\frac{T}{J} = \frac{\tau}{r} = \frac{G\theta}{L}$, where $T$ = torque, $J$ = polar moment of inertia, $\tau$ = shear stress at radius $r$, $G$ = shear modulus, $\theta$ = angle of twist, $L$ = length.

Give the polar moment of inertia $J$ for a solid circular shaft of diameter $d$, and the resulting maximum shear stress.

$J = \frac{\pi d^4}{32}$; maximum shear stress at the surface $\tau_{max} = \frac{16T}{\pi d^3}$.

What are the assumptions of uniform (pure) torsion theory for circular shafts?

The shaft is circular and prismatic; plane cross-sections remain plane and do not warp; radii remain straight after twisting; material is homogeneous, isotropic and linearly elastic; and the angle of twist is small.

Write the transformation equation for normal stress on a plane inclined at angle $\theta$ in plane stress.

$\sigma_{\theta} = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2}\cos 2\theta + \tau_{xy}\sin 2\theta$.

Write the formula for the principal stresses in a plane-stress state.

$\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$.

Write the formula for the maximum in-plane shear stress in plane stress.

$\tau_{max} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2} = \frac{\sigma_1 - \sigma_2}{2}$.

On which planes do principal stresses act, and what is the shear stress there?

Principal stresses act on the principal planes, where the shear stress is zero. The principal planes are oriented at $2\theta_p = \tan^{-1}\!\left(\frac{2\tau_{xy}}{\sigma_x - \sigma_y}\right)$.

What does Mohr's circle represent, and what are the coordinates of its centre and radius?

Mohr's circle is a graphical representation of stress transformation. Centre at $\left(\frac{\sigma_x+\sigma_y}{2},\,0\right)$ and radius $R = \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2}$, which equals $\tau_{max}$.

State Euler's critical buckling load for a column.

$P_{cr} = \frac{\pi^2 EI}{L_e^2}$, where $EI$ is flexural rigidity and $L_e$ is the effective length.

Give the effective length $L_e$ for the four standard column end conditions (in terms of actual length $L$).

Both ends pinned: $L_e = L$. Both ends fixed: $L_e = \frac{L}{2}$. One end fixed, other free: $L_e = 2L$. One end fixed, other pinned: $L_e = \frac{L}{\sqrt{2}} \approx 0.707L$.

Define slenderness ratio and critical (buckling) stress for a column.

Slenderness ratio $\lambda = \frac{L_e}{r}$, where $r = \sqrt{I/A}$ is the radius of gyration. Critical stress $\sigma_{cr} = \frac{\pi^2 E}{(L_e/r)^2}$. Euler's formula is valid only for long (slender) columns.

For combined direct (axial) and bending stress, write the resultant extreme-fibre stresses, and state the no-tension condition.

$\sigma = \frac{P}{A} \pm \frac{M\,y}{I}$. For no tension anywhere, the maximum bending stress must not exceed the direct stress: $\frac{M\,y}{I} \leq \frac{P}{A}$. For a rectangular section this gives the 'middle-third rule' (eccentricity $e \leq \frac{d}{6}$).

State the static indeterminacy formula for a plane frame (degree of static indeterminacy).

$D_s = (3m + r) - 3j - r_r$, where $m$ = members, $r$ = reaction components, $j$ = joints, $r_r$ = released reactions/condition equations. If $D_s = 0$ the frame is statically determinate; if $D_s > 0$ it is indeterminate.

Distinguish force methods from displacement (energy/stiffness) methods of structural analysis.

Force (flexibility) methods take redundant forces as unknowns and use compatibility conditions (e.g., method of consistent deformations, Castigliano's theorem, virtual work). Displacement (stiffness) methods take joint displacements/rotations as unknowns and use equilibrium (e.g., slope-deflection, moment distribution, matrix stiffness).

State Castigliano's second theorem for computing deflections in elastic structures.

The partial derivative of total strain energy $U$ with respect to a force $P$ gives the displacement at and in the direction of that force: $\delta = \frac{\partial U}{\partial P}$ (and $\theta = \frac{\partial U}{\partial M}$ for rotation).

Write the strain energy stored in a member due to bending and the unit-load (virtual work) deflection formula.

Bending strain energy $U = \int \frac{M^2}{2EI}\,dx$. Unit-load method deflection: $\delta = \int \frac{M\,m}{EI}\,dx$, where $M$ is the real moment and $m$ the moment due to a unit virtual load at the point of interest.

State the principle (method) of superposition and the condition under which it is valid.

The total response (stress, strain, deflection) due to several loads acting simultaneously equals the algebraic sum of the responses due to each load acting separately. It is valid only when the material is linearly elastic and deformations are small (linear, geometrically and materially).