SECTION A (2 × 10 = 20 Marks)
Attempt all questions in brief
(a) Load transfer mechanism for portal frames
In a portal frame, loads applied on the roof or beam are transferred to the columns through bending action. The beams mainly resist bending moments and shear forces, while the columns carry axial forces and bending moments. Finally, these loads are transferred to the foundations, which distribute them safely to the soil.
(b) Classification of trusses based on static determinacy
Trusses can be classified as statically determinate and statically indeterminate. In statically determinate trusses, member forces can be found using only equilibrium equations, whereas statically indeterminate trusses require additional compatibility conditions along with equilibrium equations.
(c) Compound truss
A compound truss is formed by combining two or more simple trusses in such a way that the overall structure acts as a single truss. It is used when spans are large and simple trusses alone are not sufficient.
(d) Methods of analysis of trusses
Trusses can be analyzed using methods such as the method of joints, method of sections, method of tension coefficients, and graphical methods. These methods help determine forces in individual members.
(e) Maxwell’s reciprocal theorem
Maxwell’s reciprocal theorem states that the displacement at point A due to a unit load applied at point B is equal to the displacement at point B due to the same unit load applied at point A, provided the structure is linearly elastic.
(f) Applications of Castigliano’s first theorem
Castigliano’s first theorem is used to calculate deflections and rotations in structures by differentiating strain energy with respect to applied loads. It is especially useful for statically indeterminate structures.
(g) Influence Line Diagram (ILD)
An influence line diagram shows the variation of a response function such as reaction, shear force, or bending moment at a specific point in a structure due to a unit load moving across the structure.
(h) Rolling loads in practice
Rolling loads are commonly encountered in bridges, crane girders, highways, and railway tracks, where loads move from one position to another.
(i) Difference between beam and arch
A beam resists loads mainly through bending action, whereas an arch transfers loads primarily through compressive forces along its curved shape, reducing bending moments.
(j) Preference between circular and parabolic arch
A parabolic arch is preferable for carrying uniformly distributed loads because it produces zero bending moment under such loading, making it more efficient than a circular arch.
SECTION B (10 × 3 = 30 Marks)
(a) Statically determinate and statically indeterminate structures
Statically determinate structures can be analyzed using only equations of equilibrium, and their internal forces are independent of material properties. Statically indeterminate structures require additional equations based on deformation compatibility, and their analysis depends on material properties and geometry.
(b) Method of tension coefficient
The method of tension coefficient is used to analyze trusses by assuming forces in members and relating them through geometry and equilibrium conditions. It is especially useful for space trusses.
(c) Application of Maxwell’s reciprocal theorem
Maxwell’s reciprocal theorem is applied to find deflections at specific points in beams, trusses, and frames. It simplifies calculations by interchanging load and displacement points.
(d) Influence line for shear force at a section
The influence line for shear force at a given section of a beam represents how shear force varies as a unit load moves across the beam. It helps determine maximum shear due to moving loads.
(e) Three-hinged parabolic arch under loads
In a three-hinged parabolic arch, internal forces such as bending moment, thrust, and radial shear are calculated by resolving forces at the section considering equilibrium. The presence of a hinge at the crown makes the structure statically determinate.
SECTION C
Q3 (a) Cable under uniformly distributed load
A cable subjected to uniformly distributed load takes a parabolic shape. The maximum tension occurs at the supports and is calculated using horizontal thrust and vertical reaction components. The inclination at supports is obtained from the slope of the cable. Forces transmitted to the pier are resolved using equilibrium principles, and bending moment in the pier is calculated using the applied forces and pier height.
Q4 (a) Method of joints
The method of joints involves analyzing each joint of a truss separately by applying equilibrium equations. Starting from a joint with only two unknowns, member forces are calculated progressively until all forces are determined.
Q5 (a) Unit load method
The unit load method is used to calculate deflection by applying a virtual unit load at the point where deflection is required. The deflection is obtained by integrating the product of real bending moment and virtual bending moment over the length of the beam.
Q6 (a) Influence line diagrams for beams
Influence line diagrams are constructed for reactions, shear force, and bending moment by placing a unit load at different positions on the beam. These diagrams help determine maximum effects due to moving loads.
Q7 (b) Classification of arches
Arches are classified based on material, shape, and structural system. Based on structural system, arches can be three-hinged, two-hinged, or fixed. A three-hinged arch is statically determinate, while a two-hinged arch is statically indeterminate but more stable.
