MZS Civil Engineering Academy: July 2025

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यह सामग्री किसी भी मौलिकता का दावा नहीं करती है और इसे निर्धारित पाठ्यपुस्तकों के विकल्प के रूप में उपयोग नहीं किया जा सकता है। मैं उन विभिन्न ओपन स्रोतों और NPTEL/ SWAYAM पाठ्य सामग्री को स्वीकार करना चाहूंगा जिनसे व्याख्यान नोट तैयार किया गया था। जानकारी का स्वामित्व संबंधित लेखकों या संस्थानों के पास है जहां ओपन स्रोत की सामग्री तैयार की गई थी। इसके अलावा, यह दस्तावेज़ व्यावसायिक उद्देश्य के लिए उपयोग करने के लिए नहीं है और ब्लॉगस्पॉट के मालिक किसी भी मुद्दों, कानूनी या अन्यथा, के लिए जिम्मेदार नहीं हैं जो इस दस्तावेज़ के उपयोग से उत्पन्न होते हैं।

Course Name: Structural Mechanics [संरचनात्मक यांत्रिकी]

Course Code: MZSCEA-SM-05

Content Creator: Dr. Mohd. Zameeruddin

Course Content

Beam Deflections: Calculations of deflection for determinate beams by double integration, Macaulay’s method, moment area method, conjugate beam method, deflection by method of superposition

Energy Principles: Strain energy and strain energy density, strain energy in traction, shear, flexure and torsion - Castigliano's and Engessor's energy theorems, principle of virtual work, application of energy theorems for computing deflections in beams, Maxwell's reciprocal theorem, Williot Mohr diagrams

Method of Consistent Deformation: Different structural systems, concept of analysis, basic assumptions, indeterminacy, choice of unknowns, Castigliano's theorem

Moment Distribution Method: Analysis of continuous beams propped cantilevers, continuous beams - theorem of three moments - analysis of continuous beams settlement effects, thermal effect, Shear Force and Bending Moment diagrams for continuous beams, portal frames with and without sway.

Slope Deflection Method: Analysis of continuous beams, analysis of rigid frames, frames without sway and with sway, settlement effects, introduction to difficulties in frames with sloping legs and gabled frames

Course Outcomes and CO Mapping with BT Levels

Course Outcome statements

Course Id	Course Outcome Statements	Planned Activities
CO1	Identify determinate and indeterminate beams and frames.	-
CO 2	Calculate slopes and deflections at various locations for different type’s beams and frames.	Demonstrative experiments
CO 3	Apply energy principles for the analysis of determinate and indeterminate structures.	-
CO 4	Demonstrates the analysis of both sway and non-sway frame structures using slope deflection and moment distribution equations	Demonstrative experiments

CO Mapping with Blooms Taxonomy Levels

CO ID	Action	POs/PSOs	Knowledge	Condition	Criteria
CO1	Identify	PO1, PO2 and PSO2	Factual, Conceptual, Procedural, Meta-cognitive and practical constraints	Concept of stability and determinacy, conditions of equilibrium and degree of freedom	Determinate and indeterminate structures
CO 2	Apply	PO1, PO2 and PSO2	Conceptual, Procedural, Meta-cognitive and practical constraints	Internal and external work done and strain resilience	Different condition of loading and deformable capabilities
CO 3	Calculate	PO1, PO2 and PSO2	Conceptual, Procedural, Meta-cognitive and practical constraints	Relationship between curvature, slope and deflection	Various type of loadings and end conditions
CO 4	Demonstrates	PO1, PO2 and PSO2	Factual, Conceptual, Procedural, Meta-cognitive and practical constraints	Continuous beams and frames	End conditions, distribution factor and rotational factors

CO-PO Mapping

CO ID	PO 1	PO 2	PO 3	PO 4	PO 5	PO 6	PO 7	PO 8	PO 9	PO 10	PO 11	PO 12
CO 1	2	2	-	-	-	-	-	-	-	-	-	-
CO 2	2	2	-	-	-	-	-	-	-	-	-	-
CO 3	3	3	-	-	-	-	-	-	-	-	-	-
CO 4	3	3	-	-	-	-	-	-	-	-	-	-
Average	2.5	2.5	-	-	-	-	-	-	-	-	-	-

Definitions -Short Question and Answer

Stiffness:

It is defined as resistance of member to deformation when subjected to load. Stiffness may be defined as a ratio of unit force [F] causing a unit displacement [∆] or a ratio of a unit moment [M] causing a unit deformation [Ɵ]. Its SI unit is kN/m

Mathematically,

K = F/∆

K = M/Ɵ

There are two types of stiffness

Absolute Stiffness

It is expressed in terms of the material's Young's modulus (E), the moment of inertia of the cross-section (I), and the length of the member (L).

For a beam with one end hinged and other end fixed, absolute stiffness is 4EI/L

For a beam with both ends hinged, absolute stiffness is 3EI/L

For a beam with one end fixed other free, absolute stiffness is zero

Relative Stiffness

The relative stiffness of a member is the ratio of a member's stiffness to the total stiffness of all members connected to a joint.

Mathematically,

K_r = K₁/K₂

Where K_r is relative stiffness, K₁ and K₂ are stiffness of two different components or members respectively.

Relative stiffness is expressed in terms of I and L, omitting the constant E.

Relative stiffness may be defined as the ratio of moment of inertia to the length of that member K = I/L.

Note: Video Lecture Link - Stiffness

Distribution factor:

When a moment is applied at a joint to create rotation without causing the members to translate, the moment is distributed among all of the members joined at that joint in proportion to their stiffness. The ratio of summative stiffness at the joint to the member stiffness is referred as Distribution Factor (DF).

Mathematically,

DF_i = K_i/∑K

DF_i is the distribution factor of i^th joint

K_i is the stiffness of i^th joint

∑K = ∑K₁+K₂+K3+K₄ is summation of stiffness of all member meeting at a joint.

Video Lecture - Distribution Factor

Carry Over Factor [COF]

It may be defined as the ratio of moment developed or induced at one end due to a moment applied at another end.

For a beam with one end hinged and other end fixed, the carry over moment at fixed end is half the applied moment at the hinge end, hence COF is 0.50

For a beam with both ends hinged, the carry over moment will be of equal magnitude, hence COF is 1.0

Flexural Rigidity

The product of Young’s Modulus (E) and Moment of Inertia (I) is called Flexural Rigidity (EI) of Beams. The unit is N / mm².

Sway

Sway is the lateral movement of the joints in a portal frame due to the unsymmetric in dimensions, unsymmetric loads, unequal moments of inertia, and different end conditions.

Video Lecture - Carry over factor

Multiple Choice Questions and Answer

1. When external and internal actions of forces are considered as a function of any two directions (coordinate axes) then the structure is idealized as.

(a) 1-D structure

(b) 2-D Structure

(d) none of the above

Answer: (d) 2-D Structure

2. Following is not an example of 1-D Structure

(a) Rod

(b) Cable

(d) Slab

Answer: (d) Slab

3. Following is not an example of 2-D structure

(a) Slab

(b) Retaining wall

(d) Multi-stranded Cable

Answer: (d) Multi-stranded Cable

4. The product of Modulus of Elasticity [E] and Moment of Inertia [I] is said to be.

(a) Flexural Rigidity

(b) Modulus of Rigidity

(d) None of the above

Answer: (a) Flexural Rigidity

5. The rotation at ends and deflection of beam cannot be evaluated using,

(a) Double Integration Method

(b) Macaulay’s Method

(d) Moment Distribution Method

Answer: (d) Moment Distribution Method

Following is not a valid boundary condition for a simply supported beam subjected to point load.

(a) at x = 0; y = 0

(b) at x = L; y = 0

(d) at x = L/2; dy/dx = 0

Answer: (c) at x = 0; dy/dx = 0

The deflection of fixed beam is ------------------ times of the deflection of simply supported beam when subjected to point load at mid-span.

a) 1/2

b) 1/5

c) 1/4

d) 1

Answer: (c) 1/4

The deflection of fixed beam is ------------- times of the deflection of simply supported beam when subjected to uniform distributed load over the entire span

a) 1/3

b) 1/5

c) 1/4

d) 1/6

Answer: (b) 1/5

Question Bank

Enlist different methods to find deflection in the beam and explain any one method. [DBATU, Summer 2025]
A cantilever beam of span 5m is subjected to a concentrated load of 100 kN applied at the free end. Assume EI, the flexural rigidity of the beam is constant. Calculate the maximum deflection at the free end of the beam in terms of EI [DBATU, Summer 2025]
Explain the concept of static and kinematic indeterminacy with examples [DBATU, Summer 2025]
Determine the deflection at free end of a cantilever beam of length 5m subjected to a point load of 100 kN at the free end. Use the strain energy method. Assume uniform flexural rigidity as "EI". [DBATU, Summer 2025]
Determine the reactions at the propped end for a propped cantilever beam of span 10m subjected to a uniformly distributed load of 5 kN/m. Also find the bending moment at the fixed end. [DBATU, Summer 2025]
A fixed beam of span "L" is subjected to a point load "W" kN at mid span. Draw shear force diagram and bending moment diagram. Locate the point of contraflexure. Assume flexural rigidity, EI is constant. Use consistent deformation method. [DBATU, Summer 2025]
A fixed beam of span 10m is subjected to uniformly distributed load of magnitude 50 kN/m. throughout the span. Construct the shear force and bending moment diagram. Locate the point of contraflexure. Flexural rigidity EI is constant. Use consistent deformation method. [DBATU, Summer 2025]
State and explain theorem of three moments. [DBATU, Summer 2025]
Explain the terms: a) Carry over moment, b) Carry over factor and c) stiffness factor [DBATU, Summer 2025]
State and explain the assumptions made in the slope deflection method while developing this method. [DBATU, Summer 2025]
Develop slope deflection equations for two span continuous beam ABC having each span (AB & BC) of length 4m and subjected to a uniformly distributed load of 10 kN/m. Support A is hinged, and B and C are rollers. Assume constant EI for both the span. [DBATU, Summer 2025]
Analyze the two span continuous beam ABC by the slope deflection method, having both ends as fixed ones. Span AB is having 4m length, moment of inertia as I and subjected to central point load of 40 kN. Span BC is having 6m length and moment of inertia as 2I and is subjected to a uniformly distributed load of 20 kN/m over the entire span BC. Draw the bending moment diagram and elastic curve/deflected profile. [DBATU, Summer 2025]
Analyze the two span continues beam ABC as shown in figure below by the Moment Distribution method and draw the bending moment diagram and deflection profile. [DBATU, Summer 2025]