Structural Engineering

Determinacy & Beam Forces

Static & kinematic indeterminacy, SF/BM diagrams and standard beam results — the analysis core of GATE CE.

PART 1

Topic Breakdown & Traps

The Engineering Principle

A structure is statically determinate when equilibrium alone fixes all reactions and internal forces. The degree of static indeterminacy counts the redundant restraints. Shear force is the algebraic sum of transverse forces on one side of a section; bending moment is the algebraic sum of their moments. BM is maximum where SF crosses zero.

The Core Formula Matrix

Static indeterminacy (beam/frame):

D_{s} = (3 m + r) - 3 n - r_{r}

(general); simple beam

\Rightarrow D_{s} = 0

.

**SSB, central point load

W

**:

M_{ma x} = \frac{W L}{4}

at midspan.

**SSB, full UDL

w

**:

M_{ma x} = \frac{w L ^{2}}{8}

, reactions

\frac{w L}{2}

.

Fixed beam, central load: end moment

= \frac{W L}{8}

.

Fixed beam, UDL: support moment

= \frac{w L ^{2}}{12}

, midspan

= \frac{w L ^{2}}{24}

The ‘IIT Traps’

⚠
BM at a free end / simple support is zero (no applied moment there).
⚠
Maximum BM at zero shear. Locate the point where SF changes sign.
⚠
Fixed-end vs simply-supported moments differ. UDL gives $w L^{2} /8$ (SS) but $w L^{2} /12$ (fixed support).

📚 Standard references

Elementary Structural Analysis — Wilbur & Norris
Basic Structural Analysis — C.S. Reddy

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · NAT

A simply supported beam of span

4 m

carries a central point load of

60 kN

. The maximum bending moment is _____ kN·m.

Q2MEDIUM2 Marks · NAT

A fixed beam of span

6 m

carries a UDL of

18 kN/m

. The magnitude of the support moment is _____ kN·m.

Q3HARD2 Marks · MCQ

A single-bay single-storey portal frame with both column bases fixed is statically indeterminate to degree: