Structural Analysis: cram pack

Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.

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1Formulas and methods

Beam support reactions

Equilibrium
ΣFx = 0, ΣFy = 0, ΣM = 0
Uniform load
resultant w·L at the middle of the load
Moment
M = F·d

Method

  1. Draw a free-body diagram and replace each support by its reactions.
  2. Replace every distributed load by a single force at its centroid.
  3. Take moments about one support to find the reaction at the other.
  4. Use ΣFy = 0 to find the remaining reaction, and check with a second moment equation.

Shear force & bending moment

Shear and moment
V = ΣF on one side, M = ΣF·d on one side
Bending stress
σ = M·y/I
Rectangle
I = bh³/12
Section modulus
W = I/ymax
Simply supported, uniform load
Mmax = wL²/8

Method

  1. Find the support reactions from equilibrium of the whole beam.
  2. Cut the beam at the section you need and keep one side.
  3. Shear force: sum of the vertical forces on that side. Bending moment: sum of their moments about the cut.
  4. For the maximum moment, find where the shear force is zero.

Truss: method of joints

Method of joints
ΣFx = 0, ΣFy = 0 at each joint
Method of sections
ΣM = 0 about a cut point
Member force
positive = tension, negative = compression

Method

  1. Find the support reactions for the whole truss.
  2. Method of joints: start at a joint with at most two unknown member forces and use ΣFx = 0 and ΣFy = 0.
  3. Method of sections: cut through the member you need and take moments about a point where the other unknown members meet.
  4. Assume tension; a negative answer means compression.

Beam deflection

Point load at midspan
δ = PL³/(48EI)
Uniform load, simply supported
δ = 5wL⁴/(384EI)
Cantilever, end load
δ = PL³/(3EI)
Cantilever, uniform load
δ = wL⁴/(8EI)
Superposition
δ = δ1 + δ2

Column buckling

Euler load
Pcr = π²EI/(KL)²
Effective length factor
K = 1 pinned–pinned, 0.5 fixed–fixed, 0.7 fixed–pinned, 2 fixed–free
Critical stress
σcr = Pcr/A
Slenderness
λ = KL/r, r = √(I/A)

2Practice problems

1.Beam support reactionsMedium
P1​ = 20 kNP2​ = 35 kNABRA​ = ?RB​1 m4 m1 m

The simply supported beam has a pin at A and a roller at B. Find the vertical support reaction RA.

L = 6 mP1 = 20 kN at 1 mP2 = 35 kN at 5 m
RA = kN
2.Beam support reactionsHard
w = 12 kN/mP = 60 kNABRA​ = ?RB​5 m3 m1 m1 m

The simply supported beam has a pin at A and a roller at B. Find the vertical support reaction RA.

L = 10 mP = 60 kN at 9 mw = 12 kN/m from 5 to 8 m
RA = kN
3.Shear force & bending momentMedium
w = 8 kN/mABsection3.5 m1.5 m

Find the bending moment Mx at the section x = 3.5 m from support A. Use sagging moments positive and upward shear on the left face positive.

L = 5 mw = 8 kN/m from 0 to 5 mx = 3.5 m
Mx = kN·m
4.Shear force & bending momentHard
w = 3 kN/mP = 45 kNABsection7 m3 m

Find the bending moment Mx at the section x = 7 m from support A. Use sagging moments positive and upward shear on the left face positive.

L = 10 mP = 45 kN at 7 mw = 3 kN/m from 0 to 10 mx = 7 m
Mx = kN·m
5.Truss: method of jointsMedium
ABBCACACB30 kN8 m5 m4 m

Use the method of joints to find the force FAC in member AC. Give tension as positive and compression as negative.

span = 8 mheight = 4 mB is 5 m from Avertical load = 30 kN
FAC = kN
6.Truss: method of jointsHard
ABBCACACB25 kN15 kN8 m7 m3 m

Use the method of joints to find the force FAC in member AC. Give tension as positive and compression as negative.

span = 8 mheight = 3 mB is 7 m from Avertical load = 25 kNhorizontal load = 15 kN ←
FAC = kN
7.Beam deflectionMedium
w = 10 kN/mP = 10 kNABδ2 m

A cantilever carries a uniformly distributed load w over its whole length and a point load P at its free end. Using superposition, find the deflection δ at the free end.

P = 10 kNw = 10 kN/mL = 2 mE = 70 GPa (aluminium)I = 57 × 10⁶ mm⁴
δ = mm
8.Beam deflectionHard
P = 27 kNABδ2.5 m1.5 m

A simply supported beam carries a point load P at b = 1.5 m from support B. Find the maximum deflection δmax of the beam.

P = 27 kNb = 1.5 mL = 4 mE = 70 GPa (aluminium)I = 12 × 10⁶ mm⁴
δmax = mm
9.Column bucklingMedium
PL = 2.5 md = 30 mmcross-section

A column of length L made of aluminium is fixed at the base and pinned at the top. Using Euler’s formula, find its critical buckling load Pcr.

L = 2.5 mE = 70 GPa (aluminium)d = 30 mm
Pcr = kN
10.Column bucklingHard
P = 100 kNL = 2.5 md = ?cross-section

A solid round column of length L made of aluminium is fixed at the base and free at the top. It must carry 100 kN with a factor of safety of 2.5 against buckling. Find the smallest diameter d.

P = 100 kNn = 2.5L = 2.5 mE = 70 GPa (aluminium)
d = mm

3Solutions

1.Beam support reactions

Answer: RA = 22.5 kN

Static equilibrium: a body at rest has ΣFx = 0, ΣFy = 0 and ΣM = 0 about any point. Taking moments about one support removes that support’s reaction from the equation.

P1​ = 20 kNP2​ = 35 kNABRA​RB​1 m4 m1 m

Free-body diagram

Replace the supports with reaction forces. A roller gives one vertical reaction; with only vertical loads the pin gives one vertical reaction too.

Equilibrium

Moments about A, counter-clockwise positive. Then vertical force balance.

ΣMA = 0: RB·6 = 20·1 + 35·5
RB = 195/6 = 32.5 kN
ΣFy = 0: RA = 55 − 32.5 = 22.5 kN

Answer

RA = 22.5 kN
2.Beam support reactions

Answer: RA = 18.6 kN

Static equilibrium: a body at rest has ΣFx = 0, ΣFy = 0 and ΣM = 0 about any point. Taking moments about one support removes that support’s reaction from the equation.

w = 12 kN/mP = 60 kNABRA​RB​5 m3 m1 m1 m

Free-body diagram

Replace the supports with reaction forces. A roller gives one vertical reaction; with only vertical loads the pin gives one vertical reaction too.

Equilibrium

Moments about A, counter-clockwise positive. Then vertical force balance.

UDL resultant: W = w·ℓ = 12·3 = 36 kN, acting at x̄ = 6.5 m
ΣMA = 0: RB·10 = 60·9 + 12·3·6.5
RB = 774/10 = 77.4 kN
ΣFy = 0: RA = 96 − 77.4 = 18.6 kN

Answer

RA = 18.6 kN
3.Shear force & bending moment

Answer: Mx = 21 kN·m

Method of sections: cut the beam where you want the internal forces and keep one side. The shear force V balances the vertical forces on that side, and the bending moment M balances their moments about the cut.

Support reactions

Equilibrium of the whole beam.

UDL resultant: W = w·ℓ = 8·5 = 40 kN, acting at x̄ = 2.5 m
ΣMA = 0: RB·5 = 8·5·2.5
RB = 100/5 = 20 kN
ΣFy = 0: RA = 40 − 20 = 20 kN
w = 8 kN/mABRA​ = 20RB​ = 20section3.5 m1.5 m

Cut at x = 3.5 m and keep the left part

Only forces left of the cut appear. A UDL portion acts as one force at the middle of that portion.

M = RA·x − (each load left of the cut × its distance to the cut)
Mx = 20·3.5 − 8·3.5·1.75 = 21 kN·m

Answer

Mx = 21 kN·m
4.Shear force & bending moment

Answer: Mx = 126 kN·m

Method of sections: cut the beam where you want the internal forces and keep one side. The shear force V balances the vertical forces on that side, and the bending moment M balances their moments about the cut.

Support reactions

Equilibrium of the whole beam.

UDL resultant: W = w·ℓ = 3·10 = 30 kN, acting at x̄ = 5 m
ΣMA = 0: RB·10 = 45·7 + 3·10·5
RB = 465/10 = 46.5 kN
ΣFy = 0: RA = 75 − 46.5 = 28.5 kN
w = 3 kN/mP = 45 kNABRA​ = 28.5RB​ = 46.5section7 m3 m

Cut at x = 7 m and keep the left part

Only forces left of the cut appear. A UDL portion acts as one force at the middle of that portion.

M = RA·x − (each load left of the cut × its distance to the cut)
Mx = 28.5·7 − 3·7·3.5 = 126 kN·m

Answer

Mx = 126 kN·m
5.Truss: method of joints

Answer: FAC = 14.06 kN

Method of joints: every pin of a truss is in equilibrium, so ΣFx = 0 and ΣFy = 0 at each joint. Start at a joint with at most two unknown member forces. A negative result means the member is in compression.

ABBCACACB30 kNAy​Cy​

Support reactions

Equilibrium of the whole truss. Moments about A, counter-clockwise positive.

ΣMA = 0: Cy·8 = 30·5 → Cy = 18.75 kN
ΣFy = 0: Ay = 30 − 18.75 = 11.25 kN

Joint C

The roller gives only a vertical reaction, so joint C has just two unknown member forces. Assume both in tension.

sin γ = 4/5 = 0.8, cos γ = 3/5 = 0.6
ΣFy = 0: Cy + FBC·sin γ = 0 → FBC = −18.75/0.8 = −23.44 kN
ΣFx = 0: −FAC − FBC·cos γ = 0 → FAC = −(−23.44)·0.6 = 14.06 kN

Answer

FAC = 14.06 kN (tension)
6.Truss: method of joints

Answer: FAC = 5.417 kN

Method of joints: every pin of a truss is in equilibrium, so ΣFx = 0 and ΣFy = 0 at each joint. Start at a joint with at most two unknown member forces. A negative result means the member is in compression.

ABBCACACB25 kN15 kNAy​Cy​Ax​

Support reactions

Equilibrium of the whole truss. Moments about A, counter-clockwise positive.

ΣMA = 0: Cy·8 = 25·7 − 15·3 → Cy = 16.25 kN
ΣFy = 0: Ay = 25 − 16.25 = 8.75 kN
ΣFx = 0: Ax = 15 kN

Joint C

The roller gives only a vertical reaction, so joint C has just two unknown member forces. Assume both in tension.

sin γ = 3/3.162 = 0.9487, cos γ = 1/3.162 = 0.3162
ΣFy = 0: Cy + FBC·sin γ = 0 → FBC = −16.25/0.9487 = −17.13 kN
ΣFx = 0: −FAC − FBC·cos γ = 0 → FAC = −(−17.13)·0.3162 = 5.417 kN

Answer

FAC = 5.417 kN (tension)
7.Beam deflection

Answer: δ = 11.7 mm

Beam deflection: deflections come from integrating the curvature M/(EI) twice. Standard cases are tabulated, for example PL³/(3EI) for a cantilever with an end load and 5wL⁴/(384EI) for a simply supported beam under a full uniform load. Deflections from separate loads add (superposition).

Consistent units

Work in N and mm: P in N, w in N/mm (kN/m = N/mm), L in mm, E in N/mm² (GPa × 1000), I in mm⁴.

P = 10000 N, w = 10 N/mm, L = 2000 mm
EI = 3.99 × 10¹² N·mm²

Each load on its own

δP = PL³/(3EI) = 10000·2000³/(3·3.99 × 10¹²) = 6.683 mm
δw = wL⁴/(8EI) = 10·2000⁴/(8·3.99 × 10¹²) = 5.013 mm

Add them

δ = 6.683 + 5.013 = 11.7 mm

Answer

δ = 11.7 mm
8.Beam deflection

Answer: δmax = 39.42 mm

Beam deflection: deflections come from integrating the curvature M/(EI) twice. Standard cases are tabulated, for example PL³/(3EI) for a cantilever with an end load and 5wL⁴/(384EI) for a simply supported beam under a full uniform load. Deflections from separate loads add (superposition).

Consistent units

Work in N and mm: P in N, w in N/mm (kN/m = N/mm), L in mm, E in N/mm² (GPa × 1000), I in mm⁴.

P = 27000 N, b = 1500 mm, L = 4000 mm
EI = 8.4 × 10¹¹ N·mm²

Position of the maximum

Setting the slope in the longer segment to zero gives the position.

x = √((L² − b²)/3) = √((4000² − 1500²)/3) = 2141 mm

Maximum deflection

δmax = Pb(L² − b²)3/2/(9√3·EIL) = 39.42 mm

Answer

δmax = 39.42 mm
9.Column buckling

Answer: Pcr = 8.97 kN

Euler buckling: a slender column becomes unstable at Pcr = π²EI/(KL)², where KL is the effective length: K = 1 for pinned–pinned, 2 for fixed–free, 0.7 for fixed–pinned and 0.5 for fixed–fixed. It buckles about the axis with the smallest I. The formula holds for slender columns, whose critical stress stays below the yield stress.

Smallest second moment of area

A round section has the same I about every axis.

I = πd⁴/64 = π·30⁴/64 = 39761 mm⁴

Effective length

KL = 0.7·2500 = 1750 mm

Critical load

Pcr = π²EI/(KL)² = π²·70000·39761/1750² = 8970 N = 8.97 kN

Answer

Pcr = 8.97 kN
10.Column buckling

Answer: d = 116.5 mm

Euler buckling: a slender column becomes unstable at Pcr = π²EI/(KL)², where KL is the effective length: K = 1 for pinned–pinned, 2 for fixed–free, 0.7 for fixed–pinned and 0.5 for fixed–fixed. It buckles about the axis with the smallest I. The formula holds for slender columns, whose critical stress stays below the yield stress.

Required critical load

Pcr = n·P = 2.5·100000 = 250000 N
KL = 2·2500 = 5000 mm

Solve for the diameter

d = (64·n·P·(KL)²/(π³E))^(1/4) = (64·250000·5000²/(π³·70000))^(1/4) = 116.5 mm

Answer

d = 116.5 mm

Check: the slenderness KL/r is 171.7, so the column is slender and Euler applies.

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