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1Formulas and methods
Vectors: dot & cross product
Magnitude
|a| = √(ax² + ay² + az²)
Unit vector
û = a/|a|
Components from an angle
Fx = F·cos θ, Fy = F·sin θ
Dot product
a·b = axbx + ayby + azbz = |a||b|cos θ
Scalar projection of a onto b
a·b/|b|
Work
W = F·d
Cross product
a × b = (aybz − azby, azbx − axbz, axby − aybx)
Areas
parallelogram |a × b|, triangle ½|a × b|
Moment about O
MO = r × F, r = A − O
Volume of a parallelepiped
V = |a·(b × c)|
Method
Write every vector in components; for a force given by a magnitude and an angle, use F·cos θ and F·sin θ.
For lengths and directions use the magnitude √(x² + y² + z²) and the unit vector a/|a|.
Use the dot product for angles, projections, perpendicularity and work: a·b = |a||b|cos θ.
Use the cross product for perpendicular vectors, areas and moments: write the three components carefully, keeping the order a × b.
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
Draw a free-body diagram and replace each support by its reactions.
Replace every distributed load by a single force at its centroid.
Take moments about one support to find the reaction at the other.
Use ΣFy = 0 to find the remaining reaction, and check with a second moment equation.
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
Find the support reactions for the whole truss.
Method of joints: start at a joint with at most two unknown member forces and use ΣFx = 0 and ΣFy = 0.
Method of sections: cut through the member you need and take moments about a point where the other unknown members meet.
Assume tension; a negative answer means compression.
Frames & machines
Equilibrium of each member
ΣFx = 0, ΣFy = 0, ΣM = 0
Two-force member
force along the line through its two pins
Pin force
equal and opposite on the two members it joins
Magnitude of a pin force
F = √(Fx² + Fy²)
Lever, pliers, bell crank
Fin·a = Fout·b
Toggle
Flink = F/(2 sin θ), Q = F/(2 tan θ)
Uniform load
resultant w·L at the middle of the load
Method
Draw the whole structure and find the support reactions if there are no more than three unknowns.
Spot the two-force members: loaded only at two pins, their force acts along the line between the pins.
Take the frame apart and draw every member on its own, with equal and opposite pin forces where members meet.
For each member, take moments about a pin with unknown forces, then use ΣFx = 0 and ΣFy = 0; combine components into a magnitude at the end.
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
Find the support reactions from equilibrium of the whole beam.
Cut the beam at the section you need and keep one side.
Shear force: sum of the vertical forces on that side. Bending moment: sum of their moments about the cut.
For the maximum moment, find where the shear force is zero.
Centroid & second moment of area
Centroid
ȳ = ΣAiyi/ΣAi
Rectangle
I = bh³/12
Parallel-axis theorem
I = Ic + A·d²
Composite section
I = Σ(Ic,i + Aidi²)
2Practice problems
1.Vectors: dot & cross productMedium
Find the scalar projection compb a of a = (4, 7, 0) onto b = (−5, −1, −6), that is, the component of a in the direction of b.
a = (4, 7, 0)b = (−5, −1, −6)
compb a =
2.Vectors: dot & cross productHard
The unit vector n̂ is perpendicular to both a = (−3, 4, 3) and b = (−1, 2, 5) and has a positive z-component. Find its x-component n̂x.
a = (−3, 4, 3)b = (−1, 2, 5)n̂z > 0
n̂x =
3.Beam support reactionsMedium
The cantilever beam is fixed to the wall at A. It carries a uniformly distributed load w from the wall to 2.5 m and a point load P at 4 m from the wall. Find the magnitude of the reaction moment MA at the wall.
L = 5 mP = 20 kN at 4 mw = 8 kN/m from 0 to 2.5 m
MA = kN·m
4.Beam support reactionsHard
The simply supported beam has a pin at A and a roller at B. A load P acts at 7 m from A, inclined at 60° to the horizontal, and a vertical load Q acts at 1 m from A. Find the magnitude of the total reaction force RA at the pin A.
L = 8 mP = 55 kN at 7 m, θ = 60°Q = 70 kN at 1 m
RA = kN
5.Truss: method of jointsMedium
Use the method of joints to find the force FAB in member AB. Give tension as positive and compression as negative.
span = 8 mheight = 4 mB is 5 m from Avertical load = 50 kN
FAB = kN
6.Truss: method of jointsHard
Use the method of joints to find the force FAB in member AB. Give tension as positive and compression as negative.
span = 5 mheight = 3 mB is 4 m from Avertical load = 30 kNhorizontal load = 5 kN →
FAB = kN
7.Frames & machinesMedium
The frame shown has pins at A, B and C; A and C are fixed to the ground 3 m apart, and B is 2 m above the ground and 0.9 m to the right of A. A vertical load of 10 kN acts on member AB at K, 0.675 m (horizontally) from A. Find the force FBC in member BC.
P = 10 kNL = 3 mH = 2 mhorizontal distance A to B = 0.9 mhorizontal distance A to K = 0.675 m
FBC = kN
8.Frames & machinesHard
The frame shown has pins at A, B and C; A and C are fixed to the ground 5 m apart, and B is 2 m high and 2.25 m to the right of A. Member AB carries a vertical load of 20 kN at K, 1.125 m (horizontally) from A, and a horizontal load of 5 kN to the right acts at B on member AB. Find the size of the horizontal reaction Ax at A.
P = 20 kNQ = 5 kNL = 5 mH = 2 mhorizontal distance A to B = 2.25 mhorizontal distance A to K = 1.125 m
Ax = kN
9.Shear force & bending momentMedium
Find the bending moment Mx at the section x = 2 m from support A. Use sagging moments positive and upward shear on the left face positive.
L = 6 mw = 5 kN/m from 0 to 6 mx = 2 m
Mx = kN·m
10.Shear force & bending momentHard
The simply supported beam carries a uniformly distributed load over part of its span. Find the maximum bending moment Mmax in the beam.
L = 4 mw = 16 kN/m from 1 to 4 m
Mmax = kN·m
11.Centroid & second moment of areaMedium
A 150 mm × 150 mm plate has a circular hole of diameter 30 mm on its vertical centre line, with its centre 45 mm above the bottom. Find the second moment of area Ix about the horizontal axis through the centroid.
b = 150 mmh = 150 mmd = 30 mmhole centre: y = 45 mm
Ix = × 10⁶ mm⁴
12.Centroid & second moment of areaHard
A 150 mm × 300 mm plate has a rectangular cut-out 50 mm wide and 20 mm high, centred on the vertical centre line with its centre 105 mm above the bottom. Find the second moment of area Ix about the horizontal axis through the centroid.
b = 150 mmh = 300 mmcut-out: 50 × 20 mmhole centre: y = 105 mm
Ix = × 10⁶ mm⁴
3Solutions
1.Vectors: dot & cross product
Answer: compb a = −3.429
Dot product:a·b = axbx + ayby + azbz = |a|·|b|·cos θ, a scalar. It gives the angle between two vectors, cos θ = a·b/(|a|·|b|), and it is zero when they are perpendicular. The scalar projection of a onto b is a·b/|b|, and a constant force F moving through a displacement d does work W = F·d.
Dot product
a·b = 4·(−5) + 7·(−1) + 0·(−6) = −27
Length of b
|b| = √((−5)² + (−1)² + (−6)²) = 7.874
Scalar projection
compb a = a·b/|b| = −27/7.874 = −3.429
Answer
compb a = −3.429
Negative: a has a component pointing against b.
2.Vectors: dot & cross product
Answer: n̂x = −0.7548
Cross product:a × b = (aybz − azby, azbx − axbz, axby − aybx) is perpendicular to both vectors, and |a × b| = |a|·|b|·sin θ is the area of the parallelogram they span (the triangle has half of it). Order matters: b × a = −(a × b). The moment of a force F applied at point A about point O is MO = r × F, with r = A − O.
Cross product
(a × b)x = aybz − azby = 4·5 − 3·2 = 14
(a × b)y = azbx − axbz = 3·(−1) − (−3)·5 = 12
(a × b)z = axby − aybx = (−3)·2 − 4·(−1) = −2
Normalise
The z-component of a × b is negative, so use −(a × b).
Cantilever equilibrium: a fixed support can push up and also resist rotation, so it gives a vertical reaction RA and a reaction moment MA. ΣFy = 0 gives RA, and moments about the wall give MA = Σ(load × distance from the wall). A distributed load acts as its resultant at the centroid of the load diagram.
Resultant of the distributed load
W = 8·2.5 = 20 kN, x̄ = 1.25 m
Moment about the wall
MA = P·a + W·x̄ = 20·4 + 20·1.25 = 105 kN·m
Answer
MA = 105 kN·m
4.Beam support reactions
Answer: RA = 72.61 kN
Static equilibrium: a body at rest has ΣFx = 0, ΣFy = 0 and ΣM = 0 about any point. Replace each distributed load by its resultant (its area) acting at the centroid of the load diagram, and take moments about a support to remove that support’s reaction from the equation.
Components of P
Pv = 55·sin 60° = 47.63 kN
Ph = 55·cos 60° = 27.5 kN
Reactions
ΣMA = 0: RB·8 = 47.63·7 + 70·1 → RB = 50.43 kN
Ay = 47.63 + 70 − 50.43 = 67.2 kN
Ax = Ph = 27.5 kN
Resultant at A
RA = √(27.5² + 67.2²) = 72.61 kN
Answer
RA = 72.61 kN
5.Truss: method of joints
Answer: FAB = −30.01 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.
Support reactions
Equilibrium of the whole truss. Moments about A, counter-clockwise positive.
ΣMA = 0: Cy·8 = 50·5 → Cy = 31.25 kN
ΣFy = 0: Ay = 50 − 31.25 = 18.75 kN
Joint A
Assume every member is in tension (pulling away from the joint). Only AB has a vertical component at A.
sin α = 4/6.403 = 0.6247
ΣFy = 0: Ay + FAB·sin α = 0
FAB = −18.75/0.6247 = −30.01 kN
Answer
FAB = −30.01 kN (compression)
6.Truss: method of joints
Answer: FAB = −5 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.
Support reactions
Equilibrium of the whole truss. Moments about A, counter-clockwise positive.
ΣMA = 0: Cy·5 = 30·4 + 5·3 → Cy = 27 kN
ΣFy = 0: Ay = 30 − 27 = 3 kN
ΣFx = 0: Ax = −5 kN
Joint A
Assume every member is in tension (pulling away from the joint). Only AB has a vertical component at A.
sin α = 3/5 = 0.6
ΣFy = 0: Ay + FAB·sin α = 0
FAB = −3/0.6 = −5 kN
Answer
FAB = −5 kN (compression)
7.Frames & machines
Answer: FBC = 3.262 kN
Frames and machines: unlike a truss, a frame contains at least one member that carries forces at more than two points, or a load between its pins. First look for two-force members: a member loaded only at two pins has its force along the line through them, so it adds one unknown, not two. Then use the whole structure for the support reactions when there are no more than three unknowns. Finally take the frame apart: draw each member separately, with the pin forces equal and opposite on the two members they connect, and write ΣFx = 0, ΣFy = 0 and ΣM = 0 for it. Taking moments about a pin removes both components of that pin force from the equation.
Two-force member BC
BC = √(2.1² + 2²) = 2.9 m
Whole frame: moments about A
The ground pushes on BC at C along CB; its horizontal component passes through A, only the vertical one has a moment.
FBC·(2/2.9)·3 = 10·0.675
FBC = 3.262 kN (compression)
Answer
FBC = 3.262 kN
8.Frames & machines
Answer: Ax = 3.938 kN
Frames and machines: unlike a truss, a frame contains at least one member that carries forces at more than two points, or a load between its pins. First look for two-force members: a member loaded only at two pins has its force along the line through them, so it adds one unknown, not two. Then use the whole structure for the support reactions when there are no more than three unknowns. Finally take the frame apart: draw each member separately, with the pin forces equal and opposite on the two members they connect, and write ΣFx = 0, ΣFy = 0 and ΣM = 0 for it. Taking moments about a pin removes both components of that pin force from the equation.
Direction of BC
BC = 3.4 m, cos = 0.8087, sin = 0.5882
Whole frame: moments about A
FBC·0.5882·5 = 20·1.125 + 5·2
FBC = 11.05 kN
Horizontal balance
Ax + 5 − 11.05·0.8087 = 0
Ax = 3.938 kN → |Ax| = 3.938 kN
Answer
Ax = 3.938 kN
Ax points to the right.
9.Shear force & bending moment
Answer: Mx = 20 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 = 5·6 = 30 kN, acting at x̄ = 3 m
ΣMA = 0: RB·6 = 5·6·3
RB = 90/6 = 15 kN
ΣFy = 0: RA = 30 − 15 = 15 kN
Cut at x = 2 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 = 15·2 − 5·2·1 = 20 kN·m
Answer
Mx = 20 kN·m
10.Shear force & bending moment
Answer: Mmax = 28.13 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. Maximum moment: since dM/dx = V, the bending moment has its peak where the shear force passes through zero, either at a point load or inside a distributed load.
Support reactions
Equilibrium of the whole beam.
UDL resultant: W = w·ℓ = 16·3 = 48 kN, acting at x̄ = 2.5 m
ΣMA = 0: RB·4 = 16·3·2.5
RB = 120/4 = 30 kN
ΣFy = 0: RA = 48 − 30 = 18 kN
Zero shear
The shear force starts at RA, stays constant over any unloaded part and falls with slope w inside the load.
Sections with holes: treat a hole as a negative area. The centroid is ȳ = (A₁y₁ − A₂y₂)/(A₁ − A₂), with A₁, y₁ for the full plate and A₂, y₂ for the hole. For the second moment of area about the centroidal axis, use the parallel-axis theorem for each part and subtract the hole: I = (Ic1 + A₁d₁²) − (Ic2 + A₂d₂²), where d is the distance from each part's own centroid to the axis through ȳ. A rectangle has Ic = bh³/12 and a circle Ic = πd⁴/64.
Centroid
ȳ = (22500·75 − 706.9·45)/(22500 − 706.9) = 75.97 mm
Full plate about ȳ
150·150³/12 + 22500·−0.973² = 42.21 × 10⁶ mm⁴
Hole about ȳ
π·30⁴/64 + 706.9·−30.97² = 0.7179 × 10⁶ mm⁴
Subtract
Ix = 41.49 × 10⁶ mm⁴
Answer
Ix = 41.49 × 10⁶ mm⁴
12.Centroid & second moment of area
Answer: Ix = 335.4 × 10⁶ mm⁴
Sections with holes: treat a hole as a negative area. The centroid is ȳ = (A₁y₁ − A₂y₂)/(A₁ − A₂), with A₁, y₁ for the full plate and A₂, y₂ for the hole. For the second moment of area about the centroidal axis, use the parallel-axis theorem for each part and subtract the hole: I = (Ic1 + A₁d₁²) − (Ic2 + A₂d₂²), where d is the distance from each part's own centroid to the axis through ȳ. A rectangle has Ic = bh³/12 and a circle Ic = πd⁴/64.
Centroid
ȳ = (45000·150 − 1000·105)/(45000 − 1000) = 151 mm