In the print window, choose “Save as PDF” to keep a copy on your device.
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 = (7, 0, 1) onto b = (4, −4, 2), that is, the component of a in the direction of b.
a = (7, 0, 1)b = (4, −4, 2)
compb a =
2.Vectors: dot & cross productHard
Find the area A of the parallelogram spanned by a = (−3, 0, −5) and b = (6, −5, 4).
a = (−3, 0, −5)b = (6, −5, 4)
A =
3.Beam support reactionsMedium
The simply supported beam has a pin at A and a roller at B. It carries a point load P at 4 m from A and a counter-clockwise couple M at 2.5 m from A. Find the vertical reaction RB, taking upward as positive.
L = 6 mP = 55 kN at 4 mM = 10 kN·m (counter-clockwise) at 2.5 m
RB = kN
4.Beam support reactionsHard
The simply supported beam has a pin at A and a roller at B. Find the vertical support reaction RB.
L = 4 mP = 45 kN at 2.5 mw = 4 kN/m from 0 to 2 m
RB = 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 = 5 mheight = 3 mB is 4 m from Avertical load = 35 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 = 4 mheight = 4 mB is 3 m from Avertical load = 50 kNhorizontal load = 25 kN ←
FAB = kN
7.Frames & machinesMedium
In the toggle clamp shown, two equal links AB and BC are pinned together at B, and each makes 5° with the line AC. A downward force of 500 N is applied at B. The slider C presses on a workpiece. Neglecting friction and weights, find the horizontal clamping force Q.
F = 500 Nθ = 5°
Q = N
8.Frames & machinesHard
A toggle press has two equal links AB and BC, each 100 mm long between pins. The joint B sits 12 mm above the line AC. Which downward force F at B produces a horizontal pressing force of 3000 N at the slider C? Neglect friction and weights.
Q = 3000 NAB = BC = 100 mmh = 12 mm
F = N
9.Shear force & bending momentMedium
The marks A to G split the beam into six equal parts. At which mark is the bending moment largest in size, sagging or hogging? Click it in the diagram.
40 kN @ 4 m4 kN/m, 0–6 m
1A (x = 0 m)
2B (x = 1 m)
3C (x = 2 m)
4D (x = 3 m)
5E (x = 4 m)
6F (x = 5 m)
7G (x = 6 m)
10.Shear force & bending momentHard
The marks A to G split the beam into six equal parts. At which mark is the bending moment largest in size, sagging or hogging? Click it in the diagram.
25 kN @ 0 m45 kN @ 6 m10 kN/m, 1–4 m
1A (x = 0 m)
2B (x = 1 m)
3C (x = 2 m)
4D (x = 3 m)
5E (x = 4 m)
6F (x = 5 m)
7G (x = 6 m)
11.Centroid & second moment of areaMedium
A circular tube has outside diameter D = 100 mm and inside diameter d = 68 mm. Find its radius of gyration r about the horizontal centroidal axis.
D = 100 mmd = 68 mm
r = mm
12.Centroid & second moment of areaHard
A 200 mm × 300 mm plate has a rectangular cut-out 60 mm wide and 40 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 = 200 mmh = 300 mmcut-out: 60 × 40 mmhole centre: y = 105 mm
Ix = × 10⁶ mm⁴
3Solutions
1.Vectors: dot & cross product
Answer: compb a = 5
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 = 7·4 + 0·(−4) + 1·2 = 30
Length of b
|b| = √(4² + (−4)² + 2²) = 6
Scalar projection
compb a = a·b/|b| = 30/6 = 5
Answer
compb a = 5
2.Vectors: dot & cross product
Answer: A = 34.26
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 = 0·4 − (−5)·(−5) = −25
(a × b)y = azbx − axbz = (−5)·6 − (−3)·4 = −18
(a × b)z = axby − aybx = (−3)·(−5) − 0·6 = 15
Area
A = |a × b| = √((−25)² + (−18)² + 15²) = 34.26
Answer
A = 34.26
3.Beam support reactions
Answer: RB = 35 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.
Moments about A
Counter-clockwise positive.
ΣMA = 0: RB·6 − 55·4 + 10 = 0
RB = 210/6 = 35 kN
Vertical equilibrium
RA = 55 − 35 = 20 kN
Answer
RB = 35 kN
A negative reaction acts downward.
4.Beam support reactions
Answer: RB = 30.13 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.
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 = 4·2 = 8 kN, acting at x̄ = 1 m
ΣMA = 0: RB·4 = 45·2.5 + 4·2·1
RB = 120.5/4 = 30.13 kN
ΣFy = 0: RA = 53 − 30.13 = 22.88 kN
Answer
RB = 30.13 kN
5.Truss: method of joints
Answer: FAB = −11.67 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 = 35·4 → Cy = 28 kN
ΣFy = 0: Ay = 35 − 28 = 7 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 = −7/0.6 = −11.67 kN
Answer
FAB = −11.67 kN (compression)
6.Truss: method of joints
Answer: FAB = −46.88 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·4 = 50·3 − 25·4 → Cy = 12.5 kN
ΣFy = 0: Ay = 50 − 12.5 = 37.5 kN
ΣFx = 0: Ax = 25 kN
Joint A
Assume every member is in tension (pulling away from the joint). Only AB has a vertical component at A.
sin α = 4/5 = 0.8
ΣFy = 0: Ay + FAB·sin α = 0
FAB = −37.5/0.8 = −46.88 kN
Answer
FAB = −46.88 kN (compression)
7.Frames & machines
Answer: Q = 2858 N
Machines: pliers, levers, linkages and clamps turn an input force into a larger or a differently directed output force. Draw one part as a free body and take moments about its pivot: the pin force drops out, and Fin·a = Fout·b gives the mechanical advantage a/b. The pin force then follows from ΣF = 0 on the same part. In a toggle, two links meet at a shallow angle θ; each link is a two-force member, so a small force F at the joint creates large link forces F/(2 sin θ) and a clamping force F/(2 tan θ).
Equilibrium of pin B
2·Flink·sin 5° = 500
Flink = 2868 N
Clamping force
Q = Flink·cos 5° = 500/(2·tan 5°) = 2858 N
Answer
Q = 2858 N
The force ratio Q/F = 5.715 grows without limit as the links straighten.
8.Frames & machines
Answer: F = 725.2 N
Machines: pliers, levers, linkages and clamps turn an input force into a larger or a differently directed output force. Draw one part as a free body and take moments about its pivot: the pin force drops out, and Fin·a = Fout·b gives the mechanical advantage a/b. The pin force then follows from ΣF = 0 on the same part. In a toggle, two links meet at a shallow angle θ; each link is a two-force member, so a small force F at the joint creates large link forces F/(2 sin θ) and a clamping force F/(2 tan θ).
Link angle
sin θ = 12/100 = 0.12, θ = 6.892°
Link force
Flink = Q/cos θ = 3022 N
Force at B
F = 2·Flink·sin θ = 2·3000·tan θ = 725.2 N
Answer
F = 725.2 N
9.Shear force & bending moment
Answer: E (x = 4 m)
Largest bending moment: M changes slope wherever the shear force changes, so its peaks sit where the shear force crosses zero: under a point load, inside a distributed load, or over a support of an overhang. At a fixed support of a cantilever the moment is largest at the wall. Compare the size of M at those places, sagging or hogging.
Support reactions
Take moments about one support, then use the vertical force balance.
R1 = 25.33 kN, R2 = 38.67 kN
Bending moment at every mark
Sum the moments of all forces on one side of the cut. Sagging is positive, hogging negative.
MA = 0, MB = 23.33, MC = 42.67, MD = 58, ME = 69.33 kN·m, MF = 36.67, MG = 0
Answer
Mark E: ME = 69.33 kN·m
10.Shear force & bending moment
Answer: E (x = 4 m)
Largest bending moment: M changes slope wherever the shear force changes, so its peaks sit where the shear force crosses zero: under a point load, inside a distributed load, or over a support of an overhang. At a fixed support of a cantilever the moment is largest at the wall. Compare the size of M at those places, sagging or hogging.
Support reactions
Take moments about one support, then use the vertical force balance.
R1 = 18.33 kN, R2 = 81.67 kN
Bending moment at every mark
Sum the moments of all forces on one side of the cut. Sagging is positive, hogging negative.
MA = 0, MB = −25, MC = −36.67, MD = −58.33, ME = −90 kN·m, MF = −45, MG = 0
Answer
Mark E: ME = −90 kN·m
11.Centroid & second moment of area
Answer: r = 30.23 mm
Second moment of area: for a rectangle about its own centroidal axis I = bh³/12, and about its base bh³/3 (parallel-axis theorem I = Ic + A·d²). A hole is handled by subtracting its I, as long as both parts share the same axis. The elastic section modulus is S = I/c, where c is the distance to the extreme fibre. The radius of gyration r = √(I/A) is the distance at which the whole area would give the same I; it is used in column buckling.
Net area and second moment of area
A = π(D² − d²)/4 = 4222 mm²
I = π(D⁴ − d⁴)/64 = 3.859 × 10⁶ mm⁴
Radius of gyration
r = √(I/A) = √(3.859 × 10⁶/4222) = 30.23 mm
Answer
r = 30.23 mm
12.Centroid & second moment of area
Answer: Ix = 444.6 × 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 = (60000·150 − 2400·105)/(60000 − 2400) = 151.9 mm