Shear Transfer at the Interface of Reinforced Concrete Members

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February 18, 2024

Table of Contents

The efficacy of shear transfer at concrete-to-concrete interfaces is very paramount to the structural integrity of numerous reinforced concrete constructions. This is very important due to the need for shear stress transfer within reinforced concrete structures especially at horizontal construction joints. These joints become necessary due to the impracticality of single-pour of concrete in most construction works or the inherent requirements of staged construction sequences.

A prime illustration of this lies in the horizontal plane of interaction between precast concrete girders and cast-in-place concrete bridge decks. The composite behaviour exhibited by the girder and deck, which ultimately dictates the bridge’s stiffness and strength, is contingent upon the interface’s capacity to effectively transmit shear forces. In essence, the transfer of shear forces from the deck to the girders plays a crucial role in determining the structure’s load-carrying capacity.

Mechanism of Shear Transfer

The mechanisms governing shear stress transfer can be broadly categorized into three main contributors:

1. Interlock between roughened surfaces (shear – friction): Intentional surface irregularities, often achieved through sandblasting or texturing, create mechanical interlock between the concrete layers. This interlock resists relative movement under shear, contributing to stress transfer.

2. Dowel action of reinforcement: Steel bars embedded in the concrete, particularly shear connectors like headed studs, act as dowels traversing the interface. When subjected to shear, these dowels experience tension and compression, contributing to interface resistance.

3. Adhesion: The inherent bond between the concrete layers, influenced by factors like material properties, curing conditions, and surface cleanliness, also plays a role in shear transfer.

The critical mechanism of shear transfer across concrete-to-concrete interfaces in reinforced concrete (RC) structures can be elucidated through the saw-tooth model (Figure 1). This model visualizes the interaction between concrete surfaces under shear force.

**Figure 1:** Interface Shear Transfer, saw-tooth model

When subjected to shear, a horizontal displacement (h) occurs between the concrete layers. This relative movement triggers a vertical displacement (v) due to interlock between the roughened surfaces. This vertical displacement, in turn, induces tension in the reinforcement crossing the interface. The generated tension translates into a clamping force, enhancing frictional resistance along the interface. Additionally, cohesion, representing the intrinsic bonding force between the concrete surfaces, contributes to shear resistance.

The contribution of each mechanism varies with the applied load. At low loads, cohesion predominates, effectively resisting the shear force. However, as the load increases, cracks develop within the interface, compromising the cohesive bond. Consequently, the burden of shear resistance shifts to a combination of shear-friction and dowel action.

Shear-friction originates from the interaction between the clamping force and the frictional resistance along the interface. Essentially, the clamping force, generated by the tensioned reinforcement, presses the concrete surfaces together, creating friction that opposes the relative movement.

Dowel action stems from the direct shear resistance offered by the steel bars traversing the interface. These bars experience tension and compression under shear, contributing to the overall interface resistance.

Steel bars (stirrups) are used for shear transfer

In essence, the shear transfer mechanism in RC interfaces operates as a dynamic interplay between cohesion, shear-friction, and dowel action. Understanding the individual contributions and their interplay under varying load conditions is crucial for ensuring the structural integrity and performance of RC structures.

Factors Affecting Shear Transfer Capacity

The magnitude of shear transfer capacity is not a static value but depends on several factors, including:

Interface characteristics: Surface roughness, presence of contaminants, and potential shrinkage gaps all influence the effectiveness of mechanical interlock and adhesion.
Concrete properties: Strength, age, and moisture content of the concrete layers affect their bond characteristics and susceptibility to cracking.
Reinforcement details: Type, spacing, and embedment depth of dowel bars significantly impact their contribution to shear transfer.
Loading conditions: Sustained or cyclic loading, along with the magnitude and distribution of shear forces, influence the interface’s response.

Shear Transfer According to the Eurocodes

Section 6.2.5 of Eurocode 2 (EC2) defines the methodology for evaluating the shear capacity of interfaces between concretes cast at different times. This approach considers the combined contributions of cohesion and friction to interface resistance.

The shear stress at the interface is calculated by the difference of the longitudinal internal force ΔF (tension or compression) in the examined part of the cross-section separated by the interface. The part of the longitudinal force (compressive or tensile) that is located within the new concrete is expressed by the coefficient β.

Following a stress-based approach, the equation provided within the code expresses the shear stress capacity (V_Rd,i) as a function of;

v_Rdi = c⋅f_ctd + μ⋅σ_n + ρ⋅f_yd ⋅ (μ⋅sinα + cosα) ≤ v_Rdi,max

where;
f_ctd is the design tensile strength of concrete;
σ_n is the stress per unit area caused by the minimum external normal force across the interface that can act simultaneously with the shear force, positive for compression, such that σ_n ≤ 0.6f_cd;
f_yd is the design yield strength of reinforcement, not more than 600 MPa;
ρ is the reinforcement ratio (As/Ac);
α is the angle between concrete interface and interface reinforcement;
c and μ are factors that depend on the roughness of the interface; values are listed in Table 2.

Condition	c	μ
Very smooth interface roughness condition; A surface cast against steel, plastic or specially prepared wooden molds.	0.025	0.5
Smooth interface roughness condition; A slip-formed or extruded surface, or a free surface left without further treatment after vibration.	0.2	0.6
Rough interface condition; A surface with at least 3 mm roughness at about 40 mm spacing, achieved by ranking, exposing of aggregate or other methods giving an equivalent behavior	0.4	0.7
Indented interface condition; A surface with indentations complying with more than 3 mm roughness and also depth of groove should be more than 5 mm and the width of the groove should be more than 10 times its depth	0.5	0.9

v_Rdi,max = 0.5 ⋅ ν ⋅ f_cd

Where ν is the strength reduction factor for shear design in accordance with EN1992-1-1 §6.2.2(6).
ν = 0.6 ⋅ (1 – f_ck / 250 MPa)
f_cd is the design compressive strength of the concrete
f_ck is the characteristic compressive strength of the concrete after 28 days

SHEAR CONNECTION REINFORCEMENT — **Figure 3**: *Shear interface between concrete cast at different times*

Design Example

Design the shear transfer between a girder of 600 mm width to a slab topping of 250 mm thickness. The shear force at the section is 655 kN and the interface between the old and new concrete is rough. Take the lever arm for the internal forces of composite section z = 0.9m. f_ck = 30 MPa, f_yk = 500 MPa.

Solution

Applied shear stress at the interface

By assuming a constant lever arm of internal forces z in the examined infinitesimal segment dx then the corresponding difference of longitudinal force ΔF is:

ΔF = β(M + dM) / z – βM/z = βdM/z

Where dM is the infinitesimal change of the bending moment. In the calculation above the variation of the normal force N is not considered significant. The shear stress at the interface v is calculated by dividing the difference of longitudinal force ΔF by the width of the interface b_i and the assumed infinitesimal length dx. According to fundamental mechanics, the shear force is calculated as the derivative of the bending moment V = dM / dx.

The aforementioned analysis leads to the design value of the shear stress at the interface v_Edi as given by EN1992-1-1 §6.2.5(1) equation (6.24):

v_Edi = β⋅V_Ed / (z⋅b_i) = (1.0 × 655 × 10³) / (900 × 600) = 1.213 MPa

Shear strength of the interface

c = 0.400 and μ = 0.700 (for rough surfaces)

The coefficient for concrete cohesion c should be reduced for the case of fatigue or dynamic loads. In general according to EN1992-1-1 §6.2.5(5) under fatigue or dynamic loads, the values for c should be halved. Specifically for bridges, according to EN1992-2 §6.2.5(105) under fatigue and dynamic loads a value of 0.0 should be considered for c. Moreover when the normal stress σ_n is tensile (i.e. negative) then a value of 0.0 should be considered for c in accordance with EN1992-1-1 §6.2.5(1).

For the examined case the adjustment factor applied to the value of c is 0.40. The adjusted value of the coefficient is c = 0.40 × 0.400 = 0.160.

Maximum shear strength that can be transferred at the interface

The maximum value of the design shear resistance v_Rdi,max of the interface is limited by the compressive strength of the concrete struts as specified in EN1992-1-1 §6.2.5(1):

v_Rdi,max = 0.5νf_cd
ν = 0.6 ⋅ (1 – f_ck / 250 MPa) = 0.6 × (1 – 30.00 MPa / 250 MPa) = 0.528
f_cd = α_cc ⋅ f_ck / γ_C = (1.00 × 30.00) / 1.50 = 20.00 MPa

Therefore the maximum value of the design shear resistance v_Rdi,max is calculated as:
v_Rdi,max = 0.5νf_cd = (0.5 × 0.528 × 20.00 MPa) = 5.280 MPa

For the examined case the applied shear stress on the interface v_Edi = 1.213 MPa does not exceed the maximum shear stress capacity of concrete v_Rdi,max = 5.280 MPa. The corresponding utilization factor is u = 0.23 ≤ 1.0 ⇒ ok.

Calculation of required shear connection reinforcement

The calculation of the required shear connection reinforcement ratio ρ can be performed by solving EN1992-1-1 equation (6.25) for ρ. The equation that defines the design shear resistance of the interface is:

v_Rdi = c⋅f_ctd + μ⋅σ_n + ρ⋅f_yd ⋅ (μ⋅sinα + cosα) ≤ v_Rdi,max

The values of the trigonometric functions when the shear connection reinforcement forms angle α = 90.0 ° with the interface plane are sinα = 1.0 and cosα = 0. The value of the design tensile strength of concrete f_ctd is calculated as specified in EN1992-1-1 §3.1.6(2)P:

f_ctd = α_ct ⋅ f_ctk,0.05 / γ_C = (1.00 × 2.03 MPa) / 1.50 = 1.35 MPa

where f_ctk,0.05 = 0.7 × f_ctm = 0.7 × 2.90 MPa = 2.03 MPa is the 5% fractile of the tensile strength of concrete as specified in EN1992-1-1 Table 3.1.

The value of the design yield strength of reinforcement steel f_yd is calculated as specified in EN1992-1-1 §3.2:

f_yd = f_yk / γ_S = 500/1.15 = 434.8 MPa

The required shear connection reinforcement is calculated when v_Edi = v_Rdi. Provided that v_Rdi ≤ v_Rdi,max the aforementioned equation can be solved for the required ratio ρ of the shear connection reinforcement:
σ_n = 0

ρ = (v_Edi – c ⋅ f_ctd – μ ⋅ σ_n) / [f_yd ⋅ (μ⋅sinα + cosα) ]
(v_Edi – c ⋅ f_ctd – μ ⋅ σ_n) = 1.213 – (0.16 × 1.35) – 0 = 0.997
[f_yd ⋅ (μ⋅sinα + cosα) ] = 434.8 × (0.7 × 1.0 + 0) = 304.36
ρ = 0.997/304.36 = 0.003275

The corresponding required shear connection reinforcement per m length of the interface area is:
Width of area = 600 mm
Length of area (per metre run) = 1000 mm

A_s = ρ × 1000 mm × 600 mm = 1965 mm²/m

Therefore provide 4legs of H12 @200 mm c/c spacing (A_sprov = 2260 mm²/m)

Conclusion

In conclusion, shear transfer in the interface of RC members serves as a critical mechanism for structural integrity. Recognizing the contributing factors, their interactions, and the limitations of current design approaches is essential for ensuring the safety and reliability of such structures. Ongoing research efforts aimed at refining analytical models and leveraging advanced experimental techniques hold promise for advancing our understanding and design capabilities, ultimately leading to safer and more efficient RC structures

Design of Cantilever Steel Carport | Monopitch Canopy Roof Design

By

Ubani Obinna

-

February 15, 2024

Cantilever steel carports have become increasingly popular due to their clean aesthetics, simplicity, minimal space requirements, and ability to span large distances without obstructing parking space. However, the unique structural demands of this design system necessitate careful consideration during the design process. The design of cantilever steel carports is consistent with the design of an open monopitch canopy roof according to EN 1991-1-4.

The design of steel carports involves the selection of adequate steel columns and beams that will be able to withstand the dead, live, and environmental loads that the structure will be subjected to without undergoing excessive deflection, vibration, or failure.

This article discusses the structural design for cantilever steel carports, exploring key principles, considerations, and design approaches.

Structural System of Cantilever Carports

Cantilever steel beams are structural systems that project outwards like outstretched arms. Technically, most steel carport frame structures fall under monopitch canopy roof systems for their wind load analysis and design. This structural system offers elegance, efficiency, and expansive coverage, finding diverse applications in bridges, balconies, and yes, even carports. But beneath their deceptively simple appearance lies a complex interplay of forces, internal stresses, and deformations.

Cantilever beams project outward from support columns without additional support at the free end. This creates a significant bending moment force at the fixed end, necessitating robust column and foundation design.

Bending Moment

Imagine a cantilever beam of a carport structure bearing a load at its free end. The beam tries to resist this bending, leading to the development of internal stresses. The top fibres experience tension, stretching as the beam deflects downwards. Conversely, the bottom fibres are compressed, pushing inwards. This stress distribution is not uniform but varies parabolically across the beam’s depth, with the maximum values occurring at the top and bottom surfaces.

Shear force

While bending usually governs overall behaviour of carport frames, shear forces also play a critical role. Imagine slicing the beam at any section. The internal forces acting across this imaginary cut represent the shear force, responsible for balancing the applied load. This force varies along the beam length, reaching a maximum value at the support and decreasing towards the free end. Understanding shear distribution is critical for selecting appropriate beam sections and preventing shear failure.

Deflection

As the beam of a carport frame bends, the free end undergoes deflection, a measure of its vertical displacement. While deflection is inevitable, excessive movement can be detrimental. Factors like beam length, material properties, load magnitude, and support conditions all influence deflection. Engineers utilize engineering mechanics principles and advanced beam theory to calculate deflections and ensure they stay within acceptable limits.

Buckling

While bending and shear are often the primary concerns, slender beams face an additional problem – buckling. Imagine pushing a long, thin ruler sideways; it bends easily. Likewise, slender beams under compression can buckle, losing their load-carrying capacity abruptly. Engineers carefully assess the risk of buckling based on beam geometry, material properties, and loading conditions, employing design techniques like increasing section depth or adding lateral supports to mitigate the risk.

Connection Details

The design doesn’t end with the beam and column structures of the carport. The connections between the structural elements play a vital role in overall behaviour. Welded, bolted, or a combination of connections transfer internal forces between the beam and the support. Improperly designed or executed connections can lead to premature failure, highlighting the importance of careful design, fabrication, and quality control during construction.

Load Analysis

Several loads and load combinations must be accounted for in the design of carports and they typically include:

Dead Loads: Weight of the steel structure, roof covering, and any attachments such as solar panels, electrical/mechanical services, and insulations (though rarely included).

Live Loads: Human access due to erection or maintenance, snow accumulation, and wind pressures. Wind pressure appears to be the most critical load in the design of carport structural systems.

Seismic Loads: Relevant in seismically active regions.

Load Path and Equilibrium: The design ensures a clear and efficient load path from the roof to the columns, foundation, and ultimately the soil. Counterbalancing is often required to achieve equilibrium, achieved through structural elements or anchor design.

Structural Analysis

For complex structural configurations or demanding loading scenarios, advanced analysis techniques like Finite Element Analysis (FEA) become invaluable. FEA software creates a digital model of the structure, discretizing it into smaller elements and applying loads. By solving complex mathematical equations, the software calculates stresses, deflections, and buckling potential at various points within the beam, providing valuable insights beyond analytical solutions.

Structural Design of Carport Structures

Each structural member in the carport frame is individually designed to resist the anticipated loads. This involves:

Beam Design: Selecting appropriate beam sections (e.g., universal beam sections) and checking for bending stress, shear stress, and deflection within allowable limits as per design codes.

Column Design: Designing columns to resist axial loads, bending moments, and potential buckling. Steel column design tables or specific software tools can be employed.

Connection Design: Designing connections between members to ensure adequate strength and stiffness. Welded, bolted, or a combination of connections are used, following code-specified design procedures.

Member Design Example

It is desired to design a monopitch canopy steel carport structure with the details provided below;

Structure data
Height of column = 2.5m
Length of beam = 3.027m
Spacing of frame members = 3.0 m c/c
Spacing of purlins = 0.605m
Angle of inclination of roof = 7.59 degrees

Dead Load
Unit weight of sheeting material = 0.02 kN/m²
Self-weight of members (calculated automatically)
Services (assume) = 0.1 kN/m²

Live Load
Imposed live load = 0.6 kN/m²

Wind Load Analysis of carport Monopitch Canopy Structures

Wind speed = 40 m/s
Basic wind velocity (Exp. 4.1); v = c_dir × c_season × v_b,0 × c_prob = 40.8 m/s
Degree of blockage under the canopy roof: φ = 0
Reference mean velocity pressure; q_b = 0.5 × ρ × v_b² = 1.020 kN/m²
Reference height (at which q is sought); z = 2900 mm
Displacement height (sheltering effects excluded); h_dis = 0 mm
A_ref = bd / cos(α) = 9.000 m ⋅ 3.000 m / 0.991 = 27.239 m²

*Pressure zones for monopitch canopy roofs, reproduced from EN1991-1-4 Table 7.6 and Figure 7.16*

Mean wind velocity
The mean wind velocity v_m(z_e) at reference height z_e depends on the terrain roughness, terrain orography and the basic wind velocity v_b. It is determined using EN1991-1-4 equation (4.3):

v_m(z_e) = c_r(z_e) ⋅ c₀(z_e) ⋅ v_b = 0.7715 × 1.000 × 40.00 m/s = 30.86 m/s

Wind turbulence
The turbulence intensity I_v(z_e) at reference height z_e is defined as the standard deviation of the turbulence divided by the mean wind velocity. It is calculated in accordance with EN1991-1-4 equation 4.7. For the examined case z_e ≥ z_min.

I_v(z_e) = k_I / [ c₀(z_e) ⋅ ln(max{z_e, z_min} / z₀) ] = 1.000 / [ 1.000 ⋅ ln(max{2.900 m, 2.0 m} / 0.050 m) ] = 0.2463

Basic velocity pressure
The basic velocity pressure q_b is the pressure corresponding to the wind momentum determined at the basic wind velocity v_b. The basic velocity pressure is calculated according to the fundamental relation specified in EN1991-14 §4.5(1):

q_b = (1/2) ⋅ ρ ⋅ v_b² = (1/2) ⋅ 1.25 kg/m³ ⋅ (40.00 m/s)² = 1000 N/m² = 1.000 kN/m²

where ρ is the density of the air in accordance with EN1991-1-4 §4.5(1). In this calculation the following value is considered: ρ = 1.25 kg/m³. Note that by definition 1 N = 1 kg⋅m/s².

Peak velocity pressure
The peak velocity pressure q_p(z_e) at reference height z_e includes mean and short-term velocity fluctuations. It is determined according to EN1991-1-4 equation 4.8:

q_p(z_e) = (1 + 7⋅I_v(z_e)) ⋅ (1/2) ⋅ ρ ⋅ v_m(z_e)² = (1 + 7⋅0.2463) ⋅ (1/2) ⋅ 1.25 kg/m³ ⋅ (30.86 m/s)² = 1621 N/m²
⇒ q_p(z_e) = 1.621 kN/m²

Calculation of local wind pressure on the canopy roof

Net pressure coefficients
The net pressure coefficients c_p,net represent the maximum local pressure for all wind directions and they should be used in the design of local elements such as roofing elements and fixings. Net pressure coefficients are given for three zones A, B, C as defined in the figure included in EN1991-1-4 Table 7.6 that is reproduced above. Zones B, C extend at the sides of the canopy and Zone A at the central region:

The inclined length of the monopitch canopy roof parallel to the wind direction is:
d’ = d / cos(α) = 3.000 m / 0.991 = 3.027 m

Zone C corresponds to the regions parallel to the windward and leeward edges having width d’/10 = 0.303 m. Zone B corresponds to the regions parallel to the side edges having width b/10 = 0.900 m, where b is the width of the canopy transverse to the wind direction. Zone A corresponds to the remaining central region.

The net pressure coefficient c_p,net for each of the zones A, B, C are defined in EN1991-1-4 Table 7.6 as a function of the roof angle α and the blockage factor φ. For the examined case: α = 7.59 ° and φ = 0.000. Therefore according to EN1991-1-4 Table 7.6 the following net pressure coefficients and overall force coefficient are obtained, using linear interpolation where appropriate:

For zone A: c_p,net,A = -1.307 or +1.007
For zone B: c_p,net,B = -1.855 or +2.255
For zone C: c_p,net,C = -1.955 or +1.455

Negative values for the external pressure coefficient correspond to suction directed away from the upper surface inducing uplift forces on the roof. Both positive and negative values should be considered for each zone.

Net wind pressure on pressure zones

The net wind pressure on the surfaces of the structure w_net corresponds to the combined effects of external wind pressure and internal wind pressure. For structural surfaces consisting of only one skin the net pressure effect is determined as:

w_net = c_p,net ⋅ q_p(z_e)

For structural surfaces consisting of more than one skin EN1991-1-4 §7.2.10 is applicable. For the different pressure zones on the canopy roof the following net pressures are obtained:

– For zone A: w_net,A = -2.119 kN/m² or +1.633 kN/m²
(zones A is the remaining central region located more than d’/10 = 0.303 m or b/10 = 0.900 m from the edges)

– For zone B: w_net,B = -3.008 kN/m² or +3.657 kN/m²
(zone B extends up to b/10 = 0.900 m from the side edges)

– For zone C: w_net,C = -3.170 kN/m² or +2.360 kN/m²
(zone C extends up to d’/10 = 0.303 m from the windward and leeward edges)

Negative net pressure values correspond to suction directed away from the external surface inducing uplift forces on the canopy roof. Both positive and negative values should be considered.

Calculation of overall wind force on the canopy roof

Overall pressure coefficient
The overall pressure coefficient c_f represents the overall wind force and it should be used in the design of the overall load bearing structure. The overall pressure coefficient c_f is defined in EN1991-1-4 Table 7.6 as a function of the roof angle α and the blockage factor φ. For the examined case: α = 7.59 ° and φ = 0.000. Therefore according to EN1991-1-4 Table 7.6 the following overall pressure coefficient is obtained, using linear interpolation where appropriate:

c_f = -0.804 or 0.452

Negative values for the overall pressure coefficient correspond to suction directed away from the upper surface inducing uplift forces on the roof. Both positive and negative values should be considered.

Structural factor
The structural factor c_sc_d takes into account the structure size effects from the non-simultaneous occurrence of peak wind pressures on the surface and the dynamic effects of structural vibrations due to turbulence. The structural factor c_sc_d is determined in accordance with EN1991-1-4 Section 6. A value of c_sc_d = 1.0 is generally conservative for small structures not-susceptible to wind turbulence effects such as buildings with heights less than 15 m.

In the following calculations, the structural factor is considered as c_sc_d = 1.000.

Overall wind force (for total roof surface)

The wind force F_w corresponding to the overall wind effect on the canopy roof is calculated in accordance with EN1991-1-4 equation 5.3:
F_w = c_sc_d ⋅ c_f ⋅ A_ref ⋅ q_p(z_e)

where A_ref = 27.239 m² is the reference wind area of the canopy roof as calculated above.

For the examined case:
– Maximum overall wind force (acting downwards):
F_w = 1.000 ⋅ (+0.452) ⋅ 27.239 m² ⋅ 1.621 kN/m² = +19.95 kN

– Minimum overall wind force (acting upwards):
F_w = 1.000 ⋅ (-0.804) ⋅ 27.239 m² ⋅ 1.621 kN/m² = -35.49 kN

Negative values correspond to suction directed away from the external surface inducing uplift forces on the canopy roof. Both positive and negative values should be considered, as explained below.

Direction and eccentricity of the overall wind force
According to EN1991-1-4 §7.3(6) and the National Annex the location of the centre of pressure is defined at an eccentricity e from the windward edge. In this calculation, the centre of pressure is considered at an eccentricity e = 0.250⋅d’ = 0.757 m, where d’ = 3.027 m is the inclined length of the canopy roof parallel to the wind direction. Two cases should be examined for the overall effect of the wind force on the canopy roof:

Maximum force F_w = +19.95 kN (i.e. acting downwards) located at a distance e = 0.757 m from the windward edge.
Minimum force F_w = -35.49 kN (i.e. acting upwards) located at a distance e = 0.757 m from the windward edge.

Structural Analysis and Results

Finite Element Analysis (FEA) software (Staad Pro) was used to model the structure and evaluate stresses, deflections, and buckling potential under various load combinations.

Structural Modelling and Loading

Some of the images from the structural model are shown below.

Finite element model of the carport structure/canopy roof

Negative wind load (suction) on the canopy roof

Support Reactions

The support reactions from the various load combinations are shown below.

Support Reactions (1.35gk + 1.5wk) – Gravity

The summary of the maximum and minimum support reactions under various load combinations are shown in the Table below.

Bending Moment and shear force diagrams

The typical bending moment and shear force diagrams from the various load combinations are shown below.

Typical bending moment diagram under gravity load

Typical bending moment diagram under wind suction

Typical shear force diagram under gravity load

Design of the Cantilever Beams

The summary of the maximum stresses occurring on the beams is shown below. An abridged design calculations are presented afterwards.

Section type; UB 254x146x37 (BS4-1)
Steel grade – EN 10025-2:2004; S275
Nominal thickness of element; t_nom = max(t_f, t_w) = 10.9 mm
Nominal yield strength; f_y = 275 N/mm²
Nominal ultimate tensile strength; f_u = 410 N/mm²
Modulus of elasticity; E = 210000 N/mm²

The section is Class 1

Check shear
Height of web; h_w = h – 2t_f = 234.2 mm; h = 1.000
h_w / t_w = 37.2 = 40.2ε/ h < 72ε / h
Shear buckling resistance can be ignored

Design shear force; V_y,Ed = 32.8 kN
Shear area – cl 6.2.6(3); A_v = max(A – 2bt_f + (t_w + 2r)t_f, hh_wt_w) = 1759 mm²
Design shear resistance – cl 6.2.6(2); V_c,y,Rd = V_pl,y,Rd = A_v × (f_y / √(3)) / γ_M0 = 279.3 kN
V_y,Ed / V_c,y,Rd = 0.118

Check bending moment
Design bending moment; M_y,Ed = 61.7 kNm
Design bending resistance moment – eq 6.13; M_c,y,Rd = M_pl,y,Rd = W_pl.y f_y / γ_M0 = 132.9 kNm
M_y,Ed / M_c,y,Rd = 0.464

Slenderness ratio for lateral torsional buckling
Correction factor – For cantilever beams; k_c = 1
C₁ = 1 / k_c² = 1
Poissons ratio; n = 0.3
Shear modulus; G = E / [2(1 + n)] = 80769 N/mm²
Unrestrained effective length; L = 1.0L_{z_s1} = 3000 mm

Elastic critical buckling moment; M_cr = C₁π²EI_z / L² × √(I_w / I_z + L²GI_t / (π²EI_z)) = 205.5 kNm
Slenderness ratio for lateral torsional buckling; λ_LT = √(W_pl.yf_y / M_cr) = 0.804
Limiting slenderness ratio; λ_LT,0 = 0.4

λ_LT > λ_LT,0 – Lateral torsional buckling cannot be ignored

Check buckling resistance
Buckling curve – Table 6.5; b
Imperfection factor – Table 6.3; α_LT = 0.34
Correction factor for rolled sections; β = 0.75
LTB reduction determination factor; φ_LT = 0.5[1 + α_LT(λ_LT – λ_LT,0) + βλ_LT²] = 0.811
LTB reduction factor – eq 6.57; c_LT = min(1 / [φ_LT + √(φ_LT² – βλ_LT²)], 1, 1 /λ_LT²) = 0.815
Modification factor; f = min(1 – 0.5(1 – k_c) × [1 – 2(λ_LT – 0.8)²], 1) = 1.000
Modified LTB reduction factor – eq 6.58; c_LT,mod = min(c_LT /f, 1, 1 / λ_LT²) = 0.815
Design buckling resistance moment – eq 6.55; M_b,y,Rd = c_LT,modW_pl.yf_y / γ_M1 = 108.3 kNm
M_y,Ed / M_b,y,Rd = 0.57

Check for Deflection

The following deflection values were obtained for the structure;

Unfactored dead load = 5.423 mm
Unfactored live load = 9.507 mm
Positive wind load (downwards) = 38.4 mm
Negative wind load (upwards) = 40.827 mm

With this information, an appropriate deflection limit can be adopted for the structure.

Design of the columns

The summary of the maximum stresses occurring on the columns is shown below. An abridged design calculations are presented afterwards.

Combined bending and axial compression (cl. 6.3.3)
Characteristic resistance to normal force; N_Rk = Af_y = 1150 kN
Characteristic moment resistance – Major axis; M_y,Rk = (W_pl.yf_y) = 132.2 kNm
Characteristic moment resistance – Minor axis; M_z,Rk = W_pl.z f_y = 16.5 kNm
Moment factor – Major axis; C_my = 0.9
Moment factor – Minor axis; C_mz = 0.9
Moment distribution factor for LTB; ψ_LT = M_y,Ed2 / M_y,Ed1 = 0.842
Moment factor for LTB; C_mLT = max(0.4, 0.6 + 0.4 ´ y_LT) = 0.937

Interaction factor k_yy;
k_yy = C_my [1 + min(0.8, λ_y – 0.2) N_Ed / (χ_yN_Rk / γ_M1)] = 0.901

Interaction factor k_zy;
k_zy = 1 – min(0.1, 0.1λ_z)N_Ed / ((C_mLT – 0.25)(χ_zN_Rk/γ_M1)) = 0.985

Interaction factor k_zz;
k_zz = C_mz [1 + min(1.4, 2λ_z – 0.6)N_Ed / (c_zN_Rk / γ_M1)] = 1.029

Interaction factor k_yz;
k_yz = 0.6k_zz = 0.617

Section utilisation;
UR_{B_1} = N_Ed / (χ_yN_Rk / γ_M1) + k_yyM_y,Ed / (c_LTM_y,Rk / γ_M1) + k_yzM_z,Ed / (M_z,Rk / γ_M1)
UR_{B_1} = 0.690

UR_{B_2} = N_Ed / (χ_zN_Rk / γ_M1) + k_zyM_y,Ed / (c_LTM_y,Rk / γ_M1) + k_zzM_z,Ed / (M_z,Rk / γ_M1)
UR_{B_2} = 0.810

Design of the Foundation

Understanding soil properties is critical for foundation design. Geotechnical investigations determine soil-bearing capacity and potential for settlement. The design of the foundation should pay good attention to uplift, sliding and overturning moment from the wind load.

Depending on soil conditions and design loads, foundations can be:

Spread Footings: Individual concrete pads for each column.
Continuous Footings: A continuous concrete strip supporting multiple columns.
Mat Foundation: A concrete slab supporting the entire structure.

Anchor Design: Anchors embedded in the foundation resist uplift forces generated by wind and seismic loads. Anchor selection and embedment depth are critical for structural stability.

Fabrication and Construction Considerations

Shop Drawings and Fabrication: Detailed shop drawings ensure accurate fabrication of steel components. Quality control during fabrication is paramount.
Erection and Field Welding: Proper erection procedures and qualified welders are necessary to ensure structural integrity and safety.
Inspection and Quality Control: On-site inspections at various stages of construction verify adherence to design specifications and ensure construction quality.

Conclusion

Cantilever steel carports offer an aesthetically pleasing and practical solution for vehicle protection. However, due to their inherent structural challenges, their design requires meticulous attention to detail. A clear understanding of the load paths, material properties, design codes, and analysis methods is essential for a safe and reliable structure. Consulting with qualified structural engineers throughout the design and construction process is crucial to ensure a successful and long-lasting cantilever steel carport.

Wind Load Analysis of Signboards and Billboards

By

Ubani Obinna

-

February 14, 2024

Signboards, with their captivating visuals and strategic placements, are very popular elements for advertisement in our towns, streets, and highways. However, the structural stability of signboards hinges on their ability to withstand the dynamic forces of wind. This requires a detailed wind load analysis from the design engineer.

Billboard advertising, despite facing competition from digital alternatives, remains a significant player in the marketing landscape. To understand its economic impact, global billboard advertising spending reached $36.8 billion in 2022, with predictions of a steady rise to $44.2 billion by 2027. The United States accounts for the largest share (around 40%), followed by China and Europe.

Wind Loads on Signboards

Wind force is the most critical action on billboards. Their cantilevered design, supported by a single column, exposes them to wind-induced stresses. Failure due to wind and hurricanes has been reported, necessitating rigorous analysis and design. Other effects, such as imperfections and the p-delta phenomenon, also impact structural performance under wind load. Wind exerts pressure on objects, generating a force proportional to the wind speed squared. This force, known as wind load, varies with factors like:

Location: Geographic location determines wind speeds within established design wind speed maps.
Terrain: Topography influences wind turbulence and local wind speeds.
Exposure category: Building codes categorize zones based on surrounding obstructions, impacting wind pressures.
Signboard geometry: Size, shape, and orientation of the signboard directly influence the wind load experienced.

Methods and Tools for Wind Load Analysis

Several methods are employed for wind load analysis of signboards:

Simplified methods: Building codes often provide simplified equations based on specific geometries and exposure categories. However, these methods may not always be suitable for complex designs.
Wind tunnel testing: Physical scale models of the signboard are subjected to simulated wind conditions in a wind tunnel, providing accurate pressure data. This method is expensive but precise, especially for unique designs.
Computational Fluid Dynamics (CFD) simulations: Numerical simulations model wind flow around the signboard using specialized software. This is a cost-effective alternative to wind tunnel testing, offering valuable insights into complex geometries.

Dynamic Considerations

While static wind loads are vital, signboards may experience dynamic effects like flutter and vortex shedding, resulting in vibrations and potential fatigue failure. Advanced analysis methods or wind tunnel testing may be necessary to assess these dynamic effects, especially for tall and slender signboards.

Wind Load Analysis Example

Let us carry out a wind load analysis on an 8m high signboard in a city centre where the basic wind speed is 35 m/s. The calculated effective wind pressure w_eff, total wind force F_W, and total wind overturning moment M_W correspond to the total wind action effects and they are appropriate for global verifications of the element according to the force coefficient method.

For local verifications, appropriate wind pressure on local surfaces must be estimated according to the relevant external pressure coefficients, as specified in EN1991-1-4 §5.2. The calculated wind action effects are characteristic values (unfactored). Appropriate load factors should be applied to the relevant design situation. For ULS verifications the partial load factor γ_Q = 1.50 is applicable for variable actions.

Input Data

Terrain category: = II
Basic wind velocity: v_b = 35 m/s
Width of the signboard wind-loaded area: b = 10 m
Height of the signboard wind-loaded area: h = 3 m
Separation height of the signboard wind-loaded area from the ground: z_g = 5 m
Orography factor at reference height z_e: c₀(z_e) = 1
Structural factor: c_sc_d = 1
Air density: ρ = 1.25 kg/m³
Additional rules defined in the National Annex for the calculation of peak velocity pressure q_p(z_e): = None
The horizontal eccentricity of the centre of pressure from the centre of the signboard as a fraction of the width b: e/b = 0.25

Calculation of peak velocity pressure

Reference area and height

The reference height for the wind action z_e is located at the centre of the signboard, as specified in EN1991-1-4 §7.4.3(3). The reference area for the wind action A_refis the wind-loaded area of the signboard, as specified in EN1991-1-4 §7.4.3(3). Therefore:

z_e = z_g + h / 2 = 5.000 m + 3.000 m / 2 = 6.500 m
A_ref = b ⋅ h = 10.000 m ⋅ 3.000 m = 30.00 m²

Basic wind velocity

The basic wind velocity v_b is defined in EN1991-1-4 §4.2(2)P as a function of the wind direction and time of year at 10 m above ground of terrain category II. The value of v_b includes the effects of the directional factor c_dir and the seasonal factor cseason and it is provided in the National Annex. In the following calculations, the basic wind velocity is considered as v_b = 35.00 m/s.

Terrain roughness

The roughness length z₀ and the minimum height z_min are specified in EN1991-1-4 Table 4.1 as a function of the terrain category. For terrain category II the corresponding values are z₀ = 0.050 m and z_min = 2.0 m. The terrain factor k_r depending on the roughness length z₀ = 0.050 m is calculated in accordance with EN1991-1-4 equation (4.5):

k_r = 0.19 ⋅ (z₀ / z_0,II)^0.07 = 0.19 ⋅ (0.050 m / 0.050 m)^0.07 = 0.1900

The roughness factor c_r(z_e) at the reference height z_e accounts for the variability of the mean wind velocity at the site. It is calculated in accordance with EN1991-1-4 equation 4.4. For the examined case z_e ≥ z_min:

c_r(z_e) = k_r ⋅ ln(max{z_e, z_min} / z₀) = 0.1900 ⋅ ln(max{6.500 m, 2.0 m} / 0.050 m) = 0.9248

Orography factor

Where orography (e.g. hills, cliffs etc.) is significant its effect on the wind velocities should be taken into account using an orography factor c0(ze) different than 1.0, as specified in EN1994-1-1 §4.3.3. The recommended procedure in EN1994-1-1 §4.3.3 for the calculation of the orography factor c₀(z_e) is described in EN1994-1-1 §A.3.

In the following calculations, the orography factor is considered as c₀(z_e) = 1.000.

Mean wind velocity

The mean wind velocity v_m(z_e) at reference height z_e depends on the terrain roughness, terrain orography and the basic wind velocity v_b. It is determined using EN1991-1-4 equation (4.3):

v_m(z_e) = c_r(z_e) ⋅ c₀(z_e) ⋅ v_b = 0.9248 ⋅ 1.000 ⋅ 35.00 m/s = 32.37 m/s

Wind turbulence

The turbulence intensity I_v(z_e) at reference height z_e is defined as the standard deviation of the turbulence divided by the mean wind velocity. It is calculated in accordance with EN1991-1-4 equation 4.7. For the examined case z_e ≥ z_min.

I_v(z_e) = k_I / [ c₀(z_e) ⋅ ln(max{z_e, z_min} / z₀) ] = 1.000 / [ 1.000 ⋅ ln(max{6.500 m, 2.0 m} / 0.050 m) ] = 0.2054

Basic velocity pressure

The basic velocity pressure q_b is the pressure corresponding to the wind momentum determined at the basic wind velocity v_b. The basic velocity pressure is calculated according to the fundamental relation specified in EN1991-14 §4.5(1):

q_b = (1/2) ⋅ ρ ⋅ v_b² = (1/2) ⋅ 1.25 kg/m³ ⋅ (35.00 m/s)² = 766 N/m² = 0.766 kN/m²

where ρ is the density of the air in accordance with EN1991-1-4 §4.5(1). In this calculation the following value is considered: ρ = 1.25 kg/m³. Note that by definition 1 N = 1 kg⋅m/s².

Peak velocity pressure

The peak velocity pressure q_p(z_e) at reference height z_e includes mean and short-term velocity fluctuations. It is determined according to EN1991-1-4 equation 4.8:

q_p(z_e) = (1 + 7⋅I_v(z_e)) ⋅ (1/2) ⋅ ρ ⋅ v_m(z_e)² = (1 + 7⋅0.2054) ⋅ (1/2) ⋅ 1.25 kg/m³ ⋅ (32.37 m/s)² = 1597 N/m²
⇒ q_p(z_e) = 1.597 kN/m²

Note that by definition 1 N = 1 kg⋅m/s².

Calculation of wind forces on the structure

Structural factor

The structural factor c_sc_d is determined in accordance with EN1991-1-4 Section 6. A value of c_sc_d = 1.0 is generally conservative for small structures not susceptible to wind turbulence effects. In the following calculations, the structural factor is considered as c_sc_d = 1.000.

Force coefficient

The force coefficient c_f is given in EN1991-1-4 Sections 7 and 8 depending on the type of structure or structural element. According to EN1991-1-4 §7.4.3, for signboards with z_g ≥ h / 4 or b / h ≤ 1, the force coefficient is c_f = 1.800.

Total wind force

The wind force on the structure F_w for the overall wind effect is estimated according to the force coefficient method as specified in EN1991-1-4 §5.3.

F_w = c_sc_d ⋅ c_f ⋅ q_p(z_e) ⋅ A_ref = 1.000 ⋅ 1.800 ⋅ 1.597 kN/m² ⋅ 30.00 m² = 86.216 kN

The total wind force F_w takes into account the overall wind effect. The corresponding effective wind pressure w_eff on the reference wind area A_ref is equal to:

w_eff = F_w / A_ref = 86.216 kN / 30.00 m² = 2.874 kN/m²

This effective pressure w_eff = 2.874 kN/m² is appropriate for global verifications of the structure according to the force coefficient method. It is not appropriate for local verifications of structural elements. For the latter case appropriate wind pressure on local surfaces must be estimated according to the relevant pressure coefficients, as specified in EN1991-1-4 §5.2.

Overturning moment

According to EN1991-1-4 §7.4.3 the resultant force normal to the signboard should be taken to act at the height of the center of the signboard. The total overturning moment M_w acting at the base of the structure is equal to:

M_w = F_w ⋅ (z_g + h / 2) = 86.216 kN ⋅ (5.000 m + 3.000 m / 2) = 560.40 kNm

The overturning moment corresponds to the wind action total effect, i.e. it is the total overturning moment for all the base supports.

Horizontal eccentricity

According to EN1991-1-4 §7.4.3 and the National Annex, the resultant force normal to the signboard should be taken to act with a horizontal eccentricity e. In this calculation, the following normalized eccentricity is considered e/b = ±0.250, where b is the width of the signboard wind-loaded area. The total torsional moment T_w acting at the base of the structure is equal to:

T_w = ±0.250 ⋅ b ⋅ F_w = ±0.250 ⋅ 10.000 m ⋅ 86.216 kN = 215.54 kNm

The torsional moment corresponds to the wind action total effect, i.e. it is the total torsional moment for all the base supports.

Conclusion

Wind load analysis is a crucial step in ensuring the safety and durability of signboards. By understanding wind forces, employing appropriate analysis methods, and considering structural design principles, engineers can guarantee structurally sound signboards that stand the test of time.

Surface Erosion on Embankments and Slopes

By

Ubani Obinna

-

February 12, 2024

Table of Contents

Soil erosion, defined as the detachment and transport of soil particles by water, wind, or other external forces, presents a significant challenge to agricultural productivity, livability, transportation, and environmental health. Surface erosion of cropland diminishes its potential yield, while the eroded sediments degrade the quality of downstream waterways such as streams, lakes, and reservoirs. Similarly, on roadside embankments, erosion creates rills and gullies, exacerbating surface runoff and eventually leading to slope instability and failure.

surface erosion 2 — Surface erosion on an embankment

The United States Department of Agriculture (USDA) developed the Universal Soil Loss Equation (USLE) in response to this pressing issue. This widely adopted model predicts the average annual rate of sheet and rill erosion on agricultural land, taking into account key factors such as rainfall patterns, soil characteristics, topographical features, crop selection, and agricultural practices. Mathematically expressed as:

A = R × K × (LS) × C × P

where:
A = average annual soil loss; it is conventionally expressed in tons/ac/yr,
R = rainfall and runoff factor; it depends on the rainfall intensity and duration,
K = soil erodibility factor; it represents a soil’s ability to resist erosion and is determined by the soil texture, soil structure, organic matter content, and soil permeability,
L = slope length,
S = steepness factor,
C = cover and management factor; it is the ratio of soil loss in an area with specified cover and management to the corresponding soil loss in a clean-tilled and continuously fallow condition. For bare ground, C = 1.0,
P = support practice factor; it is the ratio of soil loss with a support practice such as contouring, strip-cropping, or implementing terraces compared to up-and-down-the-slope cultivation. For construction sites such as roadside embankment, P is not used in the equation.

In the year 1996, the United States Department of Agriculture (USDA) recognized the limitations of the earlier Universal Soil Loss Equation (USLE) and introduced its refinement, the Revised Universal Soil Loss Equation (RUSLE). While retaining the fundamental structure and factors of the USLE, the RUSLE incorporated significant advancements in the technology used to evaluate each factor and introduced new data sources for improved accuracy.

Key improvements in the RUSLE include:

Enhanced Rainfall-Runoff Erosivity Factor (R): An expanded database for R provided a more comprehensive representation of rainfall patterns across diverse regions.
Time-Varying Soil Erodibility Factor (K): Revised K values incorporate the impact of freeze-thaw cycles and soil consolidation for a more dynamic representation of soil susceptibility to erosion.
Refined Topographic Factor (LS): The LS factor was adjusted to explicitly reflect the ratio of rill to interrill erosion, offering a more precise estimation of topographical influence.
Continuous Cover-Management Factor (C): Replacing the seasonal approach, C became a continuous function calculated by multiplying four subfactors, accounting for the time-evolving impact of crop cover and residue on erosion control.
Expanded Support Practices Factor (P): The scope of P was broadened to encompass conditions specific to rangelands, contouring, strip-cropping, and terracing, enabling a more nuanced assessment of management practices in mitigating erosion.

Surface Erosion Control Measures

There are several methods of controlling surface erosion and some of them are the use of ripraps, vegetative cover, composting, and geosynthetics. These methods are discussed in the sections below.

Riprap

The United States Federal Highway Administration (1989) defines riprap as “a flexible lining or facing for channels or banks, consisting of a well-graded mixture of rock, broken concrete, or other suitable material, typically placed by dumping or handwork, to provide erosion protection.” It finds widespread application in protecting and stabilizing embankments, side slopes of waterways (rivers, channels, lakes, etc.), dams, and drainage elements (slope and storm drains).

Riprap revetments encompass a variety of options, including:

Rock Riprap: The most common surface erosion protection method for river and channel banks. Angular, well-graded stones interlock, forming a robust unit that resists erosion due to their weight and combined mass. However, rounded rock on steeper slopes (>2:1) can be unstable, requiring alternative materials like geosynthetic matting.
Wire-Enclosed Rock: Offers enhanced stability compared to standard rock riprap, particularly on steeper slopes.
Grouted Rock: Grouting binds the stones together, creating a monolithic structure with superior erosion resistance but reduced flexibility.
Precast Concrete Block Revetments: Offer efficient placement and consistent quality but may not blend aesthetically with the environment.
Paved Lining: Provides superior erosion protection but can be costly and alter aquatic habitat.

Designing rock riprap effectively requires careful consideration of various factors, including:

Rock size
Rock gradation
Riprap layer thickness
Filter design
Material quality
Edge treatment
Construction considerations

Several government agencies offer guidelines for riprap design. Some key design criteria for ripraps are;

Gradation: Utilize a well-graded mixture of rock sizes for optimal interlocking and stability. Avoid uniform-sized stones.
Stone Quality: Select durable riprap material resistant to freeze-thaw cycles. Most igneous rocks like granite exhibit suitable durability.
Riprap Depth: Ensure the riprap layer is at least twice the thickness of the largest stone diameter used.
Filter Material: Implement a filter material, typically a synthetic fabric or gravel layer, beneath the riprap to prevent underlying soil erosion.
Riprap Limits: Extend riprap to the maximum expected flow depth or a point suitable for vegetation-based erosion control.
Curved Flow Channels: In curved sections, extend riprap five times the channel bottom width upstream and downstream of the curve’s beginning and end, covering the entire curve.
Riprap Size: Select rock size based on the anticipated shear stress from water flow. Sizes typically range from 5 cm to 60 cm in diameter.

By adhering to these recommendations, designers can ensure the effectiveness and longevity of rock riprap installations for various erosion control applications.

Compost

The strategic application of compost, a stable amendment produced by microbial decomposition of organic matter, offers valuable material for mitigating erosion in susceptible areas and accelerating vegetation establishment. Composting not only diverts waste from landfills but also transforms it into useful resources with economic and environmental benefits.

A range of compost types has been successfully employed for surface erosion control on embankments and natural slopes:

Green material compost: Derived from yard trimmings, clippings, and agricultural byproducts.
Manure compost: Sourced from dairy or poultry manure.
Co-compost: Blends biosolids with green materials.
Wood chip and forestry residual composts: Utilize wood-based waste materials.
Food scraps and municipal solid waste composts: Offer additional possibilities, though requiring careful assessment.

Due to varying sources and manufacturing processes, compost characteristics can differ significantly in terms of:

pH
Soluble salts
Moisture content
Organic matter content
Maturity
Stability
Particle size
Pathogen and physical contaminant presence

Selecting suitable compost for engineering applications necessitates considering specifications set by organizations like the USDA and the USCC. Their “Test Methods for the Examination of Composting and Compost” guide ensures the quality and suitability of the final product. By carefully selecting and applying compost, land management professionals can harness its potential to protect soil, promote vegetation growth, and contribute to sustainable infrastructure solutions.

Vegetation

The utilization of vegetative cover is a prominent method of mitigating soil erosion. Its primary functionalities encompass the protection of the soil surface from the impact of raindrops, the provision of a barrier against the erosive effects of overland flow, and the reduction of the erosive potential of flowing water through attenuation of its velocity. The complex root systems of vegetation contribute to the stabilization of the soil, thereby enhancing its resistance to erosion, fostering infiltration rates, and minimizing runoff.

Key attributes associated with vegetative cover include its cost-effectiveness, ease of establishment, and aesthetic value. While frequently employed in conjunction with alternative erosion control techniques (such as compost blankets, geosynthetic covers, and mulches), the design and implementation of vegetative cover necessitate the consideration of several critical factors:

Soil Characteristics: These encompass attributes such as acidity, moisture retention capacity, drainage, texture, organic matter content, and fertility.
Site Conditions: Factors like slope gradient and the existing extent of vegetative cover play a pivotal role.
Climate: Considerations related to temperature, wind patterns, and precipitation levels are essential.
Species Selection: The selection of appropriate plant species is contingent upon regional climate, planting season, water requirements, soil preparation, weed management, anticipated post-construction land use, and projected maintenance levels (including irrigation and associated costs).
Establishment Methods: Techniques employed for the successful establishment and growth of vegetation.
Maintenance Procedures: Practices implemented to ensure the ongoing health and effectiveness of the vegetative cover.

Erosion Control Using Geosynthetics

The utilization of geosynthetics has witnessed a significant rise in popularity in erosion and sediment control, as well as slope stabilization. This proliferation has been fueled by the development of a diverse array of methods and the ongoing emergence of innovative approaches, including degradable rolled erosion control products (RECPs), nondegradable RECPs, and hard armouring techniques.

geosynthetics in erosion control — Geosynthetics in surface erosion control

Degradable RECPs offer the potential for enhancing the establishment of vegetation on rehabilitated lakeshores, riverbanks, and newly constructed roadways. Their application is particularly suited to scenarios where vegetation alone is anticipated to provide sufficient long-term protection once the RECP has degraded.

Conversely, nondegradable RECPs offer long-term reinforcement to vegetation, making them ideal for more demanding erosion control scenarios requiring immediate and high-performance protection. These materials achieve their function by permanently reinforcing the vegetative root structure, thereby enhancing the erosion resistance of soil, rock, and other underlying materials.

It is noteworthy that geosynthetic erosion control measures often fulfil multiple functions concurrently. These functionalities encompass surface runoff collection and drainage, filtration, separation, reinforcement, and the facilitation of vegetation establishment and maintenance.

Conclusion

Surface erosion, the detachment and transportation of soil particles by wind or water, poses a significant environmental and economic challenge. Unsustainable land management practices, deforestation, and intense precipitation events exacerbate this natural process, leading to topsoil loss, decreased fertility, and downstream sedimentation.

Fortunately, various control measures can be implemented to minimize erosion’s impact. Rock riprap can be used on slopes and embankments to control surface erosion. Vegetation cover, a cost-effective and aesthetically pleasing approach, protects the soil from raindrop impact and overland flow. Additionally, geosynthetics offer reinforcement and enhanced vegetation establishment, particularly in critical areas. By employing strategies tailored to specific landscapes and climatic conditions, we can effectively mitigate surface erosion and safeguard our valuable soil resources.

Establishing Datum and Taking Levels on Site

By

Ubani Obinna

-

February 10, 2024

Establishing an accurate and reliable datum and subsequently taking precise levels on site are fundamental aspects of various construction and surveying projects. A datum acts as a fixed reference point from which all vertical measurements on a site are determined.

Establishing a clear and stable datum ensures consistency and accuracy throughout the project lifecycle, from excavation and foundation laying to building construction and finishing works. Precise levelling involves measuring the elevation of points relative to the established datum, allowing for accurate setting out of structures, drainage systems, and other critical elements. This article outlines the key principles and procedures involved in establishing datum and taking levels on construction sites.

Establishment of a Datum

For accuracy in construction, it is essential that all levels in a building are taken from a fixed point called a datum. This point must be established at the onset of the construction work, during the setting out operation. Where feasible, the datum should be related to an ordnance benchmark. This symbol resembles an arrow with a horizontal mark above the arrow as shown in Figure 1.

The centreline of the horizontal is the precise elevation denoted on an Ordnance Survey map. Ordnance benchmarks are typically located incised or embedded within the sides of walls and buildings. Where there are no benchmarks on or near the site, an appropriate reference point must be designated. A project datum or interim control point could be a concrete monument or a permanent concrete base established on the project site.

Therefore, several methods can be employed to establish a datum, depending on the project requirements and local regulations. Common approaches include:

Transferring from Ordnance Benchmarks (OBMs): Utilizing existing, fixed reference points established by national survey agencies.
Direct Connection to a Stable Structure: Referencing a permanent and stable structure on site, such as a building foundation or an established control point.
Arbitrary Datum: Setting a convenient point on site as the reference, clearly documented and protected throughout the project. Sometimes, the connecting roadway to the site is conveniently taken as the reference point.

The datum on a site, representing the reference height, can be arbitrarily assigned an elevation of 0.00 meters, regardless of its physical location with respect to ground level. This offers flexibility in assigning positive elevations to subsequent levels throughout the building. For example, if the datum is set at 0.00 m, the top of the ground floor level could be designated as +0.6 m (meaning that the ground floor level is 600 mm above the datum), followed by the first floor at +3.6 m, and so on, as the structure progresses upwards.

Conversely, any feature located below ground will have a negative elevation relative to the datum, indicated by a preceding minus sign (e.g., -1.2 m for the depth of foundation). However, it is possible to arrange the levels such that a positive value is maintained throughout the project.

It is important to note that these elevations typically refer to finished floor levels (FFL), signifying the final height of the floor surface. However, other points of interest can be designated using this system as well, such as finished structural levels (FSL).

The key considerations in the selection and adoption of a datum on site are;

Stability: The chosen datum point must be stable and unlikely to be disturbed by construction activities.
Accessibility: The point should be readily accessible for repeated measurements and transfer of levels.
Clarity: The datum definition and location must be clearly communicated and documented for all stakeholders.

Taking Levels

Levelling is a fundamental surveying technique employed to ascertain the height differential between two points. This process serves two primary purposes:

Elevation determination: Precisely establishing the elevation of a point relative to a fixed reference plane known as a datum.
Target elevation achievement: Positioning a point at a predetermined elevation with reference to the same datum.

Precise levelling involves measuring the elevation of points relative to the established datum, allowing for accurate setting out of structures, drainage systems, and other critical elements.

The primary instruments used for levelling operations are an engineer’s level (dumpy level) and a levelling staff. The engineer’s level is essentially a telescope equipped with cross-hairs for precise alignment, mounted on a tripod with a horizontal axis plate. The levelling staff, typically constructed with extendable or folding sections, typically measures 4 meters long. “E” pattern graduations, marked at 10 mm intervals, are commonly used, although some staffs may feature 5 mm graduations. To ensure accuracy, readings are estimated to the nearest millimetre.

levelling instrument — Tripod stand, dumpy level, and levelling staff

Levelling ’scope, ranging rod and ‘E’ pattern staff

Levelling procedures begin by setting up the instrument on stable ground and taking a sight to a designated benchmark. Next, staff stations are established at predetermined intervals, often following a 10-meter grid pattern. The engineer then takes instrument readings for each staff station. By combining these differential readings with calculations based on the site plan area, the volume of material required for excavation or cut-and-fill operations to level the site can be accurately determined.

Rise and fall method:
Staff reading at A = 3.2 m
Staff reading at B = 0.95 m
Ground level at A = 100 m above ordnance datum (AOD)
Level at B = 100 m + rise (− fall if declining)
Level at B = 100 m + (3.200 − 0.950) = 102.250 m.

Alternative height of collimation (HC) method:
HC at A = Reduced level (RL) + staff reading
= 100 m + 3.200 = 103.200 (AOD)
Level at B = HC at A − staff reading at B
= 103.200 − 0.950 = 102.250 m

Conclusion

Establishing a reliable datum and taking precise levels are essential aspects of ensuring project quality and accuracy. By following best practices and adhering to professional standards, surveyors and construction professionals can ensure consistent and dependable results throughout the project lifecycle.

Setting Out of Buildings: Approaches and Best Practices

By

Ubani Obinna

-

February 7, 2024

Table of Contents

In the construction of buildings, setting out is the critical first step on the path to successful construction. This process involves translating architectural plans onto the designated building site, and carefully identifying the exact locations and dimensions of foundations, walls, and other structural elements. This article discusses the process of setting out, exploring its methods, tools, and significance in building construction.

Upon gaining access to the designated building site, the contractor shall possess both the site layout plan and detailed drawings outlining the construction of the intended structure(s). Under prevalent building contract models, the onus of ensuring accurate setting out falls squarely upon the contractor. With site possession secured, preparatory measures and the critical process of setting out the building can commence. These activities can be broadly categorized into three distinct phases:

1. Site Clearance: The initial stage necessitates the removal of vegetation, debris, and any obstructive elements from the designated building area. This comprehensive clearing process ensures a level and stable platform, fostering accurate measurements and subsequent construction activities.

2. Building Setting Out: Following the precise dimensions and specifications detailed in the architectural plans, the contractor marks out the intended locations for foundation trenches. Stringent adherence to prescribed lengths, thicknesses, and angles for wall lines is achieved through the utilization of strings, pegs, or temporary structures. Rigorous verification at each step, employing measuring tools and surveying equipment, guarantees optimal alignment and dimensional accuracy.

3. Establishing a Datum Level: A crucial aspect of the setting-out process involves the establishment of a precise datum level. This reference point serves as the foundational elevation for measuring vertical distances throughout the construction process, ensuring consistent levels across the entire structure.

The Site Plan

Every construction project commences with a prepared site plan, typically drawn to a scale of 1:500. This document maps existing site features, property lines, proposed buildings, setbacks, surrounding streets, roads, services, and ancillary works like car parks, retaining walls, and landscaping. It may even show new ground levels to guide development.

typical site plan 1 — **Figure 2:** Typical generic site plan of a proposed construction

Setting Out

While the site plan serves as a drafted blueprint, its true merit lies in its practical implementation. The success hinges on the ability of onsite personnel to accurately translate the plan’s directives into physical reality. This process, known as setting out, ensures foundations are laid at the designated level and ground floors precisely match the intended height above the finished ground level. Each step relies on the accurate transfer of information from the plan to the actual construction site.

Constraints and Reference Points

The architect or engineer tasked with planning the site layout often encounters restrictions known as “building lines.” These invisible lines dictate the building’s frontage, requiring a clear depiction of the site plan with precise referencing to fixed points.

**Figure 3:** Typical site plan with complete dimensions and details

Examples include road or pavement kerb lines and extensions of existing building frontages. Figure 3 exemplifies a typical site plan layout. In the absence of designated building lines, the building’s corner positions are determined by dimensions carefully measured from at least two established fixed points.

Essential Tools for Setting Out

To effectively execute the setting-out process, a specific toolkit is indispensable:

Dumpy level, tripod, and staff: This trio enables precise levelling and height measurements.
Wooden pegs: Sturdy pegs facilitate marking key points.
Builder’s square: Useful for checking the orthogonality of angles
Hammers and nails: Securing pegs efficiently requires hammering skills.
Measuring tapes: Precise distance measurement is crucial.
Builder’s line and level: This combination ensures proper alignment and horizontal reference.
Measuring rods: Additional tools for accurate measurement tasks.
Crosscut hand saw: Cutting materials for marking purposes.
Timber boards: Useful for creating temporary structures or markers.

Process of Building Setting Out

The following established process is used in the setting out of buildings;

**Figure 4:** Elements of a proper building setting out

(1) Locating Fixed Reference Points (Baseline): The initial stage of setting out on a construction site involves identifying and verifying existing reference points employed during the preliminary survey. These points, often materialized as nails, pins, hooks, or markings, serve as crucial anchors for subsequent measurements. Re-measuring these points confirms their accuracy and ensures a reliable foundation for further site layout.

(2) Marking Building Corners and Baselines: For projects lacking predefined building lines, the initial step involves physically marking the building’s corners using wooden pegs with nails driven into their tops. Steel tape measurements taken from known fixed points, referencing dimensions provided in the site plan, guide the precise placement of these initial markers. Repeating measurements and ensuring their consistency further enhance accuracy.

(3) Utilizing Building Lines for Corner Placement: When building lines are present, their fixed positions allow for the deployment of a builder’s line stretched between nails marking the line’s ends. By measuring along this line with a steel tape, the locations of building corners can be established. Again, pegs and nails mark these corner points for easy visualization and reference.

(4) Employing Pythagoras for Precise Positioning: With two building corners accurately defined, the remaining corners can be determined using the Pythagorean theorem applied to right-angled triangles formed by the existing points (Figure 6). Once all four corners are marked with pegs, a final verification step involves diagonally measuring the resulting rectangle to ensure minimal discrepancies.

(5) Addressing Limitations of Basic Markers: The initial placement of corner markers using pegs and nails faces limitations, namely potential disturbance during subsequent construction activities and the lack of vertical alignment information. To overcome these limitations and establish robust reference points, the use of profile boards is necessary (Figure 7).

(6) Utilizing Profile Boards for Enhanced Functionality: Each profile board set is positioned at a building corner, strategically “looking” along two adjacent walls. These boards offer several advantages:

Undisturbed by Subsequent Work: They are positioned outside the excavation area, ensuring their preservation throughout construction.
Vertical and Horizontal Alignment: They provide reference points for both horizontal dimensions and vertical elevations.
Repeatability: They facilitate the precise re-establishment of corner locations, measurements, and levels if needed.

While the builder’s lines strung between profile boards might initially hinder excavation activities, alternative methods for marking out excavation boundaries are employed during the later stages of foundation construction.

Digital Setting Out Equipment

While traditional methods using pegs and profiles remain prevalent, advancements in technology offer alternative approaches for setting out on construction sites. Large companies leveraging Electronic Position and Distance Measurement (EPDM) equipment can optimize efficiency and potentially eliminate the need for extensive use of temporary markers.

In such scenarios, one or two strategically placed pegs with nails, protected from disturbance, can serve as enduring reference points throughout the construction process. These key markers, established during the initial survey, facilitate the precise layout of various site elements, encompassing roads, sewers, house drains, and even the buildings themselves. This technology-driven approach streamlines the setting-out process while upholding accuracy and efficiency.

Conclusion

Setting out, though seemingly simple, is a cornerstone of successful construction. Its proper execution requires expertise, precision, and the use of the right tools. By understanding the process, embracing its significance, and leveraging available technologies, construction professionals can ensure a good and accurate alignment and positioning for every structure they build.

Von Mises Stress and the Design of Steel Structures

By

Ubani Obinna

-

February 5, 2024

Table of Contents

In structural engineering, understanding stress and its impact on the materials to be used for construction is very important. One of the most widely used theories for assessing material failure is the Von-Mises stress criterion. In this brief article, we explore the physical significance of Von-Mises stress and its application in the design of steel structures.

Von Mises stress is a scalar value derived from the multi-axial stress state that provides a simplified metric for predicting yielding in ductile materials. The stress state within a material under load is not always a simple uniaxial tension or compression.

In most practical scenarios, a combination of normal and shear stresses act simultaneously, resulting in a multi-axial stress state. Analyzing and visualizing such complex stress states can be challenging. The von Mises yield criterion, proposed by Richard von Mises in 1913, offers a simplified approach to assess the potential for yielding in ductile materials under multi-axial loading.

Theoretical Foundation

The von Mises stress is based on the distortion energy theory, which postulates that yielding occurs when the distortion energy per unit volume reaches a critical value. Distortion energy refers to the energy stored in the material due to its deformation (excluding the volumetric change associated with hydrostatic pressure). Mathematically, the von Mises stress (σ_vm) is defined as:

σ_VM= √(σ_₁² + σ_₂² + σ_₃² – σ_₁σ_₂ – σ_₂σ_₃ – σ_₃σ_₁)

Application of von Mises Stress in the Design of Steel Structures

Steel structures experience complex, multi-axial stress states – a far cry from the uniaxial tension or compression encountered in textbooks. Analyzing and visualizing these variable stress states can be cumbersome. The genius of von Mises stress lies in its ability to simplify this complexity into a single scalar value. By focusing on distortion energy, a measure of deformation excluding volumetric changes, it essentially condenses the multi-axial stress state into a single, meaningful indicator of potential yielding.

For ductile materials like steel, exceeding the yield strength signifies the onset of plastic deformation. By comparing the calculated von Mises stress at critical points in the structure with the steel’s yield strength, engineers gain invaluable insights:

The von Mises stress plays a very important role in various engineering design and analysis scenarios:

Finite Element Analysis (FEA): In FEA software, complex structures are discretized into small elements, and stress tensors are calculated at each element point. The von Mises stress is often used as a failure criterion by comparing it to the material’s yield strength. If the von Mises stress exceeds the yield strength at any point, it indicates potential plastic deformation or yielding.
Pressure Vessel Design: Pressure vessels experience complex stress states due to internal pressure and external loads. Analyzing the von Mises stress distribution helps ensure the vessel remains within its safe operating limits and prevents catastrophic failure.
Bridges and Buildings: From towering bridges to intricate beam-column connections, engineers rely on von Mises stress to assess the load-carrying capacity of steel members, optimizing their design to withstand diverse loading conditions like wind, seismic forces, and live loads.

EN 1993-1-5 (Part 1-5 of Eurocode 3) entitled “Plated Structural Elements”, establishes the regulations for preventing local buckling in steel plated structures. This section presents designers with two primary design methodologies: the “Effective Width Method” and the “Reduced Stress Method“.

The Reduced Stress Method offers a simplified approach compared to the general form, making it particularly suitable for serviceability checks and designing non-uniform members such as tapered beams, webs with openings, and plates with non-orthogonal stiffeners. This method assumes a linear stress distribution up to the buckling limit of the first-yielding plate element. The entire cross-section remains fully effective until this stress limit is reached.

The Reduced Stress Method can be employed to determine stress limits for both stiffened and unstiffened plates. It incorporates the von Mises criterion to account for the interaction between various stress types within the plate.

Furthermore, Eurocode 3 Part 6, permits the use of ”stress design” approach for the limit state design of steel shells. When employing the stress design approach, limit states must be evaluated across three distinct stress categories: primary, secondary, and local. This categorization typically relies on the von Mises equivalent stress at specific points. However, it is very important to recognize that this value is not suitable for assessing buckling stresses.

Factor of Safety

If we define the safety factor as N = Failure Stress / Analysed Stress

For Von-Mises Stresses, the safety factor is computed using;
N = f_y / σ_vm = Yield stress/von Mises stress

For instance, if the steel plate above is to be constructed with steel grade S275 (f_y= 275 N/mm²), the factor of safety using von Mises criteria is;

Factor of Safety = (0.65 × 275)/38.5 = 4.64

In this case, the failure stress is taken as 0.65f_y.

Limitations and Considerations

It is important to remember that the von Mises stress is a simplified criterion and has certain limitations:

Material Dependence: The von Mises criterion is primarily applicable to ductile metals. It may not be accurate for brittle materials or materials with significant pressure sensitivity.
Anisotropy: The criterion assumes isotropic material behaviour, meaning the material properties are the same in all directions. If a material exhibits anisotropic behaviour, alternative yield criteria like Tresca or Hill might be necessary.
Temperature Dependence: The yield strength of materials varies with temperature. The von Mises stress should be considered in conjunction with temperature-dependent material properties for accurate failure prediction.

Conclusion

Understanding the concept of von Mises stress is fundamental for engineers and scientists working with materials under complex loading conditions. This simplified metric provides a valuable tool for assessing potential yielding and guiding design decisions. However, it is important as well to acknowledge and consider the limitations associated with this criterion for accurate and reliable engineering analysis.

Finite Element Analysis (FEA) for Structures

By

Ubani Obinna

-

February 4, 2024

Table of Contents

Finite element analysis has become an indispensable tool in structural engineering, enabling complex simulations of real-world scenarios through numerical methods. By discretizing a continuous structure into smaller finite elements, the governing equations describing its behaviour are transformed into a solvable system of algebraic equations.

The human mind, despite its remarkable capabilities, faces inherent limitations in comprehending the intricacies of complex systems in one single step. In response to this challenge, a fundamental approach emerges: decomposition or discretization of complex problems. By systematically dividing systems into their constituent elements whose behaviour is easily understood, we can then reconstruct the whole system to evaluate its overall response. This principle, employed by engineers, scientists, and even economists, forms the cornerstone of the finite element method.

At its core, the finite element method seeks to approximate solutions to complex problems by substituting them with simpler counterparts. This inherent simplification necessitates an approximate, rather than exact, solution. The method achieves this by discretizing the solution domain into smaller, interconnected subregions – the finite elements. Depending on the system, a finite number of such elements can often sufficiently represent the true system, which we classify as discrete.

However, certain systems require infinite subdivision, demanding the mathematical abstraction of infinitesimals. This leads to differential equations or their equivalent, implying an infinite number of elements, characterizing continuous systems. While digital computers excel at solving large-scale discrete problems, their finite capacity precludes exact solutions for continuous systems. Existing mathematical techniques for exact solutions are often limited to oversimplified scenarios.

In this context, the finite element method emerges as a powerful computational tool capable of addressing a wide spectrum of one, two, and three-dimensional structural problems governed by ordinary or partial differential equations. It empowers engineers and scientists to navigate the complexities of intricate systems by leveraging the power of approximate solutions derived from carefully constructed discretizations.

This article introduces the fundamental concepts of Finite Element Analysis (FEA) and guides beginners through the practical application of the method using code snippets. We explore the underlying theory, discretization techniques, and implementation considerations, equipping readers with the basic knowledge to embark on their FEA journey.

Applications of the Finite Element Method

While renowned for its impact in structural mechanics, the finite element method’s reach extends far beyond. Its potential has been successfully harnessed to address diverse engineering challenges, spanning heat conduction, fluid dynamics, seepage flow, and even the complexities of electric and magnetic fields. This widespread applicability has attracted the attention of mathematicians, who have adopted the method for tackling intricate boundary value problems and beyond.

The underlying foundation of this versatility lies in the ability to numerically solve both ordinary and partial differential equations. By recognizing the underlying similarities between seemingly disparate engineering problems, the finite element method emerges as a unifying tool capable of unlocking solutions across a vast spectrum of disciplines.

Steps in Finite Element Analysis for Structures

The complex nature of real-world materials, such as solids, liquids, and gases, necessitates their representation in the finite element method as a collection of smaller subdivisions called finite elements. These interconnected elements share specified points of contact known as nodes or nodal points, typically located on their boundaries.

Since the precise variation of a field variable (e.g., displacement, stress, temperature, pressure, or velocity) within the continuum remains unknown, the method assumes that its behavior within each element can be approximated by a simpler function. These approximating functions, also known as interpolation models, are defined based on the nodal values of the field variable.

By formulating field equations (such as equilibrium equations) for the entire continuum, we introduce new unknowns – the nodal values of the field variable. Solving these equations, typically expressed as matrix equations, yields the desired nodal values. With these values in hand, the approximating functions establish the field variable throughout the assemblage of elements. This systematic approach defines the core steps involved in applying the finite element method to diverse problems.

Specifically, considering static structural problems as an example, the step-by-step procedure are as follows:

Step 1: Discretization
The initial stage involves dividing the structure or solution domain into smaller sub-regions called elements. This forms the model representing the actual structure. Careful consideration goes into determining the number, type, size, and arrangement of these elements to ensure an accurate representation.

Step 2: Selecting the Right Displacement Model
Since the exact displacement response of a complex structure under specific loads is unknown, we rely on interpolation models to approximate this behaviour within each element. These models, typically implemented as polynomials, need to be computationally efficient while adhering to convergence requirements essential for accurate solutions.

Step 3: Building the Stiffness Matrices and Load Vectors
Leveraging the chosen displacement model, we derive the stiffness matrix [K_e] and load vector [P_e] for each element. This can be achieved either through equilibrium conditions or a suitable variational principle. The stiffness matrix captures the element’s resistance to deformations, while the load vector represents the external forces acting on it.

Step 4: Assemblage of element equations to obtain the overall equilibrium equations
Since the structure is composed of several finite elements, the individual element stiffness matrices and load vectors are to be assembled in a suitable manner and the overall equilibrium equations have to be formulated as;

[K][Φ] = [P]

where [K] is the assembled stiffness matrix, [Φ] is the vector of nodal displacements, and [P] is the vector of nodal forces for the complete structure.

Step 5: Solution for the unknown nodal displacements
To ensure our model aligns with the real-world structure’s constraints, we incorporate boundary conditions into the overall equilibrium equations. These conditions represent fixed points, support conditions, applied forces, or other restrictions on the structure’s behavior. With these adjustments, the equilibrium equations take the form:

[K]Φ = P

For linear problems, this system of equations can be readily solved using efficient numerical methods, revealing the unknown nodal displacements throughout the structure. However, for nonlinear problems, the complexities introduced by material behavior or large deformations necessitate an iterative approach. Each step in this sequence involves updating the stiffness matrix and/or load vector based on the current solution estimate until convergence is achieved.

Step 6: Computation of element strains and stresses
From the known nodal displacements (Φ), if required, the element strains and stresses can be computed by using the necessary equations of solid or structural mechanics.

Code Snippet for FEA of Trusses on Python

To bridge the gap between theory and practice, let’s explore fundamental FEA concepts through code snippets in a chosen programming language (e.g., Python). We’ll demonstrate element stiffness matrix formulation for simple elements like trusses, followed by global system assembly and solution using basic numerical libraries.

Import necessary libraries

import numpy as np

# If needed for solving linear systems:
import scipy.linalg as la

2. Define truss element properties:

def element_stiffness_matrix(E, A, L):
    """Calculates the 2x2 stiffness matrix for a truss element."""
    ke = E * A / L * np.array([[1, -1], [-1, 1]])
    return ke

def element_load_vector(q, L):
    """Calculates the 2x1 load vector for a truss element."""
    fe = q * L / 2 * np.array([[1], [1]])
    return fe

3. Assemble global stiffness matrix and load vector

def assemble_global_system(elements, nodes):
    """Assembles the global stiffness matrix and load vector."""
    K = np.zeros((nodes * 2, nodes * 2))
    P = np.zeros((nodes * 2, 1))
    for element in elements:
        node1, node2, E, A, L, q = element
        ke = element_stiffness_matrix(E, A, L)
        fe = element_load_vector(q, L)
        d = 2 * (node1 - 1)  # Global degree of freedom indices
        K[d:d+2, d:d+2] += ke
        P[d:d+2] += fe
    return K, P

4. Apply boundary conditions:

def apply_boundary_conditions(K, P, fixed_nodes, fixed_values):
    """Applies boundary conditions to the global system."""
    for node, value in zip(fixed_nodes, fixed_values):
        d = 2 * (node - 1)
        K[d:d+2, :] = 0
        K[:, d:d+2] = 0
        K[d, d] = 1
        P[d] = value

5. Solve for nodal displacements

def solve_displacements(K, P):
    """Solves the system of equations for nodal displacements."""
    U = la.solve(K, P)
    return U

6. Calculate element stresses and reactions:

def calculate_stresses(elements, U):
    # ... (Implementation for stress calculation based on element type)

def calculate_reactions(K, U, fixed_nodes):
    # ... (Implementation for reaction force calculation)

Conclusion

This article has provided a foundational understanding of FEA, its theoretical principles, and practical implementation through code snippets. By delving deeper into specific element types, advanced material models, and non-linear analysis, beginners can progressively build their FEA skillset and tackle increasingly complex engineering problems.

Form Finding in Tensile Membrane Structures

By

Ubani Obinna

-

February 2, 2024

Table of Contents

Tensile membrane structures, characterized by their lightweight and expressive forms, have captivated architects and engineers for decades. However, achieving their desired shape while ensuring structural integrity poses a unique challenge. This is where form finding, an important optimization process, becomes indispensable.

Tensile membrane structures are defined by their doubly curved surfaces and rely on inherent tension for stability. This curvature plays a critical role in distributing prestresses across the membrane, leading to its structural integrity. Without appropriate curvature, desired force distribution cannot occur, rendering the surface structurally unsound. This intrinsic quality of opposing curvatures is fundamental to membrane structures.

However, traditional architectural drawing methods fall short of capturing these complex surfaces. Instead, specialized approaches are required to model and analyze the force flow within the membrane. This process, referred to as form finding, aims to identify the optimal shape that achieves equilibrium under given boundary conditions.

Unlike conventional structures governed by bending rigidity, tensile membranes resist loads through in-plane tension. This inherent flexibility allows for diverse geometries but necessitates a form-finding process to determine a shape that satisfies both equilibrium (balance of internal forces and external loads) and architectural intent.

In essence, form finding seeks to establish an equilibrium surface within defined constraints. By iteratively adjusting boundary conditions and analyzing stress distribution, it strives to produce a surface that is not only aesthetically pleasing and functionally appropriate but also structurally viable. The boundary conditions encompass the edge elements and support points that define the membrane’s physical limitations. Finally, the achieved equilibrium form ensures that all points on the surface remain in a state of physical balance under the applied tensile load.

The application of tensile load to a membrane surface can expose areas of compression, manifesting as wrinkles. This wrinkle formation indicates an uneven distribution of prestress, hindering the ability of the surface to achieve equilibrium.

Form Finding Process

Form-finding methodology encompasses two primary approaches: physical and numerical.

Physical Modelling

The initial forays into tensile membrane structures relied heavily on physical models. Soap films and flexible fabric pieces were employed to create physical models that guided subsequent fabrication. Soap films, due to their exceptional thinness and lack of shear resistance, proved particularly adept at visualizing force distribution and arriving at optimal forms.

This practice of physical modelling in architecture remains relevant today. It offers an economical and rapid means to explore design solutions prior to delving into numerical analysis, often leading to more creatively inspired forms. Additionally, physical models can enhance comprehension of complex surfaces through three-dimensional visualization.

Numerical Modelling

Technological advancements in numerical modelling and computer capabilities have profoundly impacted the design, manufacturing, and analysis of membrane systems. In numerical form finding, the membrane surface is discretized into a mesh, upon which both manufacturing and structural analysis are conducted.

This approach offers independence from the membrane’s physical properties, such as thickness or elasticity. Various numerical methods, including Force Density, Dynamic Relaxation, and Finite Element Methods, are employed to achieve the equilibrium surface.

The design and analysis of membrane building systems witnessed a significant turning point with the Munich Olympic Stadium project in 1972. Klaus Linkwitz’s pioneering introduction of the Force Density Method in 1971 marked the first numerical approach specifically tailored to the unique needs of tensile structures.

During form finding, membrane surfaces can be conceptualized as fluid systems. Their final shape emerges from the interplay between defined boundary conditions and the applied tensile load distribution. Our control over the membrane’s geometry lies solely in manipulating these boundary conditions and the load distribution ratios. By iteratively adjusting these variables, the resulting form is refined based on various criteria, including structural capacity, functionality, and aesthetic integration.

However, it’s important to note that modifications to the form directly impact the membrane’s structural capacity. Therefore, ensuring the structural adequacy of the membrane against anticipated loads takes precedence over aesthetic considerations and functionality. This prioritization is particularly critical for snow loads, given the inherent limitations in the mechanical properties of membrane materials.

Form Finding Methods

Several form finding methods exist, each offering distinct advantages and complexities. Here, we delve into three prominent approaches:

1. Physical Soap Film Modelling: This classical technique utilizes soap films stretched across a physical boundary representing the desired supports. The minimal surface formed by the soap film, driven by surface tension, embodies the equilibrium state and serves as a direct physical analogue of the optimal shape. While elegant and intuitive, limitations arise from scalability and complex boundary conditions.

2. Force Relaxation Methods: These computational approaches, such as dynamic relaxation, mimic the relaxation process of a physical system towards equilibrium. An initial geometry is iteratively adjusted based on unbalanced forces, gradually converging towards a stable form. The method’s versatility allows for incorporating various boundary conditions and material properties, but convergence speed and numerical stability require careful consideration.

3. Energy Minimization Techniques: Based on variational calculus, these methods seek the minimum potential energy configuration of the system. By formulating the energy function encompassing membrane strain, boundary constraints, and external loads, the optimal shape can be obtained through numerical minimization algorithms. This approach offers a robust framework for complex geometries and material behaviours, but computational demands can be significant.

The choice of form-finding method depends on various factors, including project scale, complexity, desired accuracy, and available computational resources. In practice, hybrid approaches combining physical and numerical methods are often employed, leveraging the strengths of each technique.

Benefits of Form Finding

Beyond achieving equilibrium, form finding plays a crucial role in:

Optimizing material usage: By distributing tension efficiently, form finding minimizes required membrane material, leading to cost-effective and sustainable designs.
Controlling deflections: Targeted form finding can mitigate excessive deflections under wind and snow loads, enhancing structural performance and serviceability.
Integrating architectural vision: The iterative nature of form finding allows for incorporating aesthetic considerations and tailoring the shape to harmonize with the architectural intent.

Conclusion

In conclusion, form finding lies at the heart of successful tensile membrane structures. By creating the required force equilibrium in the structure, it enables the creation of lightweight, efficient, and visually striking structures that push the boundaries of architectural expression. As computational tools and design methodologies continue to evolve, form finding will remain a vital tool for shaping the future of tensile membrane architecture.

Scaffold in Building Construction: Uses, Types, Materials, and Design

By

Ubani Obinna

-

February 1, 2024

Table of Contents

In construction, scaffolds play an important role as temporary structures facilitating access to elevated work areas and providing safe work platforms. Typically assembled from steel or aluminium alloy tubes connected through clips or couplings, these structures enable workers to reach heights otherwise inaccessible while ensuring their safety during various building operations.

The design and construction of all scaffolds must adhere strictly to the established minimum requirements stipulated in both the Work at Height Regulations 2005 and BS EN 12811-1: 2003 – Temporary works equipment. Scaffolds. Performance requirements and general design. Following these regulations and standards is paramount for ensuring the structural integrity, stability, and overall safety of scaffolds, ultimately protecting the well-being of workers relying on them during construction projects.

Types of Scaffolding

There are different types of scaffolds, each suited to different construction needs and budgets. Some of the common types of scaffolds are;

1. Putlog Scaffolds: This design employs a single row of vertical supports (standards) arranged at a pre-determined distance from the wall to accommodate the desired platform width. Horizontal members (ledgers) connect the standards, further secured to the building structure with cross-members known as putlogs. Primarily utilized for brick construction, this scaffold grows incrementally alongside the building’s rising height.

2. Independent Scaffolds: Featuring two rows of standards interconnected by transoms (horizontal cross-members), independent scaffolds stand independently without relying on the building for support. This makes them ideal for framed structures. Secure ties bind the scaffold to the building at regular intervals, typically every 3.6 meters vertically and 6 meters horizontally. Attachment methods include bridles (horizontal tubes bearing on the wall’s interior), reveal pin tubes within openings, or, in the absence of suitable openings, raking tubes inclined towards the building from the ground.

**Figure 2:** Typical tubular steel independent scaffold (Chudly and Greeno, 2005)

3. Slung Scaffolds: Slung scaffolds are suspended by wire ropes or chains and lack inherent raising or lowering mechanisms. They primarily access high ceilings or undersides of elevated roofs. As a result, secure anchorage points, often utilizing the roof’s structural members above the designated work area are very important for their support.

A minimum of six evenly spaced suspension ropes or chains, securely fastened at both ends, is standard practice. Platforms, constructed similarly to conventional scaffolds with ledgers, transoms, and timber boards, require guardrails and toe boards for safety. For platforms exceeding 2.4 meters x 2.4 meters, stress analysis of supporting tubular components is recommended.

4. Truss-Out Scaffolds: This variation of the tied independent scaffold relies solely on the building for support, employed in situations where constructing a conventional ground-level scaffold is impractical or unfavourable. The projecting supporting structure, known as the truss-out, anchors to the building using adjustable struts secured internally between the floor and ceiling, from which cantilever tubes extend. Standard right-angle couplers are utilized except for securing rakers. The remaining scaffold construction follows the format of conventional independent scaffolds.

**Figure 3:** Typical truss-out scaffold details (Chudly and Greeno, 2006)

5. Suspended Scaffolds (Outrigger Cantilever Type): This configuration features a working platform suspended from supports such as outriggers cantilevering over a building’s upper edge. In this form, they serve as temporary access to the building facade for cleaning or light maintenance. Many modern tall structures incorporate suspension tracks either within the fascia or upper edge beam or a cradle suspension track is fixed to the flat roof, supporting a manual or powered trolley with retractable davit arms that hold the suspended working platform or cradle.

**Figure 4:** Typical suspended scaffold details (Chudly and Greeno, 2006)

All suspended cradles must comply with the minimum platform board, guardrail, and toe board requirements mandated by the Work at Height Regulations 2005. Cradles can be single units or grouped to form a continuous platform, connected at their abutment ends with hinges.

6. Mobile Tower Scaffolds: Primarily used by painters and maintenance personnel, these scaffolds provide quick and easy access to ceilings by offering a movable working platform. Essentially, they are square towers constructed from scaffold tubes mounted on braked wheels. Users access the platform via short opposing inclined ladders or a single inclined ladder within the tower base area.

7. Birdcage Scaffold: For extensive, high-level work areas, birdcage scaffolds offer a comprehensive solution. These structures employ a grid-like arrangement of vertical supports (standards), horizontal connectors (ledgers), and cross-members (transoms) to support a solid working platform at the desired height. Stability is paramount, necessitating close placement of standards (not exceeding 2.4 meters apart) and adequate bracing throughout the scaffold.

8. System scaffolds: This type of scaffold provides a modern alternative to traditional steel tube scaffolds. Utilizing innovative interlocking connections instead of loose couplers, they offer ease of erection, adaptability, and assembly/disassembly capabilities even for semi-skilled personnel. Notably, their design inherently adheres to the Work at Height Regulations 2005, ensuring proper handrail placement, lift heights, and other safety measures. An additional benefit is the elimination of internal cross-bracing, creating a clear walkthrough space on all levels. However, depending on the specific construction, facade bracing may still be necessary.

**Figure 7:** Systems scaffold (Chudly and Greeno, 2005)

Materials for Scaffolding

The choice of material for scaffolding is multifaceted, influenced by factors like weight, strength, deflection characteristics, and corrosion resistance. There are four prevalent types of materials used in scaffolding:

tubular steel,
tubular aluminium alloy,
timber, and
bamboo.

Tubular Steel

British Standard 1139 sets guidelines for both welded and seamless steel tubes, typically measuring 48mm in outer diameter with a 38mm bore. Galvanization offers protection against corrosion, while ungalvanized options require post-use treatments like painting or oil baths. Steel tubes are nearly three times heavier than their aluminium counterparts. Steel boasts superior strength, enabling longer spans compared to aluminium due to less deflection (approximately one-third that of aluminium).

Tubular Aluminum Alloy

Seamless tubes of aluminium alloy with a 48mm outer diameter are specified in BS 1139. Aluminium generally doesn’t require protective treatment unless exposed to specific elements like damp lime, wet cement, or seawater. In such cases, a bitumastic paint coating before use is recommended. A significant advantage of aluminium is its lightweight nature, offering easier manoeuvrability and setup.

Timber

While less frequently employed in the UK, timber remains a prevalent choice in many developing countries for temporary scaffolding structures. Structural-quality softwood is utilized in either putlog or independent configurations. Unlike metal scaffolds with coupling fittings, timber members are traditionally joined together with wire or rope.

Bamboo

Unlike its steel counterpart, bamboo scaffolding boasts an eco-friendly advantage. It is a fast-growing renewable resource, with some species reaching maturity in as little as five years. This rapid growth rate makes it a sustainable alternative to steel, which requires significant energy and resources to produce.

Additionally, bamboo scaffolding is biodegradable, decomposing naturally after use. Bamboo’s unique properties make it surprisingly well-suited for scaffolding. Its high tensile strength allows it to support significant weight, while its natural flexibility makes it adaptable to various shapes and geometries.

The selection of scaffolding material hinges on various project-specific considerations. Steel’s strength and stability make it ideal for heavy-duty tasks and longer spans, while aluminium’s lightweight properties offer advantages in portability and ease of use. Timber, though less common in some regions, presents a traditional and potentially cost-effective option in suitable settings.

Scaffold boards

Scaffold boards are important components of scaffolds used to provide safe working platforms within the scaffold structure. They are expected to adhere to the specifications outlined in BS 2482. Boards must be constructed from specified softwoods, measuring 225 mm x 38 mm in cross-section and with a maximum length of 4.800 meters.

To prevent splitting, the ends of each board are required to be bound with a minimum of 25 mm wide x 0.9 mm thick galvanized hoop iron. This reinforcement extends at least 150 mm along each edge and is secured with at least two fixings per end. The specified strength of the boards ensures they can safely support a uniformly distributed load of 6.7 kN/m² when supported at 1.2m intervals.

Scaffold Fittings

British Standard dictates the specifications for both steel and aluminium alloy scaffolding fittings, ensuring consistency and reliability across materials. These fittings typically allow for connections between various metal tubes unless otherwise specified by the manufacturer. Here’s a breakdown of key fittings used in metal scaffolding:

Double Coupler: The primary load-bearing component, essential for connecting ledgers to standards.
Swivel Coupler: Composed of two joined single couplers, enabling rotation for connecting tubes at any angle.
Putlog Coupler: Specifically designed for attaching putlogs or transoms to horizontal ledgers.
Base Plate: Distributes weight from the standard’s foot onto a sole plate or firm ground. Variations with threaded spigots cater to uneven terrain.
Split Joint Pin: Expands to grip and join tubes end-to-end.
Reveal Pin: Fits into tube ends to create adjustable struts.
Putlog End: Converts a standard tube into a putlog using a flat plate attachment.

**Figure 9:** Typical steel scaffold fittings (Chudly and Greeno, 2005)

Stabilisers, Outriggers, or Diagonal Bracings

An optional attachment that can be adjusted to ensure ground contact where the surface is uneven. They should be attached securely to enable direct transfer of loads without slipping or rotating.

Structural Design of Scaffolds

The structural design of a scaffold system is very important for its functionality and safety. It addresses the optimal size, shape, and configuration of each component to guarantee the structure’s ability to withstand anticipated loads and external forces. By meticulously considering these details, the design mitigates excessive deflection, ensures overall stability, and prevents catastrophic collapse, thereby safeguarding both workers and the surrounding environment.

Understanding Scaffold Loads

The primary objective of scaffold structural design is to ensure the structure can withstand all anticipated loads throughout its service life. These loads can be broadly categorized into two groups:

Dead loads: The weight of the scaffold itself, including all its components like standards, ledgers, braces, and working platforms.
Live loads: The weight imposed on the scaffold by workers, materials, equipment, and any environmental factors like wind or snow.

Accurately calculating both dead and live loads is paramount for designing a safe and efficient scaffold. Safety factors are then applied to these calculated loads to account for uncertainties and potential overload scenarios.

typical structural analysis of scaffolds — **Figure 10:** Typical structural analysis of scaffolds

Scaffold Stability Analysis

An important aspect of structural design is assessing the scaffold’s stability under various loading conditions. This analysis involves evaluating factors like:

Overturning: The potential for the scaffold to tip over due to uneven loading or external forces like wind.
Deflection: The amount of bending or sagging experienced by the scaffold components under applied loads. Deflection limits are established to ensure worker safety and platform functionality.
Bending, Axial, and Shear stresses: The internal forces acting within scaffold members due to applied loads. These stresses must be within the material’s capacity to prevent failure.

Sophisticated engineering software can be employed for complex scaffold stability analysis, considering factors like material properties, connection details, and geometric configurations.

Key Design Principles

Several fundamental principles guide the structural design of scaffolds:

Strength: All scaffold components must be strong enough to support the anticipated loads without exceeding their material yield strength.
Stiffness: The scaffold must be sufficiently stiff to minimize deflection and maintain platform stability under load.
Ductility: Scaffold materials should exhibit some degree of ductility to deform slightly under overload, providing a warning sign before failure.
Stability: The scaffold must resist overturning and maintain its overall stability under varying load conditions.
Safety: The design must prioritize worker safety by incorporating guardrails, toeboards, and other fall protection measures.

Sources and Citations

Chudly R. and Greeno R. (2005): Construction Technology (4th Ed.). Pearson Education Limited, England
Chudly R. and Greeno R. (2006): Advanced Construction Technology (4th Ed.). Pearson Education Limited, England