Rectangular vs Circular Columns: Strength, Design, and Uses

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April 2, 2024

Table of Contents

Columns are major structural members in buildings, with the sole purpose of transferring vertical and horizontal loads from beams and slabs to the foundation. The design of columns is very important in ensuring the stability and safety of buildings. The prevalent column geometries in building construction encompass rectangular, circular, and square cross-sections.

While rectangular and square sections are very popular in building construction due to the rectangular nature of walls, instances favouring circular or other column geometries do arise. For instance, for architectural reasons, standalone columns in the middle of halls or walkways are preferably circular due to aesthetic reasons. Furthermore, circular columns exhibit superior seismic performance compared to rectangular ones. A thorough understanding of these rationales is important for structural engineers and architects, particularly during the scheme development stage.

This knowledge facilitates informed decision-making regarding column geometry, thereby reducing the risk of inadequate construction work. It is to be emphasized that the selection of column geometry holds significant importance, similar to the determination of size, orientation, and positioning of both columns and beams within the structural system. While rectangular and circular columns satisfy the same structural functions, they offer distinct advantages and disadvantages in terms of strength, design considerations, and preferred applications.

Rectangular columns in an institutional building

Selection Criteria for Column Geometry in Building Construction

The configuration of a column’s cross-section, similar to other building elements, is governed by a multitude of factors such as architectural aesthetics, ease of construction, functional requirements, and structural demands.

Aesthetic Considerations

Architectural appeal is one of the prominent factors influencing column geometry. Square and rectangular columns can be easily placed within walls and covered with plastering, such that nobody will know that the columns were there in the first place. When square columns are not feasible due to structural size requirements, rectangular columns can be employed, such that the width will be equal to the width of the wall. This leads to a smooth flow and alignment of walls without undesirable projections.

Square columns can easily be hidden in walls

On the other hand, circular sections are often favoured for their inherent visual harmony. However, their application is frequently confined to specific contexts such as luxury mansions, monumental buildings, educational institutions, verandas, and public buildings. Circular columns are preferable when the column element is standing alone since they will rarely blend into walls without projecting out of the wall lines.

Formwork Considerations and Cost Implications

There is a significant disparity between the construction of column formwork of rectangular/square and circular sections. Achieving a smooth, curved profile for circular columns demands enhanced craftsmanship and superior formwork materials, often including new plywood or aluminium systems.

This complexity translates to challenges in attaining a high-quality concrete finish. Likewise, plastering a curved surface to the desired level is considerably more intricate compared to a flat plane, leading to increased construction time and cost. Given the budgetary constraints inherent in many medium-scale projects, clients often prioritize cost-effectiveness, making rectangular/square columns more favourable.

Wooden formwork construction for circular column

Furthermore, the pressure exerted on column formwork by fresh concrete can be more complicated when compared with square or rectangular sections. Consequently, stricter quality control measures and more elaborate formwork support systems are mandatory on-site. The increased level of workmanship required for these considerations translates to a significant cost increase compared to rectangular column construction.

Functional Requirements

The intended use of a space significantly influences column selection. In office environments, columns often serve as display surfaces for artwork, signage, or bulletin boards. Rectangular sections provide inherently planar surfaces ideal for such applications. This rationale extends to basement parking areas, where rectangular or square columns offer suitable flat surfaces for traffic flow notices, warnings, and signage – commonly observed in shopping malls and other commercial establishments.

Structural Considerations

While circular columns generally exhibit superior seismic performance, specific scenarios necessitate enhanced stiffness in a particular direction. This is particularly relevant for slender buildings with a limited base width and an extended length. To achieve this, engineers may favour a rectangular column with a larger dimension oriented towards the slender direction.

Moment of Inertia

A critical factor in column strength is the moment of inertia (I), a property that reflects a section’s resistance to bending. Circular columns boast a higher and more uniform I value across all axes compared to rectangular columns. This translates to superior resistance to bending moments and deflection, making them ideal for structures subjected to high lateral loads, such as bridges.

Shapes of rectangular and circular sections

For rectangular sections, the moment of inertia (I) is given by;
I = bh³/12

For circular sections, the moment of inertia (I) is given by;
I = πD⁴/64

Let us consider a square column of dimensions 250 mm x 250 mm. This column will have an area of 0.0625 m². A circular column that will produce a similar area will have a diameter of 282 mm.

The square section will have a moment of inertia I = bh³/12 = (0.25 × 0.25³)/12 = 3.255 × 10^-4 m⁴
The circular section will have a moment of inertia I = πD⁴/64 = (π × 0.282⁴)/64 = 3.104 × 10^-4 m⁴

Therefore for sections of similar cross-sectional area, square sections have higher moment of inertia than circular sections.

Buckling Resistance

Another important aspect of strength consideration of columns is buckling, which is the tendency of a slender column to bend under compressive loads. Circular sections, due to their uniform distribution of material around the centroidal axis, offer superior buckling resistance compared to rectangular columns, especially when the load is not perfectly centred.

Material Efficiency

Rectangular columns, however, can be more material-efficient for specific loading conditions. By strategically orienting the rectangular section with the larger dimension towards the direction of higher bending moment, engineers can achieve optimal load-carrying capacity with less material compared to a circular column of equivalent area.

Reinforcement

Circular columns typically require more reinforcement bars due to their curved shape. While this can impact material costs, it also enhances their overall compressive strength.

Applications

Rectangular Columns:

Widely used in buildings due to their ease of construction and efficient space utilization in confined areas.
Preferred for load-bearing walls where columns can be integrated with the wall structure.
Suitable for situations where higher bending moment capacity is required in a specific direction by adjusting the rectangular section.

Circular Columns:

Ideal for bridge piers and other structures subjected to high lateral loads due to their superior bending and buckling resistance.
Used in open spaces or architectural features where their aesthetic appeal is valued.
Preferred in seismic zones due to their uniform distribution of strength across all axes.

Comparative Design of Square and Circular Columns

Investigate the design requirements of the two column cross-sections analysed above;
Square section = 250 x 250mm
Circular section = 282 mm (diameter)
Axial load = 1000 kN (No bending moment considered except secondary moments)
The effective length of the columns about the major and minor axis = 3000 mm√
Compressive strength of concrete = C20/25
Yield strength of reinforcement = 500 MPa

Square section design

Column slenderness about y-axis
Radius of gyration; i_y = h/√(12) = 7.2 cm
Slenderness ratio (5.8.3.2(1)); l_y = l_0y / i_y = 41.6

Column slenderness about z-axis
Radius of gyration; i_z = b/√(12) = 7.2 cm
Slenderness ratio (5.8.3.2(1)); l_z = l_0z / i_z = 41.6

Min end moment about y-axis; M_01y= min(abs(M_topy), abs(M_btmy)) + e_iyN_Ed = 7.5 kNm
Max end moment about y-axis; M_02y = max(abs(M_topy), abs(M_btmy)) + e_iyN_Ed= 7.5 kNm
Min end moment about z-axis; M_01z = min(abs(M_topz), abs(M_btmz)) + e_izN_Ed = 7.5 kNm
Max end moment about z-axis; M_02z = max(abs(M_topz), abs(M_btmz)) + e_izN_Ed = 7.5 kNm

Design bending moment about y-axis (cl. 5.8.8.2 & 6.1(4))
Nominal 2^nd order moment; M_2y = N_Ede_2y = 14.3 kNm
M_0ey = max(0.6M_02y + 0.4M_01y, 0.4M_02y), max(0.6M_02y – 0.4M_01y, 0.4M_02y)) = 7.5 kNm

Equivalent moment from frame analysis;
M_0ey = max(0.6M_02y + 0.4M_01y, 0.4M_02y) = 7.5 kNm

Design moment;
M_Edy = max(M_02y, M_0ey + M_2y, M_01y + 0.5M_2y, N_Ed × max(h/30, 20 mm))
M_Edy = 21.8 kNm

Design bending moment about z-axis (cl. 5.8.8.2 & 6.1(4))
Nominal 2^nd order moment; M_2z = N_Ede_2z = 13.4 kNm
M_0ez = max(0.6M_02z + 0.4M_01z, 0.4M_02z), max(0.6M_02z – 0.4M_01z, 0.4M_02z)) = 7.5 kNm

Equivalent moment from frame analysis;
M_0ez = max(0.6M_02z + 0.4M_01z, 0.4M_02z) = 7.5; kNm

Design moment;
M_Edz = max(M_02z, M_0ez + M_2z, M_01z + 0.5M_2z, N_Ed× max(b/30, 20 mm))
M_Edz = 20.9 kNm

Area of reinforcement provided = 6Y20 (Asprov = 1885 mm²)

Design axial resistance of section; N_Rd = (A_cf_cd) + (A_sf_yd) = 1527.9 kN

Ratio of applied to resistance axial loads; ratio_N = N_Ed / N_Rd = 0.655
Exponent a = 1.46
Biaxial bending utilisation;
UF = (M_Edy / M_Rdy)^a + (M_Edz / M_Rdz)^a = 0.899

Description	Unit	Provided	Required	Utilisation	Result
Moment capacity (y)	kNm	34.23	21.84	0.64	PASS
Moment capacity (z)	kNm	40.43	20.88	0.52	PASS
Biaxial bending utilisation				0.90	PASS

Circular section design

Column slenderness about both axis
Radius of gyration; i_y = i_z = h / 4 = 7.1 cm
Slenderness ratio (5.8.3.2(1)); l_y = l_z = l_0y / i_y = 42.6

Ecc. due to geometric imperfections (y-axis); e_iy = l_0y /400 = 7.5 mm

Min end moment about y-axis; M_01y = min(abs(M_topy), abs(M_btmy)) + e_iyN_Ed = 7.5 kNm
Max end moment about y-axis; M_02y = max(abs(M_topy), abs(M_btmy)) + e_iyN_Ed = 7.5 kNm

Design bending moment about y-axis (cl. 5.8.8.2 & 6.1(4))
Nominal 2^nd order moment; M_2y = N_Ede_2y = 12.8 kNm
M_0ey = max(0.6M_02y + 0.4M_01y, 0.4M_02y), max(0.6M_02y – 0.4M_01y, 0.4M_02y)) = 7.5 kNm

Equivalent moment from frame analysis;
M_0ey = max(0.6M_02y + 0.4M_01y, 0.4M_02y) = 7.5 kNm

Design moment;
M_Edy = max(M_02y, M_0ey + M_2y, M_01y + 0.5M_2y, N_Ed × max(h/30, 20 mm))
M_Edy = 20.3 kNm

Design bending moment about z-axis (cl. 5.8.8.2 & 6.1(4))
Nominal 2^nd order moment; M_2z = N_Ede_2z = 12.8 kNm
M_0ez = max(0.6M_02z + 0.4M_01z, 0.4M_02z), max(0.6M_02z – 0.4M_01z, 0.4M_02z)) = 7.5 kNm

Equivalent moment from frame analysis;
M_0ez = max(0.6M_02z + 0.4M_01z, 0.4M_02z) = 7.5 kNm

Design moment;
M_Edz = max(M_02z, M_0ez + M_2z, M_01z + 0.5M_2z, N_Ed× max(b/30, 20 mm))
M_Edz = 20.3 kNm

Resultant design moment; M_Ed = √(M_Edy² + M_Edz²) = 28.7 kNm

Area of steel provided = 6Y20 (Asprov = 1885 mm²)

Description	Unit	Provided	Required	Utilisation	Result
Moment capacity (y)	kNm	30.11	20.31	0.67	PASS
Moment capacity (z)	kNm	31.68	20.31	0.64	PASS
Combined capacity	kNm	30.11	28.72	0.95	PASS

As can be seen from the design results, under similar axial loading and support conditions, square columns and circular columns of equal area will demand approximately the same area of reinforcement. However, once uniaxial and bi-axial bending moments are involved, we should expect a completely different behaviour in the quantity of reinforcements required.

Conclusion

The choice between rectangular and circular columns depends on a project’s specific requirements. When prioritizing strength and buckling resistance for structures like bridges or seismic zones, circular columns are often favoured. However, for ease of construction, space optimization, and cost-effectiveness in buildings, rectangular columns are the preferred choice. Understanding the strengths and limitations of each shape allows engineers to make informed decisions for optimal structural performance and aesthetics.

In conclusion, the selection of column cross-sectional geometry necessitates a comprehensive evaluation involving architectural intent, functional considerations, and structural demands. While aesthetics may favour circular columns in specific scenarios, the complexities and cost implications associated with formwork often render rectangular/square sections the more pragmatic choice for many building projects.

Design of Piled Raft Foundations

By

Ubani Obinna

-

March 12, 2024

Table of Contents

Piled raft foundations represent an economical and practical solution for situations where a conventional raft foundation falls short of design requirements. This type of foundation system strategically integrates a limited number of piles beneath the raft, allowing the raft itself to still contribute significantly to the load-bearing capacity.

This key distinction separates them from traditional pile foundations, where the primary responsibility for supporting the structure rests solely on the piles. Consequently, piled raft foundations occupy a unique position within the broader category of pile foundation systems, necessitating more complex analytical, design, and application considerations.

The piled raft foundation system itself comprises three key components:

Piles: These deep foundation elements transfer heavy structural loads to deeper and more stable soil layers.
Raft: This shallow foundation element distributes loads across its footprint, primarily utilizing the near-surface bearing capacity of the soil.
Supporting Soil: This plays a critical role in transmitting loads from the structure to the foundation system.

**Figure 1:** Load transfer in piled raft foundation systems

By combining the capabilities of piles and rafts, the piled raft foundation system offers a synergistic solution for supporting heavy structures. It leverages the deep load-bearing capacity of piles while simultaneously utilizing the shallow load-carrying capacity of the raft. This combined approach effectively resists both vertical and lateral loads, ensuring the serviceability and stability of the structure.

Load-settlement behaviour within piled raft systems is influenced by a multitude of factors. The physical and mechanical properties of the soil, raft foundation, and pile foundation all play a significant role. Furthermore, the construction sequence of the building itself can also exert an influence. While a range of simplified, approximate, and advanced methods exist for analyzing this complex system, the most realistic results are typically obtained through the use of advanced finite element analysis.

Piled raft foundations have become a prominent choice for a wide range of demanding construction projects. Their versatility and ability to handle challenging soil conditions make them ideal for high-rise buildings in urban centers, where maximizing footprint usage is crucial. Landmark structures like the Burj Khalifa in Dubai and the Shanghai Tower in China employed piled raft foundations to ensure stability and support their immense weight.

Burj Khalifa Tower Foundation system — **Figure 2:** Burj Khalifa tower foundation was built with piled raft

Piled rafts also find application in infrastructure projects such as offshore wind turbine installations, where they provide a stable base for these towering structures amidst wind and wave loads. Furthermore, their ability to resist buoyancy is valuable for projects on sites with high water tables, such as waterfront developments or structures built on reclaimed land.

Types of Piled Raft Foundation

Piled raft foundation can be broadly categorised into two;

Piled raft for settlement control, and
Piled raft for load transfer

Piled raft for settlement reduction
While raft foundations can offer adequate bearing capacity, they may still be susceptible to excessive settlement. Traditionally, this issue is addressed by incorporating a basement and a basement raft, which effectively reduces the total load acting on the foundation system. However, when this approach is not feasible, an alternative solution involves introducing a limited number of piles beneath the raft.

These piles function by transferring a portion of the overall load away from the raft itself. As the piles do not need to carry the entire load, the required number is significantly lower compared to a traditional piled foundation design. Additionally, due to this load redistribution, the settlement experienced by the raft is brought within acceptable limits.

Piled raft for load transfer
The second category of piled rafts, designated as “conventional,” finds application in scenarios where the underlying soil exhibits pronounced weakness and a high water table is present. In such conditions, the adoption of a raft foundation becomes essential.

These rafts serve a dual purpose: first, resisting the buoyant forces exerted by the groundwater, and second, transmitting all net structural loads to the piles for transfer to deeper, more competent soil layers. Consequently, the number of piles necessitated in this scenario will be considerably greater compared to the previous case described.

In essence, piled raft foundations offer a synergistic approach to foundation design, leveraging the strengths of both piles and rafts. The raft’s ability to share the load and reduce differential settlements, coupled with the piles’ capacity to act as “stress reducers” and “settlement reducers” while enhancing the overall bearing capacity, paves the way for efficient and reliable foundation systems, particularly in challenging soil conditions.

Load Transfer and Sharing in Piled Raft Foundation

The load distribution in foundation systems plays a crucial role in determining the interaction between the structure and the underlying soil. While a footing or raft primarily affects the shallow soil layers (approximately 1-2 times its width), pile foundations transfer loads to deeper strata. Combining these two approaches in a combined Pile and Raft Foundation (CPRF) system creates a complex interplay influenced by several factors.

These factors include:

The rigidity of the raft: A stiffer raft tends to distribute load more evenly across the foundation, while a flexible raft allows for greater load transfer to the piles.
Soil properties: The stiffness and bearing capacity of the underlying soil layers significantly impact the load transfer mechanisms.
Pile characteristics: The number, depth, and rigidity of the piles within the raft influence how the load is shared between the piles and the raft.

The piled raft foundation system, responsible for transferring a structure’s load to the underlying soil, presents a complex interaction that have captivated researchers for years. Early notions, often overly conservative, assumed that the raft, in direct contact with the soil, offered no resistance to applied loads.

However, recent research works challenge this assumption, highlighting the raft’s significant contribution, particularly in clayey soils subjected to substantial structural loads. These studies reveal that the raft bears a portion of the load, while the piles carry the remaining portion through a creep mechanism.

The analysis and design of piled raft foundations, therefore, demand careful consideration of various critical factors, including:

Raft thickness and dimensions
Pile length, diameter, and configuration within the raft
Underlying soil properties
Stiffness characteristics of both the pile and raft

These factors collectively influence the load-sharing mechanism between the piles and the raft, ultimately impacting the stability and serviceability of the structure.

Furthermore, the success of piled raft foundation hinges on understanding the two key interaction types:

Pile-to-pile interaction: This interaction depends heavily on the soil’s elastic modulus, the pile slenderness ratio (s/d), and the pile length. Ignoring this interaction can lead to underestimating settlements and bending moments in the raft, compromising structural safety.
Pile-to-raft interaction: This interaction influences the load distribution between the piles and the raft, affecting the overall settlement behaviour of the foundation system.

Nevertheless, combined pile and raft foundation systems have challenged researchers for years due to the complexities associated with load-sharing and analysis. Several researchers have endeavoured to address this challenge, notably, by proposing simplified methods that incorporate various simplifications. However, these methods should be employed with caution due to their inherent limitations.

Quantifying the load contribution of each element within a piled raft system remains a topic of ongoing investigation. While some researchers suggest piles carry 50-80% of the total load, others provide a wider range of 30-60% for the raft’s contribution, emphasizing the dependence on factors like soil conditions, pile length, and spacing. Their research also highlights a decreasing raft contribution with denser pile spacing and increased pile length.

Further research strengthens the argument for the raft’s significant role, attributing up to 50% of the structural load to its contribution.

Classical Methods of Detemining Pile-Raft Load

Some classical methods of detemining pile-raft load sharing are discussed below.

Randolph Method
In the method proposed by Randolph (1994), load sharing ratio between pile group and pile raft, pile raft stiffness and settlement of piled raft can be calculated by using Eq. 1 to Eq. 4.

α = Q_R/Q_PG = 0.2/[(1 – 0.8(K_R/K_PG)] × K_R/K_PG ——— (1)
β = Q_R/Q_PG = 1/(1 + α) ——— (2)
K_PR = [1 – 0.6(K_R/K_PG)]/[1 – 0.64(K_R/K_PG)] × K_PG ——— (3)
S = Q/K_PR ——— (4)

Where,
α = Load sharing ratio between raft and pile group,
β = Load sharing ratio between pile group and pile raft,
K_R = Stiffness of the raft
K_PG= stiffness of the pile group
K_PR = stiffness of the piled raft
S = Settlement of piled raft,
Q = Design load

Poulos-Davis-Randolph (PDR) Method
In the method proposed by Poulos, Davis, and Randolph, load sharing ratio between raft and piled raft (X) can be determined using Randolph (1994) method and piled raft settlement (S) can be established using Poulos and Davis (1980) method.

X = Q_r/Q_pr= [(1 – α_rp)k_r]/[k_pg+ (1 – 2α_rp)k_r] ——— (5)
S = Q_pr/k_pr ——— (6)
k_pr = [k_pg+ (1 – 2α_rp)k_r]/[1 – α_rp² (k_r/k_pg)] ——— (7)
α_rp = 1 – [In(r_c/r)/ς] ——— (8)
r_c = √(A/nπ) ——— (9)
ς = In(r_m/r) ——— (10)
r_m = {0.25 + ξ[2.5ρ(1 – v) – 0.25]}L ——— (11)
ρ = G_avg/G_l ——— (12)
ξ = G_l/G_b ——— (13)

where;
Q_r = Load carried by the raft
Q_pr = Load carried by piled raft,
k_r = stiffness of the raft
k_pg = stiffness of the pile group
k_pr = Stiffness of the piled raft,
α_rp = Interaction factor,
A = Raft area,
n = Number of piles
L = length of piles
r_m = Maximum radius from pile axis,
G_l= Shear modulus of soil along pile shaft
G_b = Shear modulus of soil at pile end
ν = Poisson’s ratio of soil.

Design of Piled Raft

According to Poulos (2001), the design of a piled raft foundation can be effectively divided into three distinct stages. The initial stage focuses on a preliminary analysis, estimating the impact of varying pile numbers on the overall load capacity and settlement of the structure. This analysis is typically approximate in nature.

Additionally, preliminary design stages often benefit from incorporating load-sharing ratios and settlement values derived from empirical studies and case histories. The expertise of designers familiar with piled raft systems remains an important component in achieving optimal outcomes.

The second stage goes deeper, aiming to identify the specific locations where piles are necessary and providing an initial indication of the required piling specifications.

Finally, the third stage represents the detailed design phase. Here, a more refined analysis is employed to confirm the optimal number and positioning of the piles. Additionally, this stage gathers crucial information for the structural design of the entire foundation system.

Complexities inherent to piled raft systems necessitate the use of sophisticated analytical methods during the design stage. These methods, such as the finite element method (FEM), boundary element method (BEM), equivalent element method (EEM), and plate-on-spring method (POSM), account for the numerous variables influencing the system’s behaviour.

Approximate methods like the “strip on springs” approach and the “plate on springs” offer further avenues for analysis. These methods offer simplified representations of the raft and piles (as springs) to understand their interaction.

For more detailed analysis, researchers recommend resorting to numerical methods, with the Finite Element Method (FEM) being the most prevalent choice. Software like SAP2000 and PLAXIS 3D foundation are prime examples of FEM-based solutions. FEM provides approximate solutions for various nonlinear engineering problems, including those encountered in combined piled raft foundation analysis.

It is important to note that while simplified methods can provide reasonable results for preliminary assessments, numerical methods like FEM offer a superior level of accuracy and detail for complex piled raft foundation systems. Recognizing the limitations of each approach is paramount for selecting the most appropriate analytical tool for a specific project.

Summarily, understanding the load-sharing mechanisms within piled raft foundation remains a dynamic field of research. While simplified methods offer initial insights, numerical methods like FEM provide a more robust and accurate means for analyzing these complex foundation systems. Choosing the appropriate analytical approach requires careful consideration of the project’s specific needs and complexities.

Preliminary Design Example of Piled Raft Foundation System

This section considers the preliminary design of a proposed piled raft foundation system. The raft is 750 mm thick and the superstructure load distribution on the raft is shown in Fgure 3. The initial proposed distribution of the piles, comprising of 9 number of 600 mm diameter piles is shown in Figure 4.

**Figure 3:** Load distribution on the raft foundation system

**Figure 4:** Preliminary arrangement of the piled raft system

Thickness of raft = 750 mm
Modulus of subgrade reaction k_s = 10000 kN/m²/m
Modulus of horizontal compressibility n_h (medium dense wet sand) = 4000 kN/m²/m
Pile diameter = 600 mm
Depth of pile = 10 m

Horizontal modulus of subgrade reaction = n_h(z/d) ——— (14)

The horizontal modulus of subgrade reaction was used in modelling the piles, and the spring stiffness varied with depth according to equation (14).

**Figure 5:** 3D render of the piled raft foundation on Staad Pro software

**Figure 6:** Finite element model of the piled raft foundation on Staad Pro software

**Figure 7:** Bending moment diagram of the piles

**Figure 8**: Load applied on the foundation

Total load applied on the foundation = 4(350) + 2(500) + 6(600) = 6000 kN

From the analysis results,
Total load transferred to the piles = 900 kN
Therefore, total load resisted by the raft = 6000 – 900 = 5100 kN

In this case, about 85% of the load is resisted by the raft foundation. If it is a piled raft foundation where the piles are to be used in load transfer, the arrangement of the piles will have to be changed. However, if it is a system where the piles are to be used for settlement control, the pile arrangement can be evaluated for acceptance or rejection.

In a different scenario when the number of piles was increased to 18 (additional piles were introduced along the column gridlines), 69% of the load was resisted by the raft. Therefore, preliminary analysis requires a careful consideration of the location and number of piles in the system.

Conclusion

Piled raft foundation systems offer a powerful solution for navigating complex soil conditions and supporting substantial loads. The design process involves a meticulous three-stage approach, starting with a preliminary analysis, then progressing to detailed location and quantity determination of piles, and finally culminating in a refined analysis for optimal pile placement and structural design of the entire foundation.

This staged approach ensures an efficient and cost-effective foundation that leverages the strengths of both raft foundations and pile foundations. Piled raft systems are a versatile solution for high-rise buildings, infrastructure projects, and construction on challenging sites, providing the stability and support necessary for a wide range of demanding applications.

Sources and Citations

Randolph M. F. (1994). Design methods for pile groups and piled rafts, 13th ICSMFE, New Delhi, India, 61-82.
Poulos H.G. and Davis E.H. (1980). Pile foundation analysis and design, John Willey and Sons, New York, USA.
Poulos H.G. (2001). Piled raft foundations: designs and applications. Geotechnique 51(2):95-113

A General Overview of Dynamics of Structures

By

Ubani Obinna

-

March 9, 2024

Table of Contents

Structural dynamics is a field of study that discusses the behaviour of structures subjected to dynamic loads. It encompasses the analysis, design, and evaluation of structures under the influence of various forces and vibrations. While static analysis focuses on the behaviour of structures under constant or slowly varying loads, dynamics of structures explores how structures respond to dynamic loads, which are forces that change rapidly with time.

In addition to static loads, a structural system can be subjected to variable (dynamic) loads induced by factors such as wind and wave action, earthquakes, impact, blasts, and vehicular/pedestrian traffic (which causes vibration and fatigue in bridges). Therefore, understanding the dynamic behaviour of structures is important for ensuring the safety and serviceability of structures in scenarios involving:

Earthquakes: Ground motions induced by earthquakes can cause significant dynamic forces on structures, potentially leading to failure if not properly accounted for in design.
Wind: Wind loads can create significant dynamic effects, especially on slender structures like tall buildings and suspension bridges.
Vibrations: Structures subjected to human activity, machinery operation, or traffic can experience vibrations, which can lead to fatigue, discomfort, or even damage if not managed effectively.
Blast loads: Explosions and other rapid pressure changes can create extremely dynamic forces that need to be considered in the design of structures in specific environments.

This article provides a comprehensive exploration of the key concepts and methodologies involved in the analysis and design of structures subjected to dynamic loads.

**Figure 1:** Typical Dynamics of buildings

Dynamic Analysis

The methods of analysis used for static loads are insufficient to analyze the ‘dynamic’ or ‘time-varying’ loads and their impacts. When compared to the values of displacement that are produced by static loading, the values that are produced by the response of structural members to time-varying loads will likewise be time-varying, and this can result in substantially larger values.

To make the concept of structural dynamics clearer, let us consider a structural element that is subjected to an externally applied load. By considering the equilibrium of applied forces and the internal forces that correspond to those forces, it is always possible to compute the internal stresses and displacements of a structure, regardless of whether the force that is being applied is “static” or “dynamic.”

Assuming that the structure is linearly elastic, the internal forces and the displacements are linearly proportional. If, on the other hand, the force is applied in a dynamic manner, two additional types of internal forces are generated as a consequence. The first of these is referred to as the “inertia forces,” and it is related to the acceleration. The second of these is referred to as the “damping forces,” and it is proportional to the velocity.

In this article, we are going to present a fundamental introduction to the principles of structural dynamics, and how it can be extended to the design of structures.

Importance of Structural Dynamics Analysis

A comprehensive dynamic analysis of structures can reveal the potential for serviceability failures that would be entirely undetectable through a purely static evaluation. For instance, there have been documented cases of oil rigs being decommissioned in relatively calm seas due to the initiation of oscillations that were unacceptably uncomfortable for the crew.

Similarly, electric transmission lines have been known to develop severe dynamic oscillations, referred to as “galloping,” to the extent that the lines made contact. While this phenomenon may not necessarily lead to structural collapse, it undoubtedly constitutes a serviceability failure from the perspective of electricity consumers.

Structures under construction are especially susceptible to dynamic effects. For example, temporary damping measures were deemed necessary for the towers of the Forth Road Bridge in Scotland to mitigate dynamic effects before the installation of the main cables. Even from a purely structural strength perspective, dynamic analysis can be crucial if fatigue is a primary concern.

In such scenarios, it becomes essential to predict not only the magnitude of stresses within the structure but also the frequency at which various stress levels occur. This is because a consistently applied low stress can have a more detrimental fatigue impact than an occasional instance of higher stress.

Characteristics of a Structural Dynamic Problem

A structural dynamic problem differs from a static loading problem in two significant ways. Firstly, the dynamic problem is characterized by its time-varying nature. Since both the loading and the response change over time, a dynamic problem does not have a single solution like a static problem does. Instead, the analyst must determine a series of solutions corresponding to different times of interest in the response history. As a result, dynamic analysis is inherently more complex and time-consuming than static analysis.

The second and more fundamental distinction between static and dynamic problems is illustrated in Figure 2. When a simple beam is subjected to a static load p (as shown in Figure 2a), its internal moments, shears, and deflected shape depend solely on this load and can be calculated using established principles of force equilibrium.

**Figure 2:** Basic difference between static and dynamic loads: (a) static loading; (b) dynamic loading.

However, when the load p(t) is applied dynamically (as shown in Figure 2b), the resulting beam displacements depend not only on this load but also on inertial forces that resist the accelerations causing them. Consequently, the internal moments and shears in the beam must balance not only the externally applied force p(t) but also the inertial forces resulting from the beam’s accelerations.

Inertial forces, which oppose the accelerations of the structure, are the key distinguishing characteristic of a structural dynamics problem. Generally, if the inertial forces constitute a significant portion of the total load equilibrated by the internal elastic forces of the structure, the dynamic nature of the problem must be considered in its solution.

On the other hand, if the motions are so slow that the inertial forces are negligible, the response analysis for any specific time can be conducted using static structural analysis methods, despite the load and response being time-varying.

Equations of Motion

The mass, stiffness, and damping (energy absorption capability), of a linearly elastic structural system are the basic physical parameters that define the system when it is subjected to external dynamic loading. Consider the ‘dash-pot’ model (representing a simple building with a single storey) that is presented in Figure 3. This model can be used to demonstrate the fundamental idea behind dynamic analysis.

**Figure 3:** Vibration modelling of a single storey structure

The structure is subjected to a time-varying force denoted by f(t), in which k is the spring constant that links the lateral storey deflection (x) to the storey shear force, and c is a damping coefficient that relates the dashpot’s damping force to the velocity. If it is assumed that all of the mass, m, is located at the beam, then the structure will be considered a single-degree-of-freedom (SDOF) system.

It is possible to write the equation of motion of the system as follows;

mẍ + cẋ + kx = f(t) ——– (1)

Types of Vibration

Free Vibration

While our initial discussion addressed the impact of time-varying loads on structural behaviour (dynamic behaviour), a foundational understanding of vibration in simple structures, independent of dynamic loads, proves most beneficial. This specific type of vibration, termed “free vibration,” arises whenever a structure experiences a disturbance from its state of static equilibrium. The initiation of free vibrations can be attributed to either impulsive events such as a collision or explosion, or to sudden movements in the structure’s support system.

In this case, the system is set to motion and allowed to vibrate in the absence of applied force f(t). Letting f(t) = 0, equation (1) becomes:

mẍ + cẋ + kx = 0 ——– (2)

Dividing equation (2) by the mass m, we have:
ẍ + 2ξωẋ + ω²x = 0 ——– (3)

Where;
2ξω = c/m ——– (4)
ω² = k/m ——– (5)

The solution to the equation depends on whether the vibration is damped or undamped.

Undamped Free Vibration

In the absence of not only time-dependent forces, but also any mechanisms for energy dissipation within the vibrating system, the resulting motion can be classified as both free and undamped. Realistically, energy losses due to factors such as friction and air resistance are unavoidable. Therefore, the concept of undamped vibration, while theoretically useful, represents an idealized scenario that disregards these energy-dissipating phenomena. Nevertheless, it remains a valuable tool for theoretical analysis.

In this case, c = 0, and the solution to the equation of motion may be written as:
x = Asinωt + Bcosωt ——– (6)

where ω = √(k/m) is the circular frequency. A and B are constants that can be determined by the initial boundary conditions.

Damped Free Vibration

The phenomenon of damping arises from the inevitable energy loss that occurs during vibration. This lost energy is either dissipated as heat within the structure or radiated outwards, often in the form of sound waves.

Internal friction within the structural materials themselves contributes a portion to this energy loss, with frictional losses at structural joints playing an additional role. While air resistance can also contribute to energy dissipation, it is typically considered a secondary factor.

To model the effects of damping in a simplified manner, engineers often employ a theoretical element known as a “dashpot” system.

If the system is not subjected to applied force and damping is present, the corresponding solution becomes:
x = A exp(λ₁t) + B exp(λ₂t) ——– (7)

λ₁ = ω[-ξ + √(ξ² – 1)] ——– (8)
λ₂ = ω[-ξ – √(ξ² – 1)] ——– (9)

The solution of equation (7) changes its form with the value defined as:

ξ = c/2√mk ——– (10)

Forced Vibration

When a structure experiences time-varying loads or continuous disturbances to its supports, the resulting motion is classified as forced vibration. The specific time-dependent influence that triggers this motion is termed excitation. The nature of the forced vibration – its frequency, amplitude, and overall behaviour – is directly tied to the characteristics of the excitation itself.

In essence, the excitation acts as an external “driving force” that dictates the response of the structure. This response can vary significantly depending on the excitation. For instance, a harmonic excitation (a smoothly oscillating force) will lead to a harmonic vibration with the same frequency but potentially a different amplitude. Conversely, a more impulsive excitation, like a sudden impact, can induce a transient vibration with a complex frequency spectrum.

If a structure is subjected to a sinusoidal motion such as a ground acceleration of ẍ = F sinω_ft, it will oscillate and after some time the motion of the structure will reach a steady state. For example, the equation of motion due to the ground acceleration (from equation (3)) is:

ẍ + 2ξωẋ + ω²x = Fsinω_ft ——– (11)

The solution to the equation we’ve been examining can be broken down into two key components. The first, known as the complementary solution (represented by equation 6), captures the transient behaviour of the system. If the system experiences any damping, the oscillations associated with this component will gradually diminish over time.

This decay effect eventually leads the system to reach a steady state, where it vibrates with a constant amplitude and frequency. This sustained vibration, termed forced vibration, is solely described by the second part of the solution, the particular solution, expressed as:

x = C₁sinω_ft + C₂cosω_ft ——– (12)

A key observation here is that the forced vibration occurs at the frequency of the excitation force, denoted by ω_f, rather than the natural frequency of the structure itself, ω. Essentially, the external force dictates the frequency of the vibration. The term -F/ω² within the particular solution represents the static displacement D caused by the force, essentially accounting for the inertia of the structure.

Now, let’s explore the dynamic response of the structure under varying excitation frequencies relative to its natural frequency (ω):

Low-Frequency Excitation (ω_f/ω > 1): When the applied force oscillates at a frequency significantly lower than the structure’s natural frequency, the response exhibits a characteristic termed quasi-static. In this regime, the system behaves as if it were under a constant load. The response is primarily governed by the stiffness of the structure, and the resulting displacement amplitude closely resembles the static deflection that would occur under a constant force of the same magnitude.
High-Frequency Excitation (ω_f/ω < 1): Conversely, when the excitation frequency is much higher than the natural frequency, the response becomes primarily dependent on the mass of the structure. The displacement amplitude in this case is generally less than the static deflection (D < 1). This is because the structure’s inertia can effectively resist the rapidly oscillating force.
Resonance (ω_f/ω ≈ 1): The most critical scenario arises when the excitation frequency nears the natural frequency of the structure (ω_f/ω ≈ 1). Under these conditions, a phenomenon known as resonance occurs. Resonance drastically amplifies the displacement amplitude, potentially leading to catastrophic consequences for the structure. In essence, the external force synchronizes with the structure’s natural tendency to vibrate, causing a dramatic buildup of energy within the system.

The simplest periodic motion equation can be written as;

y(t) = Asin(ωt + φ₀) ——– (13)

where A is the amplitude of vibration, φ₀ is the initial phase of vibration, and t is time. This case is presented in Fig. 6a. The initial displacement y₀ = Asinφ₀ is measured from the static equilibrium position. The number of cycles of oscillation during 2π seconds is referred to as circular (angular or natural) frequency of vibration ω = 2π/T (radians per second or s^-1), T (s) is the period of vibration. Figure 6b, c presents the damped and increased vibration with constant period.

**Figure 6:** Types of oscillatory motions

Degrees of Freedom

The concept of degrees of freedom (DOF) plays a crucial role in both statics and structural dynamics. While the definition remains the same – the number of independent parameters that uniquely define the spatial positions of all points in a structure – its interpretation differs subtly between these two fields.

In statics, the DOF is often associated with structures modeled as collections of absolutely rigid discs. Here, a DOF greater than or equal to one signifies a geometrically changeable system. Such a system wouldn’t typically be considered a realistic engineering structure, as real structures exhibit some level of deformation. Conversely, a DOF of zero implies a geometrically unchangeable and statically determinate system – a structure with a unique solution for its equilibrium under applied loads.

However, in structural dynamics, the focus shifts to the deformation of the structural members themselves. A DOF of zero in this context indicates an absolutely rigid body, incapable of any displacement in space. This scenario is purely theoretical, as all real structures exhibit some degree of flexibility.

Furthermore, structures can be broadly classified into two categories based on their DOF:

Structures with Concentrated Parameters: These represent structures where the distributed mass of individual members can be neglected compared to lumped masses concentrated at specific points along the members.
Structures with Distributed Parameters: These structures are characterized by a uniform or non-uniform distribution of mass throughout their components. Analyzing these structures often requires more complex mathematical tools compared to those used for concentrated parameter systems.

From mathematical point of view, the difference between the two types of systems is the following: the systems of the first class are described by ordinary differential equations, while the systems of the second class are described by partial differential equations.

Distributed Mass Systems

While the lumped mass model offers a valuable simplification for many structures, it’s important to recognize that all real structures are fundamentally distributed mass systems. This implies that they can be conceptually divided into an infinite number of infinitesimal particles. As a consequence, if a distributed mass system experiences repetitive motion, it theoretically possesses an infinite number of natural frequencies and corresponding mode shapes – unique vibration patterns associated with each frequency.

However, the seemingly overwhelming complexity of analyzing a distributed system can be effectively bridged once its natural frequencies and mode shapes are determined. At this point, the analysis becomes mathematically equivalent to that of a discrete system, where the structure’s behavior is represented by a finite number of lumped masses interconnected by springs or other idealized elements.

The key lies in recognizing that, in practical scenarios, only a limited number of modes, typically those associated with lower frequencies, significantly contribute to the overall dynamic response of the structure. By focusing on these dominant modes, engineers can effectively convert the problem of a distributed mass system into a more manageable discrete system. This approach allows for accurate analysis using computationally efficient methods, enabling engineers to assess the dynamic behavior of real-world structures without getting bogged down by the theoretical infinite nature of distributed systems.

Conclusion

Dynamics of structures considers the interplay between time-varying external forces, internal resistance, and the inherent flexibility of structures. This article has looked into the fundamental concepts of free and forced vibrations, recognizing the crucial role of natural frequencies and damping in shaping a structure’s response.

However, real-world forces and ground motions can be incredibly complex. To accurately predict a structure’s behavior under these conditions, engineers typically rely on numerical analysis techniques. One of the most prevalent methods for solving such complex problems is the finite element method.

The analysis of structures subjected to dynamic loads hinges on the ability to model their behaviour effectively. While lumped mass systems offer a practical approach for many structures, the underlying reality of distributed mass systems with infinite natural frequencies cannot be ignored. The key lies in identifying the dominant modes that significantly influence the dynamic response, allowing us to transform the seemingly intractable distributed system into a more manageable discrete one.

In essence, the lumped mass model serves as a powerful tool for approximating the behavior of complex distributed systems. By strategically selecting the most influential modes, engineers can achieve a high degree of accuracy while maintaining computational tractability. This balance between theoretical completeness and practical feasibility is crucial for ensuring the safety and performance of structures subjected to dynamic loads.

Understanding the dynamics of structures equips engineers with the knowledge to design and build resilient structures that can withstand the challenges of the real world. From earthquakes and windstorms to traffic vibrations and human activity, structures must be able to withstand the complex effects of time-dependent loads without compromising safety, functionality, or serviceability. By mastering the principles of dynamics of structures, engineers can ensure that these structures perform their intended function in harmony with the dynamic forces that surround them.

Bending Moment Diagrams for Frames

By

Ubani Obinna

-

March 1, 2024

Table of Contents

In the field of structural engineering, understanding the internal forces acting on framed structures is important for the design of such structures. Among these forces, bending moment plays a very important role in influencing the behaviour of beams and columns in framed structures under various loading conditions. This article discusses the concept of bending moment and its visualization through bending moment diagrams (BMDs) for framed structures.

Understanding Bending Moment

Imagine a beam supported at its ends and subjected to a transverse load (a load acting perpendicular to the beam’s axis). This load induces internal forces within the beam, causing it to bend. The bending moment at any point along the beam’s length represents the turning effect (rotational tendency) or moment created by the internal forces acting on that specific section. It is essentially the product of the force (F) acting at a perpendicular distance (d) from the point of interest, expressed mathematically as:

M = F × d

The bending moment tends to rotate the beam section about an axis perpendicular to its longitudinal axis. A positive bending moment signifies concavity downwards while a negative bending moment indicates concavity upwards.

Bending Moment Diagrams

A bending moment diagram (BMD) is a graphical representation of the bending moment throughout the length of a beam or a member in a framed structure. This diagram helps visualize the variation of the bending moment along the member, enabling engineers to identify critical sections where the moment is highest and assess the potential for bending failure. Bending moment diagrams are plotted in the tension zone of structures.

With the advent of numerous structural analysis and design software, bending moment diagrams can easily be generated using results from finite element analysis.

Steps to Construct a Bending Moment Diagram

Determine the support reactions: This involves analyzing the entire frame to calculate the forces acting at the supports due to the applied loads. For statically determinate frames, the equations of equilibrium are sufficient for determining the support reactions but for statically indeterminate structures, methods like the force method can be used.
Cut the member: Imagine isolating a specific section of the member by making a virtual cut at a chosen point.
Treat the section as a free body: Draw a free-body diagram of the isolated section, including all external forces (support reactions and applied loads) acting on it.
Apply equilibrium equations: Utilize the principles of equilibrium (summation of forces and moments equal to zero) to solve for the internal shear force (V) and bending moment (M) at the cut section.
Repeat for different sections: Choose multiple points along the member’s length and repeat steps 2-4 to determine the shear force and bending moment at each point.
Plot the values: Plot the calculated bending moments on the vertical axis and the member’s length on the horizontal axis, connecting the points to form a smooth curve. This curve represents the bending moment diagram for the member.

Interpreting Bending Moment Diagrams

Bending moment diagrams reveal valuable information about the bending behaviour of a framed structure:

Zero bending moment: Points on the BMD where the curve crosses the horizontal axis indicate locations where the bending moment is zero. These points typically occur at supports or points of contraflexure.
Maximum and minimum bending moment: The peak positive and negative values on the BMD represent the sections experiencing the highest and lowest bending moments, respectively. These sections are often critical for design considerations.
Slope of the BMD: The slope of the BMD at any point signifies the rate of change of the bending moment. A positive slope indicates an increasing moment, while a negative slope represents a decreasing moment.

Applications of Bending Moment Diagrams

Bending moment diagrams are instrumental in various aspects of structural engineering, including:

Structural design: They aid in selecting appropriate beam sizes and materials by identifying sections with high bending moments, ensuring sufficient strength and preventing failure.
Deflection analysis: By knowing the bending moment distribution, engineers can estimate the deflection of the frame using various methods, evaluating its serviceability under load.
Reinforcement detailing: In reinforced concrete structures, BMDs guide the placement of steel reinforcement to counteract the bending moment and ensure adequate structural capacity.

Typical Bending Moment Diagrams for Rigid Frames

F = Total Load
I_AB = I_CD (the moment of inertia of the columns are equal)
K =I_BCh/I_ABL
k₁ = K + 2
k₂ = 6K + 1
k₃ = 2K + 3
k₄ = 3K + 1

Rigid frame subjected to gravity uniformly distributed load on the beam

Bending moment diagram of a frame subjected to gravity uniformly distributed load on the beam (fixed support) — Bending moment diagram of a frame subjected to gravity uniformly distributed load on the beam (fixed supports)

FOR FIXED SUPPORTS
H_A = H_D = Fl/4hk₁
V_A= V_D = F/2
M_A = M_D = Fl/12k₁
M_B= M_C = Fl/6k₁

Bending moment diagram of a frame subjected to gravity uniformly distributed load on the beam (pinned supports)

FOR PINNED SUPPORTS
H_A = H_D = Fl/4hk₃
V_A = V_D = F/2
M_A = M_D = 0
M_B = M_C = H_Ah = Fl/4k₃

Rigid frame subjected to a point load on the beam

Bending moment diagram of a frame subjected to a point load on the midspan of the beam (fixed support) — Bending moment diagram of a frame subjected to a point load on the midspan of the beam (fixed supports)

FOR FIXED SUPPORTS
H_A = H_D = 3Fl/8hk₁
V_A = V_D= F/2
M_A = M_D = Fl/8k₁
M_B = M_C = Fl/4k₁

Bending moment diagram of a frame subjected to a point load on the midspan of the beam (pinned supports)

FOR PINNED SUPPORTS
H_A = H_D = 3Fl/8hk₃
V_A = V_D= F/2
M_A = M_D = 0
M_B = M_C = H_Ah = 3Fl/8k₃

Rigid frame subjected to a horizontal uniformly distributed load on the column

Bending moment diagram of a rigid frame subjected to horizontal uniformly distributed load on the column (fixed supports)

FOR FIXED SUPPORTS
H_A = F - H_D
H_D = Fk₃/8k₁
V_A = -FhK/lk₂ = -V_B
M_A = Fh/4[(K + 3)/6k₁ + (4K + 1)/k₂]
M_B= h(H_A - ½F) - M_A
M_C = H_Dh - M_D
M_D = Fh/4[(K + 3)/6k₁ - (4K + 1)/k₂]

Bending moment diagram of a rigid frame subjected to horizontal uniformly distributed load on the column (pinned supports)

FOR PINNED SUPPORTS
H_A = F/8[(6k₃ - K)/k₃]
H_D = F - H_A
V_D = -V_A = Fh/2l
M_A = M_D = 0
M_B = h(½F - H_D) = 3Fhk₁/8k₃
M_C= H_Dh = Fh/8[(2k₃ + K)/k₃]

Rigid frame subjected to a horizontal point load at the top

Bending moment diagram of a rigid frame subjected to horizontal point load at the top

FOR FIXED SUPPORTS
H_A= H_D = F/2
V_A= -V_D = -3FhK/Lk₂
M_A = M_D = Fhk₄/2k₂
M_B = M_C = 3FhK/2k₂

Bending moment diagram of a rigid frame subjected to horizontal point load at the top (pinned supports)

FOR PINNED SUPPORTS
H_A = H_D = F/2
V_D = -V_A = Fh/l
M_A= M_D = 0
M_B = M_C = Fh/2

Conclusion

Understanding bending moments and their visualization through bending moment diagrams is fundamental for structural engineers. By mastering this concept, engineers can effectively analyze framed structures, optimize designs, and ensure the safety and serviceability of their creations under various loading conditions.

Wind Load Analysis of Tank Farms and Other Cylindrical Structures

By

Ubani Obinna

-

February 28, 2024

Table of Contents

Cylindrical structures like tanks, silos, and chimneys are widely used in various industries, including oil and gas, agriculture, chemical processing, and water storage. These structures are susceptible to wind loads, which can cause significant stresses and potential failures if not properly analyzed and designed.

In industrial and agricultural settings, cylindrical above-ground vertical tank farms are commonly used for the storage of various liquids like petroleum, oil, water and fuel. These tanks are typically welded, thin-walled structures with large diameters, making them susceptible to buckling under wind loads when empty or partially filled.

The failure of such tanks can have devastating consequences, often resulting in significant financial and human losses. Additionally, these failures pose a serious threat to public safety and can have a detrimental impact on the environment.

Cylindrical structures can be affected by wind load

This article discusses the wind load analysis for tank farms and other cylindrical structures, providing a comprehensive overview of the key factors, methodologies, and considerations for engineers.

Understanding Wind Loads

Wind loads are external forces acting on a structure due to the dynamic pressure and drag exerted by moving air. The magnitude and direction of wind loads depend on various factors, including:

Basic Wind Speed (V): This is the reference wind speed, typically determined for a specific location and return period (e.g., 50-year return period). Building codes and standards like ASCE 7-16 (“Minimum Design Loads and Associated Systems for Buildings and Other Structures”) and EN 1991-1-4 provide wind speed maps for various regions.
Exposure Category: This accounts for the surrounding terrain and influences the wind speed experienced by the structure. Different exposure categories are defined in codes, ranging from open terrain to urban and suburban environments.
Topographic Effects: Local terrain features like hills and valleys can significantly influence wind speeds and turbulence intensity.
Structure Shape and Size: The shape and size of the structure play a crucial role in determining the wind pressure distribution. Cylindrical structures experience wind loads differently compared to flat or rectangular structures.

Wind Load Analysis Methods for Cylindrical Structures

Several approaches can be adopted for wind load analysis of cylindrical structures:

Simplified Methods: Building codes often provide simplified procedures for calculating wind pressures on basic shapes like cylinders. These methods typically involve applying an equivalent static wind pressure acting on the projected area of the structure. While convenient, these methods may not be suitable for complex geometries or situations with significant topographic effects.
Analytical Methods: Analytical methods utilize established formulas based on wind tunnel experiments and theoretical principles to calculate wind pressures on cylindrical structures. These methods consider factors like wind speed, exposure category, and surface roughness. However, they may involve complex calculations and require specialized knowledge.
Computational Fluid Dynamics (CFD): This advanced method employs computational software to simulate the flow of air around the structure. CFD can generate detailed pressure distributions on the entire structure, accounting for complex geometries and local effects. However, CFD analysis requires expertise and significant computational resources.

Wind Pressure Distribution, Farm (Left) vs. Solo (Right) Tank (Source: Simscale.com)

Specific Wind Load Considerations for Tank Farms

For the calculation of wind load action effects on circular cylinder elements, the total horizontal wind force is calculated from the force coefficient corresponding to the overall effect of the wind action on the cylindrical structure or cylindrical isolated element.

The calculated effective wind pressure w_eff and total wind force F_W correspond to the total wind action effects and they are appropriate for global verifications of the structure according to the force coefficient method. For local verifications, such as verification of the cylinder’s shell, appropriate wind pressure on local surfaces must be estimated according to the relevant external pressure coefficients, as specified in EN1991-1-4 §7.9.1.

For cylinders near a plane surface with a distance ratio z_g/b < 1.5 special advice is necessary. See EN1991-1-4 §7.9.2(6) for more details. For a set of cylinders arranged in a row with normalized center-to-center distance z_g/b < 30 the wind force of each cylinder in the arrangement is larger than the force of the cylinder considered as isolated. See EN1991-1-4 §7.9.3 for more details.

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.

When analyzing wind loads on tank farms, additional factors come into play:

Spacing and Interaction: The proximity of tanks within a farm can significantly influence wind pressures. Shielding effects and aerodynamic interaction between tanks need to be considered. Several empirical methods and CFD simulations are available to account for these effects.
Appurtenances: Wind loads also act on appurtenances like piping, ladders, and platforms attached to tanks. These loads can be significant and need to be included in the overall wind load analysis.
Dynamic Amplification: Tanks may experience dynamic amplification of wind loads due to their inherent dynamic properties. This can be particularly crucial for slender tanks or those with low natural frequencies.

Wind Load Analysis Example

A cylindrical structure of diameter (b) 5m and length (l) = 20 m is to be constructed in an area of terrain category II with a basic wind velocity v_b of 40 m/s. The orientation of the cylindrical element is vertical and the maximum height above ground of the cylindrical element z = 20 m. The surface of the tank is made of galvanised steel. Calculate the wind force on the tank (Take Air density: ρ = 1.25 kg/m³)

Solution

Calculation of peak velocity pressure

The reference height for the wind action z_e is equal to the maximum height above the ground of the section being considered, as specified in EN1991-1-4 §7.9.2(5). The reference area for the wind action A_ref is the projected area of the cylinder, as specified in EN1991-1-4 §7.9.2(4). Therefore:

z_e = z = 20 m
A_ref = bl = 5 m × 20m = 100 m²

Basic wind velocity v_b = 40 m/s.

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 = 0.19

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.
c_r(z_e) = k_r ⋅ ln(max{z_e, z_min} / z₀) = 0.19 × ln(max{20 m, 2 m} / 0.050 m) = 1.1384

The orography factor is considered as c₀(z_e) = 1.0

The mean wind velocity v_m(z_e)
v_m(z_e) = c_r(z_e) ⋅ c₀(z_e) ⋅ v_b = 1.1384 × 1 × 40 m/s = 45.54 m/s

The turbulence intensity I_v(z_e)
I_v(z_e) = k_I / [ c₀(z_e) ⋅ ln(max{z_e, z_min} / z₀) ] = 1.0 / [ 1.000 × ln(max{20 m, 2.0 m} / 0.050 m) ] = 0.1669

The basic velocity pressure q_b
q_b = (1/2)ρv_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.

The peak velocity pressure q_p(z_e) at reference height z_e
q_p(z_e) = (1 + 7⋅I_v(z_e)) ⋅ (1/2) ⋅ ρ ⋅ v_m(z_e)² = (1 + 7 × 0.1669) × (1/2) × 1.25 kg/m³ × (45.54 m/s)² = 2810 N/m²
⇒ q_p(z_e) = 2.810 kN/m²

The peak wind velocity v(z_e) at reference height z_e
v(z_e) = [2 ⋅ q_p(z_e) / ρ ]^0.5 = [2 × 2.810 kN/m² / 1.25 kg/m³ ]^0.5 = 67.05 m/s

Calculation of wind forces on the structure

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

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

Reynolds number
Reynolds number characterizes the airflow around the object. For airflow around cylindrical objects, Reynolds number is calculated according to EN1991-1-4 §7.9.1(1):

R_e = b ⋅ v(z_e) / ν = (5 m × 67.05 m/s) / 15.0 × 10^-6 m²/s = 22.3505 × 10⁶
where the kinematic viscosity of the air is considered as ν = 15.0 × 10^-6 m²/s in accordance with EN1991-1-4 §7.9.1(1).

Effective slenderness
The effective slenderness λ depends on the aspect ratio and the position of the structure and it is given in EN1991-1-4 §7.13(2).

For circular cylinders with length l ≤ 15 m the effective slenderness λ is equal to:
λ₁₅ = min(l / b, 70) = min(20m / 5m, 70) = 4

For circular cylinders with length l ≥ 50 m the effective slenderness λ is equal to:
λ₅₀ = min(0.7l / b, 70) = min(0.7 × 20 m / 5 m, 70) = 2.800

For circular cylinders with intermediate length 15 m < l < 50 m the effective slenderness λ is calculated using linear interpolation:
λ = λ₁₅ + (λ₅₀ – λ₁₅) ⋅ (l – 15 m) / (50m – 15m) = 4 + (2.8 – 4) × (20 m – 15 m) / (50m – 15m) = 3.829

End effect factor
The end effect factor ψ_λ takes into account the reduced resistance of the structure due to the wind flow around the end (end-effect). The value of ψ_λ is calculated in accordance with EN1991-1-4 §7.13. For solid structures (i.e. solidity ratio φ = 1.000) the value of the end effect factor ψ_λ is determined from EN1991-1-4 Figure 7.36 as a function of the slenderness λ.

The estimated value for the end effect factor is ψ_λ = 0.658

Equivalent surface roughness
The equivalent surface roughness k depends on the surface type and it is given in EN1991-1-4 §7.9.2(2). According to EN1991-1-4 Table 7.13 for surface type “galvanized steel” the corresponding equivalent surface roughness is k = 0.2000 mm.

Force coefficient without free-end flow
For circular cylinders, the force coefficient without free-end flow c_f,0 depends on the Reynolds number Re and the normalized equivalent surface roughness k/b. The force coefficient without free-end flow c_f,0 is specified in EN1991-1-4 §7.9.2. The value c_f,0 is determined according to EN1991-1-4 Figure 7.28 for the values of R_e = 22.3505 ×10⁶, k = 0.2000 mm, b = 5.000 m, k/b = 0.000040.

The estimated value for the force coefficient without free-end flow is c_f,0 = 0.803

Force coefficient
The force coefficient c_f for finite cylinders is given in EN1991-1-4 §7.9.2(1) as:
c_f = c_f,0 ⋅ ψ_λ

where c_f,0 is the force coefficient without free-end flow, and ψ_λ the end effect factor, as calculated above. Therefore:
c_f = c_f,0 ⋅ ψ_λ = 0.803 × 0.658 = 0.528

Pressure distribution for circular cylinders for different Reynolds number ranges and without end-effects

Total wind force
The total wind force on the structure F_w is estimated as:.
F_w = c_sc_d ⋅ c_f ⋅ q_p(z_e) ⋅ A_ref = 1.0 × 0.528 × 2.810 kN/m² × 100.00 m² = 148.495 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 = 148.495 kN / 100.00 m² = 1.485 kN/m²

Note:
The effective pressure w_eff = 1.485 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, such as the shell of the cylinder. For the latter case appropriate wind pressure on local surfaces must be estimated according to the relevant external pressure coefficients, as specified in EN1991-1-4 §7.9.1.

Design Implications

The results of the wind load analysis are crucial for designing safe and efficient cylindrical structures. The wind loads are translated into equivalent static forces and moments, which are then incorporated into structural analysis software to assess the stresses and deflections in the structure. Based on these results, engineers can:

Determine the appropriate wall thickness and material properties for the tank shell.
Design roof support systems capable of withstanding wind uplift and wind-induced vibrations.
Optimize the anchorage system for the tank to ensure stability under wind loads.
Evaluate the potential need for additional bracing or wind mitigation measures.

Conclusion

Wind load analysis plays a vital role in ensuring the safety and functionality of tank farms and other cylindrical structures. Understanding the wind load characteristics, utilizing appropriate analysis methods, and considering specific complexities like tank farm interaction are crucial for engineers to design robust and wind-resistant structures. Continuous advancements in software and computational techniques are expected to further enhance the accuracy and efficiency of wind load analysis in the future.

List of Top 20 Structural Analysis and Design Software in 2024

By

Ubani Obinna

-

February 27, 2024

Structural analysis and design software have revolutionized the field of civil and structural engineering. These powerful tools enable engineers to efficiently analyze, visualize, and design complex structures. They are also extensively utilised in the academia for research and development.

The ability of these software programs to model the intricacies of two and three-dimensional structures and analyse the results using finite element analysis is quite profound. A lot of analytical techniques for structures are available such as first-order linear analysis, non-linear analysis, buckling analysis, push-over analysis, time-history analysis, p-delta analysis, etc. Whether you’re designing a skyscraper, a bridge, or a residential building, having the right software can significantly impact your workflow and the safety of your designs.

In this article, we will explore the top structural analysis and design software available in 2024. These tools offer precision, speed, and compliance with a lot of international design codes. Let’s dive in:

Top 20 Structural Analysis and Design Software

1. SAP2000 (Computers & Structures Inc.)
A veteran in the field, SAP2000 boasts a long history of being used for a wide range of projects, from simple buildings to complex bridges and stadiums. Its versatility stems from its ability to handle various structural materials, nonlinear analysis capabilities, and integration with other CSI software for a comprehensive workflow. However, the software can have a steeper learning curve compared to some competitors and might be less user-friendly for beginners.

2. ETABS (Computers & Structures Inc.)
Another popular offering from CSI, ETABS focuses specifically on building analysis and design. It excels at modelling multi-story buildings, including concrete, steel, and composite structures. Its user-friendly interface and comprehensive library of codes and standards make it accessible to a broader range of engineers. However, its capabilities are primarily geared towards buildings and might not be as comprehensive for other structural types like bridges or offshore structures.

3. STAAD.Pro (Bentley Systems)
STAAD.Pro offers a user-friendly interface and extensive analysis capabilities for various structural materials and types. STAAD.Pro is very robust for structural analysis, including static, dynamic, and finite element analysis. It integrates seamlessly with other Bentley software, allowing for efficient data transfer and collaboration within a single ecosystem. However, its licensing structure can be complex, and some users may find the interface less intuitive compared to other options.

Staad pro is one of Top 20 Structural Analysis and Design Software — Typical Staad Pro user interface

4. RISA-3D (RISA Tech, Inc.)
RISA-3D is a powerful structural engineering software designed for analyzing and designing three-dimensional models of buildings and other structures. This software offers a range of features that streamline the structural design process. RISA-3D allows engineers to rapidly design structures of all types, including buildings, bridges, tanks, and culverts. It supports a wide range of materials, making it suitable for various projects.

risa 3d interface — Typical RISA-3D user interface

RISA-3D features an intuitive and user-friendly interface, making it accessible to both seasoned professionals and beginners. The software’s powerful analysis engine performs linear and non-linear static analysis, buckling analysis, modal analysis, harmonic analysis, and seismic analysis.

5. Tekla Structural Designer (Trimble)
A powerful and versatile option, Tekla Structural Designer combines robust analysis capabilities with detailed 3D modelling tools. This allows engineers to create complex structures and visualize them realistically. However, its advanced features and steep learning curve can make it less accessible to users with less experience in BIM (Building Information Modeling) workflows.

Typical Tekla Structural Designer user interface

6. Autodesk ROBOT Structural Analysis Professional
Autodesk’s ROBOT Structural Analysis Professional is a comprehensive solution for structural engineers. It offers advanced analysis capabilities, including linear and nonlinear static analysis, dynamic response, and code compliance checks. The software integrates seamlessly with other Autodesk products like Revit and AutoCAD.

7. MIDAS (MIDASoft)
MIDAS provides a comprehensive suite of structural analysis and design tools. These include dedicated programs for building analysis (Midas Civil), bridge design (Midas GTS NX), and general FEA (Midas Gen). Its nonlinear analysis capabilities, optimization features, and seamless BIM integration make it a valuable asset for engineers. It covers various materials and structural systems.

8. Dlubal RFEM (Dlubal Software GmbH)
Dlubal RFEM is a finite element analysis (FEA) software specifically designed for the analysis and design of 2D and 3D structures. RFEM can handle a wide range of structural materials, including concrete, steel, wood, masonry, and more. It also supports various structural analyses, including linear and nonlinear static, dynamic, stability, and contact analyses.

RFEM operates as a modular program, allowing users to purchase and integrate additional modules based on their specific needs, offering flexibility and cost-effectiveness for smaller or less specialized projects.

9. SOFiSTiK
SOFiSTiK is not just a single software program, but rather a suite of integrated software tools specifically designed for structural engineers. It caters to the entire structural engineering workflow, from initial concept design and analysis to detailing, formwork planning, and reinforcement generation. SOFiSTiK’s parametric capabilities allow users to quickly explore different design options and optimize their models iteratively.

The software offers a wide range of analysis capabilities, including linear and nonlinear static, dynamic, stability, and buckling analyses. SOFiSTiK integrates seamlessly with various BIM (Building Information Modeling) software, allowing engineers to collaborate effectively with other disciplines involved in the construction project.

10. Prota Structures
ProtaStructure is a structural analysis and design software specifically geared towards the needs of structural engineers. ProtaStructure combines analysis, design, detailing, and drafting capabilities within a single platform, streamlining the workflow and minimizing the need for data transfer between different software. The software offers compatibility with BIM (Building Information Modeling) workflows, allowing for data exchange with other construction software and improved collaboration.

TYPICAL PROTA MODEL — Typical Prota Structures Model

11. Lusas (Finite Element Analysis Ltd.)
Lusas is a well-established structural analysis software developed by Finite Element Analysis Ltd. It caters primarily to linear and non-linear analysis of various types of structures, making it a valuable tool for engineers tackling complex projects. Lusas boasts a long history in the structural engineering software market, earning recognition for its reliability and accuracy in analysis.

Unlike some software focused solely on linear analysis, Lusas excels in non-linear analysis, allowing engineers to consider material and geometric non-linearities for more realistic simulations. The software offers basic parametric modelling capabilities, enabling users to explore different design options efficiently. Lusas caters to a variety of structures, including buildings, bridges, offshore platforms, and other complex engineering projects.

12. Prokon
Prokon is a structural analysis and design software developed by Irish company Prokon Software Limited. While offering some global reach in about 150 countries and supporting multiple languages, Prokon has traditionally held a stronger presence in the European market.

Prokon boasts a long history in the structural engineering software industry, recognized for its reliability and accuracy in analysis. Prokon handles various structural materials like concrete, steel, wood, and timber. It offers capabilities for linear and non-linear static, dynamic, stability, and buckling analyses, making it suitable for diverse projects.

13. SCIA Engineer
SCIA Engineer is a robust structural analysis and design software widely recognized for its versatility, user-friendliness, and integration capabilities. It caters to engineers working on various projects, from simple buildings to complex bridges and stadiums. SCIA Engineer supports various materials, including steel, concrete, and composite structures. It handles both linear and nonlinear analysis.

The software provides an intuitive interface, allowing engineers to efficiently create 3D structural models. SCIA Engineer performs static and dynamic analyses, seismic analysis, buckling analysis, and more. It ensures accurate results for complex load conditions. SCIA Engineer integrates seamlessly with various BIM software like Tekla Structures and Archicad, enabling efficient data exchange and collaboration within a BIM workflow.

14. SimScale
SimScale is a cloud-native simulation platform that provides powerful tools for structural analysis and design. Developed by SimScale GmbH, it offers engineers and designers an end-to-end solution for various engineering simulation workflows. Due to its cloud-based nature, SimScale requires a stable internet connection for full functionality. This can be a limitation in areas with unreliable internet access. SimScale’s Structural Mechanics module allows users to analyze static and dynamic behaviour of structures.

15. Ansys
ANSYS Mechanical is not specifically a structural analysis software, but rather a general-purpose finite element analysis (FEA) software. However, its wide range of capabilities makes it a valuable tool for comprehensive structural analysis. Ansys Mechanical covers a wide range of structural analysis needs, including linear and nonlinear static analysis, dynamic analysis, thermal analysis, and more.

It supports various materials, boundary conditions, and loading scenarios. While offering an intuitive and customisable interface, Ansys Mechanical uses robust and accurate solvers to compute stress, deformation, and other structural responses. It handles complex geometries, contact interactions, and material nonlinearities.

16. ABAQUS
ABAQUS is a comprehensive finite element analysis (FEA) software widely used for advanced structural analysis, particularly in non-linear and complex engineering applications. While not solely dedicated to structural analysis, its powerful capabilities make it a valuable tool for specialized engineers.

ABAQUS excels in non-linear analyses, handling complex material behaviour, large deformations, and contact scenarios, crucial for simulating real-world structural behaviour accurately. The software offers a vast library of material models, allowing engineers to accurately represent the behaviour of various materials, including metals, composites, rubber, and other specialized materials.

17. PLAXIS
PLAXIS (2D and 3D) is not primarily a structural analysis software in the traditional sense. Instead, it focuses on geotechnical analysis and design, specifically for soil and rock structures and their interaction with built structures like foundations, embankments, and tunnels. PLAXIS offers dedicated tools and features tailored to analyze and design structures within the context of soil and rock mechanics. This includes considering soil behaviour, groundwater flow, and their interaction with structures.

The software incorporates various advanced constitutive models that realistically simulate the non-linear and time-dependent behaviour of soils and rocks, providing a more accurate representation of real-world performance. PLAXIS allows engineers to model the construction process in stages, enabling them to analyze the impact of each construction phase on the stability and behaviour of the soil and structure.

18. Strand7
Strand7 is a finite element analysis (FEA) software used for analyzing and designing various structures across various industries. Strand7 boasts a wide range of capabilities, allowing engineers to analyze different structural types, including buildings, bridges, offshore platforms, and other complex structures.

It supports various materials like concrete, steel, wood, and composites. Beyond linear static analysis, Strand7 offers advanced features for non-linear analysis, enabling consideration of material and geometric non-linearities for more realistic simulations of complex structural behaviour.

19. Skyciv
SkyCiv is a cloud-based structural engineering software platform offering analysis and design tools for various structures. SkyCiv provides online structural analysis capabilities, including frame analysis, section builder, buckling analysis, cable analysis, plate analysis, frequency analysis, response spectrum analysis, and non-linear analysis. The software integrates a wide range of steel, wood, and concrete design codes from organizations such as AISC, EN, AISI, NDS, AS, and CSA.

20. Oasys GSA
Oasys GSA is a comprehensive software suite catering to the structural analysis and design needs of engineers. GSA allows you to create structural models using 1D, 2D, and 3D finite elements, regardless of the structure’s size or complexity. It includes routines for form-finding in lightweight structures such as arches, cable nets, grid shells, and fabric structures.

You can assign linear and nonlinear materials to various element types, including beams, slabs, and custom shapes for static, dynamic, prestress, or thermal loads. You can evaluate buckling capacities, including tension-only fabric material models and lateral torsional buckling of plate girders. GSA offers various solvers for modal vibration, seismic response, harmonic vibrations, and more.

The summary of the software discussed above is presented in the Table below;

Software	User Friendliness and Learning Curve	Analytical Capabilities	Pricing/Subscription	BIM Integration
SAP2000	Moderate – Interface can be complex for beginners, but extensive documentation and tutorials are available.	Extensive – Suitable for static, dynamic, non-linear, and time-history analysis of various structures.	Varies based on version and modules. Can be expensive for smaller firms.	Limited – Integration with Revit and other BIM platforms requires additional tools or APIs.
ETABS	Similar to SAP2000	Similar to SAP2000, with a focus on building analysis.	Similar to SAP2000	Similar to SAP2000
STAAD.Pro	Moderate – Similar to SAP2000 in complexity.	Extensive – Offers a wide range of analysis capabilities for various structures.	Varies based on version and modules. Can be expensive for smaller firms.	Limited – Similar to SAP2000, requires additional tools or APIs for BIM integration.
RISA-3D	Relatively user-friendly – Good for beginners due to its intuitive interface.	Primarily focused on linear static analysis of buildings, but offers some non-linear capabilities.	More affordable compared to some options, with various subscription models.	Limited – Similar to SAP2000, requires additional tools or APIs for BIM integration.
Tekla Structural Designer	User-friendly – Designed for ease of use with BIM workflows.	Primarily focused on building analysis, with some non-linear capabilities.	Subscription-based pricing. May be expensive for smaller firms.	Excellent – Native integration with Tekla BIM software.
Autodesk ROBOT Structural Analysis Professional	Moderate – Similar to SAP2000 in complexity.	Extensive – Offers a wide range of analysis capabilities for various structures.	Included in the Autodesk AEC Collection subscription.	Good – Integrates with other Autodesk BIM software.
MIDAS	Moderate – Similar to SAP2000 in complexity.	Extensive – Offers a wide range of analysis capabilities for various structures.	Varies based on version and modules. Can be expensive for smaller firms.	Limited – Similar to SAP2000, requires additional tools or APIs for BIM integration.
Dlubal RFEM	Moderate – Similar to SAP2000 in complexity.	Extensive – Offers a wide range of analysis capabilities for various structures.	Varies based on version and modules. Can be expensive for smaller firms.	Limited – Similar to SAP2000, requires additional tools or APIs for BIM integration.
SOFiSTiK	Less user-friendly – More advanced software geared towards experienced users.	Highly advanced – Offers powerful analysis capabilities for complex projects.	Varies based on modules and project needs. Can be expensive for smaller firms.	Good – Offers various interfaces and plugins for BIM integration.
Prota Structures	Moderate – Similar to SAP2000 in complexity.	Primarily focused on building analysis, with some non-linear capabilities.	Varies based on version and modules. Can be expensive for smaller firms.	Limited – Similar to SAP2000, requires additional tools or APIs for BIM integration.
Lusas	Less user-friendly – More advanced software geared towards experienced users.	Highly capable – Offers advanced analysis options for complex structures and materials.	Varies based on modules and project needs. Can be expensive for smaller firms.	Limited – Primarily intended for stand-alone analysis, but some BIM integration options are available through third-party tools.
Prokon	Moderate – Similar to SAP2000 in complexity.	Extensive – Offers a wide range of analysis capabilities for various structures.	Varies based on version and modules. Can be expensive for smaller firms.	Limited – Similar to SAP2000, requires additional tools or APIs for BIM integration.
SCIA Engineer	Moderate – Similar to SAP2000 in complexity.	Extensive – Offers a wide range of analysis capabilities for various structures.	Varies based on version and modules. Can be expensive for smaller firms.	Good – Integrates with various BIM platforms like Revit and ArchiCAD.
SimScale	Relatively user-friendly – Cloud-based platform with a web interface.	Primarily focused on linear static and dynamic analysis.	Cloud-based subscription model.	Limited – Offers basic integration with some BIM platforms through APIs.
Ansys	Less user-friendly – Highly advanced software for complex engineering simulations.	Extremely powerful – Offers a wide range of capabilities for various engineering

Disclaimer: This is a general overview, and specific features and pricing may vary depending on the version and chosen modules. It is recommended to consult the respective software websites for detailed information.

Bottomline

This article presented a snapshot of the top 20 options in structural analysis and design software, but the best choice for your specific project depends on its complexity, budget, and BIM workflow needs. It’s crucial to research and compare features, pricing models, and user reviews before making a decision. Remember, the most powerful software isn’t always the best fit. Consider factors like your team’s skill level, project requirements, and budget to find the ideal partner in creating safe and sustainable structures.

The Cost of Plastering a House in Nigeria | 2024

By

Ubani Obinna

-

February 25, 2024

Plastering is the process of covering block or brick walls with cement mortar for decorative or protection purposes. Sometimes, plastering is also referred to as rendering. As an aspect of the finishing works of a building, it is a specialised item of work that must be executed with care using quality workmanship and materials. The cost of plastering a house in Nigeria depends mainly on the price of materials in the market and the labour rate in the area.

Typically, the materials that are used in plastering works are cement (grade 32.5 or grade 42.5N), plaster sand, water, and additives (where recommended). The additives could be colouring, waterproofing, bonding admixtures, etc. It is important to note that the selection of the wrong type of cement can ruin plastering works. Cement with high early strength gain is more susceptible to shrinkage cracking, and in a high-temperature zone like Nigeria, early thermal cracking can adversely affect the plastering works. The process of plastering a house must guarantee straightness and perfection.

To obtain the correct cost of plastering a house, the homeowner, quantity surveyor, and/or contractor must evaluate some important details which can affect the cost of plastering works. It is very common to express the cost of plastering per meter square (m²) of a wall. Some of the details to be evaluated are;

(1) The thickness of the rendering: Typically, the standard thickness of rendering specified by architects is usually 12 mm – 15 mm. When the thickness of the rendering exceeds 15 mm, chicken wire mesh is usually required to stop shrinkage cracking from taking place. It should also be noted that the cost of a 12 mm thick plaster is different from the cost of a 15 mm thick plaster.

(2) The mix ratio of the plaster: Mix ratios are often recommended for different reasons. Some mix ratios that can be specified are 1:1, 1:2, 1:3, or 1:4. The mix ratio of 1:3 is the commonest and it means one part of cement to three parts of plaster sand. In this case, one bag of 50kg cement will be mixed with 6 headpans of plaster sand.

(3) The area of walls to be plastered: The net area of walls to be plastered should be determined by the quantity surveyor. This is usually given by the total area of walls less the area of openings. It is important to note that the dressing of doors, windows, and other surfaces such as columns, cornices, copings, etc are usually priced separately.

dressing of doors — Dressing of doors in a building

(4) Location: In Nigeria, the labour rate for interior plastering is usually different from the labour rate for exterior plastering. Therefore, the allowance should be made as appropriate during the costing.

(5) Other ancillaries: During the plastering of framed structures, it is typical to put wire mesh at the block-column joints in order to prevent vertical cracks from occurring there in the future. Furthermore, materials like angle beads, edge trimmers, etc may be needed at the edges in order to obtain perfect straightness. All these costs should be factored in.

How to build up rate for plastering work

Let us build up the rate for plastering a wall with 15 mm thick 1:4 mix ratio plaster.
In a 1:4 mix ratio plaster, 9 bags of cement and 2035 kg of plaster sand are required to produce 1m³ of mortar.

Cement (basic cost) – ₦9,000 per 50 kg bag : ₦81,000/m³ of mortar
Plaster sand – ₦10,500 per 1000 kg: ₦21,368/m³ of mortar
Water (allow): ₦500/m³ of mortar
Labour – Production of mortar: ₦18,000/m³ of mortar
Labour – Placement: ₦18,000/m³ of mortar
Total cost = ₦138,868/m³ of mortar

Therefore, for a rendering of 15 mm thickness, the cost per square metre is (0.015 x 138,868) = ₦ 2083/m²

Allow 20% for contractor’s profit and overhead = 1.2 x 2083 = ₦2500/m²

Therefore, the cost of plastering a square metre (m²) of a wall in Nigeria using a 1:4 mix ratio plaster is about ₦2,500/m² (Two thousand five hundred Naira per square metre wall).

For a four-bedroom duplex building in Nigeria, the cost of the internal and external plastering of the walls (say 1200 m² of wall) is about ₦3,000,000.

Cost of Duplex Construction in Nigeria | 2024

By

Ubani Obinna

-

February 24, 2024

Table of Contents

In Nigeria, duplexes are the most preferred choice for private residential building construction. Duplexes normally consist of a ground floor and one suspended reinforced concrete floor with a living room (sitting room/parlour), en-suite bedrooms, dining room, kitchen, lounges, and other spaces as may be desired. Lobbies are used to connect the spaces in a duplex building, and a staircase is used for vertical circulation. Residential buildings with more than one suspended floor should not be described as duplexes.

In most duplex designs, it is very typical for the living room, dining, kitchen, laundry room, and storeroom to be on the ground floor. The living room may be double volume (with no reinforced concrete slab over it), or it can be covered with a reinforced concrete slab to create more spaces upstairs.

The former alternative is usually for aesthetics and gives a sense of a freer atmosphere in the living room. Furthermore, it reduces the cost of construction due to the void over the sitting room, however, this comes at the expense of the loss of potentially useful space on the first floor.

A traditional townhouse duplex design by Structville Integrated Services Limited

The ground floor of most duplexes in Nigeria also houses the ante-room, visitor’s toilet, and the visitor’s bedroom which is usually provided with its own convenience (en-suite with toilet and bathroom). Depending on the desire of the client or homeowner, additional bedrooms and facilities may be provided on the ground floor. While this is usually very applicable to moderate duplexes, luxury duplexes can considerably vary in arrangement.

The first floor of a moderate duplex usually contains the majority of the bedrooms. A private family lounge can be provided on the first floor with a kitchenette that can serve as a coffee room. In modern construction, all the bedrooms should be en-suite.

Luxury duplexes can contain additional spaces such as exclusive wine cellars, study rooms/private offices, gyms, saunas, library rooms, theatres/cinemas, games rooms, bigger walk-in closets, kids’ play area, exclusive pantry rooms, elevator, panic room, mechanical/electrical panel rooms, internet server rooms, indoor swimming pool, etc. Modern luxury apartments should be smart buildings with most of the facilities fully automated.

The cost of constructing a duplex building in Nigeria depends on a number of factors such as;

Size: The size of the building will significantly increase the cost of materials and labour needed to construct the building. The bigger the building, the bigger the cost of execution.

Type of land: Buildings that are founded on soft/weak soils will cost considerably more than buildings that are founded on stiff non-problematic soils. Also, the effects of the depth groundwater table can influence the cost of construction. Duplex buildings on marginal soils can be supported on a raft or pile foundation, while duplexes on good soils can be supported on pad foundations. Buildings on pad foundations are cheaper than buildings supported on a raft or pile foundation.

Method of Construction: The construction method you choose might have a big impact on the cost of your building project. Before installing the block walls and partitions, contractors can build the structural parts of duplexes as pure reinforced concrete frames consisting of beams, columns, slabs, and staircases.

On the other hand, the partition walls and the structural frames can be constructed simultaneously, an approach that is more common in low- to medium-budget duplex construction projects. The latter has the advantage of being faster while saving a significant amount of money on materials and labor. The former, however, is of higher quality due to more stringent quality control throughout the construction of the structural elements.

Construction of a building as a pure framed structure (Supervised by Engr. O. U. Ubani)

Location of the project: The cost of a project can also be influenced by its location. Sites that are close to sand and gravel supplies will have lower material costs than sites that are further away. Additionally, if the site is not accessible by truck, significant labour expenditures will be incurred in transloading the items into your site before they are used. Labour costs can also be a factor since the average cost of labour varies from location to location.

Taste of the client: While the cost of constructing the frames and carcass of a duplex should be relatively consistent among buildings of similar size and volume, the cost of finishing a duplex might vary significantly because of the wide range of alternatives available to clients. A building’s finishes might be either high-end or low-cost. This can include everything from roofing sheets to doors, windows, tiles, sanitary fittings, electrical fittings, painting, among other things. A homeowner can choose between the cheapest choice and the most expensive luxury option. The price difference can be as much as 500%.

Getting Started in Duplex Construction

Land Acquisition
As should be expected, the first step in the construction of a duplex should be the acquisition of land. In the villages or rural areas, family or communal lands can be conveniently obtained depending on the family agreements. In the towns of Nigeria, landed properties can be purchased from individuals, corporate bodies, or real estate firms.

In all cases, the land to be used for the construction should be properly surveyed by a registered surveyor, and the title of the land clearly defined. The land should be properly registered and all legal documents for ownership properly verified. All laws for property ownership in the state should be fulfilled by the client.

Architectural Design
After you have secured your plot of land for the proposed construction, the design of the building can commence. The architect designs your preferred building while also satisfying the local building regulation codes/requirements, taking into account the nature, size, and shape of your site.

The airspace and setbacks between the property lines and the building line should adhere to the local building code. Septic tanks, soakaway pits, boreholes, gatehouses, generator houses, outdoor swimming pools, and other structures should all be clearly mapped out on the site layout.

Civil Engineering Designs
The geotechnical and structural engineering designs should be carried out for the ultimate safety of the proposed building. Site investigation and sub-surface exploration should be carried out to determine the engineering properties of the sub-surface soils. This information will inform the type of foundation to be used for the duplex. At the initial stage, a cue should be taken from the type of foundation used for supporting the surrounding buildings if any.

A structural design will identify the location and design of the columns, beams, slabs, staircase, and foundation. The structural design should be carried out, checked, and sealed by a COREN-registered civil engineer. A letter of structural stability and supervision should also be issued to you by the structural engineer.

Mechanical and Electrical Designs
The mechanical design should include the building’s plumbing, HVAC, and water sprinkler systems, among other things. The design should be in harmony with the building’s architectural and structural design so that the route of pipes, ducts, and other utilities can be clearly specified during the design stage, to avoid chiselling of structural components after construction.

The arrangement of electrical pipes and light fittings, security cameras, alarm systems, internet and television cables, cable tray routes, the position of distribution boards/panels, and so on should all be included in the electrical designs. COREN-registered mechanical and electrical engineers shall design and stamp the services drawing.

Approval
Having obtained all the necessary drawings, the complete set of drawings comprising the architectural, structural, mechanical, electrical, and soil test report should be submitted to the Physical Planning Board/Agency of the locality for approval. Other documents as may be required should be submitted too. Once the drawings are approved, the construction can commence.

Cost of Constructing a Duplex in Nigeria

For the sake of convenience in costing and project management, duplex construction in Nigeria can be conveniently broken into the following stages;

Substructure (foundation)
Ground floor to overhead level
First-floor decking
First floor to roof overhead level
Parapet and roofing
Finishes

These stages are consistent with projects where the block walls and the frames are to be built up together. For buildings that are to be constructed as pure frame structures, the above breakdown will not be very adequate.

Let us now survey the cost of constructing each phase of a duplex using the four-bedroom duplex building plan below as a case study. The building is to be constructed in a semi-urban region in south-eastern Nigeria, on lateritic soil with a bearing capacity of 175 kN/m² at a depth of 1 m. Groundwater is at a great depth from the ground surface.

ground floor — Ground floor plan of the proposed duplex

first floor — First floor plan of the proposed duplex

Substructure (Foundation)

The activities usually carried out under substructure works are setting out, excavation, concrete works (blinding, column bases, strip foundation, column stubs, and ground floor slab), reinforcement works, carpentry works, and filling.

Foundation layout of a typical duplex — Foundation layout of the proposed duplex

Setting out
3 bundles of pegs @ ₦2500 = ₦7500
20 pcs of 2″ x 3″ softwood @ ₦750 = ₦15000
1 bag of 3″ and 2″ nails each @ ₦48,000 each = ₦96,000
6 rolls of twin ropes @ ₦500 = ₦3000
Labour and supervision (allow) = ₦100,000
Total for setting out = ₦221,500

Excavation works
Excavation of 82 m³of earthwork for the column bases and the strip foundation @ ₦1,800/m³ = ₦147,600
Supervision = ₦60,500
Total cost for excavation works = ₦208,100

Substructure works of a duplex by Structville Integrated Services Limited

Concrete Works (foundation)
Blinding and casting of column base and strip foundation = 26.5 m³of grade 25 concrete @ ₦108,600/m³ = ₦2,877,900
Labour and supervision cost = ₦425,000
Total cost for concrete works = ₦3,302,900

Reinforcement Works
Column base mat reinforcements = 384 kg of Y12 @ ₦1005/kg = ₦385,920
Column starter bars reinforcements = 455 kg of Y16 @ ₦1005/kg = ₦457,275
Column base links = 50 kg of Y8 @ ₦1005/kg = ₦50,250
Binding wire 1 roll @ ₦34,000 = ₦34,000
Labour and supervision cost = ₦100,000
Total cost for reinforcement works = ₦1,027,445

Block work
170 m² of 9″ hollow block work @ ₦10,150/m²= ₦1,725,500
Labour and supervision = ₦254,000
Total for block work = ₦1,979,500

Carpentry Works
Column stubs – 13.5 m² of 1″ x 10″ plank @ ₦3525/m² = ₦47,590
Slab edge formwork- 50 m length of 1″ x 9″ plank @ ₦780/m= ₦39,000
20 pcs of 2″ x 3″ bracings @ ₦750 = ₦15,000
Labour and supervision costs = ₦40,000
Total cost for carpentry works = ₦141,590

Backfilling and Compaction
150 tonnes of lateritic earth fill @ ₦6000/tonne = ₦900,000
Labour cost (allow) = ₦150,000
Total cost for backfilling and compaction = ₦1,050,000

Backfilling and Compaction of a Duplex Substructure by Structville Integrated Services Limited

BRC Mesh
160 m² of A142 BRC mesh @ ₦1400/m² = ₦224,000
Cost of installation = ₦25,000
Total cost of BRC mesh works = ₦249,000

Damp Proof Membrane
Allow ₦75,000 for purchase and installation = ₦75,000

Casting of ground floor slab
Blinding and casting of column base and strip foundation = 22.5 m³of concrete @ ₦108,600/m³ = ₦2,443,500
Labour and supervision = ₦365,000
Total cost for concreting ground floor slab = ₦2,808,500

Total cost of substructure (foundation to DPC) = ₦11,063,535

Ground Floor to Overhead Level

Block work
212 m² of 9″ hollow block work @ ₦10,150/m²= ₦2,151,800
Labour and supervision = ₦304,400
Total cost for block work = ₦2,456,200

Concrete Works (Lintel and Columns)
9m³ of concrete @ ₦108,600/m³ = ₦977,400
Labour and supervision = ₦146,000
Total cost for concrete works = ₦1,123,400

Ground floor to first floor level — Construction of a duplex (ground floor to overhead level) by Structville

Reinforcement Works
Y16 for columns – 512 kg @ ₦1005/kg = ₦514,560
Y12 for lintels – 405 kg @ ₦1005/kg = ₦407,025
Y8 for lintel and column links – 232 kg @ ₦1005/kg = ₦233,160
1 roll of binding wire = ₦34,000
Labour and supervision cost = ₦120,000
Total cost for reinforcement works = ₦1,308,745

Carpentry Works
Column formwork = 31.5 m² of 1″ x 10″ plank @ ₦3525/m² = ₦111,040
Lintel formwork (sides) = 46.5 m² of 1″ x 10″ plank @ ₦3525/m² = ₦163,915
Lintel formwork (bottom) = 12m² of sawn 1″ x 9″ plank @ ₦3525/m² = ₦42,300
Labour and supervision cost = ₦120,000
Total cost of carpentry works = ₦437,255

Total cost from ground floor to overhead level = ₦5,325,600

First Floor Decking

Formwork
1” x 9” x 12’ plank as soffit to slab and staircase = 155 m² @ ₦3525/m² = ₦546,375
1” x 9” x 12’ plank as sides to beam = 46.5 m² of 1″ x 10″ plank @ ₦3525/m² = ₦163,915
1” x 6” x 12’ plank as sides to beam and riser to staircase = 45 pcs @ ₦1200/pcs = ₦54,000
2” x 3” x 12 softwood = 120 pcs @ ₦750/pcs = ₦90,000
Bamboo props = 275 pcs @ ₦600/pcs = ₦165,000
2 bags of 2″ and 3″ nails each = ₦192,000
Labour and supervision cost = ₦300,000
Total cost of carpentry works = ₦1,511,290

Reinforcement Works
Y16 for beams – 855 kg @ ₦1005/kg = ₦859,275
Y12 for slab – 2425 kg @ ₦1005/kg = ₦2,437,125
Y10 for slab – 295 kg @ ₦1005/kg = ₦296,475
Y8 for links of beams – 190 kg @ ₦1005/kg = ₦190,950
2 rolls of binding wire @ ₦34,000 = ₦68,000
Labour and supervision cost = ₦300,000
Total cost of reinforcement works = ₦4,151,825

reinforcement works of a — Typical reinforcement works on the decking of a Duplex by Structville

Electrical and Mechanical Piping Works
Allow – ₦800,000

Concreting of the slab, beams, and staircase
33m³ of concrete @ ₦108,600/m³ = ₦3,583,800
Labour and supervision cost = ₦512,000
Total cost of concreting = ₦4,095,800

Total cost of first floor decking = ₦10,558,915

First Floor to Roof Overhead Level

Block work
223 m² of 9″ hollow block work @ ₦10,150/m²= ₦2,263,500
Labour and supervision = ₦429,100
Total cost for block work = ₦2,692,600

Concrete Works (Lintel and Columns)
9m³ of concrete @ ₦108,600/m³ = ₦977,400
Labour and supervision = ₦157,500
Total cost for concrete works = ₦1,134,900

Reinforcement Works
Y16 for columns – 512 kg @ ₦1005/kg = ₦514,560
Y12 for lintels – 405 kg @ ₦1005/kg = ₦407,025
Y8 for lintel and column links – 232 kg @ ₦1005/kg = ₦233,160
1 roll of binding wire = ₦34,000
Labour and supervision cost = ₦150,000
Total cost for reinforcement works = ₦1,338,745

Carpentry Works
Column formwork = 31.5 m² of 1″ x 10″ plank @ ₦3525/m² = ₦111,040
Lintel formwork (sides) = 46.5 m² of 1″ x 10″ plank @ ₦3525/m² = ₦169,915
Lintel formwork (bottom) = 12m² of sawn 1″ x 9″ plank @ ₦3525/m² = ₦42,300
Labour and supervision cost = ₦150,000
Total cost of carpentry works = ₦473,255

Total Cost from the first floor to roofing level = ₦5,639,500

Cost Summary

Total cost of substructure (foundation to DPC) = ₦11,063,535
Total cost from ground floor to overhead level = ₦5,325,600
Total cost of first floor decking = ₦10,558,915
Total Cost from the first floor to the roofing level = ₦5,639,500
Total cost = ₦32,587,550

Allow 25% for contractor’s profit and overhead = ₦8,146,890

Therefore, the total cost of constructing a 4-bedroom duplex from the foundation to the roofing level should range between ₦32,587,550 to ₦40,734,437. Note that different professionals and contractors may have their own scale of fees or rates for executing every item of work which may vary from what is presented here.

Residential Duplex under Construction by Structville Integrated Services Limited

For your building design and construction services, contact;
Structville Integrated Services Limited
Phone/WhatsApp: +2348060307054, +2347053638996
E-mail: info@structville.com

Analysis of Statically Determinate Frames

By

Ubani Obinna Uzodimma

-

February 24, 2024

Table of Contents

Statically determinate frames are usually two-dimensional planar structures whose unknown external reactions or internal member forces can be determined using only the three equations of equilibrium. Statically determinate frames are the basic functional frames of structural engineering, forming the backbone of countless structures from bridges to buildings. Understanding their analysis is fundamental to ensuring the stability and efficiency of most civil engineering structures.

The three equations of equilibrium sufficient for the analysis of statically determinate structures are;

∑F_x = 0; ∑F_y = 0; ∑M_i = 0

An indeterminate structure is one whose unknown forces cannot be determined by the conditions of static equilibrium alone. Additional considerations, such as compatibility conditions, are necessary for a complete analysis.

Nature of Statically Determinate Frames

As stated earlier, statically determinate frames are usually 2-dimensional structures. Three-dimensional frames are usually statically indeterminate and are more suited to computer-based finite element analysis.

The vertical members in statically determinate frames are referred to as the columns, while the horizontal members are referred to as the beams. Typically, the analysis of statically determinate frames involves the determination of the support reactions, internal stresses (bending moment, shear force, and axial force), and deflections.

Support Reactions

For simplified manual analysis, the support conditions considered in the analysis of frames are;

Fixed support (consisting of three reactions)
Pinned/hinged support (consisting of two reactions)
Roller support (consisting of one reaction)

It therefore follows that when the number of support reactions in a frame exceeds three, special conditions such as internal hinges will be required to make the frame statically determinate. Such frames with internal hinges are usually referred to as compound frames. The degree of static determinacy or indeterminacy can then be calculated as follows;

I_D = R – e – s

Where;
I_D is the degree of static indeterminacy
e is the number of equilibrium equations (typically 3)
s is the number of special conditions in the structure

Alternatively,

If (3m + R = 3j + s), the structure is statically determinate.

Here:
m represents the number of members.
R represents the number of support reactions.
J represents the number of joints.
s represents the equations of condition (e.g., two equations for an internal roller and one equation for each internal pin).

Rearranging the equation above;

R_D= (3m + R) – 3j – S

Where;
m = number of members.
r = number of support reactions.
j = number of nodes
S = number of special conditions

If the degree of static indeterminacy is less than 0, the frame is unstable. A structure is stable if it maintains its geometrical shape when subjected to external forces. Stability is important for ensuring the safety and functionality of a structure.

To obtain the support reactions in a statically determinate frame, it is usually sufficient to take the summation of moments about any of the supports or any point with a known condition (such as an internal hinge) and equate it to zero.

Loads and Actions on Frames

The skeletal framework of structural frames, composed of beams, columns, and trusses, bears the brunt of diverse forces and actions. Understanding and accurately analyzing these loads is paramount for ensuring the safety and functionality of the entire structure.

The types of loading found on frames are;

Point Loads: These concentrated forces act at discrete locations on a member. Columns sitting on a beam or the actions of secondary beams on primary beams are usually idealised as point loads. Their magnitude and position significantly influence the stress distribution within the frame.

point load on a statically determinate frame — Point load on a statically determinate frame

Uniformly Distributed Loads (UDLs): Imagine a blanket of snow uniformly accumulating on a roof. This scenario represents a UDL, where the load acts with constant intensity across the entire length of a member. Self-weight of members is also idealised as uniformly distributed loads. Analyzing such loads involves calculating their total force based on the area they cover and their intensity.

Uniformly distributed load on a statically determinate frame

Varied Distributed Loads: Not all distributed loads are uniform. Consider the wind pressure acting on a tall building, increasing in intensity towards the top due to aerodynamic effects. These loads can be modelled as linear (triangular loads) or non-linear (trapezoidal loads) functions of the member’s length, necessitating more complex analysis techniques like integration or employing equivalent uniform load representations.

Varied distributed load on a statically determinate frame

Concentrated Moments: Imagine a child swinging on a monkey bar; the force applied at the bar’s end creates a concentrated moment, twisting the bar. This can also come from torsion being transmitted from another structural member or machinery. These moments directly influence the bending stresses within the member.

Concentrated moment on a statically determinate frame

Beyond the fundamental loading idealisation, the world of structural loading extends beyond these fundamental categories:

Line Loads: Picture the weight of a cable hanging on a support. These act along a linear element, requiring specialized analysis techniques.
Hydrostatic and Earth Pressures: Retaining walls holding back water or soil experience continuous pressure from these fluids or packed earth, necessitating specialized analysis based on their specific intensity profiles.
Impact Loads: A sudden blow, like a hammer strike, can create dynamic forces requiring specialized analysis to evaluate potential damage and ensure structural integrity.

Analysis of Statically Determinate Frames

The analysis of statically determinate frames involves several key steps. Let’s break it down:

Identify the Frame: Begin by understanding the given frame’s geometry, member lengths, and support conditions. Clearly identify the structure and its supports (hinges, rollers, etc.). Determine the type and location of all applied loads (point loads, distributed loads, moments). Specify the material properties of the frame members (e.g., Young’s modulus, cross-sectional area). Confirm that the frame is statically determinate.
Reaction Forces: Determine the reaction forces at each support using global equilibrium equations (vertical, horizontal, and moment equilibrium). Check for the force and moment equilibrium of the structure.
Cut the Frame: Split the frame into separate members. Consider each member individually. Isolate each member by cutting it at a point of interest. Draw a free-body diagram of the isolated member. Apply the three equations of equilibrium to solve for the internal shear force, bending moment, and axial force (if applicable) at the point of interest. Repeat for other points of interest in each member.
Internal Forces: Calculate the bending moment, shear, and axial force at selected locations of interest (typically member ends, midpoints, and points of maximum moment). These values help us understand the internal forces within the frame.
Check for Deflections (Optional): Use beam deflection formulas or methods like the virtual work method to calculate deflections at specific points in the members. Compare the deflections to allowable limits specified in building codes or design criteria.

Solved Example

For the frame that is loaded as shown in the Figure below, find the support reactions and draw the internal stresses diagram. Internal hinges are located at points G₁, G₂, and G₃ and beam JK cantilevers out at a height of 4m from column CF

Solution

R_D= (3m + r) – 3n – S
m = 10 (ten members)
r = 6 (six reactions)
n = 11(eleven nodes)
S = 3 (three internal hinges)
R_D= 3(10) + 6 – 3(11) – 3 = 0

This shows that the structure is statically determinate and stable.

Support reactions
Let ∑MG₁^L = 0; anticlockwise negative
(A_y × 2) – ((2 × 2²)/2) = 0
A_y = 2.0 kN

Let ∑MG₃^R = 0; clockwise negative
(D_y × 4) – ((2 × 4²)/2) = 0
D_y = 4.0 kN

Let ∑MG₂^L = 0; anticlockwise negative
(A_y × 7) – ((2 × 7²)/2) + (B_y × 3) – (B_x × 6) – (4 × 2) = 0
But A_y = 2.0 kN

Hence, 7By – 6Bx = 43 ———— (a)

Let ∑M_C = 0; anticlockwise negative
(A_y × 10) – ((2 × 10²)/2) + ((2 × 7²)/2) + (4 × 4) + (B_y × 6) – (D_y × 7) + (5 × 1.5) – (4 × 7) = 0
But A_y = 2.0 kN, D_y = 4.0 kN

We then substitute the values into the above equation;
Hence By = 10.583 kN

Substituting the value of B_y into equation (a)
We obtain B_x = -1.1875 kN

Let ∑M_B = 0; clockwise negative
(C_y × 6) + ((2 × 4²)/2) – ((2 × 13²)/2) – (4 × 4) + (D_y × 13) + (7 × 4) – (5 × 7.5) – (A_y × 4) = 0
But A_y = 2.0 kN, D_y = 4.0 kN.

We then substitute the values into the above equation;
Hence C_y = 22.417 kN

Let ∑MG₂^L = 0;
(D_y × 10) + (C_y × 3) – (C_x × 6) – ((2 × 10²)/2) – (5 × 4.5) – (7 × 2) = 0
But D_y = 4.0 kN, C_y = 22.42 kN
Therefore, C_x = -4.875 kN

Internal Stresses

Section A – E^L (0 ≤ x ≤4.0)
Moment
M_x = A_y∙x – ((2x²)/2) = 2x – x²
At x = 0, M_A= 0 (simple hinged support)
At x = 2.0m, M_G1 = 2(2) – (2)2 = 0

∂Mx/∂x = Qx = 2 – 2x
When ∂Mx/∂x = 0, M = Maximum at that point
Hence, 2 – 2x = 0
x = 2/2 = 1.0m
M_max = 2(1) – (1)2 = 1.0 kNm
At x = 4m
M_E^L= 2(4) – (4)2 = -8 kNm

Shear
∂Mx/∂x = Qx = 2 – 2x
At x = 0
Q_A = 2 kN
At x = 4m

Q_E^L = 2 – 2(4) = -6 kN

Axial
No axial force in the section

Section B – I^B (0 ≤ y ≤ 4.0)
Moment
M_y = A_x.y = 1.875y
At y = 0, M_B = 0 (simple hinged support)
At y = 4m, M_I^B = 1.875(4) = 7.5 kNm

Shear
∂My/∂y = Q_y= 1.875
Q_B – Q_I^B= 1.875 kN

Axial
N_y + 10.483 kN = 0
N_B – N₁^B = -10.483 kN

Section I^UP – E^B (4 ≤ y ≤ 6.0)
M_x = A_x.y – 4(y – 4) = 1.875y – 4y + 16
M_x = -2.125y + 16
At x = 4m, M_I^UP = -2.125(4) + 16 = 7.5 kNm
At x = 6m, M_E^B = -2.125(6) + 16 = 3.25 kNm

Shear
∂My/∂y = Q_y= -2.125
Q_I^UP – Q_E^B = -2.125 kN

Axial
N_y + 10.483 kN = 0
N_B – N₁^B = -10.483 kN

Section E^R – G₁^L (4 ≤ x ≤7.0)
M_x = A_y.x – ((2x²)/2) + B_y (x – 4) + (1.875 × 6) – (4 × 2)
M_x = 2x – x² + 10.483(x – 4) + 3.25
M_x = -x² + 12.483x – 38.682

At x = 4m, M_E^R = -(4)² + 12.483(4) – 38.682 = -4.75 kNm
At x = 7m, M_G1^L = -(7)² + 12.483(7) – 38.682 = 0

∂Mx/∂x = Qx = – 2x + 12.483
When ∂Mx/∂x = 0, M = Maximum at that point
Hence, – 2x + 12.483 = 0
x = 12.483/2 = 6.2415m
Mmax = -(6.2415)² + 12.483(6.2415) – 38.682 = 0.274 kNm

Shear
∂Mx/∂x = Qx = – 2x + 12.483
At x = 4m, Q_E^R = -2(4) + 12.483 = 4.483 kN
At x = 7m, Q_G1^L = -2(7) + 12.483 = -1.517 kN

Axial
N_x – 1.875 + 4 = 0
N_x = -2.125 kN
N_E^R – N_G1^L = -2.125 kN

Coming from the right-hand side
Section D – F^R (0 ≤ x ≤ 7.0)

Moment (clockwise negative)
M_x = Dy.x – ((2x²)/2) = 4x – x²
At x = 0, M_A = 0 (simple hinged support)
At x = 4.0m, M_G3= 4(4) – (4)2 = 0
∂Mx/∂x = Qx = 4 – 2x
When ∂Mx/∂x = 0, M = Maximum at that point
Hence, 4 – 2x = 0
x = 4/2 = 2.0m
M_max = 4(2) – (2)2 = 4.0 kNm
At x = 7m, M_E^L = 4(7) – (7)2 = -21 kNm

Shear
Since the sign convention changes when we are coming from the right, we reverse the signs.
∂Mx/∂x = Qx = 4 – 2x = -4 + 2x
At x = 0, Q_D = -4 kN
At x = 7m, Q_F^R = -4 + 2(7) = 10 kN

Axial
No axial force on the section

Section C – J^B(0 ≤ y ≤ 4.0)
Moment
M_y = C_x.y = 4.875y
At y = 0, MC = 0 (simple hinged support)
At y = 4m, M_J^B = 4.875(4) = 19.5 kNm

Shear
∂My/∂y = Qy= 4.875 = -4.875
Q_C – Q_J^B = -4.875 kN

Axial
N_y + 22.417 kN = 0
N_C – N_J^B = -22.417 kN

Section K – J^R(0 ≤ x ≤1.50)
Moment
M_x = -5x
At x = 0, M_K = -5(0) = 0
At x = 1.5m, M_J^R= -5(1.5) = -7.5 kNm

Shear
∂Mx/∂x = Q_x = – 5 = 5
Q_K – Q_J^R= 5 kN

Axial
N_x = -7 kN (Compression)

Section J^UP – F^B (4 ≤ y ≤ 6.0)
Moment
M_y= C_x.y – (5 × 1.5) – 7(y – 4)
M_y = -2.125y + 20.5
At y = 4m, M_J^UP = -2.125(4) + 20.5 = 12 kNm
At y = 6m, M_F^B = -2.125(6) + 20.5 = 7.75 kNm

Shear
∂My/∂y = Q_y = -2.125 = 2.125
Q_J^UP– Q_F^B = 2.125 kN

Axial
N_x + 22.417 – 5 = 0
N_x = – 17.417 kN
N_J^UP – N_F^B = -17.417 kN

Section F^L – G₂^R (7 ≤ x ≤ 10)
M_x = Dy.x – ((2x²)/2) + Cy (x – 7) + (4.875 × 6) – (7 × 2) – 5(x – 5.5)
M_x = 4x – x² + 22.417(x – 7) – 5(x – 5.5) + 15.25
M_x = -x² + 21.417 x – 114.169

At x = 7m, M_F^L = -(7)² + 21.417(7) – 114.169 = -13.25 kNm
At x = 10m, M_G2^R= -(10)² + 21.417(10) – 114.169 = 0
∂Mx/∂x = Qx = – 2x + 21.417
When ∂Mx/∂x = 0, M = Maximum at that point
Hence, – 2x + 21.417 = 0
x = 21.417/2 = 10.7085m
Hence no point of contraflexure exists in the section.

Shear
∂Mx/∂x = Q_x = – 2x + 21.417 = 2x – 21.417
At x = 7m, Q_F^L = 2(7) – 21.417 = -7.417 kN
At x = 10m, Q_G2^R= 2(10) – 21.417 = -1.417 kN

Axial
Nx – 4.875 + 7 = 0
Nx = -2.125 kN
N_E^R – N_G1^L = -2.125 kN

Bending moment diagram

Shear force diagram

Axial force diagram

To download the full calculation sheet, click HERE

Curtain Walls: Uses and Functional Requirements

By

Ubani Obinna

-

February 19, 2024

Table of Contents

Architectural envelopes often utilize curtain walls, a type of lightweight, non-loadbearing external cladding which are attached to a framed structure to form a complete exterior sheath. They support only their own weight and wind loads, which are transferred via connectors at floor levels to the main structure.

For precision, BS EN 13830 defines curtain walling as “an external vertical building enclosure predominantly comprised of metallic, wooden, or plastic elements.” In essence, most curtain walls consist of vertical mullions (spanning floor to floor) connected by horizontal transoms. Infill panels, either glass or opaque, fill the resulting openings. Typically, such systems are constructed using proprietary systems provided by specialized metal fabricators.

view of curtain wall — Night view of a building with curtain wall system

Objectives of Curtain Wall Systems

The primary objectives of using curtain-walling systems are to:

Enclosure and Environmental Protection: Provide a comprehensive building envelope that protects the structure against external elements like wind, rain, and temperature fluctuations.
Efficient Construction: Utilize dry construction methods, potentially streamlining the building process and minimizing disruptions at the site.
Structural Optimization: Minimize the additional load placed on the building’s primary structure by the cladding system, enhancing overall structural efficiency.
Architectural Expression: Offer a versatile design element to contribute to the building’s overall aesthetic and architectural intent.

highrise building with curtain wall — Curtain walls are popular in highrise buildings due to their low self weight

Functional Requirements of Curtain Walls

The following are the functional requirements of curtain walls.

Weather Resistance

Curtain walls are expected to protect the interior of the building from the weather conditions of the exterior. While the materials of the curtain wall themselves typically offer excellent impermeability, joints within curtain walls present potential vulnerabilities. Careful design and implementation are crucial to ensure weather resistance. Therefore, achieving weather resistance relies on meticulous design and construction. Two approaches exist:

1) Impervious joints: utilizing sealants and gaskets to entirely prevent water entry, mimicking the material’s impermeability.
2) Drained joints: acknowledging potential water ingress but strategically channelling it away through dedicated drainage systems. Both methods require consideration of thermal expansion, structural shifts, and moisture movement, with appropriate materials and skilled installation being crucial for long-term success.

Internal Temperature Control

While large glass areas in curtain walls offer stunning aesthetics, they pose challenges in temperature control. The low heat resistance allows heat transfer and solar radiation to warm internal surfaces, creating uncomfortable heat build-up. Fixed louvres within the system offer limited heat gain reduction, primarily addressing glare. External louvres provide marginal improvement by absorbing and re-radiating heat outwards.

Effective solutions include:

Deep recessed windows: Coupled with external vertical fins, these create shading pockets to reduce solar heat gain.
Balanced HVAC systems: These actively manage internal temperature through heating and ventilation for year-round comfort.
Special solar control glass: Reflective glass with metallic or dielectric coatings deflects solar radiation, reducing heat gain. Tilting the glass further enhances its effectiveness.

Sufficient Structural Strength

While non-loadbearing, curtain walls require sufficient strength to withstand their own weight and varying wind pressures. Wind load intensity depends on three key factors: building height, exposure level, and location.

Curtain wall strength hinges on the rigidity of its vertical mullions and their secure anchorage to the building frame. Glazing beads and compressible materials further enhance resilience against wind damage by allowing panels to move independently within the system, minimizing stress on the overall frame.

curtain wall mullions — Vertical mullions of a curtain wall

Fire Resistance

The high percentage of unprotected areas in curtain walling systems, as defined in Building Regulations (Approved Document B4: Section 12.7), poses a significant fire resistance challenge. To achieve compliance and ensure occupant safety, architects and engineers must carefully select cladding materials or material combinations for opaque infill panels.

These materials should possess inherent fire resistance properties or be treated with fire-retardant coatings to qualify as protected areas as defined in the regulations. For further guidance on external fire spread considerations, refer to Part 7 of the same document.

Assembly and Fixing

The mullion, typically a solid or box section member, forms the backbone of a curtain wall system. It securely connects to the building’s structural frame at floor levels using adjustable anchorages or connectors, ensuring proper load transfer and stability. The infill framing and panels can be delivered as individual components requiring on-site assembly, or as prefabricated units for faster installation. When evaluating different systems, key considerations include:

Handling ease: Can the individual components or prefabricated units be safely and efficiently manoeuvred on-site, considering their size and weight?
Site assembly: Is extensive field assembly required, potentially impacting construction time and labour costs?
Access to fixing points: Can workers readily access and secure the curtain wall to the building structure at all designated anchor points?

assembly and fixing of curtain walls — Assembly and fixing of curtain walls

Sound Insulation

Curtain wall systems, due to their inherent lightweight nature, present challenges in terms of sound insulation. Both structure-borne and airborne sound transmission must be addressed to ensure a comfortable and acoustically controlled indoor environment.

Structure-borne sound: Primarily originating from machinery vibrations, this type of sound can be mitigated by isolating offending equipment with resilient pads or incorporating resilient connectors within the mullion connections. Careful equipment selection and placement can further contribute to reducing vibrations at their source.
Airborne sound: Lightweight cladding offers minimal inherent sound barrier, making glazed areas particularly vulnerable to sound transmission. Strategies to reduce airborne sound transmission include:
- Minimizing glazing area: Utilizing less glazing or opting for smaller window sections can significantly reduce sound ingress.
- Sealed windows with thicker glass: Implementing sealed windows with thicker glass panels increases the mass barrier, thereby enhancing soundproofing capabilities.
- Double-glazing: Installing double-glazed windows with an air gap of 150-200mm between the panes creates a significant barrier for sound waves, offering superior sound insulation performance.

Thermal and Structural Movements

Curtain wall systems, positioned on a building’s exterior, face heightened exposure to temperature fluctuations compared to the internal structure. This translates to significant thermal movement within the curtain wall itself, as well as potential differential settlement between the main frame and attached cladding. To accommodate these independent movements, careful design, fabrication, and fixing are crucial.

Key considerations:

Slotted bolt connections: These connections offer flexibility at attachment points between the curtain wall and the building frame, allowing for controlled thermal expansion and contraction without compromising structural integrity.
Spigot connections: Within the curtain wall system, spigot connections join components while permitting controlled movement. This flexibility mitigates stresses caused by thermal expansion and contraction within the system itself.
Mastic-sealed joints: These flexible sealant joints further accommodate movement by allowing slight shifts between individual curtain wall components while maintaining weather resistance.

Infill Panels for Curtain Wall Systems

Curtain wall infill panels, responsible for opaque areas, require specific characteristics to ensure optimal performance and longevity. These include:

Lightweight construction: Minimizes overall system weight, reducing structural loads and facilitating handling.
Rigidity: Ensures dimensional stability and resistance to deflection under wind loads and thermal stresses.
Impermeability: Prevents water ingress and maintains weathertightness of the building envelope.
Adequate fire resistance: Complies with relevant building regulations to ensure occupant safety in case of fire.
Thermal insulation: Minimizes heat transfer and contributes to energy efficiency.
Low maintenance: Requires minimal upkeep for sustained performance and aesthetic appeal.

Panel Construction and Vapour Control:

No single material possesses all these attributes, necessitating the use of composite or sandwich panels. However, such panels pose a risk of interstitial condensation, which can be mitigated by incorporating a vapour control layer near the inner panel surface. This layer, with a vapour resistance exceeding 200 MN/g, can be formed using various materials:

Adequately lapped sheeting: Aluminium foil, waterproof building papers, or polyethylene sheet
Applied materials: Two coats of bitumen or chlorinated rubber paint

Careful placement is crucial to avoid detrimental interactions between adjacent materials, such as alkali attack on aluminium when in contact with concrete or fibre cement.

External Facing Materials:

Direct exposure to the elements necessitates careful selection of external-facing materials. Plastics and plastic-coated options are viable choices if they comply with fire regulations outlined in relevant documents. One popular choice is vitreous enamelled steel or aluminium sheets (0.7-0.8mm thickness).

This process fuses a thin glass coating onto the metal surface at high temperatures, resulting in:

High hardness and impermeability: Resisting damage from acids, corrosion, and abrasion.
Crack and craze resistance: Maintaining an attractive finish with lasting strength.

Alternatively, aluminium sheeting with a silicone polyester coating can be employed. By combining these facings with insulating materials like EPS, rockwool, polyurethane, or polyisocyanurate, lightweight infill panels achieving U-values below 0.35 W/m²K can be produced.

Furthermore, both internal and external surfaces must meet fire performance requirements outlined in building regulations, often tested according to specific standards. The insulating core must also exhibit non-combustible properties. Panel dimensions can reach up to 3000mm x 1000mm.

Glazing for Curtain Wall Systems

One critical aspect of curtain walls is glazing—the use of glass in large, uninterrupted areas to create consistent and attractive facades. While protecting the building interior from the elements remains the principal objective of facade materials, the function of glazing transcends mere weather tightness. It plays a pivotal role in orchestrating two key aspects of the built environment:

1. Daylight-Driven Illumination:

Glazing serves as a conduit for natural light, not only fulfilling the basic requirement of illuminating interior spaces but also contributing demonstrably to occupant well-being and energy conservation efforts. However, solely relying on daylight to sufficiently illuminate specific tasks necessitates meticulous consideration of several factors. Window size, placement, and calculated daylight factors all come into play in ensuring adequate and appropriate natural light distribution for dedicated work areas.

2. Fostering Visual Connection with the Exterior:

Beyond illumination, glazing fulfils a psychological need by establishing a visual connection with the surrounding environment. This connection has been demonstrably linked to enhanced occupant well-being, underlining the importance of thoughtful planning when incorporating glazed areas. Size, orientation, and the quality of the view obtained through these areas are crucial aspects to consider during the design phase.

As discussed earlier in this article, other critical considerations pertaining to glazing selection include managing solar heat gain, glare control, thermal insulation performance, and acoustic properties. Therefore, these aspects shall not be revisited here.

Cleaning and Maintenance of Curtain Walls

The use of expansive glazing in high-rise structures, particularly in curtain walling systems, presents a significant challenge: safe and cost-effective access for cleaning and maintenance. While manual cleaning with tools like swabs, chamois leathers, and squeegees remains the standard method, access becomes paramount. Neglected cleaning causes the following problems on curtain wall glazings:

Aesthetic Integrity: Accumulation of dirt distorts the intended visual appearance.
Daylight Transmission: Optimal natural light penetration requires clean surfaces.
Visual Clarity: Unobstructed views are essential for occupants and aesthetics.
Material Integrity: Glazing materials are susceptible to deterioration from dirt and chemical attack.

For low- to medium-rise structures, access solutions like trestles, stepladders, and straight ladders (up to 11 meters) suffice. However, taller buildings necessitate alternative approaches:

Tower Scaffolds: While offering access, their assembly and disassembly time and cost make them impractical for frequent cleaning.
Lightweight Scaffolds: Quick-install systems can be considered for moderate heights (up to 6 meters) due to their efficiency.

High-rise curtain wall cleaning predominantly relies on suspended cradles. These come in two forms:

Temporary Cradles: These offer flexibility but must be dismantled and reassembled each time.
Permanent Systems: Integrated into the building structure, they offer readily available access but carry higher upfront costs.

The simplest permanent solution involves installing a universal beam section at roof level, extending 450 mm beyond the facade and encircling the building. A conventional cradle with castors on its lower flange runs along this beam, controlled by ropes lowered to ground level for access.

While challenges exist, a range of options ensures the cleanliness and integrity of high-rise glazed facades, contributing to both aesthetics and occupant well-being.