Design of Industrial Ground Floor Slabs

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May 17, 2022

Table of Contents

The slab-on-grade, which includes industrial ground floors, can be defined as a slab continuously supported by ground with an area of more than twice the area required to support the imposed loads. The primary design objectives in industrial ground floor slabs are to carry the intended loads and to avoid surface cracking.

The slab may be plain or reinforced and may include stiffening elements such as ribs and hidden beams. The reinforcement may be provided for structural purposes or for the control of the effects of shrinkage and temperature changes.

Industrial ground floor slabs are essential components of industrial buildings and warehouses. In order for manufacturing equipment and forklifts to function properly, the slab must be uniformly flat and joints must be relatively level without excessive movements. The growth of the industry has necessitated the use of larger and heavier machinery and storage facilities.

In addition, the evolution of construction practices, such as the trend toward larger floor slabs with fewer joints, the use of high early-strength cement, and the increase in the use of admixtures, have all contributed to the rise in the cracking and curling of these slabs.

The typical layers in an industrial ground floor slab is shown in Figure 1;

Elements of concrete industrial floor and pavement — **Figure 1:** Typical layers in an industrial ground floor slab

The manner in which uniform loads are stacked in warehouses causes some cracking in the aisles between loaded areas. The predominant types of cracking in aisles fall into two categories. The first type is longitudinal, while the second is transverse. Transverse cracking is directly attributable to shrinkage, whereas longitudinal cracking is caused by the way external loads are stacked in warehouses.

In warehouses, heavy uniform loads are typically distributed over a portion of the slab, typically around the columns, leaving clear aisles in the middle of these columns.

The requirements for concrete industrial ground floors include the following:

The floor should remain serviceable, assuming planned maintenance and no gross misuse or overloading.
The floor must be able to carry the required static point loads, uniformly distributed loads and dynamic loads, without unacceptable deflection, cracking, settlement or damage to joints.
Joint layouts should take into account the location of racking uprights or mezzanine floor columns.
Joints should be robust in both design and construction.
Joints and reinforcement should be detailed to minimise the risk of cracking.
The floor surface should have suitable surface regularity.
The floor surface should have suitable abrasion, chemical and slip resistance.
The floor should have the required type of finish.

finishes on an indutrial ground floor — **Figure 2:** Surface finishing of industrial ground floor slab

Warehouse Equipment and Floor Loading

Point loads from pallet racking, accompanying materials handling equipment (MHE), and mezzanines are frequent loads on warehouse floors. Other loads come from uniformly distributed loads (UDL) like palletized products or bulk loose materials, as well as line loads like interior walls and floor railing systems.

Typical loading on industrial ground floors — **Figure 3:** Typical loads on the floor of a warehouse

In the design of industrial ground floors, point loads are usually the most critical for design, and reliance should not be placed solely on the commonly specified uniformly distributed loads (UDLs). In all cases, the design must be based on anticipated loads from all types of equipment and other loads, and the specifier must account for the floor’s potential future uses.

It is reasonable to anticipate that taller structures will be able to support greater loads, such as those imposed by pallet racking. Point loads from pallet racking and mezzanines are treated as static loads, whereas mobile heavy equipment (MHE) is treated as a dynamic load that requires greater design safety factors.

racking in a warehouse — **Figure 4:** Back to back arrangement of a storage rack in a warehouse

Design of Industrial Ground Floor Slab

For the commonly encountered point loads on industrial ground floor slabs from storage shelving, mezzanines, and materials handling equipment (MHE), there are two possible modes of ultimate strength failure:

flexure, and
punching

The design of slabs for flexure under point loads at the ultimate limit state (ULS) is based on yield line theory, which necessitates sufficient plasticity to assume plastic behaviour. Clearly, sufficient rotation capacity of the sagging yield lines is required in order to mobilise the hogging moment capacity.

At the ULS, it is assumed that the bending moment along the sagging (positive moment) yield lines is the full plastic (or residual post-cracking) value. As the avoidance of cracks on the upper surface is a primary requirement for serviceability, the bending moment of the slab along the hogging yield lines is limited to the design cracking moment of the concrete, albeit with the partial safety factor appropriate to the ULS.

According to TR34, this is not a true ULS because the floor will not have collapsed, and the design process is meeting a serviceability requirement instead. Therefore, there are no separate checks for design serviceability. The design of the slab against punching shear around concentrated loads is based on Eurocode 2 for suspended slabs. It is taken into account that a portion of the load will be transferred through the slab to the ground.

Line loads and uniformly distributed loads are evaluated using an elastic analysis based on Hetenyi’s Beams on Elastic Foundation. The minimum recommended slab thickness for a ground-supported slab is 150 millimetres.

The designer must account for the thickness reduction caused by mat wells, induction loops, guide wires, and other features. There are practical limitations on how much concrete can be poured in one day, so most floors have joints. In most instances, the critical loading condition is a point load close to a slab panel joint.

In all designs, the load-bearing capacity of the floor alongside joints must be evaluated. This capacity will depend heavily on the joint mechanism’s ability to transfer load to the opposite side of the joint. This is especially true for MHE, which cannot be positioned away from joints, unlike static loads.

Worked Examples

Quadruple Internal Point Loads

Verify the capacity of a 200mm thick concrete industrial ground floor slab subjected to a quadruple internal 300 x 300 point load in accordance with TR34, 4th Edition 2013;

Permanent load; G_k = 45.0 kN
Variable load; Q_k = 20.0 kN
Dynamic load; D_k = 30.0 kN

Slab details
Reinforcement type; Fabric
Concrete class; C25/30
Slab thickness; h = 200 mm
Fabric reinforcement type; A393
Characteristic strength of reinforcement; f_yk = 500 N/mm²
Area of top steel provided; A_s,prov = 393 mm²/m
Diameter of reinforcement; f_s = 10 mm
Nominal cover; c_{nom_b} = 50 mm
Effective depth of reinforcement; d = 0.75h = 150 mm

Partial safety factors
Concrete (with or without fibre); γ_c = 1.50
Reinforcement (bar or fabric); γ_s = 1.15
Permanent; γ_G = 1.20
Variable; γ_Q = 1.50
Dynamic loads; γ_D = 1.60

Subgrade reaction
Modulus of subgrade reaction; k = 0.030 N/mm³

Properties of Concrete
Characteristic compressive cylinder strength; f_ck = 25 N/mm²
Characteristic compressive cube strength; f_cu = 30 N/mm²
Mean value of compressive cylinder strength; f_cm = f_ck + 8 N/mm² = 33 N/mm²
Mean value of axial tensile strength; f_ctm = 0.3 N/mm² × (f_ck)^2/3 = 2.6 N/mm²
Flexural tensile strength; f_ctd,fl = f_ctm × (1.6 – h /1000) / γ_c = 2.4 N/mm²
Design concrete compressive strength (cylinder); f_cd = f_ck / γ_c = 16.7 N/mm²
Secant modulus of elasticity of concrete; E_cm = 22 kN/mm² × [f_cm/10 ]^0.3 = 31 kN/mm²
Poisons ratio; v = 0.2

Radius of relative stiffness (Eqn. 20); l = [E_cmh³ / (12(1 – v²) × k)]^0.25 = 924 mm
Characteristic of system (Eqn. 33); l = (3k / (E_cmh³))^0.25 = 0.773 m^-1

Moment capacity
Negative moment capacity (Eqn. 2); M_n = M_un = f_ctd,fl× (h²/6) = 16.0 kNm/m
Positive moment capacity (Eqn. 2); M_p = M_un = 16.0 kNm/m

Loading – Quadruple internal 300 x 300 point load
Loading length; l_l = 300mm
Loading width; l_w = 300mm
Distance x; x = 1000mm
Distance y; y = 1000mm
Permanent load; G_k = 45.0 kN
Variable load; Q_k = 20.0 kN
Dynamic load; D_k = 30.0 kN

Contact radius ratio
Equivalent contact radius ratio; a = [(l_l × l_w) / π]^0.5 = 169.3 mm
Radius ratio; a/l = 0.183

Ultimate capacity under single internal concentrated loads
For a/l equal to 0 (Eqn. 21); P_{u_0} = 2π(M_p + M_n) = 200.6 kN
For a/l equal to 0.2 (Eqn. 22); P_{u_0.2} = 4π(M_p + M_n) / [1 – (a / (3 × l))] = 427.2 kN
Thus for a / l equal to 0.183; P_u = min(P_{u_0.2}, P_{u_0} + (P_{u_0.2} – P_{u_0}) × (a / (l × 0.2))) = 408.2 kN
4 No. individual; P_{u_4x1} = 4P_u = 1632.6 kN

Ultimate capacity under dual internal concentrated loads
For a/l equal to 0 (Eqn. 27); P_{u_0} = [2π + (1.8 × min(x, y) / l)] × [M_p + M_n] = 262.7 kN
For a/l equal to 0.2 (Eqn. 28);P_{u_0.2} = [4π / (1 – (a / (3 × l))) + 1.8 × min(x, y) / (l – (a/2))] × [M_p + M_n] = 495.7 kN
Thus for a / l equal to 0.183; P_u = min(P_{u_0.2}, P_{u_0} + (P_{u_0.2} – P_{u_0}) × (a / (l × 0.2))) = 476.1 kN
2 No. dual; P_{u_2x2} = 2 × P_u = 952.2 kN

Ultimate capacity under quadruple internal concentrated loads
For a/l equal to 0 (Eqn. 29); P_{u_0} = [2π + 1.8 × (x + y) / l] × [M_p + M_n] = 324.9 kN
For a/l equal to 0.2 (Eqn. 30); P_{u_0.2} = [4π / (1 – (a / (3 × l))) + 1.8(x + y) / (l – (a / 2))] × [M_p + M_n] = 564.1 kN
Thus for a / l equal to 0.183; P_u = min(P_{u_0.2}, P_{u_0} + (P_{u_0.2} – P_{u_0}) × (a / (l × 0.2))) = 544.0 kN
quadruple;P_{u_1x4} = P_u = 544.0 kN
Ultimate load capacity for 4 No. loads; P_u = min(P_{u_4x1}, P_{u_2x2}, P_{u_1x4}) = 544.0 kN

Check ultimate load capacity of slab
Number of loads; N = 4
Loading applied to slab; F_uls = N × [(G_kγ_G) + (Q_kγ_Q) + (D_kγ_D)] = 528.0 kN
Utilisation; F_uls / P_u = 0.971
PASS – Total slab capacity exceeds applied load

Punching shear at the face of the loaded area
Shear factor; k₂ = 0.6(1 – f_ck / 250 N/mm²) = 0.54
Length of perimeter at face of loaded area; u₀ = 8 (l_l + l_w) = 4800 mm
Shear stress at face of contact area; v_max = 0.5k₂f_cd = 4.500 N/mm²
Maximum load capacity in punching; P_p,max = v_max × u₀ × d = 3240.0 kN
Utilisation; F_uls / P_p,max = 0.163
PASS – Total slab capacity in punching at face of loaded area exceeds applied load

Punching shear at the critical perimeter
Shear factor; k_s = min(1 + (200mm / d)^0.5, 2) = 2.00
Minimum shear stress at 2d from face of load; v_Rd,c,min = 0.035k_s^3/2 × (f_ck)^0.5 = 0.495 N/mm²
Ratio of reinforcement by area in x-direction; r_x = A_s,prov / d = 0.00262
Ratio of reinforcement by area in y-direction; r_y = A_s,prov / d = 0.00262
Reinforcement ratio; r₁ = (r_x × r_y)^0.5 = 0.00262

Maximum shear stress at 2d from face of load; v_Rd,c = max(0.18 × k_s / γ_c × (100 × r₁ × f_ck )^1/3, v_Rd,c,min) = 0.495 N/mm²
Length of perimeter at 2d from face of load; u₁ = 2 × (l_w + y + l_l + x + 2π × d) = 7085 mm
Max. load capacity in punching at 2d from face; P_p = v_Rd,c × u₁ × d = 526.0 kN
Ground reaction (cl.7.10.2); R_p = 1.4 × (d / l)² × F_uls + 0.47 × (l_l + x + l_w + y) × d × F_uls / l² = 132.9 kN
Total imposed shear load; F_{uls_total} = F_uls – R_p = 395.1 kN
Utilisation; F_{uls_total} / P_p = 0.751
PASS – Total slab capacity in punching at 2d from face of loaded area exceeds applied load

Design Summary

Description	Unit	Provided	Required	Utilisation	Result
Slab capacity in flexure	kN	544.0	528.0	0.971	PASS
Shear at face	kN	3240.0	528.0	0.163	PASS
Shear at 2d	kN	526.0	395.1	0.751	PASS

Uniformly Distributed Load

Verify the capacity of a 150mm thick concrete industrial ground floor slab to support a uniformly distributed load of 45 kN/m² in accordance with TR34, 4th Edition 2013;

Slab details
Reinforcement type; Fabric
Concrete class; C25/30
Slab thickness; h = 150 mm
Fabric reinforcement type; A252
Characteristic strength of reinforcement; f_yk = 500 N/mm²
Area of top steel provided; A_s,prov = 252 mm²/m
Diameter of reinforcement; f_s = 8 mm
Nominal cover; c_{nom_b} = 50 mm
Effective depth of reinforcement; d = 0.75 × h = 112 mm

Partial safety factors
Concrete (with or without fibre); γ_c = 1.50
Reinforcement (bar or fabric); γ_s = 1.15
Permanent; γ_G = 1.20
Variable; γ_Q = 1.50
Dynamic loads; γ_D = 1.60

Subgrade reaction
Modulus of subgrade reaction; k = 0.030 N/mm³

Strength properties for concrete
Characteristic compressive cylinder strength; f_ck = 25 N/mm²
Characteristic compressive cube strength; f_cu = 30 N/mm²
Mean value of compressive cylinder strength; f_cm = f_ck + 8 N/mm² = 33 N/mm²
Mean value of axial tensile strength; f_ctm = 0.3 N/mm² × (f_ck)^2/3 = 2.6 N/mm²
Flexural tensile strength; f_ctd,fl = f_ctm × (1.6 – h / 1000) / γ_c = 2.5 N/mm²
Design concrete compressive strength (cylinder); f_cd = f_ck / γ_c = 16.7 N/mm²
Secant modulus of elasticity of concrete; E_cm = 22 kN/mm² × [f_cm/ 10 ]^0.3 = 31 kN/mm²
Poisons ratio; v = 0.2
Radius of relative stiffness (Eqn. 20); l = [E_cm × h³ / (12 × (1 – n²) × k)]^0.25 = 745 mm
Characteristic of system (Eqn. 33); l = (3 × k / (E_cm × h³))^0.25 = 0.959 m^-1

Moment capacity
Negative moment capacity (Eqn. 2); M_n = M_un = f_ctd,fl × (h² / 6) = 9.3 kNm/m
Positive moment capacity (Eqn. 2); M_p = M_un = 9.3 kNm/m

Load 1 – UDL 45 kN/m²

Working load capacity of UDL
UDL; U_k = 45.0 kN/m²
Critical aisle width; l_crit = π / (2 × l) = 1637 mm
Loaded width of single UDL (max positive moment); l_{load_p} = π / (2 × l) = 1637 mm
Loaded width of dual UDL (max nagative moment); l_{load_n} = π / l = 3275 mm
Working load capacity of slab; q = 5.95 × l² × M_n = 50.9 kN/m²
Utilisation; U_k / q = 0.884
PASS – Total slab capacity exceeds applied load

Design Summary

Description	Unit	Provided	Required	Utilisation	Result
Slab capacity in flexure	kN/m²	50.9	45.0	0.884	PASS

Joints in Concrete Pavements and Industrial Floors

By

Ubani Obinna

-

May 15, 2022

Table of Contents

Except where they join other structures, concrete pavements and industrial floors should ideally be free of joints. However, joints in concrete pavements are usually provided for a variety of reasons, including construction considerations, reducing the possibility of unanticipated shrinkage cracking, and avoiding conflict with adjacent structures and/or penetrations. The number of joints in concrete pavements should be kept to a minimum because they not only affect the evenness of the pavement in most cases, but they are also the most prone to wear and require repairs.

The type of joint, joint configuration, sealant required, and quantity of reinforcement employed in the pavement panels are all interconnected. For example, increasing the amount of reinforcement allows for wider joint spacing, but it also means that the joints will move more, increasing the possibility of random cracking inside the pavement panels.

Load Transfer Across Joints in Concrete Pavements

Load transfer mechanisms like dowels can be utilized to transfer loads across a joint to adjacent pavement panels, resulting in lower flexural stresses in the panel than at free edges with no effective load transfer. They also prevent stepping by preventing differential vertical movements of adjacent concrete floor panels.

Aggregate interlock over the rough crack faces, keyed joints, dowels, or a combination of these can enable load transfer in contraction joints. If the opening is greater than 0.9 mm, as is common when panel lengths exceed 3 m, load transfer by aggregate interlock or keyways may become ineffective, and either an effective load-transfer device for these situations should be installed, or the base thickness for a free-edge condition should be determined.

According to ACI 302.1, the base thickness of dowels should be at least 125 mm to be fully functional. It also suggests using dowels for load transfer across joints, because regulating the differential vertical movement across joints can help prevent slab edge damage from vehicles with harsh wheels, such as forklifts.

Types of Joints in Concrete Pavements

Pavement construction uses four different types of joints:

Isolation Joints
Expansion joints
Contraction Joints
Construction joints

Isolation Joints

These joints allow for horizontal and vertical movement as well as rotation between abutting pieces, allowing them to function independently. They should be installed between a pavement panel and the building’s fixed components (such as columns, walls, machinery bases, pits, etc).

To avoid the buildup of stresses due to differential movements, isolation joints should be provided at the junction when an existing pavement is being extended, as well as at connections between internal and external pavements. Typically, load transmission between the existing pavement and the pavement extension will be required in the design.

Typical joints in concrete pavements — **Figure 1:** Typical isolation joints (CCAA, 2009)

To ensure a complete separation, isolation joints in concrete pavements are often formed by casting against a compressible, prepared filler material (e.g., self-expanding cork) over the full depth of the joint. If loads are predicted to occur close to an isolation joint, the base edges adjacent to these joints may need to be thickened due to the edge loading condition. Figure 1 shows the typical characteristics of this type of joint. Note that the sealant should be applied to both the top and free edges of the joint to keep dirt and other incompressible materials out of the joint, preventing or restricting movement.

Expansion Joints

Pavements feature expansion joints to allow for temperature and moisture-induced movement of the base. These joints may, however, be required in places with significant temperature changes. Figure 2 shows the typical details of this type of joint. Note that the sealant should be applied to both the top and free edges of the joint to keep dirt and other incompressible materials out of the joint, preventing or restricting movement.

**Figure 2:** Typical expansion joint (CCAA, 2009)

Internal floors do not require expansion joints because they do not experience severe temperature variations. Internally, the expansion due to heat movement is usually less than the concrete’s initial shrinkage. Even the thermal movement of a cold-store floor will not be greater than the floor’s original shrinkage. In interior applications, AS 3600 recommends design shrinkage values of 670 x 10^–6 for 200-mm-thick slabs and 450 x 10^–6 for 400-mm-thick slabs.

According to TR 35, the overall shrinkage coefficient for pavements is around 300 x 10^–6. With a coefficient of thermal expansion of 10 x 10^–6/°C, an expansion equal to the drying shrinkage would require a temperature of around 30°C above the putting temperature. As a result of the lack of thermal ranges in industrial floors, expansion joints are not necessary.

Furthermore, if drying shrinkage is ignored and the restraint to slab lengthening due to temperature rise and/or moisture increase is ignored, the maximum compressive strain in the concrete will be 300 x 10^–6 for a temperature rise of 30°C above the placing temperature (and the slab moisture content returns to saturation), assuming no expansion joints are provided. Assuming f’_c = 40 MPa, E_c is 31975 MPa, while the compressive stress is 9.6 MPa. The absence of expansion joints in industrial floors is supported by this minimal compressive stress development.

Designers should determine whether expansion joints are necessary inside before specifying their use, as the required joint width will necessitate additional considerations.

To reduce forklift vehicle wear and accompanying high maintenance costs, treatment (armouring of joint edges) is used. Expansion joints in concrete pavements also necessitate the use of load transfer devices, such as dowels, bars, or plates, due to the separation of adjacent panels. Whenever expansion joints are not provided, alternative joints within the pavement should be sealed to avoid the intrusion of incompressible material, which could limit subsequent floor expansion.

Contraction Joints

The random drying shrinkage cracking of concrete is controlled by contraction or control joints, which cause the base to crack exclusively at the joint positions. They allow the base to move horizontally at right angles to the joint, relieving pressures that could otherwise induce random cracking. A plane of weakness is generated by shaping (with crack-inducing tapes or formers) or cutting a groove to a depth of one-quarter to one-third of the base thickness to ensure that shrinkage cracking develops at a contraction joint. Figure 3 shows the typical details.

key joint pressesed metal — **Figure 3:** Typical contraction joints – with or without reinforcement (CCAA, 2009)

However, if the groove is created early enough using an appropriate grooving tool or early-age saw cutting, the groove depth can be reduced in concrete slabs that are either plain or reinforced (mesh).

The spacing of contraction joints in jointed unreinforced pavements should be chosen to suit the shape of the pavement while ensuring that load transfer by aggregate interlock is not compromised. If this is the case, load transfer must be accomplished through other means, or the base thickness should be built as a free edge.

Contraction joints are commonly made by cutting a groove in the top of freshly laid concrete (Formed Joint) or by sawing the concrete after it has set but before it cracks uncontrollably (Sawn Joint).

Formed Joints

Forming a groove with a T-section and edge tools or inserting a prepared crack inducer into the surface while the concrete is still plastic can be used to produce formed joints. After the pavement has been completed, a sealer can be put in the contraction joints by applying an appropriate sealant reservoir and bond-breaking backing tape. The reinforcement in reinforced pavements must not obstruct the created joint. Figure 4 shows how the reinforcement could be terminated short of the joint.

joints okay — **Figure 4:** Typical transverse construction joints (CCAA, 2009)

Sawn Joints

Sawn joints are constructed just after the concrete has hardened enough that the sawing will not damage it but before shrinkage cracking develops (see Figure 5). The best time to saw depends on a variety of factors that influence concrete hardening, such as concrete strength and ambient temperature. The depth of the initial saw cut should be 3 to 5 mm. The joint can be enlarged later if necessary for the installation of a joint sealer.

Saw cut in construction joint — **Figure 5**: Sawn joint in contraction joint

When using dowels, it is important to ensure that they don’t get in the way of the joint’s opening or closing; otherwise, an uncontrolled crack could form in the joint’s surroundings. Round or square dowels cut on both ends, for example, should not be utilized because the end deformation may interfere with the joint opening or closing. Dowels should be coated with a suitable bond breaker on one side of the joint and aligned to within close tolerances parallel to the longitudinal direction of the panel and the surface of the base.

Diamond-shaped load plates can be used to replace dowels and allow the slab to travel horizontally without restraint; they also allow some differential movement in the joint’s direction, especially when shrinkage opens the joint. They can be placed within 150 mm spacings, whereas dowels should be kept at a distance of 300 mm. It is worth noting that dowels near crossings need to have expansion material on the vertical sides to allow the slab to move both parallel and perpendicular to the connection. Square dowels will almost always be required for this.

Individual panels in jointed reinforced pavements are usually joined by construction, expansion, or isolation joints, rather than contraction joints. Joint spacing should be limited to roughly 15 m for these pavements so that joint movement does not become excessive and joint sealing remains effective.

Construction Joints

Longitudinal construction joints separate portions of concrete poured at different times and form the borders of each pour. Transverse construction joints are necessary at predetermined points, such as the end of each day’s work, and at unanticipated pauses, such as those induced by bad weather or equipment breakdowns.

Simple keyed joints will frequently suffice for longitudinal construction joints if the pavement is lightly loaded, not more than 150 mm thick, and built over a strong, unyielding subgrade that is not subject to volume variations, or over a bound subbase or stabilised subgrade. Longitudinal construction joints should be provided with some type of load-transfer mechanism, such as dowels or diamond load plates, if the pavement is thicker or more heavily loaded. Figure 6 shows typical details of this sort of joint.

A keyed joint will not perform well as a load transfer device if it opens up more than 1 mm. When employing this type of joint for load transfer, it’s important to keep the joint opening to a minimum. Deformed tie bars can be used to hold keyed longitudinal joints together.

However, unless dowelled longitudinal contraction joints are additionally provided at a spacing not exceeding 15 m, such tie bars should not be utilized in panels having a total width of more than 15 m. Tie bars should be 800-mm length N12 bars with a spacing of 800 mm or 650000/DS_jmm, whichever is lesser. Where D is the base thickness in millimeters and S_j is the joint spacing in millimeters (mm).

In reinforced pavement construction, using split or slotted formwork to ensure continuity of reinforcement across the junction can provide positive control against vertical movement at the joints. However, creating and removing the formwork presents construction challenges. The reinforcement must be planned for the entire length of the pavement, not just the panel length. In jointed unreinforced pavements, transverse construction joints should be placed where a planned contraction joint would be. Unplanned construction joints should be placed in the panel’s middle third. Figure 6 shows some typical details.

A dowelled butt joint, which allows horizontal movement and performs all of the duties of a contraction joint, is recommended when a transverse construction junction is intended to coincide with the place where a contraction joint would ordinarily occur Figure 1.5. Planned construction joints are often built at regular contraction joint sites in jointed reinforced pavements. Because there is no aggregate interlock to enable weight transfer, butt joints with dowels (perhaps also with expansion material) are recommended. The dimensions, spacing, and debonding of the dowels are the same as for a transverse contraction joint.

A transverse construction joint is made in continuously reinforced pavements by putting a special header board that allows reinforcing steel to pass through. Prior to resuming concrete placement, the header is removed. Both slotted and split header boards are utilized, but the slotted form is preferred since it can be removed with the least amount of disruption to the already cast concrete.

References
Cement Concrete & Aggregates Australia (2009): Guide to Industrial Floors and Pavements – design, construction and specification CCAA T48

Some Harmful Materials in Aggregates for Concrete

By

Ubani Obinna

-

May 15, 2022

Table of Contents

Potentially harmful materials in aggregates for concrete are substances that react chemically with Portland cement concrete to produce one or more of the following:

(1) Significant volume change in the cement paste, aggregate, or both;
(2) interfering with cement’s regular hydration; and
(3) producing additional potentially hazardous byproducts

Organic impurities, silt, clay, shale, iron oxide, coal, lignite, and some lightweight and soft particles are all harmful materials in aggregates, that can affect the performance of the concrete in the fresh and/or hardened state (see Table 1). Alkali-reactive rocks and minerals include certain cherts, strained quartz, and some dolomitic limestones. Furthermore, sulfate attacks on concrete can be caused by gypsum and anhydrite. Popouts can be caused by swelling (absorption of water) or freezing of absorbed water in some aggregates, such as certain shales.

Table 1: Some harmful materials in aggregates for concrete

Substances	Effect on Concrete
Organic impurities	Affects setting and hardening, may cause deterioration
Materials finer than the 75-μm (No. 200) sieve	Affects bond, ASTM C 117 (AASHTO T 11) increases water requirement
Coal, lignite, or other lightweight materials	Affects durability, and may cause stains and popouts
Soft particles	Affects durability
Clay lumps and friable particles	Affects workability and durability, and may cause popouts
Cherts of less than 2.40 relative density	Affects durability, and may cause popouts
Alkali-reactive aggregates	Causes abnormal expansion, map cracking, and popouts

Most standards usually provide the maximum allowed levels of these harmful materials in aggregates. When defining limitations for dangerous compounds, the performance history of an aggregate should be a deciding factor.

coarse aggregates — Coarse aggregates in concrete

Organic Impurities

Organic impurities can cause concrete to take longer to set and harden, diminish strength increase, and, in rare situations, cause degradation. Organic impurities like peat, humus, and organic loam are less harmful, but they should still be avoided.

Fine Materials (Clay and Silt)

Very fine materials, such as silt and clay, may be present as loose dust and form a coating on aggregate particles finer than the 75μm (No. 200) sieve. Even tiny silt or clay coatings on gravel particles might be hazardous because they can damage the cement paste-aggregate bond. Water requirements may be greatly increased if certain types of silt or clay are present in excessive levels.

The grinding motion in a concrete mixer causes some fine aggregates to degrade; this effect, which is assessed using ASTM C 1137, can affect mixing water, entrained air, and slump requirements.

Low-density Materials (coal, lignite, wood, or fibrous materials)

Excessive volumes of coal or lignite, as well as other low-density components like wood or fibrous materials in concrete, will compromise the durability of concrete. These contaminants may dissolve, pop out, or generate stains if they exist near the surface. The ASTM C 123 standard can be used to identify potentially dangerous chert in coarse aggregate (AASHTO T 113).

Popout is an effect of harmful materials in aggregates — Popout in concrete

Soft, friable particles

Soft particles in coarse aggregates are particularly problematic because they produce popouts and can compromise concrete’s durability and wear resistance. If they are friable, they may break up when mixing, increasing the amount of water requirement. Testing may show that more inquiry or a different aggregate source is required where abrasion resistance is crucial, such as in heavy-duty industrial floors.

Clay Lumps

Clay lumps in concrete can absorb some of the mixing water, causing popouts in hardened concrete and reducing the durability and wear resistance of the concrete. They can also break up while mixing, increasing the amount of mixing water required.

Iron oxide and iron sulfide particles

Iron oxide and iron sulfide particles can occasionally be found in aggregates, causing unattractive stains on exposed concrete surfaces (see image below). When tested according to ASTM C 641, the aggregate should meet the staining standards of ASTM C 330 (AASHTO M 195); the quarry face and aggregate stockpiles should not exhibit evidence of staining.

iron stains on concrete — Iron stains on concrete

The aggregate can also be submerged in a lime slurry to help detect stained particles. If staining particles are present, a blue-green gelatinous precipitate will form within 5 to 10 minutes; when exposed to air and light, it will quickly turn brown. Within 30 minutes, the reaction should be completed. When a suspicious aggregate is placed in lime slurry and no brown gelatinous precipitate forms, there is little chance of a reaction occurring in concrete. When aggregates with no prior successful use in architectural concrete are utilized, certain tests should be required.

Buckling of Thin Plates

By

Ubani Obinna

-

May 13, 2022

Buckling of thin plates occurs when a plate moves out of plane under compressive load, causing it to bend in two directions. The buckling behaviour of thin plates is significantly different from the buckling behaviour of columns. Buckling in a column terminates the member’s ability to resist axial force, and as a result, the critical load is the member’s failure load. The same cannot be said for the buckling of thin plates due to the membrane action of the plate.

Plates under compression will continue to resist increasing axial force after achieving the critical load, and will not fail until a load far greater than the critical load is attained. As a result, a plate’s elastic critical load is not the same as its failure load. Instead, the load-carrying capability of a plate must be determined by examining its post-buckling behaviour.

A governing equation in terms of biaxial compressive forces N_x and N_y and constant shear force N_xy, as shown in Figure 1, can be developed to estimate the critical in-plane loading of a plate using the idea of neutral equilibrium.

D[δ⁴w/δx⁴ + 2(δ⁴w)/(δx²δy²) + δ⁴w)/δy⁴] + N_x(δ²w/δx²) + N_y(δ²w/δy²) + 2N_xy(δ⁴w/δxδy) = 0 ——— (1)

All the stress components are expressed in terms of the deflection w of the plate (where w is a function of the two coordinates (x, y) in the plane of the plate). D = Eh³/12(1 – v²) is the flexural rigidity of the plate per unit length; E is the modulus of elasticity; h is the thickness of the plate, and v is Poisson’s ratio.

The critical load for uniaxial compression can be determined from the differential equation:

D[δ⁴w/δx⁴ + 2(δ⁴w)/(δx²δy²) + δ⁴w)/δy⁴] + N_x(δ²w/δx²) = 0 ——— (2)

which is obtained by setting N_y = N_xy = 0 in equation (1). For example, in the case of a simply supported plate, equation (1) can be solved to give:

N_x = π²a²D/m² (m²/a² + n²/b²)² ——— (3)

Taking n equal to 1 yields the critical value of N_x (i.e. the smallest value). This means that a plate buckles in such a way that there are several half-waves in the compression direction but only one half-wave in the perpendicular direction. As a result, the formula for the compressive force’s critical value becomes:

N_x(crit) = π²D/a² [m + 1/m(a²/b²)]² ——— (4)

The Euler load for a strip of unit width and length a is represented by the first factor in this expression. The second factor denotes the proportion of greater stability gained by the continuous plate compared with that of an isolated strip. The amount of this factor is determined by the magnitude of the a/b ratio as well as the number m, which indicates how many half-waves the plate buckles into. If a is less than b, the second term in the parenthesis of equation (4) is always less than the first, and the expression’s minimum value is reached by assuming m = 1, i.e. that the plate buckles in one half-wave. N_x critical value can be written as follows:

N_cr = kπ²D/b² ——— (5)

The factor k depends on the aspect ratio a/b of the plate and m, the number of half-waves into which the plate buckles in the x-direction. The variation of k with a/b for different values of m can be plotted as shown in Figure 2. The critical value of N_x is the smallest value that is obtained for m = 1 and the corresponding value of k is equal to 4.0. This formula is analogous to Euler’s formula for buckling of a column.

plot — **Figure 2**: Buckling stress coefficients for uniaxially compressed plate (Shanmugam and Narayanan, 2008)

In the more general case in which normal forces N_x and N_y and the shearing forces N_xy are acting on the boundary of the plate, the same general method can be used. The critical stress for the case of a uniaxially compressed simply supported plate can be written as:

σ = 4π²E/[12(1 – v²)] × (h/b)² ——— (6)

The critical stress values for different loading and support conditions can be expressed in the form:

f_cr = kπ²E/[12(1 – v²)] × (h/b)² ——— (8)

in which f_cris the critical value of different loading cases. Values of k for plates with different boundary and loading conditions are given in Figure 3.

Table — **Figure 3:** Values of k for plates with different boundary and loading conditions (Shanmugam and Narayanan, 2008)

Buckling of Thin Plates in Staad Pro

plate buckling — **Figure 4:** Model example of buckling analysis of thin plate

plate model — **Figure 5:** Buckling analysis of thin plate on Staad Pro

           CALCULATED BUCKLING FACTORS FOR LOAD CASE       1

   MODE               BUCKLING FACTOR

     1                   2957.15142
     2                   3056.42462
     3                  -3102.59328
     4                  -3107.05926

References
Shanmugam N. E. and Narayanan R. (2008): Structural Analysis. In ICE Manual of Bridge Engineering. doi: 10.1680/mobe.34525.0049

Modelling of Foundation of Bridges

By

Ubani Obinna

-

May 11, 2022

Piers and abutments are typically employed as support structures for bridge decks. Basically, they transmit the superstructure load of the bridge to the foundation, which is supported on the soil. Piers, abutments, and foundations make up the substructure of any bridge. Therefore, they are influential parameters in the modelling of the foundation of bridges.

The stiffness of the piers, abutments, foundations, and soil all play a role in determining the safety and stability of a bridge. Piers and abutments are subjected to a variety of loading conditions, including live and dead loads from the superstructure; substructure dead load; soil pressure; wind load on the superstructure, substructure, and vehicles; pressure caused by streamflow and ice; and earthquake load.

Furthermore, indirect actions such as support settlement have secondary effects that cause considerable changes in the distribution of internal stresses inside the bridge deck. As a result, it is very important for a bridge designer to pay close attention to the stiffness of substructures when modelling the foundation of bridges. The relative stiffnesses of all of a bridge’s components determine how forces are distributed throughout the superstructure and substructure. The substructure of a bridge is intended to provide stability, strength, and be within the allowable bearing pressure of the soil.

Bridge foundation modelling — **Figure 1**: Typical bridge foundation modelling

Continuous structures (statically indeterminate structures) can be very efficient at redistributing internal forces, but they are also very susceptible to the effects of differential settlement. The relative stiffnesses of all the components of the bridge, including the individual bearings, have a considerable impact on the distribution of forces throughout the foundation system of the structure. A box girder bridge deck, for example, is particularly sensitive to differential settlement and bearing compression despite its high resistance to torsion and distortion.

In the design of foundation of bridges, any suitable load combination for bridges can be used. One load combination might be the most important for stability, while another might cause the most stress in the concrete or reinforcing steel, and a third might result in the most soil pressure.

Furthermore, if a portion of the pier is submerged underwater, the effect of buoyancy (uplift) must be taken into account when evaluating the pier’s stability. Bridge foundation design can be challenging because of the numerous load combinations that can occur. To assess the stability, earth pressures, and allowable soil bearing pressures, a thorough soil investigation should be carried out. Before choosing a bridge location and type in an earthquake zone, the position of fault zones, slide regions, or any other potentially big unstable ground areas should be thoroughly investigated.

Load Transmission to Foundation of Bridges

Loads from the superstructure will be transmitted to the foundation by direct compression or compression and bending, depending on the shape of the substructure. The deck loads are transmitted to the foundations by direct compression through the piers depicted in Figure 2(a) to (c).

Bridge Sections — **Figure 2:** Different bridge cross-sections (transmit load by direct compression)

Structures with a cross-head beam or cantilever, as seen in Figure 3(a) and (b), on the other hand, may have a lot of flexibility (Hambly, 1991).

flexible bridge pier 1 — **Figure 3:** Different bridge cross-sections (transmit load by direct compression, bending, and shear)

When analysing a bridge deck on such supports using a grillage analysis, it may be essential to represent both the deck and the cross-head. It might be simpler to model the structure using a space frame that mimics the pier’s shape. The foundation stiffnesses and compressible bearings with vertical members between deck and cross-head are then reasonably straightforward to model.

Bridge Foundation Stiffness and Interaction with Soil

When a foundation is subjected to dynamic loading, it will experience a vibration of the same frequency as the applied force. It is therefore critical to set a limit on the amount of dynamic force that the foundations can withstand. There is a certain amount of dynamic motion that can be allowed, as well as a defined amount of settlement that can occur.

Spring models for shallow foundations of bridges — **Figure 4:** Spring models for the stiffness of footings (after Hambly, 1991)

Researchers have presented methods for determining the stiffnesses of shallow footing foundations based on the theory of an elastic half-space (Figure 4). Hambly (1991) generated approximate stiffness estimates by simplifying the equations from these methods. For shallow footings that may slide or tilt across the shorter direction, these simplified formulae are provided below.

Shear modulus: G = E/2(1 + υ)
Vertical stiffness: K_y = 2.5GA^0.5/(1 – υ)
Horizontal stiffness: K_x = 2G(1 + υ)A^0.5
Rocking stiffness: K_m = 2.5GZ/(1 – υ)

where G is the shear modulus of soil; E is Young’s modulus of soil; υ is Poisson’s ratio of soil; A is the foundation area and Z is the foundation section modulus. Vertical and rotational stiffnesses can be more conveniently represented by two parallel springs as shown in Figure 4(e) for which the stiffness is taken as K = 0.5K_z. The parallel springs are spaced at a distance l given by l = 2(K_m/K_z)^0.5.

Although these equations are approximate, they can be used to quickly determine the order of magnitude of foundation stiffnesses. The load path and courses of reactive forces determine the stiffness of the earth beneath a foundation. Vertical soil responses will occur at vast depths as a result of the vertical loads on the foundations.

Under arching action, on the other hand, the horizontal force and moment on one foundation will be cancelled out by an equal and opposite force and moment on the second foundation, potentially increasing the horizontal stiffness of the soil if the foundations are near enough together. When horizontal breaking forces cause imbalanced reactions in the foundations, concurrent loading on neighbouring foundations reduces the horizontal stiffness of the foundation. Because of these and other difficulties, it is not always possible to account for them when choosing foundation stiffness.

Three-dimensional finite-element analysis can be used to investigate the stiffnesses of foundations for bridges across difficult ground conditions in great detail. Finite-element systems are now available that can describe soils with complex nonlinear stress-strain behaviour and concurrent pore water pressures. Soil data of adequate quality, which is required by advanced computer programmes, is usually scarce.

Pile foundation is usually employed when the stable soil stratum is at some depth below the ground level. Wooden piles, concrete piles, steel sections, and steel tubes filled with concrete have all been used in the foundation of bridges. The stiffness of pile foundations is more challenging, especially if the stiffness is derived from the interaction of pile bending and lateral soil forces rather than axial pile compression. There are various computer programmes that may be used to calculate the stiffnesses of pile groups using finite elements or the elastic half-space theory.

Stiffness moduli of soils for Foundation of Bridges

The Young’s modulus of the soil beneath the foundation is one of many elements that affect foundation stiffness. Grain size, gradation, the mineral content of soil grains, grain shape, soil type, relative density, soil particle arrangement, stress level, and prestress all affect the elastic modulus of granular soils based on effective stresses.

Table 1 shows typical elastic modulus and Poisson’s ratio values for typically consolidated granular soils (Das, 1990). In cohesive soils, the undrained elastic modulus is primarily determined by soil plasticity and overconsolidation. The slope of a stress-strain curve generated from an undrained triaxial test can be used to calculate the elastic modulus of soils (Holtz and Kovacs, 1981).

Table 1: Typical elastic modulus and Poisson’s ratio values for typically consolidated granular soils (Das, 1990)

Type of soil	Elastic modulus (MPa)	Poisson’s ratio
Loose sand	10 – 24	0.2 – 0.4
Medium dense sand	17 – 28	0.25 – 0.4
Dense sand	35 – 55	0.30 – 0.45
Silty Sand	10 – 17	0.20 – 0.40
Sand and gravel	69 – 170	0.15 – 0.35

To obtain soil stiffnesses, one must first identify the soil parameters through site inspection. This is difficult since testing on tiny samples rarely yields accurate information on huge soil masses. Large-diameter plate bearing tests and back-analyses of observations of identical structures on similar ground conditions are the best ways to determine stiffnesses. Elastic modulus and Poisson’s ratio are not constants for a soil, but rather variables that roughly represent its behaviour under a given set of stresses. For any other set of stresses, other elastic modulus and Poisson’s ratio values will apply.

When determining the values of these factors, good judgement is required. It is difficult to determine elastic modulus values with great precision, hence test data for the specific soil will be required anytime an exact estimate is required. Tangent modulus and secant modulus are phrases that are commonly used. For preliminary analyses of soil-structure interaction, textbooks provide specific direction and information. Calculations should, however, be backed up by examinations of the individual site and foundation conditions at the final design stage.

Worked Example

Let us consider for example the foundations for the portal frame shown in Figure 5. The footings for the portal frame have d = 3m (parallel to span) and b = 14m wide. Hence:

Typical 3D bridge — **Figure 5:** Model of a portal type bridge

Footing area, A = 14 × 3 = 42 m²
Section modulus of the footing, Z = bd²/6 = (14 × 3²)/6 = 21 m³

The stiffnesses, with Poisson’s ratio in the order of 0.3 to 0.5, can be calculated approximately as:
K_y = 1.5E(A)^0.5 = 1.5E(42)^0.5 = 9.721E
K_x = E(A)^0.5 = E(42)^0.5= 6.481E
K_m = 1.5EZ = 1.5E(21) = 31.5E

For an assumed value of E = 100 MPa, we find that;
K_y = 927.1 MN/m,
K_x = 648.1 MN/m and
K_m = 3150 MN/m

The total width of the foundation is 1m and stiffnesses per unit width are calculated as:

K_y = 927.1/14 = 66.22 MN/m/m
K_x = 648.1/14 = 46.29 MN/m/m
K_m = 3150/14 = 225 MN/m/rad/m

Stiffnesses K_y and K_m are represented by two parallel springs of:
K = 0.5K_y = 33.11 MN/m/m at a spacing of l = 2(K_m/K_y)^0.5 = 2(225/66.22)^0.5 = 3.686 m

Modelling of the bridge — **Figure 6:** Typical modelling of the foundation of a bridge

References

[1] Das B. M. (1990) Principles of Foundation Engineering, 2nd edn. PWS-Kent, Boston, MA.
[2] Hambly E. C. (1991) Bridge Deck Behaviour. E & FN Spon, London.
[3] Holtz R. D. and Kovacs W. D. (1981) An Introduction to Geotechnical Engineering. Prentice-Hall, Englewood Cliffs, NJ.

Introduction to Theory of Structures | Structural Analysis

By

Ubani Obinna

-

May 10, 2022

Theory of structures is a field of knowledge that is concerned with the determination of the effect of loads (actions) on structures. A structure in this context is generally regarded to be a system of connected members that can resist a load. Therefore in some programs, theory of structures is also referred to as structural analysis.

In application, theory of structures is usually interested in the computation of the deformations (displacements), internal forces, stresses, stability, support reactions, velocity, and accelerations of structures under load. The principles from applied mathematics, applied sciences (physics and mechanics), and materials science are usually employed in achieving this. The results of the analysis are used to evaluate the behaviour of a structure under the load, with the sole aim of verifying the integrity of the structure when in use. Therefore, theory of structures (structural analysis) is a key part of the engineering design of structures.

Theory of Structures and Structural Design

Buildings, bridges, and towers are prominent examples of the application of theory of structures in civil engineering. Ship and aircraft frames, tanks, pressure vessels, mechanical systems, and electrical supporting structures are examples of other engineering areas.

In the design of civil engineering structures for public use, safety, stability, serviceability, functionality, aesthetics, as well as economic and environmental constraints must be considered by the engineer. Thus, structural designs are expected to satisfy two basic limit states which are;

Ultimate Limit State (ULS) and
Serviceability Limit State (SLS)

According to limit state design, ULS requirements have to deal with issues like structural failure, overturning, buckling, instability, etc while SLS requirements have to deal with issues like deflection, vibration, cracking, etc, which affects the appearance and user experience of the building.

Usually, numerous independent evaluations of various solutions are required before a final decision on which structural scheme is best can be made. This approach to the design of structures, therefore, necessitates a basic understanding of material properties and the principles of mechanics that control material behaviour.

Following the proposal of a preliminary structural design, the structure must be examined to ensure that it has the requisite stiffness and strength. To effectively examine a structure, specific idealizations about how the components are supported and connected must be made. The loadings on a structure are calculated using codes and local requirements, and the forces and displacements in the members are calculated using theory of structures (structural analysis), which is the focus of this article.

The results of the structural analysis can then be utilised to redesign the structure, taking into consideration a more precise calculation of the self-weight and size of the members. As a result, structural design is based on a sequence of successive approximations, each of which necessitates a structural analysis.

Types of Structures

When modelling a real-life structure, it is necessary to represent the form of the structure in terms of idealized structural members, e.g. in the case of plane frames as beam elements, in which the beams, columns, etc. are indicated by line diagrams. The lines normally coincide with the centre lines of the members.

The decision as to which type of structural system to use rests with the structural designer whose choice will depend on the use of the structure, the materials to be used, and the original form of the structure as indicated by the architect. It is possible that more than one form of structural systems may be required to satisfy the requirements of the problem and the designer must then rely on experience and skill to choose the best solution. Examples of structural elements/systems are;

Beams
Columns
Trusses
Frames (portal frames, gable frames)
Arches
Cables
Shear walls
Continuum structures (shells, plates, domes, etc)

Beams

Beams are the commonest of all structural systems. They are usually made up of straight prismatic members that span in between supports that may be of any form. Beams are usually important in resisting transverse loads and achieve so by developing mainly bending moments and shear forces. Typically in structural design, beams can be made of reinforced concrete, steel, or timber.

See;
Design of reinforced concrete beams
Design of steel beams according to Eurocode 3
Design of steel beams according to BS 5950
Design of timber beam according to Eurocode 5

When a beam is fixed at one end and free at the other end, it is called a cantilever beam. It is also possible to have a beam extend beyond an external support and in this case, it is called a beam with an overhang (see Figure 3b and 3c). A beam that has intermediate supports is called a continuous beam, and when internal hinges are introduced to make it statically determinate, it is called a compound beam.

simply supported beam — **Figure 2:** Simply supported beam system for bridge

different types of beam systems — **Figure 3**: Different types of beam systems

Cantilever beams and compund beams — **Figure 4:** (a) Cantilever beam (b) Compound beam

Columns

Vertical members that are used to resist axial compressive loads are referred to as columns. Hollow sections, H-sections, and I-sections are often used in the design of steel columns. Rectangular, circular, and square cross-sections with reinforcing rods are often used in the design of reinforced concrete columns. Occasionally, columns are subjected to both axial load and a bending moment. When a column is subjected to a bending moment in one direction, it is called a uniaxially loaded column, and when it is loaded in two directions, it is called a biaxially loaded column.

Trusses

Trusses are arrangements of straight members connected at their ends. They resist loads by developing axial forces in their members but this is only true if the ends of the members are pinned together. The members in a truss are arranged to form a triangulated system so as to make them geometrically unchangeable, and also so that they will not form a mechanism. In trusses, loads are applied only at the joints.

When trusses have their ends fixed or welded together, secondary stresses develop and it is a form of analysis that favours the use of computer programs. Trusses provide practical and economical solutions to engineering problems. They can efficiently span greater lengths than beams, and hence can be found in roofs of buildings, bridges, etc. See design of steel roof trusses.

**Figure 6**: Typical Fink Truss and loading

**Figure 7:** Typical Howe Truss and loading

Pratt truss — **Figure 8:** Typical Pratt Truss and loading

Rigid Frames

Unlike trusses which are pin-jointed frames that transmit axial forces only, rigid frames are designed to transfer both axial forces, shear forces, and bending moment across the members. Rigid frames are comparatively easier to erect since their construction usually involves the connection of steel beams and columns by bolting or welding.

In reinforced concrete design, the beams and columns are usually cast monolithically and it is relatively easy to construct the formwork. Rigid frames can be statically determinate or indeterminate and may involve multi-storey or multi-bay configuration as the case may be.

Types of frames — **Figure 9:** Different forms of rigid frames

Arches

The arches are widely used in modern engineering due to their ability to cover large spans and their attractiveness from an aesthetic point of view. The greater the span, the more an arch becomes more economical than a truss. Materials of the modern arches are concrete, steel, and timber. Arches are mainly classified as three-hinged, two-hinged, and arch with fixed supports.

Arches carry most of their loads by developing compressive stresses within the arch itself and therefore in the past were frequently constructed using materials of high compressive strength and low tensile strength such as stones and masonry. Arches may be constructed in a variety of geometries; they may be semicircular, parabolic, or even linear where the members comprising the arch are straight.

types of arch structures — **Figure 10:** Different forms of arch structures

Cables

Cables are made from high-strength steel wires twisted together and present a flexible system, which can resist loads only by axial tension. The use of cables allows engineers to cover very large spans, especially in bridge design. In theory of structures, cables are deemed extremely efficient because they make the most effective use of structural material in that their loads are carried solely through tension through the wire. Therefore, there is no tendency for buckling to occur either from bending or from compressive axial loads.

A cable as a load-bearing structure has several features. One of them is the vertical load that gives rise to horizontal reactions, which, as in the case of an arch, is called a thrust. To accommodate the thrust it is necessary to have a supporting structure. It may be a pillar of a bridge or a tower or pylon. Cables are utilized in the form of suspension bridges, cable-stayed bridges, tower guy wires, etc.

suspension bridge — **Figure 11:** Suspension bridge

cable stayed bridge — **Figure 12:** Cable-stayed Bridge

Shear Walls

In tall buildings, there is a need to resist lateral forces which may arise from wind load or seismic forces, shear walls have been found to be a veritable solution. Shear walls may be planar solid or coupled and are usually provided alongside the frames of the structure. They are usually provided along the elevator shafts or stairwells. Apart from the use of computer-aided software, shear walls have been analysed using methods such as the continuous connection medium method, frame analogy method, etc.

shear core — **Figure 14:** Plan of a tall building with a shear wall in the elevator shaft

Continuum structures

Continuum structures are made from a material having a very small thickness compared to its other dimensions. Examples of continuum structures are folded plate roofs, shells, floor slabs, etc. An arch dam is a three-dimensional continuum structure as are domed roofs, aircraft fuselages, and wings. Continuum mechanics is concerned with structures that are continuous in space. The simplest elements of this type are surface elements which have a thickness of a different order from the other two dimensions. Surface elements are termed plates if they have a plane form and shells if they form a general surface.

floor slab — **Figure 15**: Reinforced concrete slab

Folded plate and shell structure — **Figure 16:** Shell structure

Loading of Structures

In theory of structures, we have different types of loading configuration and the commonest of them are;

Concentrated loads (point loads)
Uniformly distributed line loads (UDL)
Uniformly distributed area loads (full pressure loads)
Non-uniformly distributed load (e.g. trapezoidal and triangular loads)
Indirect actions

Concentrated loads

These are loads that act on a point in a structural system, and theoretically, they are like knife-edge loads acting at a précised point on a beam. However, in practice, these loads are usually distributed over a small area.

concentrated load on structures — **Figure 17:** Typical idealisation of concentrated loads

For example, a column that is supported on a suspended beam may be represented as a point load on the beam. Furthermore, the effect of secondary beams on primary beams is also treated as concentrated loads (see Figure 18). In Figure 18, Beam No 4 (axis 2-2) is a secondary beam that is supported on beams Nos 1 and 2 (axis A-A and B-B) which are primary beams that are directly supported on columns.

floor slab arrangement — **Figure 18:** Typical floor slab general arrangement

In the analysis, the loading on Beam No 4 is evaluated first, and the resultant support reactions are obtained. The reactions from Beam No 4 are applied as concentrated loads at the mid-span of beams Number 1 and 2. Note that secondary beams are usually shallower than primary beams. The equivalent loading on beams No 1 and 2 can be represented as shown in Figure 19.

loaded simply supported beams — **Figure 19:** Loading on beam No 2 (neglecting horizontal reaction)

Where;
P = Support reaction from beam No 4
q = load from the slab, self-weight of the beam, blockwork loading, and finishes, etc

Also in trusses, all loads are converted to concentrated loads acting on the nodes. It is also very usual for wind loads to be converted to concentrated loads acting at the nodes of a structure (where the nodes are at the floor slab level).

Uniformly Distributed Line Loads (UDL)

UDLs are loads that are spread with the same intensity over a span of a structural element. They are usually distributed per unit length of the element and hence their units are of the form kN/m or N/m as the case may be. For example, the load from a slab that is simply supported or clamped at all sides to a beam that is supporting it may be converted to equivalent UDL using the following relationships (British Codes) given in Table 1. However, it is important to note that the actual load distribution from slab to beams is either triangular or trapezoidal.

Table 1: Equivalent loading from slab to beams according to British Codes

Type of slab / Direction of span	Load to beam on long span (kN/m)	Load to beam on short Span (kN/m)
One-way slab	(nl_x)/5	(nl_x)/2
Two-way slab	(0.5nl_x) × (1 – 0.333k²)	(nl_x)/3

In Table 1 above, n stands for the uniformly distributed load on the slab at the ultimate limit state and is usually given by 1.4g_k + 1.6q_k (BS 8110-1:1997) or 1.35g_k + 1.5q_k (Eurocode 2). g_k stands for the dead load while q_k stands for the live load. k is the ratio between the length of the longer side (L_y) and the shorter side (L_x). This aspect ratio oftentimes is also used to define whether a slab is spanning in one direction or in two directions. Conventionally, if;

L_y/L_x > 2.0 (One way spanning slab)
L_y/L_x < 2.0 (Two way spanning slab).

You can refer to standard design textbooks for more complete information, but this is just to give us a little idea of how these loads come about.

Uniformly Distributed Area Loads (UDL)

Another variation of UDLs is pressure loads that are uniformly distributed over a unit area. These are typically the type of loads on slabs and plates, and even retaining walls. In fact, all walls are usually subjected to pressure loads, including earth intensity loads on foundations. The unit of pressure loads is kN/m². The self-weight of slabs and finishes are typical examples of uniformly distributed loads. The live loads (imposed loads) specified on codes of practice are also uniformly distributed pressure loads.

In theory of structures, the analysis of plates subjected to uniformly distributed pressure load is usually carried out per unit strip and hence, internal forces such as moment and shear are expressed per metre strip (e.g. kNm/m, kN/m). Other examples of uniformly distributed loads are the surcharge loads that act at the back of retaining walls etc.

Non-uniformly distributed loads

These are loads that have varying intensity across the element on which they act. Examples of such loads are trapezoidal loads or triangular loads. Also, beams that do not have their full length subjected to full UDL can also be brought under this category (figure 1.3a). Triangular loads can come from block work on lintels and beams, and other types of loads and trapezoidal loads can come from earth pressure or hydrostatic force that is exerted on an element that is fully immersed in a fluid at rest.

Figure 20(c) shows a gravity retaining wall that is retaining earth at the backface. The pressure due to the retained material varies with depth in a triangular manner as shown in the figure. In the figure;

q = earth pressure (kN/m²)
γ = Density of retained earth material (kN/m³)
K = Coefficient of active pressure (which may be computed using Coulomb’s or Rankine’s theory)
H = Height of the earth fill (m)

Indirect Actions (Loads)

Sometimes, other indirect actions on structures induce internal forces in the structure other than externally applied loads. The effects of these actions must be adequately assessed if the structure is to perform satisfactorily well. Some of these indirect loads are;

Relative movement of supports (differential settlement)
Temperature difference
Imperfections

It is however worth knowing that problems such as differential settlement do not affect statically determinate structures. Such problems are only critical in statically indeterminate structures.

Types of supports in Theory of Structures

In theory of structures, a lot of support systems are available and the one that is adopted is dependent on the one that significantly represents the actual physical behaviour when constructed. The common types of support conditions are;

Pinned supports
Roller supports
Fixed supports
Tension or compression springs
Elastic foundations

The first three are the commonest and they are the ones that are usually considered in theory of structures textbooks. In the selection of the types of support that will be used for analysis, the following may be used as guidelines;

Pinned Supports

These supports appear in the form of a hinge and is characterized by possessing vertical and horizontal reaction components. The joints of the support are free to rotate and as a result, pinned supports are not capable of resisting bending moment. In actual practice, situations that may be idealised as pinned supports are;

Stanchions (steel columns) that are supported on base plates (Figure 21)
Double angle cleat connection for beams and columns (Figure 22)

Column base plate — **Figure 21:** Stanchion on a base plate

double cleat supports — **Figure 22:** Double angle cleat for beams and columns

Roller supports

A single pinned support is sufficient to maintain the horizontal equilibrium of a beam and hence may not necessarily be provided at the other end. It may be advantageous to allow horizontal movement of the other end so that, for example, expansion and contraction caused by temperature variations do not cause additional stresses.

Such support may take the form of a composite steel and rubber bearing as shown in (Figure 23a) or consist of a roller sandwiched between steel plates (Figure 23b). This type of support is called roller support. It is assumed that such support allows horizontal movement and rotation but prevents movement vertically, up, or down. Roller supports resist only vertical loads.

Examples of roller support — **Figure 23:** (a) Steel laminated bearing on a bridge (b) Rocker supporting a bridge beam

Fixed Support

As the name implies, when the support of a structural member is built-in (fixed) such that no rotation or translation occurs, it is referred to as a fixed support. Fixed support resists vertical loads, horizontal loads, and moment and hence there are three reaction components. A beam-column connection that is welded with additional stiffeners can be idealised as fixed support for the beam (Figure 24a). Reinforced concrete columns and footing can also be idealised as fixed support if the base and the supporting soil are very stiff and rigid. (Figure 24b).

In theory of structures, these support conditions and their reaction components are shown in the Table below;

Types of supports in theory of structures

Tension or compression springs

Springs are also used in the idealisation of supports in theory of structures. When a structure is supported on a deformable body, springs can be used idealise the support condition. The stiffness of the spring is modelled to represent the strength of the deformable body. As a result, the interaction between the deformation of the support and the structural response of the supported member is captured in the analysis. A typical example is the use of Winkler’s springs in the idealisation of soils.

Elastic Foundations

When a structure is supported on a continuously deformable medium, it can be modelled as a structure on an elastic foundation. Beams on elastic foundations and plates on elastic foundations are popular topics in geotechnical and structural engineering. The can be used in the idealisation of raft foundations when using flexible approach. Elastic foundations are important in the modelling of soil-structure interaction.

Types of Structural Analysis

After the modelling of the structure and the supports, and assessment of the applied loads, the next step is to carry out an analysis to determine the internal forces and deformations induced in the structural members due to the applied load. Structural analysis usually encompasses one or more types of the following analytical methods.

Static linear analysis

This is the analysis that is carried out to determine the internal forces and displacements due to static loads (time-independent loads) while assuming that the structure is within the elastic state (Hooke’s law is being obeyed).

Non-Linear Static analysis

This is the analysis that is carried out to determine the internal forces and displacements in a structure when it is subjected to static loads and when we assume non-linear conditions. Non-linear analysis can be in the following forms;

Physical – The material has exceeded the yield point and Hooke’s law is no longer being obeyed.
Geometrical – Structure is subjected to large displacement, and analysis is based on the deformed structure
Structural – Analysis for structures with gaps, unbalanced constraints etc
Mixed non-linearity – A combination of the above-mentioned non-linearities

Buckling Analysis

This is the analysis that is carried out to determine the critical load (or critical load factor) including the corresponding buckling mode shapes of structures subjected to compressive loads. It is important is assessing the stability of structures.

P-delta Analysis

This analysis takes into account the additional moments due to compressive loads on the displacement caused by the lateral loads. This happens in tall and flexible structures. Thus, the analysis is also based on the deformed system (non-linear).

Free Vibration Dynamic Analysis

The purpose of this vibration analysis is to obtain the natural frequencies (eigenvalues) and the corresponding mode shapes (eigenfunctions).

Time-history Analysis

This analysis is carried out to determine the response of a structure that is subjected to arbitrarily time-varying loads.

Pushover Analysis

Pushover analysis is a non-linear approach used to determine structural capacity under static horizontal loads that grow until the structure collapses. It is a static process that estimates seismic structural deformations using a reduced nonlinear technique. During earthquakes, structures redesign themselves.

The dynamic forces on a structure are moved to other components as individual components yield or fail. A pushover analysis replicates this phenomenon by applying loads until the weak link in the structure is discovered, then changing the model to account for the structural changes generated by the weak link.

The loads are redistributed in the second iteration. The structure is pushed once more until the second weak link is found. This approach is repeated until a yield pattern for the entire structure is found under seismic loading. Pushover analysis yields several capacity curves based on the variation of base shear as a function of the displacement of a control point on the structure.

Static Determinacy of Structures

A structure is stable when it maintains balance in force and moment. As a result, we know that from statics, if a structure is to be at equilibrium,

∑F_y = 0; ∑F_x = 0; ∑M_i = 0 ——— (1)

Where;

∑F_y = Summation of the vertical forces
∑F_x = Summation of the horizontal forces
∑M_i= Summation of the moment of force components acting in the x-y plane passing through point i.

When the number of constraints in a structure permits the use of the equation of statics (equation 1) to analyse the structure, the structure is said to be statically determinate. Otherwise, it is statically indeterminate, and additional equation(s) which are derived from the load-deformation relationships are used for analysis. For the records, there are two well-known approaches to the analysis of indeterminate structures and they are;

Flexibility Methods – When the structure is analysed with respect to unknown forces
Stiffness Methods – when the structure is analysed with respect to unknown displacements

A structure may be indeterminate due to redundant components of reaction and/or redundant members. Note that a redundant reaction or member is one which is not really necessary to satisfy the minimum requirements of stability and static equilibrium. However, redundancy is desirable in structures because it can be a cheaper alternative to statically determinate structures. The degree-of-indeterminacy is equal to the number of unknown member forces/external reactions which are in excess of the equations of equilibrium available to solve for them.

Webinar Invitation: Design of Pile and Raft Foundation

By

Ubani Obinna

-

May 8, 2022

Structville Integrated Services Limited is pleased to announce the third edition of the webinar on the ‘Design of Raft and Pile Foundation’. The details are as follows;

Topic: Structural Analysis and Design of Raft and Pile Foundation
Date: Saturday, 14th of May, 2022
Time: 12:00 noon – 02:00 pm (WAT)
Fee: NGN 4000 only (USD $10.00)
Platform: Zoom Live
Facilitator: Engr. Ubani Obinna (MNSE, R.Engr)

To register for the event click HERE

Features:
(1) Theories and philosophies in the design of shallow and deep foundations
(2) Practical design of pile foundations and pile caps using real-life data
(3) Rigid and flexible approach to the design of raft foundation
(4) Structural design of raft foundation
(5) Full design material (mini textbook) with detailed drawings covering the above topics

To register for the event, click HERE

Temperature Actions on Bridge Decks | Thermal Loads on Bridges

By

Ubani Obinna

-

May 8, 2022

Table of Contents

The temperature of a bridge structure and its surroundings varies daily and seasonally, affecting the overall movement of the bridge deck as well as the stresses within it. The temperature actions on bridge decks have an impact on the design of bridge bearings and expansion joints, while the overall movement of the bridge deck and the stresses within it have an impact on the quantity and placement of structural materials such as steel.

The daily effects cause temperature variations within the depth of the superstructure which vary depending on whether it is heating or cooling. Guidance is typically provided in the form of idealized linear temperature gradients to be expected when the bridge is heating or cooling for various types of construction (concrete slab, composite deck, etc.) and blacktop surface thickness.

Internal stresses in a bridge cross-section self-equilibrate as a result of temperature gradients. There are two forms of internal stresses induced as a result of this: primary and secondary. The former is caused by temperature changes across the superstructure (whether simply supported or continuous), while the latter is caused by continuity. Both must be considered and accommodated in the design.

The temperature of a bridge deck varies throughout its mass. The variation is caused by:

the position of the sun
the intensity of the sun’s rays
thermal conductivity of the concrete and surfacing
wind
the cross-sectional make-up of the structure.

The effects of these variations on a bridge deck can be quite complex. On a daily (short-term) and annual (long-term) basis, changes occur. Daytime heat gain and nighttime heat loss occur on a daily basis. The ambient (surrounding) temperature varies from year to year.

On a daily basis, incident sun radiation controls temperatures towards the top, whereas shade air temperature controls temperatures near the bottom. In Figure 1, the whole distribution is shown. Positive indicates a rapid rise in deck slab temperature due to direct sunshine (solar radiation). Negative indicates that the ambient temperature is dropping due to heat loss (re-radiation) from the structure.

Temperature Actions on Bridge Decks — **Figure 1:** Typical temperature distributions

According to research, the distributions (or thermal gradients) for different ‘groups’ of structure as stated in Figure 9 of BD 37/01 Clause 5.4 can be idealized for analysis purposes. The four groups depend on the type of construction and are listed below;

(1) Steel deck on steel box girders
(2) Steel deck on steel truss or plate girders
(3) Concrete deck on steel truss, steel box, or plate girders
(4) Concrete deck on concrete beams or box girders

The thickness of the surface, the thickness of the deck slab, and the type of the beam are all important criteria. The curvature of the deck is caused by temperature changes, which result in internal primary and secondary stresses within the structure.

Temperature Changes on Bridge Decks

Temperature changes on bridges are expressed in terms of a uniform temperature component, a vertical difference component that contains non-linear components, and, when applicable, a horizontal difference component that can be considered to vary linearly. The temperature of the bridge is affected not only by the shade air temperature and solar radiation, but also by the scheme, cross-section, mass, and material used.

As a result, bridges can be categorised into the following categories and subcategories:

Steel bridge: steel box girder, steel truss, or plate girder;
Composite bridge
Concrete bridge: concrete slab, concrete beam, concrete box girder.

Temperature actions on bridge decks are discussed in EN 1991-1-5:2003.

Uniform Temperature Component

The maximum and minimum temperatures, T_e,max and T_e,min, that the bridge can reach over its working life determine the uniform temperature component. The uniform temperature components T_e,max and T_e,min can be calculated using the diagrams in Figure 2, where T_e,max and T_e,min, in °C, are stated in terms of T_max and T_min, in °C, for each bridge type previously mentioned.

Over the years, weather stations have recorded the maximum and minimum shade air temperatures. The maximum and minimum design temperatures that the bridge deck may encounter during its design life are predicted using these records. These temperatures are represented as isotherm maps in the codes.

temperature in bridges — **Figure 2:** Correlation between shade air temperature (T_min, T_max) and uniform
components of the bridge temperature (T_e,min, T_e,max)

To calculate how much the bridge deck will expand and contract, the maximum and minimum shade air temperatures are transformed into ‘effective’ bridge temperatures T_e,max and T_e,min and multiplied by the coefficient of thermal expansion and the deck length. The deck can be accommodated by providing joints and sliding bearings, or by restricting the movement and engineering the structure to resist the forces created. For truss or plated steel bridges (category 1), T_e,max values can be decreased by 3 °C.

A base reference temperature T₀ is used to represent the effective bridge temperature at specific stages of construction. The deck will expand from T₀ to T_e,max, and contract from T₀ to T_e,min.

In the case of a free moving deck, T₀ is used to calibrate the gap for the expansion joint and to set the sliding bearing positions when these units are installed. However, in the case of a restrained deck, it is used to determine the magnitude of movement that the supporting structure has to accommodate after it has been made integral with the deck.

If T₀ is the initial bridge temperature, i.e. the temperature of the bridge at the time when it is restrained, the variation of the uniform bridge temperature ∆T_u is given by;

∆T_u= T_e,max – T_e,min = ∆T_N,esp + ∆T_N,con

where

∆T_N,exp=T_e,max – T₀ and ∆T_N,con = T₀ – T_e,min

are the temperature variations to be considered when the bridge expands or contracts, respectively.
Assessing bearing displacements it can be assumed ∆T_N,exp = T_e,max – T₀+ 20°C and ∆T_N,con = T₀ – T_e,min + 20°C.

Vertical temperature varying component

Vertical temperature variations can arise as a result of the top and bottom surfaces of the bridge being heated and cooled differently. These differences correlate to maximum heating and cooling, respectively, when the top surface is warmer than the bottom surface and when the bottom surface is warmer than the top surface. The vertical temperature profiles can be defined under two different hypotheses, depending on whether the non-linear temperature profile ∆T_E is ignored or not: in the first case, a simplified equivalent linear profile can be considered, whereas in the second case, a non-linear profile, including ∆T_E, is considered.

Equivalent linear vertical temperature profile

Table 1 shows the maximum temperature differences corresponding to maximum heating or maximum cooling, designated as ∆T_M,heat and ∆T_M,cool, for the various bridge categories when an equivalent linear vertical temperature profile is used. The temperature differences for road and railway bridges with a 50 mm surface are represented by the upper bound values in Table 1. Table 1 values should be multiplied by the correction factors k_sur in Table 2 for varying surface thicknesses.

Table 1: Equivalent linear vertical temperature variations for bridges

Type of deck	Top warmer than bottom ∆_TM,heat [°C]	Bottom warmer than top ∆T_M,cool [°C]
Type 1: Steel deck	18	13
Type 2: Composite deck	15	18
Type 3: Concrete deck concrete box girder concrete beam concrete slab	10 15 15	5 8 8

Table 2: Adjustment factors k_sur for road, foot and railway bridges

Non-linear vertical temperature profiles

When more detailed calculations are required, the vertical temperature profile can be considered to be non-linear by using the temperature profiles shown in Figures 3, 4, and 5 for steel, concrete, and composite bridges, respectively, in the heating and cooling conditions. Tables 5.a and 5.b show two alternative profiles for composite bridges: normal and simplified.

In general, the simpler profile is safe. The temperature differences ∆T listed in tables 3 to 5 comprise both the vertical temperature component ∆T_M and the non-linear temperature component ∆T_E, as well as a small portion of the uniform component ∆T_N. The temperatures for other surfacing depths of bridge decks of type 1 to 3 are given in Tables B.1 to B.3 of EN 1991-1-5, Annex B.

Figure 3: Non-linear vertical temperature differences for steel bridges

concrete bridges temperture — **Figure 4:** Non-linear vertical temperature differences for concrete bridges

Non linear vertical temperature differences for composite bridges normal — **Figure 5.a**: Non-linear vertical temperature differences for composite bridges, normal profile

Non linear vertical temperature differences for composite bridges simplified — **Figure 5.b**: Non-linear vertical temperature differences for composite bridges, simplified profile

Horizontal temperature varying component

Horizontal temperature differences in bridges can be generally disregarded, except in special cases, for example when one side of the bridge is much more exposed to the sunlight of the other one. When the horizontal component must be taken into account, a linear variation of 5°C can be assumed.

Worked Example on Calculation of Temperature Stresses on Bridge Decks

Determine the stresses induced by both the positive and reverse temperature differences for the concrete box girder bridge shown in Figure 6 (A = 940000 mm², I =102534 × 10⁶ mm⁴, depth to NA = 409 mm, T = 12 × 10^-6, E = 34 kN/mm²). This example is adapted from Ryall (2008).

temperature stresses on a box girder bridge — **Figure 6:** Box girder dimensions and temperature distribution (Ryall, 2008)

Calculate critical depths of temperature distribution
From BD 37/88 Figure 9 this is a Group 4 section, therefore:

h₁ = 0.3h = 0.3 × 1000 = 300 > 150; thus h₁ = 150mm
h₂ = 0.3h = 0.3 × 1000 = 300 > 250; thus h₂ = 250mm
h₃ = 0.3h = 0.3 × 1000 = 300 > 170; thus h₃ = 170mm

Calculate temperature distribution
Basic values are given in Figure 9 of BD 37/01 which are modified for depth of section and surface thickness by interpolating from Table 24 of BD 37/01.

T₁ = 17.8 + (17.8 – 13.5)20/50 = 16.1°C
T₁= 4.0 + (4.0 – 3.0)20/50 = 3.6 °C
T₁ = 2.1 +(2.5 – 2.1)20/50 = 2.26 °C

Calculate restraint forces at critical points
This is accomplished by dividing the depth into convenient elements corresponding to changes in the distribution diagram and/or changes in the section (see Figure 3.2 of BD 37/01):

F = E_cβ_TT_iA_i
F₁ = 34000 × 12 × 10^-6 × (16.1 – 3.6) × 2000 × 150/1000 = 765 kN
F₂= 34000 × 12 × 10^-6 × (3.6) × 2000 × 150/1000 = 441 kN
F₃ = 34000 × 12 × 10^-6 × [(3.6 + 2.6)/2] × 2000 × (220 – 150)/1000 = 177 kN
F₄ = 34000 × 12 × 10^-6 × (2.6/2) × 2 × (250 – 70) × 250/1000 = 48 kN
F₅ = 34000 × 12 × 10^-6 × (2.26/2) × 1000 × 170/1000 = 78 kN

Total F = 1509 kN (tensile)

Element Forces — **Figure 7:** Element forces (Ryall, 2008)

Calculate restraint moment about the neutral axis
M = [765(409 – 50) + 441(409 – 75) + 177(409 – 185) + 28(409 – 270) – 78(591 – 170 × 2/3)]/1000 = 431 kNm (hogging)

Calculate restraint stresses
f = E_cβ_TT_i

f₀₁ = -34000 × 12 × 10^-6 × 16.1 = 6.56N/mm²
f₀₂ = -34000 × 12 × 10^-6 × 3.6 = 1.47 N/mm²
f₀₃ = -34000 × 12 × 10^-6 × 2.6 = 1.06 N/mm²
f₀₄ = -34000 × 12 × 10^-6 × 0 = 0.00 N/mm²
f₀₅ = -34000 × 12 × 10^-6 × 0 = 0.00 N/mm²
f₀₆ = -34000 × 12 × 10^-6 × 2.26 = 0.92 N/mm²

Calculate balancing stresses
Direct stress f₁₀= (1509 × 10³)/940000 = 1.61 N/mm²
Bending stresses f_2i = My/I:

f₂₁ = [(431 × 10⁶)/(102534 × 10⁶)] × 409 = 1.71 N/mm²
f₂₂= [(431 × 10⁶)/(102534 × 10⁶)] × 259 = 1.08 N/mm²
f₂₃ = [(431 × 10⁶)/(102534 × 10⁶)] × 180 = 0.75 N/mm²
f₂₄ = [(431 × 10⁶)/(102534 × 10⁶)] × 9 = 0.06 N/mm²
f₂₅ = [(431 × 10⁶)/(102534 × 10⁶)] × 421 = -1.76 N/mm²
f₂₆ = [(431 × 10⁶)/(102534 × 10⁶)] × 591 = -2.47 N/mm²

Calculate final stresses
The final stress distribution is shown in Figure 8. Similar calculations for the cooling (reverse) situation are shown in Figure 9.

final stress distribution — **Figure 8:** Final stress distribution (positive)(Ryall, 2008)

negative stress distribution — **Figure 9:** Final stress distribution (negative)(Ryall, 2008)

References
[1] Rryall M. J. (2008): Loads and load distribution, in ICE Manual of Bridge Engineering (2nd Edition). Edited by Parke G. and Hewson N. doi: 10.1680/mobe.34525.0023

Load Combinations for Highway Bridges

By

Ubani Obinna

-

May 2, 2022

Table of Contents

Highway bridges are subjected to a myriad of direct and indirect forces. For the purpose of obtaining the action effects on bridges, bridge designers must obtain the adequate load combination for highway bridges that will give the worst effect on any part of the bridge. The predominant action on highway bridges is gravity loads due to self-weight and the mass/dynamic effects of moving traffic. Other actions that are frequently considered are temperature loads, construction loads, snow loads, earthquake loads, and possible differential settlement of the foundation.

In this article, we are going to consider the load combinations for highway bridges according to UK standards (BD 37/01) and Eurocodes (EN 1990:2002 and EN 1991-2).

Load Combinations for Highway Bridges in the UK

In the UK, five combinations of loading are considered for the purposes of design: three principal and two secondary. These are defined in Clause 4.4 and Table 1 of BD 37/01. It is usual in practice to design for Combination 1 and to check other combinations if necessary.

Combination 1. For highway and foot/cycle track bridges, the loads to be considered are the permanent loads, together with the appropriate primary live loads, and, for railway bridges, the permanent loads, together with the appropriate primary and secondary live loads.

Combination 2. For all bridges, the loads to be considered are the loads in combination 1, together with those due to wind and, where erection is being considered, temporary erection loads.

Combination 3. For all bridges, the loads to be considered are the loads in combination 1, together with those arising from restraint due to the effects of temperature range and difference, and, where erection is being considered, temporary erection loads.

Combination 4. Combination 4 does not apply to railway bridges except for vehicle collision loading on bridge supports. For highway bridges, the loads to be considered are the permanent loads and the secondary live loads, together with the appropriate primary live loads associated with them. Secondary live loads shall be considered separately and are not required to be combined. Each shall be taken with its appropriate associated primary live load.

Combination 5. For all bridges, the loads to be considered are the permanent loads, together with the loads due to friction at bearings.

The summary of UK load combinations (BD 37/01) is shown in Table 1 below;

Table 1: UK load combinations

Combination	Description
1	Permanent + Primary Live
2	Permanent + Primary live + wind + (temporary erection loads)
3	Permanent + Primary live + temperature restraint + (temporary erection loads)
4	Permanent + Secondary live + associated primary live
5	Permanent + bearing restraint

The partial factors for load (γ_Fl) when carrying out load combination are provided in the Tables below;

Table 2: Partial factors for dead and superimposed loads

Load	Limit State	Comb. 1	Comb. 2	Comb. 3	Comb. 4	Comb. 5
Dead: Steel	ULS SLS	1.05 1.00	1.05 1.00	1.05 1.00	1.05 1.00	1.05 1.00
Dead: Concrete	ULS SLS	1.15 1.00	1.15 1.00	1.15 1.00	1.15 1.00	1.15 1.00
Superimposed: deck surfacing	ULS SLS	1.75 1.20	1.75 1.20	1.75 1.20	1.75 1.20	1.75 1.20
Superimposed dead: other loads	ULS SLS	1.20 1.00	1.20 1.00	1.20 1.00	1.20 1.00	1.20 1.00
Reduced load factor for dead and superimposed load where this has a more severe effect	ULS	1.00	1.00	1.00	1.00	1.00

The partial factors for highway live load combination are shown in Table 3 below;

Table 3: Partial factors for live load combination

Highway Live Loading	Limit State	Comb. 1	Comb. 2	Comb. 3	Comb. 4	Comb. 5
HA alone	ULS SLS	1.50 1.20	1.25 1.00	1.25 1.00
HA with HB or HB alone	ULS SLS	1.30 1.10	1.10 1.00	1.10 1.00
Footway and cycle track loading	ULS SLS	1.50 1.00	1.25 1.00	1.25 1.00
Accidental wheel loading	ULS SLS	1.50 1.20

The partial factors for wind load (on bridges) combination are shown in Table 4;

Table 4: Partial factors for wind load combination

Wind Load	Limit State	Comb. 1	Comb. 2	Comb. 3	Comb. 4	Comb. 5
During erection	ULS SLS		1.10 1.00
With dead load plus superimposed dead load only, and for members primarily resisting wind load	ULS SLS		1.40 1.00
With dead load plus superimposed dead load plus other appropriate Combination 2 loads	ULS SLS		1.10 1.00
Relieving effect of wind load	ULS SLS		1.00 1.00

The partial factors for combination involving temperature loads are given in Table 5;

Table 5: Partial factors for temperature load combination

Temperature Load	Limit State	Comb. 1	Comb. 2	Comb. 3	Comb. 4	Comb. 5
Restraint to movement, except frictional	ULS SLS			1.30 1.00
Frictional bearing restraint	ULS SLS					1.30 1.00
Effect of temperature difference	ULS SLS			1.00 0.80

Load combination factors involving earth pressures are shown in Table 6, while the partial load factors for differential settlement and erection loads are shown in Table 7.

Table 6: Partial factors for earth pressure loads

Earth Pressure Load (retained fill and/or live load)	Limit State	Comb. 1	Comb. 2	Comb. 3	Comb. 4	Comb. 5
Vertical Loads	ULS SLS	1.20 1.00	1.20 1.00	1.20 1.00	1.20 1.00	1.20 1.00
Non-vertical loads	ULS SLS	1.50 1.00	1.50 1.00	1.50 1.00	1.50 1.00	1.30 1.00
Relieving effect	ULS	1.00	1.00	1.00	1.00	1.00

Table 7: Partial factors for differential settlement and erection loads

Load	Limit State	Comb. 1	Comb. 2	Comb. 3	Comb. 4	Comb. 5
Differential settlement	ULS SLS	1.20 1.00	1.20 1.00	1.20 1.00	1.20 1.00	1.20 1.00
Erection load (temporary loads)	ULS SLS		1.15 1.00	1.15 1.00

Load Combinations for Highway Bridges (Eurocode)

Annex A2 to EN 1990:2002 specifies the rules and methods for establishing combinations of actions for serviceability and ultimate limit state verifications (except fatigue verifications), as well as the recommended design values for permanent, variable, and accidental actions and ψ factors to be used in the design of road, footbridge, and railway bridges. It also applies to actions during the execution of bridges.

General guidelines for load combination on bridges (Annex 2 EN 1990)

The effects of actions that cannot occur concurrently owing to physical or functional reasons are not required to be included jointly in combinations of actions, according to Annex 2 of EN 1990. Combinations including activities not covered by EN 1991 (for example, mining subsidence, specific wind impacts, water, floating debris, flooding, mudslides, avalanches, fire, and ice pressure) should be characterized in accordance with EN 1990, 1.1. (3).

The combinations of actions given in expressions 6.9a to 6.12b of EN 1990 should be used when verifying ultimate limit states. Expressions 6.9a is as given below;

γ_SdE{γ_g,jG_k,j; γ_pP; γ_q,1Q_k,1; γ_q,iψ_0,iQ_k,i} j ≥; i > 1

The combination of effects of actions to be considered should be based on the design value of the leading variable action, and the design combination values of accompanying variable actions. The combinations of actions given in expressions 6.14a to 6.16b of EN 1990 should be used when verifying serviceability limit states. Additional rules are given in A2A for verifications regarding deformations and vibrations.

The necessary design situations must be considered throughout execution. Specific construction loads should be taken into account in the appropriate combinations of operations where applicable. Construction loads that cannot occur at the same time due to the implementation of control measures do not need to be considered in the appropriate combinations of actions. When a bridge is put into service in phases, the appropriate design situations must be considered.

typical construction loads on a bridge deck — **Figure 1:** Typical construction load during the execution of bridges

Snow loads and wind actions do not have to be considered at the same time as construction-related loads Q_co (i.e. loads due to working personnel). During some temporary design circumstances, however, it may be required to agree on the requirements for snow loads and wind actions to be taken into account simultaneously with other construction loads (e.g., actions due to heavy equipment or cranes) for a specific project.

Thermal and water actions should be evaluated simultaneously with construction loads where possible. When determining optimal combinations with construction loads, the various parameters governing water actions and components of thermal actions should be taken into account where applicable.

In accordance with the relevant parts of EN 1991-2, variable traffic measures should be taken into consideration simultaneously where applicable. Any group of loads, as described in EN 1991-2, shall be taken into account as one variable action for any combination of variable traffic actions with other variable actions stated in other parts of EN 1991.

If the impacts of uneven settlements are regarded as considerable in comparison to the effects of direct interventions, they should be evaluated. Total and differential settlement restrictions may be specified by the specific project. Uncertainty in the assessment of these settlements should be taken into account where the structure is extremely sensitive to uneven settlements.

Uneven settlements on the structure caused by soil subsidence should be defined as a permanent action and included in the structure’s ultimate and serviceability limit state verifications as a combination of actions. G_set should be represented as a set of values relating to differences in settlements between individual foundations or segments of foundations (relative to a reference level), d_set,i (i is the number of the individual foundation or part of foundation).

Permanent loads and backfill are the main causes of settlements. For some individual projects, variable actions may need to be considered. Settlements change monotonically (in the same direction) throughout time and must be taken into account from the moment they cause structural impacts (i.e. after the structure or a part of it, becomes statically indeterminate).

Furthermore, there may be a connection between the development of settlements and the creep of concrete members in the case of a concrete structure or a structure with concrete elements. Individual foundation or pair of foundation settlement differences, d_set,i, shall be taken into consideration as best-estimate projected values in line with EN 1997, taking into account the structure’s building process.

Groups of Traffic Loads on Highway Bridges

When the simultaneous presence of traffic and non-traffic actions is considerable, the characteristic values of the traffic actions can be calculated using the five separate, mutually exclusive groups of loads listed in Table 4.4a of BS EN 1991-2:2003 (reproduced below), with the dominant component action highlighted. Each load category in the table should be seen as establishing a characteristic action for usage with non-traffic loads, but they can also be used to assess infrequent and frequent values.

group of loads for load combinations in highway bridges

To obtain infrequent combination values it is sufficient to replace in Table 4.4(a) of EN 1991-2 characteristic values with the infrequent ones, leaving unchanged the others, while frequent combination values are obtained by replacing characteristic values with the frequent ones and setting to zero all the others. The ψ-factors for traffic loads on road bridges are reported in the Table below.

Combination Rules for Highway Bridges

(1) The infrequent values of variable actions may be used for certain serviceability limit states of concrete bridges.
(2) Load Model 2 (or associated group of loads gr1b) and the concentrated load Q_fwk (see 5.3.2.2 in EN 1991-2) on footways need not be combined with any other variable non-traffic action.
(3) Neither snow loads nor wind actions need be combined with: braking and acceleration forces or the centrifugal forces or the associated group of loads gr2, loads on footways and cycle tracks or with the associated group of loads gr3, crowd loading (Load Model 4) or the associated group of loads gr4. The combination rules for special vehicles (see EN 1991-2, Annex A, Infonnative) with nominal traffic (covered LM1 and LM2) and other variable actions may be referenced as appropriate in the National Annex or agreed for the individual project.
(4) Snow loads need not be combined with Load Models 1 and 2 or with the associated groups of loads gr1a and gr1b unless otherwise specified for particular geographical areas. However, geographical areas where snow loads may have to be combined with groups of loads gr1a and gr1b in combinations of actions may be specified in the National Annex.
(5) No wind action greater than the smaller of F_w* and ψ₀F_wk should be combined with Load Model 1 or with the associated group of loads gr1a.
(6) Wind actions and thermal actions need not be taken into account simultaneously unless otherwise specified for local climatic conditions.

Partial Factors for Actions in the Limit State

(1) Design values of actions (EQU) (Set A)

For persistent design situations, the recommended set of values for γ are:
γ_G,sup = 1.05
γ_G,inf = 0.95
γ_Q= 1.35 for road and pedestrian traffic actions, where unfavourable (0 where favourable)
γ_Q=1.45 for rail traffic actions, where unfavourable (0 where favourable)
γ_Q= 1.50 for all other variable actions for persistent design situations, where unfavourable (0 where favourable).
γ_P = recommended values defined in the relevant design Eurocode.

For transient design situations during which there is a risk of loss of static equilibrium, Q_k,1 represents the dominant destabilising variable action and Q_k.irepresents the relevant accompanying destabilising variable actions.

During execution, if the construction process is adequately controlled, the recommended set of values for γ are:
γ_G,sup = 1.05
γ_G,inf = 0.95
γ_Q= 1.35 for construction loads where unfavourable (0 where favourable)
γ_Q= 1.50 for all other variable actions, where unfavourable (0 where favourable)

(2) Design values of actions (STR/GEO) (Set B)

γ_G,sup = 1.35 (This value covers the self-weight of structural and non-structural elements, ballast, soil, groundwater and free water, removable loads)
γ_G,inf = 1.00
γ_Q= 1.35 for road and pedestrian traffic actions, where unfavourable (0 where favourable)
γ_Q=1.45 for rail traffic actions, where unfavourable (0 where favourable)
γ_Q= 1.50 for all other variable actions for persistent design situations, where unfavourable (0 where favourable). This value covers variable horizontal earth pressure from groundwater, free water, and ballast, traffic load surcharge earth pressure, traffic aerodynamic actions, wind, and thermal actions, etc.
γ_P = recommended values defined in the relevant design Eurocode.
γ_Gset(partial factor for settlement) = 1.20 in the case of linear elastic analysis, and γ_Gset = 1.35 in the case of non-linear analysis, for design situations where actions due to uneven settlements may have unfavourable effects.
ξ = 0.85 (so that ξγ_G,sup = 0.85 x 1.35 = 1.15)

(3) Design values of actions (STR/GEO) (Set C)

γ_G,sup = 1.00
γ_G,inf = 1.00
γ_Gset = 1.00
γ_Q= 1.15 for road and pedestrian traffic actions where unfavourable (0 where favourable)
γ_Q = 1.25 for rail traffic actions where unfavourable (0 where favourable)
γ_Q = 1.30 for the variable part of horizontal earth pressure from soil, groundwater, free water and ballast, for traffic load surcharge horizontal earth pressure, where unfavourable (0 where favourable)
Y_Q = 1.30 for all other variable actions where unfavourable (0 where favourable)
γ_Gset= 1.00 in the case of linear elastic or non linear analysis, for design situations where actions due to uneven settlements may have unfavourable effects. For design situations where actions due to uneven settlements may have favourable effects, these actions are not to be taken into account.

Thermal Actions | How to Apply Fire Loading on Steel Structures

By

Ubani Obinna

-

April 29, 2022

The action of fire on a structure is represented by thermal actions, and EN 1991-1-2:2002 (Eurocode 1, Part 2) provides several options for considering thermal action on steel structures. Time-temperature relationships are one of the numerous ways of representing fire actions on structures. These are relationships that show the evolution of a temperature that represents the environment around the structure as a function of time. The heat flux transported from the environment to the structure can be calculated using this temperature and the relevant boundary conditions.

Relationships that directly give the heat flux impinging on the structure are another option. The temperature evolution in the structure is then determined by combining the impinging heat flux with the flux emitted by the structure. Eurocode 1 distinguishes between nominal temperature-time curves, which include the standard temperature-time curve, the hydrocarbon curve, the external fire curve, and on the other hand, the natural fire models. The thermal action to be employed is usually a legal requirement set by the country or region in which the structure is located, and it is determined by the building’s size, usage, and occupancy (Franssen and Real, 2015).

Some countries impose prescriptive standards that specify the time-temperature curve as well as the time (referred to as fire resistance) that the structure must withstand when subjected to this curve. For example, a hotel in Country A must have a 60-minute resistance to the standard curve, whereas a railway station in country B must have a 30-minute resistance to the hydrocarbon curve. In such circumstances, the designer must guarantee that the construction meets the criterion and must use the time-temperature curve that has been provided (Franssen and Real, 2015).

1117 sd 1 — **Figure 1**: Fire action in a steel building

The regulation in other countries or areas may be more flexible, allowing the designer to create a performance-based design. Although the Eurocode provides some assistance in the form of restrictions of application to some of the proposed natural fire models, it is the designer’s obligation to adopt an acceptable representation of the fire in this scenario.

Such natural fire models should ideally be utilized in conjunction with performance-based requirements, such as the time required for evacuation or intervention. Before beginning any performance-based design, it is recommended to obtain consent from the authority with jurisdiction over the design fire and design scenario.

Nominal Temperature-Time Curves for Thermal Actions

Temperature-time curves are time-dependent analytical functions that yield a temperature. Because these functions are continuous, they can be used to create a curve in a time-temperature plane. Because they aren’t supposed to represent a genuine fire, they’re called nominal. They must be regarded as standard, or arbitrary, functions (Franssen and Real, 2015).

This is why the phrase “fire curve” is a misnomer because it implies that the temperature is the same as the temperature of a fire. In fact, the temperature is on par with what is found in fires. Such relationships are to be used in a prescriptive regulatory context because they are quite conventional. As a result, any requirement defined in terms of a nominal curve is both prescriptive and, in some ways, arbitrary (Franssen and Real, 2015).

The time it takes to evacuate or intervene should not be equated to the resistance of a structure to a nominal fire. Three distinct nominal temperature-time curves are proposed by Eurocode 1. The standard temperature-time curve is the one that has been used in standard fire tests to grade structural and separating elements in the past, and it is still used today. It’s used to symbolize a compartment with a fully formed fire. Because the formula was derived from the ISO 834 standard, it is commonly referred to as the ISO curve. Equation (1) gives us this standard curve;

θ_g = 20 + 345log₁₀ (8t +1) ——– (1)

where θ_g is the gas temperature in °C and t is the time in minutes.

When a regulatory requirement is defined as Rxx, with xx equal to 30 or 60 minutes, for example, it implies that the standard fire curve must be used to evaluate the structural parts’ duration fire resistance.

The external time-temperature curve is used to describe the exterior surface of separating external walls of a building that are exposed to a fire that starts outside the building or flames that come in through the windows of a compartment below or adjacent to the external wall.

Note: This curve should not be used to calculate the effects of a fire on an exterior load-bearing structure outside the building envelope, such as steel beams and columns. Annex B of Eurocode 1 describes the thermal attack on external structural steel parts.

The external curve is given by Equation (2);

θ_g= 20 + 660(1− 0.687e^−0.32t− 0.313e^−3.8t) ——– (2)

The hydrocarbon time-temperature curve is used for representing the effects of a hydrocarbon type fire. It is given by Equation (3);
θ_g = 20 + 1080 (1 − 0.325e^−0.167t− 0.675e^−2.5t ) ——–(3)

Fire curves for thermal actions — **Figure 2:** Different fire curves for thermal actions

In Fig. 2, the standard and hydrocarbon curves are compared. It can be seen that the hydrocarbon curve rises rapidly and achieves a constant temperature of 1100 °C within half an hour, whereas the standard curve rises more slowly but steadily over time. Equation (4) should be used to simulate the heat flow at the surface of a steel element when the environment is represented by a gas temperature, as is the case for nominal curves.

h_net = α_c(θ_g − θ_m) + Φε_mε_f σ[(θ_r + 273)⁴ − (θ_m + 273)⁴] ——– (4)

where;
α_c is the coefficient of convection which is taken as 25 W/m²K for the standard or the external fire curve and 50 W/m²K for the hydrocarbon curve,
θ_g is the gas temperature in the vicinity of the surface either calculated from Eqs. (1), (2) or (3) or taken as 20 °C
θ_m is the surface temperature of the steel member (the evolution of which has to be calculated)
Φ is a configuration factor that is usually taken equal to 1.0 but can also be calculated using Annex G of Eurocode 1 when so-called position or shadow effects have to be taken into account
ε_m is the surface emissivity of the member taken as 0.7 for carbon steel, 0.4 for stainless steel, and 0.8 for other materials
εf is the emissivity of the fire, in general, taken as 1.0
σ is the Stephan Boltzmann constant equal to 5.67 × 10^-8 W/m²K⁴
θ_r is the radiation temperature of the fire environment taken as equal to θ_g in the case of fully engulfed members.

References
Franssen J. and Paulo Vila Real P. V. (2015): Fire Design of Steel Structures (2nd Edition). ECCS – European Convention for Constructional Steelwork

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