EFFECT OF FORCE INTERACTION ON PLASTIC COLLAPSE

CIVIL_ENGINEERING

Introduction:

The presence of axial and shear forces in a cross section reduces its plastic moment capacity, which, in turn, reduces the collapse load of the structure. A direct method to take account of the effect of force interaction is to modify the elastoplastic stiffness matrix in accordance with the three cases of yield condition described in Chapter 3 for the yielded section. An indirect method is to assume yielding by pure bending for all elements while the collapse load of the structure
is calculated. At the end of the analysis, the reduced plastic moment capacity due to force interaction is calculated for each element and the analysis is repeated. This is called the successive approximation method. Both methods are described here.

 

Direct Method:
This method makes use of the structure stiffness matrix modified to take account of the formation of plastic hinges. The solution of the incremental structure equilibrium.

in which KP = is the modified structure stiffness matrix. The various forms of the member elastoplastic stiffness matrix KPe ½ lead to different KP =being formulated due to the different force interaction formulations and yield conditions. Thus, this method requires special computer programming to create KPe = and hence KP = . In addition, the plastic deformation at the plastic hinge has been condensed into KPe = and the extraction of the plastic deformation has to be performed separately. The advantage of using this method is that the force interaction condition is always satisfied at any stage of calculation and the solution for the collapse load is direct.

Calculation of Load Factor:

For yield condition based on pure bending, the load factor a for predicting the formation of a plastic hinge at a section is calculated . For yield condition based on force interaction, the calculation of a is more complicated. Its calculation  in which a yield surface diagram for a section with two dimensionless forces m and b is shown. In addition, the sign of the forces has been considered so that the yield surface is symmetric and consists of four quadrants of hyperplanes. Suppose that the forces m and b in a section from a linear elastic analysis of the structure under loading F are represented by the vector OG.

 

 

Formation of a plastic hinge in the section requires an increase in the forces by a load factor a such that the vector OG is extended linearly to point H on the hyperplane CD. In practice, values of a are calculated for all hyperplanes connecting the points ABCDEF and the one with the smallest positive value is chosen. For the section to stay yielded in subsequent analysis, the force point will move along the yield surface. The following example, used previously, illustrates the various aspects of this method.

 

Successive Approximation Method:
The direct method for elastoplastic analysis requires the use of unique elastoplastic stiffness matrices pertaining to individual yield criteria. Computer software invoking the direct method must be programmed to include these unique formulations. This poses a problem for structural designers using this method as such computer software is not commonly available. As an alternative, a successive approximation method, based on yielding by pure bending in each iterative cycle, can be used to circumvent this problem.

 

It should be kept in mind that a reduction in the plastic moment capacity in members due to force interaction usually results in a reduction in the plastic collapse load of the structure. When using the successive approximation method, the collapse load factor is calculated on the basis of yielding only by pure bending. Because the total axial forces in the members are not known until the end of the analysis at collapse, the reduced plastic moment capacity as a consequence
of axial force or shear force can be calculated only when the analysis is completed.

 

 

The reduced bending moment capacity for each member can then be calculated and used in a subsequent cycle of analysis. The number of cycles of analysis to be performed depends on the degree of accuracy required for the solution. The method enables the solutions from the analysis cycles to converge to the true collapse load. However, the procedure could be tedious if the structure is complex. An alternative, but conservative, approach is to repeat the cycle of analysis only once. The result would underestimate the collapse load and err on the safe side for design. The procedure for this method is demonstrated in the following example using the structure.

WELDED JOINTS BETWEEN CHS OR RHS CHORD MEMBERS

CIVIL_ENGINEERING

General:

(1) Provided that the geometry of the joints is within the range of validity, the design resistances of welded joints between hollow section brace members and rectangular or square hollow section chord members may be determined.

(2) For joints within the range of validity, only the design criteria covered in the appropriate table need be considered. The design resistance of a connection should be taken as the minimum value for all applicable criteria.

(3) For joints outside the range of validity , all the criteria should be considered. In addition, the secondary moments in the joints caused by their rotational stiffness should be taken into account.

 

Uniplanar joints:

Unreinforced joints:
(1) In brace member connections subject only to axial forces, the design internal axial force Ni,Ed should not exceed the design axial resistance of the welded joint Ni,Rd.

(2) For welded joints between square or circular hollow section brace members and square hollow section chord members only, where the geometry of the joints is within the range of validity and also satisfies the additional conditions , the design axial resistances may be determined from the
expressions.
(3) For joints within the range of validity of  the only design criteria that need be considered are chord face failure and brace failure with reduced effective width. The design axial resistance should be taken as the minimum value for these two criteria.

NOTE: The design axial resistances for joints of hollow section brace members to square hollow section chords  have been simplified by omitting design criteria that are never decisive within the range.

(4) The design axial resistances of any unreinforced welded joint between CHS or RHS brace members and RHS chords, within the range of validity, may be determined using the expressions .

USE OF COMPUTERS FOR ELASTOPLASTIC ANALYSIS

CIVIL_ENGINEERING

Introduction:

The advent of computers has made structural analysis a good deal easier. Nowadays, commercial computer software is commonly available for linear and geometrical nonlinear elastic analyses of structural frames. Incremental elastoplastic analysis can be considered as a series of linear elastic analyses performed incrementally, perhaps using commercial computer software, in a step-by-step manner to predict plastic hinge formation. This technique, in conjunction with spreadsheet technology, can be used for elastoplastic analysis of large and complex structures for which member forces and deflections at any level of loading can be found.

 

at a load level being applied to a structure under a nominal load vector F, the common load factor corresponding to the formation of a plastic hinge is a1. The solution of for the displacement increment vector D1 is given as

Because F is directly proportional to the member forces, an increase in F by a common load factor a1 implies the same level of increase in the member forces. Hence, if D_P is the vector containing the member forces for a structure under loading given by F, member forces for the same structure under loading given by a1 F must be

If the bending moment in D_P is Mo for an arbitrary section in a structure and the corresponding bending moment in D_P 1 is M, then for a plastic hinge to occur at the section under the pure bending yield criterion, M must be equal to Mp, the plastic moment capacity of the section. Hence, the value of a1 leading to the formation of the plastic hinge is

At the load level where the load vector is a1 F, member forces in the structure can be calculated by. For any other sections not yet yielded, the remaining plastic moment capacity, Mr1, is generally given by

where Mo1 is the bending moment obtained from D_P. The remaining plastic moment capacities for all sections are then used for predicting the formation of the next plastic hinge once the yielded section has been modeled as a hinge and the structure modified according to the methods described in Chapter 3. A propped cantilever beam is used to illustrate this procedure.

Use of Spreadsheet for Automated Analysis:
Steps for the routine procedure implemented on a spreadsheet for elastoplastic analysis of general structures are as described.

1. Set up a calculation table on a spreadsheet (e.g., Microsoft Excel) with headings.
2. Perform a linear elastic analysis for the structure subjected to original loading (with any load factor set as 1); enter the values of bending moments Mo (Column 3) and deflections vo  from the results of analysis.
3. Calculate the load factor a  for each member such that plastic moment is reached at the ends of the member.
4. Choose the smallest load factor acr and calculate the cumulative bending moments  and deflections  using acr for all members. For the analysis stage i, ai ¼ acr and the values of the bending moment Mi and deflection vi are both zero when i = 1.

5. Calculate the residual plastic moments for all other sections; insert a hinge in the structure at the section where acr is obtained.
6. Repeat steps 2 to 5 until the structure collapses.

Spreadsheet table for elastoplastic analysis.

7. Theoretical collapse criterion: Determinant of structure stiffness matrix [K] = 0. When using computers, the collapse mechanism is reached when

• Run-time error occurs due to zero determinant
•  Dramatic increase in displacements occurs

The final collapse load factor acol is the sum of the load factors acr from all stages of analysis. Although the is set up for calculating bending moments and deflections, the table can be extended to include calculations of axial forces and shear forces, if desired.

SCOPE OF HOLLOW SECTION JOINTS

CIVIL_ENGINEERING

General:

(1) This section gives detailed application rules to determine the static resistances of uniplanar and multiplanar joints in lattice structures composed of circular, square or rectangular hollow sections, and of uniplanar joints in lattice structures composed of combinations of hollow sections with open sections.

(2) The static resistances of the joints are expressed in terms of maximum design axial and/or moment resistances for the brace members.

(3) These application rules are valid both for hot finished hollow sections to EN 10210 and for cold formed hollow sections to EN 10219, if the dimensions of the structural hollow sections fulfil the requirements of this section.

(4) The nominal yield strength of hot finished hollow sections and the nominal yield strength of the basic material of cold formed hollow sections should not exceed 460 N/mm2. For grades S 420 and S 460 the static resistances given in this section should be reduced by a factor 0,9.

NOTE: According to EN 10210 and EN 10219 the requirements for material is determined based on the end product, not on the base material.

(5) The nominal wall thickness of hollow sections should be limited to a minimum of 2,5 mm.

(6) The nominal wall thickness of a hollow section chord should not be greater than 25 mm unless special measures have been taken to ensure that the through thickness properties of the material will be adequate.

(7) For fatigue assessment.

 

Field of application:
(1) The application rules given in this section may be used only where all of the following conditions are satisfied.

(2) The compression elements of the members should satisfy the requirements for Class 1 or Class 2  for the condition of pure bending.

(3) The angles θi between the chords and the brace members, and between adjacent brace members, should satisfy:

(4) The ends of members that meet at a joint should be prepared in such a way that their cross-sectional shape is not modified. Flattened end connections and cropped end connections are not covered in this section.

(5) In gap type joints, in order to ensure that the clearance is adequate for forming satisfactory welds, the gap between the brace members should not be less than ( t1 t2 ).

(6) In overlap type joints, the overlap should be large enough to ensure that the interconnection of the brace members is sufficient for adequate shear transfer from one brace to the other. In any case the overlap should be at least 25%.

7) Where overlapping brace member have different thicknesses and/or different strength grades, the member with lowest ti fyi – value should overlap the other member.

(8) Where overlapping brace members are of different widths, the narrower member should overlap the wider one.

WELDED JOINTS BETWEEN CHS MEMBERS

CIVIL_ENGINEERING

Introduction:

(1) Provided that the geometry of the joints is within the range of validity. resistance of welded joints between circular hollow section members should be determined .

(2) For joints within the range of validity  only chord face failure and punching shear need be considered. The design resistance of a connection should be taken as the minimum value for these two criteria.

(3) For joints outside the range of validity  should be considered. In addition, the secondary moments in the joints caused by their rotational stiffness should be taken into account.

Uni planar joints:

(1) In brace member connections subject only to axial forces, the design internal axial force Ni,Ed should not exceed the design axial resistance of the welded joint Ni,Rd.

(2) Brace member connections subject to combined bending and axial force should satisfy:

where: Mip,i,Rd is the design in-plane moment resistance;
Mip,i,Ed is the design in-plane internal moment;
Mop,i,Rd is the design out-of-plane moment resistance;
Mop,i,Ed is the design out-of-plane internal moment.

 

Design resistance:

(1) The welds connecting the brace members to the chords should be designed to have sufficient resistance to allow for non-uniform stress-distributions and sufficient deformation capacity to allow for redistribution of bending moments.

(2) In welded joints, the connection should normally be established around the entire perimeter of the hollow section by means of a butt weld, a fillet weld, or combinations of the two. However in partially overlapping joints the hidden part of the connection need not be welded, provided that the axial forces in the brace members are such that their components perpendicular to the axis of the chord do not differ by more than 20%.

(3) Typical weld details are indicated in execution standards.

(4) The design resistance of the weld, per unit length of perimeter of a brace member, should not normally be less than the design resistance of the cross-section of that member per unit length of perimeter.

(5) The required throat thickness should be determined .

(6) The criterion given in (4) may be waived where a smaller weld size can be justified both with regard to resistance and with regard to deformation capacity and rotation capacity, taking account of the possibility that only part of its length is effective.

(7) For rectangular structural hollow sections the design throat thickness of flare groove welds.

Effective throat of flare groove welds in rectangular structural
hollow section

WELDED JOINTS BETWEEN CHS OR RHS BRASE MEMBER AND I OR H SECTION CHORDS

CIVIL_ENGINEERING

(1) Provided that the geometry of the joints is within the range of validity, the design resistances of the joints should be determined using the expressions.

Range of validity for welded joints between CHS or RHS brace members and I or H section chord members:

(2) For joints within the range of validity  only the design criteria covered in the appropriate table need be considered. The design resistance of a connection should be taken as the minimum value for all applicable criteria.

(3) For joints outside the range of validity, all the criteria  should be considered. In addition, the secondary moments in the joints caused by their rotational stiffness should be taken into account.

(4) In brace member connections subjected only to axial forces, the design axial force Ni,Ed should not exceed the design axial resistance of the welded joint Ni,Rd.

(5) Brace member connections subject to combined bending and axial force should satisfy:

where:

Mip,i,Rd is the design in-plane moment resistance;
Mip,i,Ed is the design in-plane internal moment

(6) The design internal moment Mi,Ed may be taken as the value at the point where the centreline of the brace member meets the face of the chord member.

(7) The design in-plane moment resistance Mip,1,Rd .

(8) If stiffeners in the chord  are used, then the bracing failure resistance Ni,Rd for T-, X-, Y-, K-gap and N-gap joints .

where:

beff = tw 2r 7 tf fy0 / fyi but # bi hi – 2ti
beff,s = ts 2a 7 tf fy0 / fyi but # bi hi – 2ti
beff beff,s # bi hi – 2ti

where: a is stiffener weld throat thickness, ‘2a’ becomes ‘a’ if single sided fillet welds are used; s refers to the stiffener.

(9) The stiffeners should be at least as thick as the I-section web.

Welded joints between CHS or RHS brace members and channel section chord members:

(1) Provided that the geometry of the joints is within the range of validity, the design resistances of welded joints between hollow section brace members and channel section chord members.

(2) The secondary moments in the joints caused by their bending stiffess should be taken into account.

(3) In a gap type joint, the design axial resistance of the chord cross-section N0,Rd should be determined allowing for the shear force transferred between the brace members by the chord, neglecting the associated secondary moment.

FUNCTIONS OF FOOTINGS AND FOUNDATION

CIVIL_ENGINEERING

Introduction:

To put it simply, the function of a structure is to do nothing. The most successful structures stay still. That’s the goal of the exercise. Getting slightly more technical, we can look at footings and foundations as having two functions:

 

Transfer Loads:

1. To transfer the live and dead loads of the building to the soil over a large enough area so that neither the soil nor the building will move.

2. In areas where frost occurs, to prevent frost from moving the building.

 

Dead loads are the weight of the building materials and the soil surrounding the foundations. Live loads include the weight of people, furniture, snow, rain, and wind. Wind may be a vertical force downward, a horizontal force, or an uplift force. A live load may also be exerted by water in the soil around the foundations. Wet soil exerts much more force than dry soil. Frozen soil exerts much more force than wet soil.

 

Direction of Loads:

The weight of objects is caused by gravity and results in a vertical downward load. Wind can be in any direction, as mentioned earlier. The soil exerts forces in all directions, but foundations usually see the horizontal thrust of the soil on the outside of the foundation wall. The forces of frost are also in all directions. Most frost failures in buildings include horizontal movement (foundation walls cracking, bowing, or collapsing inward) and frost heaving (upward movement of the building as the soil under the building expands due to frost).

 

Soil Pressure on Foundation Walls

Evidence of Frost Heaving

FOOTINGS AND FOUNDATION TYPE

CIVIL_ENGINEERING

Spread Footings:

This leads us to the configuration of footings. Houses may have spread footings (strip footings) that support the perimeter walls. These footings are wide pads that are continuous around the perimeter of the house. In some cases, the pads may be widened and/or thickened to accommodate concentrated loads from fireplaces, pilasters, etc.

A pilaster is a thickening of a foundation wall. It may be thickened to receive the concentrated load of a beam resting on top of the pilaster, or it may be acting as a stiffener to prevent the foundation wall from bowing inward.

Pad footings are similar to continuous footings except they are usually under a single pier or column. Pad footings spread the load out, usually in a square, with
the column or pier sitting in the middle of the square. It’s common for houses to have strip footings around the perimeter and pad footings on the building interior under columns.

Slab-on-Grade—Monolithic Slab

Spread Footings and Pad Footings

Piles:

Piles are typically used instead of footings where the soil quality is poor. They are, generally speaking, more expensive to install and have to be driven into the
ground with specialized equipment. They can work one of two ways.

1. Piles can be driven down to a point where they bear on bedrock or other sound substrate.

2. Piles can be driven into soil far enough that the friction of the soil against the sides of the pile is enough to resist any downward movement.

Incidentally, if a house is supported on piles, they probably won’t be visible and you may not know it.

Piers:

Piers are columns that may be completely concealed in the soil or may project above it. Most of you will be familiar with the piers that are commonly used to build exterior wood decks and porches. These piers may be poured concrete, often with the concrete poured into a cardboard cylinder in a hole dug in the ground. Piers usually, but not always, have footings . Piers can either be thought of as posts or columns, or can be thought of as short piles that bear on their ends.

Grade Beams:

Grade beams are usually concrete beams that are supported on footings, piles, or piers and are located at grade. In some cases they extend below grade; usually they extend only slightly above grade. Grade beams transfer the loads from the building down to the footings or piles.

Caissons:

Caissons are foundation systems created by drilling holes and filling them with concrete. A caisson pile is a cast-in-place pile that has a hollow tube driven
into the ground. The earth is excavated from the tube, and concrete is poured into the tube. Some caisson piles are flared out at the bottom to create a larger bearing surface. These are sometimes called bell caissons.

By now it should be clear that footings and foundations are:

• important to the stability of the house
• expensive
• mostly out of sight

Materials:

Footings and foundations should be strong so they can transfer loads and durable with respect to exposure from air, water, soil, and insect attack. Most modern footings are concrete (sometimes reinforced). Footings on older buildings.

DESIGN OF HOLLOW SECTION JOINTS

CIVIL_ENGINEERING

General:

(1) The design values of the internal axial forces both in the brace members and in the chords at the ultimate limit state should not exceed the design resistances of the member .

(2) The design values of the internal axial forces in the brace members at the ultimate limit state should also not exceed the design resistances of the joints  as appropriate.

(3) The stresses σ0,Ed or σp,Ed in the chord at a joint should be determined from:

 

 

 

where:

 

 

Failure modes for hollow section connections:

(1) The design joint resistances of connections between hollow sections and of connections of hollow sections to open sections, should be based on the following failure modes as applicable:

a) Chord face failure (plastic failure of the chord face) or chord plastification (plastic failure of the chord cross-section);

b) Chord side wall failure (or chord web failure) by yielding, crushing or instability (crippling or buckling of the chord side wall or chord web) under the compression brace member;

c) Chord shear failure;

d) Punching shear failure of a hollow section chord wall (crack initiation leading to rupture of the brace members from the chord member);

e) Brace failure with reduced effective width (cracking in the welds or in the brace members);

f) Local buckling failure of a brace member or of a hollow section chord member at the joint location.

 

NOTE: The phrases printed in boldface type in this list are used to describe the various failure modes in the tables of design resistances.

SPECIAL FOUNDATIONS

CIVIL_ENGINEERING

Raft or Mat Foundations:

Raft or mat foundation systems are not common, and you would not usually know that is what you’re looking at in the field (Figure 1.15). Their construction materials and failure modes are the same as what we will be looking at, in any case. So we won’t go into more detail.

Preserved Wood Foundations:

Preserved wood foundations have become popular in some areas over the last few years . Wood in a below-grade, damp soil environment has historically.

Raft and Mat Foundations

Preserved Wood Foundations

 

 

not had a long life, particularly as a structural member. As a result, there are several design challenges with respect to wood foundations.

 

They are more likely to be successful in dry soils than in wet soils. For the most part, their modes of failure will be similar to what we will look at on most other foundation systems, with a couple of exceptions. Since wood is less brittle or more flexible than concrete, for example, cracking is likely to be less common and Rot and Insects bowing may be prevalent. Rot and insect damage are obviously possibilities with wood foundations, while these are not issues with most other foundation and footing materials.

 

In most cases, the interiors of preserved wood foundations are finished as living space, and it may be difficult to identify the foundation system, let alone inspect it.

 

Post-Tensioned Foundations:

Some areas have expansive soils that make it risky to use conventional footings and foundations. A special reinforcement technique for concrete grade
beams and floor slabs is sometimes used to resist the forces of the soil and to prevent differential movement of the structure.

 

Cables or Tendons:

Post-tensioned slabs and grade beams use steel cables or tendons that are laid in place before the concrete is poured. The cables are most often surrounded by a
Cables or Tendons plastic sheathing. After the concrete is poured, jacks are used to pull the cables tight, strengthening the assembly. You may be able to see the anchors and cable ends on the exterior of foundations near grade level. These post-tensioned cables sometimes snap, and in some cases they shoot out from the foundation or come up through floor slabs. Fortunately, this problem is rare, at least so far.